Fuzzy BCC Model for Data Envelopment Analysis
Fuzzy BCC Model for Data Envelopment Analysis
SAOWANEE LERTWORASIRIKUL 1aapsal@ku.ac.th Department of Product Development,Faculty of Agro-Industry,Kasetsart University,10900,Thailand SHU-CHERNG FANG fang@https://www.sodocs.net/doc/491648529.html, Department of Industrial Engineering,North Carolina State University,Raleigh,NC 27695-7906,USA HENRY L.W.NUTTLE nuttle@https://www.sodocs.net/doc/491648529.html, Department of Industrial Engineering,North Carolina State University,Raleigh,NC 27695-7906,USA JEFFREY A.JOINES jeffjoines@https://www.sodocs.net/doc/491648529.html, Department of Textile Engineering,North Carolina State University,Raleigh,NC 27695-8301,USA
Abstract.Fuzzy Data Envelopment Analysis (FDEA)is a tool for comparing the performance of a set of activities or organizations under uncertainty environment.Imprecise data in FDEA models is represented by fuzzy sets and FDEA models take the form of fuzzy linear programming models.Previous research focused on solving the FDEA model of the CCR (named after Charnes,Cooper,and Rhodes)type (FCCR).In this paper,the FDEA model of the BCC (named after Banker,Charnes,and Cooper)type (FBCC)is studied.Possibility and Credibility approaches are provided and compared with an -level based approach for solving the FDEA https://www.sodocs.net/doc/491648529.html,ing the possibility approach,the relationship between the primal and dual models of FBCC models is revealed and fuzzy efficiency can be https://www.sodocs.net/doc/491648529.html,ing the credibility approach,an efficiency value for each DMU (Decision Making Unit)is obtained as a representative of its possible range.A numerical example is given to illustrate the proposed approaches and results are compared with those obtained with the -level based approach.
Keywords:data envelopment analysis,fuzzy mathematical programming,possibility theory,credibility measure,efficiency analysis
1.Introduction
Data Envelopment Analysis (DEA)is a non-parametric technique for evaluating the performance of many activities.DEA evaluates the relative efficiency of a set of homogenous decision making units (DMUs)by using a ratio of the weighted sum of outputs to the weighted sum of inputs.Specifically,it determines a set of weights such that the efficiency of a target DMU (DMU o )relative to the other DMUs is maximized.DEA has been used in several contexts including education systems,health care units,agricultural production,and military logistics.(Arnade (1994),Charnes et al (1994),Cooper et al (2000)).
The frequently used DEA models are the CCR,named after Charnes,Cooper,and Rhodes (Charnes et al (1978)),and the BCC models,named after Banker,Charnes,and Cooper (Banker et al (1984)).The CCR model has its production frontier spanned by the linear combination of the existing DMUs,while the BCC model has its production frontier spanned by the convex hull of the existing DMUs.Therefore,the set of feasible activities,called production possibility set,for the CCR and BCC models are different.
Fuzzy Optimization and Decision Making,2,337–358,2003#
2003Kluwer Academic Publishers.Printed in The
Netherlands.
The frontiers of the CCR model have linear characteristics,while those of the BCC model have piece-wise linear and concave characteristics as shown in Figures 1and 2.From the production frontiers of both CCR and BCC models,a DMU is inefficient if it is possible to reduce any input without increasing any other inputs and achieve the same levels of outputs or it is possible to increase any output without reducing any other outputs and use the same levels of inputs.
The relative efficiency of a DMU,which is the ratio of the weighted sum of its outputs to the weighted sum of its inputs,falls in the range of (0,1].If the efficiency of a DMU is equal to 1,the DMU is weakly efficient (technically efficient).A DMU is CCR-Efficient (Charnes et al (1985))or BCC-Efficient (Banker et al (1984)),if its efficiency value is equal to 1and its input excesses and output shortfalls are zero.Further details of the CCR and BCC models can be found in many references (Charnes et al (1994),Cooper et al (2000)).Besides from the CCR and BCC models,other well-know DEA models include the ‘‘Additive model,’’the ‘‘Free Disposal Hull’’(FDH)model,and the ‘‘Slacks-Based Measure of Efficiency’’(SBM)model.More details on other DEA models and their applications can be found in (Charnes et al (1994),Cooper et al (2000),Seiford and Thrall (1990),Sengupta (1995)).
The traditional DEA models require accurate and precise performance data since it is a methodology focused on frontiers or boundaries.A few changes in data can change efficient frontiers significantly.However,in some situations,such as in a manufacturing system,a production process or a service system,inputs and outputs are volatile and
complex.
Figure 1.Production frontier of the CCR model.
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Most of the previous studies that deal with inaccurate and imprecise data in DEA models have simply used simulation techniques like the one in Banker et al.(Banker et al (1996)).Some shortcomings of the previous methods can be found in Cooper et al.(Cooper et al (2000)).Cooper et al.(Cooper et al (1999))dealt with the problem of imprecise data in DEA models where some of input and output data are interval data.However,the data available for efficiency analysis will often be in the form of qualitative,linguistic data,e.g.,‘‘old’’equipment,‘‘high’’inventory,and cannot be represented by interval data.Fuzzy set theory,established by Zadeh (Zadeh (1965)),has been proven to be useful as a way to quantify imprecise and vague (linguistic)data in DEA models.The DEA models with fuzzy data (‘‘fuzzy DEA’’models)can more realistically represent real-world problems than the conventional DEA models.
Some approaches that have been published on solving fuzzy CCR (FCCR)problems are the defuzzification approach,the -level based approach,and the fuzzy ranking approach.In the defuzzification approach (Lertworasirikul (2002)),fuzzy inputs and fuzzy outputs are first defuzzified into crisp https://www.sodocs.net/doc/491648529.html,ing these crisp values,the resulting crisp model can be solved by an LP solver.In the -level based approach,the fuzzy DEA model is solved by parametric programming using cuts.Solving the model at a given level produces a corresponding interval efficiency for the target DMU.A number of such intervals can be used to construct the corresponding fuzzy efficiency.More detail on this approach can be found in Maeda,Entani and Tanaka (Meada et al (1998)),Kao and Liu (Kao and Liu (2000)),and Lertworasirikul (Lertworasirikul (2002)).The fuzzy ranking approach was developed by Guo and Tanaka (Guo and Tanaka (2001)).Both
fuzzy
Figure 2.Production frontier of the BCC model.
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inequalities and fuzzy equalities in the FCCR model are defined by ranking methods so that the resulting model is a bi-level linear programming model.
Even though several approaches have been developed to solve fuzzy DEA models,each of them still has shortcomings in the way it treats uncertain data in DEA models as mentioned in(Lertworasirikul et al(2003)).The key point is that fuzzy DEA models take the form of fuzzy linear programs,which requires ranking of fuzzy https://www.sodocs.net/doc/491648529.html,ing different ranking methods can result in different results.
In recent papers by Lertworasirikul et al.(Lertworasirikul et al(2003),Lertworasirikul et al(2001),Lertworasirikul et al(2003),Lertworasirikul et al(2002)),a possibility approach and a credibility approach have been proposed for solving FCCR models.The possibility approach uses the concepts of possibility measures and chance-constrained programming.It deals with the uncertainty in fuzzy objectives and fuzzy constraints through the use of possibility measures.The credibility approach uses the‘‘expected credits’’of fuzzy variables to deal with the uncertainty in fuzzy objectives and fuzzy constraints.These two approaches transform fuzzy DEA models into meaningful and unambiguous models.In this paper,we extend the possibility and credibility approaches to primal and dual models of fuzzy BCC(FBCC).Using the possibility approach,the relationship between the primal and dual FBCC models is revealed.The relationship between the primal and dual models can be extended to general fuzzy linear program-ming models.
The rest of this paper is organized as follows.Section2presents the primal and dual models of FBCC.Section3introduces the basic concepts of possibility,necessity,and credibility measures,as well as the expected credit operator of fuzzy variables.In Section4, the possibility approach is used to solve the FBCC https://www.sodocs.net/doc/491648529.html,ing this approach,the primal and dual FBCC models are transformed into PBCC P and PBCC D,respectively.Section5, the credibility approach is used to solve the FBCC models.Section6provides a numerical experiment to illustrate the possibility and credibility approaches to FBCC models in determining relative efficiencies for the case of symmetrical triangular fuzzy parameters. Finally,Section7concludes the paper,and discusses some future research directions.
2.Fuzzy BCC Models
In this paper,we focus on the input-oriented fuzzy BCC model.Suppose that there are n DMUs,each of which has m inputs and r outputs of the same type.All inputs and outputs are assumed to be nonnegative,but at least one input and one output are positive.The BCC model with fuzzy coefficients and its dual are given in Equations(1)and(2).
eFBCC PTmax
u;v; o
v T e y oà o
u T e x o?1àu T e Xtv T e Yàe T o0
u!0
v!0
e1TLERTWORASIRIKUL ET AL
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eFBCC D Tmin
; B
B
s :t : B e x
o àe X !0e Y !e y o e T
?1 !0;
e2T
where ~x o is a column vector of fuzzy inputs consumed by the target DMU (DMU o ),~
X is a matrix of fuzzy inputs of all DMUs,~y o is a column vector of fuzzy outputs produced
by the target DMU (DMU o ),~
Y is a matrix of fuzzy outputs of all DMUs,u 2R m ?1is a column vector of input weights,v 2R r ?1is a column vector of output weights, o is a scalar value being free in sign, 2R n ?1is a column vector of a linear combination of n DMUs,e is a column vector with all elements equal to 1,and h B is the objective value of the FBCC D model.
The interpretation of the fuzzy BCC model is similar to the traditional BCC model,which can be found in many references (Banker et al (1984),Cooper et al (2000)).
The constraints u T ~x o ?1and àu T ~X tv T ~
Y à o 0in the FBCC P model are used for normalization of the value v T ~y o à o .However,the objective value v T ~y o à o can now exceed one since the first and second constraints of Equation (1)are approx-imately satisfied.That is,since their parameters are fuzzy sets,u T ~x o is ‘‘approxi-mately equal to one,’’which implies that v T ~y o à o u T ~x o is ‘‘approximately less than or equal to one.’’
The FBCC D model has a feasible solution,h B ?1, i ?0for i ?o ,and o ?1.Therefore,
the optimal value B
*of the FBCC D model is not greater than 1.Also,because the amount of inputs and outputs is assumed to be nonzero,the constraints ~
Y !~y o and e T ?1force
to be a nonzero vector.This along with B ~x o à~X !0implies that B
*>0.Therefore, B *falls in the range (0,1].
The fuzzy BCC models cannot be solved by a standard LP solver like a crisp BCC model because coefficients in the fuzzy BCC model are fuzzy sets.In the next section,we provide the concepts of possibility,necessity,and credibility measures which will be useful to solve the fuzzy BCC models.
3.Possibility,Necessity and Credibility Measures
Zadeh (Zadeh (1978))proposed possibility theory in the context of the fuzzy set theory as a mathematical framework for modeling and characterizing situations involving uncertainty.A good reference can be found in Dubois and Prade (Dubois and Prade (1988)).Zadeh also introduced the ‘‘fuzzy variable,’’which is associated with a possibility distribution in the same manner that a random variable is associated with a probability distribution.In the fuzzy linear programming model,each fuzzy coefficient can be viewed as a fuzzy variable and each constraint can be considered to be a fuzzy event.
FUZZY BCC MODEL FOR DATA ENVELOPMENT ANALYSIS
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3.1.Fuzzy Event via Possibility Measure
Let (?i ,P (?i ), i ),for each i ?1,2,...,n ,be a possibility space with ?i being the nonempty set of interest,P (?i )the collection of all subsets of ?i ,and i the possibility measure from P (?i )to [0,1].
Given a possibility space (?i ,P (?i ), )with ei T e T?0; e?i T?1;and
eii T e[i
A i T?sup i
f eA i T
g wit
h each A
i 2Pe?i T;
Zadeh defined a fuzzy variable,~
,as a real-valued function defined over ?i with the membership function:
~
es T? ef i 2?i j ~ e i T?s gT?sup i 2?i
f ef i gTj ~
e i T?s g ;8s 2R :
e3T
Let (?,P (?), )be a product possibility space such that ???1??2?...??n and from possibility theory (Zadeh (1978)),
eA T?min i ?1;2;...;n
f i eA i Tj A ?A 1?A 2?...?A n ;A i 2Pe?i T
g :
Suppose ~a and ~
b are two fuzzy variables on the possibility spaces (?1,P (?1), 1)and
(?2,P (?2), 2),respectively.Then ~a ~
b is a fuzzy event defined on the product possibility space (???1??2,P (?), ),with
e~a e b T?
sup 22?2
12?1f fe 1; 2Tj ~
a e 1T e
b e 2Tgg ?sup
22?2
12?1min f ef 1gT; ef 2gTg j ~a e 1T e b e 2Tn o
Furthermore,from the definition of fuzzy variables (3),we have
e~a e b T?sup s ;t 2R
min e a e es T; b
e et TTj s t èé
:Similarly,possibilities of the fuzzy events ~a <~b ,and ~a ?~b defined on the product possibility space (?,P (?), )are given as
e~a e~a ?e b T?sup s ;t 2R f min e a e es T; b e et TTj s ?t g ; respectively.In case the right hand side ~ b becomes a crisp value b ,then the possibilities of LERTWORASIRIKUL ET AL 342 the corresponding fuzzy events are given as e~a b T?sup s 2R f a e es Tj s b g ; e~a e4T Let ~a 1;~a 2;...;~a n be fuzzy variables and f j :R n !R be a real-valued function,for j ?1,...,m .The possibility of the fuzzy event ‘‘f j e~a 1;~a 2;...;~a n T 0,j ?1,...,m ’’is given by ef j e~ a 1;~a 2;...;~a n T 0;j ?1;...;m T?sup s 1;...;s n 2R f min 1 i n f a i e es i T g j f j es 1 ;...;s n T 0;j ?1;...;m g :3.2.Necessity Measure Associated with each possibility measure is a necessity measure N .Necessity measures and possibility measures are mutually dual.In particular,given a possibility space (?,P (?), ),if A and A are two opposite events (A is the complement of A in ?),then N eA T?1à eA T: The necessity measures of a fuzzy event is defined as the impossibility of the opposite event.Thus,given fuzzy variables ~a 1;~a 2;...;~a n and f j :R n !R ,j ?1,...,m ,the necessity of the fuzzy events ‘‘f j e~a 1;~a 2;...;~a n T 0’’,‘‘f j e~a 1;~a 2;...;~a n T!0’’,and ‘‘f j e~a 1;~a 2;...;~a n T?0’’,j ?1,...,m are defined respectively by N ef j e~ a 1;~a 2;...;~a n T 0;j ?1;...;m T?1àsup s 1;...;s n 2R f min 1 i n f a i e es i Tg j 9j 2f 1;...;m g ;s :t :f j es 1;...;s n T>0g :N ef j e~ a 1;~a 2;...;~a n T!0;j ?1;...;m T?1àsup s 1;...;s n 2R f min 1 i n f a i e es i Tg j 9j 2f 1;...;m g ;s :t :f j es 1;...;s n T<0g ;N ef j e~ a 1;~a 2;...;~a n T?0;j ?1;...;m T?1àsup s 1;...;s n 2R f min 1 i n f a i e es i Tg j 9j 2f 1;...;m g ;s :t :f j es 1;...;s n T?0g :3.3.Credibility Measure Recently,Liu (Liu (2002))introduced an ‘‘expected value’’operator for fuzzy variables which is obtained by using credibility measures,which,in turn,are derived from possibility and necessity measures.In this paper,the term ‘‘expected credit’’instead of FUZZY BCC MODEL FOR DATA ENVELOPMENT ANALYSIS 343 ‘‘expected value’’is used for a fuzzy variable to distinguish it from the expected value of a random variable. Liu (Liu (2002),Liu (2001a))defines a credibility measure (Cr )of a fuzzy event as the average of its possibility and necessity measures,i.e., Cr eáT? eáTtN eáT 2 :The relationship among possibility,necessity and credibility measures is eáT!Cr eáT!N eáT: Similar to the expected value operator for a random variable in probability theory (Ross (1993),Ross (1996)),the expected credit operator for a fuzzy variable ~ on a possibility space (?,P (?), )is defined by Liu (Liu (2001a))as E e~ T? Z t10 Cr e~ !t Tdt à Z 0 à1 Cr e~ t Tdt : e5T Note that if the fuzzy variable ~ and the credibility measure Cr (á)in (5)are replaced with a random variable and a probability measure,respectively,then (5)becomes the expected value of the random variable.In addition,E ep ~ tq ~’T?pE e~ TtqE e~’ Tfor any real numbers p and q ,which is similar to the case of the expected value of random variables. All fuzzy inputs and outputs in this paper are assumed to be ‘‘normal’’and ‘‘convex.’’The definitions of normal and convex fuzzy sets (Zimmermann (1996))are given below.Definition 1(Normal fuzzy variables)Given a fuzzy variable ~a on a possibility space (?,P (?), ),the fuzzy variable ~a is normal if sup s 2R a e es T?1: Definition 2( -level sets)The -level sets of a fuzzy variable ~a is defined by the set of elements that belong to the fuzzy variable ~a with membership of at least ,i.e.,a ?f s 2R j a e es T! g : Definition 3(Convex fuzzy variables)A fuzzy variable ~a is convex if a e e s 1te1à Ts 2T!min e a e es 1T; a e es 2TTfor all s 1;s 22R ; 2?0;1 :Alternatively,the fuzzy variable ~a is convex if all -level sets are convex.LERTWORASIRIKUL ET AL 344 4.Possibility BCC Model The concept of chance-constrained programming(CCP),which was introduced by Charnes and Cooper(Charnes and Cooper(1959)),is adopted in this paper as a way to solve fuzzy BCC https://www.sodocs.net/doc/491648529.html,P deals with uncertainty by specifying the desired levels of confidence with which the constraints hold.More details about CCP can be found in (Charnes and Cooper(1959),Liu(2002)).Using the concepts of CCP and possibility of fuzzy events,the primal and dual of the FBCC models become the following primal and dual possibility BCC(PBCC)models: ePBCC PTmax u;v; o; f s:t: ev T e y oà o!fT! e1T eu T e x o?1T! oe2T eàu T e Xtv T e Yàe T o0T! e3T u!0 v!0 e6T where and o2[0,1]are prespecified acceptable levels of possibility for constraints (1)and(2),respectively,while ?[ 1,..., n]T2[0,1]n is a column vector of prespecified acceptable levels for the vector of the possibility constraints(3)in the PBCC P model. ePBCC DTmin ; B B s:t: e B e x oàe X !0T! 1e1T ee Y àe y o!0T! 2e2T e T ?1 !0; e7T where 1?? 1;...; m T2[0,1]m is a column vector of prespecified acceptable levels for the possibility constraints(1)and 2?? mt1;...; mtr T2[0,1]r is a column vector of prespecified acceptable levels for the possibility constraints(2)in the PBCC D model. In the PBCC P model,the objective value f is the maximum possible efficiency ev T~y oà oTof the targeted DMU relative to other DMUs at the‘‘possibility’’level or higher,subject to the possibility levels of constraints(2)and(3)being at least o and , respectively.In the PBCC D model,the objective value h B is the minimum possible efficiency of the targeted DMU relative to other DMUs,subject to the possibility levels of constraints(1)and(2)being at least 1and 2,respectively. Using the concept of possibilities of fuzzy events given in Section3,we have the following lemma. Lemma1Let~a1;~a2;...;~a n be fuzzy variables with normal and convex member- ship functions.LeteáTL i andeáTU i denote the lower and upper bounds of the -level FUZZY BCC MODEL FOR DATA ENVELOPMENT ANALYSIS345 set of ~a i ,i ?1,...,n .Then,for any given possibility levels 1, 2,and 3with 0 1, 2, 3 1, (1) (~a 1t...t~a n b )! 1if and only if e~a 1TL 1t...te~a n TL 1 b , (2) (~a 1t...t~a n !b )! 2if and only if e~a 1TU 2t...te~a n TU 2!b , (3) (~a 1t...t~ a n ? b )! 3if and only if e~a 1TL 3t...te~a n TL 3 b,and e~a 1TU 3t...te~a n TU 3!b.The proof for Lemma 1can be found in (Lertworasirikul et al (2003)). Given that fuzzy inputs and fuzzy outputs of the PBCC P and PBCC D models are normal and convex,it follows from Lemma 1that the PBCC P and PBCC D models can be solved by considering: ePBCC1P T max u ;v ; o ;f ev T e y o TU à o s :t : eu T e x o TU o ! 1e1Teu T e x o TL o 1e2Teàu T e X tv T e Y TL àe T o 0 e3T u !0v ! 0: e8T ePBCC1D Tmin ; B B s :t :e B e x o àe X TU 1 !0e1Tee Y àe y o TU 2 !0e2Te T ?1 !0; e9T Depending upon the membership functions of fuzzy parameters in the model,the PBCC1P and PBCC1D models may take the form of linear programming models or nonlinear programming models. From Lertworasirikul,et al (Lertworasirikul et al (2003)),for a trapezoidal fuzzy number r ?i ?eer ?i TL 0;er ?i TL 1;er ?i TU 1;er ?i TU 0T,i ?1,...,n and any given possibility level ,0 1,the following are true: e~r 1t...t~r n b T! if and only if e1à T?er ?1TL 0t...te~r n TL 0 t ?er ?1TL 1t...ter ?n TL 1 b ; e~r 1t...t~r n !b T! if and only if e1à T?er ?1TU 0t...ter ?n TU 0 t ?er ?1TU 1t...ter ?n TU 1 !b ; e~r 1t...t~ r n ?b T! if and only if e1à T?er ?1TL 0t...ter ?n TL 0 t ?er ?1TL 1t...ter ?n TL 1 b and e1à T?er ?1TU 0t...ter ?n TU 0 t ?er ?1TU 1t...ter ?n TU 1 !b : LERTWORASIRIKUL ET AL 346 FUZZY BCC MODEL FOR DATA ENVELOPMENT ANALYSIS347 Therefore,if fuzzy inputs and outputs are of trapezoidal types,PBCC1P and PBCC1D become the following linear programming models. e1à Tev T e y oTU0t ev T e y oTU1à o ePBCC2PTmax u;v; o s:t:e1à oTeu T e x oTU0t oeu T e x oTU1!1 e1à oTeu T e x oTL0t oeu T e x oTL11e1à Teeàu T e XTL0tev T e YTL0Tt eeàu T e XTL1tev T e YTL1Tàe T o0 u!0 v!0: B ePBCC2DTmin ; B s:t:e1à 1Te B e x oàe X TU0t 1e B e x oàe X TU1!0e1T e1à 2Tee Y àe y oTU0t 2ee Y àe y oTU1!0e2T e T ?1 !0; To make a reasonable efficiency comparison of DMUs,the possibility levels of constraints( , o, 1,..., n, 1;...; mtrTin the PBCC P and PBCC D models for these DMUs should be set at the same level. Given that ? o? 1?...? n? 1?...? mtr? ,the objective values of the PBCC2P and the PBCC2D models are the maximum possible efficiency and the minimum possible efficiency of the targeted DMU,respectively,relative to other DMUs at the possibility level .Therefore,interval efficiency can be constructed by solving the PBCC2P and PBCC2D models at a specified possibility level.Then,these intervals for different possibility levels can be used to build the corresponding fuzzy efficiency membership functions for each DMU. However,to determine which DMUs are more efficient,we need a method to rank the fuzzy efficiency.In the crisp BCC(the dual BCC)model,v T y oà o(h B)value of one at the optimal solution signifies that the DMU under consideration is technically efficient. Following this concept,we should use v T~y oà o(h B)to determine if a DMU is technically efficient for the FBCC P(FBCC D)model.Note that the value of v T~y oà o is greater than or equal to h B,therefore,to be optimistic v T~y oà o will be used for the ranking purpose. Correspondingly,ev T~y oTU à o in the PBCC P model is used to determine if a target DMU is technically efficient(in the possibilistic sense)at the specified possibility level( ). We define an -possibilistic efficient DMU and an -possibilistic inefficient DMU as follows. Definition4For FBCC models,a DMU is -possibilistic efficient if itsev T~y oTU à o value at the possibility level is greater than or equal to one;otherwise,it is -possibilistic inefficient. The possibility approach provides the flexibility to decision makers to set their own acceptable (possibility)levels in comparing DMUs.It also provides decision makers with fuzzy efficiency membership functions for each DMU.With these fuzzy membership functions,decision makers can see the ranges of efficiency values at any possiblity levels for each DMU. 5.Credibility BCC Model The credibility approach uses the ‘‘expected credits’’of fuzzy variables to deal with the uncertainty in fuzzy objectives and fuzzy constraints.The approach transforms fuzzy DEA models into credibility programming-DEA models.Similar to the expected value approach to stochastic programming where random variables are replaced by their expected values,in the credibility programming-DEA (CP-DEA)model fuzzy variables are replaced by ‘‘expected credits,’’which are derived by using credibility measures. Let ~ i ,i ?1,...,n ,be a normal,convex fuzzy variable on a possibility space (?i ,P (?i ), i ),and let ~ ?~ 1t...t~ n .It follows that ~ is also a normal,convex fuzzy variable on the product possibility space (?,P (?), ).From Lertworasirikul et al (Lertworasirikul et al (2003)),using the relationship among possibility,necessity,and credibility measures,the expected credit of ~ can be determined by the following lemma. Lemma 2Let ~ be a normal,convex fuzzy variable.Let eáTL and eáTU denote the lower and upper bounds of the -level set of ~ (see Figure 3).If e~ >t Tare approximated by e~ t Tand e~ !t T,respectively,then E e~ T?12 e~ TU 1te~ TL 1tZ e~ TU 0e~ TU 1 e~ !t Tdt à Z e~ TL 1e~ TL 0 e~ t Tdt 0 B @1 C A : When ~ is a normal,convex trapezoidal fuzzy variable,E e~ Tcan be obtained from E e~ T? 14 e~ TU 1te~ TL 1te~ TU 0te~ TL 0h i : This formula can be used for normal,convex symmetrical and unsymmetrical trapezoi-dal fuzzy variables.For the case that the membership functions are symmetrical,the expected credits of normal,convex trapezoidal fuzzy variables are located at the central point of their membership functions. The credibility approach treats the uncertainty in fuzzy objectives and fuzzy constraints by replacing fuzzy variables with their expected credits.In this way,the fuzzy BCC LERTWORASIRIKUL ET AL 348 models (FBCC P and FBCC D )are transformed into the following credibility programming-BCC models: eCP-BCC P T max u ;v ; o E ev T e y o à o Ts :t : E eu T e x o T?1E eàu T e X tv T e Y àe T o T 0u !0v !0: eCP-BCC D Tmin ; B B s :t : E e B e x o àe X T!0E ee Y àe y o T!0e T ?1 !0; Note that no matter what types of fuzzy variable ~ and functional form,g ,we have,for a given decision u ,g (u ,~ )is also a fuzzy variable.Thus,the fuzzy constraints can be treated by taking expected credit operators. Figure 3.The lower and upper bounds of the -level of ~a .FUZZY BCC MODEL FOR DATA ENVELOPMENT ANALYSIS 349 Using the expected credit operators,it follows that the CP-BCC models are either nonlinear or linear programming models depending upon the form of the membership functions (of fuzzy inputs and outputs).When these membership functions are of LR type,the CP-BCC models may take the form of nonlinear programming.For the special case of normal,convex trapezoidal data,the CP-BCC P and CP-BCC D models become linear programming models, max u ;v ; o 1ev T e y o TU 1 tev T e y o TL 1 tev T e y o TU 0 tev T e y o TL 0 h i s :t : 1u T e x o eTU 1tu T e x o eTL 1tu T e x o eTU 0tu T e x o eTL 0h i ?1 14àu T e X tv T e Y U 1tàu T e X tv T e Y L 1 tàu T e X tv T e Y U 0 tàu T e X tv T e Y L 0 ! e T o u !0v !0; min ; B B s :t : 14 e B e x o àe X TU 1te B e x o àe X TL 1te B e x o àe X TU 0te B e x o àe X TL 0h i !014ee Y àe y o TU 1tee Y àe y o TL 1tee Y àe y o TU 0tee Y àe y o TL 0h i !0e T ?1 !0; which can be solved by a standard LP solver (Bertsimas and Tsitsiklis (1997),Fang and Puthenpura (1993),Stanford Business Software (2001)). Since the CP-BCC P and CP-BCC D models use the same parameters (inputs and outputs),the relationship between primal and dual models of linear programming can be applied for the CP-BCC P and CP-BCC D models.To obtain the efficiency value,we can solve either CP-BCC P or CP-BCC D models. Similar to the traditional BCC model,an expected efficiency value E ev T ~ y o à o Tor h B of the target DMU falls in the range of (0,1].In the traditional BCC model,the efficiency value of one signifies that the DMU o is technically efficient.Following this concept,the E ev T ~ y o à o Tor h B in the CP-BCC P CP-BCC D models are used to determine whether a target DMU is technically efficient (in the credibilistic sense). Definition 5For FBCC models,a DMU is a credibilistically efficient DMU if its E ev T ~y o à o Tor h B value is equal to one;otherwise,it is a credibilistically inefficient DMU. LERTWORASIRIKUL ET AL 350 FUZZY BCC MODEL FOR DATA ENVELOPMENT ANALYSIS351 Table1.DMUs with two fuzzy inputs and two fuzzy outputs and the efficiency values from the fuzzy BCC model obtained by credibility approach. DMU(i)12345 input1(4.0,0.5)(2.9,0.0)(4.9,0.5)(4.1,0.7)(6.5,0.6) input2(2.1,0.2)(1.5,0.1)(2.6,0.4)(2.3,0.1)(4.1,0.5) output1(2.6,0.2)(2.2,0.0)(3.2,0.5)(2.9,0.4)(5.1,0.7) output2(4.1,0.3)(3.5,0.2)(5.1,0.8)(5.7,0.2)(7.4,0.9) E( T~y oà o)0.889 1.0000.935 1.000 1.000 The credibility approach provides an efficiency value for each DMU as a representative of its possible range in such a way that decision makers do not have to specify any parameters(confidence levels)as in the possibility https://www.sodocs.net/doc/491648529.html,paring with other approaches,this approach can be implemented more easily and requires less interactive data from decision makers than tolerance, -level based,fuzzy ranking,and possibility approaches,while it deals with uncertain data by using the expected credit instead of defuzzifying uncertain data into crisp values like the defuzzification approach. 6.Numerical Experiment In this section,a numerical experiment is presented to illustrate the possibility approach and the credibility approach for solving FBCC models.The example is taken from Guo and Tanaka(Guo and Tanaka(2001)).Table1provides the data for the example.There are two fuzzy inputs and two fuzzy outputs.These fuzzy inputs and fuzzy outputs have symmetrical triangular membership functions,a special case of trapezoidal membership functions.The membership functions are denoted by(c,d)where c is the center and d is the spread.For the case of unsymmetrical trapezoidal membership functions,it follows the same methodology that we use here. To use the possibility approach,all fuzzy constraints in PBCC P and PBCC D models should be satisfied at the same possibility level,i.e., ? o?...? n? 1?...? mtr? .Let V?[ o,..., n]T and V?? 1;...; mtr T.The efficiency valuesefTfor five different possibility levels(0,0.25,0.5,0.75,1)are provided in Tables2and3. The efficiency values obtained from the credibility approach for the primal and dual FBCC models are the same,and are given at the bottom of Table1.The credibilistic effciency values of DMUs1,2,3,4and5are0.889,1,0.935,1,1,respectively.Therefore, from Definition5DMUs2,4and5are credibilistically efficient,while DMUs1and3are credibilistically inefficient. From the possibility approach to the primal FBCC model,the possibilistic(technical) efficiency values of DMUs1,2,3,4and5are0.889,1,0.935,1,1,respectively,at the possibility level( )1.The results at other possibility levels can be interpreted in a similar https://www.sodocs.net/doc/491648529.html,ing Definition4,DMUs2,4and5are possibilistically efficient at the possibility level1,whereas DMUs1and3are possibilistically inefficient.These results are consistent with those from the credibility approach.From the results in Table2,DMUs 2,4and 5are possibilistically efficient at all possibility levels,whereas DMUs 1and 3are possibilistically efficient only at some possibility levels. From the possibility approach to the dual FBCC model,at ?1the possibilistic (technical)efficiency values of DMUs 1,2,3,4and 5are the same as those from the primal FBCC model.However,using Definition 4,only DMU 2is possibilistically efficient at all possibility levels,whereas DMUs 4and 5are possibilistically efficient only at ?1. The results from Tables 2and 3also show that the efficiency values from the primal FBCC model will be larger than or equal to those from the dual FBCC model at any given level as we mentioned in Section 4.Therefore,interval efficiency can be constructed at a given level.Then,the corresponding membership functions of fuzzy efficiency for each DMU can be constructed from these interval effciency as shown in Figure 4. Using the -level based approach (Lertworasirikul (2002)),the effficiency values from the Best-Worst and Worst-Best cases are shown in Tables 4and 5,respectively.For the Best-Worst case,the decision maker is optimistic about the target DMU (DMU o )and pessimistic about the remaining DMUs,while for the Worst-Best case,the decision maker is pessimistic about the target DMU (DMU o )and optimistic about the remaining DMUs.For the Best-Worst case,DMUs 2,4and 5are technically efficient,while DMUs 1and 3are less efficient.For the Worst-Best case,DMUs 2and 5are technically efficient,while DMUs 1,3and 4are less efficient.It can be observed that the efficiency values from the Table 3.Efficiency values from the dual BCC model obtained by the possibility approach at 5possibility levels. h B Possibility levels, V DMU1DMU2DMU3DMU4DMU500.669 1.0000.5840.8560.6340.250.712 1.0000.6470.8910.7160.50.760 1.0000.7210.9260.8040.750.819 1.0000.8190.9630.8961 0.889 1.000 0.935 1.000 1.000 Table 2.Efficiency values from the primal BCC model obtained by the possibility approach at 5possibility levels. f Possibility levels, V DMU1DMU2DMU3DMU4DMU50 1.299 1.247 1.699 1.692l 0.25 1.171 1.182 1.448 1.481l 0.5 1.062 1.119 1.243 1.300l 0.750.969 1.059 1.074 1.142l 1 0.889 1.000 0.935 1.000 1.000 LERTWORASIRIKUL ET AL 352 Best-Worst case are close to those from the possibility approach to the primal FBCC model and the efficiency values from the Worst-Best case are close to those from the possibility approach to the dual FBCC model.In addition,we can also notice that for the primal FBCC model,the efficiency values from the possibility approach are higher than those from -level based approach for the Best-Worst case.In contrast,for the dual FBCC Figure 4.Fuzzy efficiency values obtained by the possibility approach to the FBCC model. Table 4.Efficiency values from the BCC model obtained by the Best-Worst case of -level based approach. DMU1DMU2DMU3DMU4DMU50 1.000 1.000 1.000 1.000 1.0000.25 1.000 1.000 1.000 1.000 1.0000.5 1.000 1.000 1.000 1.000 1.0000.750.969 1.000 1.000 1.000 1.0001 0.889 1.000 0.935 1.000 1.000 FUZZY BCC MODEL FOR DATA ENVELOPMENT ANALYSIS 353 model,the efficiency values from the possibility approach are less than those from -level based approach for the Worst-Best case. From Kao et al.(Kao and Liu (2000)),Lertworasirikul et al (Lertworasirikul (2002))and Meada et al.(Meada et al (1998)),the minimum efficiency occurs when the outputs of DMU o and the inputs of other DMUs are set to their lower bounds,while the inputs of DMU o and the outputs of other DMUs are set to their upper bounds.The maximum efficiency occurs under the opposite conditions.Therefore,with the Best-Worst case,DMU o will have the largest possible efficiency value,while with the Worst-Best case,DMU o will have the smallest possible efficiency value.However,the Best-Worst and Worst-Best cases ignore other possible values of fuzzy inputs and fuzzy outputs of DMU o in the u T ~X and v T ~ Y terms of the normalization constraint 2of the FBCC P model and in the linear combination e~X and ~ Y Tterms of the constraints 1and 2of the FBCC D model.The other possible values of the fuzzy inputs and fuzzy outputs for DMU o may lead to a better optimal solution to the primal and dual FBCC models.This reason explains why the efficiency values from the possibility approach can be higher than and less than those from the Best-Worst and Worst-Best cases,respectively. Results from the possibility approach to the data in Table 1for primal and dual FCCR models are illustrated in Tables 6and 7.The results show that efficiency values from the Best-Worst case are close to those from the possibility approach to the primal FCCR model and efficiency values from the Worst-Best case are close to those from the possibility approach to the dual FCCR model.Membership functions of fuzzy efficiency for each DMU can also be constructed as shown in Figure 5.Efficiency values from the Table 5.Efficiency values from the BCC model obtained by the Worst-Best case of -level based approach. DMU1DMU2DMU3DMU4DMU500.669 1.0000.5850.859 1.0000.250.712 1.0000.6470.949 1.0000.50.760 1.0000.721 1.000 1.0000.750.819 1.0000.819 1.000 1.0001 0.889 1.000 0.935 1.000 1.000 Table 6.Efficiency values from the primal CCR model obtained by the possibility approach at 5possibility levels. f Possibility levels, V DMU1DMU2DMU3DMU4DMU50 1.107 1.238 1.276 1.520 1.2960.25 1.032 1.173 1.149 1.386 1.2260.50.963 1.112 1.035 1.258 1.1590.750.904 1.0550.932 1.131 1.0951 0.855 1.000 0.861 1.000 1.000 LERTWORASIRIKUL ET AL 354 Best-Worst and Worst-Best cases of the -level based approach to the FCCR models are given in Tables 8and 9,respectively.Similar to the results from the FBCC models,the possibility approach to the primal FCCR model provides the upper bound on the efficiency values,while the possibility approach to the dual FCCR model provides the lower bound on the efficiency values of DMU o at a specified possibility level. Table 7.Efficiency values from the dual CCR model obtained by the possibility approach at 5possibility levels. h B Possibility levels, V DMU1DMU2DMU3DMU4DMU500.6240.8360.5550.8550.6310.250.7020.9080.6400.8890.7100.50.7580.9570.7160.9240.7970.750.8070.9860.7910.9610.8931 0.855 1.000 0.861 1.000 1.000 Figure 5.Fuzzy efficiency values obtained by the possibility approach to the FCCR model. FUZZY BCC MODEL FOR DATA ENVELOPMENT ANALYSIS 355 What we have learned from observation from solving fuzzy DEA models with the possibility approach can be extended to other fuzzy linear programming models.For general fuzzy linear programming models,we may not know how to obtain the Best-Worst and Worst-Best cases.The possibility approach to a primal fuzzy linear programming model will give the upper bound on the objective value,while the possibility approach to a dual fuzzy linear programming model will give the lower bound on the objective value to the fuzzy linear programming problem.Then we can construct the fuzzy membership function of the objective value. 7.Concluding Remarks and Discussions Fuzzy Data Envelopment Analysis (FDEA)is a tool for comparing the performance of a set of activities or organizations under uncertainty environment.This paper has focused on FDEA models of the BCC type (FBCC).Both primal and dual forms of the FBCC model are presented.The FBCC models take the form of fuzzy linear programming models.In this paper,possibility and credibility approaches have been developed for solving primal and dual FBCC models.The possibility approach transforms FBCC models into possibility BCC models by using possibility measures of fuzzy events (fuzzy constraints).Using the possibility approach,the problem of fuzzy ranking can be handled.In addition,this approach reveals the relationship between the primal and dual FBCC models.The efficiency values from the possibility approach to the primal and dual FBCC models provide the upper bound and the lower bound of the efficiency values,respectively.The Table 8.Efficiency values from the CCR model obtained by the Best-Worst case of -level based approach. DMU1DMU2DMU3DMU4DMU50 1.000 1.000 1.000 1.000 1.0000.25 1.000 1.000 1.000 1.000 1.0000.50.963 1.000 1.000 1.000 1.0000.750.904 1.0000.932 1.000 1.0001 0.855 1.000 0.861 1.000 1.000 Table 9.Efficiency values from the CCR model obtained by the Worst-Best case of -level based approach. DMU1DMU2DMU3DMU4DMU500.6240.8360.5710.8550.6380.250.7020.9080.6420.9430.7350.50.7580.9900.716 1.0000.8450.750.807 1.0000.791 1.0000.9691 0.855 1.000 0.861 1.000 1.000 LERTWORASIRIKUL ET AL 356