Selecting and full ranking suppliers with imprecise data_
ORIGINAL ARTICLE
Selecting and full ranking suppliers with imprecise data:A new DEA method
Mehdi Toloo
Received:8August 2012/Accepted:2June 2014/Published online:24June 2014#Springer-Verlag London 2014
Abstract Supplier selection,a multi-criteria decision making (MCDM)problem,is one of the most important strategic issues in supply chain management (SCM).A good solution to this problem significantly contributes to the overall supply chain performance.This paper proposes a new integrated mixed integer programming ‐data envelopment analysis (MIP ‐DEA)model for finding the most efficient suppliers in the presence of imprecise https://www.wendangku.net/doc/6d15119344.html,ing this model,a new method for full ranking of units is introduced.This method tackles some drawbacks of the previous methods and is com-putationally more efficient.The applicability of the proposed model is illustrated,and the results and performance are compared with the previous studies.
Keyword Data envelopment analysis .Imprecise data .Supplier selection .Full ranking method .Uncertainly .Best efficient unit
1Introduction
Charnes et al.[6]introduced an innovative data oriented mathematical approach,data envelopment analysis (DEA),for evaluating numerous relative efficiency measurements of a set of peer decision making units (DMUs).Each DMU converts multiple inputs to multiple outputs.Numerous researches in various fields and various applications have quickly showed that DEA is an outstanding and straightfor-ward methodology for modeling operational process in
performance evaluations.As a result,this approach has rapid-ly gained too much attention and widespread use by the scientists.Nowadays,DEA becomes the important analysis tool and research method in management science,operational research,system engineering,decision analysis,and so on.Generally,traditional DEA models partition all DMUs into two main groups:efficient and inefficient.These models fail to discriminate between efficient DMUs.To capture this shortcoming,many different approaches are proposed:Cross-efficiency methods utilize DEA in peer evaluation in-stead of only a self-evaluation [21].Super-efficiency ranking methods eliminate the data on the under evaluation unit from the solution set [4].Benchmark ranking methods rank effi-cient DMUs by measuring their importance as a benchmark for inefficient DMUs [32].Ranking with multivariate statistics in DEA context,including linear discriminant analysis,uses the statistical techniques in alliance with DEA (Sinuany-Stern et al.[20]).In ranking inefficient DMUs approaches a new Measure of Inefficiency Dominance (MID)index is intro-duced to rank all inefficient units [5].Notwithstanding with the multi-criteria decision making (MCDM)methodologies which do not consider a complete ranking as their ultimate aim,DEA-MCDM methods deal with the use of preference information to further improve the discriminating power of the DEA models [12].
Besides all ranking studies,the problem of finding a single efficient DMU,known as the most (best)efficient DMU in DEA,has called the attention of some researchers:Karsak and Ahiska [15]proposed a MCDM-DEA model in order to select the most efficient advanced manufacturing technology (AMT).Amin and Toloo [2]and Toloo and Nalchigar [24]tried to improve and extend the approach of Karsak and Ahiska [15].Ertay et al.[9]suggested a minimax model in order to detect the most efficient layout in the facility layout design (FLD)problem.Farzipoor Saen [10]proposed an approach to deal with the supplier selection problem in supply chain system.Toloo et al.[25],Toloo and Nalchigar [27]and
M.Toloo (*)
Department of Business Administration,Faculty of Economics,Technical University of Ostrava,Sokolska tr ˇ.33,70121Ostrava 1,Ostrava,Czech Republic e-mail:mehdi.toloo@vsb.cz M.Toloo
e-mail:m_toloo@https://www.wendangku.net/doc/6d15119344.html,
Int J Adv Manuf Technol (2014)74:1141–1148DOI 10.1007/s00170-014-6035-9
Amin et al.[3]formulated an integrated MIP model for finding the most efficient discovered rules in data mining and designed an algorithm to rank all efficient discovered rules.Farzipoor Saen[11]proposed an MIP model to handle the media selection problem.Some new approaches can be found in Toloo[31],Toloo and Ertay[29]and Toloo and Kresta[30].Although ranking approaches can be applied to find the most efficient unit,it is unnecessary to rank all efficient DMUs and then find the most efficient one.More importantly,traditional DEA models usually consider variable set of optimal weights for each DMU,meanwhile the most efficient DMU should be founded in an identical condition which can be achieved by the common set of optimal weights (CSW)approaches.These approaches enjoy considerable ad-vantages in terms computational issues.Instead of solving at least one optimization problem for each DMU in variable set of optimal weights approaches,an integrated model can be formulated to find the most efficient DMU in CSW approaches.
Supplier selection is a critical managerial decision making problem,which requires considering various qualitative and quantitative factors.The goal is to find the best supplier for meeting a firm’s needs consistently and at an acceptable cost [16].This problem is a fundamental issue of supply chain area,a kind of MCDM which requires MCDM methods for an effective problem solving.A good solution to this problem significantly contributes to the overall supply chain perfor-mance.Numerous methods have been proposed to tackle the problem of supplier selection.Analytic hierarchy process (AHP),artificial intelligence(AI),analytic network process (ANP),linear programming(LP),mathematical program-ming,multi-objective programming,DEA,neural networks (NN),and fuzzy set theory(FST)are instances of methods that have been proposed in literature.A comprehensive literature review of the MCDM approaches in supplier evaluation and selection is published by Ho et al.[14].The readers are referred to this paper for further discussion on methods.
Recently,Toloo and Nalchigar[26]proposed a new DEA method for selecting and ranking suppliers in the presence of imprecise data,particularly when it is of cardinal,ordinal,or bounded form.As will be seen subsequently,their approach has some drawbacks.The aim of this study is to capture theses drawbacks.Toward this end,we propose a new integrated MIP‐DEA model for finding the most efficient unit in the presence of cardinal and ordinal data.The model improves the integrated DEA model that was proposed in Toloo and Nalchigar[26].In addition,using the suggested MIP-DEA model,we propose a new ranking method to prioritize units. This method is novel and computationally more efficient than the previous proposed methods.
The remainder of this paper is organized as follows: Section2reviews related studies and investigates the draw-backs in the previous works.Section3presents a new MIP-DEA model which tackles the drawbacks.A new method for full ranking units in the presence of cardinal and ordinal data is proposed in Section4.Section5shows the applicability of proposed method and compares its results and performance with the previous methods.Finally,the paper closes in Section6with some concluding remarks.
2Related works
Since supplier selection problem has been widely studied by many authors,various approaches have been proposed.For instance,Kilincci and Onal[16]proposed a fuzzy analytic hierarchy-based method to select the best supplier for a wash-ing machine company in Turkey.Sevkli et al.[19]applied a hybrid method,data envelopment analytic hierarchy process (DEAHP)methodology to tackle the problem of supplier selection.Lin et al.[18]proposed a hybrid methodology of ANP technique for order preference by similarity to ideal solution(TOPSIS)and LP for supplier selection process. Güneri et al.[13]developed a new approach based on adap-tive neuro-fuzzy inference system(ANFIS)to overcome sup-plier selection problem.
Although extensive research has been carried out on sup-plier selection problem,most of them focus on cardinal data. Hence,the proposed methods are not applicable for conditions in which data is imprecise,especially in the form of ordinal and interval data.In other words,in many real world supplier selection problems,the data of the alternatives is imprecise and many of the traditional methods are not applicable.To encounter these conditions,some authors have proposed sup-plier selections methods that deals with imprecise data. Among others,Farzipoor Saen[10]proposed the following model to identify a set of efficient suppliers in the presence of both cardinal and ordinal data,without considering the non-Archimedean epsilon to have positive decision variables.
π*o?max
X
r?1
s
Y ro
s:t:
X
i?1
m
X io?1
X
r?1
s
Y rjà
X
i?1
m
X ij≤0j?1;2;…;n
X ij∈e D?i i?1;2;…;m j?1;2;…;n
Y rj∈e Dtr r?1;2;…;s j?1;2;…;n
X ij≥0i?1;2;…;m j?1;2;…;n
Y rj≥0r?1;2;…;s j?1;2;…;n
e1Twhere e D?i and e Dtr are the sets of constraints to consider:
&Bounded data:y rj u r ≤Y rj ≤y rj u r ;x ij w i ≤X ij ≤x ij w i .&Weak ordinal data:Y rj ≤Y rk ,X ij ≤X ik .
&Strong ordinal data:Y rj & Ratio bounded data:L rj ≤Y rj r j o ≤U rj and G ij ≤X rj X i j o ≤H ij j ≠j o eT,where L rj and G ij represent the lower bounds,U rj and H ij represent the upper bounds.& Cardinal data:Y rj ?μr b y rj and X ij ?w i b x ij ,where b y rj and b x ij represent cardinal data. Given a set of DMUs,Model (1)finds a subset of efficient units and suggests them to decision maker.Although applicable in many situations,it has a main shortcoming since it identifies efficient suppliers and is not able to find the most efficient supplier candidates and rank them.In other words,using Farzipoor Saen [10]’s method,decision maker is not able to rank sup-pliers and chooses the best one.To overcome these drawbacks,Toloo and Nalchigar [26]proposed the following MIP model to select the most efficient supplier with the common set of weights:M *?min M s :t : M àd j ≥0j ?1;2;…;n X i ?1m X ij ≤1 j ?1;2;…;n X r ?1s Y rj à X i ?1 m X ij td j ?βj ?0 j ?1;2;…;n X j ?1 n d j ?n ?1 e2T 0≤βj ≤1;d j ∈0;1f g j ?1;2;…;n X ij ∈e D ?i i ?1;2;…;m j ?1;2;…;n Y rj ∈e D tr r ?1;2;…;s j ?1;2;…;n X ij ≥ε * i ?1;2;…;m Y rj ≥ε* r ?1;2;…;s j ?1;2;…;n where ε?is the non-Archimedean epsilon.According to Toloo and Nalchigar [26],the binary variable d j is the deviation from the efficiency of DMU j and hence their model minimizes the maximum of the deviations.Based on this assumption,they concluded that DMU j is the most efficient unit if and only if d j ?=0and introduced an algorithm for ranking the suppliers.Nevertheless,based on the efficiency definition in DEA liter-ature,DMU j is CCR-efficient if and only if there exists at least one common set of optimal variables (X ?,Y ?)∈R n (m +s )such that ∑r =1s Y rj ?=∑i =1m X ij ? .Considering the third types of con-straints in Model (2),we can infer that DMU j is a CCR-efficient DMU if and only if d j ??βj ?=0and hence d j ??βj ?is the deviation from the efficiency of DMU j .Therefore,this unit is the most efficient if and only if d j ?=βj ? .On the other hand,?1≤d j ?βj ≤1and obviously it could not be considered as the deviation from the efficiency.In ad-dition,it should be mentioned that a binary variable cannot be considered as deviation from efficiency,and this is the main drawback of Model (2).Considering these modeling drawbacks,the solution of Model (2)does not necessarily shows the most efficient unit.Moreover,Toloo and Nalchigar [26]’s algorithm for ranking units is based on Model (2);consequently,it suffers from the same drawbacks.In the next section,we propose a new MIP-DEA model to find the most efficient unit in the presence of imprecise data.In addition,a new ranking method is designed. 3Proposed model Toloo [23]proposed the following basic integrated LP model for finding a set of candidate DMUs for being the most BCC-efficient unit:z *?min d max s :t : d max ?d j ≥0 j ?1;2;…;n X i ?1 m w i x ij ≤1 j ?1;2;…;n X r ?1 s u r y rj ?u 0?X i ?1 m w i x ij td j ?0j ?1;2;…;n d j ≥0j ?1;2;…;n w i ≥ε*i ?1;2;…;m u r ≥ε* r ?1;2;…;s e3T Solving this model,the user achieves a common set of strictly positive optimal weights (u ?>0,w ?>0),which gives us a set of the most BCC-efficient unit candidates.In other words,DMU k is a candidate for the most BCC-efficient unit if and only if u ?y k ?u 0?w ?x k =0and hence in this model positive variable d k *is the deviation from efficiency of DMU k .Indeed,DMU k is a candidate for the most BCC-efficient unit if and only if d k ?=0. In addition,Toloo [23]proposed the following inte-grated MIP-DEA model to find the most efficient unit from the set of candidates:z ??min d max s :t :d max ?d j ≥0 j ?1;2;…;n X i ?1m w i x ij ≤1j ?1;2;…;n X r ?1 s u r y rj ?u 0? X i ?1 m w i x ij td j ?0j ?1;2;…;n X j ?1 n θj ?n ?1d j ≤M θj j ?1;2;…;n θj ≤Nd j j ?1;2;…;n d j ≥0j ?1;2;…;n θj ∈0;1f g j ?1;2;…;n w i ≥ε *i ?1;2;…;m u r ≥ε * r ?1;2;…;s e4T where M and N are large enough numbers.Obviously,Models (3)and (4)make an assumption that input and output data are exact numbers on a ratio scale.Hence,they are not applicable for situations in which data are imprecise.The term ‘imprecise data ’reflects the situations where some of the input and output data are only known to lie within bounded intervals (interval numbers)while other data are known only up to an order Despotis and Smirlis [8].Cooper et al.[7]and Kim et al.[17]classified imprecise data into four main groups as bound-ed data,weak ordinal data,strong ordinal data,and ratio bounded data as follows: Bounded data: y rj ≤y rj ≤y ˉrj and x ij ≤x ij ≤x ˉij for r ∈BO ;i ∈BI e5T where y rj and y rj are the lower and the upper bounds for r th bounded output,x ij and x ij are the lower and the upper bounds for i th bounded inputs,and BO and BI represent the associated sets involving bounded outputs and inputs,respectively. Weak ordinal data:y rj ≤y rk and x ij ≤x ik for r ∈DO ;i ∈DI or, y r 1≤y r 2≤…≤y rk ≤…≤y rn r ∈DO eT;e6Tx i 1≤x i 2≤…≤x ik ≤…≤x in i ∈DI eT; e7T where DO and DI represent the sets of weak ordinal outputs and inputs,respectively. Strong ordinal data:y rj for j ≠k ;r ∈DO ;i ∈DI or, y r 1 e8Tx i 1 e9T where SO and SI represent the sets of strong ordinal outputs and inputs,respectively.Clearly,SO ?DO and SI ?DI. Ratio bounded data:L rj ≤y rj y r j o ≤U rj j ≠j o eTr ∈RO eTe10TG ij ≤ x rj x i j o ≤H ij j ≠j o eTi ∈RI eT e11T where RO and RI represent the sets of ratio bounded outputs and inputs,respectively. If we incorporate Eqs.(5)–(11)to Model (3)and transform it to a linear model (by adopting similar approach to Toloo and Nalchigar [26]),we get:z *?min d max s :t :d max ?d j ≥0 j ?1;2;…;n X i ?1m X ij ≤1 j ?1;2;…;n X r ?1 s Y rj ? X i ?1 m X ij td j ?0 j ?1;2;…;n d j ≥0j ?1;2;…;n X ij ∈ e D ti i ?1;2;…;m j ?1;2;…;n Y rj ∈e D ?r r ?1;2;…;s j ?1;2;…;n X ij ≥ε*i ?1;2;…;m j ?1;2;…;n Y rj ≥ε* r ?1;2;…;s j ?1;2;…;n e12T In which e D ti and e D ?r are the same sets of con-straints as in Model (1).It should be mentioned that Model (12)finds a set of candidates for the most CCR-efficient unit with a common set of optimal variables (X ?,Y ?)∈R n (m +s )in the presence of imprecise data.The model finds the efficient unit(s)and it is easy to verify that in this model d j is the deviation from efficiency of DMU j and thus this unit is efficient if and only if d j ?=0,or equivalently ∑r =1s Y rj ?=∑i =1m X ij ? . The interest significance of the following lemma is that it allows one to formulate a simpler and more practical model than Model (4). Lemma 1In Model (12),?j d j ≤1 Proof Form the constraints in Model (12),we have d j ? X i ?1 m X ij ? X r ?1 s Y rj ≤1? X r ?1 s Y rj ≤1 which completes the proof. Let E ={j |d j ?=0j =1,…,n }results from solving Model (12).The following theorem indicates the relation between this model and Model (1)which is proposed in Farzipoor Saen [10]. Theorem 1DMU k ∈E is CCR-efficient. Proof Let k ∈E and (X ?,Y ?)∈R n (m +s )be the set of common optimal variables in Model (12).Hence,∑r =1s Y rk ?=∑i =1m X ik ? .Obviously,(X ?,Y ?)is a feasible solution of Model (1)and the related objective function value is equal to one.Therefore,DMU k ∈E is CCR-efficient. After solving Model (12),two mutually exclusive alterna-tives can be occurred for E :If k ∈E ,then DMU k is definitely the best efficient DMU with the common set of optimal variables,(X ?,Y ?).In this case,Model (12)individually is enough to determine the best efficient unit.Otherwise,this model fails to find a single efficient unit.To resolve this problem,we suggest adding some appropriate constraints to the model to impose it to find just a single efficient unit.We formulate the following MIP model to be applied in this situation:z *?min d max s :t :d max ?d j ≥0 j ?1;2;…;n X i ?1m X ij ≤1j ?1;2;…;n X r ?1s Y rj ? X i ?1 m X ij td j ?0 j ?1;2;…;n X j ?1 n θj ?n ?1 d j ≤θj j ?1;2;…;n θj ≤Nd j j ?1;2;…;n X ij ∈e D ti i ?1;2;…;m j ?1;2;…;n Y rj ∈e D ? r r ?1;2;…;s j ?1;2;…;n θj ∈0;1f g j ?1;2;…;n X ij ≥ε* i ?1;2;…;m j ?1;2;…;n Y rj ≥ε* r ?1;2;…;s j ?1;2;…;n e13T According to the model,there is only one zero auxiliary binary variable,says θk ,and the constraint d k ≤θk implies d k =0.In this case,the constraint θk ≤Nd k is redundant.On the other hand,the constraint θj ≤Nd j ,for positive binary variables (θj =1)and a large enough value for N ,leads to positive d j .Now,Lemma 1insures that the constraint d j ≤θj is redundant.Hence,the following Lemma is proved. Lemma 2In Model (13),there is only one unit with zero deviation. Table 1Data of 18suppliers Supplier no.(DMU) Inputs Output TC SR a NB 12535[50,65]226810[60,70]32593[40,50]41806[100,160]52574[45,55]62482[85,115]72728[70,95]833011[100,180]93279[90,120]103307[50,80]1132116[250,300]1232914[100,150]1328115[80,120]1430913[200,350]1529112[40,55]1633417[75,85]172491[90,180]18 216 18 [90,150] a Ranking such that 18≡highest rank,…,1≡lowest rank (x 218>x 216>…> x 217) It should be notice that Model (13)involves less parameter than Model (4)and this improvement is due to Lemma 1.As a result,Model (13)is simpler and more practical than Model (4).In a similar manner as in Toloo [31],a model for obtaining a suitable epsilon value in Model (13)can be extended. In the next section,we utilize Models (12)and (13)to propose an innovative method for full ranking of units. 4Ranking method Various authors have studied the problem of ranking DMUs in DEA.Adler et al.[1]reviewed the ranking methods in DEA and categorized them into six groups.Interested readers are referred to this paper for further discussion on ranking methods.Although there are a lot of ranking methods in DEA,most of them focus on precise data.In other words,ranking units with imprecise data have not well studied.To fill this gap,in this section,we propose a method to rank DMUs as follows: Step 0Let i =1.Replace the constraint ∑j ?1 n θj ?n ?1in Model (13)with ∑j ?1 n θj ?n ?i . Step 1Given all DMUs,solve Model (12).Let E be the set of candidates for the most efficient DMU.For each DMU j ?E ,d j ?is the deviation from the efficiency and could be used to rank DMU j ?E .DMU u has a better rank than DMU v if and only if d u ? . Step 2If |E|=1,then stop.DMU k ∈E is the most efficient unit and ranked first. Step 3Given all DMUs,solve Model (13)and suppose d k ?= 0.DMU k gets the i th rank position. Step 4If for all distinct u ,v ∈E we have d u ?≠d v ? ,then stop. DMU u has a better rank than DMU v if and only if d u ? . Step 5Let i =i +1. Step 6If i =|E |,then stop.Otherwise,go to Step 3.Indeed,in Step 1of the proposed method,a set of DMUs is selected as candidates to be the most efficient unit.In this step,those DMUs,which are marked as non-candidate,are ranked based on their deviation from the efficiency score (d j ?).In Step 2,if we have only one candidate to be the most efficient unit,we stop the algorithm since all DMUs are ranked.In this case, we have ∑r =1s Y rk ?=∑i =1m X ik ?and ?j ≠k ∑r =1s Y rj ??∑i =1m X ij ? <0and clearly DMU k is the most efficient unit.If this not were the case,then Model (13)finds the most efficient unit and this unit gets the first rank position.To rank other candidates for the most efficient unit,we must consider their deviation from the efficiency.If all distinct candidates have different devia-tion from the efficiency,then these units can be ranked based on their deviation value.Otherwise,Model (13)with new updated constraint (∑j =1n θj =n ?i )finds the next top ranked DMU.In other words,if i =t ,then this algorithm rank t th most efficient candidate.As a result,the designated algorithm must be repeated at most |E |?1times to rank all the candidates.The next section of this paper shows the applicability of proposed method and compares its result with previous methods.5Application In this section,the application of the proposed method and models is shown in a supplier selection problem previously used by Toloo and Nalchigar [26].Including 18suppliers,data Table 3Comparison of the proposed method with the previous methods Supplier Results of Toloo and Nalchigar [26]Results of the Proposed Method Supplier status DMU 414Non-candidate DMU 1427Non-candidate DMU 635Non-candidate DMU 1743Candidate DMU 1151Candidate DMU 862Candidate DMU 979Non-candidate DMU 7 – 6 Non-candidate Table 2Results of the proposed method Supplier no.(DMUs) Ranking results from Model (12) i =1 i =2 Ranking results from Model (13)Ranking results from Model (13)18882141414313131344445121212655576668––299991010101011–11121515151316161614777151717171618181817––318 11 11 11 is obtained from other previous studies that used DEA for problem of supplier selection[10,22].These suppliers con-sume two inputs as total cost of shipment(TC)and supplier reputation(SR)to produce bills received from the supplier without errors(NB)as output.The inputs are in cardinal and ordinal scale,respectively,and the output is bounded data. Table1presents the data. By adopting Zhu[33]’s approach,inputs and outputs of suppliers could be written as follows: e D? 1 ?X11?253w1;X12?268w1;X13?259w1;…;X118?216w1 f g e D? 2 ?X218≥X216≥…≥X217 f g e Dt1?50μ1≤Y11≤65μ1;60μ1≤Y12≤70μ1;40μ1≤Y13≤50μ1;…;90μ1≤Y118≤150μ1 f g Using these relations,we solve Model(12).The results indicate that E?8;11;17 f g which means DMU8,DMU11, and DMU17are the candidates to be the most efficient DMU. For the rest of DMUs,d j?E ?is the deviation from the efficiency score and is used to rank them(See Table2).Notice that based on the results of Model(1)suppliers4,6,8,9,11,14,and17 are recognized as the efficient suppliers and as we expected the most efficient supplier candidates selected among these efficient suppliers. Since the condition of Step2is not satisfied,we skip this step and directly proceed to Step3.In this step,we solve Model(13)with i=1.The result shows that d11?=0and d8?= d17?=0.001;hence,we skip Step4.In Step5,since i=2<|E|= 3,we go to Step3and solve Model(13)with i=2.The result indicates that d8?=0.As a result,DMU8gets the second rank position.Obviously,the last candidate, DMU17,is ranked third.The stop condition in Step5 is met and all DMUs are ranked.Table2presents the full ranking results. To provide further insights,we compare the performance and results of proposed method with the results of two previ-ous methods on the same data set(See Table3). The last column in Table3indicates that either a DMU is selected as the most efficient unit candidate(candidate)or not (non-candidate).This table illustrates the main drawback of Toloo and Nalchigar[26]’s method.The first three top ranked DMUs in this method(i.e.,DMU4,DMU14,and DMU6)are not even identified as the most efficient unit candidates. Apparently,a reliable approach is required to rank top DMUs among the most efficient unit candidates(i.e., DMU11,DMU8,and DMU17).This inconsistency in the method,as explained in Section2,lies in the fact that Model (2)is formulated incorrectly.As can be extracted from Table3, notwithstanding Toloo and Nalchigar[26]’s method,the pro-posed method in this paper results in reliable ranking.Indeed, the most efficient unit candidates are at the top of ranking in our new suggested method.Hence,in comparison with the previous method,the proposed method provides more deci-sion aids to decision maker and captures drawbacks of the previous studies. Finally,to compare these methods from computational efficiency,we can say that Farzipoor Saen[10]’s method requires decision maker to solve18LPs to find7efficient suppliers whereas in our approach,2(=|E|?1)MIPs must be solved.Although dealing with LPs is more computationally efficient than MIPs,Farzipoor Saen[10]’s method selects suppliers based on variable set of optimal weights meanwhile our integrated approach is based on the common set of optimal https://www.wendangku.net/doc/6d15119344.html,mon set of optimal weights lets us to select suppliers in an identical condition.As a result,the suggested method in this study is more logical and practical. 6Conclusion Supplier selection has been considered as one of the most important strategic issues in SCM.Many practitioners and researchers have emphasized that a well designed and imple-mented supply chain system play a critical role in increasing competitive advantages of companies.This paper developed a supplier selection approach based on DEA for conditions in which data of suppliers are imprecise.Although previous researches have considered the problem of imprecise data in supplier selections,their approach suffers from some draw-backs.The main contribution of this paper was to propose a new integrated MIP-DEA model to overcome these draw-backs and to use the new model to propose a new method for full ranking of units.The applicability of the new method was shown,and results were compared with the previous studies.Future research could use the proposed method of this paper,with mirror modifications,for other multi-criteria managerial decision making problems such as selection of media,technology,and international market.Moreover,the formulated models and designated algorithm in this study can be extended to find the most BCC-efficient unit in the pres-ence of imprecise data. Acknowledgment The presentation of this paper was greatly improved by numerous and most helpful remarks of anonymous referees to whom the author would like to express his sincere thanks.The research was supported by the Czech Science Foundation(GACR project14-31593S) and through European Social Fund within the project CZ.1.07/2.3.00/ 20.0296. 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