Multiple Criteria Decision Making Theory, Methods, and Applications in Engineering
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ChingSan Lin, ChenTung Chen, FarnShing Chen, WeiZhan Hung, "A Novel Multiperson Game Approach for Linguistic Multicriteria Decision Making Problems", Mathematical Problems in Engineering, vol. 2014, Article ID 592326, 20 pages, 2014. https://doi.org/10.1155/2014/592326
A Novel Multiperson Game Approach for Linguistic Multicriteria Decision Making Problems
Abstract
Game theory is considered as an efficient framework in dealing with decision making problems for two players in the competitive environment. In general, the evaluation values of payoffs matrix are expressed by crisp values in a game model. However, many uncertainties and vagueness should be considered due to the qualitative criteria and the subjective judgment of decision makers in the decision making process. The aim of this study is to develop an effective methodology for solving the payoffs matrix with linguistic variables by multiple decision makers in a game model. Based on the linguistic variables, the decision makers can easily express their opinions with respect to criteria for each alternative. By using the linear programming method, we can find the optimal solution of a game matrix in accordance with the combination of strategies of each player effectively. In addition, the expected performance value (EPV) index is defined in this paper to compare the competition ability of each player based on the optimal probability of each strategy combination. And then, numerical example will be implemented to illustrate the computation process of the proposed model. The conclusion and future research are discussed at the end of this paper.
1. Introduction
In decision science field, game theory provides an effective way for handing the interactive optimization problems. Game theory started with the publication of “the theory of games and economics behavior” in 1944 by von Neumann and Morgenstern [1]. It is a special tool to analyze the interaction result among players and has been broadly applied in business, financial, politics, education, sports, and so forth [2–5].
There are three basic components in a game such as players, players’ strategies, and the performances (payoffs) with respect to the strategies of players. Strategy form and extensive form are two main ways to describe the interaction between players [6]. In strategy form, each player executes his/her strategy simultaneously and the payoff is decided based on the strategy combination of each player. In extensive form, each player executes his/her strategy sequentially and the payoff is the final result based on the decision of each player. In general, the extensive form of a game can be transferred to the strategy form. The different classification types of game included cooperation or noncooperation games, zerosum or nonzerosum games, one round or multiround games, twoperson or multiperson games.
Recently, the development trend of game theory is to integrate multicriteria decision making (MCDM) method to deal with the decisionmaking problems in real situations. There are some literatures that have been proposed by combining multicriteria decision method (MCDM) with game theory for coping with the decision making problems.
Campos [7] proposed a twoperson zero sum fuzzy matrix game and applied fuzzy linear programming to calculate the mixed strategy probability of each player. Sakawa and Nishizaki [8] applied the maxmin concept to integrate fuzzy goals and fuzzy payoffs in a twoperson zero sum game. Song and Kandel [6] applied fuzzy set to formulate the goals of players and the strategy probability of their competitors. The mixed strategy probability is calculated based on considering their goals and the probability of each strategy of competitors simultaneously. The drawback of Song and Kandel’s model is that it is difficult to compute the game matrix with multiple persons. Wang [9] built a fuzzy linear programming model to deal with person multiattribute noncooperative game. Liao [10] applied fuzzy linear goal programming to solve a twoperson zero sum game for analyzing the wireless market in Taiwan. Angelou and Economides [11] integrated analytic hierarchy process, game theory, and real options to analyze the business alternatives of information and communication technology (ICT). Reneke [12] used the vector function to evaluate long term investment alternatives for predicting oil prices and environmental degradation under the conditions of risk and uncertainty. Madani and Lund [13] used MonteCarlo game theory (MCGT) technology to handle the uncertainty problem with the deterministic strategic games. Monroy and Fernández [14] extended the ShapleyShubik index from conventional simple games to deal with the multicriteria game problem. Barough et al. [15] used the traditional game approach to deal with two types of project construction conflict problem. Li and Hong [16] developed an effective methodology for handling the constrained matrix game with fuzzy payoffs. Monroy and Fernández [17] used voting systems to handle multicriteria simple games in socialchoice situation. The stable set and the core are the solution concept in the multicriteria simple games. Different kinds of aggregation operators such as union, intersection, marginalization, and composition can be applied in their method. Lozan and Ungureanu [18] used graphs intersection method of the best response mappings to deal with the twocriterion games. Pusillo and Stef Tijs [19] used the improvement sets for developing the equilibrium condition for noncooperative multicriteria games. Kawamura et al. [20] extended neutrally and evolutionarily stable strategies from singlecriterion game to multicriteria games. In entropy environment, Roy and Das [21] handled the multicriteria bimatrix goal game problem by determining Ggoal security strategies. They applied the real coded genetic algorithm to acquire the bounds of the objectives of the proposed game. The fuzzy programming technique is used to solve the formulated model.
Although many literatures have been proposed by applying game theory to make a decision, few of them can integrate MCDM and game theory to handle the multiperson multicriteria game in a fuzzy environment. In real environment, each player will compete with other players. A good player not only should consider his/her strategy for approaching his/her goal but also need to forecast the behaviors of competitors. There is usually more than one influenced factor that should be considered by each player in a game model. In addition, the uncertainty and fuzziness will happen in the real competitive environment because it is not easy to collect the decision information completely and the decision time is limited for making a decision. A good model must provide a mechanism such as linguistic value or fuzzy number for experts to express their opinions flexibly.
However, the original game model usually considers one dimension or criterion to make a decision and the crisp values are used to evaluate the performance with respect to the dimension by single decision maker. In order to overcome the drawbacks of original game model, the main purpose of this study is to develop a new decision making method, linguistic multiperson multicriteria game (LMPMCG) model, for dealing with the game problems under multiperson and multicriteria environment. According to the linguistic variable, the decision makers can easily express their opinions with respect to each criterion for each alternative (strategy combination). By using the linear programming method, we can find the optimal solution of a game matrix effectively.
The reminder of this study is organized as follows. In Section 2, the definitions of linguistic variables and fuzzy numbers will be introduced. After that, the linguistic multiperson multicriteria game model is presented at Section 3. And then, an example is implemented for the new mobile phone development project selection problem. Finally, conclusion and future research are discussed at the end of this paper.
2. Fuzzy Set and Linguistic Variable
Fuzzy set theory is first introduced by Zadeh in 1965 [22]. Fuzzy set theory is a very feasible method to handle the imprecise and uncertain information in a real world [23, 24]. Especially, it is more suitable for experts to express their subjective judgments and qualitative assessments in the decision making processes [25–27].
A positive triangular fuzzy number (PTFN) can be defined as , where and (shown in Figure 1). The membership function of positive triangular fuzzy number (PTFN) is defined as [28, 29]
A linguistic variable is a variable whose values are expressed in linguistic terms, in other words, variable whose values are not numbers but words or sentences in a nature or artificial language [30–32]. For example, “weight” is a linguistic variable whose values can be very low, low, medium, high, very high, and so forth. These linguistic values can also be represented by fuzzy numbers. There are two advantages for using triangular fuzzy number to express linguistic variable [33, 34]. First, it is a rational and simple method to use triangular fuzzy number to express the opinions of experts. Second, it is easy to make the arithmetic operations between fuzzy numbers when using triangular fuzzy numbers to express the linguistic variables. It is suitable to represent the degree of subjective judgment in qualitative aspect than crisp value. Some linguistic variables and their membership functions can be illustrated as Table 1 and Figure 2.

Let and be two PTFNs. The additive operation of PTFNs can be calculated as [28, 33]
Many ranking methods have been developed to transform fuzzy number into crisp value. Lee and Li [35] presented the generalized mean value method to rank fuzzy numbers. It is very easy to compare fuzzy numbers by this method. This method has been applied in decision science, personnel selection, weapon selection and supplier selection fields, and so forth [36–39]. Suppose that is a PTFN, the defuzzied value is easily computed as [35, 36] If , then .
3. Linguistic Multiperson Multicriteria Game (LMPMCG) Model
In real environment, each player must compete with other players to determine the actions. A good player not only must consider the own strategies for approaching the goals but also need to forecast the behaviors of their competitors to select the best reaction. Under this situation, many influenced criteria should be considered by each player to make a strategy decision in a game system. Because the different strategies of players will influence the performance of each other, we need to consider the strategies of each player based on a strategy combination. A strategy combination means combination of strategies of players in a specific situation. From this viewpoint, a new game model with linguistic variables for the multiperson multicriteria problem is proposed in this study.
3.1. Basic Notation of LMPMCG Model
Generally speaking, the contents of linguistic multiperson multicriteria game (LMPMCG) model can be illustrated as follows.(1)A set of players is called , where represents the number of players.(2)A set of strategies of each player is called , where , , . The is the number of strategies for player 1, player 2, …, and player . The represents the th strategy of player .(3)A set of strategies combination is . A strategy combination means a combination of the strategy of each player in a specific situation.(4)A set of criteria with respect to each strategy combination of each player is , where , , . The is the number of the criteria for player 1, player 2, …, and player . The represents the th strategy criteria of player .(5)A set of strategies combination evaluation matrix of each player is , where , , and represents the strategy performance of player with respect to criterion under strategy combination .(6)A set of decision makers or experts is , where represents the number of decision makers.
3.2. Aggregation Process
In a game model, decision makers should express their opinions for the performance of each strategy combination with respect to each criterion based on the standpoint of each player. In general, the linguistic variables are suitable for decision makers used to express their subjective judgment. For a strategy combination , the performance evaluation of decision maker with respect to criterion can be represented as a linguistic variable . This linguistic variable can be formulated as a where , , and (refer to Table 1).
For a strategy combination , the aggregated performance evaluation of decision makers with respect to criterion can be computed as where represents the aggregated performance with respect to criterion under a strategy combination based on the opinions of decision makers.
The aggregated performance evaluation of decision makers with respect to criterion under the strategy combination can be defuzzied as where .
The weight of criterion of decision maker can be represented as a linguistic variable . It can be formulated as a where , , and (refer to Table 1).
The aggregated weight of criterion can be computed as where represents the aggregated weight with respect to criterion based on the opinions of decision makers.
The aggregated weight of criterion can be defuzzified as where .
3.3. Decision Process
For the strategy combination , the weighted performance of strategy for player with respect to criterion can be calculated as where represents the weighted performance of strategy for player with respect to criterion under strategy combination and represents the normalized weight of criterion .
For the strategy combination , the integrated performance of player with respect to all criteria can be calculated as where represents the integrated performance of player with respect to all criteria under the strategy combination . It means the overall performance for player when player 1 chooses strategy , player 2 chooses strategy , player chooses strategy , and player chooses strategy .
The optimal probability of each strategy for player can be computed by considering all strategy combinations of all players. Under this situation, player will maximize the expected performance by calculating the optimal probability of each strategy. Therefore, the problem of optimal probability of each strategy for player can be formulated as a linear programming model as follows: where represents the expected performance of player and is the probability of the strategy .
4. Numerical Example
Suppose that there are three enterprises (players) , , and who can produce high technology mobile phone in the market. Each enterprise possesses its strong point and weakness. The product development strategy and manufacture strategy of each enterprise will influence the competitive performance of other enterprises in the different dimensions. Based on R&D ability and the brand impressions of each enterprise, the strategy of each enterprise will be limited to develop a new product. Enterprise invites three experts to analyze and select a suitable strategy for developing a new mobile phone. The first expert is a marketing director who comes from the marketing department in the enterprise. The second expert is an R&D manager who is the project leader of the research and development department in the enterprise. The third expert is a strategy professor who is invited from the university. Enterprise has three development strategies and also knows that competitor has three strategies and competitor has two strategies for developing a new mobile phone. Enterprise considers four criteria to make a decision. Enterprise also knows that competitor considers three criteria for making decision and competitor considers four criteria for making decision. The strategies and evaluation criteria of enterprises are shown in Table 2.

According to the computational process of linguistic multiperson multicriteria game (LMPMCG) model, the problem can be solved as follows.
4.1. Aggregating the Evaluations of Experts
Step 1. Each expert uses the linguistic variables to evaluate the performance of each enterprise with respect to each criterion based on different strategy combinations as Tables 3, 4, and 5.



Step 2. Transform the linguistic evaluation of the performance of each enterprise with respect to each criterion into PTFN as Tables 6, 7, and 8. And then, the performance of each enterprise with respect to each criterion can be aggregated as Table 9.


