Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 2874954, 13 pages

https://doi.org/10.1155/2017/2874954

## An Intuitionistic Fuzzy Stochastic Decision-Making Method Based on Case-Based Reasoning and Prospect Theory

^{1}College of Economics and Management, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu 212003, China^{2}School of Computer Science and Informatics, De Montfort University, The Gateway, Leicester LE1 9BH, UK^{3}College of Mathematic Sciences, Yangzhou University, Jiangsu 225002, China

Correspondence should be addressed to Cuiping Wei

Received 23 January 2017; Revised 23 March 2017; Accepted 30 March 2017; Published 24 April 2017

Academic Editor: Franck Massa

Copyright © 2017 Peng Li 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

According to the case-based reasoning method and prospect theory, this paper mainly focuses on finding a way to obtain decision-makers’ preferences and the criterion weights for stochastic multicriteria decision-making problems and classify alternatives. Firstly, we construct a new score function for an intuitionistic fuzzy number (IFN) considering the decision-making environment. Then, we aggregate the decision-making information in different natural states according to the prospect theory and test decision-making matrices. A mathematical programming model based on a case-based reasoning method is presented to obtain the criterion weights. Moreover, in the original decision-making problem, we integrate all the intuitionistic fuzzy decision-making matrices into an expectation matrix using the expected utility theory and classify or rank the alternatives by the case-based reasoning method. Finally, two illustrative examples are provided to illustrate the implementation process and applicability of the developed method.

#### 1. Introduction

Multicriteria decision-making (MCDM) mainly searches for the most satisfactory solutions from finite alternatives under conflicting criteria. With the increasing complexity of socioeconomic environments, it is very hard for decision-makers (DMs) to express their preferences with crisp values. Fuzzy set (FS) theory was proposed and has been deeply studied from different disciplines [1–7].

Atanassov [8] introduced the concept of intuitionistic fuzzy sets (IFSs) as an extension of FS theory. Vahdani et al. [9] proposed a novel ELECTRE method to deal with intuitionistic fuzzy group decision-making problems. Hashemi et al. [10] put forward a compromise ratio approach for water resources management based on IFS. Mousavi et al. [11] constructed a VIKOR decision-making model for addressing intuitionistic fuzzy group decision-making problems. Mousavi et al. [12] introduced the difficulties in selection issues for manufacturing firms and used grey incidence method and IFS to solve them. Chen and Chang [13] constructed a novel type of similarity measure between IFSs and applied it to pattern recognition. Song et al. [14] proposed a new distance measure for IFSs by D-S theory. Xu and Liao [15] discussed the decision-making methods for IF preference relations.

To deal with more fuzzy decision-making problems, many researchers studied interval-valued intuitionistic fuzzy decision-making methods. Hashemi et al. [16] introduced a novel compromise ratio decision-making model in an interval-valued intuitionistic fuzzy environment. Wang et al. [17] studied the defects of the present score functions for IVIFNs and proposed a novel one based on prospect theory (PT). Cao et al. [18] developed a new decision-making model for interval-valued intuitionistic stochastic information by using set pair analysis. Wang et al. [19] researched a new decision-making model for interval-valued intuitionistic linguistic numbers. Yu et al. [20] developed an extended TODIM decision-making method for intuitionistic linguistic numbers. Wan and Li [21] proposed a novel mathematical programming to solve heterogeneous decision-making information.

For stochastic fuzzy decision-making problems, some methods for deriving the weights of criteria and ranking alternatives have been proposed. Wang and Li [22] put forward a decision-making approach based on score functions. Li et al. [23] put forward some stochastic multicriteria decision-making methods for IFNs. Zhou et al. [24–26] proposed some new decision-making approaches for stochastic extended grey numbers decision problems by prospect theory and regret theory.

We note that these methods are based on the information of decision-making problems themselves to determinate the weights of criteria and rank the alternatives. However, for some very complex decision-making problems, if we can use the data from historically successful decision-making cases to help decision-making, then the method would be more reasonable. Motivated by this idea, the case-based reasoning (CBR) and prospect theory (PT) are used to realize our goal.

Case-based reasoning (CBR) is an effective approach to obtain preferential information from the DMs’ decisions on selected cases. In recent years, there has been much research on CBR methods. Chen et al. [27] discussed a new CBR method for MCDM with crisp values. Chen and Chiu [28] use CBR models to solve service-oriented value chain design making problems. Koo and Hong [29] constructed a CBR method to capture the energy performance curve. Yan et al. [30] put forward a group decision-making method using the CBR method and applied it to waste water treatment. Fan et al. [31] proposed a CBR method to solve project risk management. Evans [32] studied the business management education issues by case-based assessment.

Based on CBR and PT, this paper aims to propose a new method for stochastic intuitionistic fuzzy decision-making problems in order to overcome the aforementioned two limitations. We first put forward a new score function for IFNs. The existing score functions mainly focus on distributing the hesitancy degree by various methods without considering the decision-making environment. If the decision-making matrix is determined, the decision-making results by those traditional score functions are the same no matter what the decision-making situation is. The proposed score function is suitable for some special decision-making environments, such as investing in an R&D project. By the PT and grey incidence analysis (GIA), we construct a mathematical programming model to calculate the weights of criteria according to the idea of CBR based on the test matrices. Moreover, on the basis of the weights of criteria and threshold values obtained by CBR, we use the EU to aggregate information in all natural states and the idea of CBR to rank the alternatives. The main contributions of this study are mentioned as follows:(i)A new score function is proposed that considers the decision environment.(ii)A novel decision pattern is put forward that uses CBR method by the aid of the historical successful decision cases and PT by considering DMs’ bounded rationality.(iii)Construct a mathematical programming model according to the idea of CBR and GIA, by which the clustering or ranking of alternatives is obtained.

The rest of this paper is organized as follows. Section 2 introduces the information regarding the concepts about IFSs, the new score function, PT, and GIA. Section 3 summarizes the main idea of CBR and proposes a CBR model for IFS based on the prospect theory. Section 4 gives an illustrative example for a venture capital company investing in an R&D project. Section 5 summarizes the findings and suggests a possible extension of future research.

#### 2. Preliminaries

##### 2.1. Intuitionistic Fuzzy Set

*Definition 1 ((IFS) [8]). *Let be a universe of discourse. An intuitionistic fuzzy set in can be expressed as , where , with the condition , . are called the membership and nonmembership degree functions of to , respectively, where , .

For an IFS , we call the hesitancy degree, representing the degree of indeterminacy of to . It is obviously observed that .

For convenience, is called an intuitionistic fuzzy number (IFN). The operational laws of IFNs are presented as follows [33].

Let and be any two IFNs; then(1);(2);(3), ;(4), .

*Definition 2 ((IFWA) [33]). *Let , where is a collection of IFNs. An intuitionistic fuzzy weighted averaging (IFWA) operator is defined by where is the weighting vector of , with and .

##### 2.2. Score Functions for IFNs

Let be an IFN; the score function of , as a tool to compare two given IFNs, is proposed by Chen and Tan [34] in 1994 as follows:

Let and be IFNs; if , then is smaller than , denoted by

*Example 3. *Let and be two IFNs. By using (2), we can obtain that . It is obvious that the above score function cannot discern the values between two IFNs in some cases.

Later, Ye [35] proposed a new score function as follows:For the IFNs and in Example 3, by (3), we can obtain The main problem of using (3) lies in the acquisition of the exact value of parameter .

In this paper, we propose the following new score function.

*Definition 4. *Let be IFNs. The new score function of can be mathematically expressed aswhere .

*Remark 5. *If and , then ; if , then .

In fact, this score function considers both the ratio of membership degree and nonmembership degree for certain IFN and those of all compared IFNs.

*Example 6. *Let , , and be three IFNs.

By (5) we can obtain that , , and . Thus .

By in [35], we can obtain The score function [35] receives different ranking for IFNs with different parameters, while (5) does not involve the choice of parameters. Moreover it takes into consideration the decision-making environment. For instance, there are eleven teams () in the course of a single-recycling game. Every team has ten games. Owing to the calendar, won four matches and lost four. Meanwhile, won three and lost three and won four and lost five. The results can be described by the three IFNs as follows: , , and . All the teams have an average win rate under 50% because other teams have not played any game. will face four matches while two. Both and have a large chance to beat their opponents because their win rate is just 50%. Both and will most likely obtain the results as and , respectively; that is, is most likely bigger than the others.

##### 2.3. Prospect Theory

The prospect theory (PT) can explain individual’s bounded rationality behavior. Since the appearance of PT, more and more researchers have carried out high number of studies [36–40]. PT can be described briefly as follows.

DMs make a decision mainly according to the prospect value. The prospect value is codetermined by both value function and probability weight function:where and are called the probability weight function and the value function, respectively.

The probability weight function is a mathematical function of the probability and is shown as follows:where is the risk gain attitude coefficient.

The value function reflects the deviations from the reference point and is described by an S-shaped value function (see Figure 1). The form of the value function can be expressed aswhere and determine the concavity and convexity, respectively (), reflects the loss aversion (), and is the deviation from the reference point.