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A bi-objective generalized data envelopment analysis model and point-to-set mapping projection
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European Journal of Operational Research 190 (2008) 855–876 www.elsevier.com/locate/ejor
Interfaces with Other Disciplines
A bi-objective generalized data envelopment analysis model and point-to-set mapping projection q Quanling Wei a, Hong Yan a
b,*
, Lin Xiong
a
Institute of Operations Research and Mathematical Economics, Renmin University of China, China b Department of Logistics, The Hong Kong Polytechnic University, Hong Kong Received 3 August 2006; accepted 27 June 2007 Available online 28 August 2007
Abstract This work introduces a bi-objective generalized data envelopment analysis (Bi-GDEA) model and defines its efficiency. We show the equivalence between the Bi-GDEA efficiency and the non-dominated solutions of the multi-objective programming problem defined on the production possibility set (PPS) and discuss the returns to scale under the Bi-GDEA model. The most essential contribution is that we further define a point-to-set mapping and the mapping projection of a decision making unit (DMU) on the frontier of the PPS under the Bi-GDEA model. We give an effective approach for the construction of the point-to-set-mapping projection which distinguishes our model from other non-radial models for simultaneously considering input and output. The Bi-GDEA model represents decision makers’ specific preference on input and output and the point-to-set mapping projection provides decision makers with more possibility to determine different input and output alternatives when considering efficiency improvement. Numerical examples are employed for the illustration of the procedure of point-to-set mapping. 2007 Elsevier B.V. All rights reserved. Keywords: Data envelopment analysis (DEA); Bi-GDEA efficiency; Point-to-set mapping projection; Non-dominated solution
1. Introduction Data envelopment analysis (DEA) is a non-parametric approach to measure the relative efficiency of a decision making unit (DMU) and provide DMUs with relative performance assessment on multiple inputs and outputs. Based on different empirical axioms and corresponding to different characteristics of the production possibility set and production frontiers, different DEA models, namely the BCC model [2], the FG model [10], the ST model [16], and the CCWH model [6,7], are developed and applied in practice. Wei and Yu [22] and Yu et al. [24,25] introduce a generalized DEA model (GDEA) with three 0–1 parameters (d1, d2, d3) to give an unification of DEA models and methods. By assigning different values to these parameters, the GDEA model q
The first author is partially supported by the National Natural Science Foundation of China, NNSF 70531040 and The 985 Research Grant of Renmin University of China. The research is partially supported by The Hong Kong UGC CERG Grant PolyU5457/06H. * Corresponding author. E-mail address: [email protected] (H. Yan). 0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.06.053
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can be reduced to the classic DEA models mentioned above. Furthermore, the GDEA model involves two important concepts, which are predilection cone and preference cone. The predilection cone is used to represent the preference of the decision maker in different DMUs while the preference cone is used to describe the preference of the decision maker on different input/output categories. The research on the GDEA model can lead important properties of different DEA models and methods. A consistent task in DEA research and application is to improve the efficiency state of DMUs. In the abovementioned DEA models for an inefficient DMU, an input-oriented model obtains its efficiency projection on the production frontier by equi-proportionally decreasing its inputs while keeping its outputs fixed. Likewise, an output-oriented model increases its outputs equi-proportionally while keeping its inputs fixed to obtain the projection. Such a projection is a point-to-point mapping from the concerned DMU to a point on the production frontier. However, these models fail to describe the decision maker’s preference to achieve increasing outputs and decreasing inputs simultaneously because of the single projection feature. In addition, radial projections often lead to Pareto-inefficient portions of the frontier [13]. It is clear that multiple projections are possible if different combination of input decreasing and output increasing occur. Various models have been proposed to integrate inputs and outputs, particularly for considering the returns to scale (RTS) measurement [8,17]. Banker et al. [4] discusses and applies non-radial DEA models, e.g. additive and multiplicative models. Fukuyama [12] gives a detailed discussion on RTS measurements using non-radial DEA models. Thanassoulis and Dyson [19] proposes to assign weights to changes in the input and output element to express the decision maker’s preference structure. Lins et al. [13] uses a similar nonradial DEA model with different weights on input and output to determine alternative targets in DEA. Another way to integrate input and output measurements is the directional distance function approach. Such function is essentially a variation of the Luenberger shortage function, which measures the distance in a given direction as a shortage of a point to the production frontier [14]. Similar to that in input-oriented and outputoriented DEA models, the input distance function and output distance function are defined to scale inputs and output respectively (see [11] for a fundamental description of distance function and the production possibility set, called technology in the work). Chambers et al. [5] suggests a general directional distance function to integrate both input decrease and output increase in a simultaneous manner. This function measures the distance from a point to the production frontier in a given direction. It is especially important to estimate RTS of DMUs in the research and application of DEA. Cooper et al. [9] considers to increase outputs while decreasing inputs based on the most productive scale size in order to estimate RTS of DMUs. Their model maximizes the ratio between multiple of decreased inputs and multiple of increased outputs and uses the optimal ratio to estimate RTS. The recent work of Sueyoshi and Sekitani [18] gives a detailed introduction on the DEA based RTS measurement and discusses a range-adjusted measure (RAM) in the non-radial DEA model for return to scale (RTS) measurement. In particular, they point out that the multiple projection, if occurs, has serious influence on non-radial RTS measurement. Moreover, abundant works of using optimal solution, optimal value, and particularly the sign of l0 in an optimal solution of the DEA model to present various methods and conditions for the measurement of RTS of a DMU [1,8,17,20–22]. The multiple projection issue is noticed and investigated by Olesen and Petersen [15] based on the dual representation of a polyhedral empirical production possibility set. In the work of Thanassoulis and Dyson [19], projections are considered for sets of inputs and outputs that can be improved. As pointed out by Lins et al. [13], models using weights in the search for the most preferred efficient target have some implementation difficulties. They propose a multi-objective approach to tackle this issue. However, the projection obtained in the additive DEA model and the directional distance function is a single point projection. Additive DEA models assign all input and output components the same unit weight 1. The directional distance function is defined by assigning the same scale b, called ‘‘distance’’, to all input and output elements. Such a feature does not allow the decision maker to give his/her own preference or percentage on input and output. In this paper, based on the GDEA model, we propose a bi-objective GDEA model (Bi-GDEA). It integrates the two objectives of increasing outputs and decreasing inputs simultaneously. It is an advanced generalization of the GDEA model. In the Bi-GDEA model, in addition to the three parameters (d1, d2, d3) for unification of other models, weights k1 and k2 are used to represent the decision maker’s preference on increasing outputs and decreasing inputs. In general, the research and application of DEA is often characterized by the following three aspects, namely to measure efficiency of DMUs, to determine returns to scale, and to give
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the projection of a DMU on the production possibility set which provides foundation for efficiency improvement. We show that the Bi-GDEA model has those important properties of the GDEA model from these three aspects. To measure efficiency of DMUs, we define DEA efficiency under the Bi-GDEA model as Bi-GDEA efficiency. The GDEA efficiency under either input-oriented model [24] or output-oriented model [23] is equivalent to the non-dominated solution of a multi-objective programming defined on the production possibility set. In the Bi-GDEA model, we show such equivalency and that is to say a DMU is efficient under Bi-GDEA model if and only if it is GDEA efficient. Therefore, the RTS in the Bi-GDEA model can be directly measured by using the previous results. The most essential contribution of this work is that we define a point-to-set mapping under the Bi-GDEA model which is from the point of an inefficient DMU to a subset on the frontier of the production possibility set. We call the projection of the mapping on frontiers of the production possibility set a point-to-set mapping projection to distinguish it from the point-to-point projection in conventional DEA models. We propose an efficient approach to construct the point-to-set mapping projection for a DMU. This approach, illustrated by numerical examples, presents a simple rule for practical applications. For any point on the projection set, we show that it is weak Bi-GDEA efficient relative to original DMUs. Thus, the point-to-set mapping projection forms a subset on frontier of the production possibility set (the subset under the condition of increasing outputs and decreasing inputs simultaneously). As there is no restriction given to the weights k1 and k2, the decision maker can ‘‘freely’’ decide a preferred portion to input and output when projecting an inefficient DMU to the production frontier. Thus the projection provides decision maker with foundations and alternatives for decision making. The rest of the paper is arranged as follows. Section 2 introduces Bi-GDEA model, definition on (weak) BiGDEA efficient, and some related properties. It also outlines some particular Bi-DEA models including (CCR) Bi-DEA model, (BCC) Bi-DEA model, (FG) Bi-DEA model, and (ST) Bi-DEA model. Section 3 shows the equivalency between Bi-GDEA efficiency and non-dominated solution of multi-objective programming. Section 4 discusses returns to scale of DMU. Section 5 defines point-to-set mapping and point-to-set mapping projection, presents an approach to construct point-to-set mapping projection, and provides an example for illustration. Section 6 further summarizes the decision making process applying the point-toset mapping projection and concludes the paper. 2. Bi-objective generalized DEA model (Bi-GDEA) Consider DMU-j0, 1 6 j0 6 n. For convenience, denote x0 ¼ xj0 , y 0 ¼ y j0 . In the following model, we integrate the input factor h and the output factor z into the objective function. The model is called a bi-objective generalized DEA model (Bi-GDEA): min ðPÞs:t:
ðk 1 h k 2 zÞ ¼ h0 X k hx0 2W Y k þ zy 0
d1 ðeT k þ d2 ð1Þd3 knþ1 Þ ¼ d1 k 2 K ; knþ1 P 0 h 6 1; z P 1 and its dual: max ðDÞs:t:
ðd1 l0 b1 þ b2 Þ xT X lT Y þ d1 l0 eT 2 K x T x 0 b1 ¼ k 1 l T y 0 b2 ¼ k 2 d1 d2 ð1Þd3 l0 P 0 ðxT ; lT ÞT 2 W b1 P 0;
b2 P 0;
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where X = (x1, x2, . . ., xn) is an m · n input matrix, Y = (y1, y2, . . ., yn) is an s · n output matrix, xj = (x1j, x2j, . . ., xmj)T is the input vector for DMU-j, j = 1, . . ., n, yj = (y1j, y2j, . . ., ysj)T is the output vector for DMU-j, j = 1, . . ., n, e = (1, 1, . . ., 1)T 2 En, W Emþs is a closed convex cone with Int W 5 ;, and þ W* Em+s is the negative polar cone of W. For any (xT, lT)T 2 Wn{0}, we have xT xj > 0;
lT y j > 0;
j ¼ 1; . . . ; n;
K Enþ is a closed convex cone with Int K 5 ;, and K* En is the negative polar cone of K, (k1, k2)T is positive weight (k1 > 0, k2 > 0), which describe the decision maker’s preferences on decreasing inputs and increasing outputs, respectively. d1, d2, d3 are 0–1 binary parameters. Different values of parameters d1, d2, d3 lead to the different generalized Bi-GDEA models (where ‘‘*’’ indicates either 0 or 1): Case 1. When (d1, d2, d3) = (0, *, *), the Bi-GDEA model is reduced to the (CCR) Bi-GDEA model: min ðPCCR Þs:t:
ðk 1 h k 2 zÞ X k hx0 Y k þ zy 0 k 2 K h 6 1;
2W
z P 1;
Case 2. When (d1, d2, d3) = (1, 0, *), the Bi-GDEA model is reduced to the (BCC) Bi-GDEA model: min ðPBCC Þs:t:
ðk 1 h k 2 zÞ X k hx0 Y k þ zy 0 k 2 K ; h 6 1;
2W
eT k ¼ 1
z P 1;
Case 3. When (d1, d2, d3) = (1, 1, 0), the Bi-GDEA model is reduced to the (FG) Bi-GDEA model: min ðPFG Þs:t:
ðk 1 h k 2 zÞ X k hx0 2 W Y k þ zy 0 k 2 K ; eT k 6 1 h 6 1; z P 1;
Case 4. When (d1, d2, d3) = (1, 1, 1), the Bi-GDEA model is reduced to the (ST) Bi-GDEA model: min ðPST Þs:t:
ðk 1 h k 2 zÞ X k hx0 Y k þ zy 0
2 W
k 2 K ; eT k P 1 h 6 1; z P 1: The cone W is called a preference cone and used to describe the relative importance of different input/output categories, as viewed by the decision maker. Cone K is called a predilection cone and used to represent the decision maker’s preferences for different DMUs [22,24]. j0
T Since K Enþ , then K Enþ . Let h ¼ 1, z ¼ 1, k ¼ ð0; . . . ; 0; 1; 0; . . . ; 0Þ 2 En , knþ1 ¼ 0, then k 2 K , and ð k; knþ1 ; h; zÞ is a feasible solution of (P). Therefore the optimal value of (P) h0 satisfies h0 6 k1 k2.
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Lemma 1 (Weak duality theorem). Let (k, kn+1, h, z) be a feasible solution of (P), (x, l, l0, b1, b2) be a feasible solution of (D), then k 1 h k 2 z P d1 l0 b1 þ b2 : Proof. Let (k, kn+1, h, z), (x, l, l0, b1, b2) be feasible solutions of (P) and (D), respectively. Since ððX k hx0 ÞT ; ðY k þ zy 0 ÞT ÞT 2 W ;
ðxT ; lT ÞT 2 W ;
then xT ðX k hx0 Þ þ lT ðY k þ zy 0 Þ 6 0; that is, ðxT X lT Y Þk 6 hxT x0 zlT y 0 : T
T
ð1Þ
T
Note that x X l Y + d1l0e 2 K, k 2 T
T
K*,
we have
T
ðx X l Y þ d1 l0 e Þk P 0; that is ðxT X lT Y Þk P d1 l0 eT k:
ð2Þ
d3
Since kn+1 P 0, d1 d2 ð1Þ l0 P 0, then d
d1 d2 ð1Þ 3 l0 knþ1 P 0:
ð3Þ
From (1)–(3), we have k 1 h k 2 z ¼ ðxT x0 b1 Þh ðlT y 0 b2 Þz ¼ ðhxT x0 zlT y 0 Þ b1 h þ b2 z P ðxT X lT Y Þk b1 þ b2 P d1 l0 eT k b1 þ b2 ¼ d1 l0 þ d1 d2 ð1Þd3 knþ1 l0 b1 þ b2 P d1 l0 b1 þ b2 : Thus k 1 h k 2 z P d1 l0 b1 þ b2 :
From Lemma 1, it is easy to see the following corollary. Corollary 1. If ðx ; l ; l0 ; b1 ; b2 Þ is a feasible solution of (D) such that d1 l0 b1 þ b2 ¼ k 1 k 2 ; then ðx ; l ; l0 ; b1 ; b2 Þ is an optimal solution of (D). In a mathematical programming with cone structure, the duality theorem holds under some constraint qualifications. Therefore, we give the following assumption. Assumption 1 (Constraint qualification I). For an optimal solution of (P): ðk ; knþ1 ; h ; z Þ, set 9 80 T 1 X x Y T l þ d1 l0 e þ h1 h1 2 K; ðxT ; lT ÞT 2 W ; h2 P 0; b1 P 0; b2 P 0 > > > > > > d = > @ A xT0 x þ b1 hT1 k ¼ 0; h2 knþ1 ¼ 0 > > > > ; : T y 0 l b2 b1 ðh 1Þ ¼ 0; b2 ðz 1Þ ¼ 0 is a closed set. Under Assumption 1, we have the following theorem. Theorem 1 (Duality theorem). Let ðk ; knþ1 ; h ; z Þ be an optimal solution of (P). Accordingly, there must exist an optimal solution of (D), ðx ; l ; l0 ; b1 ; b2 Þ, such that
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k 1 h k 2 z ¼ d1 l0 b1 þ b2 : Proof. Since the dual of (P) is (D) and Assumption 1 holds, by applying Theorem A2 from [6], we obtain the result of this theorem. h In the following, we define (weak) DEA efficiency under the Bi-GDEA model. Definition 1 ((Weak) Bi-GDEA efficient). (i) If the optimal value of (P) (or (D)) is equal to k1 k2, then DMU-j0 is called weak Bi-GDEA efficient. (ii) If (D) has an optimal solution ðx ; l ; l0 ; b1 ; b2 Þ such that d1 l0 b1 þ b2 ¼ k 1 k 2 ;
T
ðxT ; lT Þ 2 Int W ;
then DMU-j0 is called Bi-GDEA efficient. The relationship of DEA efficiency under the different Bi-GDEA models: (CCR)Bi-GDEA model, (BCC)Bi-GDEA model, (FG)Bi-GDEA model and (ST)Bi-GDEA model, is similar to the GDEA model. (See Theorems 1–5 and Corollary 1 in [24]). Denote the production possibility set: Xk x d T ¼ ðx; yÞ 2 W ; d1 ðeT k þ d2 ð1Þ 3 knþ1 Þ ¼ d1 ; k 2 K ; knþ1 P 0 : Y k þ y is k1 k2, then ðx; y Þ is Definition 2 (Production frontier). Let ðx; y Þ 2 T . If the optimal value of problem ðPÞ called weak Bi-GDEA efficient, where ðk 1 h k 2 zÞ X k hx ðPÞs:t: 2 W Y k þ zy min
d
d1 ðeT k þ d2 ð1Þ 3 knþ1 Þ ¼ d1 k 2 K ; knþ1 P 0 h 6 1; z P 1: The set of weak Bi-GDEA efficient point ðx; y Þ of production possibility set T is called production frontier of T. Assumption 2. If y 5 0, then (0, y) 62 T. That is, it cannot have a non-zero output without input. From the definition of production possibility set T, (P) is rewritten into the following: min ðk 1 h k 2 zÞ ðPÞs:t: ðhx0 ; zy 0 Þ 2 T h 6 1; z P 1: Theorem 2. Let (h, z, k, knþ1 ) be an optimal solution of (P), then DMU-j0 is weak Bi-GDEA efficient if and only if h = 1, z = 1. Proof. Assume that DMU-j0 is weak Bi-GDEA efficient. Then k1h k2z = k1 k2. Thus, k1(1 h) + k2(z 1) = 0. Note that h 6 1, z P 1, k1 > 0 and k2 > 0, we have h = 1, z = 1. On the other hand, if h = 1, z = 1, then by Definition 1, DMU-j0 is weak Bi-GDEA efficient. h Let ðk ; knþ1 ; h ; z Þ be an optimal solution of (P), the conventional projection of DMU-j0 is defined as follows:
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^x0 ¼ h x0 ; ^y 0 ¼ z y 0 : Obviously, ð^x0 ; ^y 0 Þ 2 T . Theorem 3. When Assumption 2 holds, projection ð^x0 ; ^y 0 Þ is weak Bi-GDEA efficient relative to the other DMUs. Proof. Consider the program min ðk 1 h k 2 zÞ ~ ðPÞs:t: ðh^x0 ; z^y 0 Þ 2 T h 6 1; z P 1: ~ Note that ðh^x0 ; z^y 0 Þ ¼ ðhh x0 ; zz y Þ 2 T , we have that Assume that ð h; zÞ is an optimal solution of ðPÞ. 0 ð hh ; zz Þ is a feasible solution of (P). Since (h , z ) is an optimal solution of (P), then hh k 2zz P k 1 h k 2 z : k1 Since h 6 1, z P 1, and from Assumption 2, it is easy to see h > 0. Thus, 0 P k 1 h ð h 1Þ P k 2 z ðz 1Þ P 0; k 1 h ð h 1Þ ¼ k 2 z ðz 1Þ ¼ 0; then h ¼ 1, z ¼ 1 and the optimal value of ðP~ Þ is equal to k1 k2. By Definition 1, ð^x0 ; ^y 0 Þ is weak Bi-GDEA efficient. h 3. The equivalence between Bi-GDEA efficiency and non-dominated solution of the multi-objective program Consider the multi-objective program ðVPÞ min ðx; yÞ s:t: ðx; yÞ 2 T : In order to discuss the relationship between Bi-GDEA efficiency and multi-objective program, we introduce the concept of non-dominated solution of multi-objective programming problem, which is given by Yu [26] and extended from that of Pareto solution of multi-objective program. Definition 3. Let W Emþs þ . If there exists no (x, y) 2 T such that x x0 þ W n f0g; 2 y y 0 then (x0, y0) is called a non-dominated solution of (VP) associated with W*. In order to show the equivalence between Bi-GDEA efficiency and non-dominated solution of (VP), we consider the following pair of dual programming problems: max ðsT s þ ^sT sþ Þ ðP0 Þs:t: X k þ s ¼ x0 Y k sþ ¼ y 0 d1 ðeT k þ d2 ð1Þd3 knþ1 Þ ¼ d1 k 2 K ; þT T
knþ1 P 0
ðsT ; s Þ 2 W
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and min ðxT x0 lT y 0 þ d1 l0 Þ xT X lT Y þ d1 l0 eT 2 K
ðD0 Þs:t:
d
d1 d2 ð1Þ 3 l0 P 0 ððx sÞT ; ðl ^sÞT ÞT 2 W ; T where ðsT ; ^sT Þ 2 Int W . Before we proceed, we must make an assumption, which will naturally hold in a case where cone K and W are polyhedral.
Assumption 3 (Constraint qualification II). Assume that ðk ; knþ1 ; s ; sþ Þ is an optimal solution of (P0). Set 9 80 T 1 X x Y T l þ d1 l0 e þ h1 > > T T T > > > > h 2 K; ðh ; h Þ 2 W ; h P 0 4 2 3 = > @ A l þ h3 > > > > ; : h4 knþ1 ¼ 0 d3 d1 d2 ð1Þ l0 þ h4 be a closed set. It is clear that the duality theorem for (P0) and (D0) holds under the Assumption 3. That is, the following lemma holds (see [24]). Lemma 2 (Duality theorem). Let ðk ; knþ1 ; s ; sþ Þ be an optimal solution of (P0). Accordingly, there must exist l ; l0 Þ, such that an optimal solution of (D0), ðx; T x0 l T y 0 þ d1 l 0 : sT s þ ^sT sþ ¼ x l ; l 0 Þ such Lemma 3. Let (x0, y0) be a non-dominated solution of (VP) associated with W*, then there exists ðx; that TX l T Y þ d1 l 0 eT 2 K; x T x0 l T y 0 þ d1 l 0 ¼ 0; x d 0 P 0; d1 d2 ð1Þ 3 l T
T; l T Þ 2 Int W : ðx Proof. From the definition of non-dominated solution, there exists no (x, y) 2 T with x x0 2 þ W n f0g; y y 0 that is X k þ s ¼ x 0 ; Y k sþ ¼ y 0 ; d
d1 ðeT k þ d2 ð1Þ 3 knþ1 Þ ¼ d1 ; k 2 K ; T
knþ1 P 0;
þT
ðs ; s Þ 2 W n f0g: is inconsistent. Since ðsT ; ^sT Þ 2 Int W , then the optimal value of (P0) is 0. From Lemma 2, (D0) has optimal l ; l 0 Þ satisfying solution ðx;
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T x0 l T y 0 þ d1 l 0 ¼ 0; x TX l T Y þ d1 l 0 eT 2 K; x d 0 P 0; d1 d2 ð1Þ 3 l T
T T
sÞ ; ð l ^sÞ Þ 2 W : ððx T
Since ðsT ; ^sT Þ 2 Int W , then T; l sÞT ; ð T ÞT ¼ ððx ðx l ^sÞT ÞT þ ðsT ; ^sT ÞT 2 Int W : This completes the proof.
h
In the following, we give a sufficient condition for that (x0, y0) 2 T is a non-dominated solution of (VP) associated with W*. Lemma 4. If there exists (xT, lT)T 2 Int W, such that (x0, y0) is an optimal solution of min ðxT x lT yÞ; s:t:
ðx; yÞ 2 T ;
then (x0, y0) is a non-dominated solution of (VP) associated with W *. Proof. Assume that (x0, y0) isnot anon-dominated solution of (VP) associated with W*, then there exists x x0 s T (x, y) 2 T such that ¼ þ , where ðsT ; ^sT Þ 2 W n f0g. So ^s y y 0 xT x lT y ¼ xT ðx0 þ sÞ lT ðy 0 ^sÞ ¼ xT x0 lT y 0 þ ðxT s þ lT^sÞ: T
T T
T
T T
ð4Þ
Since (x , l ) 2 Int W, ðs ; ^s Þ 2 W n f0g, and since Int W 5 ; then (see [26]) xT s þ lT^s < 0:
ð5Þ
From (4) and (5), we have xT x lT y < xT x0 lT y 0 : This is a contradiction. Thus (x0, y0) is a non-dominated solution of (VP) associated with W*. h Based on above lemmas, we obtain the following theorem. Theorem 4. DMU-j0 is Bi-GDEA efficient if and only if (x0, y0) is a non-dominated solution of (VP) associated with W *. l and l 0 Proof. Let (x0, y0) be a non-dominated solution associated with W*. From Lemma 3, there exists x, such that T x0 l T y 0 þ d1 l 0 ¼ 0; x TX l T Y þ d1 l 0 eT 2 K; x d 0 P 0; d1 d2 ð1Þ 3 l T
T; l T Þ 2 Int W : ðx Denote
q ¼ max x ¼ qx;
k1 k2 ; ; T x0 l T y 0 x l ¼ q l;
b1 ¼ xT x0 k 1 ;
l0 ¼ q l0 ;
b2 ¼ lT y 0 k 2 :
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It is easy to see that ðx ; l ; l0 ; b1 ; b2 Þ is a feasible solution of (D) with d1 l0 b1 þ b2 ¼ xT x0 lT y 0 ðxT x0 k 1 Þ þ ðlT y 0 k 2 Þ ¼ k 1 k 2 ; by Corollary 1, ðx ; l ; l0 ; b1 ; b2 Þ is an optimal solution of (D) and the optimal value is k1–k2. Therefore, DMU-j0 is Bi-GDEA efficient. On the other hand, if DMU-j0 is Bi-GDEA efficient, there exists (xT, lT)T 2 Int W, l0 , b1 , b2 satisfying d1 l0 b1 þ b2 ¼ k 1 k 2 ; xT X lT Y þ d1 l0 eT 2 K; d1 d2 ð1Þd3 l0 P 0; xT x0 b1 ¼ k 1 ; lT y 0 b2 ¼ k 2 ; T
ðxT ; lT Þ 2 Int W ;
b1 P 0;
b1 P 0;
thus xT x0 lT y 0 þ d1 l0 ¼ ðk 1 þ b1 Þ ðk 2 þ b2 Þ þ d1 l0 ¼ 0: For any (x, y) 2 T, there exists (s that x X k þ s ; ¼ Y k þ sþ y
T
,s
ð6Þ d3
+T T
) 2 W*, k 2 K*, kn+1 P 0 with d1 ðeT k þ d2 ð1Þ knþ1 Þ ¼ d1 , such
thus xT x lT y ¼ xT ðX k þ s Þ lT ðY k sþ Þ ¼ ðxT X lT Y Þk þ ðxT s þ lT sþ Þ:
ð7Þ
Since (xT, lT)T 2 Int W W, (sT, s+T)T 2 W*, then xT s þ lT sþ P 0:
ð8Þ
By (7) and (8), we have xT x lT y P ðxT X lT Y Þk:
T
T
Since k 2 K ; x X l Y þ T
T
ðx X l Y þ
d1 l0 eT Þk
d1 l0 eT
ð9Þ 2 K, then
P 0:
That is, d
d
ðxT X lT Y Þk P d1 l0 eT k ¼ l0 ðd1 d1 d2 ð1Þ 3 knþ1 Þ ¼ d1 l0 þ d1 d2 ð1Þ 3 knþ1 l0 P d1 l0 :
ð10Þ
By (9), (10) and (6), for any (x, y) 2 T, we have xT x lT y P ðxT X lT Y Þk P d1 l0 ¼ xT x0 lT y 0 ; that is, (x0, y0) is an optimal solution of programming min ðxT x lT yÞ; s:t: ðx; yÞ 2 T ; where (xT, lT)T 2 Int W. By Lemma 4, we have (x0, y0) is a non-dominated solution of (VP) associated with W *. h Yu et al. [24] proved the equivalence between the DEA efficiency under GDEA model and the non-dominated solution of (VP). Theorem 4 above shows the equivalence between the Bi-GDEA efficiency and the nondominated solution of (VP). Therefore, the Bi-GDEA efficiency is parallel to the DEA efficiency under the GDEA model.
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4. Returns to scale An ideal DEA model can be used to evaluate the returns to scale for DMUs, increasing returns to scale, constant returns to scale, and decreasing returns to scale. In this section, we discuss using the Bi-GDEA model to measure returns to scale. The next theorem shows that the optimal solution of the Bi-GDEA model (D) is a positive scalar multiple of the optimal solution of the corresponding GDEA model (DGDEA) below. Theorem 5. Consider Bi-GDEA model (D) and the GDEA model in [23] given as below: min ðxT x0 þ d1 l0 Þ ðDGDEA Þs:t: xT X lT Y þ d1 l0 eT 2 K lT y 0 ¼ 1 d
d1 d2 ð1Þ 3 l0 P 0 T
ðxT ; lT Þ 2 W : Then, DMU-j0 is weak Bi-GDEA efficient if and only if it is weak GDEA efficient. In addition, the optimal solution of (D) is a positive scalar multiple of the optimal solution of (DGDEA). Proof. Assume that DMU-j0 is weak Bi-GDEA efficient, and ðx ; l ; l0 ; b1 ; b2 Þ is an optimal solution of (D). Note that k 2 þ b2 > 0, denote ¼ x
x ; k 2 þ b2
¼ l
l ; k 2 þ b2
0 ¼ l
l0 : k 2 þ b2
l ; l 0 Þ is a feasible solution of (DGDEA) with Obviously, ðx; T x 0 þ d1 l 0 ¼ x
1 1 T ðk 1 þ b1 þ d1 l0 Þ: ðx x0 þ d1 l0 Þ ¼ k 2 þ b2 k 2 þ b2
ð11Þ
Since DMU-j0 is weak Bi-GDEA efficient, then d1 l0 b1 þ b2 ¼ k 1 k 2 : That is, d1 l0 ¼ b1 þ b2 k 1 þ k 2 :
ð12Þ
By (11) and (12), we have T x 0 þ d1 l 0 ¼ x
1 ðk 1 þ b1 b1 þ b2 k 1 þ k 2 Þ ¼ 1: k 2 þ b2
l ; l 0 Þ is an optimal solution of (DGDEA). Thus, DMU-j0 Since the objective value of (DGDEA) h P 1, then ðx; is weak DEA efficient under the GDEA model. And the optimal solution of (D) is a positive scalar, k 2 þ b2 , multiple of that of (DGDEA). l ; l 0 Þ is an optimal On the other hand, assume DMU-j0 is weak DEA efficient under (DGDEA) that ðx; solution of (DGDEA). We have T x 0 þ d1 l T y 0 ¼ 1 0 ¼ l x TX l T Y þ d1 l 0 eT 2 K x d 0 P 0 d1 d2 ð1Þ 3 l T
T; l T Þ 2 W : ðx From the proof of Theorem 4, (D) has an optimal solution ðx ; l ; l0 ; b1 ; b2 Þ, where k1 k2 x ¼ qx; l ¼ q l; l0 ¼ q l0 ; q ¼ max ; > 0: T x0 l T y 0 x
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In addition, d1 l0 b1 þ b2 ¼ k 1 k 2 : Thus, DMU-j0 is Bi-GDEA efficient, the optimal solution of (D) is a positive scalar multiple, q, of that of (DGDEA). h The research and application of returns to scale can be traced back to 1984, the work of Banker et al. [2,3]. Since then, a vast amount of work has been conducted and published on this problem. (For details, one can see the survey work by Seiford and Zhu [17] and Copper et al. [8]). Among these works, one important class of methods involves measuring returns to scale by the sign of l0 in the optimal solution (see, for example [1,8,17,20,22]). In this paper, since we have proved that the optimal solution of (D) is a positive scalar multiple of the optimal solution of (DGDEA), then the results of the above works can be used in the Bi-GDEA model. In other words, we can determine returns to scale by the sign of l0 in the optimal solutions of (D). 5. Point-to-set mapping and point-to-set mapping projection This section discusses a point-to-set mapping and its projection of mapping on the frontier of the production possibility set. The point-to-set mapping of a DMU gives the projections on the frontiers of the production possibility set. Given weight (k1, k2) which describes the decision maker’s preference on increasing outputs and decreasing inputs in the Bi-GDEA model, a projection represents a subset of the production frontier. Therefore we obtain different projections for different weights (k1, k2). The union of these projections under a Bi-GDEA model forms a subset of the production frontier, which provides the efficiency improvement alternatives. We call this set a point-to-set mapping projection. In the following, using the Bi-GDEA model, we provide an approach to construct the point-to-set mapping projection. For DMU-j0, with input and output (x0, y0), a point-to-set mapping is given by F : ðx0 ; y 0 Þ ! F ðx0 ; y 0 Þ ¼ fðhx0 ; zy 0 Þjðhx0 ; zy 0 Þ 2 T ; h 6 1; z P 1g: It is clear that F(x0, y0) is a subset of the production possibility set T, which is to decrease input x0 and increase output y0. We consider the input-oriented GDEA model and the output-oriented GDEA model. The input-oriented GDEA model [24] is given by min ðPinput-GDEA Þs:t:
h
X k hx0
Y k þ y 0
2W
d1 ðeT k þ d2 ð1Þd3 knþ1 Þ ¼ d1 k 2 K ;
knþ1 P 0:
And the output-oriented GDEA model [22] is given by max ðPoutput-GDEA Þs:t:
z
X k x0 Y k þ zy 0
2W
d1 ðeT k þ d2 ð1Þd3 knþ1 Þ ¼ d1 k 2 K ;
knþ1 P 0:
If the optimal value of (Pinput-GDEA) equals to 1, then DMU-j0 is called weak DEA efficient under inputoriented GDEA model. If the optimal value of (Poutput-GDEA) is equal to 1, then DMU-j0 is called weak DEA efficient under output-oriented GDEA model. Assume that DMU-j0 is not weak efficient under both input-oriented and output-oriented GDEA models. The following algorithm defines and constructs the point-to-set mapping projection of (x0, y0).
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Algorithm (Point-to-set mapping projection) Step 1: (i) Let k1 = 1, k2 = 1, and solve min ðh zÞ ðP1 Þs:t: ðhx0 ; zy 0 Þ 2 T z ¼ 1;
h 6 1;
to obtain the optimal solution ðh1 ; 1Þ, since DMU-j0 is not weak DEA efficient under input-oriented GDEA model (Pinput-GDEA), then h1 < 1. Denote ðh1 ; z1 Þ ¼ ðh1 ; 1Þ. (ii) Let k1 = 1, k2 = 1, and solve min ðP2 Þs:t:
ðh zÞ ðhx0 ; zy 0 Þ 2 T z P 1; h ¼ 1
to obtain the optimal solution ð1; z2 Þ, since DMU-j0 is not weak DEA efficient under output-oriented GDEA model (Poutput-GDEA), then z2 > 1. Denote ðh2 ; z2 Þ ¼ ð1; z2 Þ. Obviously, h1 ¼ h1 < h2 ¼ 1;
1 ¼ z1 < z2 ¼ z2 ;
and l: = 2. Step 2: Denote k i1 ¼ ziþ1 zi ;
k i2 ¼ hiþ1 hi ;
qi ¼
k i1 ; k i2
i ¼ 1; 2; . . . ; l 1:
When l P 3, we have q1 > q2 > > ql1 : For i = 1, 2, . . ., l 1, solve min
ðk i1 h k i2 zÞ
ðPðk i1 ; k i2 ÞÞs:t: ðhx0 ; zy 0 Þ 2 T h 6 1; z P 1: to obtain its optimal solution ðh0i ; z0i Þ. If h0i hi z0 zi ¼ i ; hiþ1 hi ziþ1 zi
ð13Þ
then ðh0i , z0i Þ is abandoned. Assume that there exists l points not satisfying (13). If l ¼ 0 then go to Step 3, else continue. Sort these newly solved l optimal solutions ðh0i ; z0i Þ, i ¼ 1; 2; . . . ; l, and original l points (hi, zi) i = 1, 2, . . ., l, into a non-decreasing order: h1 ¼ h1 < h2 < < hlþl ¼ 1; 1 ¼ z1 < z2 < < zlþl ¼ z2 : Let l :¼ l þ l and go back to step 2. Step 3: At this time, we obtain h1 ¼ h1 < h2 < < hl ¼ 1; 1 ¼ z1 < z2 < < zl ¼ z2 ;
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and k i1 ¼ ziþ1 zi ;
k i2 ¼ hiþ1 hi ;
k i1 ; i ¼ 1; 2; . . . ; l 1: k i2 For convenience, denote qi ¼
k 01 ¼ þ1; k l1 ¼ 0;
k 02 ¼ 1;
k l2 ¼ 1;
q0 ¼
ql ¼
k 01 ; k 02
k l1 : k l2
Then, we have þ1 ¼ q0 > q1 > q2 > > ql1 > ql ¼ 0: See Fig. 1. We have iþ1 i k k (i) if kk12 2 ðqiþ1 ; qi Þ ¼ k1iþ1 ; k1i ; 0 6 i 6 l 1, then (hi+1, zi+1) is an optimal solution of ðPðk i1 ; k i2 ÞÞ (see The2 2 orem 6), min ðk 1 h k 2 zÞ ðPðk 1 ; k 2 ÞÞs:t: ðhx0 ; zy 0 Þ 2 T h 6 1; z P 1: Denote k1 S x0 ; y 0 ; ¼ fðhiþ1 x0 ; ziþ1 y 0 Þg; k2 (ii) if
k1 k2
¼ qi ; 1 6 i 6 l 1 then the optimal solution of problem ðPðk i1 ; k i2 ÞÞ is a set (see Theorem 7)
z zl = z20 z l −1
zi +1 (k1l −1,−k2l −1)
zi
(k1l ,−k2l )
(k1i ,−k2i ) z2
(k12 ,−k22 ) z1 = 1
(k11,−k12 )
0
θ 1 =θ10
θ2
θi
θ i +1
θ i −1
θl =1
Fig. 1. Illustration of weight coefficients in the BCC model.
θ
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fðh; zÞjz ¼ qi ðh hiþ1 Þ þ ziþ1 ; hi 6 h 6 hiþ1 g: Denote k1 S x0 ; y 0 ; ¼ fðhx0 ; zy 0 Þjz ¼ qi ðh hiþ1 Þ þ ziþ1 ; hi 6 h 6 hiþ1 g: k2 Finally, denote a point-to-set mapping projection: FWP : ðx0 ; y 0 Þ ! F WP ðx0 ; y 0 Þ; where F WP ðx0 ; y 0 Þ ¼
! ! l1 l1 [ [ [ fðhx0 ; zy 0 Þjz ¼ qi ðh hiþ1 Þ þ ziþ1 ; hi 6 h 6 hiþ1 g fðhiþ1 x0 ; ziþ1 y 0 Þg : i¼1
i¼0
Note that ðhiþ1 x0 ; ziþ1 y 0 Þ 2 fðhx0 ; zy 0 Þjz ¼ qi ðh hiþ1 Þ þ ziþ1 ; hi 6 h 6 hiþ1 g;
i ¼ 1; . . . ; l 1;
and ðh1 x0 ; z1 y 0 Þ 2 fðhx0 ; zy 0 Þjz ¼ q1 ðh h2 Þ þ z2 ; h1 6 h 6 h2 g; we have F WP ðx0 ; y 0 Þ ¼
l1 [ fðhx0 ; zy 0 Þjz ¼ qi ðh hiþ1 Þ þ ziþ1 ; hi 6 h 6 hiþ1 g: i¼1
The point-to-set mapping projection FWP is a mapping from point (x0, y0) to production frontier of production possibility set, and Fwp(x0, y0) F(x0, y0) T. i i iþ1 iþ1 Theorem 6. Assume that ð^ h; ^zÞ is an optimal solution of both ðPðk 1 ; k 2 ÞÞ and ðPðk 1 ; k 2 ÞÞ, then for any iþ1 i k k q ¼ kk12 2 ðqiþ1 ; qi Þ ¼ k1iþ1 ; k1i , we have ð^ h; ^zÞ is an optimal solution of (P(k1, k2)). 2
2
Proof. Since q ¼ kk12 2 ðqiþ1 ; qi Þ, then there must exist a 2 (0, 1) such that q ¼ aqiþ1 þ ð1 aÞqi : Since ð^ h; ^zÞ is an optimal solution of ðPðk i1 ; k i2 ÞÞ, then for any (hx0, zy0) 2 T, h 6 1, z P 1, we have h k i2^z 6 k i1 h k i2 z; k i1 ^ ^ ^z 6 q h z: qh i
ð14Þ
i
Similarly, for any (hx0, zy0) 2 T, h 6 1, z P 1, we have qiþ1 ^ h ^z 6 qiþ1 h z:
ð15Þ
By (14) and (15), for any (hx0, zy0) 2 T, h 6 1, z P 1, ðaqiþ1 þ ð1 aÞqi Þ^ h ^z 6 ðaqiþ1 þ ð1 aÞqi Þh z; ^ qh ^z 6 qh z: That is, for any (hx0, zy0) 2 T, h 6 1, z P 1, we have k1^ h k 2^z 6 k 1 h k 1 z: Hence, 8q ¼ kk12 2 ðqiþ1 ; qi Þ, ð^ h; ^zÞ is an optimal solution of (P(k1, k2)).
h
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Lemma 5. For h < h, z < z, assume that ð^ h; ^zÞ is an optimal solution of min ðk 1 h k 2 zÞ ðPðk 1 ; k 2 ÞÞs:t: ðhx0 ; zy 0 Þ 2 T h 6 1;
z P 1:
h h. where k 1 ¼ z z, k 2 ¼ ^ h z^z ðh hÞ þ z; h 6 h 6 hg is an optimal solution of ðPðk 1 ; k 2 ÞÞ, If h ¼ , then for any ðh; zÞ 2 fðh; zÞjz ¼ q zz hh
¼ kk1 . where q 2
Proof. Since
h^ h hh
z ¼ zz^ , then z
k 1 z z ^ ðh hÞ ¼ ð z ^z ¼ h^ hÞ; k2 hh thus k 1 h k 2z ¼ k 1 ^h k 2^z: ðh Note that, for any ðh; zÞ 2 fðh; zÞjz ¼ q hÞ þ z; h 6 h 6 hg k 1 h k 2 z ¼ k 1 h k 2 k 1 ðh h k 2z ¼ k 1 ^h k 2^z; hÞ þ z ¼ k 1 k 2 and since ð^ h; ^zÞ is an optimal solution of ðPðk 1 ; k 2 Þ), we have that (h, z) is an optimal solution of ðPðk 1 ; k 2 ÞÞ. h Theorem 7. Any point of set FWP(x0, y0) is weak Bi-GDEA efficient. Proof. Let ð^x; ^y Þ 2 F WP ðx0 ; y 0 Þ, then there must exist an i, 1 6 i 6 l 1, such that ð^x; ^y Þ 2 fðhx0 ; zy 0 Þjz ¼ qi ðh hiþ1 Þ þ ziþ1 ; hi 6 h 6 hiþ1 g: That is, there exists ð^ h; ^zÞ, such that ð^x; ^y Þ ¼ ð^hx0 ; ^zy 0 Þ, where ð^h; ^zÞ 2 fðh; zÞjz ¼ qi ðh hiþ1 Þ þ ziþ1 ; hi 6 h 6 hiþ1 g. From constructing of FWP(x0, y0) and Lemma 5, we have ð^h; ^zÞ is an optimal solution of ðPðk i1 ; k i2 ÞÞ. By Theorem 3, ð^x; ^y Þ is weak Bi-GDEA efficient. h Example. Consider single input and single output Bi-GDEA model:
DMU 1 x1 1 ¼ ; 1 y1
DMU 2 DMU 3 DMU 4 x2 3 x3 6 x4 5 ¼ ; ¼ ; ¼ ; 3 4 2 y2 y3 y4
Assume that (d1, d2, d3) = (1, 0, *), W ¼ E2þ , K ¼ E4þ , the model is a (BCC)Bi-DEA model. We calculate the point-to-set mapping projections of DMU-4, (x4, y4) = (5, 2) by the following problem (P). The point-to-set mapping F(x4, y4) is shown in Fig. 2: min ðPÞs:t:
ðk 1 h k 2 zÞ k1 þ 3k2 þ 6k3 þ 5k4 6 5h k1 þ 3k2 þ 4k3 þ 2k4 P 2z k1 þ k2 þ k3 þ k4 ¼ 1 ki P 0; i ¼ 1; . . . ; 4 h 6 1; z P 1:
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y C (5, 11/3) DMU2: B (3,3)
DMU3 (6,4) F(x 0 ,y 0 )
A (2, 2)
DMU4 (x 0 , y 0 )=(5,2) DMU1 (1,1)
x
0
Fig. 2. Point-to-set mapping projection of F(5,2) in the BCC model.
Step 1: (i) Let k1 = 1, k2 = 1, then min ðk 1 h k 2 zÞ ðP1 Þs:t: k1 þ 3k2 þ 6k3 þ 5k4 6 5h k1 þ 3k2 þ 4k3 þ 2k4 P 2z k1 þ k2 þ k3 þ k4 ¼ 1 ki P 0; i ¼ 1; . . . ; 4 z ¼ 1:
has the optimal solution 25 ; 1 , denote ðh1 ; z1 Þ ¼ ðh1 ; 1Þ ¼ 25 ; 1 . (ii) Let k1 = 1, k2 = 1, then h 6 1;
min ðk 1 h k 2 zÞ ðP2 Þs:t: k1 þ 3k2 þ 6k3 þ 5k4 6 5h k1 þ 3k2 þ 4k3 þ 2k4 P 2z k1 þ k2 þ k3 þ k4 ¼ 1 ki P 0;
i ¼ 1; . . . ; 4
h ¼ 1:
has the optimal solution 1; 116 , denote ðh2 ; z2 Þ ¼ ð1; z2 Þ ¼ 1; 116 . z P 1;
k1
Step 2: k 11 ¼ z2 z1 ¼ 56, k 12 ¼ h2 h1 ¼ 35. q1 ¼ k11 ¼ 25 . Solve 18 2 5 3 h z min 6 5 ðPðk 11 ; k 12 ÞÞs:t:
k1 þ 3k2 þ 6k3 þ 5k4 6 5h k1 þ 3k2 þ 4k3 þ 2k4 P 2z k1 þ k2 þ k3 þ k4 ¼ 1 ki P 0; i ¼ 1; . . . ; 4 h 6 1;
z P 1;
to obtain its optimal solution ðh01 ; z01 Þ ¼ 35 ; 32 .
z0 z h0 h Since 13 ¼ h12 h11 6¼ z12 z11 ¼ 35, then l 6¼ 0. Sort 25 ; 1 , 1; 116 , ð35 ; 32Þ into non-decreasing order: ðh1 ; z1 Þ ¼ 25 ; 1 ,
ðh2 ; z2 Þ ¼ 35 ; 32 , ðh3 ; z3 Þ ¼ 1; 116 :
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2 h1 ¼ ; 5
3 h2 ¼ ; 5
h3 ¼ 1;
z1 ¼ 1;
3 z2 ¼ ; 2
z3 ¼
11 : 6
Step 2 0 : 1 k 11 ¼ z2 z1 ¼ ; 2 q2 ¼
1 k 12 ¼ h2 h1 ¼ ; 5
q1 ¼
k 11 5 ¼ ; k 12 2
1 k 21 ¼ z3 z2 ¼ ; 3
2 k 22 ¼ h3 h2 ¼ ; 5
1 k 21 ¼ z3 z2 ¼ ; 3
2 k 22 ¼ h3 h2 ¼ ; 5
k 21 5 ¼ : k 22 6
Solve
min
1 1 h z 2 5
ðPðk 11 ; k 12 ÞÞs:t: k1 þ 3k2 þ 6k3 þ 5k4 6 5h k1 þ 3k2 þ 4k3 þ 2k4 P 2z k1 þ k2 þ k3 þ k4 ¼ 1 ki P 0; i ¼ 1; . . . ; 4 h 6 1; z P 1;
to obtain the optimal solution ðh01 ; z01 Þ ¼ 1; 116 . Solve 1 2 h z min 3 5 ðPðk 21 ; k 22 ÞÞs:t: k1 þ 3k2 þ 6k3 þ 5k4 6 5h k1 þ 3k2 þ 4k3 þ 2k4 P 2z k1 þ k2 þ k3 þ k4 ¼ 1 i ¼ 1; . . . ; 4
ki P 0; h 6 1;
z P 1;
to obtain the optimal solution ðh02 ; z02 Þ ¼ For i = 1, 2
3 3
; . 5 2
h0i hi z0 zi ¼ i : hiþ1 hi ziþ1 zi Thus l ¼ 0. Step 3: At this time, we have l = 2 and 1 k 11 ¼ z2 z1 ¼ ; 2 q2 ¼
k 21 5 ¼ ; k 22 6
(i)
1 k 12 ¼ h2 h1 ¼ ; 5
q0 ¼ þ1;
5 q1 ¼ ; 2
q1 ¼
k 11 5 ¼ ; k 12 2
5 q2 ¼ ; 6
q3 ¼ 0:
(a) if
k1 k2
(b) if
k1 k2
(c) if
k1 k2
2 ½q3 ; q2 Þ ¼ 0; 56 then S x0 ; y 0 ; kk21 ¼ 5; 113 .
2 ðq2 ; q1 Þ ¼ 56 ; 52 then S x0 ; y 0 ; kk21 ¼ fð3; 3Þg.
2 ðq1 ; q0 Þ ¼ 52 ; þ1 then S x0 ; y 0 ; kk21 ¼ fð2; 2Þg.
(a) If
k1 k2
¼ q1 ¼ 56 then
(ii)
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k2 5 2 3 : ¼ fðhx0 ; zy 0 Þjz ¼ q1 ðh h2 Þ þ z2 ; h1 6 h 6 h2 g ¼ ðhx0 ; zy 0 Þjz ¼ h þ 1; 6 h 6 S x0 ; y 0 ; 6 5 5 k1 Since (x0, y0) = (5, 2), denote x ¼ hx0 ¼ 5h;
y ¼ zy 0 ¼ 2z;
we have k2 5 2 3 S x0 ; y 0 ; ¼ fðx; yÞjx 3y þ 6 ¼ 0; 3 6 x 6 5g: ¼ ðhx0 ; zy 0 Þjz ¼ h þ 1; 6 h 6 6 5 5 k1 (b) If kk12 ¼ q2 ¼ 52 then k2 S x0 ; y 0 ; ¼ fðhx0 ; zy 0 Þjz ¼ q2 ðh h3 Þ þ z3 ; h2 6 h 6 h3 g k1 Similarly, we have k2 S x0 ; y 0 ; ¼ fðx; yÞjx ¼ y; 2 6 x 6 3g: k1 The point-to-set mapping projection of DMU-4 is the following set: [ [ 11 F WP ðx0 ; y 0 Þ ¼ fðx; yÞjx 3y þ 6 ¼ 0; 3 6 x 6 5g fðx; yÞjx ¼ y; 2 6 x 6 3g 5; ; ð3; 3Þ; ð2; 2Þ 3 [ ¼ fðx; yÞjx 3y þ 6 ¼ 0; 3 6 x 6 5g fðx; yÞjx ¼ y; 2 6 x 6 3g: In this example (see Figs. 2–4), DMU-4 has input and output (x0, y0) = (5, 4). From Fig. 2, DMU-4 is clearly not DEA efficient. The point-to-set mapping projection is given by the segments AB and BC on the production frontier. We can then adjust the input and output of DMU-4 as the following such that DMU4 can be DEA efficient.
(i) When the ratio of input and output weights, kk12 , is in 0; 56 , the projection is 5; 113 . (ii) When the ratio of input and output weights, kk12 ¼ 56, the projection is jx 3y + 6 = 0,3 6 x 6 5}.
(iii) When the ratio of input and output weights, kk12 , is in 56 ; 52 , the projection is (3, 3). (iv) When the ratio of input and output weights, (v) When the ratio of input and output weights,
k1 ¼ 52, the projection k2
k1 , is in 25 ; þ1 , the k2
z2 =
11 6
projection is (2, 2).
C' B'
3 2
2
A'
0
2
(k 1 ,k 2 )
z1 = 1
θ 1 =θ 10 =
1
1
(k 1 , k 2 )
3 2 θ2 = 5 5
set
{(x, y)
is a set {(x, y)jx = y, 2 6 x 6 3}.
z
z3 = z20 =
a
θ 3 =1
Fig. 3. Weight coefficients of F(5,2) in the BCC model.
θ
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B''
C''
A''
ρ3 = 0
ρ 0 = +∞
)(
)(
0
5
ρ2 =
ρ1 =
6
5 2
Fig. 4. Ratio of weight coefficients of F(5,2) in the BCC model.
B (5, 5)
y
DMU2 (3,3)
DMU3 (6,4) F(x 0 ,y 0 )
A (2, 2)
DMU4 (x 0 , y 0 )=(5,2) DMU1 (1,1)
x
0
Fig. 5. Point-to-set mapping projection of F(5,2) in the CCR model.
5
z
B ' = (1, 2 )
z1 = 5 2
5
A ' = ( 2 ,1) 1
z1 = 1
0
3 3
(k 11,k 2 ) = ( 2 , 5 )
θ1 =
θ
2
θ2 = 1
5
Fig. 6. Weight coefficients of F(5,2) in the CCR model.
B'' 0 ρ2 = 0
A'' ρ 0 = +∞
)( ρ1 =
5 2
Fig. 7. Ratio of weight coefficients of F(5,2) in the CCR model.
If we consider (d1, d2, d3) = (0, *, *) above, then model is a (CCR)Bi-DEA model. Similar as above, we have (i) k1 = 1, (ii) k1 = 1,
k2 = 1, k2 = 1,
ðh1 ; z1 Þ ¼ 25 ; 1 ,
ðh2 ; z2 Þ ¼ 1; 52 .
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Therefore, 3 k 11 ¼ z2 z1 ¼ ; 2
3 k 12 ¼ h2 h1 ¼ : 5
Similar to the procedure in BCC model, we further solve the linear programming problem for
k1 ðk 1 ; k 2 Þ ¼ ðk 11 ; k 12 Þ ¼ 32 ; 35 . Let q1 ¼ k11 ¼ 52. We can get the similar results as shown in Figs. 5–7. The point2
to-set mapping projection of DMU-4 is given by F WP ðx0 ; y 0 Þ ¼ fðx; yÞjx y ¼ 0; 2 6 x 6 5g: 6. Conclusion This paper investigates the issue of increasing outputs and decreasing inputs simultaneously in DEA models. It proposes a Bi-objective GDEA model (Bi-GDEA) to integrate two objectives. The Bi-GDEA model not only uses the three 0–1 parameters (d1, d2, d3) to unify other models as a generalized model, but also uses two weight parameters k1 and k2 to describe the decision maker’s preference on output increase and input decrease. The Bi-GDEA model is a non-radial model. It thus allows the decision maker to have other preferable projections. It also allows that the projection would not be in Pareto-inefficient portions of the production frontier which may occur in the radial projection. In line with the conventional DEA research, we define the Bi-GDEA efficiency of DMUs, and show the equivalency between the Bi-GDEA efficiency and the non-dominated solutions of the multi-objective programming defined on the production possibility set. Furthermore, we prove that an optimal solution of BiGDEA model is equivalent to an optimal solution of GDEA model, and the solution of one model is a positive scalar multiple of the solution of another model. Therefore, returns to scale in the Bi-GDEA model can be measured by directly using the results of GDEA model and other models. Such a generalized result extended the theoretical research in DEA area. In another line, we deal with the issue of improving efficiency of a DMU. The essential contribution of this research is that we define a mapping from a point given by a DMU’s input and output to a subset of the production possibility set. This mapping is called a point-to-set mapping. Based on the Bi-GDEA model, we further define a projection of the mapping on the frontier of the production possibility set. This projection is a subset of the production frontier and is called a point-to-set mapping projection. We propose an algorithm to construct the point-to-set mapping projection for a DMU. The projection procedure makes use of the preference structure proposed by Thanassoulis and Dyson [19]. It also provides alternative computational solutions to the decision maker for efficiency improvement. It is thus clear that the proposed model and projection procedure offer deeper management insight as well as wider foundation for managerial decision making. Our next target is to further generalize the Bi-GDEA model by assigning different weights to each individual inputs and outputs. But the point-to-set mapping algorithm proposed in this paper cannot identify the projection area on the production frontier. A new method and algorithm based on the linear form of the production possibility set would be designed. References [1] R.D. Banker, R.M. Thrall, Estimating most productive scale size using data envelopment analysis, European Journal of Operational Research 62 (1992) 74–84. [2] R.D. Banker, A. Charnes, W.W. Cooper, Some models for estimating technical and scale efficiencies in DEA, Management Science 30 (9) (1984) 1078–1092. [3] R.D. Banker, H. Chang, W.W. Cooper, Equivalence and implementation of alternative methods for determining returns to scale in data envelopment analysis, European Journal of Operational Research 89 (1996) 473–481. [4] R.D. Banker, W.W. Cooper, L.M. Seiford, R.M. Thrall, J. Zhu, Returns to scale in different DEA models, European Journal of Operational Research 154 (2004) 345–362.
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[5] R.G. Chambers, Y. Chung, R. Fa¨re, Profit, directional distance function, and Nerlovian efficiency, Journal of Optimization Theory and Applications 98 (2) (1998) 351–364. [6] A. Charnes, W.W. Cooper, Q.L. Wei, Z.M. Huang, Cone ratio data envelopment analysis and multiple associated with cone in normed linear space, International Journal of Systems Science 20 (1989) 1099–1118. [7] A. Charnes, W.W. Cooper, Q.L. Wei, Z.M. Huang, Fundamental theorems of non-dominated solutions associated with cones in normed linear space, Journal of Mathematical Analysis and Applications 150 (1990) 54–78. [8] W.W. Cooper, L.M. Seiford, K. Tone, Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-solver Software, Kluwer Academic Publishers, Boston, MA, 2000, pp. 128–131. [9] W.W. Cooper, R.G. Thompson, R.M. Thrall, Extensions and new development in DEA, Annals of Operations Research 66 (1996) 3– 45. [10] R. Fa¨re, S. Grosskopf, A nonparametric cost approach to scale efficiency, Scandinavian Journal of Economics 87 (1985) 594–604. [11] R. Fa¨re, Fundamentals of Production Theory, Springer Verlag, Berlin, Germany, 1988. [12] H. Fukuyama, Returns to scale and scale elasticity in data envelopment analysis, European Journal of Operational Research 125 (2000) 93–112. [13] M.P.E. Lins, L. Angulo-Mesa, A.C. Moreira da Silva, A multi-objective approach to determine alternative targets in data envelopment analysis, Journal of Operational Research 55 (2004) 1090–1101. [14] D.G. Luenberger, New optimality principles for economic efficiency and equilibrium, Journal of Optimization Theory and Applications 75 (1992) 221–264. [15] O.B. Olesen, N.C. Petersen, Identification and use of efficient faces and facets in DEA, Journal of Production Analysis 20 (2003) 323– 360. [16] L.M. Seiford, R.M. Thrall, Recent development in DEA, the mathematical programming approach to frontier analysis, Journal of Econometrics 46 (1990) 7–38. [17] L.M. Seiford, J. Zhu, An investigation of returns to scale in data envelopment analysis, Omega 27 (1999) 1–11. [18] T. Sueyoshi, K. Sekitani, Measurement of returns to scale using a non-radial DEA model: A range-adjusted measure approach, European Journal of Operational Research 176 (2007) 1918–1946. [19] E. Thanassoulis, R.G. Dyson, Estimating preferred target input–output levels using data envelopment analysis, European Journal of Operational Research 56 (1992) 80–97. [20] K. Tone, A simple characterization of returns to scale in DEA, Journal of the Operations Research Society of Japan 39 (1996) 604– 613. [21] Q.L. Wei, Data envelopment analysis (DEA), Chinese Science Bulletin 46 (16) (2001) 1321–1332. [22] Q.L. Wei, G. Yu, Analyzing properties of K-cones in the generalized data envelopment analysis model, Journal of Econometrics 80 (1) (1997) 63–84. [23] Q.L. Wei, G. Yu, J.S. Lu, The necessary and sufficient conditions for returns to scale properties in the generalized data envelopment analysis models, Chinese Science 45 (5) (2002) 503–517. [24] G. Yu, Q.L. Wei, P. Brockett, A generalized data envelopment analysis model: A unification and extension of existing methods for efficiency analysis of decision making units, Annals of Operations Research 66 (1996) 47–89. [25] G. Yu, Q.L. Wei, P. Brockett, L. Zhou, Construction of all DEA efficient surfaces of the production possibility set under the generalized data envelopment analysis model, European Journal of Operational Research 95 (1996) 491–510. [26] P.L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjective, Journal of Optimization Theory and Application 14 (1974) 319–377.
European Journal of Operational Research 190 (2008) 855–876 www.elsevier.com/locate/ejor
Interfaces with Other Disciplines
A bi-objective generalized data envelopment analysis model and point-to-set mapping projection q Quanling Wei a, Hong Yan a
b,*
, Lin Xiong
a
Institute of Operations Research and Mathematical Economics, Renmin University of China, China b Department of Logistics, The Hong Kong Polytechnic University, Hong Kong Received 3 August 2006; accepted 27 June 2007 Available online 28 August 2007
Abstract This work introduces a bi-objective generalized data envelopment analysis (Bi-GDEA) model and defines its efficiency. We show the equivalence between the Bi-GDEA efficiency and the non-dominated solutions of the multi-objective programming problem defined on the production possibility set (PPS) and discuss the returns to scale under the Bi-GDEA model. The most essential contribution is that we further define a point-to-set mapping and the mapping projection of a decision making unit (DMU) on the frontier of the PPS under the Bi-GDEA model. We give an effective approach for the construction of the point-to-set-mapping projection which distinguishes our model from other non-radial models for simultaneously considering input and output. The Bi-GDEA model represents decision makers’ specific preference on input and output and the point-to-set mapping projection provides decision makers with more possibility to determine different input and output alternatives when considering efficiency improvement. Numerical examples are employed for the illustration of the procedure of point-to-set mapping. 2007 Elsevier B.V. All rights reserved. Keywords: Data envelopment analysis (DEA); Bi-GDEA efficiency; Point-to-set mapping projection; Non-dominated solution
1. Introduction Data envelopment analysis (DEA) is a non-parametric approach to measure the relative efficiency of a decision making unit (DMU) and provide DMUs with relative performance assessment on multiple inputs and outputs. Based on different empirical axioms and corresponding to different characteristics of the production possibility set and production frontiers, different DEA models, namely the BCC model [2], the FG model [10], the ST model [16], and the CCWH model [6,7], are developed and applied in practice. Wei and Yu [22] and Yu et al. [24,25] introduce a generalized DEA model (GDEA) with three 0–1 parameters (d1, d2, d3) to give an unification of DEA models and methods. By assigning different values to these parameters, the GDEA model q
The first author is partially supported by the National Natural Science Foundation of China, NNSF 70531040 and The 985 Research Grant of Renmin University of China. The research is partially supported by The Hong Kong UGC CERG Grant PolyU5457/06H. * Corresponding author. E-mail address: [email protected] (H. Yan). 0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.06.053
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can be reduced to the classic DEA models mentioned above. Furthermore, the GDEA model involves two important concepts, which are predilection cone and preference cone. The predilection cone is used to represent the preference of the decision maker in different DMUs while the preference cone is used to describe the preference of the decision maker on different input/output categories. The research on the GDEA model can lead important properties of different DEA models and methods. A consistent task in DEA research and application is to improve the efficiency state of DMUs. In the abovementioned DEA models for an inefficient DMU, an input-oriented model obtains its efficiency projection on the production frontier by equi-proportionally decreasing its inputs while keeping its outputs fixed. Likewise, an output-oriented model increases its outputs equi-proportionally while keeping its inputs fixed to obtain the projection. Such a projection is a point-to-point mapping from the concerned DMU to a point on the production frontier. However, these models fail to describe the decision maker’s preference to achieve increasing outputs and decreasing inputs simultaneously because of the single projection feature. In addition, radial projections often lead to Pareto-inefficient portions of the frontier [13]. It is clear that multiple projections are possible if different combination of input decreasing and output increasing occur. Various models have been proposed to integrate inputs and outputs, particularly for considering the returns to scale (RTS) measurement [8,17]. Banker et al. [4] discusses and applies non-radial DEA models, e.g. additive and multiplicative models. Fukuyama [12] gives a detailed discussion on RTS measurements using non-radial DEA models. Thanassoulis and Dyson [19] proposes to assign weights to changes in the input and output element to express the decision maker’s preference structure. Lins et al. [13] uses a similar nonradial DEA model with different weights on input and output to determine alternative targets in DEA. Another way to integrate input and output measurements is the directional distance function approach. Such function is essentially a variation of the Luenberger shortage function, which measures the distance in a given direction as a shortage of a point to the production frontier [14]. Similar to that in input-oriented and outputoriented DEA models, the input distance function and output distance function are defined to scale inputs and output respectively (see [11] for a fundamental description of distance function and the production possibility set, called technology in the work). Chambers et al. [5] suggests a general directional distance function to integrate both input decrease and output increase in a simultaneous manner. This function measures the distance from a point to the production frontier in a given direction. It is especially important to estimate RTS of DMUs in the research and application of DEA. Cooper et al. [9] considers to increase outputs while decreasing inputs based on the most productive scale size in order to estimate RTS of DMUs. Their model maximizes the ratio between multiple of decreased inputs and multiple of increased outputs and uses the optimal ratio to estimate RTS. The recent work of Sueyoshi and Sekitani [18] gives a detailed introduction on the DEA based RTS measurement and discusses a range-adjusted measure (RAM) in the non-radial DEA model for return to scale (RTS) measurement. In particular, they point out that the multiple projection, if occurs, has serious influence on non-radial RTS measurement. Moreover, abundant works of using optimal solution, optimal value, and particularly the sign of l0 in an optimal solution of the DEA model to present various methods and conditions for the measurement of RTS of a DMU [1,8,17,20–22]. The multiple projection issue is noticed and investigated by Olesen and Petersen [15] based on the dual representation of a polyhedral empirical production possibility set. In the work of Thanassoulis and Dyson [19], projections are considered for sets of inputs and outputs that can be improved. As pointed out by Lins et al. [13], models using weights in the search for the most preferred efficient target have some implementation difficulties. They propose a multi-objective approach to tackle this issue. However, the projection obtained in the additive DEA model and the directional distance function is a single point projection. Additive DEA models assign all input and output components the same unit weight 1. The directional distance function is defined by assigning the same scale b, called ‘‘distance’’, to all input and output elements. Such a feature does not allow the decision maker to give his/her own preference or percentage on input and output. In this paper, based on the GDEA model, we propose a bi-objective GDEA model (Bi-GDEA). It integrates the two objectives of increasing outputs and decreasing inputs simultaneously. It is an advanced generalization of the GDEA model. In the Bi-GDEA model, in addition to the three parameters (d1, d2, d3) for unification of other models, weights k1 and k2 are used to represent the decision maker’s preference on increasing outputs and decreasing inputs. In general, the research and application of DEA is often characterized by the following three aspects, namely to measure efficiency of DMUs, to determine returns to scale, and to give
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the projection of a DMU on the production possibility set which provides foundation for efficiency improvement. We show that the Bi-GDEA model has those important properties of the GDEA model from these three aspects. To measure efficiency of DMUs, we define DEA efficiency under the Bi-GDEA model as Bi-GDEA efficiency. The GDEA efficiency under either input-oriented model [24] or output-oriented model [23] is equivalent to the non-dominated solution of a multi-objective programming defined on the production possibility set. In the Bi-GDEA model, we show such equivalency and that is to say a DMU is efficient under Bi-GDEA model if and only if it is GDEA efficient. Therefore, the RTS in the Bi-GDEA model can be directly measured by using the previous results. The most essential contribution of this work is that we define a point-to-set mapping under the Bi-GDEA model which is from the point of an inefficient DMU to a subset on the frontier of the production possibility set. We call the projection of the mapping on frontiers of the production possibility set a point-to-set mapping projection to distinguish it from the point-to-point projection in conventional DEA models. We propose an efficient approach to construct the point-to-set mapping projection for a DMU. This approach, illustrated by numerical examples, presents a simple rule for practical applications. For any point on the projection set, we show that it is weak Bi-GDEA efficient relative to original DMUs. Thus, the point-to-set mapping projection forms a subset on frontier of the production possibility set (the subset under the condition of increasing outputs and decreasing inputs simultaneously). As there is no restriction given to the weights k1 and k2, the decision maker can ‘‘freely’’ decide a preferred portion to input and output when projecting an inefficient DMU to the production frontier. Thus the projection provides decision maker with foundations and alternatives for decision making. The rest of the paper is arranged as follows. Section 2 introduces Bi-GDEA model, definition on (weak) BiGDEA efficient, and some related properties. It also outlines some particular Bi-DEA models including (CCR) Bi-DEA model, (BCC) Bi-DEA model, (FG) Bi-DEA model, and (ST) Bi-DEA model. Section 3 shows the equivalency between Bi-GDEA efficiency and non-dominated solution of multi-objective programming. Section 4 discusses returns to scale of DMU. Section 5 defines point-to-set mapping and point-to-set mapping projection, presents an approach to construct point-to-set mapping projection, and provides an example for illustration. Section 6 further summarizes the decision making process applying the point-toset mapping projection and concludes the paper. 2. Bi-objective generalized DEA model (Bi-GDEA) Consider DMU-j0, 1 6 j0 6 n. For convenience, denote x0 ¼ xj0 , y 0 ¼ y j0 . In the following model, we integrate the input factor h and the output factor z into the objective function. The model is called a bi-objective generalized DEA model (Bi-GDEA): min ðPÞs:t:
ðk 1 h k 2 zÞ ¼ h0 X k hx0 2W Y k þ zy 0
d1 ðeT k þ d2 ð1Þd3 knþ1 Þ ¼ d1 k 2 K ; knþ1 P 0 h 6 1; z P 1 and its dual: max ðDÞs:t:
ðd1 l0 b1 þ b2 Þ xT X lT Y þ d1 l0 eT 2 K x T x 0 b1 ¼ k 1 l T y 0 b2 ¼ k 2 d1 d2 ð1Þd3 l0 P 0 ðxT ; lT ÞT 2 W b1 P 0;
b2 P 0;
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where X = (x1, x2, . . ., xn) is an m · n input matrix, Y = (y1, y2, . . ., yn) is an s · n output matrix, xj = (x1j, x2j, . . ., xmj)T is the input vector for DMU-j, j = 1, . . ., n, yj = (y1j, y2j, . . ., ysj)T is the output vector for DMU-j, j = 1, . . ., n, e = (1, 1, . . ., 1)T 2 En, W Emþs is a closed convex cone with Int W 5 ;, and þ W* Em+s is the negative polar cone of W. For any (xT, lT)T 2 Wn{0}, we have xT xj > 0;
lT y j > 0;
j ¼ 1; . . . ; n;
K Enþ is a closed convex cone with Int K 5 ;, and K* En is the negative polar cone of K, (k1, k2)T is positive weight (k1 > 0, k2 > 0), which describe the decision maker’s preferences on decreasing inputs and increasing outputs, respectively. d1, d2, d3 are 0–1 binary parameters. Different values of parameters d1, d2, d3 lead to the different generalized Bi-GDEA models (where ‘‘*’’ indicates either 0 or 1): Case 1. When (d1, d2, d3) = (0, *, *), the Bi-GDEA model is reduced to the (CCR) Bi-GDEA model: min ðPCCR Þs:t:
ðk 1 h k 2 zÞ X k hx0 Y k þ zy 0 k 2 K h 6 1;
2W
z P 1;
Case 2. When (d1, d2, d3) = (1, 0, *), the Bi-GDEA model is reduced to the (BCC) Bi-GDEA model: min ðPBCC Þs:t:
ðk 1 h k 2 zÞ X k hx0 Y k þ zy 0 k 2 K ; h 6 1;
2W
eT k ¼ 1
z P 1;
Case 3. When (d1, d2, d3) = (1, 1, 0), the Bi-GDEA model is reduced to the (FG) Bi-GDEA model: min ðPFG Þs:t:
ðk 1 h k 2 zÞ X k hx0 2 W Y k þ zy 0 k 2 K ; eT k 6 1 h 6 1; z P 1;
Case 4. When (d1, d2, d3) = (1, 1, 1), the Bi-GDEA model is reduced to the (ST) Bi-GDEA model: min ðPST Þs:t:
ðk 1 h k 2 zÞ X k hx0 Y k þ zy 0
2 W
k 2 K ; eT k P 1 h 6 1; z P 1: The cone W is called a preference cone and used to describe the relative importance of different input/output categories, as viewed by the decision maker. Cone K is called a predilection cone and used to represent the decision maker’s preferences for different DMUs [22,24]. j0
T Since K Enþ , then K Enþ . Let h ¼ 1, z ¼ 1, k ¼ ð0; . . . ; 0; 1; 0; . . . ; 0Þ 2 En , knþ1 ¼ 0, then k 2 K , and ð k; knþ1 ; h; zÞ is a feasible solution of (P). Therefore the optimal value of (P) h0 satisfies h0 6 k1 k2.
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Lemma 1 (Weak duality theorem). Let (k, kn+1, h, z) be a feasible solution of (P), (x, l, l0, b1, b2) be a feasible solution of (D), then k 1 h k 2 z P d1 l0 b1 þ b2 : Proof. Let (k, kn+1, h, z), (x, l, l0, b1, b2) be feasible solutions of (P) and (D), respectively. Since ððX k hx0 ÞT ; ðY k þ zy 0 ÞT ÞT 2 W ;
ðxT ; lT ÞT 2 W ;
then xT ðX k hx0 Þ þ lT ðY k þ zy 0 Þ 6 0; that is, ðxT X lT Y Þk 6 hxT x0 zlT y 0 : T
T
ð1Þ
T
Note that x X l Y + d1l0e 2 K, k 2 T
T
K*,
we have
T
ðx X l Y þ d1 l0 e Þk P 0; that is ðxT X lT Y Þk P d1 l0 eT k:
ð2Þ
d3
Since kn+1 P 0, d1 d2 ð1Þ l0 P 0, then d
d1 d2 ð1Þ 3 l0 knþ1 P 0:
ð3Þ
From (1)–(3), we have k 1 h k 2 z ¼ ðxT x0 b1 Þh ðlT y 0 b2 Þz ¼ ðhxT x0 zlT y 0 Þ b1 h þ b2 z P ðxT X lT Y Þk b1 þ b2 P d1 l0 eT k b1 þ b2 ¼ d1 l0 þ d1 d2 ð1Þd3 knþ1 l0 b1 þ b2 P d1 l0 b1 þ b2 : Thus k 1 h k 2 z P d1 l0 b1 þ b2 :
From Lemma 1, it is easy to see the following corollary. Corollary 1. If ðx ; l ; l0 ; b1 ; b2 Þ is a feasible solution of (D) such that d1 l0 b1 þ b2 ¼ k 1 k 2 ; then ðx ; l ; l0 ; b1 ; b2 Þ is an optimal solution of (D). In a mathematical programming with cone structure, the duality theorem holds under some constraint qualifications. Therefore, we give the following assumption. Assumption 1 (Constraint qualification I). For an optimal solution of (P): ðk ; knþ1 ; h ; z Þ, set 9 80 T 1 X x Y T l þ d1 l0 e þ h1 h1 2 K; ðxT ; lT ÞT 2 W ; h2 P 0; b1 P 0; b2 P 0 > > > > > > d = > @ A xT0 x þ b1 hT1 k ¼ 0; h2 knþ1 ¼ 0 > > > > ; : T y 0 l b2 b1 ðh 1Þ ¼ 0; b2 ðz 1Þ ¼ 0 is a closed set. Under Assumption 1, we have the following theorem. Theorem 1 (Duality theorem). Let ðk ; knþ1 ; h ; z Þ be an optimal solution of (P). Accordingly, there must exist an optimal solution of (D), ðx ; l ; l0 ; b1 ; b2 Þ, such that
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k 1 h k 2 z ¼ d1 l0 b1 þ b2 : Proof. Since the dual of (P) is (D) and Assumption 1 holds, by applying Theorem A2 from [6], we obtain the result of this theorem. h In the following, we define (weak) DEA efficiency under the Bi-GDEA model. Definition 1 ((Weak) Bi-GDEA efficient). (i) If the optimal value of (P) (or (D)) is equal to k1 k2, then DMU-j0 is called weak Bi-GDEA efficient. (ii) If (D) has an optimal solution ðx ; l ; l0 ; b1 ; b2 Þ such that d1 l0 b1 þ b2 ¼ k 1 k 2 ;
T
ðxT ; lT Þ 2 Int W ;
then DMU-j0 is called Bi-GDEA efficient. The relationship of DEA efficiency under the different Bi-GDEA models: (CCR)Bi-GDEA model, (BCC)Bi-GDEA model, (FG)Bi-GDEA model and (ST)Bi-GDEA model, is similar to the GDEA model. (See Theorems 1–5 and Corollary 1 in [24]). Denote the production possibility set: Xk x d T ¼ ðx; yÞ 2 W ; d1 ðeT k þ d2 ð1Þ 3 knþ1 Þ ¼ d1 ; k 2 K ; knþ1 P 0 : Y k þ y is k1 k2, then ðx; y Þ is Definition 2 (Production frontier). Let ðx; y Þ 2 T . If the optimal value of problem ðPÞ called weak Bi-GDEA efficient, where ðk 1 h k 2 zÞ X k hx ðPÞs:t: 2 W Y k þ zy min
d
d1 ðeT k þ d2 ð1Þ 3 knþ1 Þ ¼ d1 k 2 K ; knþ1 P 0 h 6 1; z P 1: The set of weak Bi-GDEA efficient point ðx; y Þ of production possibility set T is called production frontier of T. Assumption 2. If y 5 0, then (0, y) 62 T. That is, it cannot have a non-zero output without input. From the definition of production possibility set T, (P) is rewritten into the following: min ðk 1 h k 2 zÞ ðPÞs:t: ðhx0 ; zy 0 Þ 2 T h 6 1; z P 1: Theorem 2. Let (h, z, k, knþ1 ) be an optimal solution of (P), then DMU-j0 is weak Bi-GDEA efficient if and only if h = 1, z = 1. Proof. Assume that DMU-j0 is weak Bi-GDEA efficient. Then k1h k2z = k1 k2. Thus, k1(1 h) + k2(z 1) = 0. Note that h 6 1, z P 1, k1 > 0 and k2 > 0, we have h = 1, z = 1. On the other hand, if h = 1, z = 1, then by Definition 1, DMU-j0 is weak Bi-GDEA efficient. h Let ðk ; knþ1 ; h ; z Þ be an optimal solution of (P), the conventional projection of DMU-j0 is defined as follows:
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^x0 ¼ h x0 ; ^y 0 ¼ z y 0 : Obviously, ð^x0 ; ^y 0 Þ 2 T . Theorem 3. When Assumption 2 holds, projection ð^x0 ; ^y 0 Þ is weak Bi-GDEA efficient relative to the other DMUs. Proof. Consider the program min ðk 1 h k 2 zÞ ~ ðPÞs:t: ðh^x0 ; z^y 0 Þ 2 T h 6 1; z P 1: ~ Note that ðh^x0 ; z^y 0 Þ ¼ ðhh x0 ; zz y Þ 2 T , we have that Assume that ð h; zÞ is an optimal solution of ðPÞ. 0 ð hh ; zz Þ is a feasible solution of (P). Since (h , z ) is an optimal solution of (P), then hh k 2zz P k 1 h k 2 z : k1 Since h 6 1, z P 1, and from Assumption 2, it is easy to see h > 0. Thus, 0 P k 1 h ð h 1Þ P k 2 z ðz 1Þ P 0; k 1 h ð h 1Þ ¼ k 2 z ðz 1Þ ¼ 0; then h ¼ 1, z ¼ 1 and the optimal value of ðP~ Þ is equal to k1 k2. By Definition 1, ð^x0 ; ^y 0 Þ is weak Bi-GDEA efficient. h 3. The equivalence between Bi-GDEA efficiency and non-dominated solution of the multi-objective program Consider the multi-objective program ðVPÞ min ðx; yÞ s:t: ðx; yÞ 2 T : In order to discuss the relationship between Bi-GDEA efficiency and multi-objective program, we introduce the concept of non-dominated solution of multi-objective programming problem, which is given by Yu [26] and extended from that of Pareto solution of multi-objective program. Definition 3. Let W Emþs þ . If there exists no (x, y) 2 T such that x x0 þ W n f0g; 2 y y 0 then (x0, y0) is called a non-dominated solution of (VP) associated with W*. In order to show the equivalence between Bi-GDEA efficiency and non-dominated solution of (VP), we consider the following pair of dual programming problems: max ðsT s þ ^sT sþ Þ ðP0 Þs:t: X k þ s ¼ x0 Y k sþ ¼ y 0 d1 ðeT k þ d2 ð1Þd3 knþ1 Þ ¼ d1 k 2 K ; þT T
knþ1 P 0
ðsT ; s Þ 2 W
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and min ðxT x0 lT y 0 þ d1 l0 Þ xT X lT Y þ d1 l0 eT 2 K
ðD0 Þs:t:
d
d1 d2 ð1Þ 3 l0 P 0 ððx sÞT ; ðl ^sÞT ÞT 2 W ; T where ðsT ; ^sT Þ 2 Int W . Before we proceed, we must make an assumption, which will naturally hold in a case where cone K and W are polyhedral.
Assumption 3 (Constraint qualification II). Assume that ðk ; knþ1 ; s ; sþ Þ is an optimal solution of (P0). Set 9 80 T 1 X x Y T l þ d1 l0 e þ h1 > > T T T > > > > h 2 K; ðh ; h Þ 2 W ; h P 0 4 2 3 = > @ A l þ h3 > > > > ; : h4 knþ1 ¼ 0 d3 d1 d2 ð1Þ l0 þ h4 be a closed set. It is clear that the duality theorem for (P0) and (D0) holds under the Assumption 3. That is, the following lemma holds (see [24]). Lemma 2 (Duality theorem). Let ðk ; knþ1 ; s ; sþ Þ be an optimal solution of (P0). Accordingly, there must exist l ; l0 Þ, such that an optimal solution of (D0), ðx; T x0 l T y 0 þ d1 l 0 : sT s þ ^sT sþ ¼ x l ; l 0 Þ such Lemma 3. Let (x0, y0) be a non-dominated solution of (VP) associated with W*, then there exists ðx; that TX l T Y þ d1 l 0 eT 2 K; x T x0 l T y 0 þ d1 l 0 ¼ 0; x d 0 P 0; d1 d2 ð1Þ 3 l T
T; l T Þ 2 Int W : ðx Proof. From the definition of non-dominated solution, there exists no (x, y) 2 T with x x0 2 þ W n f0g; y y 0 that is X k þ s ¼ x 0 ; Y k sþ ¼ y 0 ; d
d1 ðeT k þ d2 ð1Þ 3 knþ1 Þ ¼ d1 ; k 2 K ; T
knþ1 P 0;
þT
ðs ; s Þ 2 W n f0g: is inconsistent. Since ðsT ; ^sT Þ 2 Int W , then the optimal value of (P0) is 0. From Lemma 2, (D0) has optimal l ; l 0 Þ satisfying solution ðx;
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863
T x0 l T y 0 þ d1 l 0 ¼ 0; x TX l T Y þ d1 l 0 eT 2 K; x d 0 P 0; d1 d2 ð1Þ 3 l T
T T
sÞ ; ð l ^sÞ Þ 2 W : ððx T
Since ðsT ; ^sT Þ 2 Int W , then T; l sÞT ; ð T ÞT ¼ ððx ðx l ^sÞT ÞT þ ðsT ; ^sT ÞT 2 Int W : This completes the proof.
h
In the following, we give a sufficient condition for that (x0, y0) 2 T is a non-dominated solution of (VP) associated with W*. Lemma 4. If there exists (xT, lT)T 2 Int W, such that (x0, y0) is an optimal solution of min ðxT x lT yÞ; s:t:
ðx; yÞ 2 T ;
then (x0, y0) is a non-dominated solution of (VP) associated with W *. Proof. Assume that (x0, y0) isnot anon-dominated solution of (VP) associated with W*, then there exists x x0 s T (x, y) 2 T such that ¼ þ , where ðsT ; ^sT Þ 2 W n f0g. So ^s y y 0 xT x lT y ¼ xT ðx0 þ sÞ lT ðy 0 ^sÞ ¼ xT x0 lT y 0 þ ðxT s þ lT^sÞ: T
T T
T
T T
ð4Þ
Since (x , l ) 2 Int W, ðs ; ^s Þ 2 W n f0g, and since Int W 5 ; then (see [26]) xT s þ lT^s < 0:
ð5Þ
From (4) and (5), we have xT x lT y < xT x0 lT y 0 : This is a contradiction. Thus (x0, y0) is a non-dominated solution of (VP) associated with W*. h Based on above lemmas, we obtain the following theorem. Theorem 4. DMU-j0 is Bi-GDEA efficient if and only if (x0, y0) is a non-dominated solution of (VP) associated with W *. l and l 0 Proof. Let (x0, y0) be a non-dominated solution associated with W*. From Lemma 3, there exists x, such that T x0 l T y 0 þ d1 l 0 ¼ 0; x TX l T Y þ d1 l 0 eT 2 K; x d 0 P 0; d1 d2 ð1Þ 3 l T
T; l T Þ 2 Int W : ðx Denote
q ¼ max x ¼ qx;
k1 k2 ; ; T x0 l T y 0 x l ¼ q l;
b1 ¼ xT x0 k 1 ;
l0 ¼ q l0 ;
b2 ¼ lT y 0 k 2 :
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It is easy to see that ðx ; l ; l0 ; b1 ; b2 Þ is a feasible solution of (D) with d1 l0 b1 þ b2 ¼ xT x0 lT y 0 ðxT x0 k 1 Þ þ ðlT y 0 k 2 Þ ¼ k 1 k 2 ; by Corollary 1, ðx ; l ; l0 ; b1 ; b2 Þ is an optimal solution of (D) and the optimal value is k1–k2. Therefore, DMU-j0 is Bi-GDEA efficient. On the other hand, if DMU-j0 is Bi-GDEA efficient, there exists (xT, lT)T 2 Int W, l0 , b1 , b2 satisfying d1 l0 b1 þ b2 ¼ k 1 k 2 ; xT X lT Y þ d1 l0 eT 2 K; d1 d2 ð1Þd3 l0 P 0; xT x0 b1 ¼ k 1 ; lT y 0 b2 ¼ k 2 ; T
ðxT ; lT Þ 2 Int W ;
b1 P 0;
b1 P 0;
thus xT x0 lT y 0 þ d1 l0 ¼ ðk 1 þ b1 Þ ðk 2 þ b2 Þ þ d1 l0 ¼ 0: For any (x, y) 2 T, there exists (s that x X k þ s ; ¼ Y k þ sþ y
T
,s
ð6Þ d3
+T T
) 2 W*, k 2 K*, kn+1 P 0 with d1 ðeT k þ d2 ð1Þ knþ1 Þ ¼ d1 , such
thus xT x lT y ¼ xT ðX k þ s Þ lT ðY k sþ Þ ¼ ðxT X lT Y Þk þ ðxT s þ lT sþ Þ:
ð7Þ
Since (xT, lT)T 2 Int W W, (sT, s+T)T 2 W*, then xT s þ lT sþ P 0:
ð8Þ
By (7) and (8), we have xT x lT y P ðxT X lT Y Þk:
T
T
Since k 2 K ; x X l Y þ T
T
ðx X l Y þ
d1 l0 eT Þk
d1 l0 eT
ð9Þ 2 K, then
P 0:
That is, d
d
ðxT X lT Y Þk P d1 l0 eT k ¼ l0 ðd1 d1 d2 ð1Þ 3 knþ1 Þ ¼ d1 l0 þ d1 d2 ð1Þ 3 knþ1 l0 P d1 l0 :
ð10Þ
By (9), (10) and (6), for any (x, y) 2 T, we have xT x lT y P ðxT X lT Y Þk P d1 l0 ¼ xT x0 lT y 0 ; that is, (x0, y0) is an optimal solution of programming min ðxT x lT yÞ; s:t: ðx; yÞ 2 T ; where (xT, lT)T 2 Int W. By Lemma 4, we have (x0, y0) is a non-dominated solution of (VP) associated with W *. h Yu et al. [24] proved the equivalence between the DEA efficiency under GDEA model and the non-dominated solution of (VP). Theorem 4 above shows the equivalence between the Bi-GDEA efficiency and the nondominated solution of (VP). Therefore, the Bi-GDEA efficiency is parallel to the DEA efficiency under the GDEA model.
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4. Returns to scale An ideal DEA model can be used to evaluate the returns to scale for DMUs, increasing returns to scale, constant returns to scale, and decreasing returns to scale. In this section, we discuss using the Bi-GDEA model to measure returns to scale. The next theorem shows that the optimal solution of the Bi-GDEA model (D) is a positive scalar multiple of the optimal solution of the corresponding GDEA model (DGDEA) below. Theorem 5. Consider Bi-GDEA model (D) and the GDEA model in [23] given as below: min ðxT x0 þ d1 l0 Þ ðDGDEA Þs:t: xT X lT Y þ d1 l0 eT 2 K lT y 0 ¼ 1 d
d1 d2 ð1Þ 3 l0 P 0 T
ðxT ; lT Þ 2 W : Then, DMU-j0 is weak Bi-GDEA efficient if and only if it is weak GDEA efficient. In addition, the optimal solution of (D) is a positive scalar multiple of the optimal solution of (DGDEA). Proof. Assume that DMU-j0 is weak Bi-GDEA efficient, and ðx ; l ; l0 ; b1 ; b2 Þ is an optimal solution of (D). Note that k 2 þ b2 > 0, denote ¼ x
x ; k 2 þ b2
¼ l
l ; k 2 þ b2
0 ¼ l
l0 : k 2 þ b2
l ; l 0 Þ is a feasible solution of (DGDEA) with Obviously, ðx; T x 0 þ d1 l 0 ¼ x
1 1 T ðk 1 þ b1 þ d1 l0 Þ: ðx x0 þ d1 l0 Þ ¼ k 2 þ b2 k 2 þ b2
ð11Þ
Since DMU-j0 is weak Bi-GDEA efficient, then d1 l0 b1 þ b2 ¼ k 1 k 2 : That is, d1 l0 ¼ b1 þ b2 k 1 þ k 2 :
ð12Þ
By (11) and (12), we have T x 0 þ d1 l 0 ¼ x
1 ðk 1 þ b1 b1 þ b2 k 1 þ k 2 Þ ¼ 1: k 2 þ b2
l ; l 0 Þ is an optimal solution of (DGDEA). Thus, DMU-j0 Since the objective value of (DGDEA) h P 1, then ðx; is weak DEA efficient under the GDEA model. And the optimal solution of (D) is a positive scalar, k 2 þ b2 , multiple of that of (DGDEA). l ; l 0 Þ is an optimal On the other hand, assume DMU-j0 is weak DEA efficient under (DGDEA) that ðx; solution of (DGDEA). We have T x 0 þ d1 l T y 0 ¼ 1 0 ¼ l x TX l T Y þ d1 l 0 eT 2 K x d 0 P 0 d1 d2 ð1Þ 3 l T
T; l T Þ 2 W : ðx From the proof of Theorem 4, (D) has an optimal solution ðx ; l ; l0 ; b1 ; b2 Þ, where k1 k2 x ¼ qx; l ¼ q l; l0 ¼ q l0 ; q ¼ max ; > 0: T x0 l T y 0 x
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In addition, d1 l0 b1 þ b2 ¼ k 1 k 2 : Thus, DMU-j0 is Bi-GDEA efficient, the optimal solution of (D) is a positive scalar multiple, q, of that of (DGDEA). h The research and application of returns to scale can be traced back to 1984, the work of Banker et al. [2,3]. Since then, a vast amount of work has been conducted and published on this problem. (For details, one can see the survey work by Seiford and Zhu [17] and Copper et al. [8]). Among these works, one important class of methods involves measuring returns to scale by the sign of l0 in the optimal solution (see, for example [1,8,17,20,22]). In this paper, since we have proved that the optimal solution of (D) is a positive scalar multiple of the optimal solution of (DGDEA), then the results of the above works can be used in the Bi-GDEA model. In other words, we can determine returns to scale by the sign of l0 in the optimal solutions of (D). 5. Point-to-set mapping and point-to-set mapping projection This section discusses a point-to-set mapping and its projection of mapping on the frontier of the production possibility set. The point-to-set mapping of a DMU gives the projections on the frontiers of the production possibility set. Given weight (k1, k2) which describes the decision maker’s preference on increasing outputs and decreasing inputs in the Bi-GDEA model, a projection represents a subset of the production frontier. Therefore we obtain different projections for different weights (k1, k2). The union of these projections under a Bi-GDEA model forms a subset of the production frontier, which provides the efficiency improvement alternatives. We call this set a point-to-set mapping projection. In the following, using the Bi-GDEA model, we provide an approach to construct the point-to-set mapping projection. For DMU-j0, with input and output (x0, y0), a point-to-set mapping is given by F : ðx0 ; y 0 Þ ! F ðx0 ; y 0 Þ ¼ fðhx0 ; zy 0 Þjðhx0 ; zy 0 Þ 2 T ; h 6 1; z P 1g: It is clear that F(x0, y0) is a subset of the production possibility set T, which is to decrease input x0 and increase output y0. We consider the input-oriented GDEA model and the output-oriented GDEA model. The input-oriented GDEA model [24] is given by min ðPinput-GDEA Þs:t:
h
X k hx0
Y k þ y 0
2W
d1 ðeT k þ d2 ð1Þd3 knþ1 Þ ¼ d1 k 2 K ;
knþ1 P 0:
And the output-oriented GDEA model [22] is given by max ðPoutput-GDEA Þs:t:
z
X k x0 Y k þ zy 0
2W
d1 ðeT k þ d2 ð1Þd3 knþ1 Þ ¼ d1 k 2 K ;
knþ1 P 0:
If the optimal value of (Pinput-GDEA) equals to 1, then DMU-j0 is called weak DEA efficient under inputoriented GDEA model. If the optimal value of (Poutput-GDEA) is equal to 1, then DMU-j0 is called weak DEA efficient under output-oriented GDEA model. Assume that DMU-j0 is not weak efficient under both input-oriented and output-oriented GDEA models. The following algorithm defines and constructs the point-to-set mapping projection of (x0, y0).
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Algorithm (Point-to-set mapping projection) Step 1: (i) Let k1 = 1, k2 = 1, and solve min ðh zÞ ðP1 Þs:t: ðhx0 ; zy 0 Þ 2 T z ¼ 1;
h 6 1;
to obtain the optimal solution ðh1 ; 1Þ, since DMU-j0 is not weak DEA efficient under input-oriented GDEA model (Pinput-GDEA), then h1 < 1. Denote ðh1 ; z1 Þ ¼ ðh1 ; 1Þ. (ii) Let k1 = 1, k2 = 1, and solve min ðP2 Þs:t:
ðh zÞ ðhx0 ; zy 0 Þ 2 T z P 1; h ¼ 1
to obtain the optimal solution ð1; z2 Þ, since DMU-j0 is not weak DEA efficient under output-oriented GDEA model (Poutput-GDEA), then z2 > 1. Denote ðh2 ; z2 Þ ¼ ð1; z2 Þ. Obviously, h1 ¼ h1 < h2 ¼ 1;
1 ¼ z1 < z2 ¼ z2 ;
and l: = 2. Step 2: Denote k i1 ¼ ziþ1 zi ;
k i2 ¼ hiþ1 hi ;
qi ¼
k i1 ; k i2
i ¼ 1; 2; . . . ; l 1:
When l P 3, we have q1 > q2 > > ql1 : For i = 1, 2, . . ., l 1, solve min
ðk i1 h k i2 zÞ
ðPðk i1 ; k i2 ÞÞs:t: ðhx0 ; zy 0 Þ 2 T h 6 1; z P 1: to obtain its optimal solution ðh0i ; z0i Þ. If h0i hi z0 zi ¼ i ; hiþ1 hi ziþ1 zi
ð13Þ
then ðh0i , z0i Þ is abandoned. Assume that there exists l points not satisfying (13). If l ¼ 0 then go to Step 3, else continue. Sort these newly solved l optimal solutions ðh0i ; z0i Þ, i ¼ 1; 2; . . . ; l, and original l points (hi, zi) i = 1, 2, . . ., l, into a non-decreasing order: h1 ¼ h1 < h2 < < hlþl ¼ 1; 1 ¼ z1 < z2 < < zlþl ¼ z2 : Let l :¼ l þ l and go back to step 2. Step 3: At this time, we obtain h1 ¼ h1 < h2 < < hl ¼ 1; 1 ¼ z1 < z2 < < zl ¼ z2 ;
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and k i1 ¼ ziþ1 zi ;
k i2 ¼ hiþ1 hi ;
k i1 ; i ¼ 1; 2; . . . ; l 1: k i2 For convenience, denote qi ¼
k 01 ¼ þ1; k l1 ¼ 0;
k 02 ¼ 1;
k l2 ¼ 1;
q0 ¼
ql ¼
k 01 ; k 02
k l1 : k l2
Then, we have þ1 ¼ q0 > q1 > q2 > > ql1 > ql ¼ 0: See Fig. 1. We have iþ1 i k k (i) if kk12 2 ðqiþ1 ; qi Þ ¼ k1iþ1 ; k1i ; 0 6 i 6 l 1, then (hi+1, zi+1) is an optimal solution of ðPðk i1 ; k i2 ÞÞ (see The2 2 orem 6), min ðk 1 h k 2 zÞ ðPðk 1 ; k 2 ÞÞs:t: ðhx0 ; zy 0 Þ 2 T h 6 1; z P 1: Denote k1 S x0 ; y 0 ; ¼ fðhiþ1 x0 ; ziþ1 y 0 Þg; k2 (ii) if
k1 k2
¼ qi ; 1 6 i 6 l 1 then the optimal solution of problem ðPðk i1 ; k i2 ÞÞ is a set (see Theorem 7)
z zl = z20 z l −1
zi +1 (k1l −1,−k2l −1)
zi
(k1l ,−k2l )
(k1i ,−k2i ) z2
(k12 ,−k22 ) z1 = 1
(k11,−k12 )
0
θ 1 =θ10
θ2
θi
θ i +1
θ i −1
θl =1
Fig. 1. Illustration of weight coefficients in the BCC model.
θ
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fðh; zÞjz ¼ qi ðh hiþ1 Þ þ ziþ1 ; hi 6 h 6 hiþ1 g: Denote k1 S x0 ; y 0 ; ¼ fðhx0 ; zy 0 Þjz ¼ qi ðh hiþ1 Þ þ ziþ1 ; hi 6 h 6 hiþ1 g: k2 Finally, denote a point-to-set mapping projection: FWP : ðx0 ; y 0 Þ ! F WP ðx0 ; y 0 Þ; where F WP ðx0 ; y 0 Þ ¼
! ! l1 l1 [ [ [ fðhx0 ; zy 0 Þjz ¼ qi ðh hiþ1 Þ þ ziþ1 ; hi 6 h 6 hiþ1 g fðhiþ1 x0 ; ziþ1 y 0 Þg : i¼1
i¼0
Note that ðhiþ1 x0 ; ziþ1 y 0 Þ 2 fðhx0 ; zy 0 Þjz ¼ qi ðh hiþ1 Þ þ ziþ1 ; hi 6 h 6 hiþ1 g;
i ¼ 1; . . . ; l 1;
and ðh1 x0 ; z1 y 0 Þ 2 fðhx0 ; zy 0 Þjz ¼ q1 ðh h2 Þ þ z2 ; h1 6 h 6 h2 g; we have F WP ðx0 ; y 0 Þ ¼
l1 [ fðhx0 ; zy 0 Þjz ¼ qi ðh hiþ1 Þ þ ziþ1 ; hi 6 h 6 hiþ1 g: i¼1
The point-to-set mapping projection FWP is a mapping from point (x0, y0) to production frontier of production possibility set, and Fwp(x0, y0) F(x0, y0) T. i i iþ1 iþ1 Theorem 6. Assume that ð^ h; ^zÞ is an optimal solution of both ðPðk 1 ; k 2 ÞÞ and ðPðk 1 ; k 2 ÞÞ, then for any iþ1 i k k q ¼ kk12 2 ðqiþ1 ; qi Þ ¼ k1iþ1 ; k1i , we have ð^ h; ^zÞ is an optimal solution of (P(k1, k2)). 2
2
Proof. Since q ¼ kk12 2 ðqiþ1 ; qi Þ, then there must exist a 2 (0, 1) such that q ¼ aqiþ1 þ ð1 aÞqi : Since ð^ h; ^zÞ is an optimal solution of ðPðk i1 ; k i2 ÞÞ, then for any (hx0, zy0) 2 T, h 6 1, z P 1, we have h k i2^z 6 k i1 h k i2 z; k i1 ^ ^ ^z 6 q h z: qh i
ð14Þ
i
Similarly, for any (hx0, zy0) 2 T, h 6 1, z P 1, we have qiþ1 ^ h ^z 6 qiþ1 h z:
ð15Þ
By (14) and (15), for any (hx0, zy0) 2 T, h 6 1, z P 1, ðaqiþ1 þ ð1 aÞqi Þ^ h ^z 6 ðaqiþ1 þ ð1 aÞqi Þh z; ^ qh ^z 6 qh z: That is, for any (hx0, zy0) 2 T, h 6 1, z P 1, we have k1^ h k 2^z 6 k 1 h k 1 z: Hence, 8q ¼ kk12 2 ðqiþ1 ; qi Þ, ð^ h; ^zÞ is an optimal solution of (P(k1, k2)).
h
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Lemma 5. For h < h, z < z, assume that ð^ h; ^zÞ is an optimal solution of min ðk 1 h k 2 zÞ ðPðk 1 ; k 2 ÞÞs:t: ðhx0 ; zy 0 Þ 2 T h 6 1;
z P 1:
h h. where k 1 ¼ z z, k 2 ¼ ^ h z^z ðh hÞ þ z; h 6 h 6 hg is an optimal solution of ðPðk 1 ; k 2 ÞÞ, If h ¼ , then for any ðh; zÞ 2 fðh; zÞjz ¼ q zz hh
¼ kk1 . where q 2
Proof. Since
h^ h hh
z ¼ zz^ , then z
k 1 z z ^ ðh hÞ ¼ ð z ^z ¼ h^ hÞ; k2 hh thus k 1 h k 2z ¼ k 1 ^h k 2^z: ðh Note that, for any ðh; zÞ 2 fðh; zÞjz ¼ q hÞ þ z; h 6 h 6 hg k 1 h k 2 z ¼ k 1 h k 2 k 1 ðh h k 2z ¼ k 1 ^h k 2^z; hÞ þ z ¼ k 1 k 2 and since ð^ h; ^zÞ is an optimal solution of ðPðk 1 ; k 2 Þ), we have that (h, z) is an optimal solution of ðPðk 1 ; k 2 ÞÞ. h Theorem 7. Any point of set FWP(x0, y0) is weak Bi-GDEA efficient. Proof. Let ð^x; ^y Þ 2 F WP ðx0 ; y 0 Þ, then there must exist an i, 1 6 i 6 l 1, such that ð^x; ^y Þ 2 fðhx0 ; zy 0 Þjz ¼ qi ðh hiþ1 Þ þ ziþ1 ; hi 6 h 6 hiþ1 g: That is, there exists ð^ h; ^zÞ, such that ð^x; ^y Þ ¼ ð^hx0 ; ^zy 0 Þ, where ð^h; ^zÞ 2 fðh; zÞjz ¼ qi ðh hiþ1 Þ þ ziþ1 ; hi 6 h 6 hiþ1 g. From constructing of FWP(x0, y0) and Lemma 5, we have ð^h; ^zÞ is an optimal solution of ðPðk i1 ; k i2 ÞÞ. By Theorem 3, ð^x; ^y Þ is weak Bi-GDEA efficient. h Example. Consider single input and single output Bi-GDEA model:
DMU 1 x1 1 ¼ ; 1 y1
DMU 2 DMU 3 DMU 4 x2 3 x3 6 x4 5 ¼ ; ¼ ; ¼ ; 3 4 2 y2 y3 y4
Assume that (d1, d2, d3) = (1, 0, *), W ¼ E2þ , K ¼ E4þ , the model is a (BCC)Bi-DEA model. We calculate the point-to-set mapping projections of DMU-4, (x4, y4) = (5, 2) by the following problem (P). The point-to-set mapping F(x4, y4) is shown in Fig. 2: min ðPÞs:t:
ðk 1 h k 2 zÞ k1 þ 3k2 þ 6k3 þ 5k4 6 5h k1 þ 3k2 þ 4k3 þ 2k4 P 2z k1 þ k2 þ k3 þ k4 ¼ 1 ki P 0; i ¼ 1; . . . ; 4 h 6 1; z P 1:
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871
y C (5, 11/3) DMU2: B (3,3)
DMU3 (6,4) F(x 0 ,y 0 )
A (2, 2)
DMU4 (x 0 , y 0 )=(5,2) DMU1 (1,1)
x
0
Fig. 2. Point-to-set mapping projection of F(5,2) in the BCC model.
Step 1: (i) Let k1 = 1, k2 = 1, then min ðk 1 h k 2 zÞ ðP1 Þs:t: k1 þ 3k2 þ 6k3 þ 5k4 6 5h k1 þ 3k2 þ 4k3 þ 2k4 P 2z k1 þ k2 þ k3 þ k4 ¼ 1 ki P 0; i ¼ 1; . . . ; 4 z ¼ 1:
has the optimal solution 25 ; 1 , denote ðh1 ; z1 Þ ¼ ðh1 ; 1Þ ¼ 25 ; 1 . (ii) Let k1 = 1, k2 = 1, then h 6 1;
min ðk 1 h k 2 zÞ ðP2 Þs:t: k1 þ 3k2 þ 6k3 þ 5k4 6 5h k1 þ 3k2 þ 4k3 þ 2k4 P 2z k1 þ k2 þ k3 þ k4 ¼ 1 ki P 0;
i ¼ 1; . . . ; 4
h ¼ 1:
has the optimal solution 1; 116 , denote ðh2 ; z2 Þ ¼ ð1; z2 Þ ¼ 1; 116 . z P 1;
k1
Step 2: k 11 ¼ z2 z1 ¼ 56, k 12 ¼ h2 h1 ¼ 35. q1 ¼ k11 ¼ 25 . Solve 18 2 5 3 h z min 6 5 ðPðk 11 ; k 12 ÞÞs:t:
k1 þ 3k2 þ 6k3 þ 5k4 6 5h k1 þ 3k2 þ 4k3 þ 2k4 P 2z k1 þ k2 þ k3 þ k4 ¼ 1 ki P 0; i ¼ 1; . . . ; 4 h 6 1;
z P 1;
to obtain its optimal solution ðh01 ; z01 Þ ¼ 35 ; 32 .
z0 z h0 h Since 13 ¼ h12 h11 6¼ z12 z11 ¼ 35, then l 6¼ 0. Sort 25 ; 1 , 1; 116 , ð35 ; 32Þ into non-decreasing order: ðh1 ; z1 Þ ¼ 25 ; 1 ,
ðh2 ; z2 Þ ¼ 35 ; 32 , ðh3 ; z3 Þ ¼ 1; 116 :
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2 h1 ¼ ; 5
3 h2 ¼ ; 5
h3 ¼ 1;
z1 ¼ 1;
3 z2 ¼ ; 2
z3 ¼
11 : 6
Step 2 0 : 1 k 11 ¼ z2 z1 ¼ ; 2 q2 ¼
1 k 12 ¼ h2 h1 ¼ ; 5
q1 ¼
k 11 5 ¼ ; k 12 2
1 k 21 ¼ z3 z2 ¼ ; 3
2 k 22 ¼ h3 h2 ¼ ; 5
1 k 21 ¼ z3 z2 ¼ ; 3
2 k 22 ¼ h3 h2 ¼ ; 5
k 21 5 ¼ : k 22 6
Solve
min
1 1 h z 2 5
ðPðk 11 ; k 12 ÞÞs:t: k1 þ 3k2 þ 6k3 þ 5k4 6 5h k1 þ 3k2 þ 4k3 þ 2k4 P 2z k1 þ k2 þ k3 þ k4 ¼ 1 ki P 0; i ¼ 1; . . . ; 4 h 6 1; z P 1;
to obtain the optimal solution ðh01 ; z01 Þ ¼ 1; 116 . Solve 1 2 h z min 3 5 ðPðk 21 ; k 22 ÞÞs:t: k1 þ 3k2 þ 6k3 þ 5k4 6 5h k1 þ 3k2 þ 4k3 þ 2k4 P 2z k1 þ k2 þ k3 þ k4 ¼ 1 i ¼ 1; . . . ; 4
ki P 0; h 6 1;
z P 1;
to obtain the optimal solution ðh02 ; z02 Þ ¼ For i = 1, 2
3 3
; . 5 2
h0i hi z0 zi ¼ i : hiþ1 hi ziþ1 zi Thus l ¼ 0. Step 3: At this time, we have l = 2 and 1 k 11 ¼ z2 z1 ¼ ; 2 q2 ¼
k 21 5 ¼ ; k 22 6
(i)
1 k 12 ¼ h2 h1 ¼ ; 5
q0 ¼ þ1;
5 q1 ¼ ; 2
q1 ¼
k 11 5 ¼ ; k 12 2
5 q2 ¼ ; 6
q3 ¼ 0:
(a) if
k1 k2
(b) if
k1 k2
(c) if
k1 k2
2 ½q3 ; q2 Þ ¼ 0; 56 then S x0 ; y 0 ; kk21 ¼ 5; 113 .
2 ðq2 ; q1 Þ ¼ 56 ; 52 then S x0 ; y 0 ; kk21 ¼ fð3; 3Þg.
2 ðq1 ; q0 Þ ¼ 52 ; þ1 then S x0 ; y 0 ; kk21 ¼ fð2; 2Þg.
(a) If
k1 k2
¼ q1 ¼ 56 then
(ii)
Q. Wei et al. / European Journal of Operational Research 190 (2008) 855–876
873
k2 5 2 3 : ¼ fðhx0 ; zy 0 Þjz ¼ q1 ðh h2 Þ þ z2 ; h1 6 h 6 h2 g ¼ ðhx0 ; zy 0 Þjz ¼ h þ 1; 6 h 6 S x0 ; y 0 ; 6 5 5 k1 Since (x0, y0) = (5, 2), denote x ¼ hx0 ¼ 5h;
y ¼ zy 0 ¼ 2z;
we have k2 5 2 3 S x0 ; y 0 ; ¼ fðx; yÞjx 3y þ 6 ¼ 0; 3 6 x 6 5g: ¼ ðhx0 ; zy 0 Þjz ¼ h þ 1; 6 h 6 6 5 5 k1 (b) If kk12 ¼ q2 ¼ 52 then k2 S x0 ; y 0 ; ¼ fðhx0 ; zy 0 Þjz ¼ q2 ðh h3 Þ þ z3 ; h2 6 h 6 h3 g k1 Similarly, we have k2 S x0 ; y 0 ; ¼ fðx; yÞjx ¼ y; 2 6 x 6 3g: k1 The point-to-set mapping projection of DMU-4 is the following set: [ [ 11 F WP ðx0 ; y 0 Þ ¼ fðx; yÞjx 3y þ 6 ¼ 0; 3 6 x 6 5g fðx; yÞjx ¼ y; 2 6 x 6 3g 5; ; ð3; 3Þ; ð2; 2Þ 3 [ ¼ fðx; yÞjx 3y þ 6 ¼ 0; 3 6 x 6 5g fðx; yÞjx ¼ y; 2 6 x 6 3g: In this example (see Figs. 2–4), DMU-4 has input and output (x0, y0) = (5, 4). From Fig. 2, DMU-4 is clearly not DEA efficient. The point-to-set mapping projection is given by the segments AB and BC on the production frontier. We can then adjust the input and output of DMU-4 as the following such that DMU4 can be DEA efficient.
(i) When the ratio of input and output weights, kk12 , is in 0; 56 , the projection is 5; 113 . (ii) When the ratio of input and output weights, kk12 ¼ 56, the projection is jx 3y + 6 = 0,3 6 x 6 5}.
(iii) When the ratio of input and output weights, kk12 , is in 56 ; 52 , the projection is (3, 3). (iv) When the ratio of input and output weights, (v) When the ratio of input and output weights,
k1 ¼ 52, the projection k2
k1 , is in 25 ; þ1 , the k2
z2 =
11 6
projection is (2, 2).
C' B'
3 2
2
A'
0
2
(k 1 ,k 2 )
z1 = 1
θ 1 =θ 10 =
1
1
(k 1 , k 2 )
3 2 θ2 = 5 5
set
{(x, y)
is a set {(x, y)jx = y, 2 6 x 6 3}.
z
z3 = z20 =
a
θ 3 =1
Fig. 3. Weight coefficients of F(5,2) in the BCC model.
θ
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Q. Wei et al. / European Journal of Operational Research 190 (2008) 855–876
B''
C''
A''
ρ3 = 0
ρ 0 = +∞
)(
)(
0
5
ρ2 =
ρ1 =
6
5 2
Fig. 4. Ratio of weight coefficients of F(5,2) in the BCC model.
B (5, 5)
y
DMU2 (3,3)
DMU3 (6,4) F(x 0 ,y 0 )
A (2, 2)
DMU4 (x 0 , y 0 )=(5,2) DMU1 (1,1)
x
0
Fig. 5. Point-to-set mapping projection of F(5,2) in the CCR model.
5
z
B ' = (1, 2 )
z1 = 5 2
5
A ' = ( 2 ,1) 1
z1 = 1
0
3 3
(k 11,k 2 ) = ( 2 , 5 )
θ1 =
θ
2
θ2 = 1
5
Fig. 6. Weight coefficients of F(5,2) in the CCR model.
B'' 0 ρ2 = 0
A'' ρ 0 = +∞
)( ρ1 =
5 2
Fig. 7. Ratio of weight coefficients of F(5,2) in the CCR model.
If we consider (d1, d2, d3) = (0, *, *) above, then model is a (CCR)Bi-DEA model. Similar as above, we have (i) k1 = 1, (ii) k1 = 1,
k2 = 1, k2 = 1,
ðh1 ; z1 Þ ¼ 25 ; 1 ,
ðh2 ; z2 Þ ¼ 1; 52 .
Q. Wei et al. / European Journal of Operational Research 190 (2008) 855–876
875
Therefore, 3 k 11 ¼ z2 z1 ¼ ; 2
3 k 12 ¼ h2 h1 ¼ : 5
Similar to the procedure in BCC model, we further solve the linear programming problem for
k1 ðk 1 ; k 2 Þ ¼ ðk 11 ; k 12 Þ ¼ 32 ; 35 . Let q1 ¼ k11 ¼ 52. We can get the similar results as shown in Figs. 5–7. The point2
to-set mapping projection of DMU-4 is given by F WP ðx0 ; y 0 Þ ¼ fðx; yÞjx y ¼ 0; 2 6 x 6 5g: 6. Conclusion This paper investigates the issue of increasing outputs and decreasing inputs simultaneously in DEA models. It proposes a Bi-objective GDEA model (Bi-GDEA) to integrate two objectives. The Bi-GDEA model not only uses the three 0–1 parameters (d1, d2, d3) to unify other models as a generalized model, but also uses two weight parameters k1 and k2 to describe the decision maker’s preference on output increase and input decrease. The Bi-GDEA model is a non-radial model. It thus allows the decision maker to have other preferable projections. It also allows that the projection would not be in Pareto-inefficient portions of the production frontier which may occur in the radial projection. In line with the conventional DEA research, we define the Bi-GDEA efficiency of DMUs, and show the equivalency between the Bi-GDEA efficiency and the non-dominated solutions of the multi-objective programming defined on the production possibility set. Furthermore, we prove that an optimal solution of BiGDEA model is equivalent to an optimal solution of GDEA model, and the solution of one model is a positive scalar multiple of the solution of another model. Therefore, returns to scale in the Bi-GDEA model can be measured by directly using the results of GDEA model and other models. Such a generalized result extended the theoretical research in DEA area. In another line, we deal with the issue of improving efficiency of a DMU. The essential contribution of this research is that we define a mapping from a point given by a DMU’s input and output to a subset of the production possibility set. This mapping is called a point-to-set mapping. Based on the Bi-GDEA model, we further define a projection of the mapping on the frontier of the production possibility set. This projection is a subset of the production frontier and is called a point-to-set mapping projection. We propose an algorithm to construct the point-to-set mapping projection for a DMU. The projection procedure makes use of the preference structure proposed by Thanassoulis and Dyson [19]. It also provides alternative computational solutions to the decision maker for efficiency improvement. It is thus clear that the proposed model and projection procedure offer deeper management insight as well as wider foundation for managerial decision making. Our next target is to further generalize the Bi-GDEA model by assigning different weights to each individual inputs and outputs. But the point-to-set mapping algorithm proposed in this paper cannot identify the projection area on the production frontier. A new method and algorithm based on the linear form of the production possibility set would be designed. References [1] R.D. Banker, R.M. Thrall, Estimating most productive scale size using data envelopment analysis, European Journal of Operational Research 62 (1992) 74–84. [2] R.D. Banker, A. Charnes, W.W. Cooper, Some models for estimating technical and scale efficiencies in DEA, Management Science 30 (9) (1984) 1078–1092. [3] R.D. Banker, H. Chang, W.W. Cooper, Equivalence and implementation of alternative methods for determining returns to scale in data envelopment analysis, European Journal of Operational Research 89 (1996) 473–481. [4] R.D. Banker, W.W. Cooper, L.M. Seiford, R.M. Thrall, J. Zhu, Returns to scale in different DEA models, European Journal of Operational Research 154 (2004) 345–362.
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