Selecting the most preferable alternatives in a group decision making problem using DEA
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 Pareto-optimality. 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 and Lin, 1987). However, Rothenberg, 1961, 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. 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 et al., 1978, Emrouznejad et al., 2008) in ranking alternatives (Adler et al., 2002, Cook and Kress, 1990, Cook et al., 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 et al., 2007, Deng-Feng, 2007) and fuzzy clustering methodology with ordered weighted averaging operator (Chakraborty and Chakraborty, 2007, Emrouznejad, 2008) and integrated multi-objective modeling with fuzzy multi-attributive group (Ölçer et al., 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 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.
Section snippets
The most preferable alternative in a group decision making problem, a new mathematical model
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
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 pair-wise 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:Empty Cell x1 x2 x3 x4 x1 1 3 5 7 x2 1 4 5 x3 1 3 x4 1
Conclusion remarks
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
References (22)
- et al.
Review of ranking methods in the data envelopment analysis context
European Journal of Operational Research
(2002) - et al.
A fuzzy clustering methodology for linguistic opinions in group decision making
Applied Soft Computing
(2007) - et al.
A fuzzy-based military officer performance appraisal system
Applied Soft Computing
(2007) - et al.
Measuring efficiency of decision making units
European Journal of Operational Research
(1978) - et al.
Comparison among three analytical methods for knowledge communities group-decision analysis
Expert Systems with Applications
(2007) - et al.
Heuristics for ranking players in a round-robin tournament
Computers and Operations Research
(1988) - et al.
Evaluation of research in efficiency and productivity: A thirty years survey of the scholarly literature in DEA
Socio-Eco Plan Sci
(2008) - et al.
A causal analytical method for group decision-making under fuzzy environment
Expert Systems with Applications
(2008) - et al.
An integrated multi-objective optimization and fuzzy multi-attributive group decision-making technique for subdivision arrangement of Ro–Ro vessels
Applied Soft Computing
(2006) - et al.
A computing coordination based fuzzy group decision-making (CC-FGDM) for web service oriented architecture
Expert Systems with Applications.
(2008)
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