Bargaining game model in the evaluation of decision making units

Expert Systems with Applications 36 (2009) 4357–4362

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Bargaining game model in the evaluation of decision making units Jie Wu a,*, Liang Liang a, Feng Yang a, Hong Yan b a b

School of Management, University of Science and Technology of China, He Fei 230026, An Hui Province, PR China Department of Logistics, The Hong Kong Polytechnic University, Hong Kong, PR China

a r t i c l e

i n f o

Keywords: DEA Cross-efficiency Nash bargaining game R&D projects

a b s t r a c t Data envelopment analysis (DEA) has been proven an effective tool for performance evaluation and benchmarking, which can provide a relative efficiency measure for peer decision making units (DMUs) with multiple inputs and outputs. Nevertheless, its flexibility in weighted inputs and weighted outputs as well as its nature of self-evaluation has been criticized. The cross-evaluation method is developed as a DEA extensive tool being utilized to identify the best practice DMUs and to rank all DMUs using cross-efficiency scores that are associated with all DMUs, however, it still seems imperfect since the non-uniqueness of cross-efficiency measure possibly derogates the practicability of this method, and especially, the average cross-efficiency measure is not a Pareto one, so not all DMUs have the motivation to accept it. In this paper, we adopt a Nash bargaining game to improve the usual cross efficiency evaluation method. In the game, each DMU will be an independent player, and the bargaining solution between CCR efficiency and cross-efficiency can be obtained by using the classical Nash bargaining game model. The advantage of the bargaining efficiency lies on that it is a Pareto one and all the DMUs will have motivation to accept it, as well as that the adoption of common weights in evaluation will result in the equitableness of ranking all DMUs. Finally, the comparisons of those efficiency scores mentioned above and the corresponding weights are demonstrated by an example of R&D project selection. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction As a non-parametric programming technique to evaluate decision making units (DMUs) with multiple inputs and outputs, data envelopment analysis (DEA) is exhibiting more and more importance for evaluating and improving the performance of manufacturing and service operations. It has been extensively applied to performance evaluation and benchmarking of schools, hospitals, bank branches, production plants, etc. (Charnes, Cooper, Lewin, & Seiford, 1994). Recently, new ranking methods based on DEA have been emerging continually (Zilla & Lea, 2002). However, traditional DEA models are not very appropriate for ranking DMUs since they simply classify the units into two groups: efficient and inefficient (Charnes, Cooper, & Rhodes, 1978). Moreover, it is possible that some of the inefficient DMUs are in fact better in overall performers than some efficient ones. This is because of the unrestricted weight flexibility problem in DEA by being involved in an unreasonable self-rated scheme (Dyson & Thannassoulis, 1988; Wong & Beasley, 1990). The DMU under evaluation heavily weighs a few favorable measures and completely ignores other inputs and outputs in order to maximize its own DEA efficiency.

* Corresponding author. Tel.: +86 551 3491200; fax: +86 551 3600025. E-mail address: [email protected] (J. Wu). 0957-4174/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.05.001

To overcome the above limitations, the improvement in CCR model is studied more and more. Cross-efficiency evaluation method is a DEA extensive tool that can be utilized to identify good overall performers and rank DMUs (Sexton, Silkman, & Hogan, 1986). Its idea is mainly to use DEA in a peer evaluation instead of a self-evaluation. There are two principal advantages of cross-evaluation: (1) it provides for a unique ordering of the DMUs; and (2) it eliminates unrealistic weight schemes without requiring the elicitation of weight restrictions from application area experts (Anderson, Hollingsworth, & Inman, 2002). With these advantages, cross-efficiency evaluation method has been adopted in various applications, e.g., efficiency evaluations of nursing homes (Sexton et al., 1986), R&D project selection (Oral, Kettani, & Lang, 1991), preference voting (Green, Doyle, & Cook, 1996), and others. However, there are still several limitations for utilizing the average cross-efficiency measure to evaluate. One is the possible diversification of cross efficiencies generated by the non-uniqueness of the DEA optimal weights. As shown in Doyle and Green (1994), cross-efficiency scores obtained by the original DEA are generally not unique. Thus, with the use of those alternate optimal solutions to the DEA linear programs, it may be possible to improve a DMU’s (cross-efficiency) performance rating, while generally resulting in worsening that of others. Various secondary goals have been proposed for cross-efficiency calculation, such as those presented in Doyle and Green (1994). They developed aggressive (benevolent)

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model to identify optimal weights that not only maximize the efficiency of a certain DMU under evaluation, but also minimize (maximize) the average efficiency of other DMUs. The second is the losing association with the weights by averaging among the cross efficiencies (Despotis, 2002), which means that this method cannot clearly supply the weights to help decision maker to improve its performance. The third problem lies on the fact for one DMU to passively accept the weights designed for other DMUs. It implies the average cross-efficiency measure is not perfect since it is not a Pareto solution, and not all the DMUs have the incentive to accept the results. Because of the fact mentioned above, each DMU has the motivation to seek for the best efficiency score of its own which is higher than that given by the average cross-efficiency, which will surely result in the worsening of some other DMUs’ efficiency score. Such is a game process with all DMUs participating. The aim of this paper is to eliminate the limitations of the CCR model and the cross-efficiency method based on a Nash bargaining game. In the game, each DMU will be a player, and the bargaining solution can be obtained by using classical Nash bargaining game model. It’s shown that the bargaining efficiency score can be accepted as a fair evaluation measure as it is a Pareto solution and all DMUs will have the interest to admit it synchronously. The rest of this paper unfolds as follows. Section 2 introduces the CCR model and the cross-efficiency evaluation method. Section 3 presents the Nash bargaining game model. A more detailed presentation of this game process in the evaluation appears in the next section. Section 5 gives a numerical case. An illustrative example of R&D project selection is illustrated in Section 6, and finally concluding remarks are made in Section 7.

sional Euclidean space indexed by the set of individuals. A feasible set S is a subset of the payoff space, and a breakdown (or disagree* ment) point b is an element of the payoff space. A * bargaining problem can be then specified as the triple ðN; S; b Þ consisting of individuals, feasible set, and breakdown point. For the bargaining problem, Nash (1950) presented a solution characterized by four properties: Pareto efficiency (PE), invariance with respect to affine transformation (IAT), independence of irrelevant alternatives (IIA), and symmetry (SYM). Nash (1950) required that the feasible set is compact, convex, and contains some payoff vector such that each individual’s payoff is greater than the individual’s breakdown payoff. The solution is* a function F that is associated with each bargaining problem ðN; S; b Þ, * expressed as FðN; S; b Þ. Nash (1950) argued that a reasonable solution should satisfy the following four properties: 3.1. Pareto efficiency (PE) *

There exists no feasible payoff vector x such that xi > F i ðN; S; b Þ * for some individual i and xi P F i ðN; S; b Þ for all i. 3.2. Invariance with respect to affine transformation (IAT) If L is an affine transformation on RN (that is, there exist some numbers a1, . . . , an, b1, . . . , bn, where b1, . . . , bn > 0, such that . , an + bnun) for any u 2 RN), then one has L(u1, . . . , un)*= (a1 + b1u1, . . * FðN; LðSÞ; Lðb ÞÞ ¼ LðFðN; S; b ÞÞ. 3.3. Independence of irrelevant alternatives (IIA)

2. CCR model and cross-efficiency evaluation method

*

Adopting the conventional nomenclature of DEA, assume that there are n DMUs with m inputs and s outputs each to be evaluated. We denote the ith input and rth output for DMUj (j = 1, 2, . . . , n) as xij (i = 1, . . . , m) and yrj (r = 1, . . . , s), respectively. The efficiency rating for any given DMUd, can be computed using the following CCR model in the form of linear programming (LP):

Max s:t:

s X r¼1 m X

lr yrd ¼ hd xi xij 

i¼1

Xm

s X

lr yrj P 0; j ¼ 1; 2; . . . n

r¼1

ð1Þ

xx ¼1 i¼1 i id

xi P 0; i ¼ 1; 2; . . . :; m lr P 0; r ¼ 1; 2; . . . ; s For each DMUd (d = 1, . . . , n) under evaluation, we can obtain a set of optimal weights (multipliers) x1d ; . . . ; xmd ; l1d ; . . . ; lsd . Using this set, the d-cross-efficiency for any DMUj (j = 1, . . . , n), is then calculated as:

Ps

lrd yrj ; d; j ¼ 1; 2; . . . ; n  i¼1 xid xij

Edj ¼ Pr¼1 m

ð2Þ

For DMUj (j = 1, . . . , n), the average of all Edj (d = 1, . . . , n), namely n 1X Ej ¼ Edj ; n d¼1

ð3Þ

can be used as a new efficiency measure for DMUj, and will be referred to as the cross-efficiency score for DMUj. 3. Nash bargaining game model Denote the set of all individuals by N = {1, 2, . . . , n}, and a payoff vector is an element of the payoff space RN, which is the n-dimen-

If there exists another bargaining problem ðN; S0 ; b Þ such that S0 * * 0 0 is a subset of S and FðN; S; b Þ belongs to S , then FðN; S ; b Þ ¼ * FðN; S; b Þ. 3.4. Symmetry (SYM) *

then for any two individuals i, l, one has If ðS;*b Þ is symmetric, * F i ðN; S; b Þ ¼ F l ðN; S; b Þ. Since these properties are well known and discussed extensively in the literature, we will not provide any explanations here. For the traditional bargaining problem, Nash (1950) has shown that there exists a unique solution that satisfies the above four properties, called the Nash solution, and the solution can be obtained by solving the following maximization problem

Max*

*

*

u 2S;u P b

n Y

ðui  bi Þ

ð4Þ

i¼1

*

where u is the payment vector*of individuals, and ui is the ith ele* ment of u , bi is an element of b . 4. Bargaining model in the process of evaluation In this section, we will generalize the above results to our model. Suppose there are n DMUs with each using m inputs to generate s outputs. Note the cross-efficiency matrix by E, and an arbitrary element Edj in E shows the cross-efficiency of DMUj utilizing the optimal weighting scheme of DMUd. Then for each column, the P average of all row elements in the matrix, namely Ej ¼ 1n nd¼1 Edj ; j ¼ 1; . . . ; n: can be used as ultimate cross-efficiency measure. ; j ¼ 1; . . . ; n:, then we can use Nash Denote CCR efficiency as ECCR j bargaining game model (5) to get the bargaining solution, which will be surely between the CCR score and the cross-efficiency score, and the programming is shown as follows:

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J. Wu et al. / Expert Systems with Applications 36 (2009) 4357–4362

Ps

Yn

max

ðu1 ;...;us ;v1 ;...vm Þ

j¼1

ECCR  Pr¼1 m j

ur yrj

! Ps

Pr¼1 m

i¼1 vi xij

ur yrj

i¼1 vi xij

!

s:t:

Ecross 6 j ur P 0;

ð5Þ j ¼ 1    n; j–l

r ¼ 1; . . . s; vi P 0; i ¼ 1; . . . ; m: s

the corresponding CCR efficiencies, and Pr¼1 m

ur yrj

vx i¼1 i ij

Theorem. Denote S is all the restrictions in model (4), i.e. S is the feasible regions of (u1, . . . , us, v1, . . . , vm), then S is a convex set. Ps Pm ur yrj CCR P Pr¼1 and Proof. For m i¼1 vi xij > 0, the restrictions Ej Ps vx i¼1 Ps Pm i ij ur yrj cross CCR r¼1 6 Pm are equivalent to  i¼1 vi xij 6 0 Ej r¼1 ur yrj  Ej vx Ps i¼1 i ij cross Pm  i¼1 vi xij P 0 respectively. and r¼1 ur yrj  Ej Assuming both ðu01 ; . . . ; u0s ; v01 ; . . . ; v0m Þ and ðu001 ; . . . ; u00s ; v001 ; . . . ; v00m Þ 2 S, for any k 2 [0, 1], we have: s m X X ½ku0r þ ð1  kÞu00r yrj  ECCR  ½kv0i þ ð1  kÞv00i xij j r¼1

i¼1

u0r yrj  ECCR  j

m X

r¼1 s X

Ej ðj–l; l 2 ES; j ¼ 1    nÞ;

u00r yrj  ECCR  j

m X

r¼1

60

i¼1

and s m X X ½ku0r þ ð1  kÞu00r yrj  Ecross  ½kv0i þ ð1  kÞv00i xij j r¼1

¼k

u0r yrj  Ecross  j

r¼1

þ ð1  kÞ

m X

5.1. Comparisons of efficiencies We consider a simple numerical example given in Table 1 involving five DMUs, with three inputs X1, X2, X3 and two outputs Y1, Y2.

Table 1 Numerical example 1

v0i xij

s X

u00r yrj  Ecross  j

r¼1

m X

! v00i xij

P 0;

i¼1

X1

X2

X3

Y1

Y2

7 5 4 5 6

7 9 6 9 8

7 7 5 8 5

4 7 5 6 3

4 7 7 2 6

Table 2 CCR efficiency, cross-efficiency and bargaining efficiency

!

i¼1

ð7Þ

To illustrate the proposed method above, we consider two numerical examples with the data presented in Tables 1 and 4. We will compare the efficiency and weights among bargaining model, CCR model and cross efficiency evaluation method.

i¼1 s X

or ECCR ðl 2 ESÞ l

5. Numerical example

DMU1 DMU2 DMU3 DMU4 DMU5

! v00i xij

Ecross l

The advantages of Nash bargaining efficiency lie on the satisfaction of properties described above, so the evaluation approach based on Nash bargaining game model can evaluate and rank all DMUs justly, and the Nash bargaining efficiency is a Pareto efficiency, so all DMUs have motivation to accept the results. h

!

v0i xij

i¼1

þ ð1  kÞ

ð6Þ

is the efficiency of

DMUj which is obtained after bargaining. Because cross-efficiency of DMUl ðl 2 ESÞ is equal to CCR efficiency, they will not participate in the bargaining. The objective of model (5) is to maximize the Nash product of DMUs participating in the bargaining process.

s X

ðj–l; l 2 ESÞ

 i¼1 vi xij

where ES is the set of DMUs that their cross P efficiencies are equal to

¼k

ur yrj

The ultimate efficiencies of all DMUs are:

ur yrj Pr¼1 ; m vx Psi¼1 i ij ur yrj Pr¼1 m i¼1 vi xij

P

Ej ¼ Pr¼1 m

 Ecross j

j–l; l 2 ES Ps

ECCR j

Ps

DMU1 DMU2 DMU3 DMU4 DMU5

DEA efficiency

Nash bargaining efficiency

DEA cross-efficiency Arbitrary

Aggressive

Benevolent

0.6857 1.0000 1.0000 0.8571 0.8571

0.5352 0.8793 1.0000 0.5554 0.6222

0.4743 0.8793 0.9856 0.5554 0.5587

0.4473 0.8629 0.9571 0.54 0.4971

0.5845 0.9295 1 0.71 0.6386

where j – l, l 2 ES. and ku0r þ ð1  kÞu00r P 0; kv0i þ ð1  kÞv00i P 0; r ¼ 1; . . . ; s; i ¼ 1; . . . ; m:i.e.ðku01 þ ð1  kÞu001 ;...;ku0s þ ð1  kÞu00s ;kv01 þ ð1  kÞv001 ;...;kv0m þ ð1  kÞ v00m Þ 2 S. So S is a convex set. h Ps   ur yrj satisfying the Lemma. There exists only one solution Pr¼1 m  v xij

i¼1 i

j–l;l2ES

; ECCR Þj–l;l2ES i. properties above in the Nash bargaining game hS; ðEcross j j Proof. According to Nash bargaining theorem, if S is a convex set, ; ECCR Þj–l;l2ES i there exists only then in the bargaining game hS; ðEcross j j Ps   ur yrj satisfying the properties. Note that one solution Pr¼1 m  v xij

i¼1 i

Table 3 Weights of CCR model

u1 u2 v1 v2 v3

DMU2

DMU3

DMU4

DMU5

0.1714 0.0000 0.0000 0.1429 0.0000

0.1146 0.0282 0.1521 0.0140 0.0162

0.0011 0.1420 0.1355 0.0328 0.0522

0.1429 0.0000 0.1504 0.0275 0.0000

0.0000 0.1429 0.0000 0.0000 0.2000

Table 4 Data of input and outputs in Anderson et al. (2002)

j–l;l2ES

the common set of weights ðu1 ; . . . ; us ; v1 ; . . . ; vm Þ in model (5) is not unique. In our case, if ðu1 ; . . . ; us ; v1 ; . . . ; vm Þ is optimal, so is     ðku1 ; . . . ; kus ; kv1 ; . . . ; kvm Þ; k > 0. Denoting the optimal solution of model (5) as ðu1 ; . . . ; us ; v1 ; . . . ; vm Þ, using this set, Nash bargaining efficiency for any DMUj (j = 1, . . . , n), is then calculated as:

DMU1

DMU1 DMU2 DMU3 DMU4 DMU5 DMU6

X1

Y1

Y2

1 1 1 1 1 1

10.7 11.6 2.8 10.5 10.1 10.2

12.0 2.5 12.8 11.6 11.8 11.5

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Table 5 Fixed weights of DMUs to calculate DEA cross-efficiency

u1 u2 v1

DMU1

DMU2

DMU3

DMU4

DMU5

DMU6

0.0015 0.0153 0.2

0.0172 0.0000 0.2

0.0000 0.0156 0.2

0.0169 0.0016 0.2

0.0015 0.0153 0.2

0.0015 0.0153 0.2

In the algorithm, we use the regular cross-efficiency as the starting point for bargaining. Cross-efficiency is not unique and can be calculated by imposing a secondary goal. For example, we can use an aggressive strategy which not only obtains the maximum DEA efficiency for a DMU as the primary goal, but also as a secondary goal, minimizes the other DMUs’ cross efficiencies (Sexton et al., 1986). We can also use a benevolent strategy which not only obtains the maximum DEA efficiency but also maximizes the other DMUs’ cross efficiencies (Doyle & Green (1994)). The crossefficiency calculated without imposing the secondary goal is referred to as an arbitrary strategy. The results of the cross-efficiency under three strategies are reported in the last three columns of Table 2. As discussed above, we make all DMUs bargain between CCR efficiency and cross-efficiency. There is no DMU with the same cross-efficiency and CCR efficiency, so ES = U, i.e. all DMUs participate in the bargaining process, and Nash bargaining efficiency is shown in the third column. In fact, Nash bargaining efficiency and cross-efficiency of DMU2 and DMU4 are not the same, because the objective value of model (5) is a small number but not equal to 0. In the use of traditional DEA models, The DMUs under evaluation

heavily weigh some favorable measures and completely ignore other inputs and outputs in order to maximize its own DEA efficiency. So clearly, Nash bargaining efficiency lies between the CCR efficiency and the cross-efficiency, which can also be seen from the Table 2, that is, the Nash bargaining efficiencies of all the DMU are larger than the DEA cross efficiencies but smaller than the CCR efficiencies. 5.2. Comparisons of weights Based on the data in Table 1, after solving model (5), we can get the common weights are:

ðU  ; V  Þ ¼ ð0:3043931; 0:1249275; 0:01664795; 0:1212250; 0:3205015ÞT : and the weights to calculate CCR efficiency are shown in Table 3. To maximize its own CCR efficiency score, the DMU under evaluation will choose the best weights for its own, which usually implies the highlight of some measures and total ignoring of others, and usually results in the generation of some unrealistic weights. For example, the self-appraisal of DMU1considers weights u1 and v2 only, and other weights of input and output are all zero. Weights of cross-efficiency originate from the weights of CCR model. Since the solution of CCR model is not unique, the weights to determine cross efficiency are not unique either. On the other hand, cross-efficiency of a DMU is obtained by computing that DMU’s set of n scores (using the n sets of optimal weights), and then averaging those scores, which means that this method cannot clearly supply the weights to help decision maker improve his performance.

Table 6 Raw data of 37 R&D projects on five outputs and their cost (input) R&D project

Indirect economic contribution

Direct economic contribution

Technical contribution

Social contribution

Scientific contribution

Budget

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

67.53 58.94 22.27 47.32 48.96 58.88 50.10 47.46 55.26 52.40 55.13 32.09 27.49 77.17 72.00 39.74 38.50 41.23 53.02 19.91 50.96 53.36 61.60 52.56 31.22 54.64 50.40 30.76 48.97 59.68 48.28 39.78 24.93 22.32 48.83 61.45 57.78

70.82 62.86 9.68 47.05 48.48 77.16 58.20 49.54 61.09 55.09 55.54 34.04 39.00 83.35 68.32 34.54 28.65 47.18 51.34 18.98 53.56 46.47 66.59 55.11 29.84 58.05 53.58 32.45 54.97 63.78 55.58 51.69 29.72 33.12 53.41 70.22 72.10

62.64 57.47 6.73 21.75 34.90 35.42 36.12 46.89 38.93 53.45 55.13 33.57 34.51 60.01 25.84 38.01 51.18 40.01 42.48 25.49 55.47 49.72 64.54 57.58 33.08 60.03 53.06 36.63 51.52 54.80 53.30 35.10 28.72 18.94 40.82 58.26 43.83

44.91 42.84 10.99 20.82 32.73 29.11 32.46 24.54 47.71 19.52 23.36 10.60 21.25 41.37 36.64 15.79 59.59 10.18 17.42 8.66 30.23 36.53 39.10 39.69 13.27 31.16 26.68 25.45 23.02 15.94 7.61 5.30 8.38 4.03 10.45 19.53 16.14

46.28 45.64 5.92 19.64 26.21 26.08 18.90 36.35 29.47 46.57 46.31 29.36 25.74 51.91 25.84 33.06 48.82 38.86 46.30 27.04 54.72 50.44 51.12 56.49 36.75 46.71 48.85 34.79 45.75 44.04 36.74 29.57 23.45 9.58 33.72 49.33 31.32

84.20 90.00 50.20 67.50 75.40 90.00 87.40 88.80 95.90 77.50 76.50 47.50 58.50 95.00 83.80 35.40 32.10 46.70 78.60 54.10 74.40 82.10 75.60 92.30 68.50 69.30 57.10 80.00 72.00 82.90 44.60 54.50 52.70 28.00 36.00 64.10 66.40

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While Anderson et al. (2002) proved, in the single input, multipleoutput case, cross-evaluation implicitly uses a single fixed set of weights. We use the example in Anderson et al. (2002) (shown in Table 4) to compare the weights after bargaining with this fixed set of weights. DEA cross efficiencies shown in Anderson et al. (2002) are:

DMU1 ð0:9767Þ; DMU2 ð0:5063Þ; DMU3 ð0:7634Þ; DMU4 ð0:9492Þ; DMU5 ð0:9467Þ; DMU6 ð0:9342Þ: and the fixed weights of DMUs are shown in Table 5. From Table 5, we can find there still exist unrealistic weights, like the output weight u2 to evaluate DMU2 and the output weight u1 to evaluate DMU3, which means that some index was completely ignored in the process of evaluation. Via model (5) we can get the Nash bargaining efficiency of every DMU as follows:

6. R&D projects selection Green et al. (1996) illustrated their method using an example of 37 project proposals relating to the Turkish iron and steel industry by developing a DEA-based method. Each project is characterized by five output measures (direct economic contribution, indirect economic contribution, technological contribution, scientific contribution and social contribution) relative to its consumption of resources expressed in monetary terms. Table 6 reports the data. Tables 7 and 8 show CCR efficiency and corresponding weights, cross efficiency and corresponding weights respectively. Based upon the data shown in Table 6, after solving model (5), we can get the common weights are:

ðU  ; V  Þ ¼ ð0:2890024; 1:199905; 0:1439221; 0:2455516; 0:5200033; 2:893731ÞT

From the numerical cases of Tables 1 and 4, we can see all the inputs and outputs are considered when the Nash bargaining game model is used to evaluate the efficiency of any DMU, and all DMUs are evaluated with common weights which can reflect the justice of evaluation. The results satisfy the Pareto Efficiency property, so each DMU has motivation to accept it.

Table 9 shows the results along with Green et al. (1996) DEA crossefficiency scores reported in column 3.There are some ranking differences between the two approaches. For example, DMU10 is ranked at No. 21 by the Nash bargaining efficiency, whereas it is No. 22 by the DEA cross-efficiency. Based upon the Green et al. (1996) project selection rule, which chooses projects by decreasing values of DEA cross-efficiency scores, until the budget for the program is exhausted (the budget cannot exceed 1000), the same projects are selected, with no exception. But if the budget is changed, then the projects selected are different based on two kinds of efficiency, for example, if the budget is 1320, then the same 20 projects are selected, with two

Table 7 CCR efficiency and corresponding weights

Table 8 Cross-efficiency and corresponding weights

DMU1 ð1:0000Þ;

DMU2 ð0:5383Þ; DMU5 ð0:9685Þ;

DMU4 ð0:9722Þ;

DMU3 ð0:7634Þ; DMU6 ð0:9564Þ:

and the common weights are:

ðU  ; V  Þ ¼ ð0:0352399; 0:0519841; 1:000876ÞT

Project No.

v1

u1

u2

u3

u4

u5

CCR efficiency

Project No.

v1

u1

u2

u3

u4

u5

Crossefficiency

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

0.0119 0.0111 0.0199 0.0148 0.0133 0.0111 0.0114 0.0113 0.0104 0.0129 0.0131 0.0211 0.0171 0.0105 0.0119 0.0282 0.0312 0.0214 0.0127 0.0185 0.0134 0.0122 0.0132 0.0108 0.0146 0.0144 0.0175 0.0125 0.0139 0.0121 0.0224 0.0183 0.0190 0.0357 0.0278 0.0156 0.0151

0.0000 0.0000 0.0144 0.0107 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0086 0.0176 0.0020 0.0000 0.0079 0.0000 0.0000 0.0076 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0004 0.0000 0.0000

0.0075 0.0070 0.0000 0.0000 0.0083 0.0070 0.0072 0.0038 0.0065 0.0044 0.0044 0.0072 0.0107 0.0066 0.0000 0.0000 0.0012 0.0088 0.0000 0.0076 0.0055 0.0000 0.0083 0.0045 0.0060 0.0049 0.0072 0.0051 0.0057 0.0041 0.0076 0.0124 0.0065 0.0241 0.0152 0.0064 0.0095

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0049 0.0000 0.0056 0.0057 0.0092 0.0000 0.0000 0.0000 0.0000 0.0065 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0063 0.0000 0.0000 0.0000 0.0053 0.0098 0.0000 0.0083 0.0000 0.0007 0.0000 0.0000

0.0028 0.0026 0.0014 0.0011 0.0031 0.0026 0.0027 0.0000 0.0025 0.0000 0.0000 0.0000 0.0040 0.0025 0.0009 0.0000 0.0059 0.0000 0.0000 0.0000 0.0000 0.0000 0.0031 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0042 0.0000 0.0036

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0047 0.0042 0.0089 0.0021 0.0077 0.0056 0.0020 0.0000 0.0045 0.0061 0.0000 0.0073 0.0052 0.0058 0.0000 0.0000 0.0000 0.0000 0.0000 0.0028 0.0065 0.0000

0.6543 0.5512 0.3360 0.5283 0.5064 0.6148 0.5060 0.4204 0.5177 0.5431 0.5618 0.5525 0.5045 0.6539 0.6518 0.8542 1.0000 0.7618 0.5179 0.3523 0.6022 0.5068 0.6754 0.5003 0.4024 0.6633 0.7420 0.3478 0.5784 0.5505 0.9459 0.6393 0.4299 0.7973 1.0000 0.7708 0.7391

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

0.0292 0.0291 0.0532 0.0320 0.0304 0.0324 0.0322 0.0281 0.0323 0.0283 0.0283 0.0286 0.0298 0.0294 0.0338 0.0301 0.0276 0.0293 0.0294 0.0317 0.0292 0.0294 0.0282 0.0287 0.0309 0.0288 0.0285 0.0287 0.0283 0.0280 0.0302 0.0291 0.0293 0.0318 0.0274 0.0278 0.0299

0.0000 0.0000 0.0195 0.0195 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0195 0.0168 0.0017 0.0000 0.0168 0.0000 0.0000 0.0168 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0004 0.0000 0.0000

0.0170 0.0170 0.0000 0.0000 0.0170 0.0170 0.0170 0.0092 0.0170 0.0092 0.0092 0.0092 0.0170 0.0170 0.0000 0.0000 0.0010 0.0111 0.0000 0.0111 0.0111 0.0000 0.0170 0.0111 0.0111 0.0092 0.0111 0.0111 0.0111 0.0092 0.0092 0.0182 0.0092 0.0182 0.0148 0.0111 0.0170

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0118 0.0000 0.0118 0.0118 0.0118 0.0000 0.0000 0.0000 0.0000 0.0056 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0118 0.0000 0.0000 0.0000 0.0118 0.0118 0.0000 0.0118 0.0000 0.0006 0.0000 0.0000

0.0064 0.0064 0.0020 0.0020 0.0064 0.0064 0.0064 0.0000 0.0064 0.0000 0.0000 0.0000 0.0064 0.0064 0.0020 0.0000 0.0051 0.0000 0.0000 0.0000 0.0000 0.0000 0.0064 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0041 0.0000 0.0064

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0045 0.0037 0.0112 0.0045 0.0112 0.0112 0.0045 0.0000 0.0112 0.0112 0.0000 0.0112 0.0112 0.0112 0.0000 0.0000 0.0000 0.0000 0.0000 0.0027 0.0112 0.0000

0.6055 0.5124 0.1707 0.4457 0.4495 0.5121 0.4243 0.4047 0.4334 0.5184 0.5373 0.5222 0.4573 0.6015 0.5209 0.7672 0.9779 0.7020 0.4766 0.3003 0.5571 0.4667 0.6477 0.4704 0.3519 0.6234 0.7034 0.3271 0.5519 0.5304 0.8468 0.5945 0.3970 0.6774 0.9881 0.7487 0.6690

4362

J. Wu et al. / Expert Systems with Applications 36 (2009) 4357–4362

Table 9 A comparison of results to Doyle and Green (1994)

7. Conclusions

Project no.

Bargaining efficiency

Doyle et al.’s score

Doyle et al.’s selection

Our selection

Budget

35 17 31 16 36 18 27 34 37 23 26 14 1 32 21 29 11 30 15 12 10 6 2 19 24 22 13 5 4 9 7 8 33 25 28 20 3

1 1 0.8468 0.7758 0.7593 0.7177 0.7169 0.6774 0.6754 0.6545 0.6284 0.6115 0.6099 0.604 0.5706 0.5614 0.5436 0.5356 0.5317 0.5298 0.5259 0.5199 0.5183 0.4898 0.482 0.4779 0.4626 0.4538 0.4535 0.4393 0.4243 0.408 0.4016 0.363 0.3345 0.3091 0.1707

1 0.975 0.866 0.78 0.759 0.715 0.712 0.699 0.684 0.655 0.632 0.611 0.614 0.606 0.565 0.559 0.544 0.538 0.537 0.53 0.525 0.528 0.519 0.484 0.476 0.472 0.466 0.457 0.457 0.444 0.436 0.409 0.404 0.359 0.331 0.307 0.259

Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No No No Yes No No No No No No No No No No No No No No No No No

Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No No No Yes No No No No No No No No No No No No No No No No No

36 32.1 44.6 35.4 64.1 46.7 57.1 28 66.4 75.6 69.3 95 84.2 54.5 74.4 72 76.5 82.9 83.8 47.5 77.5 90

982.9

982.9

Budget sum

exceptions, namely projects No. 6 and No. 10. and the budget sum is 1303.6 and 1316.1, respectively.

In this paper we established a bargaining game model to eliminate the limitations of the CCR model and cross-efficiency evaluation model to some extent, and the bargaining efficiency scores obtained from this model satisfy the properties in Section 3, and each DMU has incentive to accept the results because the bargaining efficiency is a Pareto solution. Finally the proposed model is verified with a numerical case and the R&D projects selection problem discussed in Green et al. (1996), and the results are compared with the previous studies. Acknowledgements The research was supported by a grant from Chinese National Science Fund for Distinguished Young Scholars (70525001) and special fund for graduates of Chinese Academy of Sciences for science and social work (innovation groups). References Anderson, T. R., Hollingsworth, K. B., & Inman, L. B. (2002). The fixed weighting nature of a cross-evaluation model. Journal of Productivity Analysis, 18, 249–255. Charnes, A., Cooper, W. W., Lewin, A. Y., & Seiford, L. M. (Eds.). (1994). Data envelopment analysis: Theory, methodology, and applications. Boston: Kluwer. Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2, 429–444. Despotis, D. K. (2002). Improving the discriminating power of DEA: Focus on globally efficient units. Journal of the Operational Research Society, 53, 314–323. Doyle, J., & Green, R. (1994). Efficiency and cross-efficiency in DEA: Derivations, meanings and the uses. Journal of the Operational Research Society, 45, 567–578. Dyson, R. G., & Thannassoulis, E. (1988). Reducing weight flexibility in data envelopment analysis. Journal of Operational Research Society, 39, 563–576. Green, R., Doyle, J., & Cook, W. D. (1996). Preference voting and project ranking using DEA and cross-evaluation. European Journal of Operational Research, 90, 461–472. Nash, J. F. (1950). The bargaining problem. Economica, 5, 155–162. Oral, M., Kettani, O., & Lang, P. (1991). A methodology for collective evaluation and selection of industrial R&D projects. Management Science, 37, 871–885. Sexton, T. R., Silkman, R. H., & Hogan, A. J. (1986). Data envelopment analysis: Critique and extensions. In R. H. Silkman (Ed.). Measuring efficiency: An assessment of data envelopment analysis (vol. 32, pp. 73–105). San Francisco: Jossey-Bass. Wong, Y. H., & Beasley, J. E. (1990). Restricting weight flexibility in data envelopment analysis. Journal of the Operational Research Society, 41, 829–835. Zilla, S. S., & Lea, F. (2002). Review of ranking methods in the data envelopment analysis context. European Journal of Operational Research, 140, 249–256.