Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 9646303, 12 pages

https://doi.org/10.1155/2017/9646303

## Risky Multicriteria Group Decision Making Based on Cloud Prospect Theory and Regret Feedback

^{1}School of Economics and Management, Harbin Engineering University, No. 145, Nantong Street, Nangang District, Harbin 150001, China^{2}School of Computer Science and Technology, Harbin Institute of Technology, No. 92, Xidazhi Street, Nangang District, Harbin 150001, China^{3}College of Science, Harbin Engineering University, No. 145, Nantong Street, Nangang District, Harbin 150001, China

Correspondence should be addressed to Shuang Yao

Received 29 December 2016; Revised 12 April 2017; Accepted 26 April 2017; Published 25 May 2017

Academic Editor: M. L. R. Varela

Copyright © 2017 Yan Song 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

The assessment of risky linguistic variables has significant applications in multiattribute group decision problems. This paper focuses on risky multicriteria group decision making using linguistic variable assessment and proposes a new model which considers various and differential psychological behavior and the ambiguity of linguistic variable assessment across multicriteria risks. Based on the cloud prospect value assessment, this paper proposes a cloud prospect value aggregation method and consensus degree measurement. An improved feedback adjustment mechanism based on regret theory is employed as the consistency model, which complements prospect theory. The three theoretical methods together constitute the core elements of the proposed CPD (cloud prospect value consensus degree decision) model. The feasibility and validity of the new decision making model are demonstrated with a numerical example, and feedback performance was compared with conventional direct feedback. The proposed CPD approach satisfies given consistency threshold of 0.95 and 0.98 after three and four feedback loops, respectively. Compared to the proposed CPD method, direct feedback approach needs seven and ten feedback loops under the same threshold, respectively, which shows that the proposed model increases efficiency and accuracy of group decision making and significantly reduces time cost.

#### 1. Introduction

Multiple criteria decision making (MCDM) is an important part of modern decision science [1, 2]. Group decision making under multicriteria risk refers to decision problems whose criteria are random variables; that is, the criteria values change with the uncertain environment [3, 4]. The uncertainty is considered as risk under various states whose probabilities are known or measurable [5].

Many real-life decision making problems, such as investments, are group decision problems with multicriteria risk, often with a variety of uncertain factors and multiple uncertain states corresponding to multiple probabilities. Stock investment selection and decision making do not have exact attribute values due to the complex and uncertain environment with vagueness, ambiguity, and randomness.

Due to the complexity of the limited knowledge and decision maker (DM) perceptions, many alternative rankings occur for the uncertainty and randomness of the attribute value(s) [6]. According to their experience or related knowledge, DMs conduct fuzzy linguistic evaluations, such as “very poor,” “poor,” “fair,” “good,” and “very good.” Prospect theory [7] considers DM psychological factors, which effectively corrects their maximum subjective expected utility, and has been successfully applied to individual [8, 9] and group [10–12] decision making. Therefore, to find the most desirable alternative or rank feasible alternatives to support decision making, it is critical to transform uncertain linguistic assessment, which mainly includes crisp or fuzzy numbers and their deformation.

The traditional technique for order preference by similarity to an ideal solution (TOPSIS) proposed by Hwang and Yoon [13] is a widely used crisp number method for classical MCDM. The TOPSIS method determines a solution with the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution. Similar to TOPSIS, the VIKOR method proposed by Opricovic and Tzeng [14] considers the degree of closeness to the ideal solution using an integration function based on close to ideal solution. However, neither TOPSIS nor VIKOR considers the relative importance of the distance between two different reference points.

Fuzzy numbers, derived from fuzzy set theory, have become a main stream method to represent and handle uncertain attributes [15–18]. Criteria values include interval valued fuzzy numbers (IVFNs) [19] and the combination of the interval numbers fuzzy set and soft set [20]. The fuzzy set is well suited to dealing with vagueness [21]; for example, the hesitant fuzzy soft set based on soft and hesitant fuzzy sets has been successfully applied to group decisions [22].

The cloud theory is the innovation and development of a membership function for fuzzy set theory [23]. It transforms the uncertainty qualitative language concept into quantitative values [24] and has been successfully applied to data mining [25, 26], intelligent control, and intelligent algorithm modification [27, 28].

The different knowledge and evidence available to DMs result in inconsistent evaluations. To obtain the final group opinion, DMs should coordinate inconsistency in the group decision making process. An appropriate technique is to aggregate all the individual fuzzy preference relations and consistency processing [29, 30], repeatedly modifying the decision matrix.

Regarding the linguistic variable group decision problem, DMs usually participate in the decision making process and provide subjective evaluation in three different stages. The initial decision matrix is constructed in first stage, and the criteria weights are determined and incorporated in second stage. If the initial decision matrix fails the consistency check, the decision making process turns to the third stage and the initial decision matrix is reconstructed by the DMs. This process leads to multiple repeated subjective evaluation and increases computational complexity.

Although some current methods deal with linguistic variable assessment and provide feedback adjustment to some extent, some drawbacks remain. On one hand, current linguistic variable evaluation can only be applied to real numbers or triangular fuzzy numbers, which limits its application to other decision methods [31]. When prospect theory or related methods cannot be used directly [32], a transformation from linguistic variables to number values is required. On the other hand, feedback adjustment gives DMs little useful guidance to modify the decision matrix when the group evaluation does not satisfy the consistency threshold [33]. This modification process is time-consuming and blind, particularly for risky multicriteria group decision making. The number of DM modifications can be reduced by automatic negotiation, but this can lead to deviation from the DMs original intention to reach a consensus [22]. In addition, most group decision methods handle a single state rather than multiple states.

Group decisions based on linguistic variables are widespread in practice. Although the linguistic variables are processed through triangular fuzzy numbers or fuzzy logic, the variables are in a single state [11], not the multistate, which is not risky. For example, Wang and Lee [34] generalize TOPSIS to fuzzy multiple idea group decision making in fuzzy environment with single state criteria.

The cloud model introduced in [29] transforms the uncertainty relationship from qualitative (linguistic variable) to quantitative, accommodating fuzziness and randomness of qualitative evaluation. The model integrated two domains and constructed a mapping relationship between them. The cloud model is a continuous linguistic variable method [35] that has been widely used for multicriteria group decision making [36–38].

Therefore, we propose a method based on combining the cloud model and prospect theory, which integrates the advantages of vagueness and randomness from the cloud model and risk perception from prospect theory. The proposed model incorporates the cloud prospect value aggregation method and consensus degree measurement. An improved feedback adjustment mechanism provides the consistency model. Following Peng and Yang [19], we take full advantage of the nonexpected utility theory of regret-rejoice to compensate for the prospect theory. The new feedback adjustment rules handle DM inconsistency efficiently.

This paper proposes a new group decision making method that integrates the cloud model, prospect theory, and regret theory for risky multicriteria group decision making. Focusing on the group decision problem of multicriteria risky linguistic variables, we build a method for the cloud prospect consensus degree of risky multicriteria group decision making and feedback adjustment regulation. First, we use the cloud model to convert the uncertain linguistic variable problem to a risky multicriteria decision problem. The comprehensive prospect value is calculated, and the cloud prospect decision matrix of all DMs for all alternatives is constructed based on prospect theory. Then the decision matrixes for all DMs are aggregated and their consistency is measured. Those criteria scoring less than the acceptable threshold are returned to the DMs with guidance correction information according to the least regret value, and DMs modify their decision matrix with this correction information. Finally, the method sorts the alternatives.

#### 2. Problem Description and Theoretical Basis

Consider the group decision problem of multicriteria risk with the alternatives sets , and the associated criteria sets , where each criterion is mutually independent. The weight vector of the criteria layer is , which satisfies the constraint . The possible natural states of criteria have probability under state . For the set of DMs , represents the linguistic variable evaluation of alternative under criteria in state . We select the optimal scheme ranking under a cloud prospect decision framework with a higher level of consensus degree.

##### 2.1. Cloud Model

The cloud model reflects uncertain phenomena in the field of natural and social sciences. The usual cloud model [39] includes the complete cloud, left half cloud, and right half cloud models. It is one of the most powerful tools for characterizing the linguistic atom. The variates are assumed to be uniformly normally distributed. The complete cloud produces qualitative concepts with complete features. The derivative cloud model adds one or more parameters, generating a different form of the cloud model, such as triangular, trapezoidal, , and other clouds. The cloud model is usually denoted by a cloud droplet, , where is the expected value, which reflects the mathematical expectation of the qualitative concept; is the entropy, reflecting the fuzzy qualitative concept; and is the hyper entropy, which reflects the randomness of dispersion and the degree of certainty.

A single cloud droplet does not impact overall cloud characteristics, but the cloud droplet distribution reflects the ambiguity and randomness of cloud mapping. For effective integration of ambiguity and randomness, the universal law for most basic linguistic values in natural language can be defined as follows.

*Definition 1. *Suppose that is the qualitative concept of a quantitative universe . For , the membership degree, , of is a stochastic number with a stable random tendency. The membership degree of is called the cloud on the universe , denoted by ; namely, , for which .

*Definition 2. *Let be the qualitative concept in the universe . A random instance in , , satisfies , . When the certainty degree of can be expressed as , the distribution of in is called the normal cloud.

*Definition 3. *Let and be one-dimensional normal clouds in . The Hamming distance between and is defined by where In the decision process, the number of clouds has one-to-one correspondence with the linguistic variable value in . Therefore, most previous studies have used the golden section method to generate clouds [36]. However, this approach has some defects, so we instead use a method that can reinforce discrimination [38].

##### 2.2. Prospect Decision Theory Based on the Cloud Model

The cloud model combines uncertainty and ambiguity. Under the cloud model and prospect decision framework, DMs evaluate the alternatives against a reference point to estimate gain or loss and are more sensitive to loss. Prospect theory was proposed in 1979. The core concept is a prospect value, which includes a value and weight function, which reflects the bounded rationality of the DMs in the decision process. Many empirical studies have found that psychological behavior plays an important role in decision analysis. Compared with expected utility theory of uncertainty decision, prospect decision theory based on the cloud model proposed the reference point on the basis of the different effect from the reference point [39].

In prospect decision theory based on the cloud model (cloud prospect decision theory), a cloud prospect decision matrix , for alternative of attribute , is constructed for every DM. This matrix is composed of a cloud prospect value function, , and a cloud prospect value weight, . The prospect decision matrix of every DM is where is formed by the subjective perception of the DMs [10]:and reflects the probability weight function considering the DM’s risk attitude:where denotes the cloud evaluation of alternative of attribute for the th DM. Equation (4) suggests that DM , for the same attribute under the same state , compares the size of the cloud droplet for alternatives and () and calculates the distance between the cloud droplets, . The parameters , which represent the degree of concavity and convexity, respectively, of the regional value function for gains and losses, control the value function, , of the DM’s subjective perception. is the loss aversion coefficient. When , the DM is more sensitive to loss, that is, loss averse.

The probability weight is the subjective judgment depending on the possibility of certain events. The risk attribute coefficients and control the curvature of the prospect weight function. is the monotone increasing function of probability . When is very small, the DM could overestimate the slight probability of a given incident. When is large, , illustrating that DMs overlook large probability events [39]. Equation (5) shows that, for the same attribute under the same state , when the linguistic evaluation value of alternative is greater than that of alternative , the weight function is controlled by , which represents the risk revenue attribute coefficient. Larger means more adventurous DM behavior. In contrast, is the risk loss attribute coefficient, and Figure 1 shows the effect of on .