Sensitivity and Stability Analysis in Uncertain Data Envelopment (DEA)

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1 Sensitivity and Stability Analysis in Uncertain Data Envelopment (DEA) eilin Wen a,b, Zhongfeng Qin c, Rui Kang b a State Key Laboratory of Virtual Reality Technology and Systems b Department of System Engineering of Engineering Technology c School of Economics and anagement Science Beihang University, Beiing, , China wenmeilin@buaa.edu.cn,kangrui@buaa.edu.cn Abstract As a useful management and decision tool, data envelopment analysis (DEA) has become a pop area of research. One of the topics of interests in DEA is sensitivity and stability analysis of decision making units (DUs). Due to the uncertainty of the data in real life, this paper will give some DEA models in fuzzy environment. It is followed by a series analysis of sensitivity and stability for all DUs. Finally a numerical example will be presented to give an illustration of the sensitivity and stability analysis. Keywords: Data Envelopment Analysis; Credibility easure; Efficiency; Sensitivity; Stability 1 Introduction Data envelopment analysis (DEA), introduced by Charnes et al. [2] and extended by Banker et al. [1], Charnes et al. [3], Petersen [18], Tone [21] and Cooper [7], is a useful method to evaluate the relative efficiency of decision making units (DUs). Its goal is to classify the DUs into two classes: efficient or inefficient ones. However, uncertainty such as a measurement error should be incorporated in observed data. This indicates the necessity to assess the sensitivity of classifications in DEA. During the recent years, the issue of sensitivity and stability of DEA has been extensively studied. The first DEA sensitivity analysis paper by Charnes et al. [3] examined change in a single output. This is followed by a series of sensitivity analysis articles. Charnes and Neralic [4] gave some sufficient conditions in preserving efficiency. Charnes et al. [5][6] developed a sensitivity analysis technique on super-efficiency DEA model for the situation where simultaneous proportional change is assumed in all inputs and outputs for a specific DU under consideration. This data variation condition is relaxed in Zhu [23] and Seiford 1

2 and Zhu [19] to a situation, where inputs or outputs can be changed individually and the largest stability region is obtained. The DEA sensitivity analysis methods we have ust reviewed are all developed for the situation, where data variations are only applied to the efficient DU under evaluation and the data for the remaining DUs are assumed to be fixed. Obviously, this assumption may not be realistic, since possible data errors may occur in each DU. Seiford and Zhu [20] generalized the technique in Zhu [23] and Seiford and Zhu [19] to the case, where the efficiency of the under evaluation efficient DU is deteriorating while the efficiencies of the other DUs are improving. The original DEA models assume that inputs and outputs are measured by exact values. Recently, many researchers addressed the problem with fuzzy data (Cooper et al. [8] [9] [10], Kao & Liu [13], Entani et al. [11], Guo & Tanaka [12], Lertworasirikul et al. [14]). In this paper, we discuss a technique for assessing the sensitivity of efficiency and inefficiency classifications in DEA with fuzzy data. Having identified the efficient and inefficient DUs in fuzzy DEA analysis, we will introduce how sensitive these identifications are to possible variations in data. This paper is organized as follows: In Section 2, some basic concept and results on credibility measure will be introduced. Some introduction to one of the DEA models are given in Section 3. Section 4 will give some fuzzy models for determine the radius of the stability for all the DUs. Finally, the analysis of the sensitivity and stability is introduced through a numerical example in Section 5. 2 Preliminaries Uncertainty theory was founded by Liu [15] in 2007 and refined by Liu [17] in nowadays uncertainty theory has become a branch of axiomatic mathematics for modeling human uncertainty. In this section, we will state some basic concepts and results on uncertain variables. These results are crucial for the remainder of this paper. Let Γ be a nonempty set, and L a σ-algebra over Γ. Each element Λ L is assigned a number {Λ [0, 1]. In order to ensure that the number {Λ has certain mathematical properties, Liu [15][16] presented the four axioms: Axiom 1. {Γ = 1 for the universal set Γ. Axiom 2. {Λ + {Λ c = 1 for any event Λ. Axiom 3. For every countable sequence of events Λ 1, Λ 2,,, we have { Λ i {Λ i. Axiom 4. Let (Γ k, L k, k ) be uncertainty spaces for k = 1, 2,, n. Then the product uncertain measure is an uncertain measure satisfying 2

3 { n Λ k = min 1 k n k{λ k. If the set function satisfies the first three axioms, it is called an uncertain measure. Definition 1 (Liu [15]) Let Γ be a nonempty set, L a σ-algebra over Γ, and an uncertain measure. Then the triplet (Γ, L, ) is called an uncertainty space. Definition 2 (Liu [15]) An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, ) to the set of real numbers, i.e., for any Borel set B of real numbers, the set is an event. {ξ B = {γ Γ ξ(γ) B (1) Definition 3 (Liu [15]) The uncertainty distribution Φ of an uncertain variable ξ is defined by for any real number x. Φ(x) = {ξ x (2) Definition 4 (Liu [15]) Let ξ be an uncertain variable. Then the expected value of ξ is defined by E[ξ] = {ξ rdr {ξ rdr. (3) Example 1 An uncertain variable ξ is called linear if it has a linear uncertain distribution 0, if x a Φ(x) = (x a)/(b a), if a x b (4) 1, if x b denoted by L(a, b) where a and b are real numbers with a < b. Example 2 An uncertain variable ξ is called zigzag if it has a zigzag uncertainty distribution 0, if x a Φ(x) = (x a)/2(b a), if a x b (x + c 2b)/2(c b), if b x c 1, if x c (5) denoted by Z(a, b, c) where a, b, c are real numbers with a < b < c. 3

4 Example 3 An uncertain variable ξ is called normal if it has a normal uncertainty distribution Φ(x) = ( ( )) 1 π(e x) 1 + exp (6) 3σ denoted by N (e, σ) where e and σ are real numbers with σ > 0. Definition 5 (Liu [17]) An uncertainty distribution Φ is said to be regular if its inverse function Φ 1 (α) exists and is unique for each α (0, 1). Theorem 1 (Liu [17]) Let ξ 1, ξ 2,, ξ n be independent uncertain variables with regular uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. If f is a strictly increasing function, then ξ = f(ξ 1, ξ 2,, ξ n ) (7) is an uncertain variable with inverse uncertainty distribution Ψ 1 = f(φ 1 1 (α), Φ 1(α),, Φ 1 n (α)). (8) 2 3 Uncertain DEA odel This section will introduce one of the Uncertain DEA model, proposed by Wen et al. [?]. The symbols and notations are given as follows: DU i : the ith DU, i = 1, 2,, n; DU 0 : the target DU; x k = ( x k1, x k2,, x kp ): the uncertain inputs vector of DU k, k = 1, 2,, n; Φ ki (x): the uncertainty distribution of x ki, k = 1, 2,, n, i = 1, 2,, p; x 0 = (x 01, x 02,, x 0p ): the inputs vector of the target DU 0 ; Φ 0i (x): the uncertainty distribution of x ki, i = 1, 2,, p; ỹ k = (ỹ k1, ỹ k2,, ỹ kq ): the uncertain outputs vector of DU k, k = 1, 2,, n; Ψ k (x): the uncertainty distribution of x k, k = 1, 2,, n, = 1, 2,, q; y 0 = (y 01, y 02,, y 0q ): the outputs vector of the target DU 0. Ψ 0 (x): the uncertainty distribution of x k, = 1, 2,, q; The model [?] is given as: 4

5 max s i + q s + subect to : { x ki λ k x 0i s i α, { ỹ k λ k ỹ 0 + s + α, λ k = 1 λ k 0, k = 1, 2,, n s i 0, i = 1, 2,, p i = 1, 2,, p = 1, 2,, q (9) s + 0, = 1, 2,, q. Definition 6 (α-efficiency) DU 0 is α-efficient if i (α) and s + (α) are zero for i = 1, 2,, p and = 1, 2,, q, where i (α) and s + (α) are optimal solutions of (9). 4 Sensitivity and Stability Analysis This section will give some sensitivity and stability analysis to uncertain DEA model. Theorem 2 If DU 0 is α-inefficient, then the optimal solution satisfying λ 0(α) = 0. Proof: We assume the target DU 0 is DU 1. That is, x 0 = x 1, y 0, = y 1. We should prove λ 1 = 0. For a fixed α, suppose the optimal solution is (λ,, s + ) and the optimal obective value is p i + q s +. If λ 1 = 0, then the Theorem has been proved. Otherwise let λ 1 > 0. Since DU 1 is inefficient, there exists at least one i > 0 or s + assume 1 > 0. If λ 1 = 1, then { x 11 x 11 1 > 0, i = 1, 2,, p, = 1, 2,, q. Without loss of generality, we = 0. The contradiction implies that λ 1 1. { x ki λ k x 1i i { = x ki λ k (1 λ 1) x 1i i x ki λ k = 1 λ x 1i s i 1 (1 λ 1 ) α, i = 1, 2,, p. 5

6 Similarly we can get { ỹ k λ k ỹ 1 + s + ỹ k λ k = 1 λ ỹ s+ (1 λ 1 ) Since 1 1 λ 1 λ k 1 λ 1 i + = 1, 0, λ 2 q s + λ k >, α, = 1, 2,, q. λ 3 λ k i +,, q λ n λ k the assumption. Thus λ 1 = 0. The theorem is proved. is a feasible solution. Then the obective value is s +, since 0 < λ 1 < 1, which leads to a contradiction with Theorem 3 If a DU with ( x 0, ỹ 0 ) is inefficient after evaluating by model (9), the new DU with ( x 0, ŷ 0 ) = ( x 0, ỹ 0 + s + ) is α-efficient, in which and s + are optimal solution of (9). Proof: The efficiency of ( x 0, ŷ 0 ) is evaluated by solving the problem below: max s i + q s + subect to : { { λ k = 1 x ki λ k + x 0i λ 0 x 0i s i α, ỹ k λ k + ŷ 0 λ 0 ŷ 0 + s + α, λ k 0, k = 1, 2,, n s i 0, i = 1, 2,, p i = 1, 2, p = 1, 2,, q (10) s + 0, = 1, 2, q. Let an optimal solution be ( λ, ŝ +, ŝ ). Suppose the DU with ( x 0, ŷ 0 ) is inefficient, then λ 0 = 0 can be got by Theorem 2. By inserting the formula ( x 0, ŷ 0 ) = ( x 0, ỹ 0 + s + ) into constraints, we have x ki λk x 0i ŝ i ỹ k λk ỹ 0 + ŝ + Now we can also write the constraints as i α, i = 1, 2, p, + s+ α, = 1, 2,, q. 6

7 x ki λk x 0i s i α, i = 1, 2, p, ỹ k λk ỹ 0 + s + α, = 1, 2,, q where s + = ŝ + + s + 0 and s = ŝ + 0. Furthermore, we have s i + q s + = ( ŝ i + i ) + ( q since these constraints is a feasible solution for model (9) and that we have p ŝ + + s+ ) p i i + q q s + s + is maximal. It follows ŝ i + q ŝ + = 0 which implies that all components ŝ i and ŝ + are zero. Hence fuzzy efficiency is achieved as claimed. Theorem 3 has given the stable region when the DU is inefficient. But we also want to know the efficient radius of efficient DUs. For this purpose, the following model is proposed: min t + i + q t subect to : { {,,k 0 λ k = 1 x ki λ k x 0i + t + i α, ỹ k λ k ỹ 0 t α, λ k 0, k = 1, 2,, n t + i 0, i = 1, 2,, p i = 1, 2,, p = 1, 2,, q (11) t 0, = 1, 2,, q. Theorem 4 The α-efficient DU 0 stays α-efficient if ( x 0, ŷ 0 ) = ( x 0 + t +, ỹ 0 t ), where t + and t are optimal solution of (14). Proof: Consider the following DEA model for evaluating the relative efficiency of the adusted DU 0 : 7

8 max s i subect to : { { + q s + λ k = 1 λ k 0, x ki λ k + ( x 0i + t + i )λ 0 ( x 0i + t + i ) s i α, ỹ k λ k + (ỹ 0 t k = 1, 2,, n s i 0, i = 1, 2,, p )λ 0 (ỹ 0 t ) + s + α, i = 1, 2,, p = 1, 2,, q (12) s + 0, = 1, 2,, q. For a fixed α, let the optimal solution to be (λ, λ 0,, s + ) and assume that the DU 0 is inefficient. From Theorem 2, we get λ 0 = 0. Thus this optimal solution is a feasible solution for (14). Hence t + i i t + i and t s + t, which means leads to a contradiction with the assumption. i = 0 and s +, i = 1, 2,, p, = 1, 2,, q. This max s i subect to : + q s + λ k Φ 1 ki (α) + λ 0Φ 1 0i (1 α) Φ 1 0i (1 α) s i, λ k Ψ 1 k (1 α) + λ 0Ψ 1 0 (α) Ψ 1 0 (α) s+, λ k = 1 λ k 0, k = 1, 2,, n i = 1, 2,, p = 1, 2,, q (13) s i 0, i = 1, 2, p s + 0, = 1, 2,, q. min t + i subect to : + q t λ k Φ 1 ki (α) Φ 1 0i (1 α) + t+ i, λ k Ψ 1 k (1 α) Ψ 1 0 (α) t, λ k = 1 λ k 0, k = 1, 2,, n t + i 0, i = 1, 2,, p i = 1, 2,, p = 1, 2,, q (14) t 0, = 1, 2,, q. 8

9 From above analysis, the ranges of inputs and outputs and radius of stability of DU 0 are identified as follows: 1. If DU 0 is α-inefficient by solving model (9), then DU 0 stays α-inefficient if ( x 0, ŷ 0 ) = ( x 0 s, ỹ 0 + s + ), in which s = {(s 1,, s p ) 0 s i s +, = 1, 2,, q, where i and s + are optimal solutions of (9). < i, i = 1, 2,, p, s + = {(s + 1,, s+ q ) 0 s + < 2. If DU 0 is α-efficient by solving model (9), then we use model (14) to account the efficient radius. DU 0 stays α-efficient if ( x 0, ŷ 0 ) = ( x 0 + t +, ỹ 0 t ), in which t + = {(t 1,, t p ) 0 t i t + i, i = 1, 2,, p and t = {(t 1,, t q ) 0 t t, = 1, 2,, q, where t + i and t are optimal solutions of (14). 5 A Numerical Example This numerical example is presented to give an illustration of the sensitivity and stability analysis. Table 1 provides the information of the DUs. There are two uncertain inputs and two uncertain outputs which are all zigzag uncertain variables denoted by Z(a, b, c). Table 1: DUs with two uncertain inputs and two uncertain outputs DU i Input 1 Z(3.5,4.0,4.5) Z(2.9,2.9,2.9) Z(4.4,4.9,5.4) Z(3.4,4.1,4.8) Z(5.9,6.5,7.1) Input 2 Z(1.9,2.1,2.3) Z(1.4,1.5,1.6) Z(2.2,2.6,3.0) Z(2.1,2.3,2.5) Z(3.6,4.1,4.6) Output 1 Z(2.4,2.6,2.8) Z(2.2,2.2,2.2) Z(2.7,3.2,3.7) Z(2.5,2.9,3.3) Z(4.4,5.1,5.8) Output 2 Z(3.8,4.1,4.4) Z(3.3,3.5,3.7) Z(4.3,5.1,5.9) Z(5.5,5.7,5.9) Z(6.5,7.4,8.3) DUs (λ 1, λ 2, λ 3, λ 4, λ 5) Table 2: Results of evaluating the DUs q i + s + the result of evaluating DU 1 (0,0.74,0,0.09,0.17) 0.75 inefficiency DU 2 (0,1,0,0,0) 0 efficiency DU 3 (0,0,1,0,0) 0.19 inefficiency DU 4 (0,0,0,1,0) 0 efficiency DU 5 (0,0,0,0,1) 0 efficiency Table 2 shows the results of evaluating DUs using model (9) with confidence level α = 0.6. The results can be interpreted in the following way: DU 1 and DU 3 are inefficient, whereas DU 2, DU 4 and DU 5 are efficient. Tables 3 reports the sensitivity analysis results for inefficient DUs by model (9). In Table 3, the 9

10 Table 3: Sensitivity analysis results of inefficient DUs by model (9) DUs 1 2 s + 1 s + 2 DU DU columns 2 and 3 report lower bounds of variation ranges of inputs and the columns 4 and 5 are upper bounds of variation ranges of outputs. For instance consider DU 1 in Table 3. It stays inefficient when ( x 11, x 12, ŷ 11, ŷ 12 ) = ( x 11 r x1, x 12, ỹ 11, ỹ 12 + r y2 ), in which 0 r x1 < 0.15 and 0 r y2 < Table 4: Sensitivity analysis results of efficient DUs by model (14) DUs 1 2 s + 1 s + 2 DU DU DU Table 4 reports the sensitivity analysis results for efficient DUs by model (14). In Table 4, the columns 2 and 3 report upper bounds of variation ranges of inputs and the columns 4 and 5 are lower bounds of variation ranges of outputs. For instance consider DU 4 in Table 4. It stays efficient when ( x 41, x 42, ŷ 41, ŷ 42 ) = ( x 41, x 42 + r x2, ỹ 41, ỹ 42 r y2 ), in which 0 r x and 0 r y The similar interpretation can be stated for other rows in Table 3 and Table 4. 6 Conclusion Due to its widely practical used background, data envelopment analysis (DEA) has become a pop area of research. Since the data cannot be precisely measured in some practical cases, many paper have been published when the inputs and outputs are uncertain. Fuzzy methods are one of them. This paper has given some sensitivity and stability analysis in fuzzy DEA. Based on credibility measure, we found radius of stability for all DUs when the inputs and outputs were fuzzy variables. Finally a numerical example was presented to give an illustration of sensitivity and the stability analysis. Acknowledgments This work was supported by National Natural Science Foundation of China (No ). 10

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