Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

A New Link Between Output-oriented BCC Model with Fuzzy Data in the Present of Undesirable Outputs and MOLP

A New Link Between Output-oriented BCC Model with Fuzzy Data in the Present of Undesirable... Fuzzy Inf. Eng. (2011) 2: 113-125 DOI 10.1007/s12543-011-0070-0 ORIGINAL ARTICLE A New Link Between Output-oriented BCC Model with Fuzzy Data in the Present of Undesirable Outputs and MOLP A. Ebrahimenjad Received: 30 January 2011/ Revised: 10 April 2011/ Accepted: 15 May 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In recent years, the relation between data envelopment analysis and mul- tiple objective linear programming has received a great deal of attention from re- searchers. However, there are two difficulties in doing an objective evaluation of the performance of decision making units. The first one is how to treat undesirable factors jointly produced with the desirable factors and the second one is how to treat with im- precise data. In this paper, we establish an equivalence relation between multiple ob- jective linear programming and the output-oriented Banker, Charnes, Cooper(BCC) model in the present of undesirable factors and fuzzy data such that the decision maker’s preference can be taken into account in an interactive fashion for finding target unit. Keywords Data envelopment analysis · Multiple objective linear programming · Fuzzy numbers · Undesirable factors· Target unit 1. Introduction In recent years, the relation between data envelopment analysis (DEA) and multiple objective linear programming (MOLP) has received a great deal of attention from researchers. The structures of these two types of models have much in common, but DEA is directed to assessing past performances as part of the management con- trol function and MOLP to planning future performances. There exist some studies about the similarities between DEA and multiple criteria decision analysis (MCDA), generally, and MOLP, in particular. Doyle and Green [1] indicated that DEA is an MCDA method itself. Allen et al [2] defined value judgments as logical constructs, A. Ebrahimenjad () Department of Mathematics, Islamic Azad University, Qaemshahr Branch, Qaemshahr, Iran email: a.ebrahimnejad@srbiau.ac.ir aemarzoun@gmail.com 114 A. Ebrahimnejad (2011) incorporated within an efficiency assessment study, reflecting the decision maker’s (DM) preferences in the process of assessing efficiency. Various models have been established to actively involve DMs in the target setting process in DEA, including Golany’s [3], Thanassoulis and Dyson’s [4] and Athanassopoulos’s [5, 6] models. Golany [3] first established an interactive model involving both DEA and MOLP ap- proaches where the DM can allocate a set of input levels as resources and choose the most preferred set of output levels from a set of alternative points on the efficient fron- tier. Therefore, the effective integration of assessing past performances and planning future targets with the DM’s preferences, taken into account is of increasing inter- ests to support both management control and planning. Yang et al [7] proposed three equivalence models in MOLP, including the super-ideal point model, the ideal point model, and the shortest distance model. The super-ideal point model is proven iden- tical to the output-oriented DEA dual model, so they could assess performance and target setting values in the domain of MOLP. Hosseinzadeh Lotfi et al [8, 9] proposed a similar approach by establishing an equivalence relation between the max-min for- mulation of MOLP and the DEA models. However, there are two difficulties in doing an objective evaluation of the perfor- mance of decision making units (DMU). On the one hand, both desirable outputs and undesirable outputs may be present. For example, if inefficiency exists in production processes where final products are manufactured with a production of wastes and pol- lutants, the outputs of wastes and pollutants are undesirable and should be reduced to improve the performance. On the other hand, the original DEA models assume that inputs and outputs are measured by exact values on a ratio scale. However, this assumption may not be true, in the sense that some inputs and outputs may be only known as in forms of bounded or fuzzy data. Since the DEA model is essentially a linear programming, one straightforward idea is to apply the existing fuzzy linear programming (FLP) to the fuzzy DEA problems. In this paper, we establish an equiv- alent relation between MOLP and output-oriented BCC model [10] in the present of undesirable factors and fuzzy data such that the DM’s preference can be taken into account in an interactive fashion for finding target unit of an inefficient unit. This paper proceeds as follows. In Section 2, some preliminaries of fuzzy set the- ory and fuzzy linear programming are given. Section 3 is devoted to the description of output-oriented BCC model in the present of undesirable factors and a common technique for solving multiple objective optimization problems. In Section 4, we es- tablish an equivalent relation between fuzzy DEA model and MOLP. The conclusion is provided in Section 5. 2. Preliminaries In this section we first give some basic definitions on fuzzy numbers comparison and then review the concept of solution for fuzzy linear programming problems. 2.1. Fuzzy Numbers and Arithmetic Definition 2.1 Let X be the universal set. a ˜ is called a fuzzy set in X if a ˜ is a set of ordered pairs a ˜ = {(x,μ (x))|x ∈ X}, where μ (·) is membership function of a ˜ and a ˜ a ˜ assigns to each element x ∈ X, a real number μ (x) in the interval [0,1]. a ˜ Fuzzy Inf. Eng. (2011) 2: 113-125 115 Definition 2.2 Theα-cut or α-level of a fuzzy set a ˜ is defined as an ordinary set [˜ a] for which the degree of its membership function exceeds the level α, that is [˜ a] = {x ∈ X|μ (x) ≥ α}. a ˜ Definition 2.3 The support of a fuzzy set a ˜ is a set of elements in X for which μ (x) a ˜ is positive, that is, supp a ˜ = {x ∈ X|μ (x) > 0}. a ˜ Definition 2.4 A fuzzy set a ˜ is called convex if for each x, y ∈ X and each λ ∈ [0, 1], μ (λx+ (1−λ)y) ≥ min{μ (x),μ (y)}. a ˜ a ˜ a ˜ Definition 2.5 A fuzzy number is a convex normalized fuzzy set of the real line R; whose membership function is piecewise continuous. Definition 2.6 A fuzzy number a ˜ is said to be a triangular fuzzy number, if there exist real numbers s, l and r > 0 such that x− (l− s) ⎪ , for l− s ≤ x ≤ s, (s+ r)− x μ (x) = (1) a ˜ ⎪ , for x ≤ x ≤ s+ r, ⎪ r 0, else. We denote a triangular fuzzy number a ˜ by three real numbers s, l and r as a ˜ = (s, l, r). We also denote all triangular fuzzy numbers by F(R). Now define arithmetic on triangular fuzzy numbers. Let a ˜ = (s , l , r ) and b = (s , l , r ) be two triangular a a a b b b fuzzy numbers. Define x ≥ 0, x ∈ R; x a ˜ = (xs , xl , xr ), a a a x < 0, x ∈ R; x a ˜ = (xs , xr , xl ), a a a a ˜ + b = (s + s , l + l , r + r ). a b a b a b Here, we describe only one simple method for the ordering of fuzzy numbers (taken from Klir and Yuan [11]). We say a ˜  b if and only if: s ≤ s , s − l ≤ s − l , s + r ≤ s + r . (2) a b a a b b a a b b Definition 2.7 Let a ˜ = (s , l , r ) and b = (s , l , r ) be two triangular fuzzy numbers. a a a b b b Definitions of the relations≈ and≺ are given as follows: (i) a ˜ ≈ b i f and only i f s = s , s − l = s − l , s + r = s + r . a b a a b b a a b b (ii) a ˜ ≺ b i f and only i f s < s , s − l < s − l , s + r < s + r . a b a a b b a a b b Remark 1 We let 0 = (0, 0, 0) as the zero triangular fuzzy number. Thus any a ˜ such that a ≈ 0, is a zero too. 2.2. Fuzzy Linear Programming Problems The most popular approach for solving fuzzy linear programming problems is to con- vert the fuzzy linear program into the deterministic corresponding linear program. One important class of these methods that has been highlighted by many researches 116 A. Ebrahimnejad (2011) is based on comparison on fuzzy numbers by the use of ranking functions [11-22]. Here, we consider a special kind of fuzzy linear programming problem in which the right-hand-side numbers and the coefficient of the constraint matrix are fuzzy num- bers. In this case, the fuzzy linear programming, typically have the following from: max c x j j j=1 (3) s.t. a ˜ x  b, i = 1,··· , m, ij j i j=1 x ≥ 0, j = 1,··· , n, where a ˜ = (s , l , r ) and b = (s , l , r ) are triangular fuzzy numbers. Thus, ij a a a i b b b ij ij ij i i i this problem can be rewritten as follows: max c x j j j=1 (4) s.t. (s , l , r )x  (s , l , r ), i = 1,··· , m, a a a j b b b ij ij ij i i i j=1 x ≥ 0, j = 1,··· , n. Therefore, by relation (2), we get the following crisp linear programming that can be solved by standard methods: max c x j j j=1 s.t. s x  s , i = 1,··· , m, a j b ij i j=1 (5) (s − l )x  (s − l ), i = 1,··· , m, a a j b b ij ij i i j=1 (s + r )x  (s + r ), i = 1,··· , m, a a j b b ij ij i i j=1 x ≥ 0, j = 1,··· , n. 3. Literature Review on DEA and MOLP In this section we give a brief description of the output-oriented BCC model and MOLP technique to illustration of our approach in the next section. 3.1. Output-oriented BCC Model in the Present of Undesirable Outputs Data envelopment analysis measures the relative efficiency of decision making units with multiple performance factors which are grouped into outputs and inputs. The units are assumed to operate under similar conditions. Based on information about existing data on the performance of the units and some preliminary assumptions, DEA forms an empirical efficient surface. If a DMU lies on the surface, it is referred to as an efficient unit, otherwise inefficient. DEA also provides efficiency scores and reference set for inefficient DMU. The reference set for inefficient units consists of efficient units and determines a virtual unit on the efficient surface. The virtual unit Fuzzy Inf. Eng. (2011) 2: 113-125 117 which can be found in DEA by projecting an inefficient DMU radially to the efficient surface is regarded as a target unit for the inefficient unit. However, in accordance with the global environmental conservation awareness, undesirable outputs of productions and social activities, e.g., air pollutants and haz- ardous wastes, are being increasingly recognized as dangerous and undesirable. Thus, development of technologies with less undesirable outputs is an important subject of concern in every area of production. DEA usually assumes that producing more out- puts relative to less input resources is a criterion of efficiency. In the presence of undesirable outputs, however, technologies with more good (desirable) outputs and less bad (undesirable) outputs relative to less input resources should be recognized as efficient. Several authors have proposed efficiency measures in the presence of undesirable outputs. A conventional and traditional way to handle this problem is to shift un- desirable outputs to inputs and to apply traditional DEA models to the data set. In the VRS (Variable Returns to Scale) environment, Seiford and Zhu [23] proposed a method that first multiplies each undesirable output by −1 and then finds a proper translation vector to let all negative undesirable outputs be positive. We pursue this approach thorough this paper. Suppose we have n DMUs, each DMU produces s desirable outputs y (¯ r = j 1 rj ¯ 1, 2,··· , s ) and s undesirable outputs y (r ¯ = 1, 2,··· , s ) using m inputs (x , x , 1 2 2 1 j 2 j rj ¯ ··· , x ). The model suggested by Seiford and Zhu [23] can be expressed as follows: mj max β s.t. λ x ≤ x , i = 1, 2,··· , m, j ij ip j=1 g g λ y ≥ β y , r ¯ = 1, 2,··· , s , j p 1 rp ¯ rj ¯ j=1 (6) b b λ y ¯ ≥ β y ¯ , r ¯ = 1, 2,··· , s , j p 2 ¯ ¯ rj ¯ rp ¯ j=1 λ = 1, j=1 λ ≥ 0, j = 1, 2,··· , n, b b where y ¯ = −y + w > 0. r ¯ ¯ ¯ r ¯ rj ¯ In this model, the inverse ofβ is the efficiency score of DMU .Ifβ > 1, DMU p p p p is not efficient and the parameter β indicates the extent by which DMU has to p p increase its desirable outputs and translated undesirable outputs to become efficient. For an inefficient DMU , we define its reference set as ∗ ∗ E = {λ |λ > 0, j = 1, 2,··· , n}, j j ∗ ∗ ∗ ∗ where λ = (λ ,λ ,··· ,λ ) is the optimal solution of (6). In this case, the virtual 1 2 point on the efficient frontier (composited based on reference set) is used to evaluate the performance of DMU and can be regarded as a target unit for the inefficient unit p 118 A. Ebrahimnejad (2011) DMU . This target unit usually does not include a DM’s preference structure or value judgments. So, it needs to use an interactive method in MOLP. 3.2. Multiple Objective Linear Programming In this section, we briefly discuss basic concepts and models in multiple objective optimizations and in particular the min-ordering formulation, as a basis for the inves- tigation to be reported in the following sections. Suppose an optimization problem has t objectives reflecting the different purposes and desires of the DM. Such a problem can be represented in a general form as fol- lows: max h(λ) = [h (λ), h (λ),··· , h (λ)] 1 2 t s.t. p (λ) ≤ 0, j = 1,··· , u , (7) j 1 q (λ) = 0, e = 1,··· , u , e 2 where, h (λ)(k = 1, 2,··· , t) are continuously differentiable objective functions, and p (λ)( j = 1, 2,··· , u ) and q (λ)(e = 1, 2,··· , u ) are continuously differentiable j 1 e 2 inequality and equality constraint functions, respectively. Here h (λ), p (λ) and q (λ) k j e are all assumed to be linear functions ofλ, so Formulation (7) is referred to as multiple objective linear programming, or MOLP in short. We also denote the feasible space of (7) by Q. As in an MOLP problem we do not reach a single solution to optimize all objec- tives together, there are efficient or non-dominated solutions. Conceptually, a feasible solutionλ is said to be efficient or non-dominated if there exists no other feasible so- ∗ ∗ lution which is better than λ at least on one objective and as good as λ on all other objectives. An efficient or non-dominated solution is also referred to as a Pareto- optimal solution. In order to reach to an efficient solution, Formulation (7) can be written in min-ordering approach as follows: max min h(λ) = [h (λ), h (λ),··· , h (λ)] 1 2 t (8) s.t. λ ∈ Q. The min-ordering Formulation (8) can then be written as follows by introducing an auxiliary variableθ: maxθ (9) s.t. h (λ) ≥ θ, k = 1,··· , t, λ ∈ Q. However, this approach does not include DM’s preference structures in finding the efficient solution. Thus an interactive method requires for solving MOLP. In the fol- lowing section, we use the Zionts-Walleniuss method [24] to integrate fuzzy output- oriented BCC performance assessment and target setting such that the DMs prefer- ence can be taken into account in an interactive fashion. 4. Performance Evaluation of Fuzzy DEA Model Based on an MOLP Method In this section, we first formulate the output-oriented BCC model with fuzzy data and then establish the equivalent relation between this model and MOLP. Fuzzy Inf. Eng. (2011) 2: 113-125 119 4.1. Fuzzy Output-oriented BCC Model with Undesirable Factors In ordinary DEA models, the input and output values are assumed to be definite. In recent year, in different applications of DEA, inputs and outputs have been observed whose values are indefinite. Such data are called “inaccurate”. Inaccurate data can be probabilistic, interval, ordinal, qualitative, or fuzzy. Therefore, some papers were pre- sented on the theoretical development of this technique whit fuzzy data. To formulate g g g the fuzzy output-oriented BCC model, assume x ˜ = (s , l , r ), y ˜ = (s , l , r ) ij x x x ij ij ij y y y rj ¯ rj ¯ rj ¯ rj ¯ and y ˜ = (s b , l b , r b ) for (i = 1, 2,··· , m, r ¯ = 1, 2,··· , s , r ¯ = 1, 2,··· , s , j = y y y 1 2 rj ¯ ¯ ¯ ¯ rj ¯ rj ¯ rj ¯ 1, 2,··· , n). Thus, the fuzzy version of Model (6) is given as follows: max β s.t. λ x ˜ ≤ x ˜ , i = 1, 2,··· , m, j ij ip j=1 g g λ y ˜ ≥ β y ˜ , r ¯ = 1, 2,··· , s , j p 1 rp ¯ rj ¯ j=1 (10) b b ˜ ˜ ¯ λ y ¯ ≥ β y ¯ , r ¯ = 1, 2,··· , s , j p 2 ¯ ¯ rj ¯ rp ¯ j=1 λ = 1, j=1 λ ≥ 0, j = 1, 2,··· , n, b b where y ¯ = −y ˜ + w ˜ = (s b , l b , r b ) 0. r ¯ y ¯ y ¯ y ¯ ¯ ¯ r ¯ rj ¯ ¯ ¯ ¯ j rj ¯ rj ¯ rj ¯ This model can be rewritten as follows: max β s.t. λ s ≤ s , i = 1, 2,··· , m, j x x ij ip j=1 λ (s − l ) ≤ (s − l ), i = 1, 2,··· , m, j x x x x ij ij ip ip j=1 λ (s + r ) ≤ (s + r ), i = 1, 2,··· , m, j x x x x ij ij ip ip j=1 g g λ s ≥ β s , r ¯ = 1, 2,··· , s , j y p y 1 rp ¯ rj ¯ (11) j=1 g g g g λ (s − l ) ≥ β (s − l ), r ¯ = 1, 2,··· , s , j p 1 y y y y rj ¯ rj ¯ rp ¯ rp ¯ j=1 g g g λ (s + ry ) ≥ β (s + r ), r ¯ = 1, 2,··· , s , j p 1 y y y rj ¯ rj ¯ rp ¯ rp ¯ j=1 λ s b ≥ β s b , r ¯ = 1, 2,··· , s , j p 2 y ¯ y ¯ ¯ ¯ rj ¯ rp ¯ j=1 λ (s b − l b ) ≥ β (s b − l b ), r ¯ = 1, 2,··· , s , j y ¯ y ¯ p y ¯ y ¯ 2 ¯ ¯ ¯ ¯ rj ¯ rj ¯ rp ¯ rp ¯ j=1 120 A. Ebrahimnejad (2011) λ (s b + r b ) ≥ β (s b + r b ), r ¯ = 1, 2,··· , s , j p 2 y ¯ y ¯ y ¯ y ¯ rj ¯ ¯ rj ¯ ¯ rp ¯ ¯ rp ¯ ¯ j=1 λ = 1, j=1 λ ≥ 0, j = 1, 2,··· , n. 4.2. Relation between Fuzzy DEA Model and MOLP Consider an MOLP in the following form: 1 2 3 max f (λ), f (λ), f (λ) r ¯ r ¯ r ¯ 1 2 3 max g (λ), g (λ), g (λ) ¯ ¯ ¯ r ¯ r ¯ r ¯ s.t. λ s ≤ s , i = 1, 2,··· , m, j x x ij ip j=1 λ (s − l ) ≤ (s − l ), i = 1, 2,··· , m, j x x x x ij ij ip ip (12) j=1 λ (s + r ) ≤ (s + r ), i = 1, 2,··· , m, j x x x x ij ij ip ip j=1 λ = 1, j=1 λ ≥ 0, j = 1, 2,··· , n. It needs to point out that the above MOLP problem can be rewritten in the following form using min-ordering formulation mentioned in the last section and introducing the auxiliary variableθ: maxθ s.t. f (λ) ≥ θ, r ¯ = 1, 2,··· , s , t = 1, 2, 3, r ¯ (13) g (λ) ≥ θ, r ¯ = 1, 2,··· , s , v = 1, 2, 3, r ¯ λ ∈ Q , where, the feasible space Q is given as follows: n n λ | λ s ≤ s , λ (s − l ) ≤ (s − l ), Q = j j x x j x x x x (14) p ⎪ ij ip ij ij ip ip j=1 j=1 n n λ (s + r ) ≤ (s + r ), (i = 1, 2,··· , m), λ = 1 j x x x x j . ij ij ip ip ⎪ j=1 j=1 In this case, the fuzzy output-oriented BCC Model (11) can be equivalently rewrit- Fuzzy Inf. Eng. (2011) 2: 113-125 121 ten as follows: max β g g s.t. λ s ≥ β s , r ¯ = 1, 2,··· , s , j p 1 y y rj ¯ rp ¯ j=1 g g g g λ (s − l ) ≥ β (s − l ), r ¯ = 1, 2,··· , s , j y y p y y 1 rp ¯ rp ¯ rj ¯ rj ¯ j=1 g g g λ (s + ry ) ≥ β (s + r ), r ¯ = 1, 2,··· , s , j y p y y 1 rj ¯ rj ¯ rp ¯ rp ¯ j=1 (15) b b λ s ≥ β s , r ¯ = 1, 2,··· , s , j p 2 y ¯ y ¯ ¯ ¯ rj ¯ rp ¯ j=1 λ (s b − l b ) ≥ β (s b − l b ), r ¯ = 1, 2,··· , s , j p 2 y ¯ y ¯ y ¯ y ¯ ¯ ¯ ¯ ¯ rj ¯ rj ¯ rp ¯ rp ¯ j=1 λ (s b + r b ) ≥ β (s b + r b ), r ¯ = 1, 2,··· , s , j p 2 y ¯ y ¯ y ¯ y ¯ ¯ ¯ ¯ ¯ rj ¯ rj ¯ rp ¯ rp ¯ j=1 λ ∈ Q . In order to prove that Formulation (15) is equivalent to the min-ordering formu- lation in (12), certain conditions have to be applied. The purpose for establishing the equivalence conditions is to use the interactive techniques in MOLP to locate the most preferred solution (MPS) on the efficient frontier for target setting and resource g g g g g allocation. Suppose s , (s − l ) and (s + r ) > 0 for any r ¯ = 1, 2,··· , s and y y y y y rp ¯ rp ¯ rp ¯ rp ¯ rp ¯ t v define f (λ), (t = 1, 2, 3) and g (λ), (v = 1, 2, 3) in Formulation (12) as follows: r ¯ ¯ r ¯ λ s j y rj ¯ i=1 f (λ) = , (16) r ¯ rp ¯ g g λ (s − l ) y y rj ¯ rj ¯ i=1 f (λ) = , (17) r ¯ g g (s − l ) y y rp ¯ rp ¯ g g λ (s + r ) y y rj ¯ rj ¯ i=1 f (λ) = , (18) r ¯ g g (s + r ) y y rp ¯ rp ¯ λ s b y ¯ rj ¯ i=1 g (λ) = , (19) r ¯ s b y ¯ rp ¯ λ (s b − l b ) y ¯ y ¯ ¯ ¯ rj ¯ rj ¯ i=1 g (λ) = , (20) r ¯ (s b − l b ) y ¯ y ¯ ¯ ¯ rp ¯ rp ¯ 122 A. Ebrahimnejad (2011) b b λ (s + r ) y ¯ y ¯ ¯ ¯ rj ¯ rj ¯ i=1 g (λ) = . (21) r ¯ b b (s + r ) y ¯ y ¯ ¯ ¯ rp ¯ rp ¯ g g g g g Since s , (s − l ) and (s + r ) > 0 for any r ¯ = 1, 2,··· , s , Formulation (15) y y y y y 1 rp ¯ rp ¯ rp ¯ rp ¯ rp ¯ can be written as follows: max β λ s j y rj ¯ i=1 s.t. ≥ β , r ¯ = 1, 2,··· , s , p 1 s g rp ¯ g g λ (s − l ) y y rj ¯ rj ¯ i=1 ≥ β , r ¯ = 1, 2,··· , s , p 1 (s g −l g ) y y rp ¯ rp ¯ g g λ (s + r ) y y rj ¯ rj ¯ i=1 ≥ β , r ¯ = 1, 2,··· , s , p 1 (s g +r g ) y y rp ¯ rp ¯ m (22) λ s b y ¯ rj ¯ i=1 ≥ β , r ¯ = 1, 2,··· , s , p 2 y ¯ rp ¯ ¯ λ (s b − l b ) j y ¯ y ¯ ¯ ¯ rj ¯ rj ¯ i=1 ≥ β , r ¯ = 1, 2,··· , s , p 2 (s −l ) b b y ¯ y ¯ rp ¯ ¯ rp ¯ ¯ λ (s b + r b ) y ¯ y ¯ rj ¯ ¯ rj ¯ ¯ i=1 ≥ β , r ¯ = 1, 2,··· , s , p 2 (s +r ) b b y ¯ y ¯ ¯ ¯ rp ¯ rp ¯ λ ∈ Q . Now using Formulations (16)− (21), we get the following problem: max β s.t. f (λ) ≥ θ r ¯ = 1, 2,··· , s , t = 1, 2, 3, r ¯ (23) g (λ) ≥ θ r ¯ = 1, 2,··· , s , v = 1, 2, 3, r ¯ λ ∈ Q . This means that the fuzzy output-oriented BCC model in the present of undesirable outputs (15) is equivalent to the min-ordering Formulation (15), ifβ = θ. Alternatively, since Formulation (13) gives a special weak efficient point of For- mulation (12), then Formulation (11) or (15) also gives a special weak efficient point of following formulation: Fuzzy Inf. Eng. (2011) 2: 113-125 123 ⎡ ⎤ m m m ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ g g g g g ⎢ λ s λ (s − l ) λ (s + r )⎥ j j j ⎢ y y y y y ⎥ rj ¯ rj ¯ rj ¯ rj ¯ rj ¯ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ i=1 i=1 i=1 ⎥ ⎢ ⎥ max , , ⎢ ⎥ ⎢ g g g g g ⎥ s (s −l ) (s +r ) ⎢ ⎥ y y y y y ⎢ ⎥ rp ¯ rp ¯ rp ¯ rp ¯ rp ¯ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ m m m ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ λ s b λ (s b − l b ) λ (s b + r b )⎥ j j j ⎢ y ¯ y ¯ y ¯ y ¯ y ¯ ⎥ ⎢ rj ¯ ¯ rj ¯ ¯ rj ¯ ¯ rj ¯ ¯ rj ¯ ¯ ⎥ ⎢ ⎥ ⎢ ⎥ i=1 i=1 i=1 ⎢ ⎥ ⎢ ⎥ max ⎢ , , ⎥ ⎢ ⎥ s (s −l ) (s +r ) ⎢ b b b b b ⎥ y ¯ y ¯ y ¯ y ¯ y ¯ ⎢ ⎥ ¯ ¯ ¯ ¯ ¯ ⎢ rp ¯ rp ¯ rp ¯ rp ¯ rp ¯ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (24) s.t. λ s ≤ s , i = 1, 2,··· , m, j x x ij ip j=1 λ (s − l ) ≤ (s − l ), i = 1, 2,··· , m, j x x x x ij ij ip ip j=1 λ (s + r ) ≤ (s + r ), i = 1, 2,··· , m, j x x x x ij ij ip ip j=1 λ = 1, j=1 λ ≥ 0, j = 1, 2,··· , n, that does not include DM’s preference structures or value judgments in measuring relative efficiency and setting target values, so the efficiency score of the DMU can be generated by solving Formulation (24). Hence, an interactive MOLP method can be used to solve the DEA problem. 4.3. An Interactive Method It is an issue how decision makers decide one from the set of non-dominated solutions as the final solution. Consequently, interactive MOLP methods have been developed to this end. In this paper, we use the Zionts-Walleniuss method [24] to integrate combined- oriented CCR performance assessment and target setting such that the DMs prefer- ence can be taken into account in an interactive fashion. This method is applicable to problem in (24) where the objective functions are concave and feasible space is a convex set. The overall utility function is assumed to be unknown explicitly to the DM, but is implicitly a linear function and more generally a concave function of the objective functions. The method makes use of such an implicit function on an in- teractive basis. The first step of the method is to choose an arbitrary set of positive multipliers or weights and generate a composite objective function or utility function using these multipliers. The composite objective function is then optimized to pro- duce a non-dominated solution to the problem. From the set of non-basic variables, a subset of efficient variables is selected (an efficient variable is one which, when in- troduced into the basis, cannot increase one objective without decreasing at least one other objective). For each efficient variable a set of tradeoffs is defined by concept that some objectives are increased and others reduced. A number of such tradeoffs 124 A. Ebrahimnejad (2011) are presented to the DM, who is requested to state whether the tradeoffs are desirable, undesirable or neither. From his/her answers a new set with consistent multipliers is constructed and the associated non-dominated solution is found. The process is then repeated and a new set of tradeoffs is presented to the DM at the current solution until the DM find the most preferred solution with respect to the implicit utility. 5. Conclusion In this paper, we obtained an equivalence relation between the output-oriented BCC model with fuzzy data in the present of undesirable outputs and the min-ordering model in MOLP and showed how a DEA problem can be solved interactively by transforming it into MOLP formulation. This approach results a decrease in total undesirable outputs production and a permissible increase in total desirable output production. The proposed equivalence model provided the basis to apply interactive methods in MOLP to solve DEA problems and further locate the MPS along the efficient frontier for each inefficient DMU. We proposed the Zionts-Wallenius method to reflecting the DM preferences in the process of assessing efficiency. Acknowledgements The author would like to express their sincerest thanks to the anonymous referees and honorable editor for their valuable comments which helped in improving this paper. References 1. Doyle J, Green R (1993) Data envelopment analysis and multiple criteria decision making. Omega 21: 713-715 2. Allen R, Athanassopoulos A D, Dyson R G, Thanassoulis E (1997) Weights restrictions and value judgments in data envelopment analysis: evolution, development and future directions. Annals of Operational Research 73: 13-34 3. Golany B (1988) An interactive MOLP procedure for the extension of DEA to effectiveness analysis. Journal of Operational Research Society 39: 725-734 4. Thanassoulis E, Dyson R G (1992) Estimating preferred target input-output levels using data envel- opment analysis. European Journal of Operational Research 56: 80-97 5. Athanassopoulos A D (1995) Goal programming and data envelopment analysis (GPDEA) models for multi-level multi-unit organizations: an application to Greek local authorities. European Journal of Operational Research 87: 535-550 6. Athanassopoulos A D (1998) Decision support for target-based resource allocation of public services in multi unit and multilevel systems. Management Science 39: 1261-1264 7. Yang J B, Wong B Y H, Xu D L, Stewart T J (2009) Integrating DEA-oriented performance assess- ment and target setting using interactive MOLP methods. European Journal of Operational Research 195: 205-222 8. Hosseinzadeh Lotfi F, Jahanshaloo G R, Ebrahimnejad A, Soltanifar M, Mansourzadeh S M (2010) Target setting in the general combined-oriented CCR model using an interactive MOLP method. Journal of Computational and Applied Mathematics 234: 1-9 9. Hosseinzadeh Lotfi F, Jahanshaloo G R, Soltanifar M, Ebrahimnejad A, Mansourzadeh S M (2010) Relation between MOLP and DEA based on output-oriented CCR dual model. Expert System with Application 37: 4331-4336 10. Banker R, Charnes A, Cooper W (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science 30(9): 1078-1092 11. Klir G J, Yuan B (1996) Fuzzy sets and fuzzy logic: theory and applications. Prentice-Hall, Inc., N.J. USA Fuzzy Inf. Eng. (2011) 2: 113-125 125 12. Ebrahimnejad A, Nasseri S H, Mansourzadeh S M (2011) Bounded primal simplex algorithm for bounded linear programming with fuzzy cost coefficients. International Journal of Operations Re- search and Information Systems 2(1): 96-120 13. Ebrahimnejad A, Nasseri S H, and Hosseinzadeh Lotfi F (2010) Bounded linear programs with trape- zoidal fuzzy numbers. International Journal of Uncertainty, Fuzziness and Knowledge-Based Sys- tems 18(3): 269-286 14. Ebrahimnejad A, Nasseri S H, Hosseinzadeh Lotfi F, Soltanifar M (2010) A primal-dual method for linear programming problems with fuzzy variables. European Journal of Industrial Engineering 4(2): 189-209 15. Ebrahimnejad A, Nasseri S H (2010) A dual simplex method for bounded linear programmes with fuzzy numbers. International Journal of Mathematics in Operational Research 2(6): 762-779 16. Ebrahimnejad A, Nasseri S H (2009) Using complementary slackness property to solve linear pro- gramming with fuzzy parameters. Fuzzy Information and Engineering 3: 233-245 17. Mahdavi-Amiri N, Nasseri S H, Yazdani A (2009) Fuzzy primal simplex algorithms for solving fuzzy linear programming problems. Iranian Journal of Operational Research 1: 68-84 18. Mahdavi-Amiri N, Nasseri S H (2006) Duality in fuzzy number linear programming by use of a certain linear ranking function. Applied Mathematics and Computation 180: 206-216 19. Mahdavi-Amiri N, Nasseri S H (2007) Duality results and a dual simplex method for linear program- ming problems with trapezoidal fuzzy variables. Fuzzy Sets and Systems 158: 1961-1978 20. Nasseri S H, Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for solving flexible linear programming problem. European Journal of Industrial Engineering 4: 372-389 21. Nasseri S H, Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear program- ming problem. Advances in Fuzzy Sets and Systems 5: 81-95 22. Nasseri S H, Mahdavi-Amiri N (2010) Some duality results on fuzzy linear programming with sym- metric fuzzy numbers. Fuzzy Information and Engineering 1: 59-66 23. Seiford M, Zhu J (2002) Modeling undesirable factors in efficiency evaluation. European Journal of Operational Research 142: 16-20 24. Zionts S, Wallenius J (1976) An interactive programming method for solving the multiple criteria problem. Management Science 22: 65-63 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

A New Link Between Output-oriented BCC Model with Fuzzy Data in the Present of Undesirable Outputs and MOLP

Fuzzy Information and Engineering , Volume 3 (2): 13 – Jun 1, 2011

A New Link Between Output-oriented BCC Model with Fuzzy Data in the Present of Undesirable Outputs and MOLP

Abstract

AbstractIn recent years, the relation between data envelopment analysis and multiple objective linear programming has received a great deal of attention from researchers. However, there are two difficulties in doing an objective evaluation of the performance of decision making units. The first one is how to treat undesirable factors jointly produced with the desirable factors and the second one is how to treat with imprecise data. In this paper, we establish an equivalence relation between...
Loading next page...
 
/lp/taylor-francis/a-new-link-between-output-oriented-bcc-model-with-fuzzy-data-in-the-tmmdQr7DFN
Publisher
Taylor & Francis
Copyright
© 2011 Taylor and Francis Group, LLC
ISSN
1616-8666
eISSN
1616-8658
DOI
10.1007/s12543-011-0070-0
Publisher site
See Article on Publisher Site

Abstract

Fuzzy Inf. Eng. (2011) 2: 113-125 DOI 10.1007/s12543-011-0070-0 ORIGINAL ARTICLE A New Link Between Output-oriented BCC Model with Fuzzy Data in the Present of Undesirable Outputs and MOLP A. Ebrahimenjad Received: 30 January 2011/ Revised: 10 April 2011/ Accepted: 15 May 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In recent years, the relation between data envelopment analysis and mul- tiple objective linear programming has received a great deal of attention from re- searchers. However, there are two difficulties in doing an objective evaluation of the performance of decision making units. The first one is how to treat undesirable factors jointly produced with the desirable factors and the second one is how to treat with im- precise data. In this paper, we establish an equivalence relation between multiple ob- jective linear programming and the output-oriented Banker, Charnes, Cooper(BCC) model in the present of undesirable factors and fuzzy data such that the decision maker’s preference can be taken into account in an interactive fashion for finding target unit. Keywords Data envelopment analysis · Multiple objective linear programming · Fuzzy numbers · Undesirable factors· Target unit 1. Introduction In recent years, the relation between data envelopment analysis (DEA) and multiple objective linear programming (MOLP) has received a great deal of attention from researchers. The structures of these two types of models have much in common, but DEA is directed to assessing past performances as part of the management con- trol function and MOLP to planning future performances. There exist some studies about the similarities between DEA and multiple criteria decision analysis (MCDA), generally, and MOLP, in particular. Doyle and Green [1] indicated that DEA is an MCDA method itself. Allen et al [2] defined value judgments as logical constructs, A. Ebrahimenjad () Department of Mathematics, Islamic Azad University, Qaemshahr Branch, Qaemshahr, Iran email: a.ebrahimnejad@srbiau.ac.ir aemarzoun@gmail.com 114 A. Ebrahimnejad (2011) incorporated within an efficiency assessment study, reflecting the decision maker’s (DM) preferences in the process of assessing efficiency. Various models have been established to actively involve DMs in the target setting process in DEA, including Golany’s [3], Thanassoulis and Dyson’s [4] and Athanassopoulos’s [5, 6] models. Golany [3] first established an interactive model involving both DEA and MOLP ap- proaches where the DM can allocate a set of input levels as resources and choose the most preferred set of output levels from a set of alternative points on the efficient fron- tier. Therefore, the effective integration of assessing past performances and planning future targets with the DM’s preferences, taken into account is of increasing inter- ests to support both management control and planning. Yang et al [7] proposed three equivalence models in MOLP, including the super-ideal point model, the ideal point model, and the shortest distance model. The super-ideal point model is proven iden- tical to the output-oriented DEA dual model, so they could assess performance and target setting values in the domain of MOLP. Hosseinzadeh Lotfi et al [8, 9] proposed a similar approach by establishing an equivalence relation between the max-min for- mulation of MOLP and the DEA models. However, there are two difficulties in doing an objective evaluation of the perfor- mance of decision making units (DMU). On the one hand, both desirable outputs and undesirable outputs may be present. For example, if inefficiency exists in production processes where final products are manufactured with a production of wastes and pol- lutants, the outputs of wastes and pollutants are undesirable and should be reduced to improve the performance. On the other hand, the original DEA models assume that inputs and outputs are measured by exact values on a ratio scale. However, this assumption may not be true, in the sense that some inputs and outputs may be only known as in forms of bounded or fuzzy data. Since the DEA model is essentially a linear programming, one straightforward idea is to apply the existing fuzzy linear programming (FLP) to the fuzzy DEA problems. In this paper, we establish an equiv- alent relation between MOLP and output-oriented BCC model [10] in the present of undesirable factors and fuzzy data such that the DM’s preference can be taken into account in an interactive fashion for finding target unit of an inefficient unit. This paper proceeds as follows. In Section 2, some preliminaries of fuzzy set the- ory and fuzzy linear programming are given. Section 3 is devoted to the description of output-oriented BCC model in the present of undesirable factors and a common technique for solving multiple objective optimization problems. In Section 4, we es- tablish an equivalent relation between fuzzy DEA model and MOLP. The conclusion is provided in Section 5. 2. Preliminaries In this section we first give some basic definitions on fuzzy numbers comparison and then review the concept of solution for fuzzy linear programming problems. 2.1. Fuzzy Numbers and Arithmetic Definition 2.1 Let X be the universal set. a ˜ is called a fuzzy set in X if a ˜ is a set of ordered pairs a ˜ = {(x,μ (x))|x ∈ X}, where μ (·) is membership function of a ˜ and a ˜ a ˜ assigns to each element x ∈ X, a real number μ (x) in the interval [0,1]. a ˜ Fuzzy Inf. Eng. (2011) 2: 113-125 115 Definition 2.2 Theα-cut or α-level of a fuzzy set a ˜ is defined as an ordinary set [˜ a] for which the degree of its membership function exceeds the level α, that is [˜ a] = {x ∈ X|μ (x) ≥ α}. a ˜ Definition 2.3 The support of a fuzzy set a ˜ is a set of elements in X for which μ (x) a ˜ is positive, that is, supp a ˜ = {x ∈ X|μ (x) > 0}. a ˜ Definition 2.4 A fuzzy set a ˜ is called convex if for each x, y ∈ X and each λ ∈ [0, 1], μ (λx+ (1−λ)y) ≥ min{μ (x),μ (y)}. a ˜ a ˜ a ˜ Definition 2.5 A fuzzy number is a convex normalized fuzzy set of the real line R; whose membership function is piecewise continuous. Definition 2.6 A fuzzy number a ˜ is said to be a triangular fuzzy number, if there exist real numbers s, l and r > 0 such that x− (l− s) ⎪ , for l− s ≤ x ≤ s, (s+ r)− x μ (x) = (1) a ˜ ⎪ , for x ≤ x ≤ s+ r, ⎪ r 0, else. We denote a triangular fuzzy number a ˜ by three real numbers s, l and r as a ˜ = (s, l, r). We also denote all triangular fuzzy numbers by F(R). Now define arithmetic on triangular fuzzy numbers. Let a ˜ = (s , l , r ) and b = (s , l , r ) be two triangular a a a b b b fuzzy numbers. Define x ≥ 0, x ∈ R; x a ˜ = (xs , xl , xr ), a a a x < 0, x ∈ R; x a ˜ = (xs , xr , xl ), a a a a ˜ + b = (s + s , l + l , r + r ). a b a b a b Here, we describe only one simple method for the ordering of fuzzy numbers (taken from Klir and Yuan [11]). We say a ˜  b if and only if: s ≤ s , s − l ≤ s − l , s + r ≤ s + r . (2) a b a a b b a a b b Definition 2.7 Let a ˜ = (s , l , r ) and b = (s , l , r ) be two triangular fuzzy numbers. a a a b b b Definitions of the relations≈ and≺ are given as follows: (i) a ˜ ≈ b i f and only i f s = s , s − l = s − l , s + r = s + r . a b a a b b a a b b (ii) a ˜ ≺ b i f and only i f s < s , s − l < s − l , s + r < s + r . a b a a b b a a b b Remark 1 We let 0 = (0, 0, 0) as the zero triangular fuzzy number. Thus any a ˜ such that a ≈ 0, is a zero too. 2.2. Fuzzy Linear Programming Problems The most popular approach for solving fuzzy linear programming problems is to con- vert the fuzzy linear program into the deterministic corresponding linear program. One important class of these methods that has been highlighted by many researches 116 A. Ebrahimnejad (2011) is based on comparison on fuzzy numbers by the use of ranking functions [11-22]. Here, we consider a special kind of fuzzy linear programming problem in which the right-hand-side numbers and the coefficient of the constraint matrix are fuzzy num- bers. In this case, the fuzzy linear programming, typically have the following from: max c x j j j=1 (3) s.t. a ˜ x  b, i = 1,··· , m, ij j i j=1 x ≥ 0, j = 1,··· , n, where a ˜ = (s , l , r ) and b = (s , l , r ) are triangular fuzzy numbers. Thus, ij a a a i b b b ij ij ij i i i this problem can be rewritten as follows: max c x j j j=1 (4) s.t. (s , l , r )x  (s , l , r ), i = 1,··· , m, a a a j b b b ij ij ij i i i j=1 x ≥ 0, j = 1,··· , n. Therefore, by relation (2), we get the following crisp linear programming that can be solved by standard methods: max c x j j j=1 s.t. s x  s , i = 1,··· , m, a j b ij i j=1 (5) (s − l )x  (s − l ), i = 1,··· , m, a a j b b ij ij i i j=1 (s + r )x  (s + r ), i = 1,··· , m, a a j b b ij ij i i j=1 x ≥ 0, j = 1,··· , n. 3. Literature Review on DEA and MOLP In this section we give a brief description of the output-oriented BCC model and MOLP technique to illustration of our approach in the next section. 3.1. Output-oriented BCC Model in the Present of Undesirable Outputs Data envelopment analysis measures the relative efficiency of decision making units with multiple performance factors which are grouped into outputs and inputs. The units are assumed to operate under similar conditions. Based on information about existing data on the performance of the units and some preliminary assumptions, DEA forms an empirical efficient surface. If a DMU lies on the surface, it is referred to as an efficient unit, otherwise inefficient. DEA also provides efficiency scores and reference set for inefficient DMU. The reference set for inefficient units consists of efficient units and determines a virtual unit on the efficient surface. The virtual unit Fuzzy Inf. Eng. (2011) 2: 113-125 117 which can be found in DEA by projecting an inefficient DMU radially to the efficient surface is regarded as a target unit for the inefficient unit. However, in accordance with the global environmental conservation awareness, undesirable outputs of productions and social activities, e.g., air pollutants and haz- ardous wastes, are being increasingly recognized as dangerous and undesirable. Thus, development of technologies with less undesirable outputs is an important subject of concern in every area of production. DEA usually assumes that producing more out- puts relative to less input resources is a criterion of efficiency. In the presence of undesirable outputs, however, technologies with more good (desirable) outputs and less bad (undesirable) outputs relative to less input resources should be recognized as efficient. Several authors have proposed efficiency measures in the presence of undesirable outputs. A conventional and traditional way to handle this problem is to shift un- desirable outputs to inputs and to apply traditional DEA models to the data set. In the VRS (Variable Returns to Scale) environment, Seiford and Zhu [23] proposed a method that first multiplies each undesirable output by −1 and then finds a proper translation vector to let all negative undesirable outputs be positive. We pursue this approach thorough this paper. Suppose we have n DMUs, each DMU produces s desirable outputs y (¯ r = j 1 rj ¯ 1, 2,··· , s ) and s undesirable outputs y (r ¯ = 1, 2,··· , s ) using m inputs (x , x , 1 2 2 1 j 2 j rj ¯ ··· , x ). The model suggested by Seiford and Zhu [23] can be expressed as follows: mj max β s.t. λ x ≤ x , i = 1, 2,··· , m, j ij ip j=1 g g λ y ≥ β y , r ¯ = 1, 2,··· , s , j p 1 rp ¯ rj ¯ j=1 (6) b b λ y ¯ ≥ β y ¯ , r ¯ = 1, 2,··· , s , j p 2 ¯ ¯ rj ¯ rp ¯ j=1 λ = 1, j=1 λ ≥ 0, j = 1, 2,··· , n, b b where y ¯ = −y + w > 0. r ¯ ¯ ¯ r ¯ rj ¯ In this model, the inverse ofβ is the efficiency score of DMU .Ifβ > 1, DMU p p p p is not efficient and the parameter β indicates the extent by which DMU has to p p increase its desirable outputs and translated undesirable outputs to become efficient. For an inefficient DMU , we define its reference set as ∗ ∗ E = {λ |λ > 0, j = 1, 2,··· , n}, j j ∗ ∗ ∗ ∗ where λ = (λ ,λ ,··· ,λ ) is the optimal solution of (6). In this case, the virtual 1 2 point on the efficient frontier (composited based on reference set) is used to evaluate the performance of DMU and can be regarded as a target unit for the inefficient unit p 118 A. Ebrahimnejad (2011) DMU . This target unit usually does not include a DM’s preference structure or value judgments. So, it needs to use an interactive method in MOLP. 3.2. Multiple Objective Linear Programming In this section, we briefly discuss basic concepts and models in multiple objective optimizations and in particular the min-ordering formulation, as a basis for the inves- tigation to be reported in the following sections. Suppose an optimization problem has t objectives reflecting the different purposes and desires of the DM. Such a problem can be represented in a general form as fol- lows: max h(λ) = [h (λ), h (λ),··· , h (λ)] 1 2 t s.t. p (λ) ≤ 0, j = 1,··· , u , (7) j 1 q (λ) = 0, e = 1,··· , u , e 2 where, h (λ)(k = 1, 2,··· , t) are continuously differentiable objective functions, and p (λ)( j = 1, 2,··· , u ) and q (λ)(e = 1, 2,··· , u ) are continuously differentiable j 1 e 2 inequality and equality constraint functions, respectively. Here h (λ), p (λ) and q (λ) k j e are all assumed to be linear functions ofλ, so Formulation (7) is referred to as multiple objective linear programming, or MOLP in short. We also denote the feasible space of (7) by Q. As in an MOLP problem we do not reach a single solution to optimize all objec- tives together, there are efficient or non-dominated solutions. Conceptually, a feasible solutionλ is said to be efficient or non-dominated if there exists no other feasible so- ∗ ∗ lution which is better than λ at least on one objective and as good as λ on all other objectives. An efficient or non-dominated solution is also referred to as a Pareto- optimal solution. In order to reach to an efficient solution, Formulation (7) can be written in min-ordering approach as follows: max min h(λ) = [h (λ), h (λ),··· , h (λ)] 1 2 t (8) s.t. λ ∈ Q. The min-ordering Formulation (8) can then be written as follows by introducing an auxiliary variableθ: maxθ (9) s.t. h (λ) ≥ θ, k = 1,··· , t, λ ∈ Q. However, this approach does not include DM’s preference structures in finding the efficient solution. Thus an interactive method requires for solving MOLP. In the fol- lowing section, we use the Zionts-Walleniuss method [24] to integrate fuzzy output- oriented BCC performance assessment and target setting such that the DMs prefer- ence can be taken into account in an interactive fashion. 4. Performance Evaluation of Fuzzy DEA Model Based on an MOLP Method In this section, we first formulate the output-oriented BCC model with fuzzy data and then establish the equivalent relation between this model and MOLP. Fuzzy Inf. Eng. (2011) 2: 113-125 119 4.1. Fuzzy Output-oriented BCC Model with Undesirable Factors In ordinary DEA models, the input and output values are assumed to be definite. In recent year, in different applications of DEA, inputs and outputs have been observed whose values are indefinite. Such data are called “inaccurate”. Inaccurate data can be probabilistic, interval, ordinal, qualitative, or fuzzy. Therefore, some papers were pre- sented on the theoretical development of this technique whit fuzzy data. To formulate g g g the fuzzy output-oriented BCC model, assume x ˜ = (s , l , r ), y ˜ = (s , l , r ) ij x x x ij ij ij y y y rj ¯ rj ¯ rj ¯ rj ¯ and y ˜ = (s b , l b , r b ) for (i = 1, 2,··· , m, r ¯ = 1, 2,··· , s , r ¯ = 1, 2,··· , s , j = y y y 1 2 rj ¯ ¯ ¯ ¯ rj ¯ rj ¯ rj ¯ 1, 2,··· , n). Thus, the fuzzy version of Model (6) is given as follows: max β s.t. λ x ˜ ≤ x ˜ , i = 1, 2,··· , m, j ij ip j=1 g g λ y ˜ ≥ β y ˜ , r ¯ = 1, 2,··· , s , j p 1 rp ¯ rj ¯ j=1 (10) b b ˜ ˜ ¯ λ y ¯ ≥ β y ¯ , r ¯ = 1, 2,··· , s , j p 2 ¯ ¯ rj ¯ rp ¯ j=1 λ = 1, j=1 λ ≥ 0, j = 1, 2,··· , n, b b where y ¯ = −y ˜ + w ˜ = (s b , l b , r b ) 0. r ¯ y ¯ y ¯ y ¯ ¯ ¯ r ¯ rj ¯ ¯ ¯ ¯ j rj ¯ rj ¯ rj ¯ This model can be rewritten as follows: max β s.t. λ s ≤ s , i = 1, 2,··· , m, j x x ij ip j=1 λ (s − l ) ≤ (s − l ), i = 1, 2,··· , m, j x x x x ij ij ip ip j=1 λ (s + r ) ≤ (s + r ), i = 1, 2,··· , m, j x x x x ij ij ip ip j=1 g g λ s ≥ β s , r ¯ = 1, 2,··· , s , j y p y 1 rp ¯ rj ¯ (11) j=1 g g g g λ (s − l ) ≥ β (s − l ), r ¯ = 1, 2,··· , s , j p 1 y y y y rj ¯ rj ¯ rp ¯ rp ¯ j=1 g g g λ (s + ry ) ≥ β (s + r ), r ¯ = 1, 2,··· , s , j p 1 y y y rj ¯ rj ¯ rp ¯ rp ¯ j=1 λ s b ≥ β s b , r ¯ = 1, 2,··· , s , j p 2 y ¯ y ¯ ¯ ¯ rj ¯ rp ¯ j=1 λ (s b − l b ) ≥ β (s b − l b ), r ¯ = 1, 2,··· , s , j y ¯ y ¯ p y ¯ y ¯ 2 ¯ ¯ ¯ ¯ rj ¯ rj ¯ rp ¯ rp ¯ j=1 120 A. Ebrahimnejad (2011) λ (s b + r b ) ≥ β (s b + r b ), r ¯ = 1, 2,··· , s , j p 2 y ¯ y ¯ y ¯ y ¯ rj ¯ ¯ rj ¯ ¯ rp ¯ ¯ rp ¯ ¯ j=1 λ = 1, j=1 λ ≥ 0, j = 1, 2,··· , n. 4.2. Relation between Fuzzy DEA Model and MOLP Consider an MOLP in the following form: 1 2 3 max f (λ), f (λ), f (λ) r ¯ r ¯ r ¯ 1 2 3 max g (λ), g (λ), g (λ) ¯ ¯ ¯ r ¯ r ¯ r ¯ s.t. λ s ≤ s , i = 1, 2,··· , m, j x x ij ip j=1 λ (s − l ) ≤ (s − l ), i = 1, 2,··· , m, j x x x x ij ij ip ip (12) j=1 λ (s + r ) ≤ (s + r ), i = 1, 2,··· , m, j x x x x ij ij ip ip j=1 λ = 1, j=1 λ ≥ 0, j = 1, 2,··· , n. It needs to point out that the above MOLP problem can be rewritten in the following form using min-ordering formulation mentioned in the last section and introducing the auxiliary variableθ: maxθ s.t. f (λ) ≥ θ, r ¯ = 1, 2,··· , s , t = 1, 2, 3, r ¯ (13) g (λ) ≥ θ, r ¯ = 1, 2,··· , s , v = 1, 2, 3, r ¯ λ ∈ Q , where, the feasible space Q is given as follows: n n λ | λ s ≤ s , λ (s − l ) ≤ (s − l ), Q = j j x x j x x x x (14) p ⎪ ij ip ij ij ip ip j=1 j=1 n n λ (s + r ) ≤ (s + r ), (i = 1, 2,··· , m), λ = 1 j x x x x j . ij ij ip ip ⎪ j=1 j=1 In this case, the fuzzy output-oriented BCC Model (11) can be equivalently rewrit- Fuzzy Inf. Eng. (2011) 2: 113-125 121 ten as follows: max β g g s.t. λ s ≥ β s , r ¯ = 1, 2,··· , s , j p 1 y y rj ¯ rp ¯ j=1 g g g g λ (s − l ) ≥ β (s − l ), r ¯ = 1, 2,··· , s , j y y p y y 1 rp ¯ rp ¯ rj ¯ rj ¯ j=1 g g g λ (s + ry ) ≥ β (s + r ), r ¯ = 1, 2,··· , s , j y p y y 1 rj ¯ rj ¯ rp ¯ rp ¯ j=1 (15) b b λ s ≥ β s , r ¯ = 1, 2,··· , s , j p 2 y ¯ y ¯ ¯ ¯ rj ¯ rp ¯ j=1 λ (s b − l b ) ≥ β (s b − l b ), r ¯ = 1, 2,··· , s , j p 2 y ¯ y ¯ y ¯ y ¯ ¯ ¯ ¯ ¯ rj ¯ rj ¯ rp ¯ rp ¯ j=1 λ (s b + r b ) ≥ β (s b + r b ), r ¯ = 1, 2,··· , s , j p 2 y ¯ y ¯ y ¯ y ¯ ¯ ¯ ¯ ¯ rj ¯ rj ¯ rp ¯ rp ¯ j=1 λ ∈ Q . In order to prove that Formulation (15) is equivalent to the min-ordering formu- lation in (12), certain conditions have to be applied. The purpose for establishing the equivalence conditions is to use the interactive techniques in MOLP to locate the most preferred solution (MPS) on the efficient frontier for target setting and resource g g g g g allocation. Suppose s , (s − l ) and (s + r ) > 0 for any r ¯ = 1, 2,··· , s and y y y y y rp ¯ rp ¯ rp ¯ rp ¯ rp ¯ t v define f (λ), (t = 1, 2, 3) and g (λ), (v = 1, 2, 3) in Formulation (12) as follows: r ¯ ¯ r ¯ λ s j y rj ¯ i=1 f (λ) = , (16) r ¯ rp ¯ g g λ (s − l ) y y rj ¯ rj ¯ i=1 f (λ) = , (17) r ¯ g g (s − l ) y y rp ¯ rp ¯ g g λ (s + r ) y y rj ¯ rj ¯ i=1 f (λ) = , (18) r ¯ g g (s + r ) y y rp ¯ rp ¯ λ s b y ¯ rj ¯ i=1 g (λ) = , (19) r ¯ s b y ¯ rp ¯ λ (s b − l b ) y ¯ y ¯ ¯ ¯ rj ¯ rj ¯ i=1 g (λ) = , (20) r ¯ (s b − l b ) y ¯ y ¯ ¯ ¯ rp ¯ rp ¯ 122 A. Ebrahimnejad (2011) b b λ (s + r ) y ¯ y ¯ ¯ ¯ rj ¯ rj ¯ i=1 g (λ) = . (21) r ¯ b b (s + r ) y ¯ y ¯ ¯ ¯ rp ¯ rp ¯ g g g g g Since s , (s − l ) and (s + r ) > 0 for any r ¯ = 1, 2,··· , s , Formulation (15) y y y y y 1 rp ¯ rp ¯ rp ¯ rp ¯ rp ¯ can be written as follows: max β λ s j y rj ¯ i=1 s.t. ≥ β , r ¯ = 1, 2,··· , s , p 1 s g rp ¯ g g λ (s − l ) y y rj ¯ rj ¯ i=1 ≥ β , r ¯ = 1, 2,··· , s , p 1 (s g −l g ) y y rp ¯ rp ¯ g g λ (s + r ) y y rj ¯ rj ¯ i=1 ≥ β , r ¯ = 1, 2,··· , s , p 1 (s g +r g ) y y rp ¯ rp ¯ m (22) λ s b y ¯ rj ¯ i=1 ≥ β , r ¯ = 1, 2,··· , s , p 2 y ¯ rp ¯ ¯ λ (s b − l b ) j y ¯ y ¯ ¯ ¯ rj ¯ rj ¯ i=1 ≥ β , r ¯ = 1, 2,··· , s , p 2 (s −l ) b b y ¯ y ¯ rp ¯ ¯ rp ¯ ¯ λ (s b + r b ) y ¯ y ¯ rj ¯ ¯ rj ¯ ¯ i=1 ≥ β , r ¯ = 1, 2,··· , s , p 2 (s +r ) b b y ¯ y ¯ ¯ ¯ rp ¯ rp ¯ λ ∈ Q . Now using Formulations (16)− (21), we get the following problem: max β s.t. f (λ) ≥ θ r ¯ = 1, 2,··· , s , t = 1, 2, 3, r ¯ (23) g (λ) ≥ θ r ¯ = 1, 2,··· , s , v = 1, 2, 3, r ¯ λ ∈ Q . This means that the fuzzy output-oriented BCC model in the present of undesirable outputs (15) is equivalent to the min-ordering Formulation (15), ifβ = θ. Alternatively, since Formulation (13) gives a special weak efficient point of For- mulation (12), then Formulation (11) or (15) also gives a special weak efficient point of following formulation: Fuzzy Inf. Eng. (2011) 2: 113-125 123 ⎡ ⎤ m m m ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ g g g g g ⎢ λ s λ (s − l ) λ (s + r )⎥ j j j ⎢ y y y y y ⎥ rj ¯ rj ¯ rj ¯ rj ¯ rj ¯ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ i=1 i=1 i=1 ⎥ ⎢ ⎥ max , , ⎢ ⎥ ⎢ g g g g g ⎥ s (s −l ) (s +r ) ⎢ ⎥ y y y y y ⎢ ⎥ rp ¯ rp ¯ rp ¯ rp ¯ rp ¯ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ m m m ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ λ s b λ (s b − l b ) λ (s b + r b )⎥ j j j ⎢ y ¯ y ¯ y ¯ y ¯ y ¯ ⎥ ⎢ rj ¯ ¯ rj ¯ ¯ rj ¯ ¯ rj ¯ ¯ rj ¯ ¯ ⎥ ⎢ ⎥ ⎢ ⎥ i=1 i=1 i=1 ⎢ ⎥ ⎢ ⎥ max ⎢ , , ⎥ ⎢ ⎥ s (s −l ) (s +r ) ⎢ b b b b b ⎥ y ¯ y ¯ y ¯ y ¯ y ¯ ⎢ ⎥ ¯ ¯ ¯ ¯ ¯ ⎢ rp ¯ rp ¯ rp ¯ rp ¯ rp ¯ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (24) s.t. λ s ≤ s , i = 1, 2,··· , m, j x x ij ip j=1 λ (s − l ) ≤ (s − l ), i = 1, 2,··· , m, j x x x x ij ij ip ip j=1 λ (s + r ) ≤ (s + r ), i = 1, 2,··· , m, j x x x x ij ij ip ip j=1 λ = 1, j=1 λ ≥ 0, j = 1, 2,··· , n, that does not include DM’s preference structures or value judgments in measuring relative efficiency and setting target values, so the efficiency score of the DMU can be generated by solving Formulation (24). Hence, an interactive MOLP method can be used to solve the DEA problem. 4.3. An Interactive Method It is an issue how decision makers decide one from the set of non-dominated solutions as the final solution. Consequently, interactive MOLP methods have been developed to this end. In this paper, we use the Zionts-Walleniuss method [24] to integrate combined- oriented CCR performance assessment and target setting such that the DMs prefer- ence can be taken into account in an interactive fashion. This method is applicable to problem in (24) where the objective functions are concave and feasible space is a convex set. The overall utility function is assumed to be unknown explicitly to the DM, but is implicitly a linear function and more generally a concave function of the objective functions. The method makes use of such an implicit function on an in- teractive basis. The first step of the method is to choose an arbitrary set of positive multipliers or weights and generate a composite objective function or utility function using these multipliers. The composite objective function is then optimized to pro- duce a non-dominated solution to the problem. From the set of non-basic variables, a subset of efficient variables is selected (an efficient variable is one which, when in- troduced into the basis, cannot increase one objective without decreasing at least one other objective). For each efficient variable a set of tradeoffs is defined by concept that some objectives are increased and others reduced. A number of such tradeoffs 124 A. Ebrahimnejad (2011) are presented to the DM, who is requested to state whether the tradeoffs are desirable, undesirable or neither. From his/her answers a new set with consistent multipliers is constructed and the associated non-dominated solution is found. The process is then repeated and a new set of tradeoffs is presented to the DM at the current solution until the DM find the most preferred solution with respect to the implicit utility. 5. Conclusion In this paper, we obtained an equivalence relation between the output-oriented BCC model with fuzzy data in the present of undesirable outputs and the min-ordering model in MOLP and showed how a DEA problem can be solved interactively by transforming it into MOLP formulation. This approach results a decrease in total undesirable outputs production and a permissible increase in total desirable output production. The proposed equivalence model provided the basis to apply interactive methods in MOLP to solve DEA problems and further locate the MPS along the efficient frontier for each inefficient DMU. We proposed the Zionts-Wallenius method to reflecting the DM preferences in the process of assessing efficiency. Acknowledgements The author would like to express their sincerest thanks to the anonymous referees and honorable editor for their valuable comments which helped in improving this paper. References 1. Doyle J, Green R (1993) Data envelopment analysis and multiple criteria decision making. Omega 21: 713-715 2. Allen R, Athanassopoulos A D, Dyson R G, Thanassoulis E (1997) Weights restrictions and value judgments in data envelopment analysis: evolution, development and future directions. Annals of Operational Research 73: 13-34 3. Golany B (1988) An interactive MOLP procedure for the extension of DEA to effectiveness analysis. Journal of Operational Research Society 39: 725-734 4. Thanassoulis E, Dyson R G (1992) Estimating preferred target input-output levels using data envel- opment analysis. European Journal of Operational Research 56: 80-97 5. Athanassopoulos A D (1995) Goal programming and data envelopment analysis (GPDEA) models for multi-level multi-unit organizations: an application to Greek local authorities. European Journal of Operational Research 87: 535-550 6. Athanassopoulos A D (1998) Decision support for target-based resource allocation of public services in multi unit and multilevel systems. Management Science 39: 1261-1264 7. Yang J B, Wong B Y H, Xu D L, Stewart T J (2009) Integrating DEA-oriented performance assess- ment and target setting using interactive MOLP methods. European Journal of Operational Research 195: 205-222 8. Hosseinzadeh Lotfi F, Jahanshaloo G R, Ebrahimnejad A, Soltanifar M, Mansourzadeh S M (2010) Target setting in the general combined-oriented CCR model using an interactive MOLP method. Journal of Computational and Applied Mathematics 234: 1-9 9. Hosseinzadeh Lotfi F, Jahanshaloo G R, Soltanifar M, Ebrahimnejad A, Mansourzadeh S M (2010) Relation between MOLP and DEA based on output-oriented CCR dual model. Expert System with Application 37: 4331-4336 10. Banker R, Charnes A, Cooper W (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science 30(9): 1078-1092 11. Klir G J, Yuan B (1996) Fuzzy sets and fuzzy logic: theory and applications. Prentice-Hall, Inc., N.J. USA Fuzzy Inf. Eng. (2011) 2: 113-125 125 12. Ebrahimnejad A, Nasseri S H, Mansourzadeh S M (2011) Bounded primal simplex algorithm for bounded linear programming with fuzzy cost coefficients. International Journal of Operations Re- search and Information Systems 2(1): 96-120 13. Ebrahimnejad A, Nasseri S H, and Hosseinzadeh Lotfi F (2010) Bounded linear programs with trape- zoidal fuzzy numbers. International Journal of Uncertainty, Fuzziness and Knowledge-Based Sys- tems 18(3): 269-286 14. Ebrahimnejad A, Nasseri S H, Hosseinzadeh Lotfi F, Soltanifar M (2010) A primal-dual method for linear programming problems with fuzzy variables. European Journal of Industrial Engineering 4(2): 189-209 15. Ebrahimnejad A, Nasseri S H (2010) A dual simplex method for bounded linear programmes with fuzzy numbers. International Journal of Mathematics in Operational Research 2(6): 762-779 16. Ebrahimnejad A, Nasseri S H (2009) Using complementary slackness property to solve linear pro- gramming with fuzzy parameters. Fuzzy Information and Engineering 3: 233-245 17. Mahdavi-Amiri N, Nasseri S H, Yazdani A (2009) Fuzzy primal simplex algorithms for solving fuzzy linear programming problems. Iranian Journal of Operational Research 1: 68-84 18. Mahdavi-Amiri N, Nasseri S H (2006) Duality in fuzzy number linear programming by use of a certain linear ranking function. Applied Mathematics and Computation 180: 206-216 19. Mahdavi-Amiri N, Nasseri S H (2007) Duality results and a dual simplex method for linear program- ming problems with trapezoidal fuzzy variables. Fuzzy Sets and Systems 158: 1961-1978 20. Nasseri S H, Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for solving flexible linear programming problem. European Journal of Industrial Engineering 4: 372-389 21. Nasseri S H, Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear program- ming problem. Advances in Fuzzy Sets and Systems 5: 81-95 22. Nasseri S H, Mahdavi-Amiri N (2010) Some duality results on fuzzy linear programming with sym- metric fuzzy numbers. Fuzzy Information and Engineering 1: 59-66 23. Seiford M, Zhu J (2002) Modeling undesirable factors in efficiency evaluation. European Journal of Operational Research 142: 16-20 24. Zionts S, Wallenius J (1976) An interactive programming method for solving the multiple criteria problem. Management Science 22: 65-63

Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Jun 1, 2011

Keywords: Data envelopment analysis; Multiple objective linear programming; Fuzzy numbers; Undesirable factors; Target unit

References