robust-fuzzy model for supplier selection under uncertainty: an...

Scientia Iranica E (2018) 25(4), 2297{2311

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringhttp://scientiairanica.sharif.edu

Robust-fuzzy model for supplier selection underuncertainty: An application to the automobile industry

M. Rabieha;�, M. Modarresb, and A. Azarc

a. Department of Industrial Management, Shahid Beheshti University, Tehran, Iran.b. Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran.c. Department of Industrial Management, Tarbiat Modares University, Tehran, Iran.

Received 3 March 2016; received in revised form 22 February 2017; accepted 30 May 2017

KEYWORDSSupplier selection;Uncertainty;Robust optimization;Fuzzy programming;Robust-fuzzy model;Auto industry.

Abstract. This paper proposes an innovative robust-fuzzy method for multi-objective,multi-period supplier selection problem under multiple uncertainties. This approachintegrates robust optimization and fuzzy programming. Uncertain parameters are modeledas random variables that take value within a symmetrical interval. However, due to thecomplexity or ambiguity of some real world problems and especially the nature of some ofthe available input data, the length of interval is also highly uncertain. This ambiguitymotivated us to present a new approach, which could be applicable to multiple uncertaintiesconditions. Thus, in our approach, the half-length of these intervals is also represented byfuzzy membership function. We develop a model and a solution approach to select suppliersby considering risk. The proposed method is applied to a real case of supplier selection inautomobile industry under uncertainty and ambiguity conditions. To verify the proposedmodel, we evaluate the results by simulation technique and compare values of objectivefunction under di�erent scenarios.

© 2018 Sharif University of Technology. All rights reserved.

1. Introduction

Supplier selection is an important strategic decisionmaking for managers in supply chain management [1].One can �nd numerous studies and models in thesupplier selection literature, see Section 2. The decisionmaking models in the literature are basically developedto answer the following questions: How many suppliersare needed? Which suppliers should be selected? Whatis the optimal ordering policy to each supplier? Onecan �nd many deterministic models developed in theliterature to answer these questions.

The main disadvantage of deterministic models is

*. Corresponding author.E-mail address: m [email protected] (M. Rabieh)

doi: 10.24200/sci.2017.4456

that they fail to be accountable when encountering theuncertain nature of real world systems.

On the other hand, risk and uncertainty manage-ment is an inherent part of supplier selection. Theimportance and the e�ects of uncertainty in supplierselection have been widely emphasized in the supplychain literature [1-3].

To handle the e�ect of uncertainty of inputparameters, either probabilistic or fuzzy approachescan be applied. However, in many real cases, dueto the lack of enough historical data to �t appropri-ate probability distribution for uncertain parameters,stochastic programming cannot be applied. Further-more, computational complexity of stochastic program-ming models is another limitation on their application.Therefore, robust optimization methods can be consid-ered as an e�cient alternative for handling this typeof uncertainty [3]. Recently, some researchers have

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2298 M. Rabieh et al./Scientia Iranica, Transactions E: Industrial Engineering 25 (2018) 2297{2311

developed the robust optimization approach to handleuncertainties in supplier selection [1,2]. Moreover,robust optimization has emerged as a methodologyfor managing supply chain risks. Robustness playsan important role in sustainability of supply withuncertain input data. Robustness of supply chain hasbeen the frontier of supply chain research [4].

Fuzzy programming is another method for han-dling a type of uncertain parameters, which are usuallyexpressed in linguistic and vague terms and thentransformed into fuzzy numbers.

In real cases, due to the vagueness of the infor-mation about parameters, deterministic methods arenot suitable for e�ective supplier selection. Underthese conditions, the theory of fuzzy sets is one ofthe best tools to handle the uncertainty [5]. Fuzzyprogramming based on fuzzy sets theory is utilizedwhen the information for supplier selection is vagueand imprecise.

In this research, we develop a new method ofrobust optimization to handle uncertainties. In robustoptimization, the models are developed based on theassumption that no knowledge about the distributionof uncertain parameters exists. In these models,uncertainty of parameters is modeled as symmetric andbounded random variables. Every symmetrical intervalis characterized by two numbers: middle number ofinterval (nominal value) and half-length of interval. Itis assumed that these numbers are given. However, inreal world applications, forecasting their exact valuesseems to be di�cult or even impossible. In fact, errorsand ambiguity in forecasting and measuring modelparameters make the decision making model prone tomany resources underutilization and/or overutilizationproblems. In our real case study, we faced the problemof determining the half-length of interval of supplier'scapacity parameter. That case motivated us to providean innovative robust-fuzzy model. In this approach,the half-length of interval is presented by a fuzzynumber. In fact, the main innovation of this researchis combining the robustness and fuzziness concepts inthe presence of multiple uncertainties.

In order to illustrate our proposed Robust-Fuzzy(R-F) method, it is implemented for a real case of sup-plier selection in automobile industry under multipleuncertainties. Under various conditions, the resultsproved to be accurate enough in comparison with theresults by simulation techniques. In other words, theproposed model can properly manage a variety ofuncertain conditions and provide useful and reliableinformation for exible decision making processes.

The rest of this paper is organized as follows. InSection 2, literature review is presented. In Section 3,the case study is described. Steps of implementingthe model case are presented in Section 4. Sec-tion 5 presents the methodology of robust-fuzzy (R-

F) method. In Section 6, the mathematical model isformulated. Data collection and parameters are de-scribed in Section 7. Computational result is presentedin Section 8 and, �nally, some concluding remarks aregiven in Section 9.

2. Literature review

Due to the importance of supplier selection decision,an extensive literature review of both deterministic anduncertain models is presented to address this type ofdecision. In the section pertinent to uncertain models,three categories of the mentioned models are statedin stochastic, fuzzy, and robust forms. Moreover,relevant researches are reviewed for each category.Finally, considering the strengths and weaknesses ofeach category and the current gap in the literature,the contribution of the article is presented.

2.1. Deterministic modelsIn this section, a brief overview of relevant studies of de-terministic supplier selection models is presented. Ben-ton [6] used Lagrange relaxation to develop a non-linearprogram for supplier selection under various conditionsincluding multiple suppliers, multiple items, resourcelimitations, and quantity discount. Ghodsypour andO'Brien [7] used integrated Analytical Hierarchy Pro-cess (AHP) with mixed integer programming to developa Decision Support System (DSS) in order to reduce thenumber of suppliers. Ghodsypour and O'Brien [8] alsodeveloped a model by integration of AHP and linearprogramming model to take into account both qualita-tive and quantitative factors. In a further development,Ghodsypour and O'Brien [9] presented a mixed integernon-linear programming model by considering the totalcost of logistics for multiple sourcing cases. Basnetand Leang [10] investigated the problem of supplierselection by considering the lot-sizing. Ustun andAkhtarDemirtas [11] used an integrated approach ofAnalytic Network Process (ANP) and Multi-ObjectiveMixed Integer Linear Programming (MOMILP) forsupplier selection problem. Their approach consideredboth tangible and intangible factors in choosing thebest suppliers and de�ned the optimum quantitiesamong selected suppliers to maximize the Total Valueof Purchasing (TVP), minimize the total cost and totaldefect rate, and balance the total cost among periods.Finally, Mendoza and Ventura [12] presented two mixedinteger nonlinear programming models to select thebest suppliers and determine order quantities, whileminimizing the annual ordering, inventory holding, andpurchasing costs with respect to suppliers' capacityand quality constraints. Their research integrated theissues of inventory management and supplier selection.

The literature on deterministic models considersthe following as signi�cant items:

M. Rabieh et al./Scientia Iranica, Transactions E: Industrial Engineering 25 (2018) 2297{2311 2299

� Some of these models are multi-objective (see [10]);� Some of these models consider some important char-

acteristics such as multiple suppliers and multipleitems, simultaneously (see [12]);

� Obviously, these models are not applicable to uncer-tain environments.

2.2. Uncertain modelsIn this section, a brief overview of relevant studiesof uncertain supplier selection models (i.e., fuzzy,stochastic, and robust models) is presented.

2.2.1. Fuzzy modelsSome researchers developed fuzzy models. Kumar etal. [13] used a fuzzy mixed integer goal programmingapproach to solve the vendor selection problem in caseof multiple objectives. Chen et al. [14] presented afuzzy decision making approach to solve the supplierselection problem. They used linguistic values toevaluate the ratings for a number of quantitative andqualitative factors including quality, price, exibility,and delivery performance. Their model was shown tobe a good tool for supplier selection decision makingsituation. Kilic [15] presented a new integrated ap-proach, including fuzzy TOPSIS method and a mixedinteger linear programming model, to select the suit-able supplier in a multi-item/multi-supplier environ-ment. In the �rst stage of this approach, importancevalue of each supplier for each product was obtainedvia fuzzy TOPSIS. In the second stage, outputs ofthe �rst stage were used as inputs to the proposedmathematical model; the best suppliers were selected;and the amount of order for each product (item) toeach supplier was determined via solving the model.Nazari-Shirkohi et al. [16] proposed an interactiveFuzzy Multi-Objective Linear Programming (FMOLP)model to solve the multi-objective supplier selectionproblem and order allocation under multiple pricelevels and multiple products. The model attemptedto minimize cost as well as numbers of defective unitsand late delivered units ordered from suppliers. Finally,Roshandel et al. [17] presented a fuzzy hierarchicalTOPSIS approach to select and evaluate suppliers ofimported raw materials for the detergent productionindustry in Iran. The authors claimed that theirapproach overcame disadvantages of a prior well-knownmethod in the area of fuzzy TOPSIS (Chen's method).

Sou� Neyestani et al. [18] presented an evolution-ary algorithm for supplier order allocation with fuzzyparameters considering linear volume discount. Assome information, such as buyer demand and supplierdelivery rate, was uncertain, the authors applied fuzzysets to handle the uncertainty.

Mahmoudi et al. [19] extended fuzzy VIKORusing an e�cient fuzzy distance measure to solveapplicable Supplier Selection Problem (SSP) under

group decision-making. The authors claimed that, asSSP parameters were often imprecise, vague, uncertain,or incomplete in real-life situations, fuzzy sets theorywas the best developed approach to handle the uncer-tainties.

There are some general shortcomings in fuzzyapproaches:

� Fuzzy approaches are only applicable in the condi-tion of uncertainty in a fuzzy form;

� Determining the real membership function of thefuzzy numbers is not easy;

� In all the above-mentioned studies, the issues ofsingle uncertainty are investigated, but not fullymatched with real world cases.

2.2.2. Stochastic modelsKasilingam and Lee [20] developed a chance-constrained integer programming formulation for ven-dor selection and order allocation by minimizing costs.Their model also considered the lead time requirementsand vendors capacities, and used a chance constraintdue to the stochastic nature of demand.

Feng et al. [21] presented a Stochastic Integer Pro-gramming (SIP) method for simultaneous selection oftolerances and suppliers based on the quality loss func-tion and process capability indices. In this research, toachieve the required quality level and minimum totalcost, process capability indices and quality loss functionwere also incorporated in a mathematical model.

Bonser and Wu [22] developed a stochastic pro-gram for solving procurement problem in an electricalutilities �rm under uncertain demand and market price.

Several other works have also considered this typeof uncertainty in supplier selection problem. Seshadriet al. [23] developed a probabilistic bidding modelto represent the connections in multiple-sourcing andtheir consequences. Their bidding model investigatedthe e�ect of multiple-sourcing on competitive behav-ior prior to supplier selection. Ranjbar Tezenji etal. [24] presented an integrated model for supplierlocation-selection and order allocation under capacityconstraints in an uncertain environment. The authorsconsidered the costs to be uncertain and stochastic.

In many real cases, applications of stochasticprogramming are limited, because:

� Enough historical data about appropriate proba-bility distribution for uncertain parameters is notusually available;

� These models have some computational complexi-ties [3].

The aforementioned shortcomings motivated usto select an appropriate alternative method, especiallyfor our real case with especial characteristics. In such


cases, robust optimization methods could be adopted,because these methods (especially Bertsimas and Sim'sapproach) are more practical.

In fact, robust optimization methods try topresent solutions which are less sensitive to changesof parameters [3].

2.2.3. Robust modelsRecently, some researchers have developed the robustoptimization approach to handle uncertainties in sup-plier selection [1,2].

Li and Zabinsky [1] developed stochastic mathe-matical programming models to capture the risk asso-ciated with uncertain customer demand and suppliercapacity. They used multi-parametric programmingtechniques to analyze tradeo�s and determine a robustset of suppliers with balanced costs and risks. In fact,the �nal result of their research was robust selection ofsuppliers.

Nam et al. [2] developed a model for robust supplybase management under both demand and supply un-certainties. In fact, in supply base management underthe condition of uncertainty, having an understandingof bene�ts associated with a contractor that has anoptimal number of suppliers is important. Brie y,the model determined the optimal robust number ofsuppliers that a contractor should maintain in itssupply base in the condition of uncertainty.

The literature on the robust models shows that:

� Only the issue of single uncertainty has been inves-tigated;

� There has been no attention to robust optimizationmethods, especially Bertsimas and Sim's approachas an e�cient method.

Considering shortcomings of each section (deter-ministic and uncertain models), the literature has paidlittle attention to the following items:

� Simultaneous attention to characteristics such asmultiple items, multiple suppliers, multiple objec-tives (goals), multiple periods, and multiple prod-ucts;

� Robust optimization methods, especially Bertsimasand Sim's approach as an e�cient method;

� Multiple uncertainties.

This research aims to �ll these gaps. For exam-ple, the proposed comprehensive model incorporatesassumptions and characteristics such as multiple items(parts), multiple suppliers, multiple objectives (goals),multiple periods, multiple products (automobiles),multiple uncertainties, and two-echelon supply chainincluding suppliers and manufactures (plants). Also,in this research, the important criteria of price, quality,and delivery are considered.

The contribution of this research regarding thevalue it adds to the theory is presented as follows.

In fact, the main contribution of this researchis combining the robustness and fuzziness concepts inthe presence of multiple uncertainties. In other words,this research proposes a novel approach for multipleuncertainties.

3. Case application

By generating high employment rate, high �nancialturnover, and signi�cant rate of deals with foreignautomotive industry, the automotive industry is oneof the important industries after oil and gas indus-try. This industry signi�cantly a�ects the nationaleconomy. Therefore, its supply chain is one of themost dynamic chains in industries. This chain providesthousands of parts for each vehicle.

While some parts are supplied by a single source,many others have multiple sources, which include morethan 70 percent of the total value for each vehicle.Considering the number of suppliers, which can be fromone to four, we implement the proposed supplier selec-tion method for Iran Khodro, the largest automotivecompany in Iran.

Due to the high volume of required parts, choos-ing suitable suppliers is of great importance for thisindustry.

Some existing data (supplier's capacity and trans-portation cost) are uncertain by their nature. Thus,due to the existence of uncertain data, we should adoptrobust optimization for supplier selection planning.This motivated us to develop a new method namedrobust-fuzzy method.

Although the problem is formulated in its generalform, we collected data for two models of vehicles,namely, Peugeot 405 and Pars, only for Tehran plant,which is the largest and most important plant of thecompany.

4. Methodology of robust-fuzzy

As mentioned, one way of dealing with uncertaintyis by stochastic programming modeling. The mainshortcomings of stochastic programming approach are:(a) lack of existence of distribution function of data,and (b) computational challenges [25]. The conceptand advantages of robust approach, in general, andsupply chain planning, in particular, are illustrated inFigure 1. Deterministic approaches obtain the solutionbased on averaging or \good guess". In contrast,robust approaches provide a solution which is \near-optimal". Although the resulting cost is more than thatof deterministic approach, the solution is more reliable.In other words, taking into account the variability of


Figure 1. The e�ect of robust planning on total supplychain cost [25].

parameters in a range of values, the solution is stillreliable with high con�dence [26].

Soyster [27] took the very �rst step toward usingrobust optimization, proposing linear programmingmodel. This model produced solutions that were tooconservative. The next steps were by Ben-Tal andNemirovski [28-30], El-Ghaoui et al. [31], and El-Ghaoui and Lebert [32] to address over-conservatism.They applied robust optimization to linear program-ming problem with ellipsoidal uncertainty sets, thusobtaining conic quadratic programs.

Bertsimas and Sim [33] presented a new approachspeci�cally tailored for polyhedral uncertainty, whichled to linear robust counterparts, i.e., it retained theadvantages of linear framework of Soyster (1973) [27]while controlling conservativeness level of the solution.Thus, their method was also appropriate for solvingdiscrete optimization problems, and that is the reasonwe have adopted Bertsimas and Sim's model.

As explained before, in robust optimization ap-proach, uncertain data take value within an inter-val. Although no distribution function for a randomvariable is available, the length of interval is given.However, in some real cases, it is di�cult for thedecision maker to determine the exact interval length(or half-length if it is symmetrical). If the decisionmakers consider a high half-length of interval, then thelevel of conservatism and, consequently, the cost wouldbe increased. On the other hand, if the interval half-length is considered low, the risk would be increased.Furthermore, in some situations, the decision makersuggests ambiguous interval half-length. To overcomethis problem, we develop an innovative approach andrepresent the interval half-length in terms of fuzzynumber. In fact, we develop a Robust-Fuzzy (R-F)method, which integrates the concepts and techniquesof robust optimization (Bertsimas and Sim's approach)and fuzzy programming (Werners' approach) [34,35].

In addition to having the applicable features ofBertsimias and Sim's model, this method is properlyresponsive to the ambiguity of the interval half-lengthfor the decision maker and provides necessary exibility

for decision makers. In other words, this methodis able to solve the problems in multiple uncertaintyconditions.

Consider nominal mixed integer programming, asfollows:

Min c0x;

S.t. Ax � b; l � x � u;xi 2 Z; i = 1; :::; k: (1)

Without loss of generality, we assume that only theelements of matrices A and c0 are uncertain. In prac-tice, the mean values of coe�cients aij and its intervalhalf-length, aij , as well as cj and dj are estimated.The model of data uncertainty is represented in thefollowing [33]:

(a) Uncertainty for matrix A: let N = f1; 2; ::; ng.Each entry aij , j 2 N , is modeled as an indepen-dent, symmetric, and bounded random variable(with unknown distribution) ~~aij , j 2 N , that takesvalues in [aij � aij ; aij + aij ];

(b) Uncertainty for cost vector c: Each entry cj , j 2N , takes values in [cj ; cj +dj ], where dj representsthe deviation from the nominal cost coe�cient, cj .

Furthermore, we de�ne that jJij represents thenumber of uncertain components in the ith constraint;and �i; i = 0; 1; :::;m, takes values in [0; jJ j].

The role of parameter �i in constraint i is toadjust the robustness of the proposed method to thelevel of conservatism of the solution. Similarly, thelevel of robustness in objective function is controlledby parameter �0. Therefore, the robust counterpart ofProblem (1) is as follows:

Min z = c0x+ z0�0 +X

j2j0 p0j ;

S.t. :Xj

aijxj + zi�i +Xj2ji

pij � bi;

z0 + p0j � djyj ; zi + pij � aijyj ;pij � 0; yij � 0; zi � 0;

�yj � xj � yj8j lj � xj � uj8j xi 2 z: (2)

Model 2 is Bertsimas and Sim's robust counterpart.

4.1. Linear programming problem with fuzzyresources: Werners' approach

Consider the following general fuzzy linear program-ming model with fuzzy resources (nonsymmetrical


model):

Max Z = f(x) = CX;S.t. : (AX)i ~�bi; i = 1; 2; :::;m;X � 0

or

Max Z = f(x) = CX;S.t. : (Ax)i � ~bi; i = 1; 2; :::;m;X � 0;

(3)

where ~bi, and 8i is in [bi; bi + pi] with given pi.To make this model a symmetrical one, Wern-

ers [34,35] proposed that its objective function shouldbe of fuzzy type. Therefore, fuzzy resource's tolerancepi is assumed to be given and certain. He initiallyo�ered the following de�nitions for inf f and sup f :

Z0 = inf f = Max CX;S.t. : (Ax)i � bi;X � 0;

Z1 = sup f = MaxCX;S.t. : (Ax)i � bi + pi;X � 0:

(4)

Membership function for objective function and con-straints is shown in Figures 2 and 3.

To achieve optimal solutions, we use Max-Minoperator. Therefore, fuzzy linear programming can beconverted to a symmetrical model as follows:

Maxx�0�; where � = Min [�0(x); �1(x); :::�m(x)] :(5)

Figure 2. Membership function of the objective function.

Figure 3. Membership function of a constraint.

4.2. Innovative robust-fuzzy approachAs explained before, our robust-fuzzy approach inte-grates techniques of robust optimization (Bertsimasand Sim's approach) and fuzzy programming (Werners'approach).

In this section, we summarize the model byconsidering the uncertainty of decision makers in deter-mining interval half-length of the parameter as a fuzzynumber, ~bi, with the membership function shown inFigure 4.

The resulting model is a linear programming onewith fuzzy resources (nonsymmetrical model), whichcan be converted into a symmetric one and, �nally,into the following deterministic model:

Max z = �;

c0x+ z0�0 +X

j2j0 p0j + (Z1 � Z0)� � Z1;

S.t. :Xj

aijxj + zi�i +Xj2ji

pij + qi � bi 8i;

z0 + p0j � djyj 8j 2 J0;

zi + pij � aijyj 8i 6= 0; j 2 Ji;zi + pi0 � �(bi � bimin) � bimin 8i; j 2 Ji;pij � 0 8i; j;yij � 0 8i; j;zi � 0 8i;�yj � xj � yj 8j;lj � xj � uj 8j;xi 2 z i = 1; :::; k: (6)

Figure 4. Membership function of ~bi.


In Model 6, Pij and qi are robustness variables. Whenconverting a nominal model to its robust counterpartmodel, robustness variables are added to the model.

The proposed method is more exible than thebase model (Bertsimas and Sim's model); also, iteliminates the decision maker's uncertainty about half-length intervals. In other words, the decision makeris able to reach pessimistic, optimistic, and morebalanced results for them. If � = 1 (upper limit ofthe interval length), pessimistic results are obtainedby Model 3; and if � = 0 (lower limit of the intervallength), optimistic results are obtained.

In Section 8.1, Monte Carlo simulation exper-iments are conducted to justify robustness of solu-tions and estimate real level of conservatism againstdeterminate budget. In fact, by simulation, the

probability of constraints violation or risk rate will bedetermined. Brie y, the Monte Carlo simulation modelis implemented to test and validate the robustness ofthe solution obtained via the robust model.

5. Steps of our proposed approach

Figure 5 shows the steps of this approach.Data has been collected via intensive interviews

with managers and experts as well as from the existingdocuments of Iran Khodro Company, and SAPCOCompany, its main subsidiary, which is responsible formanaging its supplying parts and components.

Also, pro�le of experts includes: �ve people withat least 10 years of experience and expertise, includingthe logistics manager, logistics project team leader,

Figure 5. Steps of implementing the model case.


head of logistics engineering, logistics engineering ex-pert, and planning expert.

6. Formulating the model

In this section, we present the mathematical model ofsupplier selection problem.

6.1. NotationSetsI Set of products (automobiles)J Set of plantsM Set of partsN Set of suppliersT Set of planning periodsR Set of objective functions (goals)

Certain parametersPijt Production level of product i in period

t for plant jCSmn Purchase cost of part m from supplier nCmn Capacity of supplier n to supply part mChmnj Inventory cost in manufacturing plant

jPPMmn Index of returned part m to supplier nV Cim Consumption coe�cient of part m for

product iDPmn On-time delivery number of part m by

supplier nWr Importance weight of objective rGr Value of goal rLTm Lead time of part m� Safety factor for determining

permissible levels of inventory� Factor for determining the minimum

procurement from each supplier� Robust protection level

Uncertain parametersCtmnj Transportation cost of part m from

supplier n to plant jCmn Capacity of supplier n for supplying

part m

Issues related to uncertain parameters�Ctmnj Nominal value of transportation cost

of part m from supplier n to plant j�Cmn Nominal value of capacity of supplier

n to supply part m

Ctmnj Half-length of transportation cost ofpart m from supplier n to plant j

~aCmn Half-length of capacity of supplier n to

supply part m.

Decision variablesSmnjt Supply rate of part m from supplier n

in period t for plant jISmnjt Inventory level of part m from supplier

n in period t for plant jd+i ; d

�i Deviation from goal i (positive,

negative)� Degree of constraint's satisfaction

6.2. Mathematical model6.2.1. Objective functionsWe consider three objectives for the model as follows:

� Minimizing the cost of supplying parts;� Minimizing the defective parts supplied by suppliers;� Maximizing on-time delivery.

The �rst objective function consists of purchasingcost, transportation cost from supplier to manufac-turing plant, and holding cost. The second objectivefunction is related to PPM (part per million) factor.This factor is one of the most important indices inthis industry. In fact, it calculates the ratio of numberof defects to the volume of shipment. It is clear thatany suppliers that have lower PPM will attract moreorders. The third objective is an important indicator,highly emphasized by industry experts as well as in theliterature. Obviously, any suppliers that have more on-time delivery will attract more orders.

To consider all three objectives, we formulate theproblem by applying goal programming technique.

6.2.2. ConstraintsGenerally, the constraints can be classi�ed into threecategories of functional constraints, goal constraints,and robust constraints.

Functional (main) constraints of the model are:

� Parts demand: Demand in each period dependson three factors: production level per product,consumption coe�cient of part per product, andinventory.

� Inventory level, maximum and minimum: Inventorymaximum and minimum levels depend on fourfactors: production level per product, consumptioncoe�cient of part per product, lead time of part,and safety factor of determining permissible levelsof inventory;

� Suppliers' capacity;� The minimum quantity of a part which can be

purchased from each supplier.

Due to the company's policy for purchasingfrom all suppliers of a part, the minimum quantityof a part which can be purchased from each supplier


is constrained by a factor that determines theminimum procurement level from each supplier (�).

6.3. Nominal goal modelThe nominal goal model is:

Min Z =rXr=1

wr(d+r ; d

�r ) =

w1d+1

G1+w2d+

2G2

+w3d�3G3

;

S.t. :

MXm=1

NXn=1

JXj=1

TXt=1

CSmnSmntj

+MXm=1

NXn=1

JXj=1

TXt=1

CtmnjSmntj

+MXm=1

NXn=1

JXj=1

TXt=1

ChmnISmntj + d�1 � d+1 = G1;

IXi=1

MXm=1

NXn=1

TXt=1

PPMmnSmnt + d�2 � d+2 = G2;

IXi=1

MXm=1

NXn=1

TXt=1

DPmnSmnt + d�3 � d+3 = G3;

nXn=1

Smntj =iXi=1

V CimPijt �nXn=1

ISmnjt�1

+iX

n=1

ISmnjt 8j;m; t ;

nXn=1

ISmnjt �iXi=1

V Cim � Pijt=30� �� LTm

8j;m; t ;nXn=1

ISmnjt �iXi=1

V Cim � Pijt=30� (1 + �)� LTm

8j;m; t ;JXj=1

Smntj � Cmn 8m;n; t ;

JXj=1

Smnjt � � �nXn=1

JXj=1

Smnjt 8m;n; t ;

Smnjt; ISmnjt � 0; integer; 8m;n; j; t: (7)

6.4. Robust-fuzzy counterpart modelIn this section, we present the �nal model by consider-ing the uncertainty of decision makers in determininginterval half-length of parameter ci as a fuzzy number,~ac i. Thus, by considering the innovative robust-fuzzyapproach presented in Section 2.2, the nominal goalmodel is converted into the following robust-fuzzycounterpart model:

Max Z = �;

S.t. :

w1d+1

G1+w2d+

2G2

+w3d�3G3

+ (Z1 � Z0)� � Z1;

MXm=1

NXn=1

JXj=1

TXt=1

CSmnSmntj

+MXm=1

NXn=1

JXj=1

TXt=1

�CtmnjSmntj

+MXm=1

NXn=1

JXj=1

TXt=1

ChmnISmntj

+MXm=1

NXn=1

JXj=1

TXt=1

PPmnjt

+ Z � �1 � d+1 � G1;

IXi=1

MXm=1

NXn=1

TXt=1

PPMmnSmnt + d�2 � d+2 = G2;

IXi=1

MXm=1

NXn=1

TXt=1

DPmnSmnt + d�3 � d+3 = G3;

nXn=1

Smnjt =iXi=1

V CimPijt �nXn=1

ISmnjt�1

+iX

n=1

ISmnjt 8j;m; t ;

nXn=1

ISmnjt �iXi=1

V Cim � Pijt=30� �� LTm

8j;m; t ;nXn=1

ISmnjt �iXi=1

V Cim � Pijt=30� (1 + �)� LTm

8j;m; t ;


JXj=1

Smnjt+ZZmnt�2+pppmnt � �Cmn 8m;n; t ;

JXj=1

Smnjt � � �nXn=1

JXj=1

Smnjt 8m;n; t ;

PPmnjt + ZZ � CtSmnjt; 8m;n; j; t ;ZZmnt+pppmnt+�(Cmed�Cmin)�Cmin; 8m;n; t ;Smnjt; ISmnjt � 0; integer; 8m;n; j; t: (8)

7. Model parameters and data collection

Model parameters (data) can generally be divided intotwo general categories:

� Certain parameters: production level of each prod-uct, purchase cost (price), inventory cost, the num-ber of on-time deliveries, values of the goal, andgoal's importance coe�cients;

� Uncertain parameters: capacity and transportationcost.

By adopting ABC method and removing single-source parts from the list, 129 suppliers were examined.As mentioned before, the model was implementedfor only two vehicle models in manufacturing plant(Tehran). The robust model was solved based on three-periodic (monthly) production plan.

In order to determine the priority of goals (ob-jectives), we applied AHP technique. To do that, wecollected the opinions of 10 top managers and expertsof the company through AHP questionnaire, and thenanalyzed them by using Excel spreadsheets and ExpertChoice software (see Table 1).

The inconsistency ratio was set equal to 0.04,although 0.1 was usually reliable for this index.

8. Computational results

After solving the model, the following three solutions,called Normal, Pessimistic, and Optimistic, were ob-tained.

� Normal solution: solving robust-fuzzy model innormal mode, namely, considering interval half-length of supplier capacity parameter in fuzzy form;

� Pessimistic solution: solving robust-fuzzy modelwith � = 1, namely, considering maximum amount

Table 1. Importance coe�cients of goals.

Objectives (goals) First Second Third

Importance coe�cient of goal 0.364 0.327 0.309

for interval half-length of supplier capacity parame-ter;

� Optimistic solution: solving robust-fuzzy modelwith � = 0, namely, considering minimum amountfor interval half-length of supplier capacity parame-ter.

Pessimistic and optimistic solutions could showthe accuracy of robust-fuzzy model's results. Obvi-ously, Normal solution should be between Pessimisticand Optimistic solutions.

We considered �1 and �2 for 11 di�erent val-ues. Therefore, for each type of solution (Normal,Pessimistic, and Optimistic), 11 models were solved.

The role of protection level parameter, �i, is usu-ally set by model builder, according to risk preferenceof decision maker. Its value is equal the number ofuncertain parameters of which the uncertainties shouldbe considered by the model. In other words, valueof parameter, �i, is equal to the number of uncertainparameters that are allowed to violate their nominalvalues. For example, �i = 0 corresponds to an absoluteconservative manager. For more information, refer toBertsimas and Sim [33].

8.1. Verifying the model by using Monte Carlosimulation

To measure performance accuracy of the model (qualityof robust solutions), and to calculate the risk, we rana simulation model by randomly generating uncertainparameters 10000 times with di�erent deterministicparameters. In general, a total of 33 simulation runswere performed in order to determine the number ofconstraints that were violated. In other words, for eachprotection level, the ratio of the total number of vio-lated constraints to total constraints was determined.

The simulation model was implemented to testand validate the result obtained via the robust model.The outputs of simulation model are the probability ofconstraints violation or risk rate. For this research, therole of simulation and its relationship with mathemat-ical model are shown in this form.

In fact, managers could determine the values ofdecision variables after selecting a desired risk rate.

8.2. Results and analysisFrom Table 2 and Figure 6, it can be inferred thatincreasing the protection level causes the level ofconstraints satisfaction (�) to drop.

Figure 7 shows that the robust-fuzzy model values(Normal solution) are in balance compared with twoother solutions (although all the three begin from thesame point). On the other hand, it can be inferred thatwhen the protection level increases, graphs (Figure 7)are further away from each other. Therefore, in thesepoints, it is clear that interval half-lengths are fuzzy.

Table 2 and Figure 7 show that the levels of risk


Table 2. The objective function values for Normal, Pessimistic, and Optimistic solutions.

Normal solution Pessimistic solution Optimistic solutionProtection

levels�1 �2 � Sum of deviations

from goalsSum of deviations

from goalsSum of deviations

from goals

1 0 0.0 1 0.18335 0.18335 0.18335

2 39 0.1 0.905 0.223651 0.225652 0.204661

3 78 0.2 0.819 0.260208 0.267845 0.225725

4 117 0.3 0.740 0.293585 0.310022 0.24683

5 156 0.4 0.668 0.324197 0.352209 0.267928

6 195 0.5 0.601 0.352368 0.39446 0.289015

7 234 0.6 0.540 0.378407 0.436768 0.310102

8 273 0.7 0.477 0.404853 0.479143 0.331194

9 312 0.8 0.413 0.432246 0.521548 0.352286

10 351 0.9 0.349 0.459137 0.563964 0.373407

11 387 1.0 0.287 0.485414 0.607242 0.394537

Figure 6. Behavior of objective function with respect to(�) in Normal solution.

Figure 7. Behavior of sum of deviations from goals.

or conservation level highly a�ect the slopes of thelines of objective function values. Thus, the robustmodel, compared with the deterministic model, hasmore signi�cant e�ects on risk reduction.

In Table 3, columns a-c show the percentages ofdeviation from goal 1 to 3 for 3 types of solutions,respectively.

In Table 4, columns a-c show the probability of

Figure 8. The trend of the probability of constraintsviolation in di�erent modes based on Indicator 1.

�rst goal constraints violation, probability of capacityconstraints violation, and the sum of probabilities ofviolation, respectively. Table 4 indicates the riskrate (probability of constraints violation) based onindicator 1.

From Table 2, it can be seen that increasing theprotection levels decreases the objective function (�)and the probability of violation. In fact, as the levelof protection increases, the model variable values willbe chosen more stringently in the allowed interval sothat probability of limitation violation and value of theobjective function decrease. This could be a reasonwhy the performance of robust modeling is accurate.The numbers in Table 4 are the outcome of simulationand show that increasing protection level decreasesthe probability of violation. This process indicatesthe accuracy of the performance and simulation of therobust model (see Figures 8 and 9).

The �rst goal constraint has 387 uncertain param-eters and includes protection level of �1. The other


Table 3. The percentage of deviation from each goal.


levels�1 �2 a b c a b c a b c

1 0 0.0 27.90 20.05 5.24 27.90 20.05 5.24 27.90 20.05 5.242 39 0.1 28.00 20.11 18.11 28.00 20.11 18.76 28.00 20.08 12.003 78 0.2 28.02 20.18 29.85 28.02 20.19 32.31 28.02 20.11 18.764 117 0.3 28.03 20.24 40.57 28.03 20.27 45.86 28.03 20.15 25.545 156 0.4 28.04 20.30 50.41 28.04 20.35 59.42 28.04 20.19 32.316 195 0.5 28.04 20.35 59.46 28.03 20.45 73.00 28.04 20.23 39.097 234 0.6 28.04 20.41 67.84 28.00 20.56 86.61 28.04 20.27 45.878 273 0.7 28.03 22.32 74.38 27.98 20.63 100.26 28.04 20.31 52.659 312 0.8 28.01 26.21 79.15 27.98 20.68 113.95 28.05 20.35 59.4310 351 0.9 29.34 28.36 84.01 27.97 20.73 127.63 28.04 20.40 66.2211 387 1.0 32.32 28.37 89.00 27.96 20.76 141.62 28.03 20.45 73.02

Table 4. Probability of constraints violation (risk rates) in di�erent modes based on Indicator 1.


levels�1 �2 a b c a b c a b c

1 0 0.0 0.002 0.082 0.084 0.002 .082 0.084 0.002 0.082 0.0842 39 0.1 0 0.074 0.074 0 0.073 0.073 0 0.077 0.0773 78 0.2 0 0.067 0.067 0 0.066 0.066 0 0.074 0.0744 117 0.3 0 0.062 0.062 0 0.059 0.059 0 0.07 0.0705 156 0.4 0 0.056 0.056 0 0.052 0.052 0 0.066 0.0666 195 0.5 0 0.051 0.051 0 0.043 0.043 0 0.063 0.0637 234 0.6 0 0.046 0.046 0 0.035 0.035 0 0.059 0.0598 273 0.7 0 0.041 0.041 0 0.027 0.027 0 0.056 0.0569 312 0.8 0 0.037 0.037 0 0.018 0.018 0 0.052 0.05210 351 0.9 0 0.033 0.033 0 0.009 0.009 0 0.048 0.04811 387 1.0 0 0.029 0.029 0 0 0 0 0.044 0.044

Figure 9. The trend of the probability of constraintsviolation in di�erent modes based on Indicator 2.

constraints, including uncertain parameters, consist of387 capacity constraints with protection level of �2.

Simulation results indicate that only some of thecapacity constraints (up to a maximum of 73) areopen to violation. The reason is that the capacityof some suppliers is more than needed or the modelhas considered a share which is less than the existing

capacity so that the considered uncertainty for someparameters of the supplier cannot a�ect them. Accord-ing to this explanation, two indicators are consideredfor risk calculation.

� Indicator 1. To divide the total number of violatedstates by the total number of states;

� Indicator 2. To divide the total number of violatedstates by the total number of possible states whichare dependent on the maximum number of violablestates.

Figures 10 and 11 demonstrate the trade-o� be-tween sum of deviations and probability of constraintsviolation based on indicator values 1 and 2, respec-tively.

9. Conclusion

In this study, an innovative robust-fuzzy methodologywas developed and applied to a real case study ofsupplier selection problem within automobile industry


Figure 10. Trade-o� between sum of deviations andprobability of constraints violation based on Indicator 1.

Figure 11. Trade-o� between sum of deviations andprobability of constraints violation based on Indicator 2.

with multiple uncertainties. This approach was basedon the integration of techniques of robust optimization,developed by Bertsimas and Sims' approach (2004),and fuzzy programming, based on Werners' approach(1978b), to handle two modes of uncertainty. Inrobust optimization models, uncertainty of parametersis modeled as random variables that take value withina symmetrical interval. In fact, our innovative robust-fuzzy method was an improved version of robust opti-mization model. This approach was properly respon-sive to uncertainty of decision maker about intervalhalf-length and could provide the necessary exibilityfor the decision maker. In other words, this methodwas able to be responsive to both types (modes) ofuncertainty.

The performance of the results of the proposedmodel in comparison with the results obtained viasimulation techniques was proved to be reliable undervarious conditions. In other words, the proposedmodel could properly manage a variety of uncertainconditions and provide useful and reliable informationfor exible decision making processes. The exibilityof the decision making process is shown in Figures 7-9.

Although this study is the �rst attempt to sup-plier selection decision making problem through thisapproach, the results suggest that it can also beextended from both theoretical and practical points ofview.

In addition, the following managerial implicationsare of great importance: As stated in Sections 1, 2, and3, the issue of uncertainty is very important in supplierselection problem because of the inherent uncertainty

associated with various factors such as capacity andtransportation cost. Due to the high volume of requiredparts in this case study and considering multiplecriteria as well as the condition of uncertainty, choosingsuitable suppliers is of great importance.

In general, robust optimization concept and es-pecially innovative robust-fuzzy method are easy tounderstand and implement, especially by automotiveindustry managers. Robust optimization approach isespecially more practical and applicable in comparisonwith stochastic programming models. As explainedbefore, in robust optimization approach, uncertain datatake value within an interval. In some real cases, suchas our case study, it is di�cult for managers to deter-mine the exact interval length. Furthermore, in thesesituations, the decision maker suggests an ambiguousinterval half-length. To overcome this problem, wedeveloped an innovative approach and proposed theinterval half-length in terms of fuzzy number. Thenew method could help managers to achieve exibilityin decision making in case of ambiguous interval half-length. Consequently, the innovative method could beof great interest to managers as well as academics.

In addition, the following implications can beinferred from the mathematical model and computa-tional results:

1. The mathematical model considered multiple crite-ria in the form of three objectives (goals). Thiscould better match the real world. This modelhelps automotive industry supply chain managersto make integrated and comprehensive decisionsconsidering various issues such as multiple prod-ucts, multiple plants, multiple suppliers, and mul-tiple decision-making periods;

2. Computational results demonstrated that managerswith a desired risk level could make a suitabledecision. In other words, as Tables 2, 3, and 4indicate, they have 11 options to select. An optimalcombination of suppliers can exist for each selectionoption;

3. The implementation of the proposed method didnot require extensive and complex historical data;only determining an interval for uncertain parame-ters by experts was enough;

4. In this context, the improved decision making isrelated to robustness of solutions (see Figure 1).In uncertain environment, a proper decision isequivalent to a robust decision. In the proposedapproach, the main decision is the value of supplyrate of part m from supplier n in period t for plantj (value of Smnji). A robust value for this variableis equivalent to robust decision. Managers candetermine this value after selecting a desired riskrate;


5. Due to the nature of auto industry, the logisticmanger has to manage thousands of parts andcomponents. Obviously, the lack of even onepart can stop the production line and impose anincredible cost as well as some intangible costs,such as customer dissatisfaction. From this pointof view, this industry is much more sensitive toappropriate supplier selection than others. Onthe other hand, inherent uncertainty of supplierrelationship is a crucial factor, especially in the de-veloping countries. Therefore, this industry needspractical, e�cient, and reliable methods to managethe logistics of parts and components.

The contribution of this research from the view-point of the value added to companies is presented asfollows.

The innovative model is applicable and e�cientfor manufacturing companies in which the issue ofuncertainty is highly signi�cant. In other words, dueto the existence of many important sources of uncer-tainties in business environment, reliable data cannotbe found and the classical supply selection models arenot applicable any more. Thus, our proposed methodis applicable for the companies with uncertain inputdata. Although our study was planned for a large autoindustry, it could be implemented in other industriessuch as �nancial, banking, insurance, etc.

The major limitation of the research is how todetermine the membership function of fuzzy numbers.Furthermore, the model is not quite applicable to\make to order" manufacturing systems. Future re-search can extend this approach to cover other systems.Another issue in future research can be using arti�cialintelligence techniques instead of robust optimizationmethods.

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Biographies

Masood Rabieh received his PhD degree from Tar-biat Modaress University, Iran. He is an AssistantProfessor in the Department of Industrial Managementat Shahid Beheshti University, Iran. His researchinterests include operations research, system dynamicsand system thinking, supply chain management, andproject management.

Mohammad Modarres received his PhD degreefrom University of California, Los Angles (UCLA),USA. He is professor in the Department of Indus-trial Engineering at Sharif University of Technology,Iran. His research interests include operations research,mathematical modeling, and stochastic modeling.

Adel Azar received his PhD from University ofTehran, Iran. He is Professor in the Departmentof Industrial Management at Tarbiat Modaress Uni-versity, Iran. His research interests include fuzzymanagement science, managerial and economic systemsmodeling, business process reengineering, system engi-neering, performance management, and performance-based budgeting.

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