croatian operational research...

ISSN: 1848�0225 PrintISSN: 1848�9931 OnlineUDC: 519.8 (063)Abbreviation: Croat. Oper. Res. Rev.

Publisher:CROATIAN OPERATIONAL RESEARCH SOCIETY

Co-publishers:University of Osijek, Faculty of Economics in OsijekUniversity of Osijek, Department of MathematicsUniversity of Split, Faculty of EconomicsUniversity of Zagreb, Faculty of Economics and Business

Croatian Operational

Research Review

Volume 7 (2016)Number 1

Osijek, 2016

Croatian Operational Research Review is reviewed/indexed in: Current In-dex to Statistics, EBSCO host, EconLit, Genamics Journal Seek, INSPEC, Mat-hematical Reviews, Current Mathematical Publications (MathSciNet), Proquest,Zentralblatt für Mathematik/Mathematics Abstracts (CompactMath).

ISSN: 1848�0225 PrintISSN: 1848�9931 OnlineUDC: 519.8 (063)Abbreviation: Croat. Oper. Res. Rev.

Croatian Operational Research Review

Croatian Operational Research Review (CRORR) is an open access scienti�cjournal published twice per year. The main publisher of the CRORR journal is theCroatian Operational Research Society (CRORS). Starting from Volume 5, Number1, 2014, co-publishers of the journal are:

⋄ University of Osijek, Faculty of Economics in Osijek,⋄ University of Osijek, Department of Mathematics,⋄ University of Split, Faculty of Economics, and⋄ University of Zagreb, Faculty of Economics and Business.

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The aim of the Croatian Operational Research Review journal is to provide highquality scienti�c papers covering the theory and application of operations researchand related areas, mainly quantitative methods and machine learning. The scopeof the journal is focused, but not limited to the following areas: combinatorial anddiscrete optimization, integer programming, linear and nonlinear programming, mul-tiobjective and multicriteria programming, statistics and econometrics, macroeco-nomics, economic theory, games, control theory, stochastic models and optimization,banking, �nance, insurance, simulations, information and decision support systems,data envelopment analysis, neural networks and fuzzy systems, and practical ORand applications.

Occasionally, special issues are published dedicated to a particular area of ORor containing selected papers from the International Conference on Operational Re-search.

Papers blindly reviewed and accepted by two independent reviewers are publi-shed in this journal.

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Croatian Operational Research Review is reviewed/indexed by:

⋄ Current Index to Statistics⋄ EBSCO host⋄ EconLit⋄ Genamics Journal Seek⋄ INSPEC⋄ Mathematical Reviews, Current Mathematical Publications (MathSciNet)⋄ Proquest⋄ Zentralblatt für Mathematik/Mathematics Abstracts (CompactMath)⋄ Directory of Open Access Journals (DOAJ)

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Editorial O�ce and Address

Editors-in-Chief: Marijana Zeki¢-Su²acRudolf Scitovski

Co-Editors: Zoran Babi¢Snjeºana PivacKristina �ori¢Zrinka Luka£Goran Le²aja

Technical Editors: Ivan Soldo, Helena Stjepanovi¢Proofreader: Nitor usluge d.o.o. Zagreb

Contact:Croatian Operational Research Society (CRORS), Trg J. F. Kennedya 6,HR-10 000 Zagreb, CroatiaURL: http://www.hdoi.hr/crorr-journale-mail: [email protected]

Editorial Board

Zdravka Aljinovi¢, Faculty of Economics, University of Split, Croatia, e-mail:[email protected]

Josip Arneri¢, Faculty of Economics and Business, University of Zagreb, Croatia,e-mail: [email protected]

Zoran Babi¢, Faculty of Economics, University of Split, Croatia, e-mail:[email protected]

Vlasta Bahovec, Faculty of Economics and Business, University of Zagreb, Cro-atia, e-mail: [email protected]

Majda Basti£, Faculty of Economics and Business, University of Maribor, Slove-nia, e-mail: [email protected]

Valter Boljun£i¢, Faculty of Economics and Tourism �Dr. Mijo Mirkovi¢�,University of Pula, Croatia, e-mail: [email protected]

Ðula Borozan, Faculty of Economics in Osijek, University of Osijek, Croatia,e-mail: [email protected]

Mirjana �iºme²ija, Faculty of Economics and Business, University of Zagreb,Croatia, e-mail: [email protected]

Ksenija Dumi£i¢, Faculty of Economics and Business, University of Zagreb,Croatia, e-mail: [email protected]

Jose Rui Figueira, Instituto Superior Tecnico, Technical University of Lisbon,Portugal, e-mail: [email protected]

Tihomir Hunjak, Faculty of Organization and Informatics, University of Zagreb,Croatia, e-mail: [email protected]

Dragan Juki¢, Department of Mathematics, University of Osijek, Croatia,e-mail: [email protected]

Damir Kalpi¢, Department of Applied Computing, Faculty of Electrical Engine-ering and Computing, University of Zagreb, Croatia, e-mail: [email protected]

Urlike Leopold Wildburger, Institut für Statistik und Operations Research,Karl Franzens Universität Graz, Austria, e-mail: [email protected]

Goran Le²aja, Department of Mathematical Sciences, Georgia Southern Univer-sity, USA, e-mail: [email protected]

Zrinka Luka£, Faculty of Economics and Business, University of Zagreb, Croatia,e-mail: [email protected]

Robert Manger, Department of Mathematics, University of Zagreb, Croatia,e-mail: [email protected]

Marija Marinovi¢, Department of Informatics, University of Rijeka, Croatia,e-mail: [email protected]

Josip Mesari¢, Faculty of Economics in Osijek, University of Osijek, Croatia,e-mail: [email protected]

Luka Nerali¢, Faculty of Economics and Business, University of Zagreb, Croatia,e-mail: [email protected]

Snjeºana Pivac, Faculty of Economics, University of Split, Croatia, e-mail:[email protected]

Rudolf Scitovski, Department of Mathematics, University of Osijek, Croatia,e-mail: [email protected]

Paolo Serafini, Department of Mathematics and Informatics, University of Udine,Italia, e-mail: [email protected]

Darko Skorin-Kapov, School of Business, Adelphi University, Garden City, USA,e-mail: [email protected]

Jadranka Skorin-Kapov, W.A.Harriman School for Management and Policy,State University of New York at Stony Brook, USA,e-mail: [email protected]

Alemka �egota, Faculty of Economics, University of Rijeka, Croatia, e-mail:[email protected]

Kristina �ori¢, Zagreb School of Economics and Management, Croatia, e-mail:[email protected]

Greys So²i¢, Information and Operations Management Department, The MarshallSchool of Business, University of Southern California, USA,e-mail: [email protected]

Vi²nja Vojvodi¢-Rozenzweig, Zagreb School of Economics and Management,Croatia, e-mail: e-mail: [email protected]

Richard Wendell, Joseph M. Katz Graduate School of Business, University ofPittsburgh, USA, e-mail: [email protected]

Lidija Zadnik-Stirn, Biotechnical Faculty, University of Ljubljana, Slovenia,e-mail: [email protected]

Marijana Zeki¢-Su²ac, Faculty of Economics, University of Osijek, Croatia,e-mail: [email protected]

Sanjo Zlobec, Department of Mathematics and Statistics, McGill University,Canada, e-mail: [email protected]


Volume 7 (2016), Number 1

CONTENTS

Original scienti�c papers

Le²aja, G., Ozen, M., Improved full-Newton-step InfeasibleInterior-Point Method for linear complementarity problems . . . . . . . . . 1

Igbinosun, L. I., Omosigho, S. E., Tra�c �ow model at �xed controlsignals with discrete service time distribution . . . . . . . . . . . . . . . . . . . . . . 19

Shirdel, G. H., Abdolhosseinzadeh, M., The critical node problemin stochastic networks with discrete-time Markov chain . . . . . . . . . . . . 33

Tu²kan, B., Stojanovi¢, A., Measurement of cost e�ciency in theEuropean banking industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47

Janá£ek, J., Kvet, M., Min-max optimization and the radial approachto the public service system design with generalized utility . . . . . . . . 67

Turkalj, �., Markulak, D, Singer, S., Scitovski, R., Researchproject grouping and ranking by using adaptive Mahalanobisclustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

�iºme²ija, M., Erjavec, N., Can con�dence indicators forecast theprobability of expansion in Croatia? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Mileti¢, Lj., Le²aja, G., Research and evaluation of thee�ectiveness of e-learning in the case of linear programming . . . . . . 109

Ke£ek, D., �ajdela Hrustek, N., Du²ak, V., Analysis ofmultiplier e�ects of ICT sectors - a Croatian case . . . . . . . . . . . . . . . . .129

Vali¢-Vale, M., Zenzerovi¢, R., Diagnosing companies in �nancialdi�culty based on the auditor's report . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Croatian Operational Research Review 1CRORR 7(2016), 1–18

Improved Full-Newton-Step Infeasible Interior-Point Methodfor Linear Complementarity Problems∗

Goran Lesaja1,† andMustafa Ozen1

1 Department of Mathematical Sciences, Georgia Southern UniversityStatesboro, GA 30460, USA

E-mail: ⟨{goran, mo01520}@georgiasouthern.edu⟩

Abstract. We present an Infeasible Interior-Point Method for monotone Linear Comple-mentarity Problem (LCP ) which is an improved version of the algorithm given in [13]. Inthe earlier version, each iteration consisted of one feasibility step and few centering steps.The improved version guarantees that after one feasibility step, the new iterate is feasibleand close enough to the central path thanks to the much tighter proximity estimate whichis based on the new lemma introduced in [18]. Thus, the centering steps are eliminated.Another advantage of this method is the use of full-Newton-steps, that is, no calculationof the step size is required. The preliminary implementation and numerical results demon-strate the advantage of the improved version of the method in comparison with the old one.

Key words: linear complementarity problems, interior-point method, infeasible interior-point method, full-Newton-step

Invited paper, received: March 01, 2016; accepted: March 31, 2016; available online:April 04, 2016

DOI:10.17535/crorr.2016.0001

1. Introduction

In this paper, we consider a class of Linear Complementarity Problems (LCP ) inthe standard form:

s = Mx+ q

xs = 0

x ≥ 0, s ≥ 0

(1)

where x, s ∈ Rn, M ∈ Rn×n, q ∈ Rn and xs denotes Hadamard (component-wise)product of vectors x and s.

LCP is not an optimization problem, but it has robust relationship with im-portant optimization problems such as linear programming (LP ) and quadraticprogramming (QP ) problems. This strong relationship is based on the fact that

∗The paper is dedicated to Professor Luka Neralic on the occasion of his 70th Birthday.†Corresponding author.

http://www.hdoi.hr/crorr-journal c⃝2016 Croatian Operational Research Society

2 Goran Lesaja and Mustafa Ozen

Karush-Kuhn-Tucker (KKT) optimality conditions for LP and QP can be con-verted into LCP . Furthermore, many problems from, game theory, engineering,finance, transportation, etc. can be directly formulated as LCP . Therefore, efficientmethods for solving LCPs has been an important area of research in optimizationboth from theoretical and practical point of view. For a comprehensive treatmentof LCP theory and practice we refer the reader to the monographs of Cottle et al.[4], Fachinei and Pang [6] and Kojima et al. [11].

It is well known that for general matrices M , LCP is NP-complete [3]. Hence,we consider classes of matrices M for which the corresponding LCP can be solvedin polynomial time. Most common and most studied is the class of monotone-LCPs,where the matrix M is a positive-semidefinite matrix. This is largely due to the factthat the Karush-Kuhn-Tucker conditions (KKT) of the convex QP and LP problemscan be formulated as monotone− LCP . In addition, many practical problems thatcan be directly formulated as LCP are usually monotone−LCP . For these reasons,in this paper we consider a class of monotone− LCP .

Methods to solve LCPs have traditionally been connected to the methods devel-oped for LP . The generalization of the Simplex method, the Lemke’s method, wasdeveloped soon after the introduction of the Simplex Method. The trend continuedwith other simplex-type (pivot based) methods. The tradition continued after thedevelopment of Interior-Point Methods (IPMs) that has revolutionized the area ofboth linear and nonlinear, primarily convex, optimization.

The first IPM for LP was proposed by Karmarkar [9] in 1984. The main idea ofthe algorithm differs from Simplex Method. It uses projective transformations andKarmarkar’s potential function. Moreover, IPMs are iterative methods and iteratesare calculated in the interior of the feasible region. In 1986, it was proved that theKarmarkar’s algorithm is connected to barrier and Newton-type methods [8]. Soonafter in 1988, Renegar [17] developed the first path following Newton-type IPM forLP . These achievements motivated the development of Newton-based IPMs. Sincethen, many different versions of Newton-based IPMs for LP have been proposed.Many of these IPMs for LP have been generalized for LCPs, the first one beingproposed by Kojima et al. [10] in 1989. Without any attempt to being complete,see [1, 2, 13, 14, 15, 19] and the references therein for more information on thedevelopment of IPMs.

Interior-Point Methods can be classified into two groups: feasible IPMs andinfeasible IPMs. Feasible IPM requires a strictly feasible starting point and feasi-bility of each iterates. In real life, it is not easy to find a feasible starting point allthe time and finding the feasible starting point may be as difficult as solving problemitself. Hence, it is important to consider infeasible IPM which can solve problemswith infeasible starting points. The algorithm we present in this paper belongs tothe second group of IPMs. In addition, we consider full-Newton-step IPM , thatis, the step size is always one.

In the paper, an improvement of the algorithm given in [13] is proposed. In theold version of the algorithm, each iteration requires two main steps per iteration, afeasibility step and few centering steps (at most two). By suitable choice of parame-ters, it is possible to eliminate centering steps altogether and just keep the feasibilitystep at each iteration thanks to the much tighter proximity estimate which is based

Improved full-Newton-step IIPM for linear complementarity problems 3

on the Lemma 3 first proposed in [18]. The algorithm is still globally convergent anditeration bound matches the best known iteration bound for these types of IPM .

The paper is organized as follows: The outline of the algorithm is given in Section2 while the analysis and complexity of the algorithm are discussed in Section 3 andSection 4 respectively. In Section 5, numerical results are presented. Concludingremarks are given in Section 6.

2. Algorithm

In this section, an outline of the improved full-Newton-step infeasible IPM formonotone-LCP (1) is presented.

It is assumed that the LCP (1) has a solution (x∗, s∗) such that

||x∗||∞ ≤ γp, ||s∗||∞ ≤ γd and µ0 =(x0)T s0

n. (2)

The assumption basically states that the solution is in within a certain (large)box. This assumption is necessary for theoretical purposes of proving the globalconvergence of the method. However, in practice the assumption is not restrictive;if the solution is not found for a certain choice of constants γp and γd, they can beincreased. If after few increases the solution is still not found, then we conclude thatthe solution does not exist or it is too big to have any practical meaning.

For a positive random starting point (x0, s0) and for any ν with 0 < ν ≤ 1, weconsider a perturbed LCP , denoted by LCPν .

s−Mx− q = νr0

xs = 0

x ≥ 0, s ≥ 0

(3)

where r0 = s0−Mx0−q is called residual. Note that for ν = 1 LCPν has an obviousstrictly feasible point, the initial starting point (x0, s0). Often, the starting point ischosen as

x0 = γpe, s0 = γde

where e denotes an n dimensional vector of ones.The following lemma, proof of which can be found in [16], shows how LCP and

LCPν are connected.

Lemma 1. The original problem (1) is feasible if and only if the LCPν (3) is feasiblefor 0 < ν ≤ 1.

Since IPMs are Newton-based methods, the standard procedure is to perturbthe complementarity condition xs = 0 and replace it by xs = µe, with positiveparameter µ. Then the system (3) becomes

s−Mx− q = νr0

xs = µe

x ≥ 0, s ≥ 0.

(4)


It is well known that if matrix M is positive definite, then the system (4) hasa unique solution for each µ > 0. These solutions, denoted by (x(ν, µ), s(ν, µ)), arecalled µ − centers of LCPν and the set of all µ-centers is called central path. Thegeneral idea of IPM is to trace the central path by gradually reducing µ to zero.However, tracing central path exactly would be inefficient. One of the main achieve-ments of IPM is to show that it is sufficient to trace the central path approximately.As long as the iterates are in a certain neighborhood of the central path it is stillpossible to prove global convergence and, moreover, achieve polynomial complexity.In addition, we are not interested in solving LCPν , we are interested in solving theoriginal LCP , which is LCPν with ν = 0. Thus, the idea is to simultaneously reduceboth µ and ν to zero, that is to work on feasibility and optimality at the same time.

The measure of proximity to the central path is given by the norm-based measure

δ(x, s;µ) = δ(v) =1

2||v − v−1||, where v =

√xs

µ. (5)

Note that δ(x, s;µ) = 0 means (x, s) is a µ-center. It is easy to show that whenν = 1, the initial pair (x0, s0) is a µ0-center. Namely, the pair (x0, s0) is strictlyfeasible solution for LCPν=1 and δ(x0, s0, µ0) = 0.

In what follows the main idea of one iteration of the algorithm is described. Weassume that at the start of each iteration, δ(x, s;µ) ≤ τ for some threshold valueτ > 0. As we already noted above, when µ = µ0 and ν = 1, (x0, s0) is a µ-center ofLCPν . Thus, initially we have δ(x0, s0;µ0) = 0 < τ which satisfies our assumption atthe first iteration. We also assume that µ and ν are connected as follows: ν = µ/µ0.

Suppose that for µ ∈ (0, µ0], we have an iterate (x, s) feasible to the system (4)and δ(x, s;µ) ≤ τ . Then, we reduce µ to µ+ = (1− θ)µ and ν+ = µ+/µ0 = (1− θ)νusing barrier parameter θ ∈ [0, 1) and find the new iterate (x+, s+) satisfying (4)with µ+ and ν+.

In the old version of the algorithm, the new iterate was found after one feasibilitystep followed by few centering steps to satisfy δ(x+, s+;µ+) ≤ τ . The centering stepswere necessary because it was not guaranteed that the feasibility step is in the τ -neighborhood of the central path. In the new version, with the appropriate choice ofthe threshold parameter τ and barrier parameter θ and using Lemma 3 it is possibleto show tighter proximity estimate which in turns guarantees that the feasibilitystep is in the τ -neighborhood of the central path, hence, eliminating the need forcentering steps. The calculation of the feasibility step is given below.

Figures 1 and 2 present graphical representations of one iteration of the old andnew versions of the algorithm respectively.

The main part of one iteration of the algorithm consists of finding a new iterateafter the reduction of µ and ν. Let (x, s) be a starting iterate in the τ -neighborhoodof the central path of LCPν . Our goal is to find a strictly feasible point (xf , sf ) thatis in τ -neighborhood of the central path of LCPν+ . First, we need to find searchdirections △fx and △fs. A direct application of the Newtons method to the system(4) leads to the following Newton system for the search directions

M △f x−△fs = θνr0

s△f x+ x△f s = (1− θ)µe− xs.(6)


Figure 1: Graphical representation of the old version of the algorithm for LCPν

Figure 2: Graphical representation of the improved algorithm for LCPν


Given the assumptions, which include that the matrix M is positive semi-definite,the above system (6) has a unique solution for (x, s) > 0. Note that we are targetingµ+ = (1 − θ)µ center rather µ center as it is done in majority of other algorithmswhich also improves the behavior of the algorithm. Taking a full Newton-step alongthe search direction, one constructs a new point (xf , sf ) as follows

xf = x+△fx

sf = s+△fs.(7)

In the Section 3, Analysis of the Algorithm, it will be shown that with the appropri-ate choice of parameters τ and θ, the feasibility point (xf , sf ) is in the τ neighbor-hood of the central path, hence, (xf , sf ) is in fact a new iterate (x+, s+) = (xf , sf ).

Previous discussion can be summarized in the following outline of the algorithm.

Improved Full-Newton-Step Infeasible IPM for LCP

InputAccuracy parameter ϵ > 0,Barrier update parameter θ, 0 < θ < 1,Threshold parameter τ, 0 < τ < 1,Initial points:x0 > 0, s0 > 0,µ = µ0 with x0s0 = µ0e,ν = 1.

beginwhile max(xT s, ||s−Mx− q||) ≥ ϵ dobeginUpdate µ = (1− θ)µ, ν = (1− θ)ν;

Calculate △fx,△f

s by solving (6);

Update (x, s) = (x, s) + (△fx,△f

s ) as in (7);end

end

Figure 3: Algorithm

In the sequel we will refer to the algorithm described in the Figure 3 simply asthe Algorithm.

3. Analysis of the Algorithm

The main part of the analysis of the Algorithm consists of showing that with theappropriate choice of the threshold and barrier parameters, the point (xf , sf ) ob-tained by the feasibility step (6) - (7) is both feasible and close enough to thecentral path thanks to the tighter proximity estimate based on the Lemma 3, i.e.,δ(xf , sf ;µ+) ≤ τ .


The analysis will require the following transformations

v =

√xs

µ, dx =

v △ x

x, ds =

v △ s

s(8)

where the operations are component-wise operations of vectors.Using the above scaled search directions the system (6) transforms into the fol-

lowing system

Mdx − ds = Dµ−1/2θνr0

dx + ds = v−1(1− θ)− v(9)

where M = DMD and D = S−1/2X1/2. The matrices X and S represent diagonalmatrix forms of vectors x and s, i.e. X = diag(x) and S = diag(s).

Using (8) and the second equation of the system (6), we have

xf sf = (x+△fx)(s+△f s) = xs+ x△f s+ s△f x+△fx△f s = (1− θ)µe+△fx△f s

= (1− θ)µe+xs

v2dfxd

fs = (1− θ)µe+ µdfxd

fs = µ[(1− θ)e+ dfxd

fs ].

(10)The following lemma then follows.

Lemma 2. The iterates (xf , sf ) are strictly feasible if and only if (1−θ)e+dfxdfs > 0.

The more practical, however, just sufficient condition for strict feasibility is givenin the following corollary.

Corollary 3.1. The iterates (xf , sf ) are strictly feasible if ||dfxdfs ||∞ < 1− θ.

The proofs of the above lemmas can be found in [5, 16].Our goal is to find an upper bound for δ(xf , sf ;µ+). The notation ω(v) =

12 (||d

fx||2+||dfs ||2) will be useful, as well as the following relationships between norms.

||dfxdfs ||∞ ≤ ||dfxdfs || ≤ ||dfx|| ||dfs || ≤1

2(||dfx||2 + ||dfs ||2) = ω(v). (11)

Then, the above corollary assumes the following form.

Corollary 3.2. If ω(v) < (1− θ), then (xf , sf ) are strictly feasible.

The following function ξ and Lemma 3, introduced in [18], play a central role inthe subsequent analysis.

ξ(t) =1 + t

1− θ+

1− θ

1 + t− 2 =

(θ + t)2

(1− θ)(1 + t)≥ 0, t > −1. (12)

Lemma 3. Let a, b ∈ Rn, r ∈ [0, 1) and f(a, b) =∑n

i=1 ξ(aibi). If ||a||2+||b||2 ≤ 2r2,then

f(a, b) ≤ (n− 1)ξ(0) + max{ξ(r2), ξ(−r2)}.

The Lemma 3 is used to prove the following lemma which gives the upper boundfor δ(vf ).


Lemma 4. If ω(v) < 1− θ, then

4δ(vf )2 ≤ (n− 1)ξ(0) + max{ξ(ω(v)), ξ(−ω(v))}.

Proof. Using (5) and (10), (vf )2 can be written as

(vf )2 =xfsf

µ+=

(1− θ)e+ dfxdfs

1− θ

µ

µ= e+

dfxdfs

1− θ.

Since 4δ(vf )2 = ||vf − (vf )−1||2, then

4δ(vf )2 =

n∑i=1

(vfi − 1

vfi

)2

=

n∑i=1

(vfi )2 − 2 +

(1

vfi

)2

=n∑

i=1

(vfi )2 +

n∑i=1

(1

vfi

)2

−n∑

i=1

2

= −2n+n∑

i=1

((1− θ) + dfxi

dfsi1− θ

)+

n∑i=1

(1− θ

(1− θ) + dfxidfsi

)

=

n∑i=1

ξ(dfxidfsi − θ).

As it can be seen from the Figure 4, ξ(t) function is an increasing function whent > −1. Furthermore, since, θ ∈ [0, 1), we have

n∑i=1

ξ(dfxidfsi − θ) ≤

n∑i=1

ξ(dfxidfsi).

Then, using Lemma 3, the result of the lemma follows. �

Finding the upper bound on δ(vf ), reduces to finding the upper bound for ω(v)due to the Lemma 4 which essentially means finding the upper bound for ||dfx||2 +||dfs ||2. To do so we need the following lemma which was proved in [12].

Lemma 5. Given the following system

Mu− z = a

u+ z = b(13)

the following hold

(1) Du = (1 +DMD)−1(a+ b), Dz = (b−Du)

(2) ||Du|| ≤ ||a+ b||(3) ||Du||2 + ||Dz||2 ≤ ||b||2 + 2||a+ b|| ||a||

where D = S−1/2X1/2, b = Db, a = Da and M = DMD.


Figure 4: Graph of ξ(t) for different θ ∈ [0, 1)

It is easy to see that application of the Lemma 5 to the system (9), leads to

a = D(θνDr0µ−1/2) = D2(θνr0µ−1/2)

b = D((1− θ)v−1 − v)

u = dfx

z = dfs .

Then, by Lemma 5 - (3), we have

||Ddfx||2 + ||Ddfs ||2 ≤ ||D((1− θ)v−1 − v)||2 + 2||D2(θνr0µ−1/2)

+D((1− θ)v−1 − v)|| ||D2(θνr0µ−1/2)||.(14)

After applying the following norm facts to (14),

(i) ||Ddfx|| ≤ ||D|| ||dfx|| , ||Ddfs || ≤ ||D|| ||dfs ||(ii) ||D2(θνr0µ−1/2)|| ≤ ||D||2||θνr0||µ−1/2

(iii) ||D((1− θ)v−1 − v)|| ≤ ||D|| ||(1− θ)v−1 − v||

where ||D|| represents a matrix norm, we obtain

||dfx||2 + ||dfs ||2 ≤ ||(1− θ)v−1 − v||2

+ 2(||θνr0µ−1/2||+ ||(1− θ)v−1 − v||

)||D(θνr0µ−1/2)||.

(15)

To find the upper bound for ||dfx||2 + ||dfs ||2 we need to find upper bounds for||D(θνr0µ−1/2)|| and ||(1− θ)v−1 − v|| respectively.


Using the definitions of D, and v, and relationship between norms, we obtain thefollowing upper bound for ||D(θνr0µ−1/2)||

||D(θνr0µ−1/2)|| =θν√µ||Dr0||

=θν√µ||X1/2S−1/2r0||

=θν√µ||√

x

sr0||

≤θ√µ

µ

µ0||√

x

sr0||1 −→ (Since ν = µ/µ0)

=θ

µ0||√

µx

sr0||1

=θ

µ0||√

µ

xsxr0||1 −→ (Since

√µ

xs=

1

v)

≤θ

µ0

1

vmin||xr0||1 −→ (Since

∣∣∣∣ 1vi xir0i

∣∣∣∣ ≤ 1

vmin|xir

0i | ≤

1

vmin|xi| |r0i |)

≤θ

µ0

1

vmin||x||1||r0||∞.

Hence,

||D(θνr0µ−1/2)|| ≤ θ

µ0

1

vmin||x||1||r0||∞. (16)

Using initial assumption (2), we have

r0 = s0 −Mx0 − q

= γde− γpMe− q

= γd

(e− γp

γdMe− 1

γdq

).

Thus, we can bound ||r0||∞ as follows

||r0||∞ = γd||e−γpγd

Me− 1

γdq||∞

≤ γd

(1− γp

γd||Me||∞ − 1

γd||q||∞

).

By assuming max{||Me||∞, ||q||∞} ≤ γd, we have that

||r0||∞ ≤ γd(1 + 1 + 1) = 3γd. (17)

The above assumption is in addition to the assumption (2). It is not restrictive andit is used to streamline the convergence analysis.

Substituting the upper bound for ||r0|| into (16), we obtain the following upper


bound for ||D(θνr0µ−1/2)||

||D(θνr0µ−1/2)|| ≤ θ

µ0

1

vmin||x||13γd

=θ

γdγp

1

vmin3γd||x||1

=3θ

γp

||x||1vmin

.

(18)

The upper bound for ||(1− θ)v−1 − v|| is found as follows

||(1− θ)v−1 − v||2 = ||(1− θ)v−1||2 − 2(1− θ)(v−1)T v + ||v||2

= (1− θ)2||v−1||2 − 2(1− θ)n+ ||v||2

= (1− θ)2||v−1||2 − 2n+ 2θn+ ||v||2

≤ ||v−1||2 − 2n+ ||v||2 + 2θn

= ||v−1 − v||2 + 2θn

= 4δ(v)2 + 2θn.

Hence, we have

||(1− θ)v−1 − v|| ≤√4δ(v)2 + 2θn. (19)

Substituting (18) and (19) into (15), we obtain the following upper bound for ||dfx||2+||dfs ||2

||dfx||2 + ||dfs ||2 ≤(4δ(v)2 + 2θn

)+ 2

(3θ

γp

||x||1vmin

+√4δ(v)2 + 2θn

)3θ

γp

||x||1vmin

. (20)

Consequently, finding the upper bound for ||dfx||2+ ||dfs ||2, reduces to finding theupper bound and the lower bound for ||x||1 and vmin respectively. This is achievedby using the following lemma, proof of which can be found in [16].

Lemma 6. Let

q(δ) = δ +√δ2 + 1.

Then, the following inequalities hold

(i) q−1(δ) ≤ vi ≤ q(δ)

(ii) ||x||1 ≤ (2 + q(δ))nγp, ||s||1 ≤ (2 + q(δ))nγd.

Using Lemma 6, the upper bound obtained in (20) becomes

||dfx||2 + ||dfs ||2 ≤ (4δ2 + 2θn) +18θ2

γ2p

||x||21v2min

+6θ

γp

√4δ2 + 2θn(2 + q(δ))q(δ)

= (4δ2 + 2θn) + 18θ2n2(2 + q(δ))2q2(δ)

+ 6θn√4δ2 + 2θn(2 + q(δ))q(δ).

(21)


Finally, we obtain the the upper bound for δ(vf ) in terms of δ and θ

δ(vf ) ≤ 1

2

√(n− 1)ξ(0) + max{ξ(ω(v)), ξ(−ω(v))} ≤ τ, (22)

where

ω(v) ≤ (2δ2 + θn) + 9θ2n2(2 + q(δ))2q2(δ) + 3θn√4δ2 + 2θn(2 + q(δ))q(δ)

andq(δ) = δ +

√δ2 + 1.

Since δ ≤ τ , δ is replaced with τ in the above upper bounds (22). Now, we needto find specific values of τ and θ such that (22) is satisfied. Analytic estimate of thesevalues is difficult, hence, we have performed a numerical estimate using MATLABcode to find specific τ and θ values satisfying the inequality (22). The results aregiven in the Table 1.

θ τ δ(vf ) : n = 2 n = 100 n = 10001

39+n15 0.1806 0.0751 0.0468

140+n

14 0.2483 0.1060 0.0721

153+n

13 0.3272 0.1787 0.1329

1170+n

12 0.4981 0.4371 0.3708

Table 1: Proximity of new iterates to µ-center for a certain choice of τ and θ

It can be observed from the Table 1, that the estimate is tighter for smallervalues of n, where n is a dimension of the problem, while, as n increases, the newiterates get closer to the central path. The above discussion can be summarized inthe following theorem.

Theorem 3.3. Let δ and τ be one of the pairs in the Table 1 and (x, s) be acurrent iterate of the Algorithm with δ(x, s;µ) ≤ τ . Then, after the feasibility stepof the Algorithm, the point (xf , sf ) is strictly feasible and satisfies (22), that is,δ(xf , sf ;µ+) ≤ τ . Thus, (xf , sf ) is the new iterate, (x+, s+) = (xf , sf ).

The above theorem essentially states that the Algorithm is well defined.

4. Complexity analysis

In this section, we calculate an upper bound on the required number of iterations ofthe Algorithm to obtain ϵ- approximate solution of monotone−LCP (1). We beginwith the following lemma proof of which can be found in [16].

Lemma 7. The following equation and inequalities hold:

(i) △ xT △ s ≤ µδ2

(ii) xfsf = µe+△x△ s

(iii) (xf )T sf ≤ µ(n+ δ2).


The above Lemma 7 is used in the proof of the theorem below that gives the upperbound on the required number of iteration for the Algorithm to find ϵ-approximatesolution of monotone− LCP (1).

Theorem 4.1. Let θ = 139+n , τ = 1

5 and µ0 = (x0)T s0

n . Then, the Algorithm

requires at most (39 + n) log 51(x0)T s0

50ϵ iterations to reach the ϵ-approximate solutionof monotone− LCP (1).

Proof. Let xk and sk be the k-th iterates of the algorithm. Then,

xTk sk ≤ µk(n+ δ2)

≤ µk(n+1

25)

= (1− θ)kµ0(n+1

25)

= (1− θ)k(x0)T s0

n(n+

1

25)

≤ (1− θ)k(x0)T s0

n(n+

1

50n) (Since n ≥ 2)

= (1− θ)k(x0)T s0

n

51

50n

= (1− θ)k(x0)T s051

50≤ ϵ.

Then by taking the logarithm of both sides, we have

log

[(1− θ)k(x0)T s0

51

50

]≤ log ϵ

log(1− θ)k + log(x0)T s0 + log51

50≤ log ϵ

log(1− θ)k ≤ log ϵ− log(x0)T s0 − log51

50

k log(1− θ) ≤ log50ϵ

51(x0)T s0

−k log(1− θ) ≥ − log50ϵ

51(x0)T s0.

Since − log(1− θ) ≥ θ, we obtain the desired iteration bound

kθ ≥ log51(x0)T s0

50ϵ

k ≥ 1

θlog

51(x0)T s0

50ϵ

k ≥ (39 + n) log51(x0)T s0

50ϵ.

�


Similarly, we calculate the required number of iterations for other θ and τ valueslisted in the Table 1. The results are shown in the Table 2.

θ τ number of iterations1

40+n14 (40 + n) log 33(x0)T s0

32ϵ1

53+n13 (53 + n) log 19(x0)T s0

18ϵ1

170+n12 (170 + n) log 9(x0)T s0

8ϵ

Table 2: Required number of iterations for different τ and θ values

Hence, we have the following corollary.

Corollary 4.2. Let δ and τ be one of the pairs in the Table 1. Then the Algorithmis globally convergent and achieves the ϵ-approximate solution of monotone− LCP(1) in at most O(n log n

ϵ ) iterations.

Recall that the iteration bound of the old version of the algorithm is

12n log33(x0)T s0

32ϵ.

It is not hard to see that both iteration bounds have the same order of magnitudestated in the above Corollary 4.2; however, constant-wise the iteration bound of thenew version of the method is much better than the old version for sufficiently largen.

It is worth noting that the iteration bounds of both old and new versions of thealgorithm match the best known iteration bound for these type of methods as far asthe order of magnitude is concerned.

5. Numerical Results

In this section, we present preliminary numerical results of the implementation of thenew and old version of the algorithm for the set of randomly generated monotone−LCPs. Recall that the the old version of the algorithm, consists of one feasibilitystep and a few centering steps at each iteration, while the improved version of thealgorithm consists of only one feasibility step. The pseudo-code of the new versionof the algorithm is outlined in Figure 3 and the pseudo-code of the old version isoutlined in [13]. Both versions are implemented in MATLAB and run on the desktopcomputer with Intel(R) core(TM) processor and 4 Gb of RAM running Windows 7operating system.

First, the old version of the algorithm and the improved version of the methodare compared . Comparison is given in the Table 3. The number of iterations (No. ofit.) represents average number of iterations while the (CPU time) represents averageCPU time of all instances of the problems of the given dimension. For the improvedversion of the algorithm, we use θ = 1

40+n and τ = 14 , while for the old version of the

algorithm the value of θ is θ = 112n while the value of τ is the same. The accuracy

for all the cases is set at ϵ = 10−4.


Size Old version Old version Improved version Improved version

No. of it. CPU time No. of it. CPU time

3× 3 385 1.3239× 10−2 439 1.9803× 10−2

5× 5 716 3.2589× 10−2 506 2.5870× 10−2

10× 10 1626 7.7880× 10−2 676 3.7156× 10−2

100× 100 23917 16.955375 2697 1.894107

Table 3: θnew = 140+n

, θold = 112n

, τ = 14, ϵ = 10−4

As expected, Table 3 shows that the improved version of the algorithm requiresless iterations and less CPU time in almost all the cases. The exception is thelow dimensional set of problems (3x3). However, as the dimension increases theimproved version of the algorithm becomes much more efficient than the old version.The reason is twofold. The θ value essentially controls the reduction of µ (optimality)and ν (feasibility); the bigger the θ, the bigger the reduction, hence, fewer iterationsare needed. For the smaller values of n, the θ value the old version of the algorithm,θold = 1

12n is bigger than the θ value of the new version, θnew = 140+n . It starts to

be the opposite with n ≥ 4 with the gap increasing as n increases. Furthermore, inthe new version the centering steps are eliminated which significantly contributes tothe better performance of the new version in comparison to the old version of thealgorithm.

Next, we investigate the effects of different values of θ parameter on the per-formance of the improved version of the algorithm. In theory, for constant valuesof θ the convergence of the Algorithm is not guaranteed. Nevertheless, we run theMATLAB implementation of the Algorithm with θ = 0.2, 0.5, 0.9 on the set of testproblems of different dimensions with accuracy being again ϵ = 10−4. Results areshown in Table 4:

Size θ = 140+n θ = 0.2 θ = 0.5 θ = 0.9

2× 2 417 45 15 55× 5 488 54 17 610× 10 577 61 20 7100× 100 1938 87 28 91000× 1000 16795 113 37 X

Table 4: Number of iterations for different θ values

It can be observed that, although the convergence of the algorithm for the con-stant values of θ is not guaranteed, the method was able to solve almost all instancesof the problem for all values of θ except the problem of the largest dimension for thelargest value of θ = 0.9. The probable reason is that the reduction taken was tooaggressive. As expected, the number of iterations reduces dramatically as the valueof θ increases. Moreover, the number of iterations does not increase significantly asthe dimension of the problem increases.

The preliminary implementation and numerical testing indicate that the Algo-rithm has certain computational merit even for the values of θ that guarantee con-vergence and especially for the constant values of θ, if one is willing to take a low risk


that method may not converge to the solution for some instances of the problem.Additional details on the implementation of the Algorithm and numerical tests canbe found in [16].

More sophisticated implementation and more numerical testing are needed tocome to the more definite conclusions about the practical behavior of the algorithm.

6. Concluding remarks

In this paper an improved version of the Full-Newton-Step Infeasible Interior-PointMethod for monotone − LCP is considered. The method is outlined in Figure 3and throughout the text we referred to it as simply the Algorithm. The old versionof the algorithm was discussed in [13, 5]. In the old version of the method, eachiteration consisted of one feasibility step and few centering steps (at most two)per iteration. The centering steps were necessary to bring the point obtained byfeasibility step to the τ -neighborhood of the central path again. In the improvedversion of the method, with the suitable choice of the threshold parameter τ andbarrier parameter θ, it is guaranteed that after one feasibility step, the new iterateis feasible and immediately in the τ -neighborhood of the central path thanks to themuch tighter proximity estimate which is based on the new Lemma 3 introduced in[18]. Thus, the centering steps are eliminated.

The good features of the old version of the method are still preserved in the newversion. The Algorithm does not require strictly feasible starting point (infeasiblealgorithm) and it uses full-Newton-steps, thus, avoiding calculations of a step-sizeat each iteration. Furthermore, a nice feature of both versions of the method is thatthey work on simultaneously reducing infeasibility and achieving optimality.

The Algorithm is globally convergent for the values of the threshold and barrierparameters listed in the Table 1. Furthermore, the Algorithm matches the bestknown iteration complexity for these types of methods, in order of magnitude, whichis O(n log n

ϵ ). Although the order of magnitude of the iteration bounds of the oldand new version of the method is the same, constant-wise the iteration bound of theimproved version of the method is much better than the old version for sufficientlylarge n.

The disadvantage is that the Algorithm is still a short-step method, becauseθ = O

(1n

). However, the preliminary implementation and numerical testing of the

Algorithm indicate that the method has certain computational appeal even for thevalues of θ that guarantee convergence. For the constant values of θ, the Algorithmbecomes long-step method, but in that case, the global convergence of the methodis no longer guaranteed. However, the initial numerical testing shows that in mostinstances the Algorithm still converges and converges very fast, with number ofiterations reducing dramatically as the value of θ increases. Moreover, the numberof iterations does not increase significantly as the dimension of the problem increases.Furthermore, numerical testing shows that the new version of the method performsmuch better than the old version. More sophisticated implementations and morenumerical testing are needed to have a better idea about the practical behavior ofthe algorithm.


In addition to more numerical testing, some directions for further research includegeneralization of the method to more general classes of LCPs such as P∗(κ)-LCPand LCPs over symmetric cones.

References

[1] Anitescu, M., Lesaja, G. and Potra, F. A. (1997). Equivalence between different formu-lations of the linear complementarity problem. Optimization Methods and Software,7, 265–290.

[2] Bai, Y. Q., Lesaja, G. and Roos, C. (2008). A new class of polynomial interior-pointalgorithms for linear complementarity problems. Pacific Journal of Optimization, 4,19–41.

[3] Chung, S. J. (1979). A note on the complexity of LCP , The LCP is strongly NP-complete. Technical report 792, Department of Industrial and Operations Engineering,The University of Michigan, Ann Arbor, Michigan.

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[5] Drummer, A. M. (2012). Infeasible full Newton-step interior point method for thelinear complementarity problems. Master thesis, Georgia Southern University, States-boro.

[6] Facchinei, F. and Pang, J.-S. (2003). Finite-Dimensional Variational Inequalities andComplementarity Problems. Springer, New York.

[7] Ferris, M. C., Mangasarian, O. L. and Wright, S. J. (2007). Linear programming withMATLAB, SIAM, Series on Optimization.

[8] Gill, P. E., Murray, W., Saunders, M. A., Tomlin, A. J. and Wright, H. M. (1986). Onthe projected Newton barrier methods for linear programming and an equivalence toKarmarkar’s projective method. Math Program, 36, 183–209.

[9] Karmarkar, N. (1984). A polynomial-time algorithm for linear programming. Combi-natorica, 4, 373–395.

[10] Kojima, M. and Mizuno, S. and Yoshise, A. (1989). A polynomial-time algorithm fora class of linear complementarity problems. Mathematical Programming, 44, 1-26,

[11] Kojima, M. and Mizuno, S., Noma, T. and Yoshise, A. (1991). A Unified Approachto Interior Point Algorithms for Linear Complementarity problems, Lecture Notes inComputer Science 538, Springer-Verlag, New York.

[12] Lesaja, G. (1996). Interior-point methods for P*-complementarity problems. Ph.D.thesis, University of Iowa, Iowa City.

[13] Lesaja, G., Drummer, A. and Miletic, L. (2012). Infeasible full Newton-step interior-point method for linear complementarity problems. Croatian Operational ResearchReview (CRORR), 3, 163–175.

[14] Lesaja, G. and Roos (2010). C. Unified analysis of kernel-based interior-point methodsfor P∗(κ)-LCP. SIAM Journal of Optimization, 20(6), 3014–3039.

[15] R. D. C. Monteiro and S. J. Wright (1996). A superlinear infeasible-interior-pointaffine scaling algorithm for LCP. SIAM J. Optim., 6, 1–18.

[16] Ozen, M. (2015). Improved infeasible full Newton-step interior point method for thelinear complementarity problems. Master thesis, Georgia Southern University, States-boro.

[17] Renegar, J. (1988). A polynomial-time algorithm, based on Newton’s method, forlinear programming. Math Program, 40, 59–73,

[18] Roos, C. (2015). An improved and simplified full-Newton step O(n) infeasible interiorpoint method for linear optimization. SIAM J. Optim., 25, 102–114.


[19] Wright, S. J. and Zhang Y. (1996). A superquadratic infeasible-interior-point methodfor linear complementarity problems. Math. Programming Ser. A, 73, 269–289.

Croatian Operational Research Review 19 CRORR 7(2016), 19–32

http://www.hdoi.hr/crorr-journal ©2016 Croatian Operational Research Society

Traffic flow model at fixed control signals with discrete service time distribution

Lucky I. Igbinosun1,* and Sunday E.Omosigho2 1 Department of Mathematics and Statistics, University of Uyo,

Nwaniba Road, Uyo, Akwa Ibom State, Nigeria. ⟨[email protected]⟩

2 Department of Mathematics, University of Benin, Uselu, Benin City, Nigeria. ⟨[email protected]⟩

Abstract. Most of the models of road traffic flow at fixed-cycle controlled intersection assume stationary distributions and provide steady state results. The assumption that a constant number of vehicles can leave the system during the green phase is unrealistic in real life situations. A discrete time queuing model was developed to describe the operation of traffic flow at a road intersection with fixed-cycle signalized control and to account for the randomness in the number of vehicles that can leave the system. The results show the expected queue size in the system when the traffic is light and for a busy period, respectively. For the light period, when the traffic intensity is less than one, it takes a shorter green cycle time for vehicles to clear up than during high traffic intensity (the road junction is saturated). Increasing the number of cars that can leave the junction at the turn of the green phase reduces the number of cycle times before the queue is cleared.

Key words: vehicle queue size, fixed cycle, road traffic intersection

Received: January 25, 2015; accepted: March 4, 2016; available online: March 31, 2016

DOI: 10.17535/crorr.2016.0002

1. Introduction

Road traffic queues are waiting lines which occur whenever vehicles must wait to access a facility. An intersection may be controlled or signalized for a number of reasons, most of which relate to the safety and effective movement of conflicting vehicular and pedestrian flows through the intersection. The facility may be busy and therefore unavailable to render the required service, thus resulting in congestion. Road traffic congestion is a problem in many countries. It causes considerable costs due to unproductive time losses, accidents, air

* Corresponding author.

20 Lucky I. Igbinosun and Sunday E. Omosigho

pollution, fuel wastage, etc. The problem of road traffic congestion is worsened in many cities by the lack of real time information on traffic flows, lack of adequate data gathering facilities and a systematic methodology for optimizing road traffic flow through available road facilities. Given the cost of road traffic congestion to society, a comprehensive study of urban road network is necessary. A good understanding of vehicular traffic flow is a knowledge gap in modern societies.

Some initiatives such as staggered work hours, flextime, ramp metering, road widening, etc. have been proposed to alleviate problems associated with road traffic congestion, but the level and period of congestion in most major urban areas is increasing.

Viti and Van Zuylen [23] argue that “queue estimation at a controlled intersection is a problem in transportation engineering and operations research.” Queue dynamics have been modeled as deterministic phenomena, and under rather limiting assumptions of a steady state scenario. In urban networks, these assumptions are far from being deterministic or certain. Deterministic models based on fluid theory have also been studied and demonstrated as appropriate for highly oversaturated scenarios [14]. But Viti and Van Zuylen [23] state that “these models are incapable of estimating the temporal effects that occur when signals operate near the signal capacity. Modeling these effects and their probability of occurrence becomes particularly important.” The vast set of models presented in [23] indicate that there is still no clear insight into the way queues are experienced at signalized intersections.

The paper is arranged as follows: We present the problem statement in Section Two and provide a cursory review of literature relevant to the subject of study in Section Three. In Section Four, we present the discrete model for traffic flow at a road intersection with a fixed-cycle light. The results are presented in Section Five and the conclusion of the work is given in Section Six.

2. Problem statement

Viti and Van Zuylen [23] list, among others aspects, the shortcomings of the previous time-dependent models which include: (1) The assumption that arrivals and departures follow a specific distribution and have stationary rates. (2) The initial queue is assumed to be zero at the beginning of the evaluation period. In the analysis of queues at fixed-controlled intersections, Viti and Van Zuylen [23] however, provide a methodology that enables the capturing of dynamic and stochastic effects on queue length that originate from the variability of arrivals. Omosigho [13] however, shows that if the number of cars at the beginning of the green phase, i, is less or equal to the number of cars departing during the green phase, s, i.e. i ≤ s, the expression,

Traffic flow model at fixed control signals with discrete service time distribution 21

si

nn sinp

0

)( , maxasi (1)

given by Viti and Van Zuylen [23] fails to properly capture a situation, which in turn could lead to a zero state. Omosigho [13] enumerates the events that may lead to state zero for all i ≤ s and gave a corrected version as,

))(2()(2)0(00

ks

i

s

k

iAkPPQG , (2)

where )0(PQG is the probability of having no cars in the queue at the end of the green phase, P2(k) the probability of k cars in the queue at the beginning of the green phase, and A2(i) the probability of i arrivals during the green phase. However, Viti and Van Zuylen [23] and Omosigho [13] assume that the number of vehicles that can leave the system during the green phase is constant. In practice, this assumption is unrealistic. In view of the limitations, we develop a discrete time-dependent model to describe the operation of traffic flow at a road intersection with fixed-cycle signalized control and to account for the randomness in the number of vehicles that can leave the system.

3. Literature review We will discuss some existing works on road traffic queues at signalized intersections using queuing theory presented in this paper.

Darroch [8] presents a formal solution for the stationary distribution of queue length at a fixed-cycle traffic light for a fairly general distribution of arrivals and for a single stream of vehicle. Expressions for the expected queue length and the expected delay per vehicle were presented.

Rouphail et al. [14] analyse road traffic flow at signalized intersections and emphasizes the theory of descriptive models of traffic flow. Several models including the fluid theory approach and the steady-state queuing approach along with their various advantages and limitations are discussed. In studying or modeling traffic flow at controlled intersections, the average delay per vehicle, number of stopped vehicle, number of queued vehicles, expected delays and the average queue length are among the important performance measures that can be derived. Due to the minimal effect of the stochastic queuing approach in extremely congested conditions, the fluid theory approach is considered more appropriate to use for highly over-saturated conditions. In concluding, Rouphail et al. [14] agree that there are areas requiring further attention and research, and the assumption of uncorrelated arrivals found in most models is inadequate for describing flow. Secondly, an estimation of the initial overflow queue at a signal is inadequately understood and documented. Queuing models constrained by physical space available for queuing should also be developed.


Van Leeuwaarden [17] considers the fixed-cycle traffic light (FCTL) queue, where vehicles arrive at an intersection controlled by a traffic light and form a queue. He presents the queue length and delay distributions by considering two arrival Poison and geometric distributions. A method for inverting the generating functions was also presented with the conclusion that the mean delay is sensitive to the stochastic properties of the arrival distribution.

Dion et al. [9] present a summary of delay models for signalized intersections and include deterministic queuing models, a shock wave delay model, steady-state stochastic delay models, and time-dependent stochastic delay models for both under-saturated and over-saturated conditions, microscopic simulation delay models. The ITEGRATIONTM simulation package was used as a benchmark for other models with the conclusion that all compared delay models produce similar results for signalized intersections when traffic intensity is low. But under heavy traffic, noticeable differences occur.

Kakooza et al. [11] use a simple mathematical model (steady state m/m/s queuing model) to analyze different types of road intersections in terms of their performance and in relation to managing traffic congestion and establishing congestion for stabilizing road intersections after sufficiently longer periods of time (steady-state). The authors analyzed un-signalized and signalized intersections, and roundabouts. They obtained expressions for the steady state expected number and waiting time of vehicles stopping at a road intersection interrupted by delays. For a single lane system (m/m/1), Kakooza et al. [11] give an expression for the expected number of vehicles in the system as:

E X (3)

Where, λ is the average number of vehicles arriving at an intersection per unit time; r is the rate of disappearance or clearance of the delays; f is the rate of occurrence of delays; and µ0 denotes the service rate with no delays. For stability (avoiding an ever-increasing back-log of vehicles), they assert that the traffic intensity (ρ λ μ ,for a single lane) must be less than the proportion ofdelay time (τ ). The waiting time was given as:

W X (4)

They conclude that “since congestion is as a result of heavy traffic, the best option in managing traffic congestion at an intersection would be to replace un-signalized and roundabout intersections with signalized intersections in case such intersections have heavy approaching traffic’’.

Their method and results are similar to those of Baykal-Gursoy et al. [5] using probability generating functions. Excess demand for road space, irregular


occurrences such as traffic accidents, vehicle disablements, spilled loads and hazardous materials were identified by Baykal-Gursoy et al. [5] as some causes of road flow reduction.

Baykal-Gursoy and Xiao [4] consider a steady-state m/m/∞ queuing system subjected to random interruptions of exponentially distributed durations. Total system breakdown and partial failure were investigated, and in both cases present the expected number of vehicles in the system. Under the impact of the interruption, all servers work at lower efficiency until the interruption is cleared. Varying the parameters with respect to the expected number of vehicles in the system (as given in [5]), the impact of each parameter of E(X) was presented. The steady-state performance measure (expected number of vehicles in the system) was presented for m/m/2 and m/m/3 in [3] where they stated that “The analysis of the M/MSP/C queue with n server states clearly indicates that explicit solutions for the general case would be difficult to obtain’’.

Van Zuylen and Viti [18] propose a method of solving the calculation of queues and delays at a signalized intersection using a Markov chain model to derive the probability distribution of queue length within a cycle time and using any initial queue length distribution. Based on the dynamics of the expectation value for queue length, the authors derive a formula for the delay in fixed time traffic control; provide an exact probabilistic formulation of the queuing process within a cycle, which enables justification of the dynamic and stochastic character of overflow queues, especially at signals that operate near capacity. The challenge, however, is in the assumption of a constant, deterministic departure rate, while no specific distribution is assumed for arrivals.

Viti and Van Zuylen [22] propose a probabilistic model for queues at a fixed and actuated controlled signalized intersection. They showed that a probabilistic modeling approach can help to explain different traffic conditions, different signal types, etc. by simply assuming a probability distribution for the number of arrivals and departures within a cycle. However, these models do not provide insight into the way queues are experienced by drivers at signalized intersections (see also [22]). Road traffic conditions are not static; this has been observed by many authors including Cherrett et al. [6]. It is therefore desirable to develop models that incorporate realistic assumptions.

Lartey [12] presents a mathematical modeling and prediction of road congestion on an urban road in Ghanaian using queuing theory based on a stochastic process and initial value problem framework. The approach describes performance measure parameters, thus predicting the increase in queue level at a signalized intersection and subsequently providing an insight into road vehicular congestion and how the occurrence of such congestion can be managed. The author links time evolution phenomena to vehicle queues at a signalized intersection to constitute an observed stochastic processes. The work presents an analysis of vehicular traffic congestion at a signalized intersection

24 Lucky I. Igbinosun and Sunday E. Omosigho with the hope of gaining insight into certain performance measures. Lartey [12] asserts that the distribution of the busy period is the length (or duration) of time during which the server remains busy. The busy period is the interval from the moment of the arrival of a unit vehicle at an empty system to the moment that the channel becomes free for the first time. This, therefore, constitutes a random variable.

In [1], a discrete event simulation model was developed, verified and validated. The model was tested for different scenarios. The objective by Aljaafreh et al [1] was to determine the optimized timing parameters for traffic signal based through simulation. The author argues that “designing a real-time proactive adaptive control model for traffic will enhance the performance of traffic lights”. In this paper, however, we seek to show the behavior of a signalized road traffic intersection using a simple discrete model. The model will be tested for both cases of road traffic intensities ( 1 and 1).

Vasic [20] makes certain considerations in her Discrete Simulation Model for Heterogeneous Traffic Including Bicycles on Urban Road Networks. The heterogeneity of vehicle type was provided by allowing different cell sizes in the model, based on cells of different sizes and specific vehicle type parameters, such as maximal velocity.

Tiriolo et al. [15] present a traffic flow model, based on the cell transmission concept, with the aim of recording urban traffic dynamics taking into account complex flow interactions among lane groups at upstream of signalized intersections. The model is designed to simulate, at macroscopic level, more realistically the dynamic interaction of queues among neighboring lanes and intersections for large scale urban networks. The traditional intersection traffic model is extended to take into account some real aspects of traffic conditions, such as the proportion of turning and lane width with respect to different movements.

Asenime and Mobereola [2] evaluated traffic behavior at different peak periods at the Maryland Interchange in Metropolitan Lagos. The authors categorized predictable behavior into three periods namely: morning, inter-peak and evening peak periods. Asenime and Mobereola [2] reported that the signal level of traffic service controlled traffic flow effectively at the intersections, however, they agreed that land use factors had a negative impact on traffic flow.


4. Model development 4.1. Notation We will list the parameters used in this paper as follows: Let )(tPij be the probability of transition from state i to state j during a time interval of length t.

)(kAG , the probability of k arrivals during the green phase, )(iPG , the probability of i cars in a queue at the beginning of the green phase.

mksk )1(1, , the number of cars that can depart during the green phase.

kP , the probability that ks cars leave the queue during the green phase.

maxQ , the maximum number of cars that can queue at a signalized intersection. , the probability that there are cars in the queue at the end of the green

phase. , the arrival rate for the Poisson arrival process.

)(iPR , the probability of i cars in the queue at the beginning of the red phase. )(kAR , the probability of k arrivals during the red phase.

)( jPQR , the probability of j cars in the queue at the end of the red phase. , traffic intensity (road utilization factor) 4.2. Model assumptions The arrival manner is considered Markovian and independently distributed with respect to the Poisson arrival rate ( ). Where there are multiple lanes regulated by the same traffic light, we assume that all the lanes are treated as one, hence we have a single server case and no overtaking is allowed. We also assume that during the green phase, ),...3,2,1( mksk cars can be served, but no queue is allowed ahead of the phase. The initial queue size may or may not be zero at the beginning of the green phase. A maximum number of cars (

maxQ ) is allowed at any given green phase without blocking other junctions. We further assume that the light cycle is between red and green, the amber phase is considered to be either part of the red or green phase. The cars leaving the green phase is not constant, but that the green phase (green light duration) is constant and that the traffic light is 100% effective.


4.3. The model

Given that the number of cars leaving the queue during the green phase is no longer a fixed value, it is a discrete random variable. It is given the value kswith a probability of ),...3,2,1( mkPk . The goal is to determine the probability of j cars in the queue at the end of the green phase. To do so, the following conditions apply.

At the beginning of the green phase, we assume i number of cars waiting for service; i=0, 1, 2, 3,…. When i=0, no car is queuing at the beginning of the green phase. We can have 1j cars in the queue at the end of the green phase if:

ks cars depart during the green phase. i cars in the queue at the beginning of the green phase. a cars arrive during the green phase.

For j cars at the end of the green phase, we must have:

ksaij , (6) isja k , (7)

Hence the probability of having j cars at the end of the green phase can be modeled as:

max1 0

,...,3,2,1,)()()( QjPisjAiPjPGm

k

sj

ikkGG

k

(8)

If k=1, equation (1) reduces to:

1

011 )()()(

sj

iGG PisjAiPjPG

sj

iGG isjAiP

0

)()( . (9)

Where 1.),( 11 Pconstissss . This equation is consistent with that given in [13].

Meanwhile during the red phase, the probability of having j cars in the queue at the end of the phase is given by:

otherwise

QiijijARiPRjPQR

j

i

0

,...1,),()()( max

0 (10)

Where )( jPQR is the probability of j cars at the end of the red phase (see [13]).


The model proposed in this paper generates time dependent probabilities of having a j number of vehicles in the queue at the end of the discrete time periods (also known as regeneration points. See [7, 13]). Vehicles form a queue at the beginning of the red phase while waiting for service, and as the light turns green, cars leave the queue. Even though the cycle period is fixed, the rate of cars leaving the queue at different periods is not constant.

5. Solution

We used MATLAB 7.5 software to generate time dependent probabilities of having a numbers of cars in the queue at the end of the discrete time periods. For example, if the number of cars allowed to leave the system when the light turns green assumes some set of numbers, the initial number of cars in the queue is 10. Figure 1 shows that the queue gradually disappears.

Figure 1: The expected queue length when the number of cars allowed to go during the

green light is not constant. 1 , 50

Observe that when the traffic intensity is low (less than one), the queue formed during the red phase is cleared during the green phase as shown in the saw-tooth-like graph in Figure 1. This result agrees with Omosigho [13] in terms of the shape, but there are differences in the expected number of vehicles in the queue. Our focus is on the time dependent behavior of the system for a short

0 1 2 3 4 5 60

2

4

6

8

10

12

time step

Exp

ecte

d qu

eue

size

28 Lucky I. Igbinosun and Sunday E. Omosigho period of the day. For example, two to three hours is considered a short period during the day. We examine a period of free flow (when the road traffic intensity is less than one, i.e. 1 ), and a rush-hour or busy period ( 1 ). At both periods, we show that there is a relationship between the traffic intensity and .

When 1 , we observe that increasing the value of did not affect the result. This is so because for both phases of the traffic light, the number of cars is always served (see Figures 1 and 2). Figure 2 shows the various graphs indicating the expected number of cars in the queue for various values of . Notice that the queues always reach zero.

Figure 2: Expected number of vehicles in the queue for various values of .

10, 50, 100, 1000

0 1 2 3 4 5 60

2

4

6

8

10

12

green phase

Exp

ecte

d qu

eue

size

lamda=0.9

lamda=0.5lamda=0.1


Queue size ( 0.1) Queue size ( 0.5) Queue size ( 0.9) 10.00 10.00 10.0010.10 10.50 10.903.47 4.06 4.772.87 3.73 4.700.38 0.75 1.340.28 0.68 1.400.00 0.78 0.28

Table 1: Expected queue size at the turn of green phase

Table 1 shows the expected number of vehicles in the queue at the end of the green phase when the traffic intensity is less than one. Observe that the queue gradually vanishes.

When 1 , the values of and the number of vehicles in motion during the green phase becomes important. This is true because the expected number of cars in the queue will continue to grow in time if the number of vehicles permitted to head off at the onset of the green phase is small. Figure 3 demonstrates the fact that congestion results at the traffic junction. In practice, this is a problem. One way of solving this is to adjust the green phase duration to accommodate the passage of more vehicles.

Figure 3: Expected queue size when the traffic intensity is greater than one,

1,2 2, 100.

0 10 20 30 40 50 60 70 80 90 10010

12

14

16

18

20

22

24

26

28

time

E(Q

)


Figure 4: Expected queue size when the traffic intensity is greater than one,

2, 100, but with increased number of cars permitted to go during the green phase.

Figure 4 shows that increasing the number of cars permitted to go during

the green phase reduces congestion (increase of the queue size). It takes longer green circle periods for the queue to clear when a very limited number of cars are permitted to go during the green phase. 6. Conclusion A discrete time model of traffic flow at signalized intersections has been presented in this paper, where the proposed model considers the case when the number of cars departing the queue at the turn of the green light is not constant and therefore assumes a probability distribution. Constant departure rates have been presented in the literature (see [23, 13]). The results presented show that transient measures of performance (expected queue length) at different cycle times can be achieved. They also show that for very small value(s) of traffic intensity, there exists a direct relationship between the departure size and the utility function of the traffic junction. Allowing a very small number(s) of cars to depart at the turn of the green cycle phase reveals that traffic build up occurs over longer cycle times in order to clear when compared to a greater number(s)

0 2 4 6 8 10 120

5

10

15

time

E(Q

)

s=(1,2)

s=(5,6)


of departing cars. When the light turns green, and cars depart from the queue, the study has shown that the number of cars leaving the service point may not necessarily be constant. Observation shows, however, that the nature and type of vehicles (for example, it takes a longer time for a truck vehicle to leave the queue compared to a car) are a factor in determining the number of vehicles that can leave the queue at the turn of the green phase. Motor bikes also struggle to receive service at the start of green phase, [13]. Incorporating these into the model assumption can be an interesting formulation in future research. Results from the model in this paper will help road traffic engineers in building of road traffic lights to function at various green phase times in order to indicate the location and density of cars on a particular road. References [1] Aljaafreh, A., Al-Oudat, N. and Saleh, M. (2014). Adaptive traffic-signal control

using discrete event simulation model. International Journal of Computer Applications, 101(12), 7–12.

[2] Asenime, C. and Mobereola, D. (2015). Traffic behaviour at a signalised intersection in metropolitan Lagos. American Journal of Social Issues and Humanities, 5(2), 453–461.

[3] Baykal-Gursoy. M. and Duan, Z. (2006). M/M/C Queues with Markov modulated service processes. Value Tools, October 14, 2006. Pisa, Italy.

[4] Baykal-Gursoy, M. and Xiao, W. (2004). Stochastic decomposition in M/M/∞ Queues with Markov modulated service rates. Queueing Systems, 48(1), 75–88.

[5] Baykal-Gursoy, M., Xiao, W. and Ozbay, K. (2009). Modeling traffic flow interrupted by incidents. European Journal of Operational Research, 195(1), 127–138.

[6] Cherrett T., McLeod F., Bell H. and McDonald M. (2002). Journey time estimation using single inductive loop detectors on non-signalized links. Journal of Operations Research Society, 53(6), 610–619.

[7] Cox, D. R. and Smith, W. L (1963). Queues. Chapman and Hall, London. [8] Darroch, J. N., Newell, G. F. and Morris, R. W. J. (1964). Queues for vehicle-

actuated traffic light. Operations Research, 12(6), 882–895. [9] Dion, F., Hesham, R. and Youn-Soo, K. (2004). Comparison of delay estimates at

under-saturated and over-saturated pre-timed signalized intersections. Transportation Research Part B, 38(2), 99–122.

[10] Igbinosun, L. I. (2002). Road traffic queues and delays at road junction: A case study. Unpublished M.Sc Project submitted to the post graduate school, University of Benin, Benin City, Nigeria.

[11] Kakooza, R., Luboobi, L. S. and Mugisha, J. Y. T. (2005) Modeling traffic flow and management at un-signalized, signalized and roundabout road intersections. Journal of Mathematics and Statistics, 1(3), 194–202.

[12] Lartey, J. D. (2014). Predicting traffic congestion: A queuing perspective. Open Journal of Modelling and Simulation, 2, 57–66.

32 Lucky I. Igbinosun and Sunday E. Omosigho [13] Omosigho, S. E. (2011). Discrete time queuing model for queues at fixed control

signalized intersection. International Conference on Scientific Computing. Abuja, August 14th–20th.

[14] Rouphail, N., Tarko, A. and Li, J. (2000). Traffic flow at signalized intersections, Chapter 9 of the Update of Transportation Research Board Special Report 165, ‘Traffic flow theory’, 1998. Available at http://www‐

eta.ornl.gov/cta/research/trb/tft.html [Accessed on May 12, 2015]. [15] Tiriolo, M., Adacher, L. and Cipriani, E. (2014). An urban traffic flow model to

capture complex flow interactions among lane groups for signalized intersections. Procedia - Social and Behavioral Sciences 111, 839–848.

[16] Van Hinsbergen, C. P. I. and Van Lint, J. W. C. (2008). Bayesian combination of travel time prediction models. Transportation Research Record, 2064, 73–80.

[17] Van Leeuwaarden, J. S. H. (2006). Delay analysis for the fixed-cycle traffic-light queue. Transportation Science, 40(2), 189–199.

[18] Van Zuylen, H. J. and Viti, F. (2006). Queues at controlled intersections: The old theory revisited. In: Proceedings of the 2006 IEEE Intelligent Transportation Systems Conference, Toronto, Canada, September 17th–20th.

[19] Van Zuylen, H. J. and Viti, F. (2003). Uncertainty and the dynamics of queues at controlled intersections. Proceedings CTS-IFAC Conference, 6-8 August; Tokyo. Elsevier.

[20] Vasci, J. (2014). A discrete simulation model for heterogeneous traffic including bicycles on urban road networks. Unpublished PhD Thesis submitted to School of Computing, Faculty of Engineering and Computing, Dublin City University. Available at: http://www‐doras.dcu.ie/19765/1/jv_thesis.pdf [Accessed on December 16, 2015].

[21] Viti, F. and Van Zuylen, H. J. (2004). Modeling queues at signalized intersections. Transportation Research Record 1883, 68–77.

[22] Viti, F. and Van Zuylen, H. J. (2009). The dynamics and the uncertainty of queues at fixed and actuated controls: A probabilistic approach. Journal of Intelligent Transportation Systems, 13(1), 39-51.

[23] Viti, F. and Van Zuylen, H. J. (2010). Probabilistic models for queues at fixed control signals. Transportation Research Part B, 44(1), 120–135.



The critical node problem in stochastic networks with discrete-time Markov chain

Gholam Hassan Shirdel1,* and Mohsen Abdolhosseinzadeh1

1 Department of Mathematics, Faculty of Basic Science, University of Qom

Al-Ghadir Boulevard, Qom, Iran E-mail: ⟨[email protected], [email protected]⟩

Abstract. The length of the stochastic shortest path is defined as the arrival probability from a source node to a destination node. The uncertainty of the network topology causes unstable connections between nodes. A discrete-time Markov chain is devised according to the uniform distribution of existing arcs where the arrival probability is computed as a finite transition probability from the initial state to the absorbing state. Two situations are assumed, departing from the current state to a new state, or waiting in the current state while expecting better conditions. Our goal is to contribute to determining the critical node in a stochastic network, where its absence results in the greatest decrease of the arrival probability. The proposed method is a simply application for analyzing the resistance of networks against congestion and provides some crucial information of the individual nodes. Finally, this is illustrated using networks of various topologies. Key words: stochastic network, discrete-time Markov chain, arrival probability, critical node problem

Received: April 2, 2014; accepted: March 17, 2016; available online: March 31, 2016

DOI: 10.17535/crorr.2016.0003

1. Introduction The shortest path problem (SP) is one of the fundamental network optimization problems and has been studied extensively. Polynomial time algorithms can be used for the deterministic shortest path problem [5, 6, 7]. However, the stochastic nature of real world problems has led to new stochastic versions of the SP problem, especially in telecommunications and transportation networks. The stochastic shortest path problem (SSP) is defined in stochastic networks where the arc lengths are the stochastic variables or the existence of arcs or nodes in the network are defined stochastically (e.g. using uniform distribution and the probabilities that arcs are not congested are known). Our goal is to


34 Gholam Hassan Shirdel and Mohsen Abdolhosseinzadeh

determine which node causes the greatest damage, whenever absent during network routing.

The SSP problem has been developed by researchers based on stochastic programming. Liu [12] considered the SSP problem under the assumption that the arc lengths are random variables. He established indefinite programming models in line with decision criteria and converted the models into deterministic programming problems. Gao [9] presents a method for finding the α-shortest path and the shortest path in a network using a probability distribution of SP length. Pattanamekar et al. [16] consider the travel time uncertainty by incorporating two components: the individual travel time variance and the mean travel time forecasting error. Nie and Fan [14] formulated the stochastic on-time arrival problem as dynamic programming. They consider independent random travel times to be directed link lengths. Fan et al. [8] minimized the expected travel time such that each link was assumed to be congested or not, with known conditional probability density functions for link travel times.

In this paper, a stochastic process is simply applied to obtain an optimality index, rather than the stochastic programing methods. The length of the SSP is defined as the arrival probability from a source node to a destination node. A discrete-time Markov chain (DTMC) is established according to the uniform distribution of the existing arcs and the arrival probability is computed as a finite transition probability from the initial state to the absorbing state. The states of the established DTMC contain a number of traversed nodes in the original network. The proposed method provides comprehensive information on the resistance of the network against congestion during transmission from one node to another one. Kulkarni [11] developed an exact method based on the continuous-time Markov chain in order to compute the distribution function of the length of the SP. Azaron and Modarres [3] developed Kulkarni's method to queue networks. Thomas and White [19] modeled the problem of constructing a minimum expected total cost route from an origin to a destination as a Markov decision process. They wanted to respond to dissipated congestion over time according to some known probability distribution. Our model gives some crucial information on nodes, and it determines the critical network node with the greatest decrease in the arrival probability.

The uncertainty condition associated with the network topology is a clear motivation in considering the SSP problem. The conditional probabilities of leaving one node for another node are supposed to be known. A DTMC stochastic process with an absorbing state is established and the transition matrix is obtained. Two conditions at any state of the established DTMC are assumed for the absorbing state: departing from the current state to a new state whenever a larger labeled node is visited, or waiting in the current state and expecting better conditions. Subsequently, the probability of arrival at the

The critical node problem in stochastic networks with discrete-time Markov chain 35

destination node from the source node in the network is computed. Finally, we develop the proposed method by determining the critical node in the network.

The remainder of the paper is organized into a number of sections. Section 2 consists of some preliminary definitions and assumptions for the considered model of the stochastic network. The established DTMC, the proposed method for the arrival probability and its development to obtain the critical node are presented in Section 3. In Section 4, a number of implementations of the proposed method on the networks using different topologies are provided.

2. The model of an unstable stochastic network

Consider network G = (N,A) to be a directed acyclic network with node set N and arc set A. We can relabel the nodes in a topological order such that for any

Aji ),( , ji [1]. The physical topology for any Aji ),( shows a connection of nodes Nji , . Actually, the physical topology shows the possibility of communication between nodes in the network. To model the unstable topology of a network, we took into consideration communication networks, where physical connections between nodes exist but traversing any further toward the destination node is not possible due to probable congestion. Network G has an unstable topology if there are some facilities in the network but they cannot be utilized. Hence, the existence of any arc Aji ),( does not imply stable communication between nodes Nji , all the time (it may be congested). The presumption is that the existence probabilities of the arcs are made known by the uniform distribution.

Now, consider the situation where flow reaches a node but cannot progress further because of an unstable topology (some arcs are congested), and there is a waiting period for the onset of more favorable conditions. There are two options for the wait situation. First, waiting at a particular node with the expectation that some facilities will be released from their current condition, which is called Option 1. To model such conditions, we consider artificial loops (indicated by dash arcs in Figure 1) at any node except the destination node. Second, some arcs are traverse that do not lead to visiting a new node, which is called Option 2. The stochastic variable of arc ( , )i j N is shown by ijx . If

1ijx , it then becomes possible to traverse arc ),( ji (the connection exists), otherwise 0ijx (the connection does not exist). The existence probability of arc ),( ji is ]1[ ijij xPq . The existence of artificial arc ),( ii means the

decision has been made to wait at node i, so then { :( , ) }

1ii ijj i j Aq q

.


Figure 1: The example network with 5 nodes and 7 arcs

Figure 1 shows the example network with its topological ordered nodes.

This initial topology is the physical topology of the network. Node 1 is the source node and node 5 is the destination node. Arc (1, 4) cannot be traversed as it does not exist in the physical topology ( 14 0q ). However, the arcs in the physical topology might be experiencing congestion based on known probabilities. The numbers on the arcs indicates the values of ijq .

3. The established discrete-time Markov chain The discrete time stochastic process { , 1,2,3,...}rX r is called a Markov chain if it satisfies the following Markov property (see [17])

klkrlrnmrkrlr pSXSXPSXSXSXSXP ]|[],...,,|[ 1111 .

State space S1 S2 S3 S4 S5 S6 S7 S8

Current nodes {1} {1,2} {1,3} {1,2,3} {1,2,4} {1,3,4} {1,2,3,4} {1,2,3,4,5}

Table 1: The state space of the example network

Any state Sk of the established DTMC determines the traversed nodes of

the original network. For the example network (Figure 1), the created states Si, are shown in Table 1. The probability of a conditional transition to the next state depends on the current state and is independent of previous states. Let

{ , 1,2,3,...}iS S i , then the initial state }1{1 S of DTMC contains the single source node and the absorbing state | | {1, 2,3,..., | |}SS N contains all nodes of the network, and departing is not possible; hence, S is a finite state space. The transition probabilities klp satisfy the following conditions

- 10 klp for ||,...,2,1 Sk and ||,...,2,1 Sl

1

2

3

4

5

0.0579

0.3120

0.6301

0.16980.6465

0.1837

0.2470

0.4426

0.3104

0.4566

0.5434


- 1||

1

S

l klp , for ||,...,2,1 Sk .

The transition probabilities are elements of matrix |||| SSP , where klp is the transition probability in the kth row and the lth column, that is, if in some time point the chain is in state k, the probability of its one-step transition to state l is klp . The state transition diagram of DTMC for the example network is shown in Figure 2.

Figure 2: The state space diagram of the established DTMC

For the example network, the absorbing state S8= {1,2,3,4,5} contains all nodes of the network; and the instance state S4 of the state space S (Table 1) contains nodes }3,2,1{ and all connected components of the network constructed by nodes 1, 2 and 3 (see Figure 3).

Figure 3: Constructed connected components of state S4

The states of the established DTMC contain the traversed nodes of the network which are reached from some of the other nodes in a previous state. The final state contains the destination node where DTMC no longer progresses. Returning from the last traversed node is not permitted, however waiting in the current state is possible. Clearly, a new state is revealed if a leaving arc

Aji ),( is traversed such that node i is contained in the current state and the new node j is contained in the new state. As previously mentioned, the wait

S1

S2

S3

S5

S4

S6

S7

S8

0.3188

0.05790.3120

0.6301

0.0628 0.6465

0.2907

0.1699

0.0771

0.44260.3104

0.0873 0.6023

0.3104

0.13790.5434

0.60810.0983

0. 2937

0.3149

0.6851

1

2

3

1

2

3

1

2

3


states are indicated as Option 1 or Option 2. The following assumptions describe the creation of state space of the established DTMC

i. Upon arriving at the destination node, the process cannot traverse any node nor any arc (i.e. the absorbing state) ii. A new state is revealed if a new node in the network is added to the nodes of the current state iii. According to the nodes of the current state, exactly one new node during transition to a new state can be reached.

Kulkarni [11] considers an acyclic directed network and in each transition

from one state to another state, the possibly exists of adding at least one node. Nonetheless, our model is basically deferent from Kulkarni’s model, and we also extend wait states whereby traversing some arcs does not lead the creation of a new state. 3.1. Computing the transition and wait probabilities We obtain the transition matrix P of the established DTMC according to the following theorems. The transition probabilities (except for the absorbing state) are obtained by Theorem 1. Theorem 1: If klp is the kl th element of matrix P such that lk , || Sl and 1 2{1 , ,..., }k mS v v v is the current state, the transition probability from state Sk to state Sl , for all , 1, 2,..., | | 1k l S , is computed as given below. If kl then 0klp , otherwise if kl and we get

( , ) , ,( , )( , )[ ] ( (1 ))

m m mkkl vw vu v v v wv u A u w u Sv wv w

p P E q q q

.

vwE denotes the event where arc ( , )v w N of the network is traversed during the transition from kS to lS and

{( , ) : \{ }, \ ,| \ | 1}k m l k l kv w A v S v w S S S S . Proof: Since it is not allowed to traverse from one state to the previous states (Assumption (ii)), then it becomes necessary that 0klp , for kl . Otherwise, suppose kl , during transition from the current state Sk to the new state Sl, it is necessary to reach just one node other than the nodes of the current state, so | \ | 1l kS S , kv S and \l kw S S are supported by Assumptions (ii) and (iii). Two components of the klp formula should be computed.

In the last node mv of the current state Sk, it is possible to wait in mv by traversing an artificial arc ),( mm vv with a probability

mmvvq . Notice that it is not possible to wait in the other nodes \{ }k mv S v because as it should be left

The critical node problem in stochastic networks with discrete-time Markov chain 39 to create the current state. However, this is not necessary for node mv which possesses the largest label (leaving mv leads to a new node, and therefore results in a new state). If \l kw S S then one or all of the events vwE (i.e. to traverse a connecting arc between a node of the current state and another node of the new state) can occur for ( , )v w . Then the arrival probability of node lw S from the current state Sk is equal to

( , )[ ]vwv w

P E . The collection probability

should be computed because of deferent representations of the new state (e.g. see Figure 3). Subsequently, the nodes of the current state \{ }k mv S v (while waiting at mv ) should be prevented from reaching other nodes ku S and u w (Assumption (iii)), so arcs ( , )v u are not allowed to traverse and they are simultaneously excluded, thus it is equal to

( , ) , ,( , )(1 )

kvuv u A u w u Sv w

q

. The other possibility at node mv is leaving mv for the new node \l kw S S with a probability of

mv wq .□

Figure 4: The constructed states during transition from S4 to S7

For example, in the established DTMC of the example network (Figure 2),

the transition probability 47p is computed using the constructed components as shown in Figure 4, specifically 14 24 15 25 33 34( ) (1 )(1 )P E E q q q q , where

14 24 14 24 14 24( )P E E q q q q , however 14 15 25 0q q q as shown in Figure 1, so then 47 33 24 34p q q q . It is possible to wait at node 3 but at no other nodes of the current state S4={1,2,3}, where by traversing arc (2,4) or (3,4) the new state S7={1,2,3,4} occurs. Theorem 2 describes the transition probabilities to the absorbing state S|S|, which are the last column of the transition matrix P. Theorem 2: To compute the transition probability from state

1 2{1 , ,..., }k mS v v v to the absorbing state S|S|, for 1||,...,2,1 Sk , i.e., the k|S|th element of matrix P, and suppose ||Sn Sv is the given destination node of the network, then

| | ,( , )[ ]

nk nk S vvv S v v A

p P E

4

1

2

3

4

1

2

3


where nvvE denotes the event that arc ( , )nv v N of the network is traversed

during the transition from Sk to S|S|. Proof: To compute the transition probabilities ||Skp , for 1||,...,2,1 Sk , it should be evident that the final state is the absorbing state | | {1,2,3,...,| |}SS N containing all nodes of the network, and the stochastic process does not progress any further (Assumption (i)). Subsequently, leaving the arcs ( , )nv v from

kv S , the nodes of the current state, toward the destination node | |n Sv S is deemed sufficient. Then, one or all events

nvvE (i.e. traversing a connecting arc between a node of the current state and the destination node of the absorbing state) can happen and the transition probability from the current state Sk to the absorbing state S|S| is equals in total to

,( , )[ ]

nk nvvv S v v A

P E . The collection

probability should be computed because of deferent representations of the states (e.g., see Figure 3).

For state S4, transition probability 48p is obtained by 15 25 35( )P E E E , however 15 25 0q q , so then

48 35p q . The wait probabilities, the diagonal elements of the transition matrix P, are obtained by Theorem 3. Theorem 3: Suppose 1 2{1 , ,..., }k mS v v v is the current state, then the wait probability kkp is the kkth element of matrix P, which is

| |

11 if | |

1 if | | .

S

kjj kkk

p k Sp

k S

Proof: The wait probabilities kkp , for 1, 2,...,| | 1k S , are the complement probabilities of the transition probabilities from the current state kS , for

1, 2,...,| | 1k S , toward all departure states Sj, for 1, 2,...,| |j k k S .

Then, we have | |

11

S

kk kjj kp p

, for 1,2,...,| | 1k S . In other words, they

are the diagonal elements of matrix P, which are computed for any row 1, 2,...,| | 1k S of the transition matrix (see [10]). The absorbing state S|S|

does have a departure state, so | || | 1S Sp as the transition matrix P.

3.2. The arrival probability The arrival probability from the source node to the destination node in the network is analytically defined as a single or multi-step transition probability from the initial state S1 to the absorbing state S|S| in the established DTMC. According to Assumptions (i), (ii) and (iii), the state space of DTMC is directed and acyclic (otherwise returning to the previous states is possible, but contradictory). The out-degree of any state is at least one, except for the


absorbing state, so for any state Sk there is a single/multi-step transition from the initial state S1 to the absorbing state S|S| that traverses state Sk (see [1]). Consequently, the absorbing state is accessible from the initial state after finite transitions. Let ]|[)( kmlrmkl SXSXPrp denote the conditional probability that the process will be in state Sl after exactly r transitions, given that it is in state Sk now. So, if matrix P(r) is the transition matrix after exactly r transitions, then it can be shown that rPrP )( , and let )(rpkl be the klth element in matrix rP (see [10]). Thus, the arrival probability after exactly r transitions is ]|[)( 10||||1 SXSXPrp SrS and it is the 1|S|th element in matrix rP . Notice that any path from the source node to the destination node in the network needs at most |N| nodes. In other words, |N| nodes on the network could be added while DTMC progresses, requiring 1N transitions in DTMC (one node is added for each transition, initially located at the source node). Hence, we set 1| | (| | 1)Sp N as the arrival probability from the source node to the destination node. For the example network, we want to obtain the probability of arriving at node 5 from node 1. As already mentioned, probability

)(18 rp is obtained as shown by the stared line in Figure 5 for 6,5,4,3,2,1r . The arrival probability for the example network is equal to 0.6752, and is computed for 4r and does not change for 5,6r (more than four transitions did not improve the arrival probability).

Figure 5: The arrival probability and its changes

3.3. The critical node in the stochastic network

If the removal of a node causes the greatest decrease in the arrival probability, then we call it the critical node of the network. Consider that ),( iii ANG was obtained from network G when node i and its adjacent arcs are removed (except for the source and destination nodes. The following changes are sufficient:


- Set the wait probability of node i to 1, 1iiq - Set the existence probabilities of the removed arcs ),( ji to 0, 0ijq - Increase the wait probabilities of the adjacency nodes j as the existence

probabilities jiq of the removed arcs ),( ij , jijjjj qqq - Set the existence probabilities of the removed arcs ),( ij to 0, 0jiq For example, suppose node 3 is removed from the example network, then the changed probabilities are 133 q , 034 q , 035 q , 6880.011 q , 013 q ,

3535.022 q , 023 q . If follows that the arrival probability changes according to the removal of nodes (see Figure 5). Hence, failure of node 3 causes the greatest decrease in the arrival probability and it is detected as the critical node of the example network. Furthermore, Figure 5 shows the destination node is not accessible by two transitions if node 3 has failed. When either node 2 or node 4 have failed by two transitions, the arrival probability remains zero because path 1-3-5 is the only path along which destination node 5 is accessible by exactly two transitions: 31 SS and then 83 SS .

4. Numerical results

Various implementations of the proposed method on networks with different topologies have been presented. For comparison, all networks are created using nine nodes and the leaving and the waiting probabilities of nodes are random numbers produced by uniform distribution function. Node 1 is the source node and node 9 is the destination node in the all networks. They are acyclic directed networks and a path from each node to the destination node exists prior to removing any node. Subsequently, the arrival probability of the networks is computed after eight transitions in the established DTMC. All of the results were coded in MATLAB R2008a and performed on a Dell Latitude E5500 (Intel(R) Core(TM) 2 Duo CPU 2.53 GHz, 1 GB memory). Network 1 has an arbitrary topology with the arc leaving probabilities shown in Table 2. For the established DTMC on Network 1, the size of the state space is 69. The absorbing state containing the destination node is accessible by at least two transitions, even though each of the nodes (except for the source and destination nodes) has been removed.


),( ji ijq ),( ji ijq ),( ji ijq(1,2) 0.1283 (2,9) 0.7395 (4,9) 0.0923(1,4) 0.6075 (3,4) 0.1631 (5,9) 0.7980(1,6) 0.0362 (3,5) 0.5137 (6,8) 0.9518(1,8) 0.1555 (3,6) 0.3149 (7,8) 0.9340(2,3) 0.0600 (4,5) 0.1951 (8,9) 0.5290(2,4) 0.0460 (4,7) 0.0604(2,5) 0.1043 (4,8) 0.6100Table 2: Arc leaving probabilities of Network 1

The arrival probability of Network 1 is equal to 0.7275. As shown in Figure 6, node 4 is the critical node of Network 1.

Figure 6: The arrival probability of network 1

Network 2 is a grid network and the leaving probabilities of its arcs are shown in Table 3. The size of the state space for the established DTMC on network 2 is 76.

),( ji ijq ),( ji ijq ),( ji ijq(1,2) 0.3957 (3,5) 0.7956 (5,8) 0.5195(1,4) 0.2432 (3,6) 0.0524 (5,9) 0.1709(1,5) 0.2605 (4,5) 0.5506 (6,8) 0.9118(2,3) 0.7250 (4,7) 0.1617 (6,9) 0.0552(2,4) 0.0154 (4,8) 0.2663 (7,8) 0.9413(2,5) 0.1392 (5,6) 0.0182 (8,9) 0.6796(2,6) 0.0844 (5,7) 0.1996Table 3: Arc leaving probabilities of network 2

Figure 7 shows that node 5 is the critical node of Network 2 in first five transitions, and according to the arrival probability, both nodes 5 and 8 are the critical nodes of the network at the end of nine transitions. The destination node


of Network 2 is accessible upon three transitions where the arrival probability is equal to 0.7366.

Figure 7: The arrival probability of Network 2

Network 3 is a complete graph with leaving arc probabilities shown in

Table 4. The size of the state space for the established DTMC on Network 3 is 129.

),( ji ijq ),( ji ijq ),( ji ijq ),( ji ijq ),( ji ijq

(1,2) 0.0348 (2,3) 0.4192 (3,5) 0.1177 (4,8) 0.5340 (6,9) 0.1135(1,3) 0.1000 (2,4) 0.0560 (3,6) 0.0162 (4,9) 0.1590 (7,8) 0.2530 (1,4) 0.4603 (2,5) 0.1308 (3,7) 0.1584 (5,6) 0.0297 (7,9) 0.5592(1,5) 0.1197 (2,6) 0.1082 (3,8) 0.4912 (5,7) 0.6488 (8,9) 0.6505(1,6) 0.1196 (2,7) 0.0051 (3,9) 0.1158 (5,8) 0.1638 (1,7) 0.0197 (2,8) 0.1371 (4,5) 0.0980 (5,9) 0.0690 (1,8) 0.0790 (2,9) 0.1334 (4,6) 0.1293 (6,7) 0.7289 (1,9) 0.0159 (3,4) 0.0462 (4,7) 0.0763 (6,8) 0.0746

Table 4: Arc leaving probabilities of Network 3

The obtained arrival probability of network 3 in Figure 8 shows the arrival

probability of network 3 is equal to 0.7893 and node 8 is the critical node of the network.


Figure 8: The arrival probability of Network 3

5. Conclusion We have considered an established discrete-time Markov chain stochastic process over directed acyclic networks. The arrival probability from a given source node to a given destination node was computed by a single/multi-step transition probability from the initial state to the absorbing state. Numerical results have shown the efficiency of the proposed method in obtaining the arrival probability and the transition when the destination node is accessible for the first time. The critical node that causes the largest reduction in the arrival probability was determined. Hence, this method can be applied to rank nodes of a network when computing their criticality probability, separately. Future considered research may include extending the described model to continuous-time varying networks, using the discrete nature of the proposed model when applying meta-heuristic methods and reducing the associated number of computations. References [1] Ahuja, R. K., Magnanti, T. L. and Orlin, J. B. (1993). Network Flows: Theory,

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Measurement of cost efficiency in the European banking industry

Branka Tuškan1,* and Alen Stojanović1

1 Faculty of Economics and Business, University of Zagreb

J. F. Kennedy Sq. 6, 10000 Zagreb, Croatia E-mail: ⟨{bjurcevic, astojanovic}@efzg.hr⟩

Abstract. In this paper we analysed and compared efficiency results in the banking industry using two different approaches: financial indicators and the Data Envelopment Analysis (DEA) methodology. In the indicator-based approach, we used chosen accounting ratios (Return on Assets - ROA, Return on Equity – ROE and Cost to Income Ratio - CIR) and the descriptive statistics methodology to conduct analysis. In the case of DEA, a nonparametric linear programming methodology approach, expenses as input data and income as output data are used for measuring efficiency using the CCR DEA model, BCC DEA model and window analysis DEA technique. The objective of this research is ascertain whether a correlation exists between the results of the different ways of measuring efficiency. In that sense, the main purpose of this research is to draw a more precise conclusion about the efficiency of the banking industry, as tested for the period 2008–2012 on a sample of 28 European banking systems.

The main difference in the obtained results is a lag of values of average accounting ratios in comparison to the results of the DEA methodology. Such a finding suggests that the DEA methodology can be useful in detecting early signs of inadequate business strategies, which can lead to the slowdown of business activity or poorer efficiency results. This can be especially important in times of an unstable financial or macroeconomic environment, as it can assist in detecting early signs of a crisis. In general, the results of both approaches suggest that banking systems in post-transition countries have a higher cost efficiency. Such systems continue to be dominantly financed through long-term deposits and are also exposed to a specific risk. They do business in a specific competitive, financial and macroeconomic environment that significantly influences the prices of financial services (i.e. higher margins), and as a consequence, leads to potentially higher banking sector earnings.

Key words: banking industry, efficiency, data envelopment analysis, accounting ratios

Received: September 29, 2014; accepted: March 17, 2016; available online: March 31, 2016

DOI: 10.17535/crorr.2016.0004

1. Introduction

This paper uses a mathematical tool to measure banking industry efficiency, in addition to descriptive statistics and analysis of chosen associated accounting * Corresponding author.

48 Branka Tuškan and Alen Stojanović

indicators. Accounting profitability ratios in general are common tools for measuring efficiency of bank performance and are simple to calculate and interpret. Those ratios give us valuable information about the financial performance of the banking sector when compared to previous periods, but they are also limited due to variables included in the calculations. In general, the main weakness of ratio analysis is a lack of agreement on the relative importance of various types of input and output. Banking sectors may appear to be operating well even if they have an inadequate business strategy and poor management in certain areas, but as long as they compensate for it by operating better in other areas [22]. Furthermore, accounting ratios fail to take into account a greater number of outputs and inputs, and also fail to assess management actions and investment decisions that affect future as opposed to current performance [22]. Accounting ratios are short-run measures and may be inappropriate for describing the actual efficiency of a bank in the long run. Consequently, simple statistical analysis of accounting ratios and results obtained from such analysis may be insufficient.

Average accounting profitability ratios observed in this paper in the period during and just after the onset of the last financial crisis continued to have relatively high values at the end of 2008, even though the crisis had obviously already influenced the business performance of the banking sector and variables used to calculate accounting profitability ratios. On the other hand, inputs and outputs used in the mathematical approach to measuring efficiency by application of the DEA methodology can be different pairs of variables. In this paper, inputs and outputs used in the DEA approach and the specificity the methodology resulted in lower efficiency scores as early as 2008. Thus, we find that the mentioned disadvantages of Return on Assets (ROA) and Return on Equity (ROE) as the primary observed accounting ratios in this paper can be removed using the DEA methodology to measure efficiency. For this reason, the methodology can be a useful independent, alternative and/or complementary tool for gaining a better understanding and drawing a more precise conclusion about banking sector efficiency. This research is originally because in that past researches on banking efficiency used DEA or an indicator-based approach, both approaches have not been used simultaneously on a sample of entire banking systems as the decision making units (DMUs). Samples from existing researches on banking efficiency generally take into account individual banks in certain banking system(s) as DMUs. This research is also comprehensive since it takes into account all EU member states.

The paper is organized as follows. Following the Introduction in Section 2 (Literature), a few common approaches to measuring the efficiency of financial institutions is given, while noting which of them were used in the research. Firstly, the approach using indicators is briefly explained, along with its advantages and limitations. For the purpose of measuring a bank’s efficiency in general, the parametric and non-parametric approach may also be used. For this research the non-parametric approach as being more appropriate based on the

Measurement of cost efficiency in the European banking industry 49 assumptions given here is selected and briefly explained. In the second part of Section 2, literature is reviewed and the main conclusions from previous research are presented. In Section 3 (Methodology), the chosen indicators (3.1.) and DEA methodology and models including techniques used for this research (3.2) are explained in more detailed. Section 4 (Data) presents the data used in the analysis, while Section 5 identifies and discusses the results of efficiency measurements, first in regard to the indicators (5.1.), secondly the DEA approach (5.2.) and finally a comparative analysis (5.3). Conclusions and proposals for future research are provided in Section 6 (Conclusion).

2. Literature review There are a few common approaches to measuring the efficiency of financial institutions. The first approach to measuring efficiency is to use indicators (ratio) analysis from among several financial institutions and to calculate numerous accounting ratios, providing a measurement of the overall financial soundness of financial institutions and the operating efficiency of its management [15, 30]. Financial statements are the main source of accounting information used in measuring the operating efficiency of a financial institution. Accordingly, an analysis of the relationship between specific values by calculating the efficiency indicators was carried out. Financial ratios as indicators of a bank’s efficiency are divided into four main groups: (1) accounting indicators calculated on the basis of the data from a balance sheet, (2) accounting indicators calculated on the basis of the data from a profit and loss statement, (3) accounting profitability indicators, and (4) market profitability indicators, i.e. investment indicators. Accounting ratios as measures of efficiency in general are easier to calculate given that they are calculated using readily available information in financial statements: net profit or profit before taxes, total or average asset and equity, total income, total expenditure and the like. The most significant profitability indicators (ratios) calculated for banks are ROA, ROE and cost to income ratio (CIR).

The second approach in measuring efficiency is parametric programming, and is generally concerned with the production or expense function base. It is used to estimate the characteristics of the function and measures economies of scale, while assuming all decision-making units (DMUs) are operating efficiently. The analysis conducted in this paper does not include parametric programming in measuring efficiency, as it does not assume that all DMUs are operating efficiently. A parametric approach to efficiency measurement includes the Stochastic Frontier Approach (SFA), the Thick Frontier Approach (TFA), and Distribution Free Approach (DFA) [4].

The third approach uses DMUs efficiency frontiers to construct measures of efficiency and is labelled as a non-parametric programming approach. The approach considers the degree to which total efficiency in the financial sector


can be improved, and ranks the efficiency scores of DMUs. This efficiency measurement is derived from analysing empirical observations obtained from DMUs for defining productive units, which are characterized by common multiple outputs and common designated inputs [5]. In such terms, certain insufficiencies in the accounting indicators approach to efficiency measurement can be eliminated and conclusions about the efficiency can be improved using the DEA approach. Application of the DEA methodology in efficiency measurement has a wide scope and has been used extensively to assess school efficiency [23, 28]; hospital efficiency [13]; bank efficiency [1, 4, 12, 14, 15, 17, 19, 20, 21, 22, 24, 25, 26, 27]; insurance company efficiency [9, 10, 15, 18], etc. In reviewing the literature, the conclusion is that most studies deal with similar approaches in measuring the relative efficiency of banks over a period of time. However, several differences in specific DEA models and the techniques used do exist, i.e. considered aspects and goals of analyses [27]. DEA Window analysis and Malmquist Index analysis techniques are more specific than, for example, the DEA Charnes-Cooper-Rhodes (CCR) and Banker-Charnes-Cooper (BCC) models, due to analysis of panel data.

In the past, researching banking efficiency has not utilising the DEA approach or accounting indicators analysis simultaneously and/or a sample of the entire banking systems of EU member states in the form of decision-making units (DMUs). A similar research that used the DEA approach as opposed to the accounting approach for measuring the efficiency of Croatian financial institutions was conducted on a sample of 30 individual banks and 19 insurance companies operating in the country in the period from 2005-2009 [15]. The results suggest that the DEA approach for calculating efficiency was better (for CCR and BCC models) and the chosen accounting ratios as they were influenced by the boom in Croatian financial sector prior to the recent crisis [15]. The main difference in the results between those two approaches relates to lag scores after 2008 from efficiency measurement using the accounting approach, i.e. after the onset of the financial crisis. Another research was made on the sample of 26 individual banks operating in Bosnia and Herzegovina in the period 2008-2010 by simultaneously applying the DEA approach and financial indicators approach [19]. Bank efficiency varied throughout the observed period and not all banks performed negatively in the banking sector during the crisis. No significant difference between performance of banks in different entities of Bosnia and Herzegovina and those smaller and larger banks were noticed. A comparative analysis of banking sector efficiency indicators in the Republic of Macedonia in the period from 2007 to 2012 and the 16 CSEE countries for the period 2003-2012 shows that the values of observed indicators shifted around the average value for the entire analyzed sample, suggesting that countries with banking sectors that exhibit lower operating costs had a higher level of financial deepening and greater degree of financial intermediation [20]. At the same time, results of DEA approach suggest that the group of large banks with the highest efficiency was in the Macedonian banking sector. The

Measurement of cost efficiency in the European banking industry 51 comparison of the results of the DEA and accounting approach in the efficiency testing of the Bulgarian banking system in the period 1999-2006 showed that foreign banks perform better than domestic and state-owned banks due to technological and managerial improvements; and secondly, the large banks are more efficient than small banks due to smaller operating costs and economies of scale [21]. Samples in other researches mostly covered individual banks and were based only on DEA. Efficiency scores (using a sample of 125 large banks from 14 emerging European countries) prior to the crisis suggest a strong correlation with a country’s level of development, and also that bank efficiency suffered in the period prior to the crisis when credit activity was expanding, even though efficiency scores increased [1]. Foreign ownership positively influences a banks’ efficiency in less developed countries, and there exists a direct correlation between a bank’s cost efficiency (sample of 20 emerging European countries, 1993-2004) and economic reforms and stability, capital regulation and market structure in the banking sector [25]. Banking sectors (sample of 289 banks in 15 East European countries, 1994-2001) in which foreign owned banks have a larger share of total assets operate with lower expenses and also progress in banking reform exhibits a non-linear association with cost efficiency [12]. An average-sized bank in the sample operated at a point close to constant returns to scale, while smaller banks operated with significant unrealised economies of scale. Consolidation of smaller banks contributes to greater cost efficiency in banking, with private banks more cost efficient than state-owned banks, while privatised banks in majority foreign ownership are the most efficient whereas those in domestic ownership are the least efficient [12]. Smaller banks (sample of Croatian banks, period 1995-2000) are globally efficient, but large banks are locally efficient; foreign owned banks on average are the most efficient, while new banks are more efficient than previously established banks [14].

3. Methodology 3.1. Indicators approach The accounting bank profitability indicators calculated for research criteria are ROA, ROE and CIR. ROA is determined by model that includes the profit before taxes divided by average asset [30]. It is the most important ratio in comparing the efficiency and operating performance of banks as it indicates the returns generated from the assets financed by them [16]. For that reason, ROA can also be observed as a measure of a bank’s management quality. To calculate the indicator ROE, the following model is applied: profit before taxes is divided by the average shareholder's equity [30]. Subsequently, ROE measures the return on investment made by an investor’s equity. In other words, ROE measures how much profit (in %) is earned by the unit of shareholder's equity. ROA is a commonly used accounting ratio and a key measure of a bank's


operating efficiency [15]. The commonly accepted efficiency frontier for a value of ROA is at least 1% and for ROE at least 15% in a boom period, whereas in a recession period each bank with a ROE value of at least 10% is considered profitable, i.e. efficient. CIR is calculated as a share of a bank’s operating expenses (general administrative expenses and amortization) in net income [30]. For the purpose of this research values of ROA, ROE and CIR accounting efficiency indicators on the level of EU were analysed as a result of applying simple descriptive statistics methods (minimum - MIN, maximum-MAX, average-MEAN and median-MEDIAN). In the statements from the Croatian banking sector, Return on Average Assets (ROAA) and Return on Average Equity (ROAE) were analysed. They take into account the average values of assets/equity instead of total values, as in case of ROA and ROE. ROAA represents pre-tax profit as a % of average assets. ROAE represents the after-tax profit as a % of average equity. Accounting indicators, in general, are short-run measures and may be inappropriate for describing the actual efficiency of a bank in the long run. Consequently, simple statistical analysis of accounting ratios and results obtained from such analysis may have shortages, in terms of operating efficiency measurement. 3.2. DEA approach DEA is specifically a mathematical nonparametric linear programming methodology used to measure efficiency and enjoys a number of advantages over other traditional parametric efficiency measurement approaches. In that sense, shortages of the accounting indicators approach for efficiency measurement mentioned earlier are eliminated by the DEA approach. If the goal is to calculate the indicator of business performance representing the efficiency of an organizational unit within a financial institution, then the input and output ratio is used. Thus, if we want to calculate the efficiency measure that takes into account a number of inputs and outputs, it becomes necessary to make a selection of those that will be included in the calculation and assign them certain weights in order to determine a unique efficiency measure. The methodology that enables this is data envelopment analysis. Unlike statistical approaches that derive estimations on the basis of the average production unit, DEA is an extreme point methodology in which each DMU is compared only to the best one. For observed inputs and outputs of the DMU’s, the assumption is that a connection exists between them, but the shape of that connection is not defined, which is the case with statistical approaches. DEA is most useful in cases where accounting and financial ratios are of little value, multiple outputs are produced through the transformation of multiple inputs, and the input-output transformation relationships are not known [5]. The results of DEA can help DMUs to improve their business results. The basic idea behind DEA is to identify the most efficient DMU from among all DMUs, where the set of best practice or frontier observations are those for which no other DMU or linear

Measurement of cost efficiency in the European banking industry 53 combination of units has as much or more of every output (given inputs) or as little or less of every input (given outputs) [4].

As a result, different DEA models give different types of efficiency. The CCR-model [6, p.p. 23, 24, 43] assumes constant returns to scale (CRS) and provides a measure of global technical efficiency [2, 6]. On the other hand, the BCC-model [6, p.p. 87] assumes variable returns to scale (VRS), and as a result gives pure local technical efficiency [2, 6]. If the DMU is perfectly efficient based on both models, then its efficiency with respect to returns to scale is the highest. When output is increasing by the same factor as input, then it is CRS. If an increase in inputs does not result in a proportional change in the outputs, then it is VRS. A particular version of the CCR-model aims to minimize inputs while satisfying at least the given output levels; this is called the input-oriented model.

Another version is called the output-oriented model, and attempts to maximize outputs without requiring more of the observed input values [6, p.p. 41]. Assumptions of the output-oriented model are chosen for the purpose of this research as well. An optimal solution to the linear problem in the CCR-model is (θ *, v*, u*), where v and u represent input and output weight vectors. In that sense, DMU0 is CCR-efficient if θ*=1 and there exists at least one optimal (v*, u*) with v*>0 and u* >0, otherwise DMU0 is CCR-inefficient [6, p.p. 24, 25]. An optimal solution ( *, s-*, s+*) of the Phase II linear programming problem used to discover the possible input excesses and output shortfalls is called the max-slack solution [6, p.p. 44, 45]. “Slack” vectors (s-, s+) represent input excesses and output shortfalls. The objective of Phase II therefore is to find a solution that maximizes the sum of input excesses and output shortfalls while maintaining θ = θ*. If the max-slack solution satisfies s-* = 0 and s+* = 0, then it is called zero-slack. Further, a DMU0 is called CCR-efficient (Radial Efficiency or Technical Efficiency) if an optimal solution (θ*, *, s-*, s+*) of the two-phase linear programming problem of discovering the possible input excesses and output shortfalls satisfies both, θ* = 1 and is zero-slack (s-* = 0 and s+* = 0), otherwise it is called CCR-inefficient [6, p.p. 45]. A DMU is called weakly efficient if θ* = 1 and s-* ≠ 0 and (or) s+* ≠ 0. In that sense, a DMU is fully efficient if and only if any input or output cannot be improved without worsening some other input or output (Pareto-Koopmans Efficiency) [6, p.p. 45]. If an optimal solution (θ*B, *, s-*, s+*) obtained in the two-phase linear programming problem for BCCo satisfies θ*B = 1 and has no slack (s-* = 0, s+* = 0), then the DMU0 is called BCC-efficient, otherwise it is BCC-inefficient [6, p.p. 87, 88].

The window analysis technique is specific, when compared to the BCC or CCR models for instance, due to analysis of panel data. Due to the reason that window analysis is based on panel data, it is better to capture the variations of efficiency over time and use them as the more appropriate tool for efficiency measurement of the European banking systems in this research as well, but the


results are compared also with the results of CCR and BCC models. Window analysis is a DEA technique for examining changes in the efficiencies of a set of DMUs over time. A set of time periods (1..t) is chosen and the efficiency of each DMU (1..n) is computed separately for each period so that the efficiency of a given DMU over each period is treated like a new DMU resulting in the total number of tn DMUs. Window analysis assesses the performance of a DMU over time by treating it as a different entity in each time period. For example, if there are n units with data on their input and output measures in k periods, then a total of nk units need to be assessed simultaneously to capture the efficiency variations over time. For n = number of DMUs, k = number of periods, p = length of window (p less or equal k) the number of “different“ DMUs is np(k-p+1) [6, p.p. 295]. In classical window analysis, when a new period is introduced into the window the earliest period left out.

However, some critical factors must be considered when applying the DEA models. The efficiency scores might be very sensitive to data changes and depend heavily on the number and type of input and output factors considered. In general, inputs can include any resources utilized by a DMU, and the outputs can range from actual products produced to a range of performance and activity measures [29]. In that regard, there are a few different approaches that can be used for measuring relative efficiency in the banking sector:

• The Production Approach views a bank as a producer of services and products using labour and other resources as inputs and providing deposits, loans and other services (in value or number of transactions) as outputs [7];

• The Intermediation Approach studies the intermediary role of a bank in order to examine how efficient the bank is in collecting deposits and other funds from customers (inputs) and then lending out the money in various forms of loans, mortgages, and other assets (i.e., investments, etc.) [7];

• The Profitability Approach examines the process of how well a bank uses its inputs (expenses) to produce revenues [24].

This research uses the profitability approach (analysis of bank profit

efficiency) on a set of input and output data for estimating efficiency. This is the basis for choosing output-oriented DEA models in this analysis, which in that regard provides efficiency trends and ranks each banking system of EU member states, in terms of profit and operating effectiveness. 4. Data The selected general data from consolidated financial statements for the EU-27 are shown in Table 1, and for the Croatian banking sector in Table 2, this in light of the fact that the Republic of Croatia was not an EU member state in the observed period of this research. However, country is also included in the

Measurement of cost efficiency in the European banking industry 55 sample for DEA analysis because the assumption is that comparing the performance of the Croatian banking sector with other EU banking sectors would be interesting, given that about 90% of assets in the Croatian banking sector are in foreign ownership (mostly Austrian and Italian) and compliance by Croatian banks with regard to business performance (legal and practical) was already at a very high level in that period when compared with banks in EU countries (such compliance was a prerequisite for finalising accession negotiations). As a consequence, it is assumed that input and output data for Croatia are comparable to those in the EU banking sector and can be used in DEA analysis (in all, 28 DMUs).

2008 2009 2010 2011 2012 Number of credit institutions, EU-27 Stand-alone credit institutions 4,500 4,448 4,350 4,296 4,213 Banking groups 484 452 418 417 398 Domestic credit institutions 3,928 3,859 3,730 3,691 3,609 Foreign-controlled subsidiaries/branches 1,056 1,041 1,038 1,022 1,003 Total 4,984 4,900 4,768 4,713 4,612 Total assets of credit institutions (EU-27, in EUR billions) Domestic credit institutions 37,770.8 34,663.5 34,638.0 35,901.6 35,471.9 Foreign subsidiaries and branches 7,045.2 7,860.6 8,234.2 8,916.2 8,038.6

Table 1: EU-27 consolidated banking data [11]

The Republic of Croatia 2008 2009 2010 2011 2012

Number of credit institutions 43 43 38 37 36 Number of commercial banks 35 34 32 31 30 Total assets of CI (in EUR billions) 50.3 51.7 53.7 54.9 54

Table 2: General banking data for the Republic of Croatia (HR) [8]

The data in the above two tables serve primarily to show size-linked

relations between EU-27 countries and the Republic of Croatia, the relative importance of the Republic of Croatia in light of the European banking industry as a whole, and consequently, its influence on the comparability of results from both approaches. The sample in this study involves 28 European banking systems (EU-27 and the Republic of Croatia) and data sets for the period 2008–2012, given that the crisis strongly affected the business performance results of banks especially in this period. An analysis was carried out of the values of ROA, ROE and CIR accounting efficiency indicators at the EU level, resulting from the application of simple descriptive statistics on data for all EU-27 countries (in %, Appendix 1). When average values were calculated, (excluded) minimum and maximum values (outliers - countries with values significantly


higher/lower than average) were considered, in order to obtain more objective and comparable results. When performing statistical analysis of accounting indicators for the Republic of Croatia, only ROAA and ROAE were available as Croatia was observed separately (excluded from the calculation of values at the EU level). Given that the relative importance of Croatia in terms of compared data (Table 1 and 2) is obviously very small, the assumption is that does not exert a greater influence on the comparability of results.

All input and output data used in the DEA analysis models and previously calculated, official accounting indicators analysed by using descriptive statistics methods are taken from the European Central Bank database (Consolidated banking data; for EU-27 [11]) and from Croatian National Bank (official data, for the Republic of Croatia [8]). In the case of the DEA approach, 28 European banking systems were viewed as 28 DMUs in the period 2008-2012. The data set used for DEA analysis (in the CCR model, BCC model and in window analysis, DEA-Solver-Pro software used) is given in Appendix 2. For each j-th DMU (i.e. for each banking system) inputs (xij) include (in EUR billions): Input 1, (x1j) interest expenses Input 2, (x2j) total operating expenses [sum of two positions: fee and commission expenses and other operating expenses (labour-related and capital-related administrative expenses and other expenses from the business activity of a bank)]. On the other side, output data (yij) include: Output 1, (y1j) interest income Output 2, (y2j) total operating income (fee and commission, and other operating income).

The core business activity of banks is assumed to be the collection of deposits, lending transactions and payment operations. Such activities provide the main categories of incomes and expenses. The interest income category assumes income from interest earned in a bank’s lending activities and related revenues while total operating income includes income from fees and commissions, other related revenues and other incomes from business activity. On the other hand, expenses from a bank’s business activity include: interest expenses that arise from collection of deposits and related expenses and operating expenses which include expenses for fees and commissions, other related expenses, and the category other operating expenses which includes: labour-related administrative expenses (employee costs), capital-related administrative expenses (depreciation, office supplies, etc.) and other expenses from the business activity of a bank.

Measurement of cost efficiency in the European banking industry 57 5. Results 5.1. Indicators approach The worst average and median values of accounting indicators ROA and CIR at the level of EU-27 occurred in 2009 (see Table 3). In the case of ROE, the lowest value occurred in 2012 due to the negative effect of the widest spread between MIN (Cyprus) and MAX (Estonia) values in that very year. If we exclude country outliers from the calculation, the absolute lowest ROE was also recorded in the year 2009.

MIN MAX MEAN MEDIAN 2008

ROA -1.5 (BE)

2.0 (BG)

0.44 (HR 1.60)

0.40

ROE -44.8 (BE)

18.8 (RO)

4.40 (HR 9.91)

5.20

CIR -186.2 (NL)

-40.5 (EE)

-63.37 (HR 52.40)

-55.20

2009

ROA -4.0 (LV)

1.8 (MT)

-0.10 (HR 1.13)

0.20

ROE -70.1 (LT)

22.5 (HU)

-2.83 (HR 6.40)

4.00

CIR -76.7 (BE)

-18.8 (MT)

-53.36 (HR 49.50)

-55.00

2010

ROA -3.1 (IE)

1.3 (CZ)

0.2 (HR 1.12)

0.40

ROE -65.2 (IE)

15.2 (CZ)

1.9 (HR 6.46)

5.99

CIR -412.2 (IE)

-26 (MT)

-69.0 (HR 48.00)

-57.45

2011

ROA -4.0 (CY)

3.1 (EE)

0.20 (HR 1.17)

0.23

ROE -86.0 (CY)

25.5 (EE)

0.31 (HR 6.88)

4.20

CIR -72.1 (AT)

-30.4 (MT)

-55.84 (HR 42.80)

-55.84

2012

ROA -3.4 (CY)

2.0 (EE)

-0.01 (HR 0.86)

0.20

ROE -90.3 (CY)

14.2 (EE)

-4.51 (HR 4.84)

3.42

CIR -92.7 (IE)

-25,1 (MT)

-59.62 (HR 50.30)

-58.59

* AT-Austria, BE-Belgium, BG-Bulgaria, CY-Cyprus, CZ-Czech Rep., EE-Estonia, IE-Ireland, GR-Greece, HR-Croatia, HU-Hungary, MT-Malta, RO-Romania Table 3: The results of statistical analysis of chosen accounting efficiency indicators

for the EU-27 and average values for the Republic of Croatia (HR) [8, 11]


The Republic of Croatia recorded its lowest average values for accounting indicators in 2012 (Tables 3 and 4). This lag in comparison to the EU-27 can be explained due to the great recession and specific political, macroeconomic and financial conditions in Croatia, which annually deteriorated between 2008 and 2012, and even after that. As a consequence, there also exists a significant lag of effects on banking sector accounting ratios. The best indicators of average values were recorded in 2008, given that the crisis had not yet affected results. 5.2. The DEA approach The results of the DEA approach (Table 4) suggest different conclusions about EU banking sector efficiency, depending on the model/technique used and on the type of results observed, but they also differ in comparison to the results of the accounting indicators approach.

2008 2009 2010 2011 2012 Statistical analysis of the accounting efficiency indicators approach

EU-27 (Average values) ROA (%) ROA (%) without outliers

0.44 0.51 (BE)

-0.10

0.20 0.32 (IE)

0.20 0.37 (CY,GR)

-0.01 0.24 (CY,GR)

ROE (%) ROE (%) without outliers

4.40 6.22 (BE)

-2.83 -0.25(LT)

1.87 4.45 (IE)

0.31 3.76 (CY,GR)

-4.51 1.81 (CY,GR)

CIR (%) CIR (%) without outliers

-63.37 -58.64(NL) -53.40

-69.00 -55.73 (IE) -55.84

-59.62 -57.88 (IE)

The Republic of Croatia ROAA (%) 1.60 1.13 1.12 1.17 0.86 ROAE (%) 9.91 6.40 6.46 6.88 4.84 CIR (%) 52.40 49.50 48.00 42.80 50.30

DEA approach window analysis, output oriented, CRS (length of window=5, average by term) No. of DMUs 28 28 28 28 28 No. of efficient DMUs 0 1 1 2 1 Average relative efficiency 0.659 0.715 0.727 0.732 0.726 CCR – model, output-oriented, CRS No. of DMUs 28 28 28 28 28 No. of efficient DMUs 5 3 4 4 4 Average relative efficiency 0.833 0.794 0.789 0.769 0.766 SD 0.125 0.109 0.126 0.134 0.134 No. of DMUs - efficiency lower than average 15 16 15 16 18

BCC - model, output-oriented, VRS No. of DMUs 28 28 28 28 28 No. of efficient DMUs 15 13 12 14 11 Average relative efficiency 0.966 0.920 0.944 0.955 0.931 SD 0.046 0.091 0.084 0.061 0.075 No. of DMUs - efficiency lower than average 11 12 9 11 11

Table 4: European banking system efficiency measurement summary results

Measurement of cost efficiency in the European banking industry 59

According to the CCR-model (output-oriented, CRS), the worst average relative efficiency was recorded in 2012 and the highest in 2008. The smallest number of efficient DMUs was recorded in 2009. In the case of the BCC-model (output-oriented, VRS), the lowest average efficiency was recorded in 2009 and the highest level as well as the number of efficient DMUs in 2008 was identical to that of CCR-model. For the case of DEA window analysis, efficiency results suggest that 2008 was the worst year with respect to average relative efficiency and the number of efficient DMUs, whereas 2011 was the best year. 5.3. Comparative analysis of the results The results of DEA efficiency analysis acquired from CCR and BCC output-oriented models and window analysis, including the statistical analysis results of accounting efficiency indicators are compared in the Table 4. The main difference in the observed results indicates a lag of the average accounting indicators with respect to the results of DEA window analysis approach for efficiency measurement. In that regard, efficiency scores from DEA window analysis, the average for EU-28 (table 4) and for each country in the sample separately observed as well (Appendix 6) generally had the lowest values in 2008. This can be explained due to the onset of the financial crisis in that year, which affected input and output data used in DEA approach more than variables used in calculating the already existing accounting ratios in that year. The average in window order by rank suggests also that the best results (greater than the average, Appendix 5) occur most often in the banking systems of post-transition and other EU-12 countries (Croatia, Czech Republic, Slovakia, Hungary, Poland, Romania, Bulgaria, etc.), where financing based on deposits prevails, whereas other sources of financing have a relatively low significance, most often up to 10%.

Similar results for each country in the sample observed separately (relative efficiency = 1 or an above-average efficiency; Appendices 3 and 4) are provided by the CCR and BCC models. However, those systems also exhibit a specific competition environment, risk exposure, a financial and macroeconomic environment, significantly influencing the price of financial services (i.e. higher margins) and as a consequence, potentially higher banking sector earnings. Such results are also explained by the fact that the last global financial and economic crisis had a significantly higher impact on well-developed banking systems, exposure to mortgage securities instruments as relatively significant sources of financing in terms of long-term lending activities. The Republic of Croatia, which was not EU member state in the observed period, had in absolute terms the best DEA efficiency results (relative efficiency = 1 with a rank of 1) in all years in the CCR and BCC models in comparison to the other countries in the sample observed (with the exception of 2009 when it was ranked fourth). The


window analysis also showed that it accomplished the best results when observing the average based on window results (Appendix 5).

When considering all these facts, a more precise interpretation of such results cannot be given. CCR and BCC models did not provide comparable results. The results indicating the number of efficient DMUs and average relative efficiency scores for the BCC model are better due to the fact that BCC model is based on VRS. However, the CCR model based on CRS is better when observing the efficiency of financial institutions because under VRS, the majority of large banks (banking sectors in this case) might appear as fully efficient, due to the lack of truly comparable efficient banks [1, 3]. Given that the DEA window analysis is based on panel data, the assumption is that it provides a better efficiency measurement, analysis results and conclusions when compared to the CCR and BCC models. This is especially true when comparing DEA results with the efficiency results of the average accounting indicators approach, and also when analysing a sample similar to that observed in this research, and observing entire banking systems as DMUs. 6. Conclusion One of the main aims of the paper is to determine whether there exists a correlation between the average efficiency results at the EU level and individual results for countries included in the sample. Another aim was to observe trends of results in a given period and based on different efficiency measurement approaches. Based on such analysis, the paper provided a more precise conclusion about the efficiency of the EU banking industry. The main difference in the research results indicates a lag of average accounting indicators over the results of the DEA window analysis approach for efficiency measurement. The DEA window analysis was recognized as the better tool in comparison to the CCR and BCC models for efficiency measurement, as it assumes analysis of panel data. Further, it was also recognized as the better tool when comparing DEA results with the results of indicator-based approach.

Efficiency scores from the DEA window analysis had their lowest values in 2008. On the other hand, average ROA and ROE accounting ratios had the lowest values in 2009. This may be explained by the fact that although banks operate more efficiency in times of crisis, were expense and income results are used as inputs/outputs in the DEA approach to efficiency measurement, variables in accounting ratio calculations did not achieve results as good as in boom times, due to deteriorating market conditions and a more conservative business strategy by banks in crisis periods. If the input and output values used in the window analysis of this research are observed as accounting efficiency indicators, the same conclusion can be deduced in the case of ROA and ROE. Such a finding suggests that the DEA methodology can be a useful alternative

Measurement of cost efficiency in the European banking industry 61 or complementary analytical tool for detecting early signs of inadequate business strategies, which in turn can lead to a slowdown of business activity or poorer efficiency results. Importantly, this is also true in times of an unstable financial or macroeconomic environment, as it may facilitate detecting early signs of a crisis, earlier than when using accounting indicators.

Finally, a comparison of the results of both approaches may facilitate drawing more precise conclusions about banking industry efficiency. In general, results from efficiency measurement using those two different approaches suggest that greater cost efficiency are attributed to deposit-oriented banking systems of post-transition countries, where banks operate in specific environments and therefore are able to theoretically achieve higher earnings. Such results can be explained by the fact that the last global financial and economic crisis had a significantly higher impact on banking systems exposed to mortgage securities instruments as relatively significant sources of financing long-term lending activities. This research is important because the DEA methodology and statistical analysis of accounting indicators were previously used for ascertaining the efficiency measurement and comparing efficiency results of banks solely with entire banking systems acting as decision making units. Suggestions for future research include comparing window analysis results with the results of Malmquist Index analysis, and considering other pairs of inputs and outputs. Further research might be able to classify banking sectors (or individual banks) into two groups, depending on whether they are developed or post-transitional, or into additional groups depending on whether they are “large”, “medium” or “small” in terms of the asset size of DMUs as a percentage of total assets. In addition, ongoing research could also compare the efficiency scores of each one of these groups with the results of other groups, and their total average efficiency scores. The result of this multi-group classification, in addition to more variables and more advanced methodologies, could become an even better analytical tool for determining key factors of efficiency with respect to accounting indicators. References [1] Anayiotos, G., Toroyan, H. and Vamvakidis, A. (2010). The efficiency of emerging

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APPENDICES Appendix 1: Data set for the indicators approach (in %)

Appendix 2: Data set for DEA approach (amounts in EUR billions)

Measurement of cost efficiency in the European banking industry 65 Appendix 3: CCR DEA model results - output oriented – CRS (excel output from DEA-Solver-Pro)

Source: author’s calculation Appendix 4: BCC DEA model results - output oriented – VRS (excel output from DEA-Solver-Pro)

Source: author’s calculation


Appendix 5: Output-orientated results of the window analysis DEA technique –CRS (excel output from DEA-Solver-Pro)

Source: author’s calculation



Min-max optimization and the radial approach to the public service system design with generalized utility

Jaroslav Janáček1,* and Marek Kvet2

1 Faculty of Management Science and Informatics, University of Žilina

Univerzitná 8215/1, 010 26 Žilina, Slovakia E-Mail: ⟨[email protected]⟩

2 University Science Park, University of Žilina Univerzitná 8215/1, 010 26 Žilina, Slovakia

E-mail: ⟨[email protected]⟩ Abstract. The paper deals with the min-max public service system design, where the generalized utility is considered. In contrast to the formulations presented in the literature, the generalized utility defined for a public service system assumes that the user’s utility comes generally from more than one located service center and the individual contributions from relevant centers are weighted by reduction coefficients depending on a center order. Given that commercial IP-solvers often fail due to enormous computational times or extreme memory demands when resolving this issue, we suggested and compared several approaches based on a bisection process with the purpose of developing an effective max-min approach to the public service system design with a generalized utility. Key words: public service system, generalized disutility, min-max approach

Received: September 29, 2014; accepted: March 17, 2016; available online: March 31, 2016

DOI: 10.17535/crorr.2016.0005

1. Introduction The design of almost any public service system [3, 5, 9, 11] includes determining center locations, from which the associated service is distributed to all users of the system. Thus the public service system structure is formed by the deployment of a limited number of service centers in a finite set of possible locations and the objective in the standard formulation is to minimize some form of disutility, which is proportional to the distance between serviced objects and the nearest service centers. This assumption of being serviced by the nearest center is not fully true, when a rescue service system is designed for random


68 Jaroslav Janáček and Marek Kvet

service demand and limited capacity at the service centers. During momentary demand for a service, another user may occupy the nearest service center. In such situations, momentary demand is usually serviced from the second or even the third nearest center, if the second nearest center is also occupied. This randomly restricted capacity of a service center can be generalized so that the r nearest centers influence the disutility perceived by a user, where r is a parameter of the generalized disutility model. In this paper, the generalized disutility perceived by a user is modelled by a sum of weighted disutility contributions from the r nearest centers. The weights will depend only on the order of distances from the user to the r nearest centers. The k-th weight can be proportional to the probability that the k-1 nearest centers are occupied and the k-th nearest center is available [15]. In contrast to the min-sum public service system design, when average user disutility is to be minimized, we focus on the fair-optimal system design.

The fairness in general emerges whenever limited resources are to be fairly distributed among participants [2, 12, 13]. The strongest scheme is the so called lexicographic min-max criterion. By applying this scheme, the disutility perceived by the worst situated user is minimized first, and then disutility of the second worst situated user is minimized, unless the previously achieved disutility of the worst situated users is worsened. This approach is applied step by step to the remaining users [14]. The effective use of the approach is based on partitioning the range of all possible disutility values, which can be perceived by a user. The initial phase of the process is called the min-max optimal public service system design. In this paper, we focus on a problem-solving method for the initial phase, when the generalized disutility is considered. Based on our experiences in designing the min-sum optimal public service system, we found that the radial formulation of the problem can considerably accelerate the associated problem-solving process [1, 4, 6, 8]. Furthermore, we can start from our previous research [7, 10], where we developed and successfully tested radial formulation of the min-sum service system design problem with the generalized disutility. We want to ascertain whether the radial approach contributes considerably to more effective problem-solving of the min-max optimal public service system design. The remainder of the paper is organized as follows. Section 2 introduces the generalized model of an individual user’s disutility by considering more than one contributing center and provides a mathematical formulation of the problem based on location-allocation and radial formulations. A possible reduction of the set of relevant disutility values is also discussed. Section 3 contains a description of suggested approaches to the min-max problem. Next, Section 4 presents the numerical experiments, a comparison of the suggested approaches and finally, Sections 5 draws the final conclusions.

Min-max optimization and the radial approach to the public service 69 system design with generalized utility

2. Generalized disutility and min-max criterion in the public service system 2.1. Location-allocation formulation The generalized model of the public service system disutility for an individual user is based on the summation of weighted disutility contributions from a given number of located service centers. The nonnegative disutility contribution dij from a given service center i to disutility perceived by a user located at a location j grows with increasing distance between the center and user locations. Let us introduce the mapping : Rr Rr such that a vector [u1, u2, …, ur] is mapped to the vector [uτ(1), u τ(2), …, u τ(r)] using the permutation τ, where the values do not decrease uτ(1) ≤ u τ(2) …. ≤ u τ(r). Based on this notation, the symbol k(ui : i=1, … , r) denotes the k-th component of the resulting r-tuple. If I1 denotes the set of all located service centers in the public service system and dij denotes the disutility contribution from service center location i to the customer j, then the disutility of the system for the user j can be expressed by (1), where r denotes the given number of service centers, which take part on the utility for the user. The coefficients qk for k=1...r are positive real values, which fulfil the following inequalities q1 ≥ q2 ≥ … ≥ qr. According to [15], the coefficients can be proportional to the probabilities that only the k-th nearest center is available.

r

kijkk Iidq

11):( (1)

The problem behind the min-max optimal public service system design with generalized disutility for users is determining the service centers by minimizing perceived disutility and restricting the total number of located centers up to a given number p. To describe the problem, we denote J as the set of user locations and I as the set of possible center locations. The basic decisions in any problem-solving process relates to the location of centers for possible locations from the set I. These decisions will be modelled by binary variables yi for iI, where yi takes the value of 1 if a center is to be located at the location i, and takes the value of 0 otherwise. Further, we introduce binary variables xijk for iI, jJ, k=1 … r. The variable xijk takes the value of 1 specifically for the case, when user j obtains the k-th smallest disutility contribution from the service location i. The associated model can be written according to [15] as follows:


hMinimize (2)

(3)

rkIiJjforyx iijk ,...,1,, (4)

rkJjforxijkIi

,...,1,1

(5)

IiJjforxijk

r

k

,11

(6)

Jjforhxdqr

kijkij

Iik

1

(7)

Iiforyi }1,0{ (8)

rkIiJjforxijk ,...,1,,}1,0{ (9)

0h (10)

Constraint (3) limits the number of located service centers whereas constraint (4) link up the allocation variables xijk and the location variables yi, preventing the variable xijk from assigning a place i without a service center to the user j. Constraints (5) ensure that exactly one contribution dij will be assigned to the user j as the k-th smallest contribution. Constraints (6) ensure that the contribution dij will be assigned to the given user j at most once. The link-up constraints (7) ensure that each perceived disutility is less than or equal to the upper bound h. 2.2. Radial formulation We assume that the range of disutility contribution value covers only non-negative integers for the range [d0, dm] of all possible disutility values d0<d1<…<dm from the matrix {dij}. The values partition the range into m = v+1 intervals. The interval s has the form (ds, ds+1]. The length of the s-th interval is denoted by es for s = 0 … v. To describe the homogeneous system of radii determined by the values d0<d1<…<dm for individual users’ locations, a system of binary constants is defined so that the constant aij

s is equal to 1 if and only if the disutility contribution dij for a user from location j from the possible center location i is less than or equal to ds, otherwise aij

s is equal to 0. Let the location variable yi have the same meaning as above. Further, we introduce auxiliary binary variables xjsk for jJ, s = 0… v, k = 1 … r in order to model the disutility contribution value of the k-th nearest service center to the user j. The variable xjsk takes the value 1 if the k-th smallest disutility contribution for the

pytoSubjectIi

i


customer j J is greater than ds and it takes the value 0 otherwise. Then the expression e0 xj0k + e1 xj1k + e2 xj2k + e3 xj3k +…+ ev xjvk constitutes the k-th smallest disutility contribution dk

j* for customer j. Under above-mentioned preconditions, we can describe the min-max-optimal public service system design problem (11) - (17) using the variables and other notations introduced above.

hMinimize (11)

pytoSubjectIi

i

(12)

vsJjforryaxIi

isijjsk

r

k

,...,0,1

(13)

Jjforhxeq jsks

v

s

r

kk

01

(14)

Iiforyi }1,0{ (15) rkvsJjforx jsk ,...,1,...,,0,}1,0{ (16)

0h (17)

Constraint (12) defines an upper bound p on the number of located centers. Since the second term of the left-hand-side of (13) gives the number of centers located within radius ds from the user location j, constraint (13) ensures for a given j that the sum of variables xjsk over k = 1 … r expresses the complement of that number for the value r. The link-up constraints (14) ensure that each perceived disutility is less than or equal to the upper bound h. Validity of the assertion that the expression on the left-hand side of (14) expresses the sum q1di1,j+ q2di2,j +…+ qrdir,j of weighted relevant disutility values from the r nearest service centers i1, i2, …, ir to j, is based on the following reasoning. It can be easily found that the minimal sum of the variables xjsk over k=1 … r completes the number of located service centers in the radius s from user location j to the number r. In this way, the sum gives the number t of the nearest service centers, whose disutility contribution is greater than or equal to the value ds. As the sequence of qk decreases, only xjsk for k=r-t+1, r-t+2 … r must be equal to one for the given j and s. This implies that the biggest disutility contribution is assigned the smallest value of qk. The left-hand-side of (14) is pushed down by the optimization process, and subsequently the constraints xjsk ≤ xjs-1,k for s=1 … v must hold due to the construction of aij

s and constraints (13) and furthermore, the constraints xjsk ≤ xjsk+1 for k=1 … r-1 must hold due to convexity given by a decreasing sequence of qk.


2.3. Discussion on the location-allocation and radial approaches Both the above mentioned approaches were broadly tested and compared. The studies were performed for the case of min-sum optimization, which is used when designing the min-sum optimal public service system. In that case, the sum of all disutility values perceived by all users is minimized. That means that the link-up constraints (7) and (14) were absent in models (2)-(10) and (11)-(17), respectively. It was found [4, 10] that the radial approach considerably (in orders) outperformed the location-allocation approach in terms of computational time when designing the min-sum optimal public service system. Nevertheless, our preliminary experiments suggested that the link-up constraints for the upper bound definition significantly spoil the convergence of the computational process based on the branch and bound principle. This deterioration was so strong that it discharged the better convergence characteristic of the radial approach. We tried to improve convergence of the location-allocation and radial approaches using a trick and which was used in fuzzy optimization to avoid solving a non-linear problem. This approach consists in fixing the upper bound h at some chosen value and solving the problem with some surrogate objective function to find whether the original problem has a feasible solution with the fixed objective function value. This process is repeated for decreasing values of h until no feasible solution exists.

Furthermore, the generalized disutility concept is accompanied by another disadvantage comparing to the classical disutility coming only from the nearest located center. When designing the min-max public service system, and only a classical disutility model. Then, the minimal disutility value of the worst situated user can be used as a threshold and all disutility values exceeding the threshold can be excluded from the process. The general disutility model does not have this useful property, as we will show in the next sub-section. 2.4. Impossibility of reducing general disutility

The perceived disutility model is based on the matrix {dij} of integer contributions from a possible center location i to a user located at a place j. The set of all values contained in the matrix {dij} can be represented by an ordered sequence of unique values d0< d1< d2<… dm. In such cases, when the classical disutility model is considered, i.e. r =1, and a solution of the problems (2) – (10) or (11) – (17) is found for h = dt at step t of the bisection process, then both problems can be considerably reduced. The problem (2) – (10) can be reduced by excluding all the allocation variables, associated with disutility value dij>dt. The problem (11) – (17) can be reduced in the parameter v so that v can be set at the value dt-1 instead of the original value dm-1. This reduction can be used in further steps, even if only the initial phase of the lexicographic


minimization or the complete process is continued. We will show in the next example that this straightforward reduction is impossible, when using the generalized utility with r ≥ 2.

Let us consider the network graph depicted in Figure 1. The graph consists of the vertex set V = {1, 2, …, 6, j, j’} and the set of weighted edges, where the lengths of the edges are placed at individual edges as depicted in the figure.

Figure 1: Example network

The counter example can be defined on the network, where J = {j, j’} represents the set of user locations and I = {1, 2 … 6} is the set of possible center locations. The matrix {dij} of the potential disutility contributions from the center locations is presented in Table 1.

Center locations dij dij’

1 7 72 7 73 7 74 6 85 10 46 10 4

Table 1: Potential disutility contributions from the center locations to the user

locations j and j’.

The disutility contributions from individual centers to a user are defined

here as the lengths of the shortest paths from the user location to the center

j j‘

1

2

3

4

5

6

7

7

7

6

1

3

3

4

4


locations. We consider the counterexample parameters r = 3 and p = 3. The reduction coefficients qk for k=1, 2, 3 are 1, 0.2 and 0.1, respectively. The objective is to find three locations from I so that the maximal value from generalized disutility perceived by users j, and j’ is minimal. It can be easy found that three centers located at locations from I1 = {1, 2, 3} represent a feasible solution, where the generalized disutility value for users j and j’ are q1d1j+ q2d2j+ q3d3j =1*7+0.2*7+0.1*7=9.1 and q1d1j‘+ q2d2j‘+ q3d3j‘= 1*7+0.2*7+0.1*7=9.1, respectively. Since the complete sequence of the disutility contribution values consists of the values 4, 6, 7, 8 and 10, and the two highest values are not used in the discussed solution, it may seem that the values 8 and 10 could be excluded from the next steps of the computational process, similarly to the case of r=1. However, in contrast to the classical case, the reduction of these values in the presented example excludes the better solution given by I1 = {4, 5, 6}, where generalized disutility values for j and j’ are q1d4j+ q2d5j+ q3d6j

=1*6+0.2*10+0.1*10=9 and q1d4j‘+ q2d5j‘+ q3d6j‘=1*4+0.2*4+0.1*8=5.6, respectively. The maximum of these general disutility values for j and j’ is 9, but this solution would be unattainable if the disutility values 8 and 10 were excluded.

3. Min-max problem solving methods We have to note that the min-max problem solution can be obtained directly by using a common IP-solver to solve the problems described by models (2)-(10) or (11)-(17).

Another more promising approach is based on a bisection search covering the range of the generalized disutility values. This search tries to find the minimal value G* for a feasible service system design, where no user perceives a disutility higher than G*.

The searching process consists of individual steps, where step t answers the question as to whether there is a feasible solution with maximal perceived disutility less than or equal to a given value Gt determined at step t of the bisection process.

If the location-allocation formulation of the public service system design problem is considered and a common IP-solver is used, then there are two different formulations of the particular problem solved at step t. The first formulation consists in minimizing expression (18) subject to (3)-(6), (8), (9) and (19).

ijkijIi

r

kk

Jj

xdq 1

(18)


JjforGxdq tijkijIi

r

kk

1

(19)

If the optimization process of the used IP-solver leads to an optimal

solution for the above problem, a feasible solution exists and the lower value Gt+1 can be tested. In the opposite case, the searched value G* is higher than Gt.

In the second formulation, the auxiliary variables hj ≥ 0 for jJ are introduced and the minimization problem (20) subject to (3)-(6), (8), (9), (21) and (22) is solved.

jJj

h

(20)

JjforhGxdq jtijksIi

r

kk

1

(21)

Jjforh j 0 (22)

If the optimization process of the used IP-solver leads to an optimal solution of the above problem and the optimal objective function value is zero, a feasible solution for the t-th step problem exists and the lower value Gt+1 can be tested. In the opposite case, the searched value G* is higher than Gt.

A similar construction can be developed for radial formulation, where the first formulation consists of minimizing expression (23) subject to (12), (13), (15), (16) and (24).

jsks

v

s

r

kk

Jj

xeq 01

(23)

JjforGxeq tjsks

v

s

r

kk

01

(24)

In the second formulation, the auxiliary variables hj ≥ 0 for jJ are also

introduced and the minimization problem (25) subject to (12), (13), (15), (16), (26) and (22) is solved.

jJj

h

(25)

JjforhGxeq jtjsks

v

s

r

kk

01

(26)


Further, the zero value of the objective function value indicates that a feasible solution for the t-th step problem exists.

4. Computational study To compare the four approaches mentioned in the Section 3 and the two approaches in the Section 2, several initial experiments were performed. The benchmarks were obtained by modifying a model of a real emergency health care system, which was originally designed for a self-governing region of Žilina. This system covers the demands of 315 communities - towns and villages, over a region covered by 36 ambulance vehicles, where each represents a service center. These communities were considered as elements of the set J of users’ locations and as elements of the set I of possible service center locations. The disutility contribution from a possible location i to a user location j was represented by the road network distance between the two locations. In the benchmarks, the generalized disutility perceived by a user sharing a given location was the sum of three distances from the user’s location to the three nearest vehicle locations. The distances are multiplied by the reduction coefficients so that the largest coefficient multiplies the smallest distance etc. The four triples q1, q2, q3, q4 of the reduction coefficients define the individual benchmarks, and the symbols of the triples are used for distinguishing the results obtained using individual approaches on the benchmarks. The used triples were q1 = [1, 0.2, 0.1], q2 = [1, 0.1, 0.05], q3 = [1, 0.5, 0.2] and q4 = [1, 0.8, 0.5].

The tested approaches are called LA_EX, RA_EX, LA_BG, RA_BG, LA_BGh and RA_BGh. The prefixes LA and RA denote location-allocation and radial formulation, respectively. The suffix EX denotes the exact approach, when IP-solver solves the problem described by program (2)-(10) or (11)-(17). The suffixes BG and BGh denote bisection approaches, where the IP-solver solves the optimization problem for a fixed value Gt at each step t. The denotation BG corresponds to the models (18), (3)-(6), (8), (9), (19) or (23), (12), (13), (15), (16), (24), while the denotation BGh is used for the models (20), (3)-(6), (8), (9), (21), (22) or (25), (12), (13), (15), (16), (26), (22) depending on location-allocation or radial formulations. To solve the problems described by the mentioned models, optimization software FICO Xpress 7.3 (64-bit, release 2012) was used and the experiments were run on a PC equipped with the Intel® Core™ i7 3610 QM processor, at 2.3 GHz and with 8 GB RAM. The preliminary experiments showed that the IP-solver required unpredictable computational time. When the middle-size integer programming problem is solved to optimality, we decided to test each approach during a one-hour period. The LA_EX and RA_EX approaches were run for an hour to solve the problem for each triple of the reduction coefficients and the objective function values of the best found feasible solutions are presented in Table2. The best found


objective function values in the computation study were put in the row denoted as “Best found solution”.

Approaches q1 q2 q3 q4 LA_EX 100.4 96.3 77.2 87.5 RA_EX 25.2 18.95 32.2 43.5 Best found solution 20.8 17.25 30.6 43.5

Table 2: The best found objective function values reached by the LA˙EX and RA˙EX approaches over a one-hour period.

Comparison of the approaches are based on bisection met with several

technical obstacles due to the fact that optimization procedures of the IP-solver are run at each step of the bisection process, and furthermore, only the first feasible solution was searched in order to complete the step, when LA_BG and RA_BG are tested. Since the bisection process needs at most seven steps to finish the optimization process, the performance of one step was limited to 450 seconds, to prevent the solver from doing a long search for verifying unnecessary optimality. On the other hand, this restriction may cause premature stopping of the particular search before the first feasible solution is found, even if a feasible solution exists. The bisection process may then fail in separating the correct interval containing the searched minimal value. That is why two parameters of the bisection processes are studied. The first parameter is the computation time CT in seconds and the second one is the number CS of steps, which do not terminate prematurely. The symbol G* denotes the best found value of the generalized disutility, which corresponds to the maximal disutility perceived by the most exposed users of the designed public service system. The results of experiments are presented in Table 3.

q1 q2 q3 q4 Approaches CS CT [s] G* CS CT [s] G* CS CT [s] G* CS CT [s] G*

LA_BG 7 1535 20.8 7 1030 17.3 7 1883 30.6 6 1237 46.0 LA_BGh 5 2548 53.3 3 2922 40.3 5 2704 48.8 4 2587 59.8 RA_BG 7 2186 20.8 6 1709 23.0 6 2211 34.0 6 1999 46.0 RA_BGh 5 2640 24.7 6 1687 23.0 5 1779 35.7 4 1933 52.9

Table 3: Comparison of the bisection approaches

As can be seen, the bisection approaches were able to obtain a better

solution than the exact approaches in the limited time, but they were not too reliable as concerns the possibility to fail at the particular steps of the bisection process. The approaches BG (meaning LA_BG and RA_BG) perform better than the BGh approach. Surprisingly, the LA_BG approach outperformed the radial formulation approach. The unreliability of the bisection approaches


evoked the idea to start the bisection process with an initial pre-search based on strengthening of the constraints (24). We replaced constraints (24) with (27).

1

0 1 0

/ , 1,...,fv r

s jsk t t ss t s

e x G q e for j J k r

(27)

This approximation of the original constraints enabled solving the problem (25), (28), (30), (12) and (15) instead of (23), (12), (13), (15), (16) and (24) at each step of the bisection process.

JjforryahIi

if

ijj

(28)

This initial phase gives the result presented in Table 4.

q1 q2 q3 q4

CT [s] G* CT [s] G* CT [s] G* CT [s] G*

0.39 32.5 0.38 28.8 0.39 42.5 0.36 57.5Table 4: Results of the initial phase

5. Conclusions This paper suggests and compares solving techniques for the min-max public service system design with the generalized disutility model. The generalized disutility model impacts the complexity of the problems, where such problems are resolved using the presented techniques. The consequence of using the generalized disutility model when designing the min-max optimal public service system is that the suitability of the common approaches to the min-sum public service system design changes considerably. We suggested several bisection approaches to the problem and explored their effectiveness. In contrast to previously obtained results for min-sum designs, the radial approach seems not to outperform the location-allocation approach. Furthermore, we suggested improving the bisection process requiring the insertion of so a called initial phase, which in turn provides a good starting solution with good objective function value in very short time. This initial phase algorithm is much faster than the continuing bisection process. The min-max design, treated and solved in this paper, plays not only important role in lexicographic optimization, but also considerably reduces the set of effective general disutility values.

Acknowledgement This work was supported by the research grants VEGA 1/0518/15 “Resilient rescue systems with uncertain accessibility of service”, VEGA 1/0463/16 “Economically efficient charging infrastructure deployment for electric vehicles in smart cities and communities”, APVV-0760-11 “Designing of Fair Service


Systems on Transportation Networks” and also by the project University Science Park of the University of Žilina (ITMS: 26220220184) supported by the Research & Development Operational Program and funded by the European Regional Development Fund.

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Croatian Operational Research Review 81CRORR 7(2016), 81–96

Research project grouping and ranking by using adaptiveMahalanobis clustering

Zeljko Turkalj1, Damir Markulak2, Slavica Singer1 andRudolfScitovski3,∗

1 Faculty of Economics, Josip Juraj Strossmayer University of OsijekTrg Ljudevita Gaja 7, 31 000 Osijek, Croatia

E-mail: ⟨{turkalj, singer}@efos.hr⟩2 Faculty of Civil Engineering, Josip Juraj Strossmayer University of Osijek

Crkvena 21, 31 000 Osijek, CroatiaE-mail: ⟨[email protected]⟩

3 Department of Mathematics, Josip Juraj Strossmayer University of OsijekTrg Ljudevita Gaja 6, 31 000 Osijek, Croatia

E-mail: ⟨[email protected]⟩

Abstract. The paper discusses the problem of grouping and ranking of research projectssubmitted for a call. The projects are grouped into clusters based on the assessmentobtained in the review procedure and by using the adaptive Mahalanobis clustering methodas a special case of the Expectation Maximization algorithm. The cluster of projectsassessed as best is specially analyzed and ranked. The paper outlines several possibilitiesfor the use of data obtained in the review procedure, and the proposed method is illustratedwith the example of internal research projects at the University of Osijek.

Key words: adaptive Mahalanobis clustering, multi-criteria decision making, evaluation,project clustering

Received: February 11, 2016; accepted: March 21, 2016; available online: March 31, 2016

DOI:10.17535/crorr.2016.0006

1. Introduction

Differences and similarities between certain phenomena are always an intriguingstarting point not only for researchers, but also for decision makers, in whichargumentation of the similarities and differences is important, e.g., for achievingas equitable allocation of limited resources (financial, human, material, etc.) aspossible.

Why are the elements of some set more compact and separated better for somevalues of their features and how to group them better? For example, grouping a setof interested buyers of sports shoes with respect to age, education and purchasingpower can be used to define the promotion policy of a manufacturer of sports shoesor grouping university students depending on the type of their previous education

∗Corresponding author.

http://www.hdoi.hr/crorr-journal c⃝2016 Croatian Operational Research Society

82 Zeljko Turkalj, Damir Markulak, Slavica Singer and Rudolf Scitovski

and the achieved GPA can be used to define the admissions policy of that university[4, 8, 14].

By including the criteria referring to limiting resources and expectations of anequitable distribution of such resources to mutually competitive activities, theanswer to such question becomes exceptionally important from the application pointof view in a situation where there is certain homogeneity of phenomena or activitiesthat need to be assessed.

The issue of quality/excellence assessment of one’s scientific achievements orresearch proposal is very topical and important not only for researchers, but also forthe wider community the individual belongs to. There are many discussions referringthereto that have been published in scientific and professional papers and variouspublications (see e.g., [8, 12, 10, 2]), but also in the daily newspapers, which in factis not surprising if one knows that the consequence of this process is the distributionof financial resources for the purpose of research, which are always limited. Hence,the debate is most often about whether the method of distribution of funds availablefor research corresponds to the actual scientific excellence of the respective research.And here we come to the basic problem - how someone’s scientific performance orresearch proposal can be assessed in a clear, unambiguous, transparent and fair way?

Josip Juraj Strossmayer University of Osijek administration faced such situationwhen they decided to encourage research of young researchers through internal fund-ing programs. This will be used as an example to illustrate the proposed methodfor project grouping and ranking.

A fairly large body of literature is dedicated to the assessment and ranking ofresearch projects (see, for example, [3, 9, 11]) and ranking of departments, institutesand universities closely associated therewith (see, for example, [4, 8, 14]). Mostapproaches use different multi-criteria decision making methods, most frequently thewell-known Analytic Hierarchy Process (AHP) [15]. In our paper, we have combinedthe AHP method and the adaptive Mahalanobis clustering (AMC) algorithm pro-posed in [13]. First, the set of projects that have passed the administrativeverification was grouped into several clusters depending upon the features used. Af-ter that, ranking was conducted within the cluster of projects assessed as best bymeasuring the relative ranking “distance” from the perfectly assessed project, i.e.,the project that has achieved the maximum grade possible.

The paper is organized as follows. The description and the structure of datathat characterize the projects concerned are given in Section 2. This section alsodescribes in more detail an example of internal competition for research projectsat the University of Osijek. Section 3 outlines basic facts about cluster analysisand gives a short description of the AMC algorithm. Different approaches to theconstruction of the data set on the basis of which projects are grouped and rankedas well as appropriate examples are presented in Section 4.

2. Data

Suppose that N project proposal applications with full documentation weresubmitted in reply to a call for project proposals. Let us denote this set by PN .Projects will be assessed on the basis of features f1, . . . , fn describing the quality

Research project grouping and ranking by using adaptive Mahalanobis clustering 83

of both the applicants and the project (the quality and relevance of the researchproposal, the quality of the applicants, etc.) and the general impression F of theproject.

By using the well-known AHP method (see, e.g., [3, 4, 15]), to each feature fs

we associate the weights ws > 0, s = 1, . . . , n with the conditionn∑

s=1ws = 1.

Suppose further that for each project pi ∈ PN two independent, blind reviewsRi

#1 and Ri#2 are obtained in which features f1, . . . , fn were assessed by grades

uis, v

is ∈ [1, 5], s = 1, . . . , n and the general impression of the project by grades

U i, V i ∈ [1, 10]. Grades (uis, v

is, U

i, V i) do not have to be integers. If for someproject pi0 ∈ PN one of the grades referring to the general impression U i0 or V i0 isless than 6, such project is considered to be negatively assessed and it will not beconsidered for further evaluation.

For project pi ∈ PN assessed by grades((ui

s, vis), s = 1, . . . , n; U i, V i

)we define

the vector f i of the GPA of features f is = 1

2 (uis + vis), s = 1, . . . , n and the GPA of

the general impression Fi= 1

2 (Ui + V i).

In this way, for every project pi ∈ PN we have the following data:

f i1, . . . , f

in – the GPA of features based upon reviews Ri

#1 and Ri#2,

Fi

– the GPA of the general impression of reviewers Ri#1 and Ri

#2,(1)

taking into account corresponding weights of features w1, . . . , wn > 0.

Example 1. Josip Juraj Strossmayer University of Osijek administration decidedto encourage research of young researchers by internal funding programs and createda unique fund for that particular purpose. The main goal of this concept is to helpyoung researchers, who have yet to acquire their own scientific recognition, in theimplementation of their ideas as this is a difficult time if we take into account thereduced scope of financing scientific research on the national level and the relatedlower likelihood of approval of funding. Thus, the second call for internal scientificresearch project proposals was opened in the 2014-2015 academic year. In orderto make the process more transparent, detailed conditions of the call as well as theassessment criteria were clearly defined on the website of the University‡. Two areasof scientific research were identified, i.e., the STEM fields and the fields of arts,humanities and social sciences. The maximum possible score for research, use offunds, etc. were defined for each field. Following standard administrative checks, allproject proposals are supposed to undergo a peer-review process with two independentreviewers one of whom is from the specific field the project proposal refers to, and theother covers a broader project proposal research area. Both reviewers had to fill outan appropriate peer-review form which was the basis for establishing project proposalassessment criteria.

However, for each of these fields, there is an open question of equity of the limitedfinancial resources available to the University for this purpose, which is based on anequal comparison of all project proposals taking into account the same features.

‡http://news.unios.hr/research/projects/open-calls/research-projects/


In this and the following examples we will work out the problem of grouping andranking research projects in the fields of arts, humanities and social sciences. N = 61project applications with full documentation were submitted (that have passed theadministrative verification). Each project was reviewed by at least two independent,blinded reviewers: Ri

#1 was selected from the field the respective project topic belongs

to, and Ri#2 was selected from another related field. Reviewers Ri

#1 and Ri#2 had

similar but adapted forms in which they assessed n = 6 project features (see Table 1).

Features Reviewer Ri#1 Reviewer Ri

#2 Weights ws

f1 The quality and relevance ofthe research proposal

The quality and relevance ofthe research proposal

0.30

f2 The quality of applicants The quality of applicants 0.20f3 Research feasibility study Dissemination and utilization

of research results 0.15f4 Financial plan Financial plan 0.10f5 Institutional support Institutional support 0.10f6 Inclusion of students Inclusion of students 0.15

Table 1: The elements assessed by reviewers from Example 1 with corresponding weights

Each reviewer also rated the general impression of the proposed project. Wherethe respective assessments of reviewers Ri

#1 and Ri#2 differed substantially,

additional reviews were requested. If for some project the grade referring to thegeneral impression was less than 6, such project was considered to be negativelyassessed and it was not considered for further evaluation.

The set of all positively assessed project proposals with corresponding data ofthe form (1) will be denoted by P. The set P needed to be grouped according totheir quality and a decision should be made on which projects shall be financed.

3. Data clustering

Clustering or grouping a data set A = {ai ∈ Rn : i = 1, . . . ,m} with n featuresin several compact and well-separated clusters has practical importance in a widevariety of applications, such as biology, medicine, physics, economy, environmentalscience, energy management, business, social sciences, etc. (see e.g. [1, 14, 17, 18, 19,22]). A general problem is as follows: the set A should be partitioned into 1 ≤ k ≤ mnonempty disjoint subsets π1, . . . , πk, such that

k∪i=1

πi = A, πr ∩ πs = ∅, r = s, |πj | ≥ 1, j = 1, . . . , k. (2)

Subsets π1, . . . , πk are called clusters in Rn and the set of all clusters is called apartition, which will be denoted by Π = {π1, . . . , πk}. The collection of all suchpartitions will be denoted by C(A, k).

If components ais, s = 1, . . . , n of the data point ai lie in intervals [αi, βi] whichare not of equal range, i.e., if numbers β1−α1, . . . , βn−αn, are mutually significantlydifferent, they should first be normalized [13]. This can be achieved by transforming


the set A into the set B = {bi = T (ai) : ai ∈ A} ⊂ [0, 1]n by using the mappingT : [α, β] → [0, 1]n, where

T (x) = D(x− α), D = diag(

1β1−α1

, . . . , 1βn−αn

). (3)

After clustering the set B, the obtained results will be transformed again into [α, β]by the inverse mapping T−1 : [0, 1]n → [α, β], T−1(x) = D−1x+ α.

If we introduce some distance-like function (see e.g. [1]) d : Rn × Rn → R+,R+ := [0,+∞⟩, then to each cluster πj ∈ Π we can associate its center cj defined by

cj = argminx∈Rn

∑a∈πj

d(x, a). (4)

After that, a globally optimal k-partition Π⋆ ∈ C(A, k) can be defined as a solutionof the following global optimization problem

Π⋆ = argminΠ∈C(A,k)

F(Π), F(Π) =k∑

j=1

∑a∈πj

d(cj , a), (5)

where F : C(A, k) → R+ is the objective function (see e.g. [13, 16]).

3.1. Adaptive Mahalanobis clustering

Given the structure of the data set in this paper, the set A will be grouped intoellipsoidal clusters. An efficient algorithm for searching for a locally optimalpartition with ellipsoidal clusters is the Adaptive Mahalanobis k-means (see [13]),which can be carried out as a special case of the well-known Expectation Maximiza-tion algorithm (see [24]), but its efficiency is significantly greater than the standardExpectation Maximization algorithm. The adaptive Mahalanobis k-means algorithmcan be described by two steps which are iteratively repeated:

Step A: Based on the set of mutually different assignment points c1, . . . , ck ∈ Rn,the set A should be divided into k disjoint clusters π1, . . . , πk by using theminimum distance principle

πj = {a ∈ A : djM (cj , a;Sj) ≤ dsM (cs, a;Ss), ∀s ∈ J}, j ∈ J,

where

djM (x, y;Sj) =n√

detSj (x− y)TS−1j (x− y), (6)

is the adaptive Mahalanobis distance-like function, and

Sj =1

|πj |

∑ai∈πj

(cj − ai)(cj − ai)T , (7)

is a covariance matrix (see e.g. [1], [13, 20, 21]);


Step B: For each cluster of the partition Π = {π1, . . . , πk} of the set A, one candefine the corresponding cluster centers

cj =1

|πj |

∑ai∈πj

ai. (8)

Remark 1. According to [20] the covariance matrix Sj is positive definite if andonly if the set of vectors

φs = (a1s − csj , . . . , a|πj |s − csj)

T ∈ R|πj |, s = 1, . . . , n,

is linearly independent. The matrix Sj can become singular in some cases mentionedin [20]. That problem can than be solved by taking S = I (identity matrix) or byintroducing the small perturbation of the component of all data points in πj. Formore details see [20].

Searching for a globally optimal partition Π⋆ is a complex global optimizationproblem for the solution of which there is generally no effective method. An efficientincremental partitioning algorithm is proposed in the paper written by [13], whichis able to find either a globally optimal partition or a locally optimal partitionof the set A ⊂ Rn close to the global one. By knowing an optimal r-partition(r ≥ 1), the algorithm searches for the following additional cluster by using the well-known DIRECT algorithm for global optimization [6, 5, 7], and after that by using theadaptive Mahalanobis k-means algorithm it determines the optimal (r+1)-partition.

This algorithm successively gives optimal partitions (consisting of elliptical shapeclusters that are ascompact and relatively strongly separated as possible) for k = 2, . . . , kmax, wherekmax is the maximum number of clusters that makes sense to be calculated. There-fore, this algorithm is also very suitable for searching for a partition with the mostappropriate number of clusters by using some known indexes (see Section 3.2).

3.2. Choosing of a partition with the most appropriate numberof clusters

In some cases, the number of clusters k is determined by the nature of the problemitself and therefore it is known in advance. If the number of clusters is not knownin advance, then it is natural to search for an optimal partition which consists ofclusters that are as compact and relatively strongly separated as possible. This canbe done by using some of the well-known validity indexes (see e.g. [13, 23]). Inour paper, we will use the Calinski-Harabasz (CH) index and the Davies-Bouldin(DB) index. More compact and better separated clusters in an optimal partitionwill result in a greater CH index and a smaller DB index, respectively.

4. Project clustering and ranking

The given set of positively assessed projects P should be grouped into k ≥ 1 as com-pact and well-separated clusters as possible. The very nature of the data implies theneed for searching for ellipsoidal clusters by applying the AMC algorithm describedin Section 3.1.


4.1. Project clustering and ranking based upon the assessedproject features

For each project pi ∈ P, first the vector f i = (f i1, . . . , f

in)

T ∈ [1, 5]n of the GPA andthe vector ai = (w1f

i1, . . . , wnf

in)

T ∈ Rn of the weighted GPA (WGPA) of features(1) as well as the set

A ={ai = (w1fi1, . . . , wnf

in)

T ∈ Rn : i = 1, . . . ,m} ⊂ [α, β], (9)

α = (w1, . . . , wn)T , β = 5(w1, . . . , wn)

T , (10)

should be defined (see Fig. 1a).Since there is a bijection between the set P of all projects and the set A, in order

to group projects into groups by their quality, we will find an optimal partitionof the set A (see Section 3.1) with the most appropriate number of clusters (seeSection 3.2).

In order to ensure the same influence of such weighted grades, the data pointsshould first be normalized by using the mapping T : [α, β] → [0, 1]n given by (3),where

T (x) = Dx− 14e, D = 1

4 diag(

1w1

, . . . , 1wn

), e = (1, . . . , 1)T ∈ Rn. (11)

(a) Non-normalized data ai (b) Normalized data bi

Figure 1: Non-normalized data and normalized data

This yields a normalized set B = {bi = T (ai) : ai ∈ A} ⊂ [0, 1]n (see Fig. 1b).Applying the AMC algorithm described in Section 3.1, by using validity indexesmentioned in Section 3.2 we obtain an optimal partition Π⋆ = {π⋆

1 , . . . , π⋆k} with

the most appropriate number of clusters π⋆1 , . . . , π

⋆k with centers ζ⋆1 , . . . , ζ

⋆k . After

clustering the set B, the obtained results can be transformed again into [α, β] by theinverse mapping T−1 : [0, 1]n → [α, β].

Project ranking will be carried out based upon measuring the “weightedEuclidean distance” to the perfectly assessed project p⋆, i.e., the project which thevector f⋆ = (5, . . . , 5)T ∈ Rn is associated to. In this way, we will achieve a fineranking structure in which all GPAs achieved as well as their weights will be takeninto account. In this regard, we introduce the following definition.

Definition 1. Let f i = (f i1, . . . , f

in)

T be a vector of the GPA of features of theproject pi and let f⋆ = (5, . . . , 5)T be a vector associated to the perfectly assessedproject p⋆. The quality measure of the project pi ∈ P is the weighted Euclidean


distance d(pi, p⋆) defined by

d2(pi, p⋆) = ∥f i − f⋆∥2w =

n∑s=1

ws(fis − 5)2. (12)

Remark 2. If the quality measure of the project pr is less than the quality measureof the project ps, i.e., if d(pr, p⋆) < d(ps, p⋆), then the project pr is “closer” to theperfectly assessed project p⋆ and it will be ranked higher than the project ps.

Furthermore, note that the quality measure of the project given by (12) can beexpressed by using normalized data. Specifically, by using (11) we obtain

bi = T (ai) = D(ai − α) = 14 (f

i − b⋆),

where b⋆ = (1, . . . , 1)T ∈ Rn (a normalized representant of the perfectly assessedproject). Hence

d(pi, b⋆) := ∥bi − b⋆∥w = 14∥f

i − f⋆∥w = 14d(p

i, p⋆). (13)

This means that the same quality measure of the project can also be obtained suchthat we measure the weighted Euclidean distances of normalized data bi to the vectorb⋆, what will be used below.

Specially, the ranking of clusters within a partition can also be performed bymeasuring the weighted Euclidean distances of their centers to the vector of theperfectly assessed project. In this regard, the quality measure of the cluster π⋆

j withthe centers ζ⋆j is defined as

d(π⋆j , b

⋆) := ∥ζ⋆j − b⋆∥w. (14)

Example 2. The set of all positively assessed projects P from Example 1 containsm = 47 projects. These projects should be grouped on the basis of the WGPA,w1f

i1, . . . , w6f

i6 of these projects obtained based upon reviews by independent, blinded

reviewers Ri#1 and Ri

#2 and weights of features w1, . . . , wn that can be seen in Ta-ble 1.

After defining the set A = {ai = (w1fi1, . . . , w6f

i6)

T ∈ R6 : i = 1, . . . ,m}, onthe corresponding normalized set B = {bi = T (ai) : ai ∈ A} ⊂ [0, 1]6 the AMCalgorithm is carried out as described in Section 3.1. By using indexes specified inSection 3.2 it was shown that an optimal partition with the most appropriate numberof clusters has four clusters. The obtained results were transformed again into [α, β].Characteristics of the optimal cluster partition (the number of projects per cluster|π⋆

j |, the standard deviation of the cluster σ⋆j , the cluster center c⋆j = T−1(ζ⋆j ) and

the quality measure d(π⋆j , b

⋆) of the cluster) are given in Table 2.Note that the quality measure d(π⋆

1 , b⋆) of the cluster of projects assessed as best

is significantly lower than the quality measures of other clusters, whereby there is aninsignificant difference between standard deviations by clusters (see Table 2). Thismeans that the cluster of projects π⋆

1 assessed as best is significantly separated fromother clusters.


Cluster |π⋆j | σ⋆

j c⋆j = T−1(ζ⋆j ) d(π⋆j , b

⋆)

π⋆1 17 0.69 (4.527, 4.801, 4.713, 4.779, 4.956, 4.846) 0.077

π⋆2 11 0.98 (4.432, 4.159, 4.182, 3.977, 4.364, 4.591) 0.178

π⋆3 10 1.26 (3.908, 4.725, 3.95, 4.25, 4.65, 3.75) 0.229

π⋆4 9 1.08 (3.519, 3.722, 3.5, 4.167, 4.528, 4.5) 0.301

Table 2: Properties of clusters of the optimal partition

The cluster π⋆1 which consists of projects assessed as best contains 17 projects

ranked according to the achieved quality measures d(pi, b⋆) as defined by (13) (seeTable 3). GPA of all features of these projects, average weighted grades (AWG) of

all features f i =n∑

s=1wsf

is, the GPA of the general impression F

iand the quality

measures d(pi, b⋆) of these projects are also given in Table 3.

pi f i1 f i

2 f i3 f i

4 f i5 f i

6 f i F i d(pi, b⋆) Rank

p35 5 5 5 5 5 5 5 10 0 1p40 5 5 5 5 5 5 5 9.5 0 2p6 4.833 4.75 5 5 5 4.875 4.881 9.75 0.038 3p36 5 5 4.5 4.5 5 5 4.875 9.5 0.062 4p10 4.5 4.75 5 4.75 5 5 4.775 10 0.076 5p1 4.417 5 5 5 5 5 4.825 9.25 0.080 6p33 4.5 5 4.5 5 5 5 4.775 10 0.084 7p7 4.5 5 5 5 5 4.5 4.775 9.5 0.084 8p8 5 4.5 5 4.5 5 4.5 4.775 9.5 0.084 9p26 4.5 4.5 5 5 5 5 4.750 9.5 0.088 10p45 4.5 4.5 5 5 5 5 4.750 8.5 0.088 11p28 4.875 4.875 4.125 4.5 5 5 4.756 9.875 0.096 12p9 4.5 4.75 4 4.5 5 5 4.600 9 0.128 13p30 4.333 5 4.75 4.5 5 4 4.563 8.75 0.141 14p3 4 5 4.5 4.5 5 5 4.575 9 0.151 15p24 4 4.5 4.25 4.5 4.75 4.5 4.337 8.5 0.177 16p19 3.5 4.5 4.5 5 4.5 5 4.325 9 0.222 17

Table 3: Cluster of projects π⋆1 assessed as best

Theoretically, it may happen that for some project pr ∈ π1 and for some projectps ∈ π2 holds d(pr, b⋆) > d(ps, b⋆), but application of ellipsoidal clusters reducessuch possibility significantly.

Note that the ranking of projects in Table 3 does not follow the ranking of theseprojects by the AWG of all features (see column f i in the table). For example,project p1 has a higher average score than project p10, but it is still ranked lower.The reason for that lies in its relatively low grade given to feature f1

1 of project p1,which is much more important than other features (w1 = 0.30).

Thus, the proposed method accepts better a fine structure of project featureratings than the ordinary ranking obtained on the basis of the AWG of all features.


Since the well-known AHP method is in the background of ranking projectsaccording to the AWG of all features (see e.g. [3, 9, 14]), in Table 3 it is possible torecognize advantages of the method we propose in relation to the AHP method.

4.2. Project clustering and ranking based upon assessedfeatures and the general impression of the projects

Similarly to the previous section, for every project pi ∈ P define (n+1)-dimensional

vector (f i1, . . . , f

in, F

i)T ∈ [1, 5]n × [1, 10]. The first n components of this vector

are the GPAs of features, and the last component represents a GPA of the generalimpression of the project. A set of data A will be defined by means of these grades

such that the grade of the general impression of the project Fihas the same impact

as all weighted features w1fi1, . . . , wnf

in together. This means that we have the

following set of data

A ={ai = (w1fi1, . . . , wnf

in, F

i) ∈ Rn+1 : i = 1, . . . ,m} ⊂ [α, β], (15)

α = (w1, . . . , wn, 1)T , β = (5w1, . . . , 5wn, 10)

T . (16)

Please note that the ratio of the impact of the general impression of the projectand the weighted features w1f

i1, . . . , wnf

in could also be defined in a different way.

Since there is a bijection between the set P of projects and the set A, in orderto group projects into groups by quality, we will find an optimal partition of the setA (see Section 3.1) with the most appropriate number of clusters (see Section 3.2).

In order to ensure the same impact of grades weighted in this way, the datapoints should first be normalized by using the mapping T : [α, β] → [0, 1]n+1 givenby (3), where

T (x) = D(x− α), D = diag(

14w1

, . . . , 14wn

, 19

), α = (w1, . . . , wn, 1)

T ∈ Rn. (17)

Thus, this yields the set B = {bi = T (ai) : ai ∈ A} ⊂ [0, 1]n+1 on which we appliedthe AMC algorithm described in Section 3.1 by using indexes given in Section 3.2and obtained the optimal partition Π⋆ = {π⋆

1 , . . . , π⋆k} with the most appropriate

number of clusters π⋆1 , . . . , π


⋆k . After clustering the set B,

the obtained results can be transformed again into [α, β] by the inverse mappingT−1 : [0, 1]n → [α, β].

In accordance with Remark 2, project ranking can be carried out on the basis ofthe quality measures of the projects defined by (13).

Example 3. The set P of m = 47 projects from Example 1 will be grouped equally,on the basis of the WGPA of 6 project features w1f

i1, . . . , w6f

i6 and on the basis of

the GPA of the general impression Fi, which were obtained based upon reviews by

independent, blinded reviewers Ri#1 and Ri

#2.

After defining the set A = {ai = (w1fi1, . . . , w6f

i6, F

i) ∈ R7 : i = 1, . . . ,m}, on

the corresponding normalized set B = {bi = T (ai) : ai ∈ A} ⊂ [0, 1]7 the AMCalgorithm is carried out as described in Section 3.1. By using indexes specified


in Section 3.2, it was shown that an optimal partition with the most appropriatenumber of clusters has four clusters. The obtained results were transformed into[α, β]. Characteristics of the optimal cluster partition (the number of projects percluster |π⋆

j |, the standard deviation of the cluster σ⋆j , the cluster center c

⋆j = T−1(ζ⋆j )

and the quality measure d(π⋆j , b

⋆) of the cluster defined by (14)) are given in Table 4.

Cluster |π⋆j | σ⋆

j c⋆j = T−1(ζ⋆j ) d(π⋆j , b

⋆)

π⋆1 18 0.37 (4.512 4.785 4.674 4.736 4.958 4.797 9.326) 0.250

π⋆2 8 0.51 (4.187 4.656 4.031 4.094 4.187 4.250 8.347) 0.599

π⋆3 3 0.34 (3.778 4.917 4.833 4.500 4.833 2.833 9.000) 0.559

π⋆4 18 0.58 (3.912 3.889 3.681 4.083 4.569 4.556 7.982) 0.755

Table 4: Properties of the optimal partition

Note that the cluster π⋆1 which consists of projects assessed as best is ranked

significantly higher than other clusters, whereby there is an insignificant differencebetween standard deviations by clusters (see Table 4). This means that the clusterof projects π⋆

1 assessed as best is significantly separated from other clusters.

pi f i1 f i

2 f i3 f i

4 f i5 f i

6 f i Fi ∥bi − b⋆∥ Rank

p35 5 5 5 5 5 5 5 10 0 1p6 4.833 4.75 5 5 5 4.875 4.881 9.75 0.047 2p40 5 5 5 5 5 5 5 9.5 0.055 3p10 4.5 4.75 5 4.75 5 5 4.775 10 0.076 4p36 5 5 4.5 4.5 5 5 4.875 9.5 0.0829 5p33 4.5 5 4.5 5 5 5 4.775 10 0.0834 6p28 4.875 4.875 4.125 4.5 5 5 4.756 9.875 0.096 7p8 5 4.5 5 4.5 5 4.5 4.775 9.5 0.1000 8p7 4.5 5 5 5 5 4.5 4.775 9.5 0.1001 9p26 4.5 4.5 5 5 5 5 4.750 9.5 0.104 10p1 4.417 5 5 5 5 5 4.825 9.25 0.115 11p9 4.5 4.75 4 4.5 5 5 4.600 9 0.169 12p3 4 5 4.5 4.5 5 5 4.575 9 0.187 13p45 4.5 4.5 5 5 5 5 4.750 8.5 0.188 14p30 4.333 5 4.75 4.5 5 4 4.563 8.75 0.197 15p37 4 4.5 4.25 4.5 4.75 4.5 4.275 8.5 0.240 16p24 4 4.5 4.25 4.5 4.75 4.5 4.337 8.5 0.243 17p19 3.5 4.5 4.5 5 4.5 5 4.325 9 0.248 18

Table 5: Cluster of projects π⋆1assessed as best

The cluster of projects π⋆1 assessed as best in this case contains 18 projects. It is

interesting to notice that these are all projects selected as best in the previous section(Example 2), but their order is modified under the influence of grades referring tothe general impression of projects. For the very same reason, project p37 becamepart of the cluster projects assessed as best (data referring to this project can beseen in Table 5).


The ranking of projects in the cluster of projects π⋆1 assessed as best is determined

on the basis of the quality measures d(pi, b⋆) of the projects defined by (13) inaccordance with Remark 2 and shown in Table 5. Since in this approach the impactof the GPA of the general impression of the project and the WGPA of all featuresis assumed to be equal, the quality measures of projects is primarily determined

by the grades Fi, but there are also fine corrections. For example, project p6 was

rated higher than project p40 due to a higher general impression grade, but projectp19 is ranked lower than projects p45, p30, p37, p24, although it has a higher generalimpression grade. The reason for that lies in a relatively low grade given to featuref1 of these projects, which is much more important than other features (w1 = 0.30).

4.3. A possibility of simplification

A set of data (15) can have a great number of components, from which seriousnumerical problems in the implementation of data clustering may arise. Namely, inthis case, there is a high possibility of the singularity of the covariance matrix (see[20]). That is why it makes sense to observe the following simplification: instead

of the vector (f i1, . . . , f

in, F

i)T we could observe the vector (f i, F i)T , where f i =

n∑s=1

wsfis are the AWG of all features of the i-th project. In this way, the problem

would be reduced to the problem of grouping data with two features consideredequally, i.e., we consider the set

A = {ai = (f i, Fi) ∈ R2 : f i =

n∑s=1

wsfis, i = 1, . . . ,m}. (18)

In this approach we should be aware of the fact that we have lost a fine structureof grades, but obtained a simpler set of data that can also be displayed graphically(see Fig. 2a).

Due to a disproportionate range of numbers f i and Fi, when grouping the set A,

the general impression would be preferred to the AWG of all features. In order toeliminate this discrepancy, in accordance with Section 3, the set of data A should befirst transformed into the set B = {bi = T (ai) : ai ∈ A} ⊂ [0, 1]2 in the unit square[0, 1]2. In this case, T : [1, 5]× [1, 10] → [0, 1]2,

T (x) = D(x− α), D = diag(14 ,

19

), α = (1, 1)T . (19)

Thus we ensure a balanced simultaneous impact of the AWG of all features andthe GPA of the general impression of the project.

Applying the AMC algorithm (see Section 3.1), by using indexes specified inSection 3.2 we obtain the optimal partition Π⋆ = {π⋆

1 , . . . , π⋆k} with the most appro-

priate number of clusters π⋆1 , . . . , π


⋆k . After clustering the set

B, the obtained results can be transformed again into [1, 5] × [1, 10] by the inversemapping T−1.

In accordance with Remark 2, in this case project ranking can also be carriedout by comparing the quality measures d(pi, b⋆) of respective projects.


Example 4. The set P of m = 47 projects from Example 1 will be grouped on thebasis of the AWG of all project features f i and the GPA of the general impression

Fiobtained on the basis of reviews by independent, blinded reviewers Ri

#1 and Ri#2.

(a) Non-normed data ai

f⋆

(b) Normed data bi

b⋆

Figure 2: Distances of non-normed data ai to the vector f⋆ (ellipses) and distances of normeddata bi to the vector b⋆ (circles)

After defining the corresponding set of data A = {ai = (f i, F i) ∈ R2 : i =1, . . . ,m}, first the corresponding normalized set B = {bi = T (ai) : ai ∈ A} is definedby means of mapping (19). The set A is shown in Fig. 2a, and the correspondingset of normalized data is given in B in Fig. 2b. The AMC algorithm is applied tothe set B, as described in Section 3.1 (see also Fig. 3b). After clustering the set B,the obtained results will be transformed again into [1, 5]× [1, 10].

(a) Non-normalized data

π1

π2

π3

π4

(b) Normalized data

Figure 3: Clusters of non-normalized and normalized data

It was shown that the optimal partition with the most appropriate number ofclusters has four clusters, too. Characteristics of the optimal cluster partition (thenumber of data per cluster |π⋆

j |, the standard deviation of the cluster σ⋆j , the cluster

center c⋆j = T−1(ζ⋆j ) and the quality measure d(π⋆j , b

⋆) of the cluster) are given inTable 6.

The cluster of projects π⋆1 assessed as best in this case contains 13 projects (see

Table 7 and Fig. 3). It is interesting to notice that these are all projects selected asbest in the previous sections, but projects p19, p24, p30, p37 and p45 are missing.


Cluster |πj | σ⋆j c⋆j = T−1(ζ⋆j ) d(π⋆

j , b⋆)

π1 13 0.21 (4.82 9.57) 0.066π2 19 0.62 (4.30 8.71) 0.226π3 12 0.82 (4.22 7.85) 0.308π4 3 1.27 (3.62 7.08) 0.473

Table 6: Properties of clusters of the optimal partition

Ranking clusters within the partition was carried out on the basis of the qualitymeasures of the clusters d(π⋆

j , b⋆), as defined in (14). The order of projects in the

cluster of projects π⋆1 assessed as best is determined based upon the quality measures

d(pi, b⋆) of the projects defined by (13).

pi f i F i d(pi, b⋆) Rank

p35 5 10 0. 1p6 4.877 9.75 0.0209 2p33 4.825 10 0.0239 3p40 5 9.5 0.0248 4p28 4.8 9.875 0.02808 5p36 4.9 9.5 0.02837 6p10 4.7875 10 0.0291 7p3 4.65 9 0.0690 8p7 4.8 9.5 0.0369 9p1 4.8542 9.25 0.0422 10p26 4.75 9.5 0.0423 11p8 4.75 9.5 0.0423 12p9 4.6625 9 0.0678 13

Table 7: Cluster of projects assessed as best

Please note that this ranking is not the same any more as it was in previoussections, and it is shown in Table 7. Note also that the cluster of projects π⋆

1

assessed as best is ranked significantly higher than other clusters, whereby thereis an insignificant difference between standard deviations by clusters (see Table 6).This means that the cluster of projects π⋆

1 assessed as best is significantly separatedfrom other clusters.

A balanced simultaneous impact of the AWG of all features and the GPA of thegeneral impression of the project determined the project ranking list.

5. Conclusions

The problem of a fair, equitable and transparent selection of research projects to befinanced from a fund is important for both the institution that allocates financialresources and researchers, i.e., potential users. To tackle this problem, numerousapproaches can be found in the literature, which are most often based on the AHPmethod. The combination of the AHP method and the AMC algorithm proved to


be a very reasonable approach to solve this problem because the proposed methodfor the formation of the cluster of projects assessed as best optimally connects allfeatures of the data set, i.e., grades obtained in the review procedure. Note thatthe well-known Expectation Maximization Algorithm lies in the background of themethod [22, 24].

The quality measure of projects in the cluster of projects assessed as best isdefined such that it takes into account the weighted structure of grades obtained inthe review procedure.

Based upon this grouping and ranking of positively assessed projects from Exam-ple 1, the obtained results were presented to University of Osijek constituent unitsand a list of projects to be financed was published on the University of Osijek web-site. It was observed that the reactions of applicants in the call for project proposalsto this transparent and clear assessment process are generally very positive. In thisway, we have maximally avoided possible objections and dissatisfaction of applicantswhose project proposals were not selected for funding.

Acknowledgement

This work was supported by the Ministry of Science, Education and Sports, Re-public of Croatia, through research grant 235-2352818-1034. The authors wouldlike to thank the administration of Josip Juraj Strossmayer University of Osijek forpreparing the data for this paper.

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Can confidence indicators forecast the probability of expansion in Croatia?

Mirjana Čižmešija1,* and Nataša Erjavec1

1 Faculty of Economics and Business, University of Zagreb

J. F. Kennedy 6, 10000 Zagreb, Croatia E-mail: ⟨{mcizmesija, nerjavec}@efzg.hr⟩

Abstract. The aim of this paper is to investigate how reliable are confidence indicators in forecasting the probability of expansion. We consider three Croatian Business Survey indicators: the Industrial Confidence Indicator (ICI), the Construction Confidence Indicator (BCI) and the Retail Trade Confidence Indicator (RTCI). The quarterly data, used in the research, covered the periods from 1999/Q1 to 2014/Q1. Empirical analysis consists of two parts. The non-parametric Bry-Boschan algorithm is used for distinguishing periods of expansion from the period of recession in the Croatian economy. Then, various nonlinear probit models were estimated. The models differ with respect to the regressors (confidence indicators) and the time lags. The positive signs of estimated parameters suggest that the probability of expansion increases with an increase in Confidence Indicators. Based on the obtained results, the conclusion is that ICI is the most powerful predictor of the probability of expansion in Croatia. Key words: confidence indicators, expansion, business survey, probit models

Received: March 4, 2016; accepted: March 21, 2016; available online: March 31, 2016

DOI: 10.17535/crorr.2016.0007

1. Introduction Many researches and papers incorporate the psychological sentiment (obtained by Business and Consumer Surveys; BCS) in macroeconomic modelling [19, 11, 13, 7, 4, 21, 17]. Namely, BCS offer direct assessments of the otherwise intangible factors such as economic agents’ perceptions and expectations. They measure the agents’ willingness to consume/invest/save, as opposed to their ability to do the same. Hence, these psychological factors are crucial to understanding the underlying market forces and agents’ behaviour. The BSC researches published so far (both for Croatia and the majority of developing countries) mostly rely on well-known econometric methods such as linear time series models, or simple regression/correlation analysis [8, 1, 2, 16]. However, the


98 Mirjana Čižmešija and Nataša Erjavec

role of economic agents’ sentiment in governing actual economic developments can be explained using probability models.

2. Business and consumer survey – Croatian experience and literature review Business and consumer survey data represents certain timely economic information derived from qualitative questions asking managers and consumers about assessments and expectations on the state of the economy. To summarize managers' subjective assessment of economic variables, there are different approaches in using and interpreting business survey results as leading or coincidence indicators [7]. These indicators are used to explain and predict changes in macroeconomic variables.

Since 1995, Business surveys in Croatia have been conducted on a quarterly basis in manufacturing industry, construction and retail trade, and in the services sector since 2008. The surveys are conducted using the harmonized EU methodology, adjusted for specific characteristics of the Croatian economy. The surveys are financed by the Croatian Chamber of Commerce and carried out by the Research Centre of the business journal Privredni vjesnik, which regularly publishes the survey findings [5]. Variables as measures derived from the business surveys are expressed as a difference (balance) between weighted percentages of the positive (good) and negative (bad) responses of managers to questions.

The questions in the business surveys are (in essence) of a qualitative nature with three reply options: positive (increase, more than sufficient, etc.), equal (remain unchanged, sufficient, etc.) and negative (decrease, not sufficient, etc.). For each answer option (positive, equal or negative) relative frequencies are calculated. The common way of presenting business survey data is the balance. If P, E and M denote percentages of respondents’ chosen options: positive, equal and negative, respectively, with the sum equalling 100 for each variable, the balance is defined as the difference between P and M (difference between the percentages of respondent’s positive and negative replies). In accordance with the Harmonized European Business Survey methodology [6], a weighted counting of answers is used in the Croatian business survey. This means that the answers of each respondent are weighted with the coefficient in line with a business’s turnover. Balance (B) is calculated for all questions (variables). In general, time series of balances are seasonally adjusted and then used in calculating composite indicators. The European Commission is using Dainties as the seasonal-adjustment algorithm, as originally developed by Eurostat. The main advantage of Dainties is the absence of revision of past data when adding data at the end of a time series. Croatia's survey data are seasonally adjusted using Dainties, as well.

Can confidence indicators forecast the probability of expansion in Croatia? 99

In the simplest way, the time series of seasonally adjusted balances can be used as a basis for explaining and predicting changes in a national economy. The next level in aggregation of business survey data is calculating composite (confidence) indicators for the three different sectors: manufacturing industry, construction and retail trade.

Each confidence indicator is calculated as the simple arithmetic average of the seasonally adjusted balances of answers to specific questions chosen from the full set of questions in each survey. The selection of questions was guided by the aim of achieving as high as possible coincident correlation of the confidence indicator with a reference series [6], such as year-on-year growth in industrial production.

Only three variables are used in calculating the composite indicator in the manufacturing industry, i.e. the Industrial Confidence Indicator (ICI), namely: order books, stock of finished products (with an inverted sign) and production expectation. Balances are seasonally adjusted. The Construction Confidence Indicator (BCI) is the arithmetic average of seasonally adjusted balances of answers to the questions on order books and employment expectations. The Retail Trade Confidence Indicator (RTCI) is the arithmetic average of seasonally adjusted balances of answers to the questions on the present and future economic situation, and on stocks (the last with an inverted sign).

Confidence indicators effectively predict changes in the macroeconomic reference series (at national and regional level) up to six months in advance [9]. There have been many research results, studies and papers on this subject (for a list of results see: [4, 8]. Some empirical researches have shown the existence of a relationship between business survey variables and confidence indicators with the reference series [11, 12].

According to research results in Croatia during the last 10 years (as in: [1, 2, 16, 21], the Croatian Industrial Confidence Indicator correctly predicts changes in Croatian industrial production one or two quarters ahead in almost 60% of cases. However, the latest results have not been so good. Moreover, Croatia’s business survey indicators in construction and in retail trade have been shown to be weak predictors of national economy.

However, some researches [14, 19, 5] indicate that in the recession period (after 2008), harmonized indicators used in an official sense, do not exhibit good predictive properties. Therefore, these indicators should be revised, new ones considered or certain methodological improvements implemented when using indicators as coincidence of leading indicators, in order to accurately predict changes in the macroeconomic reference series one or two quarters ahead. The survey research conducted in this paper is one of the methodological improvements that aims to investigate the potential forecasting property of BCS Indicators.


3. Business cycle turning points for Croatia To distinguish the periods of expansion from periods of recession in the Croatian economy, we applied the non-parametric BBQ algorithm proposed by Pagan and Harding, [17]. The BBQ algorithm is a quarterly implementation of the original Bry and Boschan, [3] BB monthly algorithm. The potential turning points are identified as the local minima (troughs) and maxima (peaks) for real seasonally adjusted quarterly GDP data. Between a peak and trough of economic activity an economy is in a contractionary phase (a recession), whereas between a trough and peak of activity an economy is in an expansionary phase (a recovery). Although the algorithm is wildly used to analyse business cycles, the drawback is that it cannot identify turning points at the beginning of a sample (the first two observations) nor at the end of a time series (the last two observations), as there are no previous or subsequent observations for these observations [17, 10].

Since quarterly GDP data are available for the period 1997/Q1-2014/Q1, the BBQ dating algorithm was applied to that period. In the analysed period, the BBQ algorithm identified three recession periods in Croatia’s economic activity (1998/Q2-1999/Q2, 2009/Q1-2010/Q2 and 2011/Q3-2013/Q1). The post-war recession ended in the second quarter of 1999 (trough), which was followed by a long period of expansion, ending in 2008/Q4 (peak). The Croatian economy subsequently entered another recession in the first quarter of 2009, and ended in 2010/Q2. The expansion which followed lasted for almost a year. Another recession is detected for the period 2011/Q3-2013/Q1.

The algorithm detected the business cycle peak in the third quarter of 2013 indicating the end of the expansion period and beginning of another recession in the Croatian economy. The algorithm applied on the data prior to the first quarter of 2014 shows that the recession, which began in the fourth quarter of 2013, had not ended by the first quarter of 2014.

4. Data selection To separate periods of expansion from periods of recession, as a relevant measure of economic activity, quarterly data on GDP (real GDP at market prices of the previous year) are used. Data provided from Eurostat are available from the first quarter of 1997 to the first quarter of 2014, and are seasonally adjusted and adjusted for working days using the Demetra statistical software Tramo/Seats seasonal adjustment method. Data for predictor variables, i.e. the Croatian Business Survey’s confidence indicators (ICI, BCI and RTCI), are taken from Privredni vjesnik and are available from the first quarter of 1999 to the first quarter of 2014. The availability of data for the business survey

Can confidence indicators forecast the probability of expansion in Croatia? 101 indicators limited the empirical analysis for the period from the first quarter of 1999 to the first quarter of 2014.

5. Methodology The empirical analysis consists of two parts. In the first part, the variable expansion is defined based on results obtained from the BBQ algorithm. In the second part, various probit models were estimated [18, 15]. The models differ in the predictor variable (Business Survey Indicator) and in its time lag included in the model.

In all probit models the dependent variable is a binary expansion indicator which takes a value of one to indicate that an economy is in an expansionary state at time t, and a value of zero, if economy is in a recession state at time t. Based on the predictor variable, we consider a set of Croatian Business Survey indicators such as; Industrial Confidence Indicator (ICI), Construction Confidence Indicator (BCI) and Retail Trade Confidence Indicator (RTCI). The indicators are calculated in line with the harmonized European Union methodology. The standard probit model that quantifies expansion probabilities is given by the equation (1).

P(Yt=1)=F(b0+ b1Xt) (1)

where P(Yt=1) is the probability that the economy is in expansion at time t. Xt is a predictor variable. Predictor variables in the estimated models are Croatia’s business survey indicators: ICI, BCI or RTCI. b0 and b1 are unknown parameters and F(·) denotes the cumulative distribution function of the standard normal distribution.

The probit models generate a probability of expansion from information in a set of leading indicators. The closer the probability is to 1, the more likely the economy will be in expansion; the closer the probability is to 0.5 the less likely the economy will be in expansion. These probabilities can be easily used to predict the turning point from expansion to recession. When the probability of expansion exceeds 0.5, the economy is more likely to be headed toward expansion than remaining in recession, and thus a business cycle turning point is signalled.

After the estimation of probit models for each predictor, the various measures are employed to discriminate between models. Model selection criteria include two types of measures; measures of model fit and measures of model classification ability. For measures of model fit, we calculated the pseudo R-squared (pseudo R2) and the information criteria (AIC and BIC). Additionally, a Pearson goodness of fit test and Hosmer-Lemeshow goodness of fit test are performed.


The pseudo R-squared is McFadden R-squared, known as the “likelihood ratio index”, and compares a model without any predictor to a model being estimated. It is defined as one minus the ratio of the log likelihood with intercepts only, and the log likelihood for the model being estimated.

The second group of measures refers to the measures of model classification ability, such as the overall rate of correct classification and the area under the ROC curve (AUROC). The ROC curve (receiver operating characteristic curve) is a useful tool to select a possibly optimal model. It captures the ability of each model to accurately categorize recessions and expansions. The area under the ROC curve (AUROC) quantifies the overall ability of the model to discriminate between two states (in our case expansion and recession). A larger value of AUROC is preferred. The model with no predictive power has an area of 0.5 whereas the perfect model has an area of 1.

6. Empirical results

Changing the predictor variable and its lag length included in the model, different probit models are estimated. Based on economic theory and empirical results [9, 16], it is evident that Business Survey indicators can predict changes in referent macroeconomic series instantaneously and for one or two quarters ahead. For each predictor (business survey indicator: ICI, BCI and RTCI), we analyse its instantaneous predictive performance for Croatian expansion, as well as at horizons of one and two quarters.

The estimates from models with two predictors (Industrial Confidence Indicator (ICI) and Construction Confidence Indicator (BCI)) are presented in Table 1 and Table 3. The signs of the estimated parameters in all models are positive, as expected, (Table 1 and Table 3) implying that the probability of expansion is higher as the values of indicators increase.

ICIt ICIt-1 ICIt-2

Parameter estimate* 0.0867 0.1303 0.0766Standard error 0.0232 0.0335 0.0211p-value, P>|z| <0.0001 <0.0001 <0.0001

* An estimate of a constant is not reportedTable 1: Estimates of probit models (predictor ICI variable)

Based on the results of both chi-square goodness of fit tests (Pearson and

Hosmer-Lemeshow), all models are correctly specified and cannot be rejected (Table 2 and Table 4). Additionally, the area under the ROC curve (exceeding 0.8) points to a high predictive power for all the estimated models.

Can confidence indicators forecast the probability of expansion in Croatia? 103

Measures of model fit

ICIt ICIt-1 ICIt-2

Pseudo R2 0.2986 0.4610 0.2767AIC 54.6322 41.5079 52.3855BIC 58.8540 45.6966 56.5406Pearson test

chi2(59) =48.11 Prob > chi2 =

0.8437

chi2(58) = 38.61Prob > chi2 = 0.9766

chi2(57) = 51.10 Prob > chi2 =

0.6948Hosmer - Lemeshow test

chi2(8) =10.64 Prob > chi2 =

0.2227

chi2(8) = 6.97Prob > chi2 = 0.5403

chi2(8) = 4.81 Prob > chi2 =

0.7778Measures of model classification abilityCorrectly classified

77.05% 85.00% 79.66%

Area under the ROC curve

0.8676 0.9261 0.8621

Table 2: Measures of model fit (predictor ICI variable)

The overall model selection evidence (Table 2) indicates that the model

with a one quarter lag of ICI is superior to other models. The overall rate of correct classification for that model is estimated to be 85%. The value of the ICI indicator in the previous quarter is a strongly statistically significant predictor of expansion period in Croatia. The obtained results confirm the conclusions drawn from previous researches, i.e. on the basis of changes in ICI, changes in the whole national economy can be correctly predicted with a one quarter lag.

BCIt BCIt-1 BCIt-2

Parameter estimate* 0.0345 0.0284 0.0185Standard error 0.0093 0.0082 0.0070p-value, P>|z| <0.0001 0.0010 0.0080* An estimate of a constant is not reported

Table 3: Estimates from the probit models (predictor BCI variable)

In analysing the relative model performance with the BCI indicator as a predictor variable (Table 4), the appropriate model appears to be a model without lag (instantaneous impact of BCI indicator). However, measures of model fit and measures of model classification ability may lead to different conclusions.

The values of parameter estimates in all models (Table 3) are positive and decrease with an increase in a time lag of BCI. The highest value is obtained in the model without a lagged predictor. If we know that business survey results are available approximately one quarter before the publishing of official


statistical GDP data, BCI can be used to predict changes in a national economy and in signalising expansion.

Measures of model fit

BCI t BCI t-1 BCI t-2

Pseudo R2 0.2820 0.2244 0.1152AIC 55.8324 57.97373 63.1907BIC 60.0542 62.1624 67.3458Pearson test

chi2(59) = 46.51Prob > chi2 =

0.8809

chi2(58) = 58.38Prob > chi2 =

0.4615

chi2(57) = 61.12 Prob > chi2 =

0.3303Hosmer-Lemeshow test

chi2(8) = 5.59Prob > chi2 =

0.6933

chi2(8) = 6.41Prob > chi2 =

0.6017

chi2(8) = 10.30 Prob > chi2 =

0.2443Measures of model classification abilityCorrectly classified

73.77% 75.00% 74.58%

Area under the ROC curve

0.8275 0.8139 0.7364

Table 4: Measures of model fit (predictor BCI variable)

The Retail Trade Confidence Indicator (RTCI) proved to be an

insignificant predictor of Croatian expansion in all models (regardless of the lag length). The estimated parameters in all models were statistically insignificant as can be seen in Table 5.

RTCIt RTCIt-1 RTCIt-2

Parameter estimate* 0.0054 -0.0055 -0.0113Standard error 0.0093 0.0138 0.0114p-value, P>|z| 0.562 0.594 0.325

* An estimate of a constant is not reportedTable 5: Estimates from the probit models (predictor RTCI variable)

Can confidence indicators forecast the probability of expansion in Croatia? 105 7. Conclusions This paper examines the performance of various probit models in forecasting the probability of expansion in Croatia based on a set of Croatian Business Survey indicators such as the Industrial Confidence Indicator (ICI), the Construction Confidence Indicator (BCI) and the Retail Trade Confidence Indicator (RTCI). Nonlinear probit models applied in this paper are a methodological improvement for examining and explaining predictive properties of Croatian Business survey indicators. In accordance with well-known theoretical and empirical findings, the probit models were estimated for the business survey indicators instantaneously (without lags) and for one or two quarters of lag.

In predicting the direction of changes in industrial production and in an entire national economy, ICI proved to have the best predictive abilities with a one quarter lead. That means that an increase of one point in ICI in a current quarter, signals an increase of economic activity in the next quarter. Our research results based on the new methodological basis has confirmed this conclusion. The highest (positive) value of the estimated parameter is obtained in the probit model with ICI predictor with a one quarter lead (in that case, all measures of model fit are optimal).

Following the evidence obtained in our study, the BCI has been found to be a useful predictor variable for the Croatian expansion period as well. Ambiguous conclusions can be derived on the basis of measures of model fit and measures of model classification ability. Values of estimated parameters in all probit models are positive, thus indicating a positive relationship between the indicator and the probability that the economy is in expansion. The highest value of the estimate is obtained for the model without lag, i.e. for the model with instantaneous impact of the BCI indicator. It shows that BCI is a coincidence (not leading) indicator. Since the business survey results are available approximately one quarter before the publishing of official statistical GDP data, our results show that BCI can (nevertheless) be used as a leading indicator of change in the national economy.

The Retail Trade Confidence Indicator (RTCI) proved to be an insignificant predictor of Croatian expansion in all models (regardless of the lag length of a predictor included in the model).

Research on the Croatian business cycle relies mostly on Business and Consumer Surveys (BCS). The recent global financial crisis has opened a variety of economic fields which could greatly benefit from BCS results. Namely, the BCS offer a direct empirical assessment of otherwise “intangible” factors, such as economic agents’ perceptions and expectations. Nonlinear probit models applied in this paper are one of the methodological improvements in explaining BCS results in short-term macroeconomic forecasting. However, apart from Croatia, it would be interesting to analyse and compare (using the nonlinear


probit methodology applied in paper) the forecast properties of BCS indicators for countries that recently joined the EU. Acknowledgement This work has been fully supported by the Croatian Science Foundation under the project No. 3858.

Some of the survey results published in the paper were presented at “The second International Conference on Business Management and Economics (ICBME) 2016”, Colombo (Sri Lanka) 19-20 February 2016.

References [1] Bahovec, V., Čižmešija, M. and Kurnoga Živadinović, N. (2006). The forecasts of

some macroeconomic variables for Croatia using the information provided by business surveys. Proceedings of the 11th International Conference on Operational Research KOI 2006, Boljunčić, V. (editor), Pula, Croatia, September 27-29, 2006, 77–86.

[2] Bahovec, V., Čižmešija, M. and Kurnoga Živadinović, N. (2007). Testing for granger causality between economic sentiment indicator and gross domestic product for the Croatian economy. Proceedings of the 9th International Symposium on Operational Research SOR '07, Zadnik Stirn, N., Drobne, S. (editors), Nova Gorica, Slovenia, September 26-28, 2007, 403–408.

[3] Bry, G. and Boschan, C. (1971). Cyclical Analysis of Time Series: Selected Procedures and Computer Programs. NBER, New York.

[4] Claveria, O., Pons, E. and Ramos, R. (2005). Business and consumer expectations and macroeconomic forecasts. Proceedings of Conference on Survey data in Economics – Methodology and Applications, Cess IFO, Munich, 1–20.

[5] Čižmešija, M., Erjavec, N. and Bahovec, V. (2014). The role of business survey measures in forecasting Croatian industrial production. International Journal of Social, Human Science and Engineering, 8, 3, 67–72.

[6] European Commission, (2014). The joint harmonized EU programme of business and consumer surveys. User guide, European Economy, Directorate-General for Economic and financial affairs.

[7] Frale, C., Marcellino, M., Mazzi, G. L. and Proietti, T. (2009). Survey data as coincidence and leading indicators. European University Institute Working Paper ECO 2009/19, Florence.

[8] Fusari, A and Pellissier, M. (2008). Some new indicators and procedures to get additional information from the Business Tendency Surveys. Proceedings of the 29th CIRET Conference, Santiago.

[9] Gayer, C. (2004). Forecast evaluation of European commission survey indicators. Proceedings of the 27th CIRET Conference, Warsaw.

[10] Krznar, I. (2011). Identifying recession and expansion periods in Croatia. Working Papers W-29, Croatian National Bank. Zagreb.

Can confidence indicators forecast the probability of expansion in Croatia? 107 [11] Kurnoga, N., Erjavec, N. and Bahovec, V. (2015). The analysis of association

between the variables in Croatian business survey for services sector. The Business & Management Review, Datta, P. R. (editor), 263–268.

[12] Kurnoga, N. and Čižmešija, M. (2015). Has the long-term recession changed managers’ assessments and expectations in the Croatia’s retail trade sector? Proceedings of the 13th International Symposium on Operational Research SOR '15, Zadnik Stirn, L., Žerovnik, J., Kljajić Borštnar, M., Drobne, S. (editor.), Ljubljana: Slovenian Society Informatika, Section for Operational Research, 508–513.

[13] Liu, W. and Moench, E. (2014). What Predicts U.S. Recessions? Federal Reserve Bank of New York, Staff Report, No. 691.

[14] Lolić, I., Sorić, P. and Čižmešija, M. (2015). Redefining the Croatian economic sentiment indicator. International Journal of Social, Behavioural, Education, Economic and Management Engineering, 9(8), 2337–2340

[15] Long, J. S. and Freese, J. (2014). Regression Models for Categorical Dependent Variables Using Stata. 3rd ed. New York. College Station, TX Stata Press.

[16] Nikić, G., Šošić, I. and Čižmešija, M. (2002). Business and investment surveys in Croatia – a case study of an economy in transition. Proceedings (CD) 26th CIRET Conference in Taipei, 16-19, October 2002, 12.

[17] Pagan, A. and Harding, D. (2002). Dissecting the cycle: A methodological investigation. Journal of Monetary Economics, 49, 2, 365–381.

[18] Powers, D. A., and Xie, Y. (2008). Statistical Methods for Categorical Data Analysis. 2nd ed. Bingley. Emerald.

[19] Sorić, P. and Čižmešija, M. (2015). Economic sentiment and the recession depth in Croatia: A structural break analysis. The Business & Management Review, Datta P. R. (editor), New York: The Academy of Business & Retail Management, 192–200.

[20] Sorić, P. (2013). Assessing the sensitivity of consumption expenditure to inflation sentiment in post-communist economies, Post-Communist Economies, 15/4, 529–538.

[21] Šošić, I. and Čižmešija, M. (2003). A note about forecasting accuracy of business survey in Croatia. Bulletin of the International Statistical Institute, 54th Session, Contributed Papers, Vol. LX, Book 2, 465–466.



Research and evaluation of the effectiveness of e-learning in the case of linear programming

Ljiljana Miletić1,* and Goran Lešaja2

1 Gimnazija Požega

Street dr. F. Tudmana 4.A ⟨[email protected]⟩

2 Georgia Southern University

1332 Southern Drive, Statesboro, Georgia 30458, USA ⟨[email protected]⟩

Abstract. The paper evaluates the effectiveness of the e-learning approach to linear programming. The goal was to investigate how proper use of information and communication technologies (ICT) and interactive learning helps to improve high school students’ understanding, learning and retention of advanced non-curriculum material. The hypothesis was that ICT and e-learning is helpful in teaching linear programming methods. In the first phase of the research, a module of lessons for linear programming (LP) was created using the software package Loomen Moodle and other interactive software packages such as Geogebra. In the second phase, the LP module was taught as a short course to two groups of high school students. These two groups of students were second-grade students in a Croatian high school. In Class 1, the module was taught using ICT and e-learning, while the module was taught using classical methods in Class 2. The action research methodology was an integral part in delivering the course to both student groups. The sample student groups were carefully selected to ensure that differences in background knowledge and learning potential were statistically negligible. Relevant data was collected while delivering the course. Statistical analysis of the collected data showed that the student group using the e-learning method produced better results than the group using a classical learning method. These findings support previous results on the effectiveness of e-learning, and also establish a specific approach to e-learning in linear programming. Key words: linear programming, simplex method, interior point-method, Loomen Moodle - course management system, GeoGebra - dynamic geometry program, action research

Received: October 10, 2014; accepted: March 26, 2016; available online: March 31, 2016

DOI: 10.17535/crorr.2016.0008

1. Introduction

Information and communication technology is important in the development of various scientific disciplines and mathematics, enabling their widespread use. * Corresponding author.

110 Ljiljana Miletić and Goran Lešaja

Mathematical content is largely abstract and difficult to understand. Computers can assist in visualizing such content. "While physical objects become more abstract when modelled onscreen (e.g. science simulation), mathematical objects, already inherently abstract, become concrete" [8].

The effectiveness of e-learning was investigated using an LP lesson module as an example. The less module provides an overview of the development of the linear programming problem, linear programming model, theory of linear programming, counting vertices – enumeration method, geometric method, 2_ D, simplex method, Charnes’ M–method [15], the two-phase simplex method [15], and finally, the method of interior points [9, 10]. Software that aided learning linear programming included Excel Solver, Lindo, Winqsb and Simplex apple. The basics of running GeoGebra was also explained. The module implemented in the software package for online learning is Loomen Moodle. To study the effectiveness of e-learning, action research methodology was used, bringing about a different approach to teaching. The empirical part of the research was carried out in the two sophomore classes, Class1 and Class2 at Požega High School as already mentioned. The learning outcomes, assessment criteria and evaluation of student achievement were defined. Students participated freely in the research which was organized into three cycles, where each cycle required a knowledge test and a final exam. Both classes were taught the same linear programming content using action research. Action research results are usually directly applicable in teaching practice as they are obtained performed “on the spot” [11, 14]. Most modern professions, especially teaching, are faced with complex practical requirements where problems cannot always be predicted in advance, and furthermore, offer clear solutions and solutions that can be applied. Nowadays, school teachers are active participants in the research process. Action research is a systematic process of observation and description, planning, action, reflection, evaluation, modification, where a combination of action and research can provide new insights for improving the quality of educational practices [4, 11]. Although previous research has clearly proven the advantage of action research and e-learning over traditional learning [1, 7, 17], the effectiveness of e-learning in operational research topics, such as linear programming methods, have not been investigated adequately to date. Besides addressing this gap, this paper also aims to analyze the manner in which different linear programming methods are incorporated by students in an e-learning environment.

The material covered in the module is not part of the regular educational curriculum. An important premise of the research was to establish that there were no statistically significant differences in prior knowledge and previous successes between the two student groups. This was determined by collecting data and testing the knowledge of students in areas relevant to the material taught in class. The only difference was that students in class 2a received all the necessary materials in digital form via Loomen Moodle and used computers in the classroom, whereas students in class 2b received their materials in paper

Research and evaluation of the effectiveness of e-learning in the case of linear programming 111 form and were taught in the classroom using a board and exercise books. After each cycle, all the students were given tasks. Students in class 2a submitted these digitally via Loomen Moodle from home, whereas students in class 2b brought their homework to school and submitted it in paper form. The results were analyzed statistically. The t- test was used to compare the results. Given that action research was used, the qualitative analysis of results was more important than the quantitative statistical analysis. Action research included three critical friends who followed the course, and a number of useful tips was obtained from them.

At the end of the course, statistical analysis of learning effectiveness was performed and the result compared for both classes in order to test the starting hypothesis which states: Effective usage of information and communication technology for interactive learning of linear programming improves the understanding and adoption of materials and student learning outcomes.

2. Previous Research E-learning and education provides temporally and spatially (time-wise and space-wise) flexible access to up to date multimedia and interactive teaching materials and allows dynamic use of the Croatian and international repository of educational content, digital libraries, archives and museums. ICT enables personalization of educational content to the individual needs of students, and supports collaborative learning and team work. Accordingly, availability is increased for a wider range of participants (participants with special needs, students in remote locations, foreign students, etc.). The growth of e-learning and education increases the role and importance of teachers as mentors, coordinators and instigators of the educational process. E-learning shifts the center of the learning process to the student, who then assumes an active role and responsibility for learning outcomes [12]. The positive outcomes of e-learning, especially web-based learning in teaching international students in Russia were highlighted by Yanuschik et al. [17]. Dečman [7] tested and evaluated the applicability of the Unified Theory of Acceptance and Use of Technology (UTAUT) in a specific mandatory e-learning environment in higher education in Slovenia. His research has shown that a student’s gender and previous education influence the acceptance and use of ICT in learning. The effectiveness of e-learning is also point out in [1]. In Croatia, part of the e-content has been developed to facilitate educational processes in primary and secondary schools, and made available on the national portal Nikola Tesla, where the Croatian Academic and Research Network (CARNet) provides information and infrastructural support for e-learning, as well as a system for managing educational content and a videoconferencing system. Besides the existing e-aids, systematic support should be given to developing other e-tools, for example, peer-reviewed e-books, multimedia demonstration lessons, exercises, simulations, expert teaching


systems, development of e-learning software tools at all levels of education [12]. In secondary schools, e-learning is still not utilized at a satisfactory level and very often even it is void of an appropriate form. According to the results of an input survey conducted upon commencement of research, none of the participating students had attended an online course before. Research conducted at the Marko Marulić Secondary School in Slatina [16] addressed three specific problems encountered by the school The first problem is the lack of teacher training for working with students incorporating a modern teaching approach and modern technologies. Another problem is insufficient knowledge of English by teachers that is required for modernizing teaching and more intensive use of the Internet. The third problem is the lack of material resources needed to support modern teaching methods. A discussion concerning problems relating to the introduction of e-learning in secondary schools is given in [2]. Some of the conclusions are: (1) For now, only a relatively small number of secondary school teachers have

expressed interest in introducing technology in the educational process. The reasons are twofold: first, the introduction of new technologies and methods is not mandatory, and secondly, there is a lack of readily available appropriate multi-media materials that teachers can use. Consequently, they would have to create the materials themselves which requires a lot of effort and time, time they do not have or which is insufficiently valued [2].

(2) If viewing e-learning as a learning continuum, the conclusion is that computers are only used as a teaching aid in secondary schools [2]. In some rare cases, there is an indication of combined teaching techniques in a way that students use materials on web or on CD-ROM, while communicate with the teacher by e-mail or on-line forum. For more substantive shift to online teaching in secondary education teachers should be more educated in methodics and technology.

The second important element of the research was the use of action research methodology which has recently been gaining popularity. Most modern professions, especially teaching, face complex practical requirements where problems cannot always be predicted in advance. Nowadays, school teachers have become active participants in research processes. Action research is the systematic process of observation and description, planning, action, reflection, evaluation, and modification. By combination action and research, new insights into improving the quality of educational practices are obtained [5, 6, 14]. Although action research is under-represented in teaching practice, the situation has recently changed. The teachers are trying to get to know students better and involve them in becoming action researchers, and to start utilizing increasingly online courses and e-learning. In Croatia, the biggest contribution to the development of action research has come from Bognar [4, 5, 6]. Under his mentorship, students at the Teacher Training Faculty in Osijek have been exploring active learning methods in classroom teaching, including action

Research and evaluation of the effectiveness of e-learning in the case of linear programming 113 research [6]. Lately, an increasing number of teachers have begun using action research in their teaching as noted in [5]. Bjelanović Dijanić, a mathematics and computer science teacher at Čazma High School, has been specifically incorporating action research in teaching mathematics [3]. The author in [3] claims that learning mathematics aided by computers and the dynamic geometry software GeoGebra and using methodically designed, interactive educational materials in digital form guarantees full participation of all students in classes, increases motivation in learning mathematics, encourages self-discovery and cooperation among students. Importantly, students should work alone or in pairs on computers, and endeavor to gain new insights. Retention of knowledge acquired in this manner will be greater, students will better understand the material rather than rely on memorization, and they will also have better problem-solving skills.

3. Methodology

3.1 . Learning cycles in the Linear Programming module of lectures

The first phase of the project involved designing a module of linear programming lessons. The module was divided into three lesson groups that served as the basis for three cycles within the course, as recommended by action research method [3].

In both classes, students were given the same learning materials. In Class1, the material was in digital form and provided via Loomen Moodle, whereas Class 2 received the materials in paper form as worksheets. In all cycles, the students were given homework, and were also required to solve crossword puzzles and quizzes in conjunction with each taught unit. At the end of each cycle, students took a knowledge test. The students in each cycle were awarded a certain number of points based on content learning activities. Students were informed of the evaluation criteria at the beginning of the course.

The first cycle of the course referred to the linear programming problem and theory. Topics included a brief overview of the development of LP, the LP model, basic LP theory in addition to the main theorems, duality, geometric representation and solution of two-dimensional (2D) LP problems. Students in Class 2 drew a feasible region in an exercise book and the objective function. Students in Class 1 drew the feasible region using GeoGebra, dynamic geometry software. While working in GeoGebra, students could see the feasible region and move parallel level lines through the vertices of the feasible region. This enabled them to readily see in which vertex the objective function achieved a minimum or maximum for a given problem. Examples of the completed GeoGebra applets can be viewed at the following link: http://www.geogebratube.org/student/m88378.


The second cycle of the course relates involves the simplex method. Students were taught how to solve problems using simplex tables. In addition, they also learnt the Charne’s M-method, and the two-phase simplex method. The students in Class 1 who used ICT were introduced to several popular LP solvers such as Excel Solver, Lindo, and Winqsb.

The third cycle of the course describes a relatively new method known as the method of interior points. The method is fundamentally based on Newton's method. The algorithm is presented and explained, and then applied in solving a few simple examples.

The LP lecture module was implemented using the Loomen Moodle e-learning system. Students in Class1 were added as course users. The course was made available at all times and at any place, and can still be found at the following link: https://loomen.carnet.hr/course/view.php?id=2746. After going through all three cycles, students solved a demo exam which prepared them for the final exam given at the end of the course. 3.2. Interior-Point Method used in the third learning cycle Since the interior–points method (IPM) for solving LP problems are not as common used as the simplex method (SM), here we will present a novel modification of IPM that ties well with simplex method.

IPM is based on solving a Newton-type system at each iteration. At first, the system is much larger than the system used in SM. However, it can be reduced to a system of the same size as in SM. This reduction is known as normal equations reduction. It is very important as it illustrates to students that systems of comparable size are solved in both methods.

Specifically, the original Newton-type system is given by

0 0

0 .

0

kx P

T ky D

k k k ks k

A d r

A I d r

S X d X s e

(1)

The system can be reduced to the system

rMd y , (2) where

).()(

,)(1

1

erXSAbr

AXSAM

kk

Dkk

Tkk

(3) The size of the system that leads to the solution of yd is comparable to

the size of the existing system when using SM. Since sd and xd are obtained from backward substitutions

Research and evaluation of the effectiveness of e-learning in the case of linear programming 115

),()(

,1

sk

kkk

x

yTk

ds

dXeSxd

dArd

(4)

the number of computations per iteration in IPM and SM are comparable. The second modification involves calculating the step size along the

Newton direction. Originally, the step size is calculated to insure that the new iterate lies within a certain neighborhood of the central path. Theoretically, this importantly ensures convergence of the IPM. We simplified the calculation of the step size by enforcing only the positivity requirement for the new iterate in which case the calculation of the step size is almost exactly the same as the minimal ratio test in SM. Based on this simplification, convergence is not guaranteed, but it works well in practice, and problems rarely occur for certain ill-conditioned problems [9].

The step size is chosen so that the positivity of x and s are preserved when updated. As in SM, max is a maximum possible step size until one of the variables becomes 0. Hence,

0,0:0maxmax sk

xk dsdx . (5)

In practice max is calculated as follows:

max

max

max max max

min : ( ) 0, 1,..., ,( )

min : ( ) 0, 1,..., ,( )

min , ,

iP x i

x i

iD s i

s i

P D

xd i n

d

sd i n

d

(6)

which is similar to a minimal ratio test for SM. Since we do not allow any of the variables to be 0, we take

max,1min k , (7)

where )1,0( . The usual choice of is 9.0 or 95.0 .

The outline of the IPM is presented below.


Algorithm (Simplified IPM) Initialization:

1. Choose )1,0(, and 0 . Choose ),,( 000 syx such that esx 00 and 00 y .

2. Set 0k . Step:

3. Set kkP Axbr , kkTk

D syAcr , n

sx kTk

k

)( .

4. Check the termination. If kTkk

Dk

P sxrr )(,, , then terminate.

5. Compute the direction by using (A1) – (A4). 6. Compute the step size by using (A5) – (A7). 7. Update

,1xk

kk dxx

,1yk

kk dyy

.1sk

kk dss

8. Set 1 kk and go to step 3.

3.3. Action research An important and integral part in implementing the course was action research methodology [4, 5]. First, activities that led to desired changes in improving practice based on set objectives were identified. Subsequently, systematic data collection was needed to improve the quality of the teaching process, which had to be included in the action research plan. As noted above, the collected data covered student learning outcomes in form of grades. The important part was self-evaluation through questionnaires handed out upon completion of each course cycle.

The self-evaluation questionnaire posed questions regarding student autonomy in solving assignments throughout all three cycles, general satisfaction with the course, evaluation of their contribution to the research, quizzes on linear programming methods and the final test. Action research has an elasticity property, meaning that the research plan (outline) may change while implementing the action, and whenever circumstances require change [13, 14]. This is the reason why research was planned to take place in cycles, giving participants enough time for problem solving and improving the flow of planned activities.

Another characteristic of action research is that it improves teaching. Action research is carried out in a researcher’s professional or life context and there is almost universal consensus that action research is not conducted on people, but instead with people. Teachers are often unable to recognize problems in their work processes, preventing them from improving their teaching. Therefore, the recommendation in action research is to seek

Research and evaluation of the effectiveness of e-learning in the case of linear programming 117 critical friends in other colleagues. A critical friend is a person who usually shares a professional interest and helps the practitioner in achieving the action research, while providing advice and feedback. A critical friend can also be an immediate colleague at work. However, feedback from critical friends should critically accepted and evaluated. Furthermore, action research is a combination of action and research. The school teaching process is action whereas data collection and analysis, including the preparation of reports is considered research. Action involves data collection, and that specific action is followed by data analysis and preparation of reports. When envolved in action research, students become familiar with the objectives, learning outcomes, and the activities expected of them.

In our project, there were three learning cycles as detailed in the previous section. Classes were filmed by video camera and photographed during each cycle. Examples of the teaching activities carried out in Class1 can be seen on YouTube: https://www.youtube.com/watch?v=9Hta7CqhHPg. 3.4. Samples of teaching material in learning cycles Shown in Table 1 below are samples of work by students in learning cycles for both groups. As already noted for Class 1, the course was taught using ICT tools for distance learning, specifically Moodle Loomen, whereas Class2 was taught the course in a traditional manner, using a board and assignments in paper form.

In the first learning cycle, Class 1 used the dynamic geometry software tool GeoGebra to help draw a feasible region and level lines for a particular two-dimensional LP problem. A nice feature of GeoGebra is the ability to move a level line of an objective function and track the maximal or minimal value in which vertex of the feasible region.

In the second cycle, students studied how to solve LP problems using the simplex method in tabular form. The tables were custom built in MS Word for students to use in Class 1. In the third cycle, students studied a newer method of solving LP problems called the interior-point method (IPM). In Class 1, they again used MS Word to help them solve the systems and present the results. The reason why MS Word was used was to assure that the amount and nature of the work for both groups was at the same level. Calculating the tables and systems for both the simplex method and the interior-point method can be streamlined and “automated” using other software tools such as Excel or MathLab. We plan to incorporate this option into the module for more advanced student groups that possess a good background in linear algebra and linear system solving, and therefore skip this section and concentrate on the main concepts of both methods.


Class1 Class2Cycle 1- Problem and theory of linear programming

Cycle 2- Simplex method

Cycle 3- Interior – Point method

Table 1: Samples of learning material in learning cycles

a) Class 1

b) Class 2Figure 1: Photographs of the teaching process in Class 1 and Class 2

Research and evaluation of the effectiveness of e-learning in the case of linear programming 119 3.5. Sampling procedure and variables As already stated, the created course on linear programming was taught in two classes, Class1 and Class 2 at the Požega High School. Both classes were approximately the same size; Class 1 had 31 students and Class 2 had 29 students. The course was taught in Class 1 with the aid of an ICT tool for distance learning called Moodle Loomen. Class2 was taught the course in a traditional manner, using a board and assignments in paper form. In addition, action research, described in more detail below, was carried out in both classes. Thus, the work in these two classes differs only in regards to one particular component, and that is, the manner in which the course was taught. As the research demanded a certain level of quality, two “equal” groups of students were required. Accordingly, we recorded the initial knowledge of students. The data included their grades in mathematics, computer science, and overall success in the first year, as well as their grades in mathematics and computer science at the end of the first semester of their sophomore year. The additional important components were knowledge of informatics and linear functions. After conducting a statistical test of difference in means and proportions, the conclusion was that the difference between these two student groups is negligible, meaning and we could proceed with the research.

The tracked measurable variables included points (grades) from the students’ homework, quizzes, and tests. After completing each teaching cycle, students filled out a questionnaire on assessing the respective cycle, and they also participated in a group interview. Lickert scale was used to obtain values for the questionnaire responses, and relevant data was analyzed. 3.6. Statistical methods Statistical analysis was performed on the collected quantitative data. The method incorporated basic statistical measures such as mean, median, standard deviation, and covariance. Statistical hypothesis testing was also performed using t-test. In assessing the teaching, the Likert scale with five degrees of intensity was applied. A research diary was updated each day. Group interviews were conducted with the students who were asked to complete the questionnaires. Further, the teaching process was photographed and videotaped with the permission of parents.

Action research uses more qualitative data. The main goal of action research is to promote the teaching practiceand it is essential that it be carried out with students. During the action research, data was to be deliberately and systematically collected on everything that happens. In particular, an important source of information in action research is the research diary. Action research is


carried out in a spiral form amending the basic stages: planning, action, observation and reflection.

A range of points for obtaining a certain grade for each test and final exam was determined. Assignments were graded equally for both groups. Based on the number of obtained points for each test, students received their individual grades. A final grade was then calculated as the arithmetic mean of the grades from three tests after the completion of each cycle and the final exam.

4. Results

The final grades for students from both groups are shown in Table 2, as well as the arithmetic means and standard deviations.

Grades Class 1 Class 25 11 84 14 43 6 82 0 91 0 0

Number of students 31 29Arithmetic Mean 4.16 3.38

Standard Deviation 0.723 1.187Table 2: Final evaluations after research

In order to test whether there is a significant difference between the two

arithmetic means in Table 1, the t-test was performed such that:

29,31,187.1,723.0,38.3,16.4 212121

2

22

1

21

21

NNssXXwhereN

s

N

s

XX

s

Xt

d

d

The degree of freedom is determined as 5822931221 nn . The resulting t value was 3.058, which is larger than the theoretical value

of t = 2.00 for student distribution, while the p value was 0.004. Thus, the arithmetic mean of the two samples differ significantly at the 5% level, i.e. the difference between the means of the two grades is statistically significant. Figure 2 and 3 provide a graphical presentation of the grade distribution.


Figure 2: Distribution of grades in two classes

Figure 3: Box-Whisker graph for the average grades between two classes

0123456789

1011121314

1 2 3 4 5

no of students

Grades

Class 1

Class 2


Distribution of student responses in the final questionnaire is shown in Table 2.

Statement

Does not agree at

all

Mostly does not agree Indifferent Mostly

agree Fully agree

Cycle 1

1.

I solved homework by myself- First cycle

Class 1: 0.00%

Class 2: 10.34%

Class 1: 3.26%

Class 2: 0.00%

Class 1: 12.90% Class 2: 6.90%

Class 1: 29.03% Class 2: 38.71%

Class 1: 54.89% Class 2: 38.71%

Cycle 2

2.

I solved homework by myself- Second cycle

Class 1: 3.24%

Class 2: 10.34%

Class 1: 0.00%

Class 2: 3.45%

Class 1: 16.13% Class 2: 20.69%

Class 1: 29.03% Class 2: 27.59%

Class 1: 51.61% Class 2: 37.93%

Cycle 3

3.

I solved homework by myself- Thrid cycle

Class 1: 3.23%

Class 2: 6.90%

Class 1: 0.00%

Class 2: 3.45%

Class 1: 16.13% Class 2: 10.34%

Class 1: 22.58% Class 2: 37.93%

Class 1: 58.06% Class 2: 41.38%

Final Exam

4. I solved Final exam by myself

Class 1: 0.00%

Class 2: 6.90%

Class 1: 0.00%

Class 2: 0.00%

Class 1: 6.45%

Class 2: 17.24%

Class 1: 29.03% Class 2: 34.48%

Class 1: 64.52% Class 2: 41.38%

5.

Asses your way of following the action research

Class 1: 0.00%

Class 2: 6.90%

Class 1: 0.00%

Class 2: 6.90%

Class 1: 32.26% Class 2: 17.23%

Class 1: 58.06% Class 2: 41.38%

Class 1: 9.69%

Class 2: 27.59%

6.

Asses your contribution to the action research

Class 1: 0.00%

Class 2: 17.24%

Class 1: 12.90% Class 2: 3.45%

Class 1: 29.03% Class 2: 41.38%

Class 1: 48.39% Class 2: 24.14%

Class 1: 9.68%

Class 2: 13.79%

8.

Are you satisfied with the course?

Class 1: 3.23%

Class 2: 6.90%

Class 1: 3.23%

Class 2: 10.34%

Class 1: 16.13% Class 2: 17.24%

Class 1: 48.39% Class 2: 31.04%

Class 1: 29.02% Class 2: 34.48%

Statement 1 2 3 4 5

7.

If allowed, what grade you would give to yourself

Class 1: 0.00% Class 2: 0.00%

Class 1: 3.23% Class 2: 10.34%

Class 1: 3.23% Class 2: 27.59%

Class 1: 41.94% Class 2: 24.14%

Class 1: 51.60% Class 2: 37.93%

Table 3: Distribution of responses after the conducted questionnaires


As is evident in Table 3, a higher percentage of students from Class 1 than Class 2 solved the assignments and the final exam independently. Also, more students in Class 1 evaluated themselves with higher grades (4 or 5) than students in Class 2.

Class 1 expressed a percentagewise stronger satisfaction with the course (77.41% of students in Class 1 mostly agreed or fully agreed that they were satisfied with the course compared to 65.52% of students in Class 2). This conforms to the fact that their success in the course was better than that of Class 2. In Class 1, 90.32% of students would participate again in a similar course with the same format and action research methodology, while only 79.31% of students in Class 2 would do the same. In addition to the above questions, students were asked to write down their general opinion of the way they were exposed to learning. Students in Class 1 commented that they liked online learning, emphasizing the ease of accessing the teaching materials even out of class, and they liked the online exams. Students in Class 2 generally liked the opportunity to learn new methods, and they liked to work in teams, but they felt stressed by the limitations of having a lot of paper material which was hard to learn.

5. Comments and Discussion

Both student groups felt that the course was presented in a useful and interesting manner, and that they learned a lot about a subject which otherwise is not part of their standard curriculum. Most of them liked the action research methodology and thought the approach enabled them to participate more actively in the learning process.

Students in both classes thought the second cycle of the course was the hardest and most time consuming. Students in Class 1 liked the first cycle the most and the main reason was the use of the GeoGebra software package used to solve problems. Students in Class 2 liked the second cycle the most and the main reason was the interactive use of simplex tables drawn on the board.

In addition, the student group that used e-learning (Class 1) expressed additional satisfaction in using e-learning methods due to the ease of accessing the content, flexibility and convenience in writing homework, doing the quizzes and tests using computers and the paperless way of communicating. They claimed that e-learning facilitated learning and increased success in class. Even the group that used classical learning techniques (Class 2) expressed a desire to use e-learning.

Critical friends provided positive comments on the design and execution of the project. The first critical friend had doubts at the beginning of the project as to whether it was possible to teach the advanced topic effectively, especially


to high school sophomores. However, during the course of the project, the friend was surprised as to how well students responded and learned, at times, new challenging material. Success was attributed to the exceptionally well designed course, the novel use of ICT, and creative use of the action research methodology. The second critical friend noticed methodological challenges in setting up two parallel versions of the course and making sure they differ only in one component, that is, the use of ICT. The friend also noticed the challenge posed in forming two sample student groups with statistically negligible differences at the beginning of the project for all categories relevant for the research. The friend observed that these challenges were resolved diligently and effectively in accordance with scientific guidelines for designing the test samples. The opinion was that these appropriate methodological solutions, coupled with well-planned and consistent action research during execution of the project, contributed to the success of the project and validity of the findings.

The experiences and findings of the project were presented at several seminars and workshops for high school teachers. The feedback was positive. Several teachers expressed an opinion that the project could possibly serve as a model for teaching advanced, nonstandard topics, not part of the standard curriculum, and not only mathematics but in other subjects as well. Some of the teachers were interested in getting access to the online module and possibly experimenting with a similar course in their own schools.

The issue that was brought up in discussion at some of the workshops and seminars was a concern that the majority of teachers in Croatia are not trained and ready to use ICT on a regular basis. Transition and adoption of ICT and e-learning may be easier for students than for some teachers, which is understandable given that the younger generation is growing up surrounded by ICT to which they adopt quickly. Comments and our own assessment suggest two directions in which the project and future research can be improved. The test samples, that is, the groups of students participating in the project, were limited due to various circumstances. This was mainly because research was not sponsored by any organization, rather it was an individual effort. Certainly, larger test samples would have been more desirable; however, mechanisms that were used in forming two student groups with equal background knowledge and learning potential were implemented very carefully to assure the validity of the findings at the end of the project. We are confident that the results of our research would stand, and possibly be even more transparent, with larger test samples. Nevertheless, having larger test samples and more groups of students would allow for consideration of additional research questions. For example, it would be interesting to compare the performance of two groups of students who use e-learning but one group of students have face-to-face access to the instructor while the other group does not. Another way in which the project can be improved is expanding the on-line module of lectures with additional material, innovative examples and problems, and the inclusion of more software

Research and evaluation of the effectiveness of e-learning in the case of linear programming 125 tools such as Excel and Matlab. Furthermore, it would be helpful to work on the design of a better, more user-friendly interface. The goal is to develop an on-line course that would be self-sustained and easy to use for other instructors as well as students.

6. Concluding remarks In this paper, we have described and evaluated a project to research the effectiveness of e-learning in linear programming. The goal of the project was to investigate whether and how proper use of information and communication technologies, as well as interactive learning helps students to understanding, learning and retain taught material. The initial hypothesis was that it does. In the first phase of research, a module lesson for linear programming (LP) was devised using the software package Loomen Moodle and other interactive software packages such as Geogebra. In addition to traditional content, such as the simplex method, the module also contained a chapter on new LP methods called the interior points method (IPM).

In the second phase, research was carried out to examine the effectiveness of e-learning using the created LP module as an example. The LP module was taught as a short course to two groups of students. These two groups of students were second-graders in a Croatian high school. In Class 1, the module was taught using e-learning, whereas Class 2 was taught using traditional methods.

During the course of the research, the data were collected, analyzed and evaluated. The methodology for the project and data collection is called “action research”, which recently has been gaining in popularity. This methodology emphasizes the importance of both quantitative and qualitative data collection. The qualitative data collection relies on the research diary, interviews, questionnaires, Likert’s evaluation scale, systematic observations, collection of visual data from photos and videos taken of the teaching process. The quantitative data collection include the results of homework, quizzes and tests. Statistical analysis of quantitative data was performed utilizing basic statistical tools such as mean, median, standard deviation, covariance and statistical hypothesis testing using t-test.

Statistical analysis showed that the group of students who used e-learning showed better results than the group that used traditional learning methods. Thus, the initial hypothesis was validated. Furthermore, in their feedback students expressed satisfaction with the project and the way it was conducted. As far as we are aware, this research is novel in several aspects. It showed that advanced topics, such as linear programing, can be taught effectively and successfully to high school students. An appropriate “model” is suggested, developed and tested. The integral part of the design and the delivery of the


course is action research, which in our opinion contributed greatly to the success of the project. Basic but rigorous statistical analysis was used to validate the hypothesis that appropriate use of ICT and e-learning is more effective than the traditional learning methods in such settings.

The proposed model serves as an example for developing on-line courses in mathematics and other areas, especially for topics that are not an integral part of standard high-school curriculum. This research supports a more general thesis, which is that the future of successful learning by new generations of students will depend on a more intensive use of ICT and e-learning. This will definitely require additional efforts by teachers and the entire educational system, but we are adamant that the investment will eventually pay off and lead us into a better future. References [1] All, A., Nuñez Castellar E. P., Van Looy, J. (2016). Assessing the effectiveness of

digital game-based learning: Best practices. Computers & Education, 92–93, 90–103. [2] Babic, S. (2009). Introduction to e-learning, look through the window. Available at:

https://pogledkrozprozor.wordpress.com/2009/08/29/uvod‐u‐e‐%E2%80%93‐

learning‐1‐dio/ [Accessed on July 13, 2014]. [3] Bjelanović Dijanic, Ž. (2011). Learning mathematics detection with the help of

dynamic geometry software GeoGebra - Action Research. The Third International Scientific Colloquium "Mathematics and Children" - The Math Teacher, Osijek. Available at: http://free‐bj.t‐com.hr/zbjelanovic/radovi/akcijsko_istrazivanje_GeoGebra.pdf [Accessed on July 13, 2014].

[4] Bognar, B. (2006). Action research in school. Educational sciences, 8(1), 209–227. [5] Bognar, B. (2013). Initiating teacher’s action research: Empowering teachers' voices.

Educational Journal of Living Theories. 6(1), 1–39. [6] Bognar, B. (2008). Possibility of becoming the teacher - action researcher through

electronic learning (doctoral dissertation, University of Zagreb). Available at: http://kreativnost.pedagogija.net/file.php/1/Dokumenti/ddisertacija_kraj.p

df [Accessed on July 13, 2014]. [7] Dečman, M. (2015). Modeling the acceptance of e-learning in mandatory

environments of higher education: The influence of previous education and gender, Computers in Human Behavior, 49, 272–281.

[8] Lester, J. Designing interactive mathematics. Available at: http://www.cecm.sfu.ca/~jalester/DesignIntMath.pdf [Accessed on July 13, 2014].

[9] Lesaja, G. (2009). Introducing Interior-Point Methods for introductory operations research courses and/or linear programming courses. Open Operational Research Journal, 3: 1-12. Available at: http://digitalcommons.georgiasouthern.edu/math‐sci‐facpubs/77 [Accessed on July 13, 2014].

[10] Lešaja, G., Drummer A. Miletic, Lj. (2012). Full infeasible Newton-Step Interior-Point Method for linear complementarity problems. Croatian Operational Research Review (CRORR), 3(1), 163–176.

Research and evaluation of the effectiveness of e-learning in the case of linear programming 127 [11] Matijevic, M. (1990). Action research and the reform of elementary school. In the

monograph: Action research in educational process, Slovenian pedagogical society, Ljubljana, pp. 78–87.

[12] Matijevic, M., Radovanovic, D. Teaching focused on students. The School Newspaper, Zagreb, 2011.

[13] Ministry of Science, Education and Sports. The full text of the strategy for education, science and technology. Available at: http://public.mzos.hr/fgs.axd?id=22355 [Accessed on July 13, 2014].

[14] Muzic, V. (1999). Introduction to Research Methodology Education. Educa, Zagreb. [15] Neralic, L. (2012). Introduction to Mathematical Programming 1, Fourth Edition.

Element, Zagreb. [16] Secondary school Marko Marulic Slatina. Uvođenje e-učenja (Introducing e-

learning). Available at: http://ss‐mmarulica‐slatina.skole.hr/uvodjenje_e_ucenja [Accessed on July 13, 2014].

[17] Yanusik, O. V., Pakhomova, E. G., Batbold, K. (2015). E-learning as a way to improve the quality of educational for international students, Procedia - Social and Behavioral Sciences, 215, 147–155.



Analysis of multiplier effects of ICT sectors – a Croatian case

Damira Keček1,*, Nikolina Žajdela Hrustek2 and Vesna Dušak2

1 University Centre Varaždin, University North 104. brigade 1, 42 000 Varaždin, Croatia

⟨[email protected]⟩

2 Faculty of Organization and Informatics, University of Zagreb Pavlinska 2, 42 000 Varaždin, Croatia ⟨{nikolina.zajdela, vesna.dusak}@foi.hr⟩

Abstract. The impact of Information and Communication Technology (ICT) on economic growth and development has nowadays proven to be significant for almost all national economies. In this paper, a quantitative analysis of the impact of ICT on Croatian economic growth is performed using the input-output (IO) method. The direct and indirect effects of ICT sectors are analysed. Subsequently, simple output multipliers and simple value added multipliers are then calculated. The results indicate that there are no significant differences between the ICT multipliers for the Croatian economy in 2010 and multipliers of other sectors. The largest values of multipliers of all ICT sectors are attributed to the one of ICT service sector. Moreover, significant changes were also not observed when comparing ICT multipliers for the Croatian economy in 2004 and 2010. In addition to Croatia, multiplier analysis of ICT sectors was conducted for the group of new and long-standing European Union member states. The conclusion is that, in new member states, the implementation and usage of ICT has a lower contribution to economic growth and development.

Key words: ICT sectors, input-output analysis, simple multipliers

Received: February 9, 2016; accepted: March 30, 2016; available online: April 04, 2016

DOI: 10.17535/crorr.2016.0009

1. Introduction

Information and Communication Technology (ICT) has become the most important driver of continued growth and development of any economy, stimulating the creation of new and more efficient models of organizational


130 Damira Keček, Nikolina Žajdela Hrustek and Vesna Dušak structures in the global economy. ICT can, among other things, play a significant role in the globalization of production, as well as in the transfer of technologies, enabling constant interaction via the Internet, mobile communications, digital television and e-commerce, regardless of geographical position or time zone.

1.1. Contribution of ICT to macroeconomic growth and development

The latest research on ICT indicates that the dynamics of ICT has indeed become a major driving force for productivity, competitiveness, collaboration, and superposition of resources at both a national and international level [33], [41]. The influence of ICT on productivity, economic growth and overall development is evident in ICT investments and its utilization, contributing to an increase in human capital, higher efficiency, rapid technological progress in the production of ICT goods and services, including faster growth and development of the productive ICT sector [1, 35, 51]. The importance of manufactured goods is decreasing in the leading industrial economies, whereas the importance of production data and information processing is rapidly increasing. Generally speaking, one can presume that, due to the transition from an industrial to an information society at a macroeconomic level, production and consumption of ICT goods and services is significantly increasing, which in turn exerts a positive influence on economic growth and development [9, 27]. This strategy was adopted by the United States of America which intends to invest 7.2 billion dollars in the expansion of ICT and the Internet, as declared in the American Recovery and Reinvestment Act of 2009. Europe has taken a similar path as reflected in the goals of its Digital Agenda, where by the year 2020, at least 50% of Europe's population will have Internet access through the implementation of specific projects.

Many studies have confirmed that ICT can also be classified as a general purpose technology, because ICT is a generic technology that leads to the expansion of all productive economic and social systems [2]. The main importance of a general purpose technology is that it leads to fundamental changes in production processes, as well as fast expansion, technological dynamics and innovative complementarity, meaning that productivity in various sectors increases specifically due to the constant progress of ICT technology. The advancement of general purpose technologies reflects directly on the entire economy in terms of increased productivity [19]. Accordingly, ICT is classified as a general purpose technology, given that today, computers and the associated peripheral equipment are used in all economic and social sectors [5]. The

Analysis of multiplier effects of ICT sectors – a Croatian case 131 contribution of ICT as a general purpose technology has been supported by Jalava and Pohjola [24] whose study shows that the contribution of ICT sector to Finland’s GDP between 1990 and 2004 was three times greater than the contribution of electricity from 1920 to 1938.

Apparently, ICT has an even greater impact on the economy, as it supports numerous service sectors. This is particularly evident at the present time, when new forms of services are gaining importance, such as e-banking, e-commerce, e-learning, e-health, and others services [15, 45]. Since the mid-1990s, a majority of researchers have identified a positive correlation between ICT investment and economic performance at aggregation levels, e.g. companies, industries, and countries [38, 39, 52, 4, 25, 28]. In the last decade, ICT investment has contributed 0.2 – 0.6 percent to annual GDP growth in Organisation for Economic Cooperation and Development (OECD) countries [37]. Moreover, studies at a macro-level conducted by van Ark and other researchers clearly demonstrate the existence of an increasing productivity gap between Europe and the USA, caused by a less effective and widespread adoption of ICT by European companies [47, 11, 12, 48, 49, 50, 17, 46, 20, 21, 22].

1.2. IO analysis for determining the contribution of ICT to macroeconomic growth and development

Numerous studies noted in the previous sections clearly show that the expansion of ICT and the Internet contributes to positive results, both on a micro and macro level. This section provides an overview of relevant research related to the use of IO analysis in order to determine whether and to what extent intensive investment and use of ICT can generate desired results for national economies that have recognized the role of ICT in growth and development. Using a hypothetical extraction method in the IO framework based on the aggregated six-sector IO table in 2001 in which ICT is a main sector, Bazzazan [3] provides an analysis of the economic importance of ICT in Iran at the national level. The results of analysis show that, in terms of demand, the ICT sector is ranked fourth from among six sectors and accounts for 8.6 percent of total output, whereas in terms of supply, it is also ranked fourth with 9.5 percent of total output. Similar positive results for the impact of expansion and use of ICT on economic growth is also shown in the analysis conducted for the economy of Italy by Di Carlo and Santarelli [14]. Their aim was to evaluate of the impact of ICT investments on the Italian national economy by analysing production and demand multipliers which were calculated using IO matrices released by ISTAT for the years 1995, 2000, 2005. The results have shown that ICT has a greater multiplicative effect on the productive system than the non-ICT sectors and, thus, is a key sector for economic growth.

132 Damira Keček, Nikolina Žajdela Hrustek and Vesna Dušak

An attempt to analyse the impact of ICT at a global world level was done by Mattioli and Lamonica [31]. In their research, the authors used the World Input Output Table for the period 1995–2009, which measured the interaction of the ICT sector with other productive sectors for 27 European countries and 13 other major countries classified as the highest industrialized countries in the world. Rasmussen forward and backward linkage indices were used for data analysis. The study confirmed that the ICT sector has a multiplier effect on other sectors, leading to the conclusion, as with numerous other previous studies, that the ICT sector plays a significant role in the economic systems of the highly developed countries.

On the other hand, some research results, especially for developed countries, do not show positive but instead negative or stagnant results from the expansion of ICT based on development indicators. A study conducted by Rohman and Bohlin [43] and based on a sectoral approach using the IO methodology investigates the contribution of ICT sectors in driving economic performance in European economies. The authors used a decomposition analysis confirming that some countries (e.g. Germany and Spain) experienced a decline in the output growth of their ICT sectors in the period 2000–2005 when compared to the period 1995–2000. The study also notes that, at a country level, in time ICT sectors lost the advantage of export and impact of technological change, and that the impact of technological change is reduced due to lack of integration among ICT and other sectors in terms of manufacturing. On the other hand, the same analysis has shown that the technological change effect in France remains stable. Decomposition analysis conducted by Rohman [44] shows similar results for 10 European economies. Data analysis provided the same results as did previous studies, indicating that the multiplier effect of ICT sectors on the rest of the economy decreased during the period 2000–2005, when compared to 1995–2000, and finally, a decline in the output of ICT sectors was linked to the loss of export advantages and technical change gains in the said sectors. Much research on European economies on the regional level has also shown a low contribution to economic growth and development by ICT sectors.

Furthermore, decomposition analysis using IO tables performed by Zuhdi et al. [54], for Indonesia in the period 1990–2005 and Japan for the period 1995–2005, whose purpose it was to analyze the role of ICT sectors in contributing to structural changes in the national economies has indicated that ICT sectors played an important role in changing Japan’s economy, but did not have a significant influence on structural changes in the economy of Indonesia. In later research, Zuhdi [55] endeavoured to obtain another perspective on the role of the ICT sector in Indonesia’s national economy by applying IO analysis for the

Analysis of multiplier effects of ICT sectors – a Croatian case 133 period 1990-2005. Here, Zuhdi used the simple output multipliers method to achieve his purpose. Similar results also appear in this study. The same author analyzes in [56] the impacts of final demand changes on total output of Indonesian ICT sectors applying demand-pull IO quantity model. “Whole sector change” and “pure change” conditions are considered in this study. The results of calculation show that, in both conditions, the biggest positive impact on the total output of the sectors is attributed to a change in household consumption, while the change of import has a negative impact. One of the recommendations of this study is the drafting of import restriction policies for ICT products.

A similar study of Japan’s economy was also conducted by Zuhdi and Prasetyo [57] using the IO table for 2005. The purpose of the study was to analyse total output trends of the Japanese ICT sector as influenced by final demand changes. The study also employs IO analysis to the interdependence of industries in an economy. The results show that the Japanese ICT sector exhibits a similar pattern. The authors of the study suggest the following: (1) export activity from the ICT sector should be enhanced, (2) import activity of ICT products should be restricted, and (3) more ICT domestic market should be captured.

An interesting study was conducted by Irawan [23] from the perspective of developed and developing countries. The author performed comparative analysis based on the IO Table for 2005 from four ASEAN Member States – i.e. Indonesia, Singapore, Malaysia and Thailand. The conclusion drawn from the analysis results shows that the size and structure of ICT sector is important in national economies, that ICT sectors have a positive impact on growth and development and finally, that countries which are more developed benefit much more from ICT than countries which are less developed.

The study of relevant literature shows that IO methodology was used in Croatian economy in analyzing the impact of forestry and wood industry to the economic growth, as well as on the problems of the impact of export in food industry and textile industry. The 2004 IO table for Croatia was used as reference data in all of these reviews. Lovrinčević and Mikulić in [29] quantified the importance of the Croatian forestry and wood industry using IO analysis. The obtained data indicated that the multiplier effects of the forestry and wood industry were significant. The calculated multipliers indicated high values, especially the output multiplier in section 20 – Wood products, which was also the highest multiplier among all other industries. The IO model in [8] was the main method for obtaining new findings about the state and position of exports from the Croatian food industry and its effects on the national economy. Type I multipliers and type II multipliers of gross output, value added and employment were calculated. Multiplier values indicate the strategic importance of the food sector for the national economy. The results also showed that food industry

134 Damira Keček, Nikolina Žajdela Hrustek and Vesna Dušak exports had the highest multiplicative effect on agricultural production and trade. In the paper [7], authors Buturac et al. measured the overall importance of textile industry for the Croatian economy in terms of gross output, value added and employment by applying IO analysis. The obtained output and value added multipliers for the textile industry were low, while the change in final demand had the strongest direct effect on employment. Relatively low multipliers of the Croatian economy reflect that this economy is service-based, and that the international competitive position of its manufacturing industry has decreased.

Led by the same aim, the authors of this article will analyse the impact a growing ICT sectors using the IO table for the Croatian economy for the years 2004 and 2010, and will compare IO multipliers between the new and long-standing European Union (EU) member states. The reason behind this is that no similar studies have used the IO methodology to investigate impact of ICT sectors on national economic growth and development. The following chapters explain the methodology and present the respective research results, conclusions and recommendations. 1.3. Definition and classification of the ICT sectors

Many definitions and classifications of the ICT sectors were encountered when reviewing relevant literature. The main data sources for analyzing the impact of ICT on growth and development of the national economy were symmetric IO tables for 2004 and 2010 from the Croatian Bureau of Statistics and Eurostat [10, 16]. Importantly, the symmetric IO tables for the year 2004 and 2010 were not designed using the same methodology and classification. Symmetric IO tables for the year 2010 were prepared according to the 2007 National Classification of Activities, that correspond in content and structure to the Statistical Classification of Economic Activities in the European Community, NACE Rev. 2. This classification differs in structure from the 2002 National Classification of Activities, which is used to prepare symmetric IO tables for 2004. The 2002 National Classification of Activities corresponds in content and structure to NACE Rev. 1.1.

The definition used in this paper to identify ICT economic activities is [13]: “The production (goods and services) of a candidate industry must primarily be intended to fulfil or enable the function of information processing and communication by electronic means, including transmission and display”. This definition of ICT provides a statistical basis for measuring economic activity generated by the production of ICT goods and services and is comparable internationally. There are three main groups of ICT activities: manufacturing

Analysis of multiplier effects of ICT sectors – a Croatian case 135 industry, ICT trade industry and ICT services industry. Those groups include the following productive sectors according to the 2007 National Classification of Activities in the symmetric IO table for 2010: CPA_C26 - Computer, electronic and optical products; CPA_G46 - Wholesale trade services, except for motor vehicles and motorcycles; CPA_J58 - Publishing services; CPA_J61 - Telecommunications services; CPA_J62_J63 - Computer programming, consultancy and related services, information services; CPA_S95 - Repair services of computers and personal and household goods.

According to the OECD definition [36]: “ICT products must primarily be intended to fulfil or enable the function of information processing and communication by electronic means, including transmission and display. Content corresponds to an organized message intended for human beings published in mass communication media and related media activities. The value of such a product to the consumer does not lie in its tangible qualities but in its information, educational, cultural or entertainment content.” An ICT sectors consists of two large groups: ICT products plus content and media products. Based on this definition, ICT sectors have been detected in the symmetric IO table for the year 2004 and according to [30] those sectors that correspond to the above mentioned manufacturing, trade and services ICT sectors have been identified.

Thus, sector CPA_C26 - Computer, electronic and optical products corresponds to sectors 30 - Office machinery and computers and 32 - Radio, television and communication equipment and apparatus, sector CPA_G46 - Wholesale trade services, except of motor vehicles and motorcycles corresponds to sector 51 - Wholesale trade and commission trade services, except for motor vehicles and motorcycles, sector CPA_J61 - Telecommunications services corresponds to sector 64 - Post and telecommunication services, while sectors CPA_J58 - Publishing services, CPA_J62_J63 - Computer programming, consultancy and related services, information services and CPA_S95 - Repair services of computers and personal and household goods correspond to sector 72 - Computer and related services.

2. Research methodology Input-output analysis is considered a practical method for quantitative macroeconomic analysis. Its importance has been recognized in various aspects of planning the economic development, and in investigating complex quantitative effects of certain economic policy measures and emergency interventions in the economic development of the country [42, 53, 34]. The statistical basis of IO analysis are IO tables. In the IO table, the production system of an economy is broken down into a number of productive sectors,

136 Damira Keček, Nikolina Žajdela Hrustek and Vesna Dušak indicating how outputs from each sector of the economy are used as inputs by other sectors.

The basic equation in the IO model shows the impact of cross-sector flows on the total production of each sector in the IO table [34]. For sector , the equation expressing this dependence is given as follows:

∑ (1)

where is total output of sector , the amount of a product from sector used as an intermediate input in production by sector , and is the final demand of sector , where , 1, . . . , . In defining the technical coefficient

as a ratio of a product from sector that is required by sector in

order to produce one unit of its product, the system of equations (1) for the entire economy in matrix form can be rewritten as:

(2)

where ⋮ , ⋯

⋮ ⋱ ⋮⋯

and ⋮ .

Matrix is called the technology matrix. A solution to the system (2), where is an -by- identity matrix is:

(3)

The information about conditions for the matrix and the matrix is given in more detailed in [40]. The Leontief Inverse matrix , also known as a multiplier matrix, measures how the total output is changed as a result of the change in final demand. Elements of the multiplier matrix represent the output of sector directly and indirectly required per unit of final demand from sector .

In this paper, IO analysis is used to calculate simple multipliers. The open IO table, consisting of all production sectors of the national economy with households excluded, is used to calculate simple multipliers. In the case of the open IO table, elements of the Leontief Inverse matrix indicate the direct and indirect effects per unit of final demand. Contrary to this, inclusion of households makes the IO table closed. Households are therefore included in the calculation of Inverse Leontief matrix elements, thus indicating direct, indirect and induced effects per unit of final demand. [32, 34, 6].

The authors in [18] argue about the output multiplier and employment multiplier are derived from an open and a closed IO model. They do not recommend using the multiplier results derived from a closed IO table as they

Analysis of multiplier effects of ICT sectors – a Croatian case 137 yield exaggerated estimates of the impact of program expenditures on the economy. Several simple multipliers are possible, the ones for output and value added are provided here. Simple multipliers are calculated as the ratio of the direct and indirect effect to the initial effect alone. The sum of the j-th column of the multiplier matrix indicates the output of each sector of the economy directly and indirectly as required per unit of final demand of sector j. The initial output effect on the economy is defined as the initial monetary unit’s worth of sector j output needed to satisfy the additional final demand. Hence, the simple output multiplier for the sector is defined as:

∑ , 1, … , (4)

Simple value added multipliers express the value added of an economy directly and indirectly required per unit of final demand. For the sector , a simple value added multiplier is computed by multiplying the -th column of the multiplier matrix by the value added generated per unit of its output, i.e.:

∑ ∙ , 1, … , (5)

where represents value added of sector .

3. Research results The total output of all sectors that include ICT activities at basic prices was determined to be approx. 23.1 billion kuna based on the Croatian symmetric IO table for domestic production for 2010 [10]. Total intermediate consumption of domestic products from the Croatian ICT sectors was approx. 7.6 billion kuna, while the total gross value added amount approx. 13.4 billion kuna. Of all ICT sectors, sector CPA_J61 - Telecommunications services had the largest share in the total output of all ICT sectors (around 43.1 percent), the largest intermediate consumption of domestic products (around 3.2 billion kuna) and the largest gross value added (around 6.1 billion kuna), while sector CPA_S95 - Repair services of computers and personal and household goods had the lowest share in the total output of the ICT sectors (around 4.4 percent), the lowest intermediate consumption of domestic products (around 312.3 million kuna) and the lowest gross value added (around 628.5 million kuna).


Activity sections codes

Description Output multiplier

Value added

multiplier A Agriculture, forestry and fishing 1.598 1.557

B+C+D+E

Mining and quarrying; Manufacturing; Electricity, gas, steam and air

conditioning supply, sewerage, waste management, remediation activities

1.647 1.848

F Construction 1.716 1.948

G+H+I

Wholesale and retail trade; repair of motor vehicles and motorcycles;

Transportation and storage; Accommodation and food service

activities

1.589 1.577

J Information and communication 1.580 1.530 K Financial and insurance activities 1.506 1.413 L Real estate activities 1.140 1.105

M+N Professional, scientific and technical

activities; Administrative and support service activities

1.571 1.549

O+P+Q Public administration and defence;

compulsory social security; Education; Human health and social work activities

1.410 1.304

R+S+T+U

Arts, entertainment and recreation; Other service activities; Activities of households as employers; Activities of extraterritorial

organisations and bodies

1.500 1.437

Table 1: Output and value added multipliers for the Croatian economy

Source: Author’s calculations based on data from the Croatian Bureau of Statistics [10].

Table 1 shows the values of the output and value added multipliers for 2010 for various sectors of the Croatian economy according to activity sections in the 2007 National Classification of Activities. Based on multiplier values, it becomes evident that activity F - Construction had the highest contribution to the economy while activity L - Real estate activities had the lowest contribution. For the case of the ICT sectors in 2010 (Table 2), the largest output multiplier of 1.691 is attributed to sector CPA_J58 - Publishing services, meaning that a unit increase in final demand was expected to increase national output by around 1.691 units. Medium output multipliers were identified in sector CPA_C26 - Computer, electronic and optical products and in sector CPA_G46 - Wholesale trade services, except for motor vehicles and motorcycles with values of 1.581 and 1.601 respectively. The lowest output

Analysis of multiplier effects of ICT sectors – a Croatian case 139 multiplier of 1.456 was in sector CPA_S95 - Repair services of computers and personal and household goods, due to the fact that ICT technology is rapidly and daily changing, and affordability of this technology is increasing. Therefore, more users decided to purchase new technology rather than repair or upgrade existing technologies.

The two sectors: CPA_J61 - Telecommunications services, CPA_J62_J63 - Computer programming, consultancy and related services, and information services, including the previously mentioned sector CPA_S95 - Repair services of computers and personal and household goods, belong to ICT services industries. These had relatively low output multipliers with values below 1.5. A similar conclusion relating to the above mentioned sectors can also be drawn for value added multiplier.

Year 2004 Year 2010

Sector code

Output multiplier

Sectorcode

Output multiplier

30 1.694 CPA_C26 1.581

32 1.798 51 1.676 CPA_G46 1.601

64 1.536 CPA_J61 1.484

72 1.479

CPA_J58 1.691

CPA_J62_J63 1.475

CPA_S95 1.456

Table 2: Output multipliers for ICT sectors for 2004 and 2010


Table 3: Value added multipliers for ICT sectors


When comparing the output multiplier and value added multiplier values

in the Croatian ICT sectors for 2004 and 2010, it is observed that the values in 2010 decreased, but not significantly (see Table 2 and Table 3).

Similarly, based on calculations of total sectoral multipliers using symmetric IO tables for 2004 and 2010 for all sectors of the Croatian economy, the authors in [26] conclude that significant changes in cross-sectoral relations during these two periods did not occur and that the overall sectoral multiplier decreased in the most comparable activities.

Subsequently, a discussion on output and value added multipliers for new and long-standing EU member states is carried out (Table 4, Appendix 1 and Table 5, Appendix 2). By comparing the minimum and maximum values of the

Year 2004 Year 2010

Sector code

Value added

multiplier Sector code

Value added

multiplier30 1.682

CPA_C26 1.631 32 1.875

51 1.651 CPA_G46 1.600

64 1.489 CPA_J61 1.404

72 1.426

CPA_J58 1.835

CPA_J62_J63 1.407 CPA_S95 1.418

140 Damira Keček, Nikolina Žajdela Hrustek and Vesna Dušak output and value added multipliers for almost all ICT sectors, a gap between the minimum and maximum values of new and long-standing EU member states was observed, indicating that the long-standing EU member states utilized ICT more efficiently for economic growth and development .

In fact, for sector CPA_C26 - Computer, electronic and optical products, which is covered by manufacturing, output multipliers were the lowest and highest among new EU member states, with 1.131 for Hungary and 1.595 for Slovenia, respectively. The lowest and highest output multiplier values for the same sector among long-standing EU member states were 1.524 for Belgium and 1.849 for France, respectively. In the ICT trade sector denoted by CPA_G46 - Wholesale trade services, except for motor vehicles and motorcycles, the difference in the maximum (0.011) and minimum (0.028) output multiplier values for new and long-standing EU member states is almost negligible. Among ICT service sectors, differences were found to exist between sectors CPA_J58 - Publishing services and CPA_J61 - Telecommunications services and sectors CPA_J62_J63 - Computer programming, consultancy and related services, information services and CPA_S95 - Repair services of computers and personal and household goods. New EU member states had lower minimum and maximum multiplier values for sectors CPA_J58 - Publishing services and CPA_J61 - Telecommunications services than long-standing members. On the other hand, sectors CPA_J62_J63 - Computer programming, consultancy and related services, information services and CPA_S95 - Repair services of computers and personal and household goods had lower minimum multiplier values in new EU member states, but greater maximum multiplier values in long-standing EU member states. Differences for the minimum and maximum values of the value added multiplier in the above analyzed sectors when comparing new and long-standing EU member states do exist, but they are not significant.

4. Conclusion Rapid technological progress in the production of ICT goods and services, and faster growth and development of the ICT productive sector, has a significant impact on the productivity and efficiency of all the other sectors of national economies, as well as on the growth and overall development of social and economic systems as a whole. This research analyses the impact of ICT on Croatia’s economic growth using the IO method to calculate simple output and value added multipliers. A comparative analysis of multipliers based on accessible symmetric IO tables was performed for ICT production, service and trade sectors of the Croatian economy. The analysis results indicate that the differences in multiplier values in the mentioned sector for the period in question

Analysis of multiplier effects of ICT sectors – a Croatian case 141 were not significant. The values of the output multiplier for ICT sectors ranged from 1.479 to 1.798 in 2004 and from 1.456 to 1.691 for 2010, while the values of the value added multiplier ranged from 1.426 to 1.875 in 2004 and from 1.407 to 1.835 for 2010. This leads to conclusion that the contribution of ICT to growth and development of the Croatian economy decreased during those years.

By using the latest data for 2010, output and value added multipliers were analysed for all sectors of Croatian economy, and a comparison to the multipliers of the ICT sectors was also performed. The results show that the average value of output multipliers, as well as that of value added multipliers for all ICT sectors is around 1.5, which is consistent for the majority of multiplier values in all other activity sections with the exception of the respective four sections. Two of those four activity sectors had the lowest multiplier values: L (Real estate activities) and O+P+Q (Public administration and defence; compulsory social security; education; human health and social work activities, while the other two sectors: F (Construction), and B+C+D+E (Mining and quarrying; manufacturing; electricity, gas, steam and air conditioning supply, sewerage, waste management, remediation activities) had the highest multiplier values. Moreover, a comparative multiplier analysis was conducted for a set of EU countries which were, for the purpose of this paper, classified into two groups: new and the long-standing EU members, in order to establish whether ICT contributes equally to growth and development in long-standing EU member states, as opposed to new member states. The obtained results indicate a divide in the multiplier values between the new and long-standing EU member states leading to the conclusion that long-standing EU member states have capitalized on the potential of ICT sector more for the purpose of growth and development, as opposed to the new member states that subsequently joined the EU.

The unavailability of data necessary for calculating the remaining multipliers associated with IO analysis (e.g. employment multiplier) is one of the main limitations of undertaking this kind of research in Croatia, as is the case in other observed European Union countries. Another limitation of this analysis is due to the lack of data and the impossibility of performing long-term continuing analysis of the impact of ICT on the growth and development of the Croatian economy. The cause may be that, since becoming independent, the Republic of Croatia has not given much importance to the creation of IO tables, as is evident by the availability of only two IO tables, those for 2004 and 2010. Such information would enable recording and observing significant changes. Emphasis should be placed on the fact that a direct inter-sectoral comparison based on data available in IO tables from 2004 and 2010 cannot be carried out given that the methodology for creating IO tables for the respective years has not been consistent. Guidelines for future research should move towards more detailed research on data availability in order to collect additional data for

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APPENDICES

Table 4: Output and value added multipliers for ICT sectors for new member states of EU for 2010

Source: Author’s calculations based on Croatian Bureau of Statistics [10] andEurostat [16] data

Sectors codes

CPA_C26 CPA_G46 CPA_J58 CPA_J61 CPA_J62_J63 CPA_S95

Long-standing member states of

EU

Output multiplier

Value added

multiplier

Output multiplier

Value added

multiplier

Output multiplier

Value added

multiplier

Output multiplier

Value added

multiplier

Output multiplier

Value added

multiplier

Output multiplier

Value added

multiplier

Germany 1.654 1.812 1.773 1.756 1.840 1.937 2.016 2.396 1.493 1.451 1.316 1.205

France 1.849 2.430 1.769 1.781 1.818 1.836 1.881 1.933 1.609 1.511 1.488 1.387

Italy 1.689 1.885 1.873 1.843 1.904 2.012 1.811 1.731 1.698 1.654 1.652 1.480

Belgium 1.524 1.714 1.592 1.546 1.799 1.815 1.685 1.667 1.782 1.814 1.699 1.707

United Kingdom

1.710 1.720 1.786 1.816 1.683 1.638 1.465 1.450 1.461 1.396 1.462 1.419

Table 5: Output and value added multipliers for ICT sectors for long-standing member states of EU for 2010

Source: Author’s calculations based on Eurostat [16] data

Sectors codes

CPA_C26 CPA_G46 CPA_J58 CPA_J61 CPA_J62_J63 CPA_S95

New member states of

EU

Output multiplie

r

Value added

multiplier

Output multiplie

r

Value added

multiplier

Output multiplie

r

Value added

multiplier

Output multiplie

r

Value added

multiplier

Output multiplie

r

Value added

multiplier

Output multiplie

r

Value added

multiplier

Croatia 1.581 1.631 1.601 1.600 1.691 1.835 1.484 1.404 1.475 1.407 1.456 1.418

Slovenia 1.595 1.693 1.683 1.625 1.868 2.250 1.852 2.009 1.552 1.539 1.402 1.451

Czech Republi

c

1.328 2.824 1.862 1.831 1.897 2.085 1.618 1.563 1.808 1.793 1.782 1.815

Slovakia 1.463 2.558 1.567 1.493 1.548 1.569 1.654 1.599 1.571 1.519 1.174 1.155

Hungary 1.131 1.692 1.563 1.635 1.654 1.694 1.431 1.394 1.365 1.332 1.438 1.442



Diagnosing companies in financial difficulty based on the auditor’s report

Martin Valić-Vale1 and Robert Zenzerović2,*

1 Valamar Riviera d.d. Stancija Kaligari 1, 52440 Poreč, Croatia E-mail: ⟨[email protected]⟩

2 Faculty of Economics and Tourism "Dr. Mijo Mirković", Juraj Dobrila University of Pula

Preradovićeva 1, 52100 Pula, Croatia E-mail: ⟨[email protected]⟩

Abstract. The approach used in this paper expands on existing research that focuses on devising prediction models for companies experiencing financial difficulties and which in turn serves as a criteria-based diagnosis tool for distinguishing healthy companies from those facing seriously financial difficulties. It draws on auditors’ reports on company financial statements that emphasize a company’s ability to continue as a going concern as the main criterion used to distinguish companies experiencing financial difficulties from companies that are not. Two closely-related hypotheses were tested in this paper. First, the authors tested the hypothesis that an auditor’s report accompanied by an explanatory paragraph pointing out issues associated with the going concern assumption is the proper criterion for differentiating companies experiencing financial difficulties from those that are not. Second, the central assumption that is tested relates to a combination of financial ratios whereby authors presume that an appropriate combination of financial ratios is a good analytical tool for distinguishing companies experiencing serious financial difficulties from those that are not. Research results conducted among 191 companies listed on the Zagreb Stock Exchange confirm both hypotheses. The LRA model – a diagnosis tool for identifying companies with financial problems, was also derived using logistic regression analysis. The statistical adequacy and quality of the model was tested using measures like Nagelkerke R2, type 1 and type 2 errors that appear when calculating the classification ability of the model. All measures indicated that model was statistically sufficient and validated its use as a diagnosis tool in recognizing the companies facing financial difficulties.

Key words: diagnosis of financial difficulties, auditor reports, logistic regression analysis

Received: September 26, 2014; accepted: March 31, 2016; available online: April 04, 2016

DOI: 10.17535/crorr.2016.0010


148 Martin Valić-Vale and Robert Zenzerović

1. Introduction

Forecasting the financial stability or instability of a business is always a current topic. It becomes a researcher’s particular focus during times of economic crisis – and noticing that forecasting financial stability or instability is subject tocyclical effects, just like national economies.

Research on financial (in)stability forecast focuses on historical data as the basis for estimating model(s) that present an acceptable level of classification and accuracy, and consequently are a good diagnostic tool for financial (in)stability. Past experience provides the basis for make future decisions. In that regard, financial (in)stability model(s) are not only oriented towards the past, but are tools for prognosis of future financial health.

Most research on diagnosing financial (in)stability and prognosis enables researchers to derive models that distinguish stable companies from those experiencing financial difficulties based on whether a company is faced with bankruptcy (or not), to what extent is it paying off its due debts and other financial characteristics that are drawn from financial reports. This paper employs another approach that distinguishes financial stable companies from those facing serious financial difficulties based on the auditors’ opinion on companies’ financial statements. The second chapter elaborates on previous studies, and subsequently, chapter three presents research methodology and two main hypotheses. Chapter four presents research results together with an estimation model and the indicators of a quality model.

2. Previous studies

Diagnosis and prediction of financial instability dates from the period when loans were first given out, regardless of whether in the form of monies or goods. During the evolution of business relationships, lending money became more sophisticated and included various (quantitative and qualitative) financial and nonfinancial inputs. The scientific approach to researching financial (in)stability began after the Great Depression in 1930s thirties when simpler models existed. The application of quantitative statistical methods began some 20 years later with univariate statistics [3]. The biggest impulse for applying a more complex statistical method known as multiple discriminant analysis was given by Edward I. Altman, who developed the Z-score model using data from U.S. companies [2] – the most cited financial (in)stability model in the literature [24]. Among the other authors most cited in literature who used similar techniques are Deakin, Ohlson, Edmister and Kralicek.

Contemporary research in predicting financial instability uses increasingly sophisticated statistical techniques like logistic regression analysis, multidimensional scaling, survival analysis, decision trees, neural networks,

Diagnosing companies in financial difficulty based on the auditor’s report 149

fuzzy rules based analysis, cumulative sum models, chaos theory, linear goal programming, multi-criteria decision aid approach, rough set analysis, expert systems, self-organizing maps, etc. [25].

Research has uncovered that country-specific models provide poor classification when applied to companies from other countries. This is particularly true in situations when the model was developed using data from a company operating in a developed country and applied to a company operating in a less developed economic environment [19]. The recommendation then is to develop and use a model designed using the data from the host country as it includes much of the particular economic and related variables specific for the economic environment that is analyzed. Škeljo [19] has shown that the Altman Z-score did not perform well in transitional countries such as Croatia. Šarlija [18] has developed a credit scoring model for small companies using various statistical techniques and was among the first to neural networks for developing credit scoring models in the Croatian business environment. Šarlija [17], in her later works with Šorić, Vlah and Vojvodić Rosenzweig, used logistic regression and multicriteria decision making in credit scoring. Šarlija, Penavin and Harc [16] have developed a model that forecasts short term (one year) insolvency from data provided by companies operating in Croatia. Streitenberger, M. and Miloš Sprčić [15] have conducted research to find financial ratios that best distinguish companies regularly defaulting on their liabilities as opposed to those that remain solvent. Another relevant insolvency prediction model was developed by Pervan and Filipović [13] using the financial data of companies meeting their liabilities within 90 days of the maturity date, and those that do so after 90 days. A similar model was developed by Novak and Crnković [11] who utilised a bank’s experience with clients in order to classify clients into categories of good, medium and bad. Zenzerović [23], [24] have used logistic regression analysis to develop models for estimating the going concern assumption for companies of various sizes. Last but not least, important work was carried out by Belak and Aljinović Barać [6]. They have developed a business excellence model using data from listed companies that suggests six levels of business excellence. As given above, most of this research has been directed towards company bankruptcy or a company’s inability to settle its liabilities prior to maturity as a main criterion for discriminate financially stable from unstable companies.

3. Formulation of a theoretical model – hypothesis andmethodology

A scientific approach to predicting financial (in)stability begins with the study of economic and financial theory which is the basis of the theoretical model. The authors of this research place their focus on new criterion – the auditor’s report.


For the purpose of this research the selected company was viewed as unstable if the auditor issued an explanatory paragraph in the report pointing out that there existed substantial doubt about company’s ability to continue as a going concern. This meant that the auditor asserted as questionable the company’s ability to continue operating without needing to significantly reducing its operations or to continue operating at all in next 12 months. This leads to the first hypothesis requiring testing: H1 – the auditor’s report accompanied by an explanatory paragraph pointing out the issues relating to the going concern assumption is an appropriate criterion for classifying the company as financially unstable. The second hypothesis is subsequently tested if the first is validated as true. It is presented as follows: H2 – a combination of financial ratios are a good analytical tool for distinguishing companies experiencing serious financial difficulties from those that are not.

Having defined the theoretical model, research then focused on data collection. Research was performed on companies listed on the Zagreb Stock Exchange. The sample included financial statements along with auditor reports for 191 companies from the nonfinancial sector for the year 2009. The sample was divided in two subsamples. The first included financial statements of 22 companies for which the auditor report with explanatory paragraph that point out the problems with going concern assumption was issued. Companies included in this subsample are viewed as financially unstable. The second subsample covered the financial statements of 169 companies that did not have an explanatory paragraph referring to the going concern assumption in the auditor’s report. The reason the authors selected the year 2009 is that it was the first year of the recession in the Republic of Croatia with a decline in BDP of 6.9%, leading to an expected higher proportion of opinions on the going concern assumption.

For each company, 28 financial ratios were calculated. The financial ratios included liquidity ratios (3), solvency ratios (4), activity (3) and profitability ratios (9), cash flow ratios (8) and economic value added.

The research methodology incorporated two closely-related statistical methods. To test hypothesis H1, simple descriptive statistics was used that included group mean comparisons and onward carried arithmetic mean and standard deviation analysis. Testing hypothesis H2 required applying a more sophisticated statistical method known as logistic regression analysis. It is a form of regression analysis used when the dependent variable is a dichotomy and the independent variables are of any type. The dependent variable were dichotomous, where a value of 0 is given to financially unstable companies (where the auditor’s report included an explanatory paragraph pointing out the problems associated with the going concern assumption), whereas those viewed as financially stable (no explanatory paragraph on the going concern assumption


in the auditor’s report) had a value of 1. The independent variables were the abovementioned 28 financial ratios.

Logistic regression is used to predict a dependent variable on the basis of continuous and/or categorical independents and to determine the percentage of variance in the dependent variable as given by the independents. It also used to rank the relative importance of independents, assess interaction effects, and to understand the impact of covariate control variables [8]. The advantages it provides are primarily its robustness as is evident in the following:

Logistic regression analysis does not assume linear relations betweendependent and independent variables;

Logistic regression analysis does not assume normally distributedvariables;

The subsamples or groups in the sample could possibly be of differentsizes;

Logistic regression analysis does not assume homoscedasticity [24].

The logarithmic form of the logistic regression function is given by equation 1. [9]

nni

ii XXX

P

PL

...1

ln 22110 (1)

This form is transformed by antilogarithming giving equation 2 [9], which in turn are used to calculate the prognostic probability of financial problems appearing in the prediction model. The prognostic probability is the probability that there is a low possibility of financial difficulties occurring in a company. Logically, this raises the following question: When should a company be treated as having a low appearance or no financial problems or vice versa? Generally, if the prognostic probability is higher than or equal to 0.5, the company is viewed as having a low appearance of financial problems and vice versa.

)...( 221101

1nn XXXi e

P (2)

In the fourth step of the scientific approach to predicting the appearance of financial problems the estimation of statistical adequacy of the model is required. If the statistical parameters are appropriate, the model should be theoretically examined once more (the fifth step), and it can be used on real world cases. If the parameters indicate that the model is not statistically adequate, it should be theoretically reformulated and the scientific approach starts from beginning again [24].


4. Research results

The central part of the research is the analysis of research results in order to test the hypothesis. The first hypothesis reads as follow: An auditor’s report accompanied by an explanatory paragraph pointing out the problems with the going concern assumption is the appropriate criterion for classify whether a company is faced with financial difficulties. This is tested using the group means analysis.

Table 1 shows the group means of companies with no financial difficulties and those facing financial difficulties i.e. for two subsamples. Research results clearly indicate that there is a significant difference between independent variables (financial ratios) of companies with and without financial difficulties, where the distinction between them is done based on the auditor’s report. Financially unstable companies were those in which the auditor has emphasised the problems related to liquidity, high indebtedness and negative profitability. According to research results, companies with financial difficulties i.e. their financial statements are accompanied by an auditor’s report with an explanatory paragraph pointing out problems with the going concern assumption have significantly worse values of ratios than those without financial difficulties, i.e. those where no emphasis on problems with the going concern assumption was made. The t-test of differences in means is conducted on each of the 27 input variables from the auditor's report, with the results are shown in Table 1. It is evident from Table 1 that a significant difference in means (at a 10% level) between the financially stable group and the financially unstable group of companies was found in 17 of the 27 input variables. This proves the first hypothesis. Consequently, the hypothesis that an auditor’s report accompanied by an explanatory paragraph pointing out the problems with the going concern assumption is an appropriate criterion for classifying whether a company is faced with financial difficulties is confirmed and the research proceeds to testing hypothesis H2.

Hypothesis H2 – Combination of financial ratios represent a good analytical tool for distinguishing companies experiencing serious financial difficulties from those that are not which is tested using the logistic regression analysis as explained previously. Logistic regression analysis starts with analysing the statistical relation of 28 financial ratios - independent variables which show if companies are facing with financial difficulties. According to the assumption of no multicollinearity which has to be fulfilled, correlated independent variables were omitted as well as statistically insignificant variables. The backward stepwise method was used to omit correlated independent variables. The final result comes from logistic regression analysis and is a logistic regression function given by Equation 3, where the characteristic of independent variables are shown in Table 2.


Financial ratios / Companies With financial problems Without financial problems T-test of

differences in means

Difference significant at the

10% level Mean Std. Deviation Mean Std. Deviation

Short term assets/Short term liabilities 0.6211 0.7743 1.6823 2.7104 0,0349 yes

Cash and cash equivalents/Short term

liabilities 0.0834 0.1948 0.6092 1.7814 0,0846 yes

Working capital/Total assets -0.364 0.3998 0.03818 0.2226 0 yes

Total debt/Total assets 0.8536 0.3375 0.4522 0.2251 0 yes EBIT/Interests’ costs -200.8159 932.412 3.6392 15.5989 0,021 yes Total debt/(Retained

earnings+Depreciation) 0.356 18.4602 4.3085 14.0617 0,1172 no

Retained earnings/Total assets -0.3126 0.4407 -0.0094 0.2498 0 yes

Total revenues/Total assets 0.4960 0.392 0.5607 0.51 0,2837 no Sales/Accounts receivables 6.6883 10.7352 6.3941 7.4276 0,4345 no

365/( Sales/Accounts receivables) 2232.6977 9961.7425 612.4318 4227.7358 0,0849 yes

Total revenues/Total expenses 0.75322 0.1833 0.998 0.3844 0,019 yes

Sales/Costs of sales 0.8335 0.1885 1.0275 0.4222 0,0175 yes Revenues from financing activities/Expenses from

financing activities 0.2562 0.469 1.6633 6.8026 0,1672 no

EBI/Total revenues -6.4228 29.021 -0.0179 0.4216 0,001 yes EBIT/Total revenues -6.4194 29.0218 -0.012 0.4242 0,002 yes

Net sales/Total revenues -0.2876 0.4926 -0.0654 0.414 0,0108 yes Return on assets (ROA) -0.1157 0.2149 0.0222 0.052 0 yes

Return on common equity (ROCE) -0.71789 1.1347 0.1053 0.5305 0 yes

(Cash flow from operations+interests+tax)/I

nterests 0.0136 2.7707 8.7969 36.16 0,1286 no

Cash flow from operations/Total liabilities -0.0843 0.242 0.3328 2.137 0,1813 no

Cash flow from operations/Short term

liabilities -0.1026 0.2783 0.5329 2.3057 0,0995 yes

(Cash flow from operations+interests+tax)/

EBIT -1.2767 5.7056 2.5733 14.5154 0,1102 no

Cash flow from investments/(Cash flow

from operations + financing activities)

-0.7865 1.4226 -0.8871 1.8516 0,4032 no

Cash flow from investments/Cash flow from

financing activities 4.0204 20.28845 -109.3774 1391.4496 0,3517 no

(Cash flow from operations+interests+tax)/

Total assets 0.0038 0.1224 0.0697 0.1094 0,0047 yes

Cash flow from operations/Equity -0.096 0.5447 0.3244 1.0677 0,0357 yes

Economic value added -55016.948.7381 49985736.0281 -52253188.6706 178050053.134 0,4712 no

Table 1: Group means for financially stable and unstable companies


The logistic regression model given in Equation 3 includes five independent variables. Two of them are solvency ratios (Total debt/Total assets and Retained earnings/Total assets), two are profitability ratios (Return on assets and Return on common equity) and one is the cash flow ratio (Cash flow/Total assets).

LRA = 1 (3)

1+e –(5,758– 5,241TD/TA + 1,707RE/TA + 13,841ROA + 1,573ROCE + 9,089CF/TA)

B S.E. Wald df Sig. Exp(B) Total debt/Total assets (TD/TA) -5.241 1.726 9.218 1 0.002 0.005

Retained earnings/Total assets (RE/TA) 1.707 0.778 4.810 1 0.028 5.511

ROA 13.841 6.544 4.474 1 0.034 1026326.572 ROCE 1.573 1.319 1.423 1 0.233 4.822 (Cash flow from operations+interests+tax)/ Total assets (CF/TA)

9.089 4.107 4.899 1 0.027 8858.534

Constant 5.758 1.208 22.711 1 0.000 316.804 Table 2: Characteristics of independent variables

The quality of the logistic regression model is tested using the Hosmer–Lemeshow test. A high significance of the Hosmer–Lemeshow test (0.994) indicates that the hypothesis for a difference between real and prognostic values of dependent variables is acceptable i.e. the model is statistically adequate. Another quality indicator is Nagelkerke R2 which evaluates goodness of fit for the model. According to statistical analysis, Nagelkerke R2 indicates that the LRA model explains 61.3% of variations confirming the representativeness of the LRA model.

-2 Log likelihood

Cox & Snell R2

Nagelkerke R2

Hosmer-Lemeshow Test Hi square df Significance

63.909 0.321 0.613 1.406 8 0.994 Table 3: The quality coefficient of the LRA model

Another approach to testing the statistical adequacy model is to analyse its classification ability. Classification results from the LRA model (Table 4) show that the model correctly classify 91.7% of companies included in the sample. The theory usually considers model classification ability as acceptable when it correctly classifies more than 62.5% of companies, hence the model classification ability can be estimated as being relatively high [9]. The overall model classification ability should be broadened and examined in more detail.


Specifically, the overall classification results included in statistics are two well-known types of errors: type 1 error and type 2 error. Type 1 error occurs in situations where the model is classifying a financially unstable company as a stable one, while type 2 error appears in opposite situations i.e. when the model classifies financially stable companies as unstable. In classification error analysis, the occurrences of type 1 errors are less eligible because misclassifying a financially unstable company as a stable one most often results in significant additional costs than the costs incurred in the case of a financially stable company characterized as being unstable which is type 2 error. According to model classification results, type 1 error occurs in 31.8% of cases (or 7 cases as shown in Table 4) while type 2 error appears in only 5.1% (or 8 cases as shown in Table 4). The most probable explanation of the relatively high occurrence of type 1 error is that the explanatory paragraph in the auditor’s report is issued not only in cases when a company exhibits poor financial indicators, but also in some situations when financial ratios remain acceptable. In other words, there are some cases when auditors point out problems with the going concern assumption based on certain qualitative explanatory variables that will influence a company’s financial difficulties in the near future. The identification and influence of the qualitative independent variables could very well be the focus of future research and further development of this model.

Real state of financial difficulties

Predicted state of financial s difficulties Financial problems Percentage of

correct classification YES NO

Financial difficulties YES 15 7 68.2 NO 8 150 94.9

Overall classification accuracy in % 91.7 a. The cut value is 0.650

Table 4: Classification results of the LRA model

The abovementioned quality measures indicate that the LRA model can be used as a tool for diagnosing the financial health of companies by potential investors, customers, suppliers, creditors, employees and other stakeholders. Auditors could find this approach interesting when estimating a company’s ability to continue as a going concern keeping in mind the approach used in this research.


5. Conclusion

This paper extends existing research such that it uses the auditor’s report on the financial statements of companies that points out a company’s ability to continue as a going concern as the main criterion to distinguish companies experiencing serious financial difficulties from those that are not. The research results confirm the first hypothesis i.e. they validate the approach used. The authors suggest that auditors are among the few external stakeholders most familiar with a company’s financial health. Auditing standards stipulate that auditors publicly disclose whether a company is experiencing difficulties in its ability to continue as a going concern, hence their reports are, or should be, a valuable source of information. Research results indicate that the combination of financial ratios is a good analytical basis for distinguishing companies experiencing serious financial difficulties from those that are not. The combination of five financial ratios that best distinguish companies experiencing serious financial problems from those that are not includes two solvency ratios, two profitability ratios and a cash flow ratio. This combination, represented as the LRA model, has shown a high degree of statistical adequacy, making it an appropriate diagnosis tool when estimating a company’s financial health. Despite its wide scope of application, analysts should direct their attention to the type 1 error i.e. the model’s incorrect classification of companies experiencing serious financial difficulties into the group of companies that are not. The authors consider this a consequence of the criterion used for distinguishing companies experiencing serious financial difficulties from companies that are not. Specifically, pointing out the ability of a company to proceed as a going concern could possibly be noted in the auditor’s report as a consequence of some qualitative variable that does not exert an immediate influence on a quantitative or financial variable. Consequently, the derived LRA model classifies the company into the group of companies that are not experiencing financial difficulties, though in fact the company is unstable considering the auditor added an explanatory paragraph in the audit report. This opens up new questions that should be addressed scientifically. For instance, what qualitative variables/measures influence a company’s ability to continue as a going concern and what is the direction and degree of their influence; or is it possible to calculate not only two degrees of financial stability, but multiple levels of financial stability. No matter what the answers are to the previous questions, the complexity and stochastic character of economics as a field of social sciences will always require the attention of professional judgment based on experience as well as intuition which should be intensively utilised when deciding whether a company is stable or not.

Possibly the main criticisms of the research and its results in this paper might be the size of the subsample of companies experiencing financial


difficulties. However, this is reflected in the situation for the selected year involving the biggest companies in Republic of Croatia which have or should have the highest possible quality of financial statements. In future work, the derived model should be tested in order to gain more insight into its classification ability and the subsample extended to check and improve the conclusions drawn from this paper.

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Since 1994, CRORS is a member of the The International Federation of OperationalResearch Societies (IFORS), an umbrella organization comprising the national Oper-ations Research societies of over forty five countries from four geographical regions:Asia, Europe, North America and South America. CRORS is also a member ofthe Association of European Operational Research Societies (EURO) and activelyparticipates in international promotion of operational research.

If you are a researcher, academic teacher, student or practitioner interested indeveloping and applying operations research methods, you are welcome to becomea member of the CRORS and join our OR community.

Former Presidents: Prof. Luka Neralic, PhD, University of Zagreb (1992–1996)Prof. Tihomir Hunjak, PhD, University of Zagreb (1996–2000)Prof.Kristina Soric, PhD, University of Zagreb (2000–2004)Prof. Valter Boljuncic, PhD, University of Pula (2004–2008)Prof. Zoran Babic, PhD, University of Split (2008–2012)

President: Prof.Marijana Zekic-Susac, PhD, University of Osijek (since 2012)

Vice President: Prof. Zrinka Lukac, PhD, University of Zagreb

Secretary: Ivan Soldo, PhD, University of Osijek

Treasurer: Adela Has, University of Osijek

If you want to become a member of the CRORS, please fill in the application formavailable at http://www.hdoi.hr.

Banking Information:

Recepient name: Croatian Operational Research SocietyRecepient address: Trg J. F. Kennedyja 6, HR-10 000 Zagreb, CroatiaRecepient ID (OIB): 19764968282IBAN (Account number): HR9323600001101418273Bank address: Zagrebacka banka d.d., Paromlinska 2, HR-10 000 Zagreb, CroatiaRef. number: 00Bank code: 2360000-1000000013SWIFT: ZABAHR2X

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