Full Length ArticleValue determination method based on multiple reference points under a trapezoidal intuitionistic fuzzy environment
Introduction
Utility measurement is an important tool for decision analysis to help with informed decision making. Utility measurement is a descriptive task in most practical applications, and a descriptively valid theory is needed to construct valid and consistent utility measurement [4]. Traditionally expected utility theory is adopted by many decision makers for utility measurement because it is the dominant prescriptive theory for decision making under uncertainty. However, it is widely acknowledged that expected utility theory is not valid as a descriptive theory for individual decision making under risk [5]. The descriptive deficiencies of expected utility theory cause it to incorrectly reflect individual values and preferences in decision analyses and lead to biased utilities; the biased utilities, as a consequence, may lead to incorrect decisions.
Compared with expected utility theory, prospect theory is a descriptive model that can effectively describe and explain violations of expected utility. Currently, prospect theory has attracted widespread attention in the decision-making field [1], [10], [17], [21], [33], [34], [37]. In prospect theory, the reference point choice is undoubtedly the most important factor in utility measurement, and a basic assumption is that there is a single, fixed reference point. The reference point is often defined by the status quo [20]. However, empirical studies show that in addition to status quo, there are many other alternative reference points in the decision-making process, such as goals and aspirations [7], [20], [41], minimum requirements [25], and expected level of other schemes [6], [45].
Decision makers can give value judgments with one of the above reference points according to the specific decision-making environment and personal preferences. However, this is limited to the situation where decision makers have less information such as investor judgment regarding stock returns [2]. Under a complex environment, however, more than one reference point may be available, so decision makers may be confronted simultaneously with multiple reference points (mRPs) in a single decision setting [2], [6], [31], [44]. For example, in company strategy selection decisions, decision makers should consider three dimensions in the choice of the reference points as inter-organization and intra-organization aspects and time [11]. In a branded goods selection process, consumers may also synthetically consider multiple brand commodity prices to make informed decisions according to their experience [46]. Although Kahneman and Tversky [18] acknowledge that in some cases, gains and losses are coded relative to an expectation or aspiration level rather than status quo, they have not given the value determination method under mRPs.
For utility measurement under mRPs, there are two measurement patterns, the composite measurement model and the independent measurement model. In the composite measurement model, outcomes are compared with a single composite point integrated by mRPs; in the independent measurement model, outcomes are compared separately with each reference point, and the overall utility is obtained based on the above comparison results [12], [19].
Early theories, such as research by Olson, Roese and Zanna [30], and Ordóñez [32], suggested that mRPs should be combined into a single composite point. However, under complex environment, the complexity of decision problems increases and may result in poor comparability between each reference point; therefore, in reality, utility measurement under mRPs follows the independent measurement model [31]. Many empirical studies present evidence for the independent measurement model. Recently, Ordóñez, Connolly and Coughlan [31] examined the effects of two referents on ratings of salary satisfaction and fairness, and the results show that the focal salary is compared separately to each reference point rather than to a single composite point. Similarly, Koop and Johnson [20] presented evidence that individuals can utilize the minimum requirement, status quo and goal within a single risky decision task. In addition, the study of animal foraging behavior also supports the simultaneous impact of such mRPs [16].
To support decision making under the independent measurement model, Diener, Sandvik and Pavot [9] proposed a frequency model, in which individuals obtain positive experience and negative experience by comparing outcomes with different reference points. If the frequency of positive experience is higher than the frequency of negative experience, individuals can produce satisfied emotions; otherwise, the emotions are unsatisfied. However, this model is simply based on the frequency of the positive and negative feelings rather than specific utility values and cannot meet the need of effective decision making [19].
Further, the value determination method in prospect theory is only applicable to decision-making problems where information is represented with crisp numbers. In real life, however, due to the complexity of the decision-making problem itself, the limited knowledge of the decision makers, as well as the high cost and other constraints of obtaining accurate information, it is difficult or even impossible to represent decision information with crisp numbers. Intuitionistic fuzzy sets [3] provide a powerful tool to address the uncertainty and fuzzy information. Intuitionistic fuzzy numbers [13] are special form of the intuitionistic fuzzy sets. The trapezoidal intuitionistic fuzzy numbers [14], [23], [29], [42], [43] are the newest form of intuitionistic fuzzy numbers. Thus, it is more flexible and practical to construct a prospect value determination method using trapezoidal intuitionistic fuzzy numbers. Recently, various trapezoidal intuitionistic fuzzy multi-attribute decision-making methods based on prospect theory have been developed [22], [24], but only one reference point is used for utility measurement, which limits their use in real-life applications. Additionally, other scholars combined prospect theory with uncertain information such as Qin, Liu and Pedrycz [33], and Krohling and de Souza [21]. These studies provide motivation for this study.
In summary, this paper aims to re-examine the value determination methods of existing prospect theory under a complex environment and make a scientific, reasonable criticism and improvement by combining the changes. In particular, a value determination method based on mRPs under trapezoidal intuitionistic fuzzy environment is constructed.
This paper is organized as follows. In Section 2, a basic framework of the value determination based on mRPs under trapezoidal intuitionistic fuzzy environment is constructed. In Section 3, trapezoidal intuitionistic fuzzy numbers are defined and basic concepts of prospect theory and Dempster-Shafer theory based-induced ordered weighted average (DS-IOWA) operator are introduced. In Section 4, prospect theory is extended into a trapezoidal intuitionistic fuzzy environment. Based on Section 4, the condition of mRPs is considered, and a value determination method based on mRPs under trapezoidal intuitionistic fuzzy environment is constructed in Section 5. In Section 6, an illustrative example shows the practicality and effectiveness of the proposed method. Finally, concluding remarks are drawn in Section 7.
Section snippets
Research design and basic framework
To construct the value determination method under a trapezoidal intuitionistic fuzzy environment, prospect theory needs to be extended to the trapezoidal intuitionistic fuzzy environment. Prospect theory [18] suggests that the value should be determined based on the gains and losses rather than the absolute levels of wealth; therefore, the independent variable in the value function is the amount of change with respect to a reference point. The trapezoidal intuitionistic fuzzy distance
Preliminaries
In this section, some basic definitions of trapezoidal intuitionistic fuzzy numbers, cumulative prospect theory and DS-IOWA operator are introduced.
Trapezoidal intuitionistic fuzzy value function
A trapezoidal intuitionistic fuzzy prospect can be seen as a function from natural state set S = {s1, s2, ⋯, sn} to outcome set , and any state si∈S is assigned a corresponding outcome , such that , where , is the ith possible outcome, pi is the occurrence probability corresponding to , and is represented by a trapezoidal intuitionistic fuzzy number.
In prospect theory, determination of the value function is closely related to the chosen
Trapezoidal intuitionistic fuzzy prospect value determination method based on mRPs and DS-IOWA operator
In the previous section, the trapezoidal intuitionistic fuzzy prospect value determination method with a single preference point was discussed. However, this is only limited to the decision-making problems where decision makers have less decision information. Many real-world decision-making problems are not so simple under a complex environment, and it is necessary to consider the problem of mRPs [2], [6], [31].
According to the research design and basic framework of the value determination
Illustrative example
Suppose that a venture capital company wants to select the most appropriate alternative plan for investment. For each alternative, the investment result depends on the future economic conditions. Decision makers need to assess the prospect values of each alternative and select the best alternative with the greatest prospect value.
Step 1. Assume there are four alternatives A1, A2, A3, A4 for further selection after pre-evaluation, and there are four possible economic conditions {s1,s2,s3,s4}
Conclusion
Utility measurements are an important tool for effective decision analyses. Utility measurements are a descriptive task influenced by the psychological characteristics and behavioral patterns of the decision makers. In the complex decision-making environment, uncertainty and mRPs make it more difficult to obtain utility measurement. To solve this problem, this paper constructs a trapezoidal intuitionistic fuzzy prospect value function based on mRPs and DS-IOWA operator. It extends utility
Acknowledgments
This research work was supported by the National Natural Sciences Foundation of China (Nos. 71401184, 71431006), Key Project of Philosophy and Social Sciences Research, Ministry of Education PRC [No. 16JZD013], Research Fund for Innovation-driven Plan of Central South University (2015CX010), China Postdoctoral Science Foundation (No. 2014M552169).
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