Int. J. Production Economics 128 (2010) 248–260

Contents lists available at ScienceDirect

Int. J. Production Economics

0925-52

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ijpe

A new methodology for multi-echelon inventory management in stochasticand neuro-fuzzy environments

Alev Taskin Gumus a, Ali Fuat Guneri a, Fusun Ulengin b,n

a Department of Industrial Engineering, Yildiz Technical University, Istanbul, Turkeyb Department of Industrial Engineering, Dogus University, Istanbul, Turkey

a r t i c l e i n f o

Article history:

Received 2 April 2008

Accepted 28 June 2010Available online 17 July 2010

Keywords:

Multi-echelon inventory management

Stochastic cost model

Neuro-fuzzy approximation

73/$ - see front matter & 2010 Elsevier B.V. A

016/j.ijpe.2010.06.019

esponding author. Tel.: +902165445555; fax

ail address: fulengin@dogus.edu.tr (F. Ulengin

a b s t r a c t

Managing inventory in a multi-echelon supply chain is considerably more difficult than managing it in a

single-echelon one. A strategy that optimizes inventory one echelon at a time results in excess

inventory without necessarily improving service to customer. In this paper, a methodology for effective

multi-echelon inventory management is proposed. Subsequently; a neural network simulation of the

model is then presented with the support of neuro-fuzzy demand and lead time forecasting, and finally

its performance is calculated using performance metrics selected from the SCOR model. The results

show that, the inventory is efficiently deployed and uses realistic breakdowns. The proposed

methodology aims to provide an important tool for the management of general N-echelon tree-

structured supply chains that overcomes some of the deficiencies of competing methodologies.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

A manufacturing supply chain is a network of suppliers,factories, subcontractors, warehouses, distribution centers, andretailers, through which raw materials are acquired, transformed,produced, and delivered to the end customers. Such a supplychain network must meet customers’ service goals at specifiedservice levels at the lowest possible cost (Bhaskaran and Leung,1997; Tsiakis et al., 2001; Taskin and Guneri, 2005).

Supply chain inventory management (SCIM) is an integratedapproach to the planning and control of inventory throughout theentire network of cooperating organizations from the source ofsupply to the end user. SCIM goals are to improve customerservice, increase product variety, and decrease costs (Giannoccaroet al., 2003). Fig. 1 shows a multi-echelon system consisting of anumber of suppliers, plants, warehouses, distribution centers andcustomers (Axsater, 1990; Andersson and Melchiors, 2001;Axsater, 2003).

An important issue in supply chains (SCs) is the need tomake decisions in the face of uncertainty. In addition to demandand lead time fluctuations for a product, the uncertainty iscompounded by information delays associated with the manu-facturing and distribution processes that characterize SCs(Giannoccaro et al., 2003). To incorporate uncertainty in a supplychain model a suitable means of representing it must be found(Gupta and Maranas, 2003). Three methods are frequently used

ll rights reserved.

: +902163279631.

).

for representing uncertainty (Gupta and Maranas, 2003; Hameriand Paatela, 2005): First, the uncertain parameters are assumed tohave normal distribution with a specified mean and standarddeviation; second, the forecast parameters are described as fuzzynumbers defined by their degree of membership; and third,expected outcomes are captured by several discrete scenarios andtheir associated probability levels (Chen and Lee, 2004).

Examples of all three approaches can be found in the literature(Taskin Gumus and Guneri, 2007). For example, Gupta andMaranas (2000), Gupta et al. (2000) incorporate uncertaindemand in their supply chain model via a normal distributionfunction, and propose a two-stage solution to the restockingproblem. They have recently published a generalization of theirmodel that can handle multi-period and multi-customer problem(Gupta and Maranas, 2003). Tsiakis et al. (2001) use the scenarioplanning approach to describe demand uncertainties. Petrovicet al. (1999) use fuzzy sets to handle uncertainties in customerdemand and the supply of raw materials for manufacturingproducts. Giannoccaro et al. (2003) also apply fuzzy set theory tomodel the uncertainties associated with both market demand andinventory costs.

In this paper, Mitra and Chatterjee’s (2004) deterministic andstochastic models of supply chain inventories are improved by theaddition of neuro-fuzzy demand and lead time forecasting, theincorporation of variable costs and generalization to N-echelons.The goal of the work is to improve supply chain performance byminimizing uncertainty.

Mitra and Chatterjee (2004) examine De Bodt and Graves’model (1985), that they developed this model in their papernamely ‘‘Continuous-review policies for a multi-echelon inventory

Nomenclature

Ki fixed ordering cost at echelon i (includes transporta-tion cost)

Ai variable cost at echelon i

Hi installation holding cost at echelon i, which is the costof keeping a stock unit per time unit at that echelon

hi echelon holding cost at echelon i, which is theincremental cost of keeping a stock unit at a givenechelon rather than at the upstream one

pi expediting cost at echelon i

b backorder cost at echelon i (same for all retailersbecause of identity)

li lead time at ith echelonRi reorder point at ith echelonki safety factor at ith echelon (stock surplus kept to

hedge against demand fluctuations)Di demand per unit time at ith echelons standard deviation of demand per unit timeQi order quantity at echelon i

IPi echelon inventory position at echelon i

BO expected backorders at echelon 1BO0 transferred backorders from preliminary cycleN cycle number for retailers, integerX number of retailers at 1st echelonyi number of installations in the ith echelon

A.T. Gumus et al. / Int. J. Production Economics 128 (2010) 248–260 249

problem with stochastic demand’’, for fast moving items from theimplementation point of view. The proposed modification of themodel leads to a reduction of the expected total cost of the systemunder certain conditions. They assume that the end-item demand isnormally distributed and lead time is deterministic. Also their modelhas a restriction of being appropriate for only two echelon supplychains. Their model can be extended to multi-stage serial and two-echelon assembly systems (Taskin Gumus and Guneri, 2007).

2. Literature on multi-echelon inventory management

The analysis of the multi-echelon inventory systems thatpervade the business world has a long history (Chiang andMonahan, 2005; Routroy and Kodali, 2005), extending back to thedevelopment of the economic order quantity (EOQ) formula byHarris in 1913. Research on multi-echelon supply chain modelshas gained importance over the last decade mainly becausemodern information technology has made it feasible to studythem (Rau et al., 2003; Diks and de Kok, 1998; Kalchschmidt et al.,2003). Clark and Scarf (1960) were the first to study a two-echelon inventory model (Diks and de Kok, 1998; Bollapragadaet al., 1998; van der Vorst et al., 2000; Tee and Rossetti, 2002; Rauet al., 2003; Dong and Lee, 2003; Chiang and Monahan, 2005).Bessler and Veinott (1965) extended the Clark and Scarf (1960)model to include general tree structures. Eppen and Schrage(1981) analyzed a model with a stockless central depot thatimmediately shipped orders to the warehouses (van der Heijden,1999). Sherbrooke (1968) constructed the multi-echelon techni-que for recoverable item control (METRIC) model, which identifies

Suppliers Plants Warehous

Fig. 1. A multi-echelon

the stock levels that minimize the expected number of backordersat lower echelons, subject to a budget constraint. For detailedliterature review of multi-echelon models please see TaskinGumus and Guneri (2007).

Mitra’s (2009) work that we are extending relaxes severalassumptions available in literature on multi-echelon systems withreturns base, as non-existence or non-relevance of set-up andholding costs at different levels. A two-echelon system with returnsunder more generalized conditions is considered. A deterministicmodel as well as a stochastic model under continuous review forthe supply system with returns is developed.

In the papers cited above and also in Mitra’s studies (Mitra,2009; Mitra and Chatterjee, 2004), demand and lead times areassumed to have probabilistic distribution functions, productcosts are assumed to be fixed and a two-echelon serial system isanalyzed generally. The basic difference between our model andthe others seen in the literature is that our model is generalized toN-echelons. Additionally, the variable and expediting costs areincluded in cost computations, and a neuro-fuzzy forecasting isused in order to treat demand and lead time uncertainties in amore realistic way. We analyze a three-echelon system as anillustrative example. The basic differences and contributions ofthe proposed methodology can be seen in Table 1.

3. The proposed methodology and related models

This section describes our methodology for developing totalcost models for supply chains. The technologies our method-ology exploits, such as artificial neural networks, neuro-fuzzy

es Distribution Centers Customers

inventory system.

Table 1Literature review.

Author, year Research technique Number of echelons Inventory system/policy

Demand assumption Lead time assumption Exact/approximatesolution

Objective

Bollapragada

et al.’s study

(1998)

Mathematic modeling and

simulation

2—A single depot which

supplies several

warehouses

Base stock policy-

Optimal allocation

policy at the depot

Stochastic Fixed Approximate To generalize the earlier work by Eppen and Schrage,

to allow for non-identical warehouses

Diks and de

Kok (1998)

Mathematic modeling Divergent N echelon Periodic review

order-up-to policy

i.i.d.a Fixed Exact To minimize the expected holding and penalty costs

per period

van der Vorst

et al. (2000)

Discrete event simulation,

Petri net modeling, scenario

analysis

3—A producer, a

distribution centre and 2

retailer outlets

Order policy based

on forecasts

Derived from data

recorded by the point-

of-sale systems in

practice for 20

consecutive weeks

Deterministic Approximate To present a method for modeling the dynamic

behaviour of food supply chains and evaluate

alternative designs of the supply chain by applying

discrete-event simulation

van der

Heijden

(1999)

Mathematic modeling and

simulation

2—A central depot and

multiple (nonidentical)

local warehouses

Order-up-to (R, S)

policy

Stochastic and

stationary in time

Constant and

deterministic

Approximate To present a computational method to derive the

control parameters in a two echelon distribution

system with different shipment frequencies at both

levels

Tee and

Rossetti

(2002)

Simulation 2—one-warehouse,

multiple retailer system

(R,Q) inventory

policies

Stochastic-non-

stationary Poisson

demand process

One day for all

situations

Approximate To examine the robustness of a standard model of

multi-echelon inventory systems

Rau et al.

(2003)

Mathematic modeling 3—Single supplier, single

producer and single

retailer

Not specified Demand rate is

deterministic and

constant

Negligible Exact To develop a multi-echelon inventory model for a

deteriorating item and to derive an optimal joint total

cost from an integrated perspective among the

supplier, the producer, and the buyer

Kalchschmidt

et al. (2003)

An algorithmic solution is

provided through

probabilistic forecasting and

inventory management

1 and 2—Central

warehouse serves on one

side, a one-echelon chain

and, a two-echelon

supply chain

Order-up-to-policy Stochastic-variable and

lumpy

Not specified Approximate To describe an integrated system for managing

inventories in a multi-echelon spare parts supply

chain, in which customers of different size lay at the

same level of the supply chain

Dong and Lee

(2003)

Mathematic modeling M-echelon serial periodic

review inventory system

and 3 echelons for

numerical example

An echelon base-

stock inventory

policy, order-up-to

S policy

An autoregressive

demand model;

assumed to be i.i.d.

Variable to see the

impact of lead times

and autocorrelation on

the performance of the

system

Approximate To extend the approximation to the time correlated

demand process of Clark and Scarf (1960)

Mitra and

Chatterjee

(2004)

Mathematic modeling 2—Stage 1 is facing

demand and is supplied

by stage 2, which in turn

is supplied by an outside

source

Continuous review

(R,Q) system under

nested and echelon

stock based policy

Stochastic Joint and deterministic Approximate To examine De Bodt and Graves’ model (1985), that

they developed this model in their paper namely

‘‘Continuous-review policies for a multi-echelon

inventory problem with stochastic demand’’, and

suggest a modification

Routroy and

Kodali

(2005)

Mathematic modeling and

differential evolution

algorithm

3—A retailer, a

warehouse and a

manufacturer

A continuous

review policy (Q, r)

Stochastic-normally

distributed

Constant Approximate To minimize the total system wide cost i.e. supply

chain inventory capital, supply chain ordering/set-up

cost and supply chain inventory stock out cost

Mitra (2009) Mathematic modeling,

simulation

2—A depot and a

distributor (with returns)

Echelon-stock-

based continuous

review policy

Stationary and

uniformly occurring

Deterministic Approximate To relax the assumptions in the literature and

consider a two echelon system with returns under

more generalized conditions

Our methodo-

logy

Mathematic modeling,

neuro-fuzzy forecasting,

artificial neural network

simulation, SCOR

performance evaluation

N-echelon model and 3

echelons for numerical

example

Echelon-stock-

based continuous

review policy

Neuro-fuzzy forecast Neuro-fuzzy forecast Approximate To propose a methodology for effective multi-echelon

inventory management using a neural network

simulation of its application to a supply chain with

the support of neuro-fuzzy demand and lead time

forecasting, and an effective performance evaluation

via SCOR model.

a i.i.d.¼ independent and identically distributed.

A.T

.G

um

us

eta

l./

Int.

J.P

rod

uctio

nE

con

om

ics1

28

(20

10

)2

48

–2

60

25

0

A.T. Gumus et al. / Int. J. Production Economics 128 (2010) 248–260 251

integration and the SCOR model for evaluating supply chainperformance are explained and their algorithms are given.

3.1. The methodology

The methodology developed here attempts to optimize an N-echelon supply system, where all echelons contain one or moresites, or installations. Each echelon is modeled as a stocking pointthat feeds lower echelons and is fed by upper echelons. It isassumed that the last echelon (the central depot) is fed withlimitless stock. If the demand at the retailers rises above athreshold value, the additional product units are assumed to bebackordered. The market demand at echelon 1 (the retailers) andthe supply lead times between echelons are uncertain, and arecalculated by neuro-fuzzy computation.

The first step of the methodology is to develop deterministic(Model-G1) and stochastic (Model-G2) models of the supplychain. Model-G2 includes Model-G1 and the purpose of Model-G1is to calculate order quantities and cycle numbers that are used asinputs to Model-G2. Demand and lead times for Model-G2 arederived by neuro-fuzzy calculations. Finally some input data forModel-G2 may be obtained by the simple expedient of askingreal-world supply chain experts. Model-G2 contains neuro-fuzzyforecasting; it cannot be simulated with a classical simulationpackage without incorporating simplifying assumptions. For thisreason we simulated Model-G2 by means of artificial neuralnetworks (ANN). Then the model’s performance was evaluatedusing performance criteria derived from supply chain operationsreference (SCOR) model. Finally, we performed sensitivity analysison the simulation to investigate how the changes in demand andlead time affected the model output.

The aim of our methodology is to ensure more effective costand inventory computations and arrangement system for supplychains via the integration of deterministic and stochastic models.For this purpose, neuro-fuzzy and neural network techniques areused for forecasting of uncertain variables and model simulation.The performance of the proposed methodology is also evaluatedvia SCOR model.

According to the proposed methodology Model-G1 and G2 aresimulated via ANNs. Then, in order to evaluate the simulation

Determination of demand and lead time variables of the models by Neuro-

Fuzzy computations

Simulation of the proby Neural Networks a

of the total supply

Performance evaluatiomodels by SCO

Gaining the methodolmaking sensitivit

Development of MModel-G

Fig. 2. The steps of the de

results on the hypothetical supply chain example, severalperformance metrics are proposed and the supply chain perfor-mance is measured via SCOR model according to these metrics.The results obtained before and after the use of the proposedmethodology are compared.

These steps of the proposed methodology mentioned abovecan be seen in Fig. 2.

3.2. The proposed models

Here we present the mathematical models that make up oursupply chain model. The order quantity and cycle number valuesare calculated by a deterministic model (Model-G1), while thedemand and lead times are forecasted based on neuro-fuzzycalculations. Subsequently, all the generated data are used asinputs to a stochastic model (Model-G2).

Our objects are to determine customer demand and lead timesin a more realistic way and to derive policy variables thatminimize the total cost of the system. The cost components of thesystem are fixed ordering costs, variable costs, inventory holdingcosts and the costs of expediting delivery of product to the nextlevel of the echelon in the case of a shortfall. The mathematicalnotation we use in the models is defined below:

3.2.1. The deterministic-neuro-fuzzy model: model-G1

Model-G1 assumes that all elements in the same echelon areidentical (retailers with one another, distributors with oneanother, etc.). The model consists of transformations amongvariables that together compute the total cost (TC) of the supplychain:

TC ¼ xK1D

Qþ

Xn

i ¼ 2

yiKi

nQ

!þ

Q

2

h1A1

xþXn

i ¼ 2

nhiAi

yi

!ð1Þ

Q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2D xK1þ

Pni ¼ 2ðyiKi=nÞ

� �A1h1þ

Pni ¼ 2 nAihi

sð2Þ

n¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni ¼ 2 yiKi

� �ðA1h1Þ

xK1Pn

i ¼ 2 Aihi

� �s

ð3Þ

Determination of other variables of the models by mathematical calculations and/or learning from the SC members

posed models nd calculation chain cost

n of proposed R Model

ogy results and y analysis

odel-G1 and 2.

veloped methodology.

Table 2The variable relations and transformations.

Order quantity Demand Holding cost Fixed ordering cost Flexible cost Lead time

Q11¼Q12¼Q13¼Q1 D11¼D12¼D13¼D1 H01¼H02¼H03¼H0 K11¼K12¼K13¼K1 A11¼A12¼A13¼A1 l11¼ l12¼ l13¼ l21¼ l22¼ l3¼ l

Q21¼Q22¼Q2 D21¼D22¼D2 H11¼H12¼H13¼H1 K21¼K22¼K2 A21¼A22¼A

Q2¼3Q1/2 D2¼3D1/2 H21¼H22¼H2

3Q1¼2Q2¼Q3¼Q 3D1¼2D2¼D3¼D h1¼H0�2H2, h2¼2H2�H3, h3¼H3

H0¼3(bH1/(b+H1))

A.T. Gumus et al. / Int. J. Production Economics 128 (2010) 248–260252

In Table 2 we give the variable relations and transformationsfor a three-echelon supply chain that has three retailers, twodistributors and one depot. Customer demand is assumed to beequal for all retailers. The total demand from echelon 1 istransmitted to the distributors at echelon 2 and divided equallybetween them. Then the total demand goes to the depot. Orderquantity (Q) is handled similarly.

In this model, customer demand (D1) is determined by neuro-fuzzy calculations. Each retailer follows a stationary orderingpolicy, where stock is ordered in cycles, and therefore the modelcan only handle integer values of n, the cycle number.

In the model, lead times (l) between the echelons, like thedemand, are derived from neuro-fuzzy calculations. The prepara-tion time at each unit time for each retailer at echelon 1 is D/Q

(D1/Q1¼(D/3)/(Q/3)). For each distributor at echelon 2 and depotat echelon 3 it is D/nQ. Mean echelon inventories for echelon 1, 2and 3 are Q1/2 (for each retailer), nQ2/2 (for each distributor) andnQ/2(for the depot). Given these transformations, the total cost(TC), order quantity (Q) and cycle number (n) for a three-echelonsupply chain can be calculated as follows:

TC ¼D

Q3K1þ

2K2

nþ

K3

n

� �þ

Q

2

H0A1

3þH2 nA2�

2

3A1

� �þnH3 A3�

A2

2

� �� �ð4Þ

Q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Dð3K1þ2K2=nþK3=nÞ

A1H0þ2ðnA2�A1ÞH2þnðA3�A2ÞH3

sð5Þ

n¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2K2þK3ÞðA1H0�2A1H2Þ

3K1ð2A2H2þðA3�A2ÞH3Þ

sð6Þ

If n is not integer, then TC(n�) and TC(n+) are calculated, and(n�) or (n+) is considered for subsequent calculations that makeTC(.) minimum. Since TC is a convex function in Q and n, thiswould ensure optimality of the solution (Mitra, 2009).

3.2.2. The stochastic-neuro-fuzzy model: model-G2

In this stochastic model the standard deviation (s) of demandis based on a neuro-fuzzy computation and there is a safety factor(k). k is the safety factor to achieve a desired service level (e.g.k¼1.64 for 95% service level). The required safety factor dependson the variability of demand and of lead times.

Model-G2, which calculates the expected total cost (ETC), canbe written as

ETC ¼ xK1D

Qþ

Xn

i ¼ 2

yiKi

nQ

!þ

Q

2

h1A1

xþXn

i ¼ 2

nhiAi

yi

!" #

þ xn�1

nk1sx

ffiffiffiffil1

p� �þ

1

nk1

Xn

i ¼ 2

syi

ffiffiffili

p !H1A1

"

þ

ffiffiffiffiffiffiffinQ

D

r Xn

i ¼ 2

kisHiAi

!#þ fðk1Þ�k1

þk1Fðk1Þðxp1

D

Q

n�1

n

sx

ffiffiffiffil1

pþ

1

n

Xn

i ¼ 2

syi

ffiffiffili

p !" # !

þD

nQ

ffiffiffiffiffiffiffinQ

D

r Xn

i ¼ 2

pisffðkiÞ�kiþkiFðkiÞg

!ð7Þ

To clarify the structure of the model, we adapted it for a three-echelon supply chain consisting of three retailers. For this exercisewe assumed all echelons followed a continuous review policy. Thepolicy variables the model incorporates are the reordering point(R1) and fixed order quantity (Q1) for the retailers at echelon 1,and reordering points based on echelon stock (R2) and (R3) forechelon 2 and 3, respectively. The reordering point reflects thesafety factor needed to reach the service goal as well as the fixedorders for that echelon.

The echelon inventory position (IP2) at echelon 2 includes on-hand stock at echelon 2, on-hand stock at echelon 1, andbackorders. In De Bodt and Graves’ model (1985), the safety stockwas distributed between two stages. Mitra and Chatterjee (2004)suggested a modification to De Bodt and Graves’ model allowingsafety stock only at echelon 1 (Mitra, 2009). We follow Mitra andChatterjee’s model, keeping the safety stock only at echelon 1.When the inventory position at echelon 2 reaches R2, an order issent to the depot, which in its turn, sends an order to an externalsupplier. After replenishment, if echelons 2 and 3 have less thannQ stock on hand, the shortfall is made up by expedited deliveryof additional stock. If the stock at echelon 1 is insufficient to fillcustomer orders, the shortfall is considered to be backordered.

The expressions for R1, R2 and R3 are given by Eqs. (8)–(10). Tocalculate the echelon inventory positions IP1, IP2 and IP3, weassumed the backorder quantity of each retailer was 0.5% ofdemand. Then, IP1, IP2 and IP3 are defined by Eqs. (11)–(13):

R1 ¼1

3

l

30Dþk1s

ffiffiffiffiffiffil

30

r !ð8Þ

R2 ¼1

2

2l

15Dþk2s

ffiffiffiffiffiffil

15

r !ð9Þ

R3 ¼l

10Dþk3s

ffiffiffiffiffiffil

10

rð10Þ

IP1 ¼ 3Q

3þSS1�BO

� �þBO0 ð11Þ

IP2 ¼ 2 nQ

2þSS2

� �þ IP1 ð12Þ

IP3 ¼ nQþSS3þ IP2 ð13Þ

The next step is to calculate the expected total cost (ETC) foreach unit time. Expected total cost includes the expected fixedordering cost, the expected total variable cost, the expected totalaverage cycle stock cost, the expected total safety stock cost andthe expected total expediting cost. The sum of the first threevalues is as the TC given in Model-G1. The safety stock (SS)and expected shortage (ESPRC) can be calculated with theexpressions given below, where k2 and k3 are safety factors forechelons 2 and 3.

A.T. Gumus et al. / Int. J. Production Economics 128 (2010) 248–260 253

Safety stock (SS):For each retailer at echelon 1:

SS1 ¼k1s

n

n�1

3

ffiffiffiffiffiffil

30

rþ

1

2

ffiffiffiffiffiffil

15

rþ

ffiffiffiffiffiffil

10

r !ð14Þ

For each distributor at echelon 2:

SS2 ¼1

2k2s

ffiffiffiffiffiffiffinQ

D

rð15Þ

For the depot at echelon 3:

SS3 ¼ k3sffiffiffiffiffiffiffinQ

D

rð16Þ

Expected shortage per replenishment cycle (ESPRC):

For each retailer:

ESPRC1 ¼n�1

3ns

ffiffiffiffiffiffil

30

rþ

1

2ns

ffiffiffiffiffiffil

15

rþ

1

ns

ffiffiffiffiffiffil

10

r" #fðk1Þ�k1þk1Fðk1Þ

ð17Þ

For each distributor:

ESPRC2 ¼s2

ffiffiffiffiffiffiffinQ

D

rjðk2Þ�k2þk2Fðk2Þ

ð18Þ

For the depot:

ESPRC3 ¼ sffiffiffiffiffiffiffinQ

D

rjðk3Þ�k3þk3Fðk3Þ

ð19Þ

The optimal values of k1, k2 and k3 can be obtained fromEqs. (20)–(22), respectively, where f( � ) and Fð:Þ are theprobability density function (pdf) and cumulative density function(cdf) of standard normal distribution:

1�Fðk1Þ ¼QH1A1

p1Dð20Þ

1�Fðk2Þ ¼nQH2A2

p2Dð21Þ

1�Fðk3Þ ¼nQH3A3

p3Dð22Þ

Then, for a three-echelon SC, the equation for ETC can beprinted as below:

ETC ¼ 3K1D

Qþ2

K2D

nQþ

K3D

nQþ

Q

2ðH0�2H2ÞA1

�

þnQ

2ð2H2�H3ÞA2þ

nQ

2H3A3

�

þ 3n�1

3nk1s

ffiffiffiffiffiffil

30

rþ

1

2nk1s

ffiffiffiffiffiffil

15

rþ

1

nk1s

ffiffiffiffiffiffil

10

r !H1A1

"

þk2sffiffiffiffiffiffiffinQ

D

rH2A2þk3s

ffiffiffiffiffiffiffinQ

D

rH3A3

#

þ3p1D

Qjðk1Þ�k1þk1Fðk1Þ

�n�1

3ns

ffiffiffiffiffiffil

30

rþ

1

2ns

ffiffiffiffiffiffil

15

rþ

1

ns

ffiffiffiffiffiffil

10

r" #

þp2D

nQs

ffiffiffiffiffiffiffinQ

D

rjðk2Þ�k2þk2Fðk2Þ

þp3D

nQs

ffiffiffiffiffiffiffinQ

D

rjðk3Þ�k3þk3Fðk3Þ

ð23Þ

Obtaining optimal values of Q, n, k1, k2 and k3 requires theiterative solution of five simultaneous equations. For the sake ofsimplicity, the optimal values of Q and n can alternatively beobtained from the deterministic model (Eqs. (5) and (6)), andoptimal k1, k2 and k3 from Eqs. (20)–(22). Using optimal values ofQ and n from a deterministic model is a common approach to the

solution of forward logistic models, and is known to work wellthere. The validity of this approximation will be tested in the nextsection on numerical experimentation.

3.3. Neuro-fuzzy integration

This section of the paper describes adaptive-network-basedfuzzy-inference system (ANFIS), the neuro-fuzzy model we usedto forecast demand and lead time. ANFIS is a neural networkrepresentation of a fuzzy system that uses neural networklearning algorithms and fuzzy reasoning to map an input spaceto an output space (Jang, 1993; Escoda et al., 1997). Hayashi andBuckley (1994) proved that any rule-based fuzzy system may beapproximated by a neural net and any feedforward multilayeredneural net may be approximated by a rule-based fuzzy system(Mitra and Hayashi, 2000).

ANFIS is a frequently used neuro-fuzzy system with provenability to model different processes. It infers fuzzy rules fromnumerical data or expert knowledge, and can restructure a rulebase adaptively (Chang and Chang, 2006).

For simplicity, we assume the fuzzy inference system underconsideration has two inputs, x and y, and one output, z.According to Takagi and Sugeno (1985), a rule set with two fuzzyif-then rules can be expressed as

Rule 1. If xA1 and yB1, then f1¼p1� x+q1� y+r1

Rule 2. If xA2 and yB2, then f2¼p2� x+q2� y+r2

where pi, qi and ri (i¼1 or 2) are linear parameters in theconsequent part of the rules. The ANFIS architecture consists offive layers (Jang, 1993; Escoda et al., 1997; Esen et al., 2008).

Layer 1: input nodes. Each node of this layer generates a degree-of-membership score of each input in the appropriate fuzzy sets

O1,i ¼ mAiðxÞ for i¼ 1,2 ð24Þ

O1,i ¼ mBi�2ðyÞ for i¼ 3,4 ð25Þ

where x, y are crisp inputs to node i, and Ai, Bi are the linguisticlabels (small, large, etc.) characterized by membership functionsmAi

and mBi.

Layer 2: rule nodes. In the second layer, the AND operator isapplied to obtain one output that represents the result of thedegree to which the antecedent part of a fuzzy rule is satisfied interms of a firing strength. Hence the outputs O2,k of this layer arethe products of the degrees of membership from Layer 1:

O2,k ¼wk ¼ mAiðxÞ � mBj

ðyÞ k¼ 1,. . .,4; i¼ 1,2; j¼ 1,2 ð26Þ

Layer 3: average nodes. In the third layer, the main objective isto calculate the ratio of each rule’s firing strength to the sum ofthe firing strengths of all the rules. This is the normalized firingstrength (wi):

O3,i ¼wi ¼wi

X4

k ¼ 1

wk

!, i¼ 1,. . .,4

,ð27Þ

Layer 4: consequent nodes. The function of the fourth layernodes is to compute the contribution of each rule toward the totaloutput using the function defined as

O4,i ¼wifi ¼wiðpixþqiyþriÞ, i¼ 1,. . .,4 ð28Þ

where wi is the input from the previous layer and pi, qi, ri are theparameters in the consequent part of the Sugeno-model rules.

Layer 5: output nodes. A single node computes the overalloutput by summing all the incoming signals. This defuzzification

A.T. Gumus et al. / Int. J. Production Economics 128 (2010) 248–260254

process transforms each rule’s fuzzy result into a crisp output:

O5,i ¼X4

i ¼ 1

wifi ¼X4

i ¼ 1

wifi

! X4

i ¼ 1

wi

!,ð29Þ

3.4. Artificial neural networks

Model-G1 and G2, that Model-G2 has a neuro-fuzzy module tocompute demand and lead times, are simulated through anartificial neural network (ANN).

ANNs are heavily used in the engineering and scientific fieldsto model systems ranging from control systems to artificialintelligence (Taskin and Guneri, 2006; Guneri and Taskin Gumus,2007). ANNs are networks of simple processing elements capableof processing information in response to external inputs (Hecht-Nielsen, 1989; Freeman and Skapura, 1991; Badiru, 1992; Haykin,1999).

The ANN we used is a multi-layer perceptron (MLP) network,the most common neural network model. The network consists ofan input layer, one or more hidden layers, and an output layer.Each layer computes a nonlinear activation function of a weightedsum of the layer’s inputs. The learning algorithm is the general-ized delta rule, which ‘‘learns’’ by performing gradient descent onthe error surface (Rumelhart et al., 1987; Vysniauskas et al., 1993;Zurada, 1995; Jondarr, 1996).

3.5. The SCOR model

The performance of the simulated models are evaluated usinga reference model published by the supply chain council (SCC).The supply chain operations reference (SCOR) model (SCC, 1999;Wang et al., 2004; Huang et al., 2005) is a process reference modelused to benchmark the operations of supply chains (Huang et al.,2005; Persson and Araldi, 2009).

SCOR enables the companies to communicate supplychain issues, measure their performance objectively, identifyperformance improvement objectives, and influence future SCM

Fig. 3. The SCOR p

software development. It links process elements, metrics, bestpractices and the so forth in a unique format (Wang et al., 2004).

The SCOR model is originally founded on five distinct manage-ment processes, namely, Plan, Source, Make, Deliver and Returnwhich are called Level 1 processes. The processes are furtherdecomposed into process categories (Level 2) depending on thetype of environment in is the third and the last Level in the SCORmodel. At Level 3, the elements model which the SCOR model isapplied. The process categories further contain process elements,which contains performance attributes, metrics, best practicesand software features required for that element. Fig. 3 shows theprocess levels of the SCOR model (Stephens, 2001; Huang et al.,2005; Persson and Araldi, 2009).

The SCOR model endorses 12 performance metrics, which fallinto four defining categories (Wang et al., 2004):

1.

roc

Delivery reliability: 1.1. delivery performance (DR1), 1.2. fill rate(DR2), 1.3. order fulfillment lead time (DR3), 1.4. perfect orderfulfillment (DR4),

2.

Flexibility and responsiveness: 2.1. supply chain response time(FR1), 2.2. production flexibility (FR2),

3.

Cost: 3.1. total logistics management cost (CT1), 3.2. value-added productivity (CT2), 3.3. warranty cost or returnsprocessing cost (CT3),

4.

Assets: 4.1. cash-to-cash cycle time (AT1), 4.2. inventory daysof supply (AT2), 4.3. asset turns (AT3).

Our performance measures are especially related to thecategories of ‘‘flexibility and responsiveness’’ and ‘‘cost’’. Becauseour paper mostly aims to reveal a flexible and responsive supplychain having the capability of answering demand and lead timeuncertainties fastly, and also, to realize this aim with effectivecost management via proposed cost models.

It should be noted that our methodology was not designedwith these metrics in mind and its performance could bemeasured with others. In other words, the same methodologycan be used if a company decides to either remove or add metrics.

ess levels.

A.T. Gumus et al. / Int. J. Production Economics 128 (2010) 248–260 255

4. A numerical application in food sector

To test our methodology we simulated a three-echelon foodsupply chain consisting of three retailers, two distributors andone depot (it can be seen from Fig. 4).

4.1. The supply chain structure and assumptions of the application

Models-G1 and G2 were adapted for a three-echelon foodsupply chain, and then the input data are calculated or obtainedfrom a real company. Demand and lead time data were obtainedby neuro-fuzzy calculations and Model-G2 was simulated by anANN and used to calculate approximate lowest-cost inventorypositions. Finally, the performance of the model was measuredagainst performance criteria from the SCOR model.

A single product was considered (ketchup in 30 cl. packages,12 units in a package). The data used in forecasting marketdemand and lead times were monthly data for the period from1996 to 2005.

Our simulation assumes distributors and retailers are identicalat the same echelon, the central depot is fed by an externalsupplier with limitless stock, only one product is moving throughthe supply chain, expediting costs fall into two categories,representing two different customers, and backorders at theretail level are 0.5% of demand.

4.2. Demand and lead time forecasting by ANFIS

Monthly demand and lead times were forecast by the neuro-fuzzy Model-G1, and then the statistical model ARIMA was usedto calculate the same data. The neuro-fuzzy and statistical datawere compared to show the advantages of the neuro-fuzzymethod.

4.2.1. Demand forecasting

For the neuro-fuzzy computations Matlab 7.0 Fuzzy LogicToolbox and the ANFIS module were used. Eviews 3.0, a statistical

Depot(Echelon 3)

Distributors(Echelon 2)

Retailers(Echelon 1)

Fig. 4. The three echelon supply chain used as a hypothetical example.

Table 3Demand data parameters and membership functions.

Parameters Membership functions

Ketchup demand (unit/month) Very low [1200]

Low [3600]

Normal [4800]

Above normal [5200]

High [5600]

Very high [6000]

Product unit price ($) Low [0 0.5 1]

Normal [0.95 1.250 1.300]

High [1.250 1.350 1.400]

package that includes an automated routine for estimating anARIMA model, was used for the statistical calculations.

4.2.1.1. Neuro-fuzzy calculations for demand forecasting. The neu-ro-fuzzy network’s output was demand, calculated using unitprice, accessibility, macaroni demand (macaroni is a correlatedproduct) and freshness as inputs. The membership functions forthe fuzzy sets to which the data were assigned are shown inTable 3. The membership functions are determined to betriangular via expert opinions from our example food supplychain, wherein the notation about c¼triangle [x,l,c,u] stands for

(Giannoccaro et al., 2003; Taskin Gumus and Guneri 2009):

triangle x,l,c,u� �

¼

0 xr1

ðx�lÞ=ðc�lÞ lrxrc

ðu�xÞ=ðu�cÞ crxru

0 xZu

8>>><>>>:

ð30Þ

The network was trained on 108 data sets and the trainednetwork was then used to forecast monthly demand for the year2005. Comparing the forecast values to the actual values (suppliedby the retailer) revealed an average error of 0.000619.

4.2.1.2. Statistical analysis for demand forecasting. To forecastdemand statistically we used EViews 3.0. The correlogram of theoriginal ketchup data set showed that the time series’ auto-correlations (auto correlation function—ACF) are dying downslowly and partial auto-correlations are going out at 1, whichmeans that the time series is not stationary (its statistical prop-erties are not constant).

AR, MA and ARIMA models assume stationary (constant meanand variance) data. Non-stationary time series must be trans-formed to stationarity before one can fit an ARIMA model. First,the Augmented Dickey Fuller (ADF)/unit root test was applied.The results at first difference (I¼1), at the equation with trendand intercept and at lag 3, showed that there was no unitroot of the series, and it is stationary. The appropriate AR(I¼1)MAmodel was: D(LOG(X))¼0.03647808392+[AR(4)¼�0.2307175838,AR(12)¼0.2338745721, MA(1)¼0.4563239329, BACKCAST¼1997:02].

The model result showed that each coefficient of AR andMA was meaningful. Also, the residuals correlogram showedthat no value went out to the band and all ACF and PACF valueprobabilities were higher than 0.15. This supports beingstationary.

Then the Wald coefficient test was applied to the January–December 2005 test period, and it was determined that H0

hypothesis is approved for a¼0.05 (H0¼model is correct,H1¼model is not correct).

The real values of central depot demand (Y), the ARMA modelforecast values (Y) and the neuro-fuzzy forecast values (YNF) arecompared in Fig. 5. It can be clearly seen that the YNF values are

Parameters Membership functions

Product accessibility Low [0 2 4]

Normal [3 6 7]

High [6 8 10]

Macaroni demand (unit package/month) Low [0 250 500]

Normal [400 1500 2500]

High [2400 3500 5000]

Freshness Low [0 2 4]

High [3 7 10]

4000

4500

5000

5500

6000

6500

1

Y Ŷ YNF

2 3 4 5 6 7 8 9 10 11 12

Fig. 5. Real data and ARIMA and ANFIS forecasting data comparison for demand.

Table 4Lead time data parameters and membership functions.

Parameters Membership functions Parameters Membership functions

Lead time (day) Short [1.5] Demand(unit/month) Low [0 500 1200]

Normal [4] Normal [1000 2500 3600]

Long [7] High [3400 4800 6000]

Too Long [12]

Urgency of demand Low [0 2 4] Product quantity at depot Low [0 2 4]

Normal [3 6 7] High [3 7 10]

High [6 8 10]

2

3

4

5

6

1

Y Ŷ YNF

2 3 4 5 6 7 8 9 10 11 12

Fig. 6. Real data and ARIMA and ANFIS forecasting data comparison for lead time.

A.T. Gumus et al. / Int. J. Production Economics 128 (2010) 248–260256

more realistic. Also the mean square error (MSE) value of YNF issmaller than Y0s: MSEYNF¼0.00298oMSEY ¼0.00689.

4.2.2. Lead time forecasting

Lead time is the time that elapses between the retailer sendingan order and receiving the ordered product. The lead time is thesame for the distributors and the central depot, because it isassumed that the order will reach the distributors and theretailers on the same day. As in demand forecasting, Matlab 7.0Fuzzy Logic Toolbox and the ANFIS module were used for neuro-fuzzy computations and Eviews 3.0 for statistical analysis. Thedata set is the average lead time in days needed to fill ketchuporders between January 1996 and December 2005.

4.2.2.1. Neuro-fuzzy calculations for lead time forecasting. The leadtime was the output of the network and urgency of demand,product quantity at the depot, and demand were its inputs. Theirmembership functions are given in Table 4. Also, the membershipfunctions of the lead time are assumed to be triangular based onexpert opinions (see Eq. (30)).

The network was trained on 108 data sets and the lead timevalues for the year 2005 were forecast in days. The average errorfor forecasted values is 0.000203.

4.2.2.2. Statistical analysis for lead time forecasting. The same stepswere followed here as in the case of demand series forecasting.The original lead time series’ correlogram, showed that its auto-correlations were dying down slowly and partial auto-correla-tions were going out at 1, which meant that the time series is notstationary, and there was a trend.

Then, the ADF/unit root test was applied. The results at firstdifference (I¼1), at the equation with trend and intercept and atlag 3, showed that there is no unit root of the series, and it wasstationary.

The ARIMA model was structured and reached meaningfulAR and MA coefficients (AR(1) and MA(4), MA(12)). Also, the

residuals correlogram showed that no value went out to the bandand all ACF and PACF value probabilities were higher than 0.15.This supports being stationary. Then the Wald coefficient test wasapplied to the January–December 2005 test period, and it wasdetermined that the H0 hypothesis was approved for a¼0.05(H0¼model is correct, H1¼model is not correct).

The real lead-time values of supply chain members (Y), ARMAmodel forecast values (Y) and neuro-fuzzy forecast values (YNF)are compared in Fig. 6. It can be clearly seen that the YNF valuesare more realistic. Also the MSE value of YNF is smaller than Y’s:MSEYNF¼0.00450oMSEY¼0.00597.

4.3. Model simulation by artificial neural networks

In the next step, the models are simulated and used tocalculate the approximate minimum costs for twelve months, aswell as the reorder points, the echelon inventory positions, thesafety stocks and the expected shortages. Our goal was to see howthe expected total cost responded to different demand and leadtime combinations and to find the approximate minimum costsby using the forecasted demand and lead-time values and variable

Yes

Define the inputs and outputs of the ANN

Ask the current values of the variables

Call ANFIS forecast values of demand and lead times

Calculate H0

Run Model-G1

n=integer

Calculate Q and TC

Calculate F

Ask k to the user

Calculate Ø

Calculate SS, R, IP andES

Ask the customer priority to the user

Priority=0

Consider the group of p21, p22, p23

Consider the group of p11, p12, p13

Run Model-G2

Calculate monthly ETC

Save this value and calculate the ETC of the

next month

Determine the n value that makes TC minimum

No

No

Yes

Fig. 7. The neural network simulation algorithm.

Table 5The cost data gained through mathematical calculations and/or learned from SC memb

K ($/package/month) A ($/package/month) H ($/package

K1¼400 A1¼175 H0¼600

K2¼500 A2¼100 H1¼400

K3¼600 A3¼090 H2¼100

H3¼200

A.T. Gumus et al. / Int. J. Production Economics 128 (2010) 248–260 257

expediting costs (p11, p12, p13 or p21, p22, p23). The simulationalgorithm is given in Fig. 7.

Matlab 7.0 Neural Network Toolbox was used to simulate ourmodel. The input and output variables of the network are asbelow:

Inputs:

Fixed ordering cost : {K1, K2, K3}, Expediting cost group 1 : {p11,p12, p13},Variable cost : {A1, A2, A3},Expediting cost group 2 : {p21, p22,p23},Installation holding cost : {H1, H2, H3}, Emergency factor : {k1,k2, k3}Backorder cost : b,Output:

Expected total cost : {ETC1, ETC2, ETC3, ETC4, ETC5, ETC6, ETC7,ETC8, ETC9, ETC10, ETC11, ETC12}.

The modeled network had 19 input nodes, a hidden layer with15 nodes and 12 output nodes. The nodes in the hidden layerimplemented tangent hyperbolic functions. The output nodes hadlinear activation functions. The network was trained for 1000cycles, and the training coefficient was determined to be 0.1.Training took 25 s. The error reached was 3.4e�27. The data used inthe models and the simulation results are shown in Tables 5 and 6.

There are two groups of expediting costs (p11, p12, p13 and p21,p22, p23).The choice of a group depends on several factors,including the urgency of demand, the customer’s importanceand priority, etc. Table 6 shows how the expected total cost of thesupply chain reflects the expediting cost group chosen during thesimulation.

After 12 months of the operation of the supply chain wassimulated, the minimum cost values were calculated underdemand and lead time uncertainty and expediting cost variability.The simulation results are given in Table 6. For example, forexpediting cost group (p11, p12, p13) and forecasted demand andlead time values of 4800 packages and 3.25 days, the total orderquantity is calculated to be 700 packages, TC from Model-G1 isfound to be $ 2486.43, ETC from Model-G2 is found to be $9907.976. If the customer is a preferred one, then the expeditingcost group is (p21, p22, p23). Given the same demand and lead timeforecast values, the TC does not change but the ETC increases to $11 756.3.

4.4. Performance evaluation by SCOR

In order to evaluate the performance of our methodology,appropriate performance measures are selected from the SCORmodel and additional ones are revaluated from the literature. Theperformance measures used are given and the related perfor-mance results are given in Table 7.

Here, the terminology, that is used in performance evaluationand Table 7, needs to be clarified. By ‘‘classical’’ values we mean

ers.

/month) b ($/package/month) p ($/package/month)

b¼400 p11¼400 p21¼500

p12¼250 p22¼300

p13¼125 p23¼150

Table 6The simulation results of the Model-G2 via ANNs for 12 months.

Months Expediting costgroup

Demand(package)

Lead time(day)

Order quantity(package)

Model-G1 —totalCost ($)

Model-G2 —expectedtotal cost ($)

1 p11,p12,p13 4800 325 700 2486.430 9907.976

p21,p22,p23 4800 325 700 2486.430 11756.30

2 p11,p12,p13 4907 340 705 2514.035 10059.03

p21,p22,p23 4907 340 705 2514.035 12053.22

3 p11,p12,p13 5070 345 719 2555.467 10248.22

p21,p22,p23 5070 345 719 2555.467 12211.69

4 p11,p12,p13 5232 355 731 2595.956 10275.91

p21,p22,p23 5232 355 731 2595.956 12386.11

5 p11,p12,p13 5286 370 734 2609.299 10486.15

p21,p22,p23 5286 370 734 2609.299 27186.88

6 p11,p12,p13 5232 380 731 2595.956 10535.48

p21,p22,p23 5232 380 731 2595.956 12685.56

7 p11,p12,p13 5298 375 735 2612.276 10527.63

p21,p22,p23 5298 375 735 2612.276 12580.76

8 p11,p12,p13 5385 395 738 2633.619 10807.96

p21,p22,p23 5385 395 738 2633.619 12901.48

9 p11,p12,p13 5370 400 740 2629.978 10829.83

p21,p22,p23 5370 400 740 2629.978 12944.97

10 p11,p12,p13 5535 410 750 2670.082 11086.28

p21,p22,p23 5535 410 750 2670.082 13088.21

11 p11,p12,p13 5706 450 763 2711.032 11483.55

p21,p22,p23 5706 450 763 2711.032 13690.25

12 p11,p12,p13 5982 483 781 2775.794 12028.02

p21,p22,p23 5982 483 781 2775.794 14006.51

Table 7Performance evaluation results.

Performance measure Classical (C) Model results (MR) Higher performance state

A1 – No significant difference between real values of demand and

lead time and neuro-fuzzy forecasting values of them

MR

A2 – Extension of A1 MR

B1-OFLT 0.0110 days 0.0046 days MR

C1-SCRT 0.0131 days 0.0067 days MR

C2-OFR 0.17 1.05 MR

D1-FOC $ 294 $ 66 MR

A.T. Gumus et al. / Int. J. Production Economics 128 (2010) 248–260258

those calculated by classical inventory theory. By ‘‘model results’’values mean those calculated by the proposed model.

A. Forecast accuracyA1. We have shown that the results of neuro-fuzzy forecasts

for demand and lead times are more accurate than those obtainedby statistical analysis (Section 4.2). Here, we apply the t-test toascertain whether the forecast values are significantly differentfrom the real values (McGhee, 1985):

sDx ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs2

1=n1Þþðs22=n2Þ

qð31Þ

tcalc ¼ ðx1�x2Þ=sDx ð32Þ

v¼ ðs21=n1Þþðs

22=n2Þ

� �2= ðs2

1=n1Þ2=ðn1�1Þ

n oþ ðs2

2=n2Þ2=ðn2�1Þ

n oh ið33Þ

Here, s1 and s2 are the standard deviations of each forecastingtechniques, n1 and n2 are the number of forecasted values and v isthe degree of freedom. The tcalc value is calculated and comparedwith the ttable value at the 5% reliance level. The outcome of thiscomparison determines whether the H0 hypothesis is accepted ornot. The hypotheses are: H0: x1�x2 ¼ 0; H1: x1�x2a0.

For demand: n1¼12; n2¼12; s1¼260.36; s2¼337.64; x1¼5246.5;x2¼5316.917; sDx¼123.08; tcalc¼0.586; v¼20.004¼21; At 5%reliance level, t-test critical value is: ttable¼1.721.

tcalc¼0.586ottable¼1.721, so the H0 hypothesis is accepted:there is no significant difference between the means of thecalculated and real values.

For lead time: n1¼12; n2¼12; s1¼0.435; s2¼0.4597; x1¼3.743;x2¼3.856; sDx¼0.1827; tcalc¼0.618; v¼21.933¼22. At 5% reliancelevel, t-test critical value is: ttable¼1.717.

tcalc¼0.618ottable¼1.717, then H0 hypothesis is accepted.Again, there is no significant difference between the two means.

Hence, it can be said the forecasts of both demand and leadtimes are accurate.

A2. This measure is an extension of A1 and an indirect measureof model performance. Because of forecast accuracy, the compu-tations of reorder point, echelon inventory position, safety stock,total cost, etc. are also more accurate than those of models thatuse statistically based forecasts.

B. ResponsivenessB1. Order fulfillment lead time (OFLT): Average actual lead time

encloses the time interval from the customer approval forpurchasing order to fulfillment of the last order. Then the leadtime for a single package at a single order is gained. The maindepot lead time was forecasted as 3.25 days, before. According toclassical theory, the order quantity for the main depot iscalculated as below:

Q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2KD=H

pð34Þ

1

DETC-DLTETC-LT

2 3 4 5 6 7 8 9 10 11 12

Fig. 8. The sensitivity analysis results.

A.T. Gumus et al. / Int. J. Production Economics 128 (2010) 248–260 259

then Q¼294 packages. According to our methodology, theorder is aggregated at the depot and it is calculated to be 700 (seeTable 6). Then, the order fulfillment lead times for C and MR arecalculated by: OFLT¼ l/Q. It can be seen from Table 7 that the orderfulfillment lead time for a single package is lower now, afterdeveloped multi-echelon structure. And former, it was 2.39 timeshigher.

C. FlexibilityC1. Supply chain response time (SCRT): This measure is

intimately related to the order fulfillment lead time calculatedin B1. Here, response time for a single package is calculated. Theconsumption rate is calculated (d¼Q/t) and then the SCRT can befound by: SCRT¼OFLT+(1/d). The model SCRT is shorter thanformer SCRT. In other words, the SCRT performance has improved,as can be seen in Table 7.

C2. Order fulfillment rate (OFR): The ratio of the reorder point tothe inventory position (OFR) should be as close to 1 as possible. Ifit is less than 1, either there is too much inventory and/or ordercosts increase. If it is above 1, customer demand is not being met.

According to classical inventory theory, the reorder point canbe calculated as: R¼ ððdÞ � ðlÞÞ. For the C state, inventory positionis only depot’s inventory position, but for the MR it represents theechelon inventory position calculated by ANNs simulation. Theformula for OFR is as: OFR¼R/IP. The former OFR is found to be0.17, while the actual OFR is 1.05. The actual OFR is closer to 1than the former OFR.

D. CostD1. Fixed ordering cost (FOC): The fixed order cost depends on

the order numbers. Order number is (D/Q) in classical inventorytheory, but in our models it is (D/nQ). The FOC is K(D/Q) in classicaltheory, and K(D/nQ) in actual state. It can be seen from Table 7that the fixed ordering costs at the depot are 445 times lower thanbefore.

4.5. Results and sensitivity analysis

We have shown that the new methodology improves severaldifferent metrics of supply chain performance. Since novel meansof estimating demand and lead time were keys to our model, wealso performed sensitivity analyses on these variables, based onModel-G2.

If the demand is changed while other variables are heldconstant, there is little change in ETC (ETC-D) (see Fig. 8). Whiledemand fluctuates from 14 400 to 18 000, the ETC-D fluctuatesbetween 9900 and 11 400. For example, when the demandchanges from 14 400 units to 14 721, then ETC-D changes from$ 9907.976 to $ 9901.854. A 2,23% increase in demand results in a0.06% decrease in ETC-D from one month to another. In this three-echelon supply chain, at any rate, higher demand results in lowertotal chain cost, reflecting economies of scale. But if the demandcontinues to increase, then the expected total cost starts toincrease, too. As the demand changes from 14 721 to 14 810, theETC-D increases from $ 9901.854 to $ 10 060.66. So ETC-D doeschange as demand increases or decreases.

If the lead time changes while other variables are heldconstant, ETC (ETC-LT) increases. For example, if lead timeincreases from 3.25 to 3.5, a change of 7.69%, then the ETC-LT

increases from $ 9907.976 to $ 10 164.27, a change of 2.59%. Leadtime appears at several points in the model, affecting several ofthe costs that contribute to the estimated total cost. As the modelsuggests lead time should be held to a minimum because anyincrease in lead time results in additional costs and a higherexpected total cost.

5. Conclusion

The supply chain model developed in this paper remediesseveral deficiencies of similar models found in the literature. Thebasic deficiency is that demand and/or lead times are assumed tobe constant or to fit a probabilistic distribution. In our model,demand and lead time are determined by neuro-fuzzy calcula-tions, a method that gives realistic results (Section 4.2).

Another deficiency of other models is that they apply only toshallow chains with a few echelons and assume that theconnections between units at different echelons are serial. Themodel we have developed is generalized for N-echelons and atree-structured supply chain.

Another assumption of published models is that late orders aredelayed until the next order cycle arrives. Our model allowsorders that arrive out of phase to be expedited. In this sense itmore accurately reflects the real world, where expediting costsare tolerated to provide high service levels.

Another innovation of our paper is that we employed a neuralnetwork to simulate our neuro-fuzzy forecast based supply chainmodel. Conventional simulation software (ARENA, SLAM II, etc.)could not be used because our model incorporated neuro-fuzzyforecasting (to see the summary of the innovations and differenceof our paper, please look at Table 1, as mentioned in Section 2).

Many published models have been subjected to little or noperformance analysis. In contrast, we used an industry referencemodel, the SCOR model, to test the performance of our supplychain model.

By building accurate forecast data and realistic cost figures intoa general N-echelon tree-structured supply chain, our model andmethodology eliminate several deficiencies in published models.In future research, we plan to eliminate the identity assumptionfor retailers and distributors and to expand the model to handlemore than one product type. Finally, we plan to calculate morevariables calculated by neuro-fuzzy approximation to furtherincrease the model’s fidelity to the real world.

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