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Selecting the most preferable alternatives in a group decision making problem using DEA
Expert Systems with Applications 36 (2009) 9599–9602
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Selecting the most preferable alternatives in a group decision making problem using DEA M. Zerafat Angiz L. a,c, A. Emrouznejad b,*, A. Mustafa c, A. Rashidi Komijan a a
Department of Mathematics, Islamic, Azad University, Firooz-Kooh, Iran Aston Business School, Aston University, Birmingham B4 7ET, UK c School of Mathematical Sciences, Universiti Sains Malaysia, Penang Malaysia b
a r t i c l e
i n f o
Keywords: Group decision making Preferential voting system Data envelopment analysis Most preferable alternative
a b s t r a c t Group decision making is the study of identifying and selecting alternatives based on the values and preferences of the decision maker. Making a decision implies that there are several alternative choices to be considered. This paper uses the concept of Data Envelopment Analysis to introduce a new mathematical method for selecting the best alternative in a group decision making environment. The introduced model is a multi-objective function which is converted into a multi-objective linear programming model from which the optimal solution is obtained. A numerical example shows how the new model can be applied to rank the alternatives or to choose a subset of the most promising alternatives. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Social functions can be categorized into social choice and welfare functions. Each method for vote counting is assumed as a social function but if Arrow’s possibility theorem is used for a social function, social welfare function is achieved. Some specifications of the social functions are decisiveness, neutrality, anonymity, monotonocity, unanimity, homogeneity and weak and strong Paretooptimality. No social choice function meets these requirements in an ordinal scale simultaneously. Some of the social choice functions are minimum deviations measure (Goddard, 1983), Condercet’s, Borda’s, Nanson’s, Dodgson’s function (Hwang & Lin, 1987), Kemeny’s function (Asgharpour, 2004), priority ranking (Cook & Seiford, 1978), Eigenvector (Goddard, 1983) and Fishburn’s function (Asgharpour, 2004). On the other hand, social welfare functions face similar problems in defining a function that fully meets fairness measures. In other words, no social welfare function meets two Arrow’s theorems and five conditions in an ordinal scale simultaneously (Asgharpour, 2004; Hwang & Lin, 1987). However, Rothenberg (1961) and Fishburn (1973) proved that there is a paradox in the conditions of Arrow’s possibility theorem and called it Arrow’s impossibility theorem. As a result, it is essential to revise fairness measures and social welfare functions. The revision may be based on the logical conception of decision makers thinking process in voting alternatives and their expectations from aggregating votes. * Corresponding author. E-mail address: [email protected] (A. Emrouznejad). 0957-4174/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.07.011
Some of the social welfare functions in ordinal scale are Bowman and Clantoni’s approach (Bowman & Clantoni, 1973), Black’s single-peaked preferences and Goodman and Markowitz’s approach (Goodman & Markowitz, 1952). In recent years, researchers have used Data Envelopment Analysis technique (Charnes, Cooper, & Rhodes, 1978; Emrouznejad, Parker, & Tavares, 2008) in ranking alternatives (Adler, Friedman, & Sinvany- Stern, 2002; Cook & Kress, 1990; Cook, Doyle, Green, & Kress, 1988). Adler et al. (2002) considered different ranking methods using DEA technique and divided them into the six categories that overlap with each other. It is worth noting that preferential voting system differs from DEA in structure. In a preferential voting system, the known votes of decision makers are used as inputs of the model and the output is an aggregate ranking that is unknown. In the common DEA models, the objective is to determine technical efficiency. Cook and Kress (1990) specifically considered the use of DEA in aggregating preferential votes. Some authors have recently attempted to use fuzzy multi-attribute (Chang, Cheng, & Chen, 2007; Deng-Feng, 2007) and fuzzy clustering methodology with ordered weighted averaging operator (Chakraborty & Chakraborty, 2007; Emrouznejad, 2008) and integrated multi-objective modeling with fuzzy multi-attributive group (Ölçer, Tuzcu, & Tura, 2006;Emrouznejad, 2008) for similar problem in group decision making, however, the developed model uses optimization technique to find the best weights for selecting the most favorable alternative. The weights then can be used to define a social function that can fairly solve a voting problem. Other recent developments in group decision-making include a study of causal analytical method for group decision-making under fuzzy
9600
M. Zerafat Angiz L. et al. / Expert Systems with Applications 36 (2009) 9599–9602
environment (Lin & Wu, 2008), computing coordination based fuzzy group decision-making (Tsai & Wang, 2008), application of grey relation analysis with fuzzy group decision-making (Yu-Jie Wang, 2009) and a comparative study of several analytical methods for knowledge communities group-decision analysis (Chu, Shyu, Tzeng, & Khosla, 2007). As a result, ranking alternatives in a preferential voting system is done in a more realistic manner. The proposed model is presented in Section 2. Illustration with some numerical examples is given in Section 3. Finally, the concluding results are presented. 2. The most preferable alternative in a group decision making problem, a new mathematical model
In summary, the new objective functions are added in order to calculate weights of a rank in such a way that to be close to the majority of votes. Considering the last objective function and the P first n constraints of model (1) and also assuming nj¼1 wj vpj ¼ 1, a CCR model with one input (equal to one) and n outputs is achieved. In developing a CCR model, this conversion is proved. In order to convert model (1) to a linear one, the following process is done: Assume that w0jh ¼ wj wjh (j = 1, 2, . . . , n) and h 2 H. So, þ w jw0jh j ¼ wþ jh . Applying this substitution, we could conclude jh model (2) that may be solved using multi-objective methods:
Min
X
ah ðwþjh þ wjh Þ ðj ¼ 1; 2; . . . ; nÞ
h
The most important characteristic of a social function is identification of the interactive effect of alternatives and creating a logical relation with the ranks. Meeting decision makers’ expectations and achieving the optimal ranking are among other characteristics of social functions. Assume that there are m decision makers who are supposed to rank n alternatives using preferential voting system. The best and the worst alternatives are given ranks 1 and n, respectively. Each decision maker ranks the alternatives and also compares the ranks using pair-wise comparisons. As the preferential voting system is based on value function, decision makers should have sufficient information about alternatives and weights of the ranks so that they can definitely priories alternatives and ranks. Assume Vij is the number of votes that gives the rank j to the alternative i. In the preferential voting system, it is important to know the weights of the ranks as well as comparing alternatives. So, each decision maker is asked to express how much rank j is preferred to rank j + 1. This is done using pair-wise comparison and eigenvector method. Assume that wjk (k = 1, 2, . . ., m, j = 1, 2, . . ., n) is the weight of rank j in view of decision maker k. A bounded variable is determined for each rank. The upper and lower limits of this variable are the most and the least weights that decision makers assign to the rank. We propose the following mathematical model for decision maker p:
min
X
max s:t:
Max
wj vpj
j¼1 n X
wj vpj P
n X
j¼1
lj 6 wj 6 uj n X wj ¼ 1
wj vij
ði ¼ 1; 2; . . . ; nÞ
ð1Þ
wj vpj
j¼1
s:t:
n X
wj vpj P
j¼1
n X
wj vij
ði ¼ 1; 2; . . . ; nÞ
ð2Þ
j¼1
lj 6 wj 6 uj
ðj ¼ 1; 2; . . . ; nÞ
wj wjh ðwþjh wjh Þ ¼ 0 ðj ¼ 1; 2; . . . ; n & h in HÞ n X
wj ¼ 1
j¼1
3. Illustration with numerical examples This section illustrates the use of Model (2) for selecting the most preferable DMU in a set of 20 decision makers and 4 alternatives. Decision makers vote the importance of the ranks using pairwise comparison matrices. As the weight of the rank j is definitely greater than j + 1, the values of the rows and columns are sorted in ascending and descending order. Assume four decision makers express their comparisons according to the following matrix:
ah jwj wjh j ðj ¼ 1; 2; . . . ; nÞ
h2H n X
n X
x1 x2 x3 x4
x1
x2
x3
x4
1
3 1
5 4 1
7 5 3 1
1 3 1 5 1 7
1 4 1 5
1 2
j¼1
ðj ¼ 1; 2; ::; nÞ
j¼1
where wjh is the weight of rank j due to at least 50% of decision makers. For example, if there are 100 (people) decision makers and 30, 15 and 10 of them believe that the weight of rank one is 0.6, 0.5 and 0.65 and the rest give different weights, then w11, w12 and w13 will be 0.6, 0.5 and 0.65, respectively. This means that H = {1, 2, 3}. These weights are calculated using eigenvector method. lj and uj are upper and lower limits of the weights of the ranks. The last constraint expresses the fact that weights summation in a pair-wise comparison is equal to one. The coefficient of ah ensures that the votes of the most decision makers are applied in the model more precisely. In the above-mentioned example, 30 decision makers (among 55 = 30 + 15 + 10 ones) assign the weight 0.6 to the rank one. So, 0.6 is more probably consider than 0.5 that is assigned by only 15 decision makers. ah for a specific weight is calculated by dividing the number of decision makers who assigned that weight to the number of majority of decision makers. For the specific weight 0.6, ah ¼ 30=55 ¼ 0:55 and for 0.5, ah ¼ 15=55 ¼ 0:27.
Using eigenvector method, the weights vector is calculated as w = (0.57, 0.27, 0.10, 0.06). It means that four decision makers believe that the weights of the ranks one to four are 0.57, 0.27, 0.1 and 0.6, respectively. Assume that three decision makers express their comparisons according to the following matrix:
x1 x2 x3 x4
x1
x2
x3
x4
1
2 1
4 3 1
7 5 2 1
1 2 1 4 1 7
1 3 1 5
1 2
According to these three decision makers and using eigenvector method the weights vector is w = (0.57, 0.27, 0.10, 0.06). Assume also that the following matrix is associated with five decision makers and the resulted weights vector is w = (0.52, 0.29, 0.11, 0.07).
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x1 x2 x3 x4
x1
x2
x3
x4
1
2 1
5 3 1
6 4 2 1
1 2 1 5 1 6
1 3 1 4
1 2
Table 1 Ranking alternatives by 20 decision makers
The weights vectors in view of the other decision makers have also been calculated and summary of the weights are listed in the following table. Number of decision makers
Weight 1
Weight 2
Weight 3
Weight 4
Five decision makers Four decision makers Three decision makers Two decision makers Two decision makers Two decision makers One decision makers One decision makers
0.52 0.57 0.57 0.51 0.49 0.54 0.55 0.52
0.29 0.27 0.27 0.34 0.31 0.25 0.29 0.28
0.11 0.10 0.10 0.09 0.14 0.13 0.11 0.12
0.07 0.06 0.06 0.06 0.05 0.08 0.05 0.08
Max weight Min weight
0.57 0.49
0.34 0.25
0.14 0.09
0.08 0.05
Decision maker
Rank
1
1
2
3
4
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
3 3 2 4 4 2 1 1 3 4 1 1 1 4 4 4 3 2 2
1 2 1 2 2 3 4 3 2 3 2 3 4 2 3 2 4 4 3
4 1 3 3 1 1 2 4 4 3 3 2 3 3 1 1 2 3 1
2 4 4 1 3 4 3 2 1 1 4 4 2 1 2 3 1 1 4
Table 2 The number of assigning each rank to each alternative
Using the resulted weights vectors, a weigh interval is determined for each rank. Assume that w1, w2, w3 and w4 are weight intervals for rank one to four. So:
w1 ¼ ½0:49; 0:57
Alternatives
Rank
1
1
2
3
4
1 2 3 4
5 5 4 6
2 7 7 4
6 4 6 4
7 4 3 6
w2 ¼ ½0:25; 0:34 w3 ¼ ½0:09; 0:14 This should be done for each alternative, assuming that
w4 ¼ ½0:05; 0:08 If the weights proposed by all decision makers are the same, a Borda function type is resulted which combines with different weight. The only difference is that we have weights resulted from pairwise comparisons instead of scores resulted from Borda technique. Table 1 shows the ranks assigned to alternatives by decision makers. The number of rank j assigned to alternative i is easily calculated (see Table 2): In order to select the best alternative, we could construct a multi objective model based on model (2) for each alternative. The resulted model for the first alternative is as follows:
wþ11 þ w11 ¼ jw1 0:57j wþ12 þ w12 ¼ jw1 0:50j wþ13 þ w13 ¼ jw1 0:52j wþ21 þ w21 ¼ jw2 0:27j wþ22 þ w22 ¼ jw2 0:31j wþ23 þ w23 ¼ jw2 0:29j wþ31 þ w31 ¼ jw3 0:10j wþ32 þ w32 ¼ jw3 0:12j wþ33 þ w33 ¼ jw3 0:11j wþ41 þ w41 ¼ jw4 0:06j wþ42 þ w42 ¼ jw4 0:07j
Min Z 1 ¼ 0:33jw1 0:57j þ 0:25jw1 0:50j þ 0:42jw1 0:52j Min Z 2 ¼ 0:33jw2 0:27j þ 0:25jw2 0:31j þ 0:42jw2 0:29j
With the above-mentioned substitutions, model (3) is converted to a multi-objective model (4):
Min Z 3 ¼ 0:33jw3 0:10j þ 0:25jw3 0:12j þ 0:42jw3 0:11j
Min Z 1 ¼ 0:33ðwþ11 þ w11 Þ þ 0:25ðwþ12 þ w12 Þ þ 0:42ðwþ13 þ w13 Þ
Min Z 4 ¼ 0:33jw4 0:06j þ 0:25jw4 0:07j þ 0:42jw4 0:07j
Min Z 2 ¼ 0:33ðwþ21 þ w21 Þ þ 0:25ðwþ22 þ w22 Þ þ 0:42ðwþ23 þ w23 Þ
Max Z 5 ¼ 5w1 þ 2w2 þ 6w3 þ 7w4
Min Z 3 ¼ 0:33ðwþ31 þ w31 Þ þ 0:25ðwþ32 þ w32 Þ þ 0:42ðwþ33 þ w33 Þ
s:t
Min Z 4 ¼ 0:33ðwþ41 þ w41 Þ þ 0:67ðwþ42 þ w42 Þ
5w1 þ 2w2 þ 6w3 þ 7w4 P 5w1 þ 7w2 þ 4w3 þ 4w4 5w1 þ 2w2 þ 6w3 þ 7w4 P 4w1 þ 7w2 þ 6w3 þ 3w4
Max Z 5 ¼ 5w1 þ 2w2 þ 6w3 þ 7w4
5w1 þ 2w2 þ 6w3 þ 7w4 P 6w1 þ 4w2 þ 4w3 þ 6w4
s:t:
5w1 þ 2w2 þ 6w3 þ 7w4 P 5w1 þ 7w2 þ 4w3 þ 4w4
w1 þ w2 þ w3 þ w4 ¼ 1
5w1 þ 2w2 þ 6w3 þ 7w4 P 4w1 þ 7w2 þ 6w3 þ 3w4
0:49 6 w1 6 0:57
5w1 þ 2w2 þ 6w3 þ 7w4 P 6w1 þ 4w2 þ 4w3 þ 6w4
0:25 6 w2 6 0:34
w1 þ w2 þ w3 þ w4 ¼ 1
0:09 6 w3 6 0:14 0:05 6 w4 6 0:08
0:49 6 w1 6 0:57 ð3Þ
0:25 6 w2 6 0:34
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M. Zerafat Angiz L. et al. / Expert Systems with Applications 36 (2009) 9599–9602
0:09 6 w3 6 0:14
4. Conclusion remarks
0:05 6 w4 6 0:08 wþ11 w11 ¼ w1 0:57 wþ12 w12 ¼ w1 0:27 wþ13 w13 ¼ w1 0:52 wþ21 w21 ¼ w2 0:27 wþ22 w22 ¼ w2 0:31 wþ23 w23 ¼ w2 0:29 wþ31 w31 ¼ w3 0:10 wþ32 w32 ¼ w3 0:12 wþ33 w33 ¼ w3 0:11 wþ41 w41 ¼ w4 0:06 wþ42 w42 ¼ w4 0:07 The same procedure should be done for other alternatives. In order to obtain full ranking model (2) should be solved for each alternative. If a model has feasible solution, the associated alternative will be placed in Group one. The Group one alternatives are put away and the rest of alternatives are assigned lower ranks. In other words, model (2) is developed for each alternative that has not been placed in Group one. This process is repeated until all alternatives are placed in a Group. At least one of the mathematical models that are developed in each stage should have a feasible solution. It worth noting that alternatives in lower Groups can never be assigned the ranks of upper Groups alternatives. For example, consider an alternative in Group two. If this alternative is compared with others using any arbitrary weights vector, it can never be placed in Group one. The result of the first stage of ranking alternatives in a preferential voting system is summarized in Table 3. As the only feasible model is associated with alternative 2, the first Group only includes this alternative. After omitting this alternative, the procedure is repeated for the rest of alternatives (Table 4). Group 2 includes alternatives 3 and 4. As the optimal solution of the model associated with alternative 4 is more desirable (due to the minimum objective function), rank 2 is assigned to alternative 4. Finally, rank 4 is assigned to alternative 1.
Table 3 Result of solving the first stage model Alternatives
Optimal solution of model (4)
1 2 3 4
Infeasible 0.792 Infeasible Infeasible
Table 4 Result of solving the second stage model Alternatives
Optimal solution of model (4)
1 3 4
Infeasible 0.8361 0.1188
Preferential voting system is a type of multi-criteria decision making problem in which decision makers rank alternatives. Then, their votes are collected and considered. There are different methods to solve such a problem, e.g. social functions based on weighting ranks. Borda and Eigenvector functions are well known social functions. Some of the methods such as Cook and Kress (1990) approach consider the weights of the ranks in a different way. It is important that a social function to be fair in different approaches. One approach that leads to a more realistic solution is to determine the importance of ranks by decision makers. In this paper, a new mathematical model is introduced that could determine the importance of ranks by decision makers in order to reach a more realistic solution. Although large scale mathematical models should be solved in a large scale decision making problem, new technologies and PCs may justify using this method. It is worth noting that increasing the number of decision makers does not affect the dimension of the problem but the number of alternatives may cause more constraints and variables. Further research is needed to simplify the new multi-objective program to a linear programming. References Adler, N., Friedman, L., & Sinvany- Stern, Z. (2002). Review of ranking methods in the data envelopment analysis context. European Journal of Operational Research, 140, 249–265. Asgharpour, M. J. (2004) Group decision making and game theory. Tehran University Publication. Bowman, V. J., & Clantoni, C. S. (1973). Majority rule under transitivity constraints. Management Science, 19(9), 1029–1041. Chakraborty, C., & Chakraborty, D. (2007). A fuzzy clustering methodology for linguistic opinions in group decision making. Applied Soft Computing, 7(3), 858–869. Chang, J. R., Cheng, C. H., & Chen, L. S. (2007). A fuzzy-based military officer performance appraisal system. Applied Soft Computing, 7(3), 936–945. Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring efficiency of decision making units. European Journal of Operational Research, 2, 429–444. Chu, Mei-Tai, Shyu, Joseph, Tzeng, Gwo-Hshiung, & Khosla, Rajiv (2007). Comparison among three analytical methods for knowledge communities group-decision analysis. Expert Systems with Applications, 33(4), 1011–1024. Cook, W. D., Doyle, J., Green, R., & Kress, M. (1988). Heuristics for ranking players in a round-robin tournament. Computers and Operations Research, 15(2), 135–144. Cook, W. D., & Kress, M. (1990). A data envelopment model for aggregating preference rankings. Management Science, 36(11), 1302–1310. Cook, W. D., & Seiford, L. M. (1978). Priority ranking and consensus formation. Management Science, 24(16), 1721–1732. Deng-Feng, Li. (2007). Compromise ratio method for fuzzy multi-attribute group decision making. Applied Soft Computing, 7(3), 807–817. Emrouznejad, A. (2008). MP-OWA: The most preferred OWA operator, KnowledgeBased Systems, in press. doi:10.1016/j.knosys.2008.03.057. Emrouznejad, A., Parker, B., & Tavares, G. (2008). Evaluation of research in efficiency and productivity: A thirty years survey of the scholarly literature in DEA. SocioEco Plan Sci, 42(3), 151–157. Fishburn, P. C. (1973). The theory of social choice. Princeton, New Jersey: Princeton University Press. Goddard, S. T. (1983). Ranking in tournaments and group decision making. Management Science, 29(12), 1384–1392. Goodman, L., & Markowitz, H. (1952). Social welfare functions based on individual rankings. American Journal of Sociology, 58, 257–262. Hwang, C. L., & Lin, M. J. (1987). Group decision making under multiple criteria. Lecture notes in economics and mathematics system (No. 281). Berlin: Springer. Lin, Chi-Jen, & Wu, Wei-Wen (2008). A causal analytical method for group decisionmaking under fuzzy environment. Expert Systems with Applications, 34(1), 205–213. _ Tuzcu, C., & Tura, O. (2006). An integrated multi-objective optimization Ölçer, A. I., and fuzzy multi-attributive group decision-making technique for subdivision arrangement of Ro–Ro vessels. Applied Soft Computing, 6(3), 221–243. Rothenberg, J. (1961). The measure of social welfare. New Jersey: Prentice-Hall Inc. Tsai, Min-Jen., & Wang, Chen-Sheng. (2008). A computing coordination based fuzzy group decision-making (CC-FGDM) for web service oriented architecture. Expert Systems with Applications., 34(4), 2921–2936. Yu-Jie, Wang (2009). Combining grey relation analysis with FMCGDM to evaluate financial performance of Taiwan container lines. Expert Systems with Applications, 36(2P1), 2424–2432.
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Selecting the most preferable alternatives in a group decision making problem using DEA M. Zerafat Angiz L. a,c, A. Emrouznejad b,*, A. Mustafa c, A. Rashidi Komijan a a
Department of Mathematics, Islamic, Azad University, Firooz-Kooh, Iran Aston Business School, Aston University, Birmingham B4 7ET, UK c School of Mathematical Sciences, Universiti Sains Malaysia, Penang Malaysia b
a r t i c l e
i n f o
Keywords: Group decision making Preferential voting system Data envelopment analysis Most preferable alternative
a b s t r a c t Group decision making is the study of identifying and selecting alternatives based on the values and preferences of the decision maker. Making a decision implies that there are several alternative choices to be considered. This paper uses the concept of Data Envelopment Analysis to introduce a new mathematical method for selecting the best alternative in a group decision making environment. The introduced model is a multi-objective function which is converted into a multi-objective linear programming model from which the optimal solution is obtained. A numerical example shows how the new model can be applied to rank the alternatives or to choose a subset of the most promising alternatives. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Social functions can be categorized into social choice and welfare functions. Each method for vote counting is assumed as a social function but if Arrow’s possibility theorem is used for a social function, social welfare function is achieved. Some specifications of the social functions are decisiveness, neutrality, anonymity, monotonocity, unanimity, homogeneity and weak and strong Paretooptimality. No social choice function meets these requirements in an ordinal scale simultaneously. Some of the social choice functions are minimum deviations measure (Goddard, 1983), Condercet’s, Borda’s, Nanson’s, Dodgson’s function (Hwang & Lin, 1987), Kemeny’s function (Asgharpour, 2004), priority ranking (Cook & Seiford, 1978), Eigenvector (Goddard, 1983) and Fishburn’s function (Asgharpour, 2004). On the other hand, social welfare functions face similar problems in defining a function that fully meets fairness measures. In other words, no social welfare function meets two Arrow’s theorems and five conditions in an ordinal scale simultaneously (Asgharpour, 2004; Hwang & Lin, 1987). However, Rothenberg (1961) and Fishburn (1973) proved that there is a paradox in the conditions of Arrow’s possibility theorem and called it Arrow’s impossibility theorem. As a result, it is essential to revise fairness measures and social welfare functions. The revision may be based on the logical conception of decision makers thinking process in voting alternatives and their expectations from aggregating votes. * Corresponding author. E-mail address: [email protected] (A. Emrouznejad). 0957-4174/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.07.011
Some of the social welfare functions in ordinal scale are Bowman and Clantoni’s approach (Bowman & Clantoni, 1973), Black’s single-peaked preferences and Goodman and Markowitz’s approach (Goodman & Markowitz, 1952). In recent years, researchers have used Data Envelopment Analysis technique (Charnes, Cooper, & Rhodes, 1978; Emrouznejad, Parker, & Tavares, 2008) in ranking alternatives (Adler, Friedman, & Sinvany- Stern, 2002; Cook & Kress, 1990; Cook, Doyle, Green, & Kress, 1988). Adler et al. (2002) considered different ranking methods using DEA technique and divided them into the six categories that overlap with each other. It is worth noting that preferential voting system differs from DEA in structure. In a preferential voting system, the known votes of decision makers are used as inputs of the model and the output is an aggregate ranking that is unknown. In the common DEA models, the objective is to determine technical efficiency. Cook and Kress (1990) specifically considered the use of DEA in aggregating preferential votes. Some authors have recently attempted to use fuzzy multi-attribute (Chang, Cheng, & Chen, 2007; Deng-Feng, 2007) and fuzzy clustering methodology with ordered weighted averaging operator (Chakraborty & Chakraborty, 2007; Emrouznejad, 2008) and integrated multi-objective modeling with fuzzy multi-attributive group (Ölçer, Tuzcu, & Tura, 2006;Emrouznejad, 2008) for similar problem in group decision making, however, the developed model uses optimization technique to find the best weights for selecting the most favorable alternative. The weights then can be used to define a social function that can fairly solve a voting problem. Other recent developments in group decision-making include a study of causal analytical method for group decision-making under fuzzy
9600
M. Zerafat Angiz L. et al. / Expert Systems with Applications 36 (2009) 9599–9602
environment (Lin & Wu, 2008), computing coordination based fuzzy group decision-making (Tsai & Wang, 2008), application of grey relation analysis with fuzzy group decision-making (Yu-Jie Wang, 2009) and a comparative study of several analytical methods for knowledge communities group-decision analysis (Chu, Shyu, Tzeng, & Khosla, 2007). As a result, ranking alternatives in a preferential voting system is done in a more realistic manner. The proposed model is presented in Section 2. Illustration with some numerical examples is given in Section 3. Finally, the concluding results are presented. 2. The most preferable alternative in a group decision making problem, a new mathematical model
In summary, the new objective functions are added in order to calculate weights of a rank in such a way that to be close to the majority of votes. Considering the last objective function and the P first n constraints of model (1) and also assuming nj¼1 wj vpj ¼ 1, a CCR model with one input (equal to one) and n outputs is achieved. In developing a CCR model, this conversion is proved. In order to convert model (1) to a linear one, the following process is done: Assume that w0jh ¼ wj wjh (j = 1, 2, . . . , n) and h 2 H. So, þ w jw0jh j ¼ wþ jh . Applying this substitution, we could conclude jh model (2) that may be solved using multi-objective methods:
Min
X
ah ðwþjh þ wjh Þ ðj ¼ 1; 2; . . . ; nÞ
h
The most important characteristic of a social function is identification of the interactive effect of alternatives and creating a logical relation with the ranks. Meeting decision makers’ expectations and achieving the optimal ranking are among other characteristics of social functions. Assume that there are m decision makers who are supposed to rank n alternatives using preferential voting system. The best and the worst alternatives are given ranks 1 and n, respectively. Each decision maker ranks the alternatives and also compares the ranks using pair-wise comparisons. As the preferential voting system is based on value function, decision makers should have sufficient information about alternatives and weights of the ranks so that they can definitely priories alternatives and ranks. Assume Vij is the number of votes that gives the rank j to the alternative i. In the preferential voting system, it is important to know the weights of the ranks as well as comparing alternatives. So, each decision maker is asked to express how much rank j is preferred to rank j + 1. This is done using pair-wise comparison and eigenvector method. Assume that wjk (k = 1, 2, . . ., m, j = 1, 2, . . ., n) is the weight of rank j in view of decision maker k. A bounded variable is determined for each rank. The upper and lower limits of this variable are the most and the least weights that decision makers assign to the rank. We propose the following mathematical model for decision maker p:
min
X
max s:t:
Max
wj vpj
j¼1 n X
wj vpj P
n X
j¼1
lj 6 wj 6 uj n X wj ¼ 1
wj vij
ði ¼ 1; 2; . . . ; nÞ
ð1Þ
wj vpj
j¼1
s:t:
n X
wj vpj P
j¼1
n X
wj vij
ði ¼ 1; 2; . . . ; nÞ
ð2Þ
j¼1
lj 6 wj 6 uj
ðj ¼ 1; 2; . . . ; nÞ
wj wjh ðwþjh wjh Þ ¼ 0 ðj ¼ 1; 2; . . . ; n & h in HÞ n X
wj ¼ 1
j¼1
3. Illustration with numerical examples This section illustrates the use of Model (2) for selecting the most preferable DMU in a set of 20 decision makers and 4 alternatives. Decision makers vote the importance of the ranks using pairwise comparison matrices. As the weight of the rank j is definitely greater than j + 1, the values of the rows and columns are sorted in ascending and descending order. Assume four decision makers express their comparisons according to the following matrix:
ah jwj wjh j ðj ¼ 1; 2; . . . ; nÞ
h2H n X
n X
x1 x2 x3 x4
x1
x2
x3
x4
1
3 1
5 4 1
7 5 3 1
1 3 1 5 1 7
1 4 1 5
1 2
j¼1
ðj ¼ 1; 2; ::; nÞ
j¼1
where wjh is the weight of rank j due to at least 50% of decision makers. For example, if there are 100 (people) decision makers and 30, 15 and 10 of them believe that the weight of rank one is 0.6, 0.5 and 0.65 and the rest give different weights, then w11, w12 and w13 will be 0.6, 0.5 and 0.65, respectively. This means that H = {1, 2, 3}. These weights are calculated using eigenvector method. lj and uj are upper and lower limits of the weights of the ranks. The last constraint expresses the fact that weights summation in a pair-wise comparison is equal to one. The coefficient of ah ensures that the votes of the most decision makers are applied in the model more precisely. In the above-mentioned example, 30 decision makers (among 55 = 30 + 15 + 10 ones) assign the weight 0.6 to the rank one. So, 0.6 is more probably consider than 0.5 that is assigned by only 15 decision makers. ah for a specific weight is calculated by dividing the number of decision makers who assigned that weight to the number of majority of decision makers. For the specific weight 0.6, ah ¼ 30=55 ¼ 0:55 and for 0.5, ah ¼ 15=55 ¼ 0:27.
Using eigenvector method, the weights vector is calculated as w = (0.57, 0.27, 0.10, 0.06). It means that four decision makers believe that the weights of the ranks one to four are 0.57, 0.27, 0.1 and 0.6, respectively. Assume that three decision makers express their comparisons according to the following matrix:
x1 x2 x3 x4
x1
x2
x3
x4
1
2 1
4 3 1
7 5 2 1
1 2 1 4 1 7
1 3 1 5
1 2
According to these three decision makers and using eigenvector method the weights vector is w = (0.57, 0.27, 0.10, 0.06). Assume also that the following matrix is associated with five decision makers and the resulted weights vector is w = (0.52, 0.29, 0.11, 0.07).
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x1 x2 x3 x4
x1
x2
x3
x4
1
2 1
5 3 1
6 4 2 1
1 2 1 5 1 6
1 3 1 4
1 2
Table 1 Ranking alternatives by 20 decision makers
The weights vectors in view of the other decision makers have also been calculated and summary of the weights are listed in the following table. Number of decision makers
Weight 1
Weight 2
Weight 3
Weight 4
Five decision makers Four decision makers Three decision makers Two decision makers Two decision makers Two decision makers One decision makers One decision makers
0.52 0.57 0.57 0.51 0.49 0.54 0.55 0.52
0.29 0.27 0.27 0.34 0.31 0.25 0.29 0.28
0.11 0.10 0.10 0.09 0.14 0.13 0.11 0.12
0.07 0.06 0.06 0.06 0.05 0.08 0.05 0.08
Max weight Min weight
0.57 0.49
0.34 0.25
0.14 0.09
0.08 0.05
Decision maker
Rank
1
1
2
3
4
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
3 3 2 4 4 2 1 1 3 4 1 1 1 4 4 4 3 2 2
1 2 1 2 2 3 4 3 2 3 2 3 4 2 3 2 4 4 3
4 1 3 3 1 1 2 4 4 3 3 2 3 3 1 1 2 3 1
2 4 4 1 3 4 3 2 1 1 4 4 2 1 2 3 1 1 4
Table 2 The number of assigning each rank to each alternative
Using the resulted weights vectors, a weigh interval is determined for each rank. Assume that w1, w2, w3 and w4 are weight intervals for rank one to four. So:
w1 ¼ ½0:49; 0:57
Alternatives
Rank
1
1
2
3
4
1 2 3 4
5 5 4 6
2 7 7 4
6 4 6 4
7 4 3 6
w2 ¼ ½0:25; 0:34 w3 ¼ ½0:09; 0:14 This should be done for each alternative, assuming that
w4 ¼ ½0:05; 0:08 If the weights proposed by all decision makers are the same, a Borda function type is resulted which combines with different weight. The only difference is that we have weights resulted from pairwise comparisons instead of scores resulted from Borda technique. Table 1 shows the ranks assigned to alternatives by decision makers. The number of rank j assigned to alternative i is easily calculated (see Table 2): In order to select the best alternative, we could construct a multi objective model based on model (2) for each alternative. The resulted model for the first alternative is as follows:
wþ11 þ w11 ¼ jw1 0:57j wþ12 þ w12 ¼ jw1 0:50j wþ13 þ w13 ¼ jw1 0:52j wþ21 þ w21 ¼ jw2 0:27j wþ22 þ w22 ¼ jw2 0:31j wþ23 þ w23 ¼ jw2 0:29j wþ31 þ w31 ¼ jw3 0:10j wþ32 þ w32 ¼ jw3 0:12j wþ33 þ w33 ¼ jw3 0:11j wþ41 þ w41 ¼ jw4 0:06j wþ42 þ w42 ¼ jw4 0:07j
Min Z 1 ¼ 0:33jw1 0:57j þ 0:25jw1 0:50j þ 0:42jw1 0:52j Min Z 2 ¼ 0:33jw2 0:27j þ 0:25jw2 0:31j þ 0:42jw2 0:29j
With the above-mentioned substitutions, model (3) is converted to a multi-objective model (4):
Min Z 3 ¼ 0:33jw3 0:10j þ 0:25jw3 0:12j þ 0:42jw3 0:11j
Min Z 1 ¼ 0:33ðwþ11 þ w11 Þ þ 0:25ðwþ12 þ w12 Þ þ 0:42ðwþ13 þ w13 Þ
Min Z 4 ¼ 0:33jw4 0:06j þ 0:25jw4 0:07j þ 0:42jw4 0:07j
Min Z 2 ¼ 0:33ðwþ21 þ w21 Þ þ 0:25ðwþ22 þ w22 Þ þ 0:42ðwþ23 þ w23 Þ
Max Z 5 ¼ 5w1 þ 2w2 þ 6w3 þ 7w4
Min Z 3 ¼ 0:33ðwþ31 þ w31 Þ þ 0:25ðwþ32 þ w32 Þ þ 0:42ðwþ33 þ w33 Þ
s:t
Min Z 4 ¼ 0:33ðwþ41 þ w41 Þ þ 0:67ðwþ42 þ w42 Þ
5w1 þ 2w2 þ 6w3 þ 7w4 P 5w1 þ 7w2 þ 4w3 þ 4w4 5w1 þ 2w2 þ 6w3 þ 7w4 P 4w1 þ 7w2 þ 6w3 þ 3w4
Max Z 5 ¼ 5w1 þ 2w2 þ 6w3 þ 7w4
5w1 þ 2w2 þ 6w3 þ 7w4 P 6w1 þ 4w2 þ 4w3 þ 6w4
s:t:
5w1 þ 2w2 þ 6w3 þ 7w4 P 5w1 þ 7w2 þ 4w3 þ 4w4
w1 þ w2 þ w3 þ w4 ¼ 1
5w1 þ 2w2 þ 6w3 þ 7w4 P 4w1 þ 7w2 þ 6w3 þ 3w4
0:49 6 w1 6 0:57
5w1 þ 2w2 þ 6w3 þ 7w4 P 6w1 þ 4w2 þ 4w3 þ 6w4
0:25 6 w2 6 0:34
w1 þ w2 þ w3 þ w4 ¼ 1
0:09 6 w3 6 0:14 0:05 6 w4 6 0:08
0:49 6 w1 6 0:57 ð3Þ
0:25 6 w2 6 0:34
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0:09 6 w3 6 0:14
4. Conclusion remarks
0:05 6 w4 6 0:08 wþ11 w11 ¼ w1 0:57 wþ12 w12 ¼ w1 0:27 wþ13 w13 ¼ w1 0:52 wþ21 w21 ¼ w2 0:27 wþ22 w22 ¼ w2 0:31 wþ23 w23 ¼ w2 0:29 wþ31 w31 ¼ w3 0:10 wþ32 w32 ¼ w3 0:12 wþ33 w33 ¼ w3 0:11 wþ41 w41 ¼ w4 0:06 wþ42 w42 ¼ w4 0:07 The same procedure should be done for other alternatives. In order to obtain full ranking model (2) should be solved for each alternative. If a model has feasible solution, the associated alternative will be placed in Group one. The Group one alternatives are put away and the rest of alternatives are assigned lower ranks. In other words, model (2) is developed for each alternative that has not been placed in Group one. This process is repeated until all alternatives are placed in a Group. At least one of the mathematical models that are developed in each stage should have a feasible solution. It worth noting that alternatives in lower Groups can never be assigned the ranks of upper Groups alternatives. For example, consider an alternative in Group two. If this alternative is compared with others using any arbitrary weights vector, it can never be placed in Group one. The result of the first stage of ranking alternatives in a preferential voting system is summarized in Table 3. As the only feasible model is associated with alternative 2, the first Group only includes this alternative. After omitting this alternative, the procedure is repeated for the rest of alternatives (Table 4). Group 2 includes alternatives 3 and 4. As the optimal solution of the model associated with alternative 4 is more desirable (due to the minimum objective function), rank 2 is assigned to alternative 4. Finally, rank 4 is assigned to alternative 1.
Table 3 Result of solving the first stage model Alternatives
Optimal solution of model (4)
1 2 3 4
Infeasible 0.792 Infeasible Infeasible
Table 4 Result of solving the second stage model Alternatives
Optimal solution of model (4)
1 3 4
Infeasible 0.8361 0.1188
Preferential voting system is a type of multi-criteria decision making problem in which decision makers rank alternatives. Then, their votes are collected and considered. There are different methods to solve such a problem, e.g. social functions based on weighting ranks. Borda and Eigenvector functions are well known social functions. Some of the methods such as Cook and Kress (1990) approach consider the weights of the ranks in a different way. It is important that a social function to be fair in different approaches. One approach that leads to a more realistic solution is to determine the importance of ranks by decision makers. In this paper, a new mathematical model is introduced that could determine the importance of ranks by decision makers in order to reach a more realistic solution. Although large scale mathematical models should be solved in a large scale decision making problem, new technologies and PCs may justify using this method. It is worth noting that increasing the number of decision makers does not affect the dimension of the problem but the number of alternatives may cause more constraints and variables. Further research is needed to simplify the new multi-objective program to a linear programming. References Adler, N., Friedman, L., & Sinvany- Stern, Z. (2002). Review of ranking methods in the data envelopment analysis context. European Journal of Operational Research, 140, 249–265. Asgharpour, M. J. (2004) Group decision making and game theory. Tehran University Publication. Bowman, V. J., & Clantoni, C. S. (1973). Majority rule under transitivity constraints. Management Science, 19(9), 1029–1041. Chakraborty, C., & Chakraborty, D. (2007). A fuzzy clustering methodology for linguistic opinions in group decision making. Applied Soft Computing, 7(3), 858–869. Chang, J. R., Cheng, C. H., & Chen, L. S. (2007). A fuzzy-based military officer performance appraisal system. Applied Soft Computing, 7(3), 936–945. Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring efficiency of decision making units. European Journal of Operational Research, 2, 429–444. Chu, Mei-Tai, Shyu, Joseph, Tzeng, Gwo-Hshiung, & Khosla, Rajiv (2007). Comparison among three analytical methods for knowledge communities group-decision analysis. Expert Systems with Applications, 33(4), 1011–1024. Cook, W. D., Doyle, J., Green, R., & Kress, M. (1988). Heuristics for ranking players in a round-robin tournament. Computers and Operations Research, 15(2), 135–144. Cook, W. D., & Kress, M. (1990). A data envelopment model for aggregating preference rankings. Management Science, 36(11), 1302–1310. Cook, W. D., & Seiford, L. M. (1978). Priority ranking and consensus formation. Management Science, 24(16), 1721–1732. Deng-Feng, Li. (2007). Compromise ratio method for fuzzy multi-attribute group decision making. Applied Soft Computing, 7(3), 807–817. Emrouznejad, A. (2008). MP-OWA: The most preferred OWA operator, KnowledgeBased Systems, in press. doi:10.1016/j.knosys.2008.03.057. Emrouznejad, A., Parker, B., & Tavares, G. (2008). Evaluation of research in efficiency and productivity: A thirty years survey of the scholarly literature in DEA. SocioEco Plan Sci, 42(3), 151–157. Fishburn, P. C. (1973). The theory of social choice. Princeton, New Jersey: Princeton University Press. Goddard, S. T. (1983). Ranking in tournaments and group decision making. Management Science, 29(12), 1384–1392. Goodman, L., & Markowitz, H. (1952). Social welfare functions based on individual rankings. American Journal of Sociology, 58, 257–262. Hwang, C. L., & Lin, M. J. (1987). Group decision making under multiple criteria. Lecture notes in economics and mathematics system (No. 281). Berlin: Springer. Lin, Chi-Jen, & Wu, Wei-Wen (2008). A causal analytical method for group decisionmaking under fuzzy environment. Expert Systems with Applications, 34(1), 205–213. _ Tuzcu, C., & Tura, O. (2006). An integrated multi-objective optimization Ölçer, A. I., and fuzzy multi-attributive group decision-making technique for subdivision arrangement of Ro–Ro vessels. Applied Soft Computing, 6(3), 221–243. Rothenberg, J. (1961). The measure of social welfare. New Jersey: Prentice-Hall Inc. Tsai, Min-Jen., & Wang, Chen-Sheng. (2008). A computing coordination based fuzzy group decision-making (CC-FGDM) for web service oriented architecture. Expert Systems with Applications., 34(4), 2921–2936. Yu-Jie, Wang (2009). Combining grey relation analysis with FMCGDM to evaluate financial performance of Taiwan container lines. Expert Systems with Applications, 36(2P1), 2424–2432.