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Applied Soft Computing 13 (2013) 2045–2056

Contents lists available at SciVerse ScienceDirect

Applied Soft Computing

j ourna l ho mepage: www.elsev ier .com/ locate /asoc

ynthesis of individual best local priority vectors in AHP-group decision making

ojan Srdjevic ∗, Zorica Srdjevicaculty of Agriculture, University of Novi Sad, Trg. D. Obradovica 8, 21000 Novi Sad, Serbia

r t i c l e i n f o

rticle history:eceived 17 September 2011eceived in revised form 27 March 2012ccepted 9 November 2012vailable online 7 December 2012

eywords:roup decision makingHPest priority vectors

a b s t r a c t

An assessment of the individual judgments and AHP-produced priority vectors for involved decision-makers indicates that the individual consistencies of decision makers may vary significantly, thus makingthe final group decision less reliable. In this paper, an approach is proposed as to how to combine decisionmakers’ local priority vectors in AHP synthesis and reduce so-called group inconsistency. Instead of aggre-gating individual judgments (AIJ), or aggregating individually derived final priorities (AIP), we proposeto perform an AHP synthesis of the best local priority vectors taken from the most consistent decisionmakers. The approach and related algorithm we label as MGPS after the key terms ‘multicriteria groupprioritization synthesis.’ The concept is analogous to the one proposed by Srdjevic [1] for individual AHPapplications where the best local priority vectors are selected based on the consistency performance

onsistency measures of several of the most popular prioritization methods. Here, decision makers are combined instead ofprioritization methods, and group context is fully implemented. After completing an evaluation of thedecision makers inconsistencies in each node of the hierarchy, the selected best local priority vectors aresynthesized in a standard manner, and the final solution is declared to be an AHP-group decision. Twonumerical examples indicate that the developed approach and algorithm generate the final priorities ofalternatives with the lowest overall inconsistency (in the multicriteria sense).

. Introduction

Group decision making problems arise from many real-worldituations in many fields such as human spaceflight mission plan-ing, water management, and selection of advanced technologyChoudhury et al. [2], Srdjevic [3], Tavana and Hatami-Marbini [4]).n recent times, most of these problems are attacked and success-ully solved with the application of the analytic hierarchy processAHP), developed by Saaty [5]. In fact, the so-called AHP-grouppplication means that the standard AHP-individual application isxtended to provide for certain types of aggregations, consensusrocedures, etc. Rich, pertinent literature in this regard is refer-nced appropriately in the remaining part of this text.

Three mainstream theories and applications comprise AHP. Therst theory applies to the preference relations - linguistic, numer-

cal, and fuzzy (e.g. Saaty [5], Chiclana et al. [6], Fu and Yang [7],errera and Martínez [8], Herrera et al. [9]) - that are widelysed in individual and group decision making. The second onepplies to the prioritization methods for extracting cardinal infor-

ation (weights of the decision elements, usually the criteria

nd alternatives) from so-called judgment matrices (i.e. multi-licative preference relations) at each node of a hierarchy. The

∗ Corresponding author. Tel.: +381 21 4853 337; fax: +381 21 455 713.E-mail address: bojans@polj.uns.ac.rs (B. Srdjevic).

568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.asoc.2012.11.010

© 2012 Elsevier B.V. All rights reserved.

third mainstream theory applies to consensus building, aggregationmethods, and measuring (in)consistency in group decision mak-ing. This paper falls within this third theory and focuses on theAHP-group synthesis of the best local priority vectors identified bymulticriteria evaluation of inconsistency contained in individualvectors obtained for participating members of a group. To our bestknowledge, the approach we propose is novel and the related com-putational procedure is labeled as MGPS algorithm, after the keyterms ‘multicriteria group prioritization synthesis.’

The AHP offers a variety of options in supporting the decisionmaking processes, including group contexts. Being a soft computingtechnique as well as a multicriteria analysis method, the proposedapproach is aimed to contribute to the soft computing commu-nity by introducing an objective method of synthesizing locallycomputed priority vectors for all involved individuals in a group.In general, computations within AHP are inherently simple. How-ever, an issue of (in)consistency in decision makers in modern timespresents the challenge of employing evolution strategies, geneticalgorithms, particle swarm optimization and other soft computingtechniques and heuristics in deriving priorities from inconsistent(or low consistency) matrices. In a way, a new paradigm aims toprovide more efficient solving of the spectrum of decision mak-

ing problems where individual judgments of decision makers playa leading role. The MGPS algorithm we propose is inspired by anidea for making objective the process of prioritizing alternativesvs. a global goal. We developed it to be straightforward and easy

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046 B. Srdjevic, Z. Srdjevic / Applied

o program and implement in real life applications. Being fullyrogrammed in FORTRAN, on a proof-of-concept level it was suc-essfully used in many test applications as well as in two describedxamples presented in the second part of this paper.

Most of the research papers dominantly report theoreticalorks and application results concentrating on just one ratio-scaleatrix at a time. Much fewer research reports relate to the whole

ierarchy where the original AHP philosophy belongs to and theoncept of AHP synthesis is fully implemented. This paper relates tohe latter case only; that is, we consider a complete AHP hierarchy,ot just one local matrix. In addition, the presented work is aimedo cover several topics related to AHP application in group contextshere decision makers demonstrate different consistencies.

The core of our approach is to perform AHP synthesis of the bestocal priority vectors taken from the most consistent decision mak-rs. Without losing generality, if we consider a three level hierarchygoal-criteria-alternatives), the number of matrices is 1 + na, wherea is the number of alternatives. If there are K decision makers thenhere are K(1 + na) matrices, and in every node of a hierarchy there is

set of K matrices with their priority vectors. Some vectors are moreonsistent than others, and it is possible to identify the best one ifertain criteria are adopted for assessing their quality regardingonsistency. On the other hand, consistency measures applicableo AHP can be divided into two groups: (a) general (e.g. Euclideanistance or Minimum Violation), and (b) specific (e.g. prioritiza-

ion method related, such as CR for eigenvector method or GCI forogarithmic least square method). If consistency measures are useds criteria for assessing the quality of priority vectors derived fromecision makers at a given node of hierarchy, then the best (optimal

n multicriteria sense) priority vector can be identified and prop-gated to the final AHP synthesis. Obviously, the final synthesis iserformed with the best vectors and therefore the final (group)riorities of alternatives are objectively the best possible.

The paper is organized as follows: in Section 2, we briefly presenteviewed related research, basic preliminary knowledge of AHP,ncluding its most commonly used prioritization method known ashe eigenvector (EV) method. In Section 3, we develop an approacho AHP-group synthesis following an idea presented in Srdjevic [1],nd instead of AIJ or AIP aggregations, we propose to combine theocal priorities derived from group members based on a multicrite-ia evaluation of their demonstrated consistencies. In Sections 4 and

two illustrative examples from real-life AHP-group applicationsre provided, and the results of MGPS application are discussed.ection 6 presents concluding remarks and an agenda for futureesearch in the subject area.

. Related research

.1. AHP

AHP (Saaty [5]) is one of the most popular decision supportools because of its powerfulness, simplicity, and potential of beingtilized for the group decision-making process that involves mul-iple actors, scenarios, and decision elements (criteria, sub criteria,nd alternatives). As correctly stated by Altuzarra et al. [16], AHPs ‘one of the methods that best captures changes in philosophyfrom mechanistic reductionism to evolutionist holism), method-logy (from the search for truth to the search for knowledge), andechnology (communication networks) that took place in the latterears of the 20th century.’

The AHP requires a well-structured problem represented as a

ierarchy with the goal at the top. The subsequent levels containhe criteria and sub-criteria, while alternatives lie at the bottom ofhe hierarchy. The AHP determines the preferences among the setf alternatives by employing pairwise comparisons of the hierarchy

mputing 13 (2013) 2045–2056

elements at all levels following the rule that at a given hierarchylevel, elements are compared with respect to the elements in thehigher level by using the fundamental importance scale (Saaty [5]).

The synthesis is performed by multiplying the criteria-specificpriority vector of the alternatives with the corresponding criterionweight and summing up the results to obtain the final compositeof the alternatives’ priorities with respect to the goal. The highestvalue of the priority vector indicates the best-ranked alternative.

2.2. Prioritization in AHP

Over the years, several methods have been proposed for esti-mating the weights from a matrix of pairwise comparisons,including additive normalization (AN), eigenvector (EV), logarith-mic least squares (LLS), weighted logarithmic least square (WLS),logarithmic goal programming (LGP), fuzzy preference program-ming (FPP), and others. A brief description of these competingmethods is provided by Harker and Vargas [10], and Srdjevic [1].Herein, we present the main features of the EV method becausewe consider it to be competitive with the other methods and it iscommonly used in practice as well as in our research.

The EV method generates a priority vector for the given pair-wise comparison matrix obtained from the decision maker. Themethod, originally proposed by Saaty [5], solves an eigenvalueproblem associated with a matrix of size n and is nicely describedin Chandran et al. [11]. The mathematical notation that follows isintroduction to the next section where the MGPS algorithm will bedescribed in detail.

Let A = (aij), for i, j = 1, 2, . . ., n, denote a square pairwise com-parison matrix, where entry aij gives the importance of element irelative to element j. Each entry is a positive value (aij > 0) with areciprocal aji = 1/aij for all i, j = 1, 2,. . ., n). The decision maker wantsto compute a vector of weights (w1, w2, . . ., wn) associated with A.

If the matrix A is consistent (that is, aij = aikakj for all i, j, k = 1, 2,. . .,n), then A contains no errors. Therefore, the weights are alreadyknown, and we have

aij = wi

wj, i, j = 1, 2, . . . , n. (1)

Summing over all j, we obtain

n∑

j=1

aijwj = nwi, i = 1, 2, . . . , n. (2)

In matrix notation, the result is equivalent to

Aw = nw, eT w = 1 (3)

The vector w is the principal right eigenvector of matrix A corre-sponding to the eigenvalue n. If the vector of weights is not known,then it can be estimated from the pairwise comparison of matrix A′

generated by the decision maker and solving

A′w′ = �′w′, eT w′ = 1 (4)

for w′. The matrix A′ contains the pairwise judgments of the deci-sion maker and approximates the matrix A whose entries areunknown. In Eq. (4), �′ is an eigenvalue of A′, and w′ is the esti-mated vector of weights. Saaty [5] uses the largest eigenvalue �max

of A′ when solving for w′ in

A′w′ = �maxw′, eT w′ = 1 (5)

Saaty has shown that �max is always greater than or equal to n,and if its value is close to n then the estimated vector of weights w′

solves Eq. (3) approximately.A good estimate of the principal eigenvector for an inconsistent

matrix is obtainable by consecutively squaring the matrix, normal-izing the row sums each time, and stopping the procedure when

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B. Srdjevic, Z. Srdjevic / Applied

he difference between the normalized sums in two consecutivealculations is smaller than a prescribed value.

It was shown by various researchers that for small deviationsround the consistent ratios wi/wj, the EV method gives a reason-bly good approximation of the priorities’ vector. However, whenhe inconsistencies are large, it is generally accepted that the solu-ions are not so satisfactory.

.3. AHP in group context

In general, group decision making problems use preference rela-ions and apply an aggregation rule on individual preferences tobtain the collective preference. The two most useful methodsor AHP-group decision making are the aggregation of individualudgments (AIJ), and the aggregation of individual priorities (AIP)Dong et al. [12], Forman and Peniwati [13]). A review of the litera-ure shows that researchers have some disagreement on the use ofIJ and AIP if the standard eigenvector (EV) prioritization method

s used. Forman and Peniwati [13] argue that whether AIJ or AIPre used depends on whether the group intends to behave as aynergistic unit or as a group of individuals, respectively. Barzilaind Golany [14] have shown that if the logarithmic least squareLLS) method is used for prioritization, then there is an equivalenceetween AIJ and AIP, while Ramanathan and Ganesh [15] suggestsing AIP only because AIJ violates the Pareto principle of socialhoice theory (Arrow [16]). Dong et al. [17] show that the effects ofV and LLS methods are very similar, and because computationalimes are of o(n2) and o(n), respectively, suggest that LLS is moreppropriate in building a consensus model in AHP-group decisionaking (Dong et al. [17]).An interesting approach in the AHP-group application is pre-

ented in Altuzarra et al. [18] where a prioritization procedure isroposed that does not require intermediate filters for the indi-idual’s initial judgments. The procedure is based on a Bayesiannalysis of the problem. It is argued that in general, the analysis pro-ides more efficient estimates than the techniques conventionallypplied in the literature for AHP-group applications based on AIJnd AIP. Moreover, authors use a case study example and claim thathe proposed procedure naturally extends to the analysis of incom-lete and/or imprecise pairwise comparison matrices and enhanceealism in decision making itself.

Regarding the prioritization problem in AHP (i.e. extracting theriority vector from judgment matrix), there are many methods

n use. To mention but a few, Saaty proposed the EV and additiveormalization (AN) methods [5], Crawford and Williams proposedhe logarithmic least square (LLS) method [19], while Mikhailovescribed the fuzzy preference programming (FPP) method [20].rdjevic [1] showed that combining the different prioritizationethods in AHP synthesis, based on their consistency performance

t the local nodes of a hierarchy, can produce a better final resulthan if only one prioritization method is used in AHP. In addition,t is argued that none of the prioritization methods have any a pri-ry advantage in relation to other methods if global criteria, suchs Euclidean distance and the minimum violation of ordering botheing used to compare methods of consistency.

In a group decision making process, both consensus and con-istency need to be pursued and sought after. A solution with aigh level of consensus is desirable. Many researchers focus onow to define an acceptable level of consensus and, in turn, how tochieve it. den Honert [21] proposed a model for the representationf a consensus-seeking group decision making process under theultiplicative variant of AHP. In particular, he proposed a group

reference model that expresses the group’s preference inten-ity judgments as random variables with associated probabilityistributions, and multiplicative AHP is used to derive interval judg-ents for the alternatives’ final impact scores as perceived by the

mputing 13 (2013) 2045–2056 2047

group. Parreiras et al. [22] propose a flexible consensus scheme forgroup decision making that obtains a consistent collective opinionfrom information provided by each expert in terms of multigranu-lar fuzzy estimates. The scheme is based on a linguistic hierarchicalmodel with multigranular sets of linguistic terms, and the choiceof the most suitable set is a prerogative of each expert. Alonsoet al. [23] present a web-based consensus support system that isable to help, or even replace, the moderator in a consensus processwhere experts are allowed to provide their preferences using fuzzy,linguistic, or multi-granular linguistic and incomplete preferencerelations. The system is based on both consistency and consen-sus measures and is designed to provide advice to the experts toincrease the group consensus level while maintaining the individ-ual consistency of each expert. Dong et al. [12] consider judgmentmatrices and define the consensus indexes to measure the degree ofconsensus among judgment matrices (or decision makers) for AHP-group decision making using the logarithmic least square (LLS)method. They presented two consensus models (under LLS) byusing the Chiclana et al. [24] consensus framework and by extend-ing Xu and Wei’s [25] individual consistency improvement method.Both models look promising; however, the two models depend onthe LLS prioritization method that restricts the application of thesemodels to just this prioritization method. In Moreno-Jimenez et al.[26], a tool labeled the consistency consensus matrix is proposed toencourage the search for consensus in group decision making whenusing AHP. The procedure exploits one of the characteristics of AHP:the possibility of measuring consistency in judgment elicitation.

If consistency among the decision makers is analyzed, thereare obviously cases where the members of a group are individu-ally consistent and the group solution is not in accordance withany individual final preference order (Blankmeyer [27]). Literaturereview shows that there are many recently published papers thataddress consistency of the judgment matrices (i.e. multiplicativepreference relations) (Srdjevic [1], Dong et al. [12], Chiclana et al.[24], Liu [28], Sun et al. [29]).

2.4. Group synthesis of AHP judgments or priorities

There are various variants of how to synthesize the results ofindividual AHP applications. Considering the most general casewhere the hierarchy is specified and agreed upon by all the decisionmakers that participate in the group decision making process, themost common aggregations are a geometrical averaging of indi-vidual judgments at each hierarchical node (known as AIJ – theaggregation of individual judgments), or a geometrical averagingof the final individual priority vectors (known as AIP – the aggre-gation of individual priorities). For example, the later aggregationis performed by applying Eq. (6):

zGi =

K∏

k=1

[zi(k)]˛k (6)

where K stands for the number of decision makers, zi(k) for thepriority of the ith alternative for the kth decision maker, ˛k for theweight of the kth decision maker, and zG

ifor the aggregated group

priority value. Weights ˛k are additively normalized prior to theiruse in Eq. (12), and the final additive normalization of priorities zG

iis required.

For details on these two most common aggregation procedures,see Dong et al. [12] and Forman and Peniwati [13].

Leaving aside the discussion on the influence and modes ofdefining the weights of the decision makers before aggregation is

undertaken, or issues such as consensus building and compromisesolution frameworks, we concentrate in the remainder of thispaper on the so-called ideal (benchmark) aggregation. At eachnode of the hierarchy, the best priority vector is taken from the

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048 B. Srdjevic, Z. Srdjevic / Applied

et of vectors derived by individuals in the group and propagatedoward the end point of the AHP-final synthesis. A mutual impor-ance of the decision makers is intentionally neglected becauset is extremely sensitive to manipulation and misleading in thenterpretation of final results.

.5. Consistency measures in AHP

To measure the quality of a given priority vector extracted fromultiplicative preference relations, several measures, such as the

onsistency ratio (CR) used with the EV method (Saaty [5]), the geo-etric consistency index (GCI) used with the LLS method (Aguarón

t al. [30]), and the fuzzy intersection measure (�) used with FPPMikhailov [20]), are used. More general and applicable to all pri-ritization methods are error of estimate measures such as theeneralized L2 Euclidean distance (ED) and the order reversal indi-ator known as the minimum violations (MV) criterion (Golany andress [31]). They measure both the accuracy of the solution and

he ranking of the order properties, and are widely accepted byesearchers. Some other measures of accuracy, such as the Frobe-ius L1 norm, are in use as well.

In our approach, we use three consistency measures as criterionsor assessing the prioritization results at local levels in AHP acrossll involved decision makers. The first two (ED and MV) are general,hile the third one (CR) is method (i.e. eigenvector) specific.

. A MGPS algorithm for synthesizing local prioritieserived from group members

In brief, the MGPS algorithm is developed to realize a conceptnalogous to the one proposed in Srdjevic [1] for individual AHPpplications where the best local priority vectors are selectedased on the consistency performance of several of the most pop-lar prioritization methods. Here, in all nodes of a given hierarchy,

udgment matrices (i.e. multiplicative preference relations) aslicited from the decision makers in a group are firstly subjectedo selected prioritization methods to compute priority vectors. Inurn, several consistency measures are computed for each matrixnd for each decision maker. Assuming that consistency measuresre evaluation criteria with associated weights, evaluation of thendividually obtained priority vectors enables to identify a singleecision maker with the best demonstrated consistency in a mul-icriteria sense. Selected best local priority vectors are synthesizedn a standard manner, and the final solution is declared as theHP-group decision.

Our approach is based on several major assumptions and claims:

The decision makers demonstrate their best knowledge and will-ingness to preserve the internal logic of their judgments;

The decision makers achieve a consensus in the group to allowan objective algorithm to use their judgments and derive localpriority vectors from AHP synthesis based on the quality of theirconsistency as demonstrated in each particular node of a hierar-chy;

To select the best local priority vectors that are the most accept-able in the final synthesis, various criteria can be used, (e.g.total Euclidean distance (ED), minimum violation of orders (MV),Saaty’s consistency ratio (CR) (if eigenvector method is used forprioritization), geometric consistency index (GCI) (if logarithmicleast squares methods is used), � (if fuzzy preference program-ming method is used), etc.);

In Srdjevic [1], it is argued that a combination of locally best pri-

ority vectors have the best overall performance in the final AHPsynthesis. In the MGPS approach and algorithm, the most con-sistent decision makers at the local nodes produce the best localvectors, and, analogously to our earlier works [1], their use in AHP

mputing 13 (2013) 2045–2056

synthesis is represented by the additive utility function; that is,the best final priority vector.

For multicriteria evaluation of individual priority vectors in eachnode of a hierarchy, we propose to use the weighted sum method(WSM), with higher weights associated with ED and CR in favorof MV. A good choice in this regard would be to use the adaptiveweighted sum method for Pareto front generation as proposed inKim and de Weck [32]. In our analyses, we also used the weightedproduct method (WPM), as well as CP and TOPSIS methods in var-ious tests with larger groups of more than ten individuals. In themajority of tests performed, the best local priority vectors wereidentically identified by all methods; for this reason, we proposeto rely on the simplest method, WSM, although the choice can bedifferent, as stated before.

In its central computational part, AHP manipulates the quadraticreciprocal positive judgment matrices, also called multiplicativepreference relations in more recent literature, to derive the relatedpriority vectors. Once the priority vectors are extracted from allthe matrices, they are additively synthesized in the standard AHPmanner to come up with the final priorities (relative weights) ofthe alternatives with regard to a global goal on the top of thehierarchy.

The number of judgment matrices depends on the size of theproblem. At a given level of hierarchy, the size of the matrices isthe same and is equal to the number of elements at that level. Thesizes of the matrices usually vary between the levels of a hierarchy.If a matrix is of the size 3 × 3 or 4 × 4, it is possible that consistentjudgments would occur, i.e. that the condition aij = wi/wj is satisfiedfor all i, j = 1, 2,. . ., n, where n is the size of a matrix. The possibilityof inconsistencies while eliciting judgments from the DM generallyrises with the size of the judgment matrix. Then, only approximatesof aij = wi/wj are possible to obtain. How good the approximationof the true (ideal) priority vector depends on which prioritizationmethod is used, and the resultant vector is therefore faced withunavoidable and unpredictable errors. A need for a compromisesolution is obvious (Saaty [5]).

To achieve this, the following three-stage algorithm is proposed,which is analogous to the algorithm reported in Srdjevic [1]. Themathematics are adjusted to a group context while the key dif-ferences in the content of the steps will be explained in turn andsupported by novel remarks.

3.1. MGPS (multicriteria group prioritization synthesis) algorithm

• Stage 11.1. Set up a hierarchy H of the problem. Without loss of generality,

assume a 3-level hierarchy with the goal on the top first level, nccriteria at the second level, and na alternatives at the third level(bottom level).

1.2. Let G be a group of K decision makers. Make all necessaryjudgments and create a set of matrices for all decision makers:Pk = {P1k, P2k, . . . , PIk}, k = 1, 2, . . . , K (7)where I is the number of matrices in H. For a 3-level hierarchy, wehaveI = 1 + nc (8)and the total number of matrices isM = K · I = K · (1 + nc). (9)

1.3. Define a set of evaluating criteria for assessing the quality ofestimates for priority vectors:E = {E1, E2, . . . , EJ } (10)where J is the number of criteria. The evaluating criteria mayinclude, for example, E1 = ED (the Euclidean distance), E2 = MV (theminimum violation), and E3 = CR (the consistency ratio), ifeigenvector (EV) method is used. Notice that if another

measure should be used instead of CR. For example, if thelogarithmic least squares (LLS) method is employed forprioritization, then GCI (the geometric consistency index) shouldbe used instead of CR.

B. Srdjevic, Z. Srdjevic / Applied Soft Co

Table 1Performance matrix Ri for comparison matrix Pi .

Decision maker Evaluating criteria

E1 E2 . . . EJ

˛1 ˛2 . . . ˛J

DM1 ri11 ri

12 ri1J

DM2 ri ri . . . ri

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2

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K1 riK2 ri

KJ

Stage 2.1. Apply the eigenvector (EV) method to a given judgment matrix Pik

in the hierarchy (for each individual k), and obtain a set of I · Kpriority vectors, one for each Pik (i = 1, 2, . . ., I; k = 1, 2, . . ., K).

.2. Compute the estimate error for all matrices Pik for each evaluatingcriterion Ej . Insert these values into the performance matrixRi = {ri

kj}, Table 1.

.3. Normalize the columns of matrix Ri .

.4. Compute additively the total utility (minimum estimate error) foreach individual across the criteria set E. Before doing so, associatethe weights (˛1, ˛2, . . ., ˛J) to the criteria to express the group’spreferences in minimizing the estimate errors (see Table 3); ifnecessary, normalize weights to have the criteria weightssumming to 1. The association rule is as follows: the higher theweight of a criterion, the more important it is for the criterion(estimate error) to be minimized.

.5. Identify an individual with the highest total utility. Consider apriority vector obtained for this individual as the optimal one in agroup multicriteria sense and record it for final Stage 3 where theAHP synthesis takes place.

.6. Repeat steps 2.1–2.5 for all matrices in H.

Stage 3.1. Get all the priority vectors identified in Stage 2 as local optima and

perform AHP synthesis to obtain the final priorities of thealternatives. This vector can be considered the group optimalsolution for the decision problem in the multicriteria sense.

.2. Compute the aggregate (overall) estimate errors for the synthesisperformed in 3.1 over all evaluating criteria E, and compare thesevalues with the corresponding values obtained for each individualover the complete hierarchy H, including the final AHP synthesis.

The following remarks provide more details on the proposedpproach.

emark 1. In Stage 1, a set of evaluating criteria are defined forssessing the quality of the estimates of the priority vectors to beerived in Stage 2. Although there are various error criteria, moreeneral and applicable to all standard AHP ratio matrices are threerror-of-estimate measures known as generalized L2 Euclidean dis-ance (ED), minimum violations (MV) criterion, and consistencyatio (CR). For a consistent matrix, all these errors are zero. Fornconsistent matrices, ED and MV should be as small as possible,nd CR should be less than 0.10 (Saaty [5]).

The use of CR is justified only if the eigenvector method is appliedor prioritization. If another prioritization method is used thenxclude this error measure from the criteria set, or replace this errory another measure. For example, if the geometric mean methodknown also as the logarithmic least squares method) is used forrioritization, then replace CR with GCI (Aguarón et al. [30]).

Notice that the values of the criterion MV in the proposed MGPSlgorithm are scaled to the size of the analyzed comparison matrixby dividing MV by n2 where n is the size of matrix) in order to pre-erve consistent comparisons of individuals on a global hierarchycale.

emark 2. In Stage 2, the eigenvector method is applied for all

udgment matrices Pik (i = 1, 2, . . ., I; k = 1, 2,. . .,K) to estimate the

et of priority vectors wik = {wik1 , wik

2 , . . . , wikm}T

. Index m representshe dimension of a given judgment matrix Pik and can take onlywo values: nc (number of criteria) and na (number of alternatives).

mputing 13 (2013) 2045–2056 2049

Note that m varies with respect to i, and all judgment matrices ata given hierarchy level have a dimension equal to the number ofelements at this level while their sizes are equal to the number ofelements at the upper level.

Remark 3. The central part of the algorithm is the repeated activ-ity 2.5 by which a complete set W* of locally optimal vectors isderived. This set is given as:

W∗ = {wik∗|i = 1, 2, . . . , I; k∗ ∈ K} (11)

Each vector element corresponds to one judgment matrix (Pi)and to the particular individual (k*) for which the estimate error isminimal in the multicriteria sense. Note that k* corresponds to thebest individual (individual whose vector is selected) and thereforemay take repeatedly the same integer values from the set (1, 2, . . .,K).

Remark 4. For obtaining W* the procedure is as follows: the per-formance matrix Ri is created for each particular matrix Pi ∈ P, withentries (ri

kj) representing estimate errors with regard to each eval-

uation criterion Ej ∈ E. To assess the quality of the estimate, i.e.to compute (ri

kj), a priority vector wik is used as obtained by the

decision maker k* ∈ K (best in a set of all decision makers K).

Obviously, entry rikj

is the performance score of a decision makerk with respect to Ej evaluation criterion for matrix Pi. Weights ˛1,˛2, . . ., ˛J represent the relative importance of elements in the crite-ria set E and are defined in advance. Weights of criteria can bedefined in different ways, e.g. by consensus of the decision makerswhen any scientifically sound consensus model is applicable.

After additive column normalization of matrix Ri is performed,an additive weighting model given by Eq. (18) may be used to com-pute a set of utilities U i = (Ui

1, Ui2, . . . , Ui

k, . . . , Ui

K ) for each row ofmatrix Ri.

Uik =

J∑

j=1

˛jrikj (12)

The minimum utility, representing the minimal error in a mul-ticriteria sense, is given by

Uik∗ = min

k=1,...,KUi

k (13)

and because it is obtained for decision maker k*, the correspondingpriority vector

wik∗ = (wik∗1 , wik∗

2 , . . . , wik∗m )

T(14)

is locally optimal in a multicriteria sense for judgment matrix Pi.Notice that m is the dimension of this matrix.

Remark 5. The final AHP synthesis in Stage 3 (task 3.1) consists ofthe additive aggregation of the optimal priority vector for criteria

w1k∗ = (w1k∗1 , w1k∗

2 , . . . , w1k∗nc )

T(15)

locally optimal priority vectors for the alternatives:

wik∗ = (wik∗1 , wik∗

2 , . . . , wik∗na )

T, i = 2, 3, . . . , I (16)

and is given by:

zq =nc∑

p=1

w1k∗p w(1+p)k∗

q , q = 1, . . . , na. (17)

2050 B. Srdjevic, Z. Srdjevic / Applied Soft Computing 13 (2013) 2045–2056

CHOO SE THE BE ST WETLAND SEGMENTATION

AREA COST EFF ECT COND COVER LOA D

ment

at

∑

apca

Rt(b(isp

debvc

mh

C

wvsscM

tetcictm

H-D-V H-D-D H-V-V

Fig. 1. Hierarchy of the wetland seg

The final priority of alternative q is value zq obtained by (17),nd the final priority vector z is the final synthetic result. It is easyo show that the sum of the alternatives’ priorities is 1:

na

q=1

zq = 1 (i.e. eT z = 1). (18)

Recall that nc is the number of criteria, na is the number oflternatives, and I is the number of matrices in H. Since I = 1 + nc,-indexing in [23] is adjusted to account for one matrix at theriteria level and nc matrices (indexed as 1 + p, p = 1, . . ., nc) at thelternatives level.

emark 6. To enable cross-comparison of individual final solu-ions with the solution obtained by the proposed MGPS algorithmtask 3.2), and to complete an assessment of the solution processy AHP, we propose to use the conformity (C) index and compare:1) the final results obtained in standard AHP synthesis for eachndividual exclusively and (2) the final result obtained when AHPynthesis is performed by combining the individuals’ locally bestriority vectors (MGPS).

If the conformity of a given decision maker with the rest of theecision makers in a group in a given node of a hierarchy is consid-red as a measure of deviation between the priority vector obtainedy this individual and an average priority vector across all indi-iduals, then this definition can be referred to as matrix-contextonformity.

In a global multilevel hierarchy-context where more than oneatrix exists to account for the hierarchy-context, as is the case

ere, conformity can be expressed as:

k =na∑

q=1

|zkq − zS

q |, k = 1, . . . , K (19)

here k is the individuals’ index and K is the number of indi-iduals in a group used in a combined group AHP synthesis. Theuperscript S stands for the priority vector obtained by combinedynthesis (MGPS). Obviously, a lower value of Ck indicates betteronformity of decision maker k with the group decision obtained byGPS.Conformity defined in this way indicates a global similarity of

he final result obtained by a particular individual assuming itsxclusive use of AHP, with the ideal result obtained by combininghe best individuals in the proposed group synthesis. Obviously, Cannot be used as criterion in the same way as ED or MV because

t applies only after all computations in AHP are concluded. Itan be compared with just the CR error measure (if the eigenvec-or method is used for prioritization), because this comparison is

ethodologically consistent with the concept of AHP.

H-V-H D-D-D V-D-V

ation problem (Srdjevic et al. [33]).

4. Numerical examples

This section presents the main results of the application of theMGPS algorithm on two examples taken from practice. In the firstexample, local priority vectors are synthesized from five decisionmakers who assessed the same hierarchy with six criteria and sixalternatives. In the second example, the MGPS synthesized localpriority vectors are from the two decision makers who evaluatedfive alternatives against seven criteria.

4.1. Example 1 (constructed wetland)

This problem is taken from Srdjevic et al. [33] and relates toa real-life problem of constructing a wetland area for purifyingwastewater discharged from a small rural community. The hier-archy of the problem is given in Fig. 1 and a description of thedecision-making elements is as follows:

• GoalGiven the set of optional orderings of three segments in con-

structed wetland, identify the best ordering, respecting typicalcriteria defined by experts in wastewater management andhydraulics.

• CriteriaThere are six criterions adopted for the assessment of alter-

native segmentation of the wetland: AREA – size of wetlandarea needed; COST – cost of constructing the wetland; EFFECT– effectiveness of the wetland segmentation alternative; COND– hydraulic conductivity of the soil; COVER – planned/availableplant coverage; and LOAD – amount of collected water in thefuture.

• AlternativesTo achieve good purification of wastewater, solutions that com-

bine vertical and horizontal flows in shallow gravel ditches areconsidered as alternatives in constructing wetland. For example,alternative A1, identified in Fig. 1 as H–D–V, relates to wet-land construction in which the primary purification is within asegment with horizontal groundwater flow (H); the secondarypurification occurs in a segment with shallow gravel ditches (D);and the tertiary purification occurs in the final segment wherevertical water flow (V) is dominant.

• Decision makersThe five decision makers (DMi, i = 1, 2, . . ., 5) participated in

a real-life decision-making process, all with specific experiencein the planning and implementating of several constructed wet-lands in Serbia, mainly for purification of wastewaters in smallrural municipalities in the Vojvodina Province: DM1 – ExpertUniversity Professor #1 (mainly planner); DM2 – Expert Uni-

versity Professor #2 (mainly analyst); DM3 – Project Engineer(experienced professional); DM4 – Project Engineer and Manager(consultant and team leader); and DM5 – System Analyst (youngresearcher).

B. Srdjevic, Z. Srdjevic / Applied Soft Co

Table 2Priorities of criteria vs. goal for all decision makers (wetland problem).

Criteria Decision makers

DM1 DM2 DM3 DM4 DM5

AREA 0.092 0.043 0.024 0.069 0.105COST 0.115 0.062 0.066 0.122 0.042EFFECT 0.402 0.350 0.302 0.169 0.439COND 0.122 0.251 0.245 0.409 0.073

ampt

mDio

pa˛mtifa

piti˛

dwtd

TP

COVER 0.112 0.145 0.176 0.103 0.101LOAD 0.157 0.148 0.187 0.128 0.241

Considering the hierarchy in Fig. 1, each decision maker cre-ted one judgment matrix of size 6 × 6 for the criteria level and sixatrices for the alternatives level. All matrices and corresponding

riority vectors (computed by the EV method) are local and subjecto consistency evaluation.

Tables 2 and 3 present all local priorities obtained by the decisionakers: criteria vs. goal and alternatives vs. criteria, respectively.ifferent backgrounds and attitudes of decision makers expectedly

mplied differences in their judgments, and consequently prioritiesf decision elements, namely criteria and alternatives.

Table 4 presents consistency measures ED, MV, and CR com-uted for all local matrices and all decision makers. Based onssigned weights to consistency measures ˛ED = 0.45, ˛MV = 0.1, andCR = 0.45, the MGPS algorithm identified the sequence of localatrices and their corresponding priority vectors for the final syn-

hesis. That is, priority vectors should be taken from the followingndividuals and corresponding local matrices: DM5 (priority vectoror matrix P1), DM3 (P2), DM3 (P3), DM1 (P4), DM1 (P5), DM1 (P6),nd DM3 (P1).

Given the best sequence of local vectors, the final AHP synthesis-roduced MGPS solution was presented in Table 5 together with

ndividually computed final solutions by the decision makers. Wor-hy to mention is that the same selection of local priority vectorss obtained if a single criterion (CR) is used, that is: ˛ED = 0.00;MV = 0.00; and ˛CR = 1.00.

Data presented in Table 5 shows that the MGPS algorithm pro-

uced the final priority vector with the best overall performanceith regards to both consistency measures CR and MV, while the

otal ED is only slightly higher than the lowest one obtained for oneecision maker (here, DM1). Therefore, the MGPS priority vector

able 3riorities of alternatives vs. criteria for all decision makers (wetland problem).

Cri/Alt Decision makers

A1 A2 A3 A4 A5 A6

DM1

AREA 0.095 0.144 0.061 0.063 0.459 0.17COST 0.180 0.117 0.256 0.268 0.050 0.12EFFECT 0.130 0.173 0.074 0.069 0.381 0.17COND 0.155 0.080 0.230 0.363 0.051 0.12COVER 0.151 0.085 0.230 0.375 0.051 0.10LOAD 0.142 0.078 0.230 0.385 0.053 0.11

DM3

AREA 0.148 0.068 0.359 0.205 0.049 0.17COST 0.142 0.274 0.049 0.081 0.371 0.08EFFECT 0.114 0.058 0.382 0.273 0.042 0.13COND 0.131 0.064 0.370 0.228 0.041 0.16COVER 0.155 0.082 0.325 0.258 0.046 0.13LOAD 0.110 0.060 0.343 0.309 0.044 0.13

DM5AREA 0.130 0.107 0.219 0.465 0.031 0.04COST 0.103 0.053 0.284 0.471 0.023 0.06EFFECT 0.097 0.193 0.043 0.030 0.505 0.13COND 0.103 0.299 0.135 0.364 0.054 0.04COVER 0.088 0.054 0.262 0.213 0.177 0.20LOAD 0.224 0.285 0.073 0.287 0.089 0.04

mputing 13 (2013) 2045–2056 2051

and final ranking, shown in the last two columns of Table 5, can beconsidered as the group decision based on the best consistency ofdecision makers demonstrated during their judgments of decisionelements within the complete hierarchy.

4.2. Example 2 (walnut selections)

This problem is also taken from practice [34]. Two national wal-nut experts participated in a scientific experiment and individuallyevaluated five walnut varieties against seven typical selection crite-ria, Fig. 2.

The decision-making elements were as follows:

• GoalSelect the best one among five candidate walnut varieties given

the set of criteria commonly used in official selection and stan-dardization processes applicable to nut fruits.

• Criteria setThere are seven typically used criteria in assessing walnut vari-

eties, namely: Color – kernel’s color; Portion – kernel’s portion;Weight – nut’s weight; Taste – taste of the kernel; Shell – shellshape; Storage – preserved quality in storage; and Trade – tradevalue.

• Alternatives (walnut varieties)Five walnut varieties are evaluated as decision alternatives:

RASNA – national variety (candidate for approval as selection);KRODNI – national variety (candidate for approval as selection),MACVA – national variety (candidate for approval as selection);SEJNOVO – selection from Bulgaria (orig. known as Sejnovo); andFRANKET – selection from France (orig. known as Franquette).

• Decision makersTwo university professors in viticulture, both nationally recog-

nized experts in walnut production and selection, participated asdecision makers DM1 and DM2.

After a brief introduction on the AHP methodology, each deci-sion maker created one judgment matrix of size 7 × 7 at the criteria

level, and seven matrices of size 5 × 5 at the alternatives level(cf. Fig. 2). Once judgments were made, prioritization through theeigenvector method produced eight priority vectors for each deci-sion maker as shown in Tables 6 and 7.

A1 A2 A3 A4 A5 A6

DM29 0.095 0.053 0.277 0.148 0.070 0.3579 0.211 0.220 0.066 0.260 0.188 0.0542 0.102 0.087 0.304 0.066 0.110 0.3322 0.077 0.074 0.283 0.054 0.102 0.4109 0.153 0.271 0.062 0.282 0.183 0.0501 0.085 0.090 0.301 0.058 0.099 0.366

DM42 0.142 0.027 0.216 0.522 0.013 0.0803 0.107 0.347 0.015 0.035 0.227 0.2681 0.112 0.031 0.315 0.376 0.018 0.1486 0.153 0.040 0.277 0.425 0.018 0.0874 0.154 0.040 0.277 0.397 0.019 0.1134 0.159 0.041 0.284 0.386 0.019 0.112

872562

2052 B. Srdjevic, Z. Srdjevic / Applied Soft Computing 13 (2013) 2045–2056

Table 4Consistency measures and selected best priority vectors for the final synthesis (wetland problem).

Decision makers ED MV CR

P1

DM1 5.571 0.17 0.089DM2 5.042 0.06 0.090DM3 7.305 0.11 0.093DM4 19.977 0.14 0.830DM5 6.001 0.00 0.066

Selected vector from: DM5

DM ED MV CR ED MV CR ED MV CR

P2 P3 P4

DM1 3.810 0.06 0.073 2.771 0.08 0.051 1.923 0.08 0.027DM2 5.883 0.03 0.095 2.881 0.11 0.049 3.863 0.08 0.063DM3 4.592 0.00 0.060 3.917 0.03 0.036 4.298 0.00 0.038DM4 37.004 0.03 0.522 24.363 0.06 0.615 18.925 0.08 0.202DM5 5.444 0.00 0.062 13.928 0.00 0.095 9.701 0.00 0.043

Selected vector from: DM3 Selected vector from: DM3 Selected vector from: DM1

P5 P6 P7

DM1 2.725 0.00 0.021 3.254 0.00 0.029 3.255 0.03 0.031DM2 4.801 0.08 0.072 3.315 0.08 0.036 4.390 0.06 0.084DM3 3.731 0.00 0.027 3.376 0.06 0.029 3.359 0.03 0.028DM4 19.564 0.06 0.190 16.452 0.08 0.173 15.982 0.08 0.163DM5 7.010 0.00 0.128 4.738 0.08 0.144 6.018 0.08 0.088

Selected vector from: DM1 Selected vector from: DM1 Selected vector from: DM3

Note: Selected vectors obtained for ˛ED = 0.45, ˛MV = 0.10, and ˛CR = 0.45.

Table 5Final priorities of alternatives obtained for decision makers acting individually, and priorities obtained by the MGPS algorithm (wetland problem).

Alternatives Decision makers/MGPS algorithm

DM1 DM2 DM3 DM4 DM5 MGPS Rank

A1 0.140 5 0.107 6 0.127 4 0.140 3 0.131 4 0.131 5A2 0.128 6 0.118 4 0.079 5 0.075 5 0.194 3 0.123 6A3 0.155 3 0.247 2 0.339 1 0.248 2 0.108 5 0.195 2A4 0.221 1 0.109 5 0.251 2 0.368 1 0.199 2 0.194 3A5 0.211 2 0.120 3 0.065 6 0.044 6 0.269 1 0.207 1A6 0.145 4 0.299 1 0.139 3 0.139 4 0.100 6 0.149 4

TotalED 23.308 30.175 30.577 152.267 55.285 25.771MV 0.060 0.070 0.030 0.080 0.020 0.020CR 0.061 0.078 0.063 0.546 0.070 0.048

N DM1

2 he sa

r˛sm(

p

ote: 1. Local vectors are taken from decision makers DM5, DM3, DM3, DM1, DM1,. Best local priority vectors are identified for ˛ED = 0.45, ˛MV = 0.10, and ˛CR = 0.45. T

Computed consistency measures ED, MV, and CR are summa-ized in Table 8. For assigned weights ˛ED = 0.45, ˛MV = 0.10, andCR = 0.45, the MGPS algorithm indicated that in the final AHPynthesis the local priority vectors should be taken from decision

akers as follows: DM1 (for matrix P1), DM2 (P2), DM1 (P3), DM1

P4), DM1 (P5), DM2 (P6), DM2 (P7), and DM2 (P8).Individual AHP results and the MGPS generated group result are

resented in Table 9. Again, the MGPS produced the final priorities

Selec tion of tWALNU T VA

Color Portion

RASNA

Weight

KRODN I

Taste

MAC VA

Fig. 2. Hierarchy of the walnut selecti

and DM3, in that order which corresponds to matrices P1–P7.me selection of vectors is obtained for ˛ED = 0.00, ˛MV = 0.00, and ˛CR = 1.00.

of alternatives differently from both decision makers. It can benoted that the group-computed ED value is closer to the DM2’sED value, while both the group-computed MV and CR values arecloser to the respective values obtained for the DM1. The com-

pensation effect among the three consistency measures/criteriain the multicriteria part of MGPS is obvious (cf. Table 8), andthe final result is considered better than either individualresult.

he bestRIETY

Shell

SEJNOVO

Storage

FRANKET

Trade

on problem (Srdjevic et al. [34]).

B. Srdjevic, Z. Srdjevic / Applied Soft Computing 13 (2013) 2045–2056 2053

Table 6Priorities of criteria vs. goal for all decision makers (walnut problem).

Criteria Decision makers

DM1 DM2

Color 0.321 0.176Portion 0.180 0.278Weight 0.321 0.142Taste 0.059 0.054Shell 0.053 0.206

5

awaeal(dr

pmd

Table 9Final priorities of alternatives obtained for decision makers acting individually, andpriorities obtained by the MGPS algorithm (walnut problem).

Alternatives Decision makers/MGPS procedure

DM1 DM2 MGPS Rank

A1 0.422 1 0.306 1 0.396 1A2 0.097 5 0.095 5 0.068 5A3 0.239 2 0.208 3 0.242 2A4 0.134 3 0.211 2 0.127 4A5 0.108 4 0.180 4 0.167 3

TotalED 32.885 19.383 22.382MV 0.020 0.090 0.050CR 0.042 0.060 0.040

Note: 1. Local vectors are taken from decision makers DM1, DM2, DM1, DM1, DM1,DM2, DM2 and DM2, in that order which corresponds to matrices P1–P8.

TP

TC

N

Storage 0.023 0.047Trade 0.042 0.098

. Discussion of results

The selected two examples serve to illustrate how the MGPSlgorithm performs in an AHP-group decision making frameworkith (1) a normal-sized group of five decision makers coming from

cademia, management, and engineering, who made fairly differ-nt judgments and demonstrated quite different (in)consistencies,nd (2) with a small group of two decision makers of simi-ar academic background and expertise in the same subject areanut-fruits selection) who made locally different judgments butemonstrated very high consistency and produced similar finalanking of walnut alternatives.

In both examples, group synthesis by the MGPS algorithm

roduced satisfactory results with respect to three consistencyeasures adopted as criteria set for comparing local matrices from

ecision makers as internal decision alternatives in all nodes of a

able 7riorities of alternatives vs. criteria for all decision makers (walnut problem).

Cri/Alt Decision makers

A1 A2 A3 A4 A5

DM1Color 0.449 0.174 0.174 0.030 0.174Portion 0.406 0.038 0.232 0.285 0.038Weight 0.493 0.046 0.312 0.103 0.046Taste 0.273 0.091 0.273 0.091 0.273Shell 0.284 0.061 0.145 0.450 0.061Storage 0.129 0.343 0.129 0.055 0.343Trade 0.279 0.082 0.348 0.210 0.082

able 8onsistency measures and selected local priority vectors for final synthesis (walnut probl

DM ED MV

P1

DM1 8.322 0.04

DM2 6.011 0.10

Selected vector from: DM1

DM ED MV CR ED

P2 P3

DM1 6.231 0.00 0.020 3.060

DM2 2.268 0.04 0.025 2.797

Selected vector from: DM2 Selected vecto

P5 P6

DM1 0.000 0.00 0.000 3.970

DM2 1.944 0.24 0.044 0.978

Selected vector from: DM1 Selected vecto

DM ED MV

P8

DM1 3.286 0.04

DM2 1.087 0.04

Selected vector from DM2

ote: Selected vectors obtained for ˛ED = 0.45, ˛MV = 0.10, and ˛CR = 0.45.

2. Best local priority vectors are identified for ˛ED = 0.45, ˛MV = 0.10, and ˛CR = 0.45.The same selection of vectors is obtained for ˛ED = 0.00, ˛MV = 0.00, and ˛CR = 1.00.

hierarchy. Interesting to note is that when the final weights andranks of alternatives are presented to the decision makers, in bothcases they coincided with the groups’ expectation. Decision mak-ers were not informed about either which matrices were used andwhy, nor which decision makers ‘participated’ more or less in thesynthesis process. In other words, the MGPS results are approved

as correct without any aggregation (such as AIJ or AIP), associatingthe weights to the decision makers within a group, or searching forany consensus during the decision making process.

A1 A2 A3 A4 A5

DM2 0.360 0.093 0.185 0.048 0.315

0.263 0.100 0.234 0.355 0.048 0.524 0.076 0.192 0.134 0.074

0.243 0.130 0.243 0.192 0.192 0.109 0.100 0.179 0.295 0.317

0.307 0.130 0.281 0.141 0.141 0.457 0.070 0.209 0.070 0.194

em).

CR

0.0470.084

MV CR ED MV CR

P4

0.08 0.025 5.919 0.00 0.0550.04 0.039 3.550 0.08 0.077

r from: DM1 Selected vector from: DM1

P7

0.00 0.046 2.098 0.00 0.0120.08 0.010 0.748 0.12 0.004

r from: DM2 Selected vector from: DM2

CR

0.0940.004

2054 B. Srdjevic, Z. Srdjevic / Applied Soft Computing 13 (2013) 2045–2056

Table 10Global consistencies of decision makers and their conformity with the group decision derived by the MGPS algorithm (wetland problem).

Evaluating criteria Decision makers/MGPS algorithm

DM1 DM2 DM3 DM4 DM5 MGPS

ED 23.308 30.175 30.577 152.267 55.285 25.771MV 0.060 0.070 0CR 0.061 0.078 0Conformity (C) 0.089 0.403 0

Table 11Global consistencies of decision makers and their conformity with the group deci-sion derived by the MGPS algorithm (walnut problem).

Evaluating criteria Decision makers/MGPS algorithm

DM1 DM2 MGPS

ED 32.885 19.383 22.382MV 0.020 0.090 0.050

EpdMoa(phca

mas(tvtaDp

TM

TM

CR 0.042 0.060 0.040Conformity (C) 0.124 0.248 –

The results presented in Tables 5 and 9 show that the totaluclidean distance (ED) obtained by MGPS algorithm in both exam-les was close to the corresponding value of the most consistentecision maker(s) while the Consistency ratio (CR) obtained byGPS in both cases was lower, i.e. better, than any individually

btained CR. This effect is a direct consequence of balanced weightsssociated to those two evaluation criteria in the multicriteria parttask 2.4) of the algorithm and their joint high preference of 0.9 vs.reference of 0.1 given to order violation criterion (MV). Generatedigh quality solutions with respect to the two most common usedlassical consistency measures indicate that preferences 0.9 (EDnd CR) and 0.1 (MV) defined to perform the task 2.4 are justified.

Tables 10 and 11 summarize individually obtained consistencyeasures for involved decision makers in two example problems

nd the same consistency measures obtained for virtual MGPS deci-ion makers representing corresponding groups. By using Eq. (19)cf. Remark 5) in the first example problem (wetland), conformity ofhe final priority vectors obtained by five decision makers with theirtual one, Table 11, indicates that it was DM1 who demonstrated

he best overall performance within a group (lowest values of EDnd CR). The second best was DM5 (lowest MV) and the third wasM3 (low CR and MV). Recall that in generating the final MGPS’sriority vector, matrices are taken from the decision makers in

able 12GPS selection of local priority vectors if single consistency measures are used (wetland

Criteria weights Selected local priority vectors for the final MGPS sy

ED MV CR

˛1 ˛2 ˛3 P1 P2 P3 P4

1.00 0.00 0.00 DM2 DM1 DM1 DM1

0.00 1.00 0.00 DM5 DM3 or DM5 DM5 DM3 o0.00 0.00 1.00 DM5 DM3 DM3 DM1

0.45 0.10 0.45 DM5 DM3 DM3 DM1

able 13GPS selection of local priority vectors if single consistency measures are used (walnut p

Criteria weights Selected local priority vectors for the fina

ED MV CR

˛1 ˛2 ˛3 P1 P2 P3

1.00 0.00 0.00 DM2 DM2 DM2

0.00 1.00 0.00 DM1 DM1 DM2

0.00 0.00 1.00 DM1 DM1 DM1

0.45 0.10 0.45 DM1 DM2 DM1

.030 0.080 0.020 0.020

.063 0.546 0.070 0.048

.401 0.457 0.274 –

order: DM5 (priority vector for matrix P1), DM3 (P2), DM3 (P3),DM1 (P4), DM1 (P5), DM1 (P6), and DM3 (P1). Decision maker DM1participated with three local vectors in the final synthesis, DM3with 2, and DM5 with one. Obviously those three decision mak-ers performed best in the group (in terms of consistency) and thealgorithm ‘selected’ them to participate in generating the best finalsolution for the group.

Similar analysis of the final priority vectors obtained in thesecond example problem (walnuts) shows that DM1 is better con-formed than DM2 with the virtual decision maker (0.124 vs. 0.248,Table 11). Recalling how the final vector is obtained by MGPS, thatis: DM1 (for matrix P1), DM2 (P2), DM1 (P3), DM1 (P4), DM1 (P5),DM2 (P6), DM2 (P7), and DM2 (P8), balanced participation of bothdecision makers in the final synthesis is a logical outcome becausethey demonstrated satisfactory consistency at all instances of thedecision masking process (cf. Table 11).

The MGPS algorithm also permits performing simplified assess-ments of local priority vectors derived for involved decision makersby applying any single consistency criterion. Tables 12 and 13summarize what would happen if just one consistency measureis used to identify local matrices in two presented examples.Brief inspection of the consistency data in Tables 4 and 8 indi-cates different orderings of decision makers (i.e. their involvementwith associated priority vectors) as presented in the first threerows of Tables 12 and 13. All orderings are different from thoseobtained as a result of multicriteria assessment (last row in bothTables 12 and 13). From a methodological point of view, orderingand synthesis by MGPS is only fully justified while single criterionorderings (and corresponding final vectors) may serve only as aMGPS’s performance control in its full implementation, as well as

a possible aggregation scheme based on propagation of the single-criterion best local vectors.

The results presented in Tables 12 and 13 indicate that differ-ences in the consistency of decision makers influence the selection

problem).

nthesis

Matrices

P5 P6 P7

DM1 DM1 DM1r DM5 DM1 or DM3 or DM5 DM1 DM1 or DM3

DM1 DM1 or DM3 DM3DM1 DM1 DM3

roblem).

l MGPS synthesis

Matrices

P4 P5 P6 P7 P8

DM2 DM1 DM2 DM2 DM2DM1 DM1 DM1 DM1 DM1 or DM2DM1 DM1 DM2 DM2 DM2DM1 DM1 DM2 DM2 DM2

Soft Co

ootfisbC(

ttfe

6

cviticoatapdcrtnfioav

fmocatwgttiiatppbur

coietds

B. Srdjevic, Z. Srdjevic / Applied

f corresponding local priority vectors and therefore the final pri-ritization of alternatives in the hierarchy. For example, althoughhere are opinions that ED and CR correlate highly, one may easilynd from Tables 12 and 13 that the selection of vectors can be a sen-itive decision across the whole hierarchy (cf. rows 1 and 3). Fromoth tables, it is easy to see that the MGPS selection is similar to aR single criterion selection if ED and CR receive the same weightscf. rows 3 and 4 in both tables).

Table 12 indicates that decision maker DM1 conforms better tohe AHP-group solution obtained by MGPS, while Table 13 showshat the single-criterion selection of local priority vectors differsrom the three-criterion selection, and in this way reproduces theffect obtained in the first example.

. Conclusions and agenda for future research

In this paper, we present a novel approach to AHP-group appli-ations based on the synthesis of individually optimal local priorityectors obtained by participating decision makers. The approachs called the MGPS (group-related) algorithm and is analogouso the MPS (prioritization method-related) algorithm developedn the authors’ earlier work. In every node of a given hierar-hy, decision makers perform judgments independently from eachther and create multiplicative preference relations, known alsos judgment or pairwise comparison matrices. After a prioritiza-ion method is applied to derive weights from these matrices forll decision makers (members of a group), a multicriteria analysis iserformed to identify the best priority vector across the group. Theecision matrix for multicriteria analysis consists of individuallyomputed priority vectors as alternatives, and a criteria set rep-esented by consistency measures that are applicable when ratinghe performance of individual priority vectors. The identified bestode-dependent (local) priority vectors are used in the standardnal AHP synthesis. This way, the final result is obtained in anbjective way, avoiding the application of any consensus model orssociating weights to the decision makers that can be subjected toarious manipulations that can be both positive and negative.

Our approach is based on use of the eigenvector (EV) methodor local prioritization, and we selected three relevant consistency

easures that go with the EV method for multicriteria analysisf individual priority vectors obtained in all nodes of a hierar-hy. Multicriteria analysis is performed locally for one node at

time. Regarding the adopted consistency measures, recall thathe consistency ratio (CR) applies to the eigenvector method only,hile Euclidean distance (ED) and minimum violation (MV) are

eneral consistency measures applicable to whichever prioritiza-ion method is used. Future research agenda will include the otherwo well-known consistency measures: (a) geometric consistencyndex (GCI) applicable if the logarithmic least squares (LLS) methods used for prioritization, or (b) fuzzy consistency measure (�) thatpplies to the fuzzy preference programming (FPP) method. Notehat these two measures cannot be used when EV prioritization iserformed, as is the case in our approach; when prioritizations areerformed with either method, LLS or FPP, in such cases CR shoulde replaced accordingly with GCI or �, both computed as global val-es for the whole hierarchy; the other two measures, ED and MV,emain eligible evaluation criteria in both cases.

The issue of consistency measures is a hot topic in sophisti-ated research environments. Further research will probably relyn recent works such as Wang et al. [35] and the presentedntuitionistic fuzzy AHP (IF-AHP) approach which synthesizes the

igenvectors of an intuitionistic fuzzy comparison matrix and athe same time handles consistency and satisfactory consistencyefined after some basal knowledge is introduced. IF-AHP has beenhown successful in combination with the extend analysis method

mputing 13 (2013) 2045–2056 2055

and modified fuzzy logarithmic least squares method. Moreover, inFallahi et al. [36] there is an interesting application of ant colonyoptimization (ACO) which is used to obtain optimal solutions sat-isfying some path planning criteria. Fuzzy AHP is then employedin the decision making judgments due to their inherent vaguenessand uncertainty. Finally, a bi-criteria evolution strategy has beenused in Srdjevic and Srdjevic [37] to identify priority vectors froma given matrix.

There are other works in the subject area and discussion onthe tocic can be considered as still open. In general, various con-sistency measures can be used as criteria when evaluating localpriority vectors derived by participating decision makers. Some ofthe measures can be highly correlated (e.g. ED and CR), and this issuedeserves additional research. Also, different multicriteria methodscan be used for evaluating local vectors, such as the additive weight-ing method (used in MGPS) and the product weighting method,or, in the case of many decision makers, the ideal-point methodsTOPSIS or CP. In all cases, it is necessary to allocate weights toselected consistency measures (criteria), and this is subject to ananalyst’s decision. If the number of decision makers is high (say,more than ten), the entropy method can be additionally employedto assess the decision matrix, recognize the objective importanceof the consistency measures, and in turn generate the weights formulticriteria analysis.

Finally, in the presented approach, we commented on the resultswhen a single consistency measure (criterion) is used for the eval-uation of local vectors. This is also a possible direction for futureresearch and a challenge to combine our approach with goal pro-gramming optimization aimed at identifying target (benchmark)points in group syntheses based on consensus models.

The algorithm itself is independent of a prioritization method,but the same method should be used at all nodes and for alldecision makers. In fact, this is no limitation of the MGPS algo-rithm. Either ‘classical’ or ‘evolution based’ prioritization methodis applicable uniquely over the whole hierarchy. Related consis-tency measures should also be used uniquely in regards to whichprioritization method is used (e.g. CR with EV, and GCI with LLS).General consistency measures such as Euclidean distance or a rankreversal indicator are directly applicable with the MGPS algorithmregardless of the prioritization method used. The next step in devel-opment could be letting the algorithm identify at each node of ahierarchy the best vector possible by computing priority vectorsof all individual matrices with all approved prioritization methodsand identify the one with best consistency indicators.

Acknowledgments

Authors acknowledge the financial support from the Ministry ofEducation and Science of Serbia under the Fundamental scientificresearch program in Mathematics, Computer Science and Mechan-ics; Grant No. 174003 (2011–2014): Theory and application of theanalytic hierarchy process (AHP) in multicriteria decision makingunder conditions of risk and uncertainty (individual and group con-text). Authors are also grateful to Lorrie Carter (USA) for proofreading the manuscript.

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