Computational Intelligence and Neuroscience

Volume 2019, Article ID 8389035, 16 pages

https://doi.org/10.1155/2019/8389035

## A Competitive Multiattribute Group Decision-Making Approach for the Game between Manufacturers

School of Information Engineering, ShenYang University of Technology, ShenYang, China

Correspondence should be addressed to Quan Zhang; nc.ude.tus@gnahzqsi

Received 1 August 2019; Accepted 1 October 2019; Published 11 November 2019

Academic Editor: Carmen De Maio

Copyright © 2019 Quan Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Under the competitive market environment, the game between manufacturers comes down to the competitive multiattribute group decision-making problem. In this study, the evaluation information of experts is given in the form of 2-dimension 2-tuple linguistic variables, and an approach is proposed for the competitive multiattribute group decision-making problem based on game theory and evidence theory. Firstly, based on the evidence theory, the attribute values of each situation are obtained by aggregating the 2-dimension 2-tuple linguistic evaluation information given by experts. Secondly, according to the attribute values of every situation, the evidence theory is applied for the second aggregation to obtain the overall values of every situation, and then the game matrix of competitive multiattribute group decision problem is formed. Then, according to the bivariate game matrix, the Nash equilibrium point of competitive multiattribute group decision-making problem is determined based on game theory. Finally, a practical case about the alternative selections for a duopoly problem is used to illustrate the effectiveness and applicability of the proposed approach for the competitive multiattribute group decision-making problem.

#### 1. Introduction

With the economic globalization and market competition intensification, the game between manufacturers has become very common, which can be viewed as a multiple attribute group decision-making problem (MAGDM). MAGDM is the process to select the most desirable alternative with respect to multiple conflicting attributes based on the evaluation information given by the invited experts [1]. In a competitive multiattribute group decision-making problem, facing a decision matrix, two decision makers usually have to compete with each other when selecting their most desirable alternative [2]. In [2], a competitive multiattribute decision-making framework was proposed. The alternative selected by one decision maker affects the benefits of the other one. The competing decision makers must search a stable situation to ensure their benefits to be maximum or their loss to be minimum. In the course of competitive multiattribute group decision making, the stable situation is searched with respect to the alternatives by the two competing decision makers, which is called the Nash equilibrium. The evaluation information of every alternative is given by experts to determine the Nash equilibrium.

Because of the complexity of multiple attribute decision-making problems, the evaluation information of every alternative given by experts is always uncertain. In competitive multiattribute group decision-making problems, the uncertain evaluation information from the experts can be expressed in various forms, such as fuzzy linguistic information [3–7] and linguistic term sets [8–12]. Compared with linguistic term sets, the fuzzy linguistic information lacks precision in the final results. But, the original linguistic term sets cannot describe the situation where the evaluation information proposed by experts is between two standard linguistic terms. In order to solve this problem, Herrera and Martínez proposed the concept of 2-tuple linguistics and added the unitary information to the original linguistic variables in order to describe the degree to which the evaluation information deviated from the standard linguistic terms [13]. However, 2-tuple linguistic information does not take into account the degree of uncertainty of evaluation information. For this reason, Zhu et al. [14] proposed the 2-dimension 2-tuple linguistic concept by adding one-dimensional information on the basis of 2-tuple linguistic information to describe the degree of uncertainty of evaluation information. The 2-dimension 2-tuple linguistic information can describe the evaluation information of experts more accurately than original linguistic information.

In order to improve the accuracy of decision-making results as compared with the study in [2], this paper proposes an approach for the competitive multiattribute group decision-making problem of the game between manufacturers, where the experts’ evaluation information is in 2-dimension 2-tuple linguistics. To solve the competitive multiattribute group decision-making problem with 2-dimension 2-tuple linguistic evaluation information, it is necessary to aggregate experts’ evaluation information. The weighted sum is a common approach to aggregate 2-dimension 2-tuple linguistic evaluation information. However, the evidence theory possesses the ability to describe uncertainty and ignorance. Thus, an approach for aggregating 2-dimension 2-tuple linguistic evaluation information is proposed in this paper based on the evidence theory (e.g., Dempster–Shafer theory). In addition, based on the game matrix of competitive multiattribute group decision problem, the Nash equilibrium point of the problem is determined based on game theory.

The organization of this paper is as follows. Section 2 reviews the current research on the multiple attribute group decision-making problems and the methods for aggregating 2-dimension 2-tuple linguistic information as well as the applications of evidence theory and game theory in multiple attribute decision-making problems. Section 3 gives preliminary knowledge about 2-tuple linguistic information, 2-dimension 2-tuple linguistic information, and evidence theory. The competitive multiattribute decision-making problem with two decision makers is described in Section 4. Section 5 proposes an approach for the competitive multiattribute group decision-making problem with 2-dimension 2-tuple linguistic evaluation information. In Section 6, the application of the proposed approach is illustrated by an example. Section 7 summarizes the results of this study.

#### 2. Literature Review

In order to determine the best solution, in [15], two intuitionistic fuzzy aggregation operators were proposed to aggregate the experts’ information and attribute information. Garg and Nancy [16] proposed some new linguistic priority aggregation operators, based on which a multiattribute group decision-making method was proposed too. By considering the priority of attributes and experts at the same time, Verma studied the multiattribute group decision-making problems under the trapezoidal fuzzy linguistic information and proposed a new weighted average priority operator [17].

Xu and Cai [18] established a nonlinear programming model to determine the experts’ weights based on the principle that the experts’ weights should produce the highest consistency of the experts. On this basis, the authors of [19] pointed out the shortcomings of applying genetic algorithm to solve the nonlinear programming model and proposed a new method to determine the experts’ weights by linearizing the nonlinear programming model. The authors of [20] proposed the concept of probabilistic uncertain linguistic term sets and the method for probabilistic uncertain linguistic multiattribute group decision-making problems.

In recent years, many scholars studied the 2-dimension 2-tuple linguistic information. In [21], some power aggregation operators were proposed, including 2-dimension uncertain linguistic power generalized aggregation operator (2DULPGA) and 2-dimension uncertain linguistic power generalized weighted aggregation operator (2DULPGWA), and some properties were discussed, as well as the special cases of them.

The authors of [22] proposed a 2-dimension linguistic lattice implication algebra (2DL-LIA) as a linguistic evaluation set with lattice structure. Motivated by the ideas of dependent aggregation operator, in [23], some 2-dimension linguistic dependent aggregation operators were developed. Yu et al. [24] developed a 2-dimension linguistic weighted averaging (2DLWA) operator and a 2-dimension linguistic ordered weighted averaging (2DLOWA) operator. In [25], some density aggregation operators were proposed based on 2DULVs. Liu and Wang [26] proposed three operators: 2-dimension uncertain linguistic weighted averaging (2DULWA) operator, 2-dimension uncertain linguistic weighted geometric (2DULWG) operator, and 2-dimension uncertain linguistic generalized weighted average (2DULGWA) operator. Wu et al. [27] put forward 2-dimension interval type-2 trapezoidal fuzzy ordered weighted average (2DIT2TFOWA) operator and quasi-2-dimension interval type-2 trapezoidal fuzzy ordered weighted average (quasi-2DIT2TFOWA) operator. However, most of the research on the aggregation methods for 2-dimension linguistic variables as discussed above relied on the weighted average operator. The weighted average operator mainly combines the 2-dimension linguistic information by taking the smallest method. Then, the fusion of the highest-level 2-dimension linguistic information with the lowest-level 2-dimension linguistic information is the lowest level of fusion information, which is very unreasonable. Therefore, the approach proposed in this paper is to use the 2-dimension linguistic information as the reliability of the experts, and the above unreasonable situation will not occur.

Because of the advantages of 2-dimension 2-tuple linguistic information, it is very meaningful to study the competitive multiattribute group decision-making problems with 2-dimension 2-tuple linguistic evaluation information.

Evidence theory was widely used in multiattribute group decision-making problems. The authors of [28] proposed a DS-AHP approach for the multiattribute decision-making problem with incomplete information. The authors of [29] proposed an evidence reasoning approach for the multiattribute decision-making problem with incomplete decision matrix. The authors of [30] applied the Dempster–Shafer theory of evidence to develop a new 2-tuple linguistic representation model for decision making and then put forward some linguistic aggregation operators.

Game theory is widely used in modern management science and produces fruitful research results. Chen and Larbani [31] applied game theory in fuzzy multiattribute decision-making problems, by assuming that fuzzy multiattribute decision making is a game between decision makers and nature. Zhou et al. [32] applied game theory to solve multiattribute decision-making problems and thought that the multiattribute decision-making problem was a game between the attributes. And sometimes, manufacturers are often in a fiercely competitive environment when making decisions. Therefore, this paper proposes a multiattribute group decision-making approach for manufacturers to conduct alternative selections in a competitive environment.

#### 3. Preliminaries

Some background knowledge about 2-tuple linguistic information and 2-dimension 2-tuple linguistic information is introduced in this section, as well as evidence theory [13, 14, 33].

##### 3.1. 2-Tuple Linguistic Variables

*Definition 1. *Let be the result of an aggregation of the indexes of a set of labels assessed in a linguistic term set , i. e., the result of a symbolic aggregation operation. , with being the cardinality of . Let and be two values such that and ; then, is called symbolic translation [13].

The symbolic translation of a linguistic term , is a numerical value assessed in that supports the “difference of information” between counting of information obtained after a symbolic aggregation operation and the closest value in that indicates the index of the closest linguistic term in [13].

A 2-tuple is usually used to represent the linguistic information, , and , as follows: represents the linguistic label center of the information is a numerical value expressing the value of the translation from the original result to the closest index label, , in the linguistic term set , i.e., the symbolic translation [13]

*Definition 2. *Let be a linguistic term set and be a value representing the result of a symbolic aggregation operation; then, the 2-tuple that expresses the equivalent information to is obtained with the following function [13]:where is the usual round operation, has the closest index label to “,” and “” is the value of the symbolic translation.

*Definition 3. *Let be a linguistic term set and be a 2-tuple. There is always a function such that from a 2-tuple it returns its equivalent numerical value [13]; :

*Definition 4. *Let and be two linguistic term sets; then, their distance is obtained with the following function [33]:

##### 3.2. 2-Dimension 2-Tuple Linguistic Variables

The definition of 2-dimension 2-tuple linguistic variables is given as follows.

*Definition 5. *Let be the set of linguistic information evaluation in the first dimension. Let be the set of linguistic information evaluation in the second dimension. Then, is defined as a 2-dimension 2-tuple linguistic term [14].

##### 3.3. Evidence Theory

*Definition 6. *Let be a set of mutually exclusive and collectively exhaustive events indicated bywhere set is called a frame of discernment. The power set of is indicated by :Dempster’s rule of combination, denoted by , is defined as follows [2]:with

#### 4. Problem Descriptions

Let and be two manufacturers involved in the competitive multiattribute group decision problems; then, their collections of alternatives, attributes, and experts are stated as follows.

The set of alternatives for manufacturers and is stated as and , respectively.

The set of attributes for manufacturers and is and , respectively.

The set of experts is and for manufacturers and , respectively. In this study, the experts express their evaluation information on the alternatives against every attributes in the form of 2-dimension 2-tuple linguistic variables, according to the different strategies of opponent. The standard linguistic term set of the first dimension is . The standard linguistic term set of the second dimension is . The utility value of the first dimension linguistic term set is . Thus, the competitive multiattribute group decision matrix is obtained as follows:where is the evaluation value of the alternative under the attribute given by expert for different strategies chosen by manufacturer *B* and is the evaluation value of the alternative under the attribute given by expert for different strategies chosen by manufacturer *A*.

As shown in Figure 1, the problem focused in this study is about the game between manufacturer *A* and manufacturer *B*, where experts of both sides give their evaluation information on the alternatives against the corresponding attributes in the form of 2-dimension 2-tuple linguistic variables. The purpose is to determine the Nash equilibrium for manufacturer *A* and manufacturer *B*.