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An Interval-Valued Three-Way Decision Model Based on Cumulative Prospect Theory
An Interval-Valued Three-Way Decision Model Based on Cumulative Prospect Theory
Zhou, Hongli;Tang, Xiao;Zhao, Rongle
Article An Interval-Valued Three-Way Decision Model Based on Cumulative Prospect Theory Hongli Zhou, Xiao Tang * and Rongle Zhao School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, China * Correspondence: email@example.com Abstract: In interval-valued three-way decision, the reﬂection of decision-makers’ preference under the full consideration of interval-valued characteristics is particularly important. In this paper, we propose an interval-valued three-way decision model based on the cumulative prospect theory. First, by means of the interval distance measurement method, the loss function and the gain function are constructed to reﬂect the differences of interval radius and expectation simultaneously. Second, combined with the reference point, the prospect value function is utilized to reﬂect decision-makers’ different risk preferences for gains and losses. Third,the calculation method of cumulative prospect value for taking action is given through the transformation of the prospect value function and cumulative weight function. Then, the new decision rules are deduced based on the principle of maximizing the cumulative prospect value. Finally, in order to verify the effectiveness and feasibility of the algorithm, the prospect value for decision-making and threshold changes are analyzed under different risk attitudes and different radii of the interval-valued decision model. In addition, compared with the interval-valued decision rough set model, our method in this paper has better decision prospects. Keywords: three-way decisions; accumulative prospect theory; risk attitude; interval value; threshold method MSC: 91B05; 91B06; 91B16; 91B86 Citation: Zhou, H.; Tang, X.; Zhao, R. An Interval-Valued Three-Way 1. Introduction Decision Model Based on Cumulative The three-way decision theory (TWD), proposed by Yao  in 2009, was applied to Prospect Theory. AppliedMath 2023, 3, address uncertain information based on the rough set theory. As an extension of two-way 286–304. https://doi.org/10.3390/ decisions in acceptance or rejection, it took the boundary region as the third decision rule appliedmath3020016 on the basis of the positive region and the negative region, that is, to make decision of Academic Editor: Tommi Sottinen non-commitment. In real life, people are often faced with a signiﬁcant amount of decision- making problems, and how to effectively evaluate decision risk for reducing decision loss Received: 21 February 2023 becomes an important research question. When information is insufﬁcient or inadequate, Revised: 18 March 2023 huge losses are produced if we reject a good decision or accept a bad one. Therefore, Accepted: 20 March 2023 increasing the non-commitment decision rules is can minimize the losses of decisions in Published: 3 April 2023 the three-way decision theory. In recent years, the three-way decision theory has gradually become an important decision-making method, which has been widely applied in the ﬁelds of information management, medical treatment, risk insurance investment, etc. [2–5]. Copyright: © 2023 by the authors. In the decision-theoretic rough sets (DTRSs), with the aid of the loss function, the Licensee MDPI, Basel, Switzerland. expected loss under three different decision rules was calculated according to Bayesian This article is an open access article decision procedure, and then the threshold was obtained from the principle of minimum distributed under the terms and expected loss . Xu et al.  analyzed the characteristics of the loss function in DTRSs conditions of the Creative Commons and the logical relationship between the loss function and threshold, and then proposed Attribution (CC BY) license (https:// a threshold calculation method based on the logical relationship between decision loss creativecommons.org/licenses/by/ objective functions. Considering that the difference in equivalence classes will affect the 4.0/). AppliedMath 2023, 3, 286–304. https://doi.org/10.3390/appliedmath3020016 https://www.mdpi.com/journal/appliedmath AppliedMath 2023, 3 287 decision result, Xie et al.  proposed an adaptive threshold calculation method based on similarity measure. Certain prior knowledge was used to presuppose the loss function, which led to some limitations in the application of the three-way decision theory. Without the loss function, Chen et al.  proposed an optimal threshold algorithm based on grid search, aiming at minimizing the sum of decision losses. Jia et al.  proposed a simulated annealing algorithm to address the optimal threshold problem, and veriﬁed the advantage of the algorithm in running time. Two thresholds, a and b, which are calculated according to the principle of minimum risk loss in decision-making, cannot reﬂect the subjective initiative of decision-makers well. Zhang et al.  introduced the utility theory, by replacing the loss function with the utility function and proposing a utility three-way decision model (UTWD) in order to reﬂect the decision-maker ’s attitude toward risk better. The prospect theory (PT), established by Kahneman and Tversky in 1979 , reveals the reason and essence of people’s decision-making behavior deviating from rationality under uncertainty. It can better reﬂect the decision-making preferences of decision-makers, supplement the deﬁciency of the expected utility theory, and has been widely applied in multi-attribute decision-making [13–16]. The cumulative prospect theory (CPT) was proposed in 1992 , considering that the PT cannot solve the stochastic dominance prob- lem, and it has a wider application range compared with the PT [18,19]. Wang et al.  thought that the utility theory, which relies on intuitive decision-making, can reduce the complexity of decision-making, but cannot reﬂect the attitude toward the loss when facing risks. Therefore, they introduced the PT into a three-way decision model and proposed the prospect theory-based three-way decision model (PTWD). On the basis of the PTWD, Wang et al.  introduced the CPT to linearize the weight function further and proposed a three-way decision model based on the cumulative prospect theory (CPTWD). The data involved in prospect theory are all in the form of single value, while the complexity of the environment and the existence of irrational factors, such as decision- makers’subjective preference, emotional thinking, etc, lead to the uncertainty of decision- making risk. Therefore, it is closer to real life by describing the outcome function in prospect theory with interval numbers characterized by multi-value. Yin et al.  transformed the interval number into the form of the score function, and introduced the prospect value function to describe the subjective feeling of decision-making. Hu et al. , ﬁrstly, dispersed the interval number into different ﬁnite data, and described the distribution law of the values in the interval value by using the normal distribution function, then obtained the total decision prospect through the weighted average method. Xiong et al.  reserved the features of interval value to directly calculate the interval-valued prospect, and then derived the synthetic foreground value of each decision-making rule based on the determination factor rule library. Fan et al.  treated the reference point as single value according to the positional relationship between the reference point and the attribute value, and calculated the loss value and the gain value on the basis of the prospect value function. This method only considered the upper bound or the lower bound of the interval in the treatment of the reference point. In addition, when the reference point is included in the attribute value, the loss value and the gain value are both regarded as 0, but in fact, the attribute values including the reference point are also different. Besides interval numbers, Wang et al.  used Z-numbers to describe uncertainty in decision-making, and proposed a three-way decision model combined with Z-numbers and the third-generation prospect theory. Inspired by the above observation, we use interval values to describe the cumulative prospect theory. In order to address interval values, we adopt the interval-valued distance measurement method  characterized by similarity to describe the loss value and the gain value. The advantages of the proposed model are summarized as follows. (1). Interpret the distance between the two interval values as the beneﬁt of taking action. Since the decision-makers have different attitudes toward loss and gain, the distance between two interval values is studied from two angles, namely the gain distance and AppliedMath 2023, 3 288 the loss distance. It can measure the prospect value more accurately when generated by taking action. (2). Combining the value function and interval-valued distance with similar characteristics can better distinguish the difference between different interval values, especially when the two interval values have the same expectation. (3). On the basis of , using interval values to describe the outcome matrix is more in line with the actual situation. At the same time, the model proposed in this paper can also address the outcome matrix in the form of single values. Thus, it has a wider range of application. The remainder of this paper is detailed below. In Section 2, some basic concepts of interval value, classical three-way decision model, and cumulative prospect theory are presented. In Section 3, a new method of measuring the prospect value based on the interval value is proposed, and then an interval-valued three-way decision model based on the cumulative prospect theory is constructed. The thresholds and simpliﬁed decision rules are further analyzed in Section 4. In Section 5, an example is given to illustrate the effectiveness of our model in distinguishing different interval values; then, the proposed model is compared with the interval number three-way decision model. The whole methods and experiments’ results conclude in Section 6. 2. Preliminaries 2.1. Basic Theory of Intervals + + Deﬁnition 1 (). Let R denote the set of real numbers. For 8a , a 2 R and a a , then + + a ˜ = [a , a ] is called an interval value, where a and a represent the lower and upper bounds of the interval value, respectively. In particular, if a = a , the interval value degenerates to a + + + real number. Supposing b = [b , b ] is another interval value, if a = b and a = b , we have a ˜ = b. + a +a Deﬁnition 2 (). Let a ˜ = [a , a ] is an interval value, then is called the expectation of a a the interval value a ˜, denoted by m(a ˜). Furthermore, is called the radius of the interval value a ˜, denoted by r(a ˜). Evidently, a ˜ = [m(a ˜) r(a ˜), m(a ˜) + r(a ˜)], that is, the expected value and the radius of the interval can exactly describe an interval value. + + Deﬁnition 3 (). Given two interval values a ˜ = [a , a ], b = [b , b ], and a real number k, then deﬁne the operational relationship between them as follows: + + (1). a ˜ + b = [a + b , a + b ]; + + (2). a ˜