A three-way decision model based on cumulative prospect theory

A Three-Way Decision Model Based on Cumulative Prospect Theory

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A Three-Way Decision Model Based on Cumulative Prospect Theory Tianxing Wang, Huaxiong Li, Libo Zhang, Xianzhong Zhou, Bing Huang PII: DOI: Reference:

S0020-0255(20)30032-3 https://doi.org/10.1016/j.ins.2020.01.030 INS 15161

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Information Sciences

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21 September 2019 13 January 2020 16 January 2020

Please cite this article as: Tianxing Wang, Huaxiong Li, Libo Zhang, Xianzhong Zhou, Bing Huang, A Three-Way Decision Model Based on Cumulative Prospect Theory, Information Sciences (2020), doi: https://doi.org/10.1016/j.ins.2020.01.030

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A Three-Way Decision Model Based on Cumulative Prospect Theory Tianxing Wanga , Huaxiong Lia,∗, Libo Zhanga,d , Xianzhong Zhoua,c , Bing Huangb a Department

of Control and Systems Engineering, School of Management and Engineering, Nanjing University, Nanjing 210093, PR China b School of Information Engineering, Nanjing Audit University, Nanjing 211815, PR China c Research Center for New Technology in Intelligent Equipment, Nanjing University, Nanjing 210093, PR China d School of Artificial Intelligence, Southwest University, Chongqing 400715, PR China

Abstract In three-way decision, the description on the risk attitude of decision-makers is a focus topic. In this paper, we propose a novel three-way decision model based on cumulative prospect theory. First, with the aid of a reference point, the value functions are utilized to describe different risk appetites of decision-makers towards gains and losses. Second, the weight functions incorporate the nonlinear transformation of the conditional probability. The cumulative decision weights are decided by taking into account the increasing order of value functions. Then, with value functions and weight functions, the new decision rules of the proposed model are deduced based on the principle of maximizing the cumulative prospect value rather than minimizing the cost. Further, we analyze and prove the existence and uniqueness of thresholds of our model. Then, the decision rules are simplified based on the conditional probability and numerical solutions of thresholds, and the algorithm for deriving three-way decision rules is constructed. Finally, an illustrative example and a series of relevant comparisons are presented to illustrate and validate the effectiveness and feasibility of our model. Keywords: Three-way decision, cumulative prospect theory, risk attitude, decision rules

1. Introduction Three-way decision (3WD), as an uncertain decision approach, has developed rapidly in the fields of decisionmaking, granular computing and incomplete data analysis, since it was proposed by Yao in [38, 39]. Three-way decision can be viewed as a generalized theory of traditional rough set, proposed by Pawlak [23], which received much attention in recent literatures [1, 5, 12, 27, 36]. The main concept of three-way decision is to divide the universe into three disjoint regions (positive, boundary and negative regions) by two thresholds α and β. Three types of decision rules, including acceptance, non-commitment and rejection, are generated from three disjoint regions based on the principle of Bayesian risk minimization. In the case that the information is insufficient, the non-commitment is taken as the delayed decision, which is in line with the human decision process. In recent years, three-way decision has been widely applied in diverse territories, such as recommendation systems [19], image recognition [11], cognitive concept learning [14], cost-sensitive learning [6], medical diagnosis [37] and incremental clustering [41]. The theory and development of three-way decision have received a constant attention and achieved many research results [2, 7, 24, 44]. In summary, the study of three-way decision has two main aspects. First, many researchers focused on the extension of the model and notion of three-way decision [22, 31, 40, 48]. Lang et al. [10] presented the concepts of probabilistic conflict, neutral and allied sets of conflicts, and discussed the mechanism for computing the thresholds. Xu et al. [34] proposed a new aggregation method of interval loss function based on the principle of justifiable granularity. Sun et al. [28] deduced the three-way decision based on the multigranulation fuzzy ∗ Corresponding

author: Huaxiong Li Email addresses: [email protected] (Tianxing Wang), [email protected] (Huaxiong Li), [email protected] (Libo Zhang), [email protected] (Xianzhong Zhou), [email protected] (Bing Huang)

Preprint submitted to Information Sciences

January 18, 2020

decision-theoretic rough set over two universes. The second aspect is the development of the sequential three-way decision and granular computing [8, 13, 42]. Li et al. [11] proposed a sequential three-way decision method for cost-sensitive face recognition. Yang et al. [35] proposed a unified dynamic framework of decision-theoretic rough sets for incrementally updating three-way probabilistic regions. Zhang et al. [43] defined a new concept of attribute ratio and established a dynamic three-way decision model. Qian et al. [25] proposed a generalized multigranulation sequential three-way decision model based on multiple different thresholds. In recent years, many scholars have tended to pay more attention to the description on the risk attitude of decision-makers [17, 20, 26, 46]. For example, Liang et al. [18] involved the risk appetite of the decision-maker into three-way decisions. Zhang et al. [45] utilized the utility function as a new risk measurement, and introduced the utility theory into 3WD theory. Thus, how to reflect the risk attitude of decision-makers has become a challenge and attracted increasing attention. However, most of the aforementioned literatures described the risk attitudes of decision-makers from the perspective of loss functions. The risks are mainly measured in terms of losses, while ignoring the impact of gains on decision-makers. In many real-life applications, the risk attitude and behavior of decision-makers are determined by both gains and losses. Some studies have shown that decision-makers may have different risk appetites towards gains and losses [21]. In the actual decision process of decision-makers, the pain of losses can be far greater than the pleasure of gains [47]. Cumulative prospect theory (CPT), proposed by Tversky and Kahneman [30], is an important theory to describe and predict the decision behaviors under risk and uncertainty. It is stated in CPT that the outcomes are represented as gains and losses in comparison with the reference point of decision-makers instead of the final state of wealth [50]. The decision-makers are risk aversion towards gains and risk seeking towards losses [16]. Moreover, two nonlinear transformations of the probability are taken into account towards gains and losses, which overweight small probabilities, while underweight moderate and high probabilities. Instead of transforming each single probability, the cumulative distribution function of the decision weight is also considered. CPT has been verified by a large number of psychological experiments, and it can correctly predict the real decision behavior of decision-makers. In recent years, CPT has attracted growing attention in many diverse decision-making areas. For example, Hu et al. [4] proposed a cumulative prospect theory-based model to describe the charging behavior of battery electric vehicle drivers. Keskin [9] studied cumulative prospect theory preferences in rent-seeking contests. Zhao et al. [47] proposed an optimal execution model with transient price impact and permanent price impact under cumulative prospect theory. Zhou et al. [50] solved the problem of portfolio selection by a group decision-making process and considered the psychological behavior of experts with CPT. Wu et al. [32] established a comprehensive criteria system for the evaluation of renewable power sources based on CPT. Following CPT, Metzger et al. [21] investigated a framework for non-cooperative games in normal form where players have behavioral preferences. Li et al. [16] proposed an integrated cumulative prospect theory based hybrid-information multi-criteria decision making approach, which could be utilized to appraise and identify sustainable third party reverse logistics provider. Xu et al. [33] studied psychological behaviors of decision-makers and considered bounded rationality by using CPT. Motivated by the success of CPT in the researches on decision-making problems, we introduced CPT into 3WD theory to construct a new three-way decision model. A summary of the work completed in this study is shown as follows. First, a 3WD model based on CPT is proposed. Different risk attitudes of decision-makers and appetites towards gains and losses are considered on the basis of the reference point. The conditional probability is generalized into two different weight functions and the decision weight is performed over the cumulative distribution function. With the aid of value functions and weight functions, decision rules are determined based on the principle of maximizing the cumulative prospect value. Then, since the analytic solutions of thresholds of our model cannot be calculated directly, the existence and uniqueness of thresholds are analyzed and proven. The simplified form of decision rules and the algorithm for deriving three-way decision rules are presented. Finally, the proposed model is utilized to support an illustrative emergency decision-making problem. The illustrative example and comparative analysis show the effectiveness of our model. The remainder of the paper is presented as follows. In Section 2, some basic notions of three-way decision

2

Table 1: The loss function matrix

aP

C

¬C

λP P

λP N

aB

λBP

λBN

aN

λN P

λN N

and cumulative prospect theory are presented. Then, in Section 3, the novel three-way decision model based on cumulative prospect theory is proposed. Section 4 provides a further analysis of the thresholds and simplified decision rules. An illustrative example is given to elaborate the effectiveness of our model, and the proposed model is compared with some other 3WD models in Section 5. Section 6 concludes the paper and elaborates on future studies. 2. Preliminary The basic ideas and concepts of three-way decision [38, 39] and cumulative prospect theory [30] are briefly reviewed in this section. 2.1. The classical three-way decision model Suppose Ω = {C, ¬C} is a state set, which represents the object belongs to C or not, respectively. With respect to the three regions, the set of actions is given by A = {aP , aB , aN }, where aP , aB and aN represent three actions for classifying an object into three disjoint regions: POS(C), BND(C) and NEG(C). The loss function λij (i = P, B, N , j = P, N ) in different states is given by a 3 × 2 matrix, as shown in Table 1. In the matrix, λP P , λBP , λN P denote the losses incurred for taking actions aP , aB and aN on an object in C, respectively. λP N , λBN , λN N denote the losses incurred for taking the same actions on an object in ¬C. Usually, the equivalence class [o] is utilized to depict the object o, and P r(C|[o]) represents the probability of o belongs to C. Thus, the expected loss R(ai |[o]) for each object o can be calculated as follows: R(aP |[o]) = λP P P r(C|[o]) + λP N P r(¬C|[o]),

R(aB |[o]) = λBP P r(C|[o]) + λBN P r(¬C|[o]),

(1)

R(aN |[o]) = λN P P r(C|[o]) + λN N P r(¬C|[o]). According to Bayesian decision procedure, the minimum-cost decision rules can be induced as: (P0) If R(aP |[o]) ≤ R(aB |[o]) and R(aP |[o]) ≤ R(aN |[o]), decide o ∈ POS(C);

(B0) If R(aB |[o]) ≤ R(aP |[o]) and R(aB |[o]) ≤ R(aN |[o]), decide o ∈ BND(C);

(N0) If R(aN |[o]) ≤ R(aP |[o]) and R(aN |[o]) ≤ R(aB |[o]), decide o ∈ NEG(C). Considering (λP N − λBN )(λN P − λBP ) > (λBP − λP P )(λBN − λN N ) [38, 39], we can rewrite the decision rules (P0) − (N0) as: (P1) If P r(C|[o]) ≥ α1 , decide o ∈ POS(C);

(B1) If β1 < P r(C|[o]) < α1 , decide o ∈ BND(C);

(N1) If P r(C|[o]) ≤ β1 , decide o ∈ NEG(C);

3

v( x )

v ( x1 )

x2

x0

x1

x

v ( x2 )

Figure 1: The value function of cumulative prospect theory.

where α1 =

λP N − λBN λBN − λN N , β1 = . (λP N − λBN ) + (λBP − λP P ) (λBN − λN N ) + (λN P − λBP )

(2)

Otherwise, the decision rules (P0) − (N0) are expressed as: (P2) If P r(C|[o]) ≥ γ1 , decide o ∈ POS(C);

(N2) If P r(C|[o]) ≤ γ1 , decide o ∈ NEG(C);

where γ1 =

λP N − λN N . (λP N − λN N ) + (λN P − λP P )

(3)

2.2. Cumulative prospect theory Cumulative prospect theory, proposed by Tversky and Kahneman [30], provides an excellent description and explanation for the decision behavior and risk attitude of decision-makers under risk and uncertainty. The main idea of CPT can be summarized into three aspects. First, the outcomes are regarded as gains or losses relative to the reference point instead of the final status of wealth. Second, decision-makers are risk aversion towards gains, risk seeking towards losses, and are more sensitive on losses than gains. Third, the nonlinear transformation of the probability is incorporated with two decision weights towards gains and losses, which are performed over the cumulative distribution function. Suppose xh is defined as the h-th outcome and x ¯ is defined as the reference point selected by decision-makers. In CPT, the outcomes are represented as gains and losses in comparison with the reference point of decision-makers instead of the final state of wealth. If xh ≥ x ¯, the outcome is viewed as a gain. Otherwise, the outcome is deemed as a loss (xh < x ¯). Thus, the outcomes can be transformed into decision values, which is shown in Fig. 1. In this way, the value function curve is steeper in the loss than in the gain domain, since decision-makers are more sensitive on losses than gains. In the light of an experimental validation, Tversky and Kahneman [30] provided a detailed formula of the value function as follows.

4

w( p) 1

0

1

p

Figure 2: The weight function of cumulative prospect theory.

v(xh ) =

(

(xh − x ¯)µ , ν

−θ(¯ x − xh ) ,

xh ≥ x ¯,

(4)

xh < x ¯,

where µ = ν = 0.88, θ = 2.25. CPT illustrates that decision-makers may overweight small probabilities and underweight moderate and high probabilities, which is shown in Fig. 2. Besides, the weight functions may be different towards gains and losses, such that two nonlinear transformations of the probability are taken into account. Suppose ph is the probability of the outcome xh . By the means of psychological experiments, the weight functions towards gains and losses of the single probability ph are given respectively as follows [30]. ph σ , (ph σ + (1 − ph )σ )1/σ ph δ w− (ph ) = , (ph δ + (1 − ph )δ )1/δ w+ (ph ) =

(5)

where σ = 0.61, δ = 0.69. Thereby, σ = 0.61 and δ = 0.69 are the factors that distinguish between the domain of gains and losses, and mainly control the curvature. In CPT, cumulative probabilities need to be weighted rather than single probability. All the outcomes are sorted in ascending order and decision weights are computed based on the cumulative distribution function. First, assume there are totally m + n + 1 outcomes and let us sort all the outcomes in ascending order as: x−m < · · · < x0 < · · · < xn , where the corresponding outcomes and probabilities are obtained as: x = (x−m , . . . , xn ), p = (p−m , . . . , pn ). Hence, the cumulative weight function πh for each outcome can be expressed as: πh =

(

w+ (ph + . . . + pn ) − w+ (ph+1 + . . . + pn ),

w− (p−m + . . . + ph ) − w− (p−m + . . . + ph−1 ),

h ≥ 0, h < 0,

(6)

where −m ≤ h ≤ n. For two special cases: h = n and h = −m, the cumulative weight function πh for the outcome is also given as: ( w+ (pn ), h = n, πh = (7) − w (p−m ), h = −m. It is stated in CPT that decision-makers prefer the decision option with the maximum cumulative prospect

5

Table 2: The outcome matrix

aP

C

¬C

xP P

xP N

aB

xBP

xBN

aN

xN P

xN N

value. With the aid of the value function v(xh ) and decision weight πh , the cumulative prospect value function V is expressed as: n X V = πh v(xh ). (8) h=−m

Many experiments have shown that CPT can explain and predict the behavior of decision-makers well under risk and uncertainty [15, 16, 33]. Therefore, this paper introduces cumulative prospect theory into three-way decision to describe and reflect the risk attitude and risk appetite of decision-makers. 3. Three-way decision based on cumulative prospect theory Cumulative prospect theory suggests that both gains and losses may affect the decision behaviour and risk attitude of decision-makers under risk and uncertainty. Decision-makers tend to choose the decision option with the maximum cumulative prospect value when facing the actual decision problems. On this basis, a new 3WD model can be constructed to properly describe and reflect the risk attitude of decision-makers. Suppose Ω = {C, ¬C} is a state set, which represents the object belongs to C or not, respectively. With respect to the three regions, the set of actions is given by A = {aP , aB , aN }. In CPT, the outcomes are utilized to describe the final state of wealth of decision-makers. Suppose the outcome xij (i = P, B, N , j = P, N ) in different states can be given by a 3 × 2 matrix, as shown in Table 2. In the matrix, xP P , xBP , xN P denote the outcomes incurred for taking aP , aB and aN on an object in C, respectively. xP N , xBN , xN N denote the outcomes incurred for taking the same actions on an object in ¬C. The outcome xij may be positive or negative, which is different from the loss function. Considering an actual decision-making situation, where the reasonable condition of outcomes satisfies the following restrictions: xN P
(10)

In this case, the outcomes of classifying an object o belonging to C into the negative region NEG(C) are strictly fewer than these into the positive region POS(C). The outcomes of classifying an object o belonging to ¬C into the positive region POS(C) are strictly fewer than classifying it into the negative region NEG(C). Suppose the decision-maker set is given as D = {d1 , d2 , . . . , dm }, where dk represents the k-th decision-maker. According to cumulative prospect theory, decision-makers will select the reference point on the basis of their decision 6

Table 3: The value function matrix

C

¬C

aP

vPk P

vPk N

aB

k vBP

k vBN

aN

k vN P

k vN N

preference. For simplicity, the reference point of the k-th decision-maker is defined as x ¯k . In light of CPT, if the outcome xij ≥ x ¯k , it is regarded as a gain; otherwise, it is regarded as a loss. With the aid of the reference point and outcome matrix, the value function in different states of each decision-maker can be calculated, as shown in formula k k (11). The value function matrix is presented in Table 3. In Table 3, vPk P , vBP , vN P denote the decision values of k k the k-th decision-maker incurred for taking aP , aB and aN on an object in C, respectively. vPk N , vBN , vN N denote the decision values of the k-th decision-maker incurred for taking the same actions on an object in ¬C. Similar to k the outcome xij , the value function vij may be positive or negative, which depends on the reference point x ¯k . vPk P = v(xP P ),

vPk N = v(xP N ),

k k = v(xBN ), vBP = v(xBP ), vBN k vN P

= v(xN P ),

k vN N

(11)

= v(xN N ).

On the basis of the restrictions of outcomes, we can also consider another reasonable condition and obtain the restrictions of the value functions. At the beginning, let us give the following proposition of the value function of cumulative prospect theory. Proposition 1. Let v(xh ) be the value function of CPT. Then v(xh ) is a monotonically increasing function of xh . Proof. Because the value function v(xh ) is given by formula (4), the derivative of v(xh ) can be calculated as follows. 0

v (xh ) =

(

µ(xh − x ¯)µ−1 , xh ≥ x ¯,

θν(¯ x − xh )ν−1 , xh < x ¯.

As stated in Section 2, we can obtain that µ = ν = 0.88, θ = 2.25. Then, we calculate that v 0 (xh ) > 0 always holds. Since v(xh ) is a continuous function, we can prove v(xh ) is a monotonically increasing function of xh . After the monotonicity of the value function of CPT is verified, we have the following proposition of the restrick tions of the value function vij . Proposition 2. Let xij be the outcome in different states. Then, if xN P < xBP ≤ xP P , xP N < xBN ≤ xN N , k xP N < xP P , xN P < xN N , the value function vij satisfies: k k k vN P
k k vPk N
vPk N < vPk P , k k vN P < vN N .

7

(12)

(13)

Proof. According to Proposition 1, if xN P < xBP ≤ xP P , for the k-th decision-maker, we have: ⇔

⇔

xN P < xBP ≤ xP P ,

v(xN P ) < v(xBP ) ≤ v(xP P ), k k k vN P < vBP ≤ vP P .

k k k k k k k k k k Thus, vN P < vBP ≤ vP P holds. Similarly, vP N < vBN ≤ vN N , vP N < vP P and vN P < vN N also hold.

In accordance with CPT, the probability is generalized into two different weight functions towards gains and losses. Besides, the decision weight is performed over the cumulative distribution function by ordering the outcomes. As mentioned in Section 2, all the outcomes need to be sorted in ascending order and identified as a gain or loss. Because the value function of CPT is proved to be a monotonically increasing function in Proposition 1, we can order the value functions of our model to obtain the weight function associated with each ack k tion. Thus, it indicates that the values of viP and viN need to be compared and ordered. For example, when k k 0 ≤ viN ≤ viP , the calculative process of the weight function πik (P r(C|[o])) at this condition is presented based on formula (7): πik (P r(C|[o])) = w+ (P r(C|[o])). Furthermore, since P r(C|[o]) + P r(¬C|[o]) = 1 and w+ (1) = 1, the calculative process of πik (P r(¬C|[o])) can be also expressed based on formula (6): πik (P r(¬C|[o])) = w+ (P r(C|[o]) + P r(¬C|[o])) − w+ (P r(C|[o])) = 1 − w+ (P r(C|[o])). k k , all the conditions of the weight function πik (P r(C|[o])) (i = P, B, N , and viN By comparing the values of viP k = 1, 2, . . . , m) of the k-th decision-maker are written as follows:  w+ (P r(C|[o])),       1 − w+ (P r(¬C|[o])),      1 − w− (P r(¬C|[o])), πik (P r(C|[o])) =  w− (P r(C|[o])),       w+ (P r(C|[o])),     w− (P r(C|[o])),

k k 0 ≤ viN ≤ viP ,

k k 0 ≤ viP ≤ viN , k k viN ≤ viP < 0,

k k viP ≤ viN < 0,

(14)

k k viN < 0 ≤ viP ,

k k viP < 0 ≤ viN .

Similarly, all the conditions of the weight function πik (P r(¬C|[o])) can be written as:  k k 1 − w+ (P r(C|[o])), 0 ≤ viN ≤ viP ,      k k  w+ (P r(¬C|[o])), 0 ≤ viP ≤ viN ,      w− (P r(¬C|[o])), v k ≤ v k < 0, iN iP πik (P r(¬C|[o])) = − k k  1 − w (P r(C|[o])), viP ≤ viN < 0,     k k   w− (P r(¬C|[o])), viN < 0 ≤ viP ,     + k k w (P r(¬C|[o])), viP < 0 ≤ viN .

(15)

Example 1. Let xP P = 5, xBP = 0, xN P = −6, xP N = −5, xBN = 1, xN N = 4. Suppose the reference point x ¯1 of the first decision-maker is 2. Based on formula (11), the value functions of the decision-maker d1 can be calculated

8

as follows: vP1 P = (5 − 2)0.88 ≈ 2.63,

1 vBP = −2.25 ∗ (2 − 0)0.88 ≈ −4.14,

1 0.88 vN ≈ −14.02, P = −2.25 ∗ (2 − (−6))

vP1 N = −2.25 ∗ (2 − (−5))0.88 ≈ −12.47,

1 vBN = −2.25 ∗ (2 − 1)0.88 = −2.25,

1 0.88 vN ≈ 1.84. N = (4 − 2)

1 1 1 1 1 1 To compare the values of viP and viN , we can obtain: vP1 N < 0 < vP1 P , vBP < vBN < 0, vN P < 0 < vN N . Based on formulas (14) and (15), the weight functions πi1 (P r(C|[o])) and πi1 (P r(¬C|[o])) can be calculated:

πP1 (P r(C|[o])) = w+ (P r(C|[o])), πP1 (P r(¬C|[o])) = w− (P r(¬C|[o])), 1 πB (P r(C|[o])) = w− (P r(C|[o])), 1 πB (P r(¬C|[o])) = 1 − w− (P r(C|[o])), 1 πN (P r(C|[o])) = w− (P r(C|[o])),

1 πN (P r(¬C|[o])) = w+ (P r(¬C|[o])).

With the aid of value functions and weight functions, the cumulative prospect value V k (ai |[o])(i = P, B, N, k = 1, 2, . . . , m) associated with different actions in A = {aP , aB , aN } can be expressed as follows: V k (aP |[o]) = vPk P πPk (P r(C|[o])) + vPk N πPk (P r(¬C|[o])),

k k k k V k (aB |[o]) = vBP πB (P r(C|[o])) + vBN πB (P r(¬C|[o])), k

V (aN |[o]) =

k k vN P πN (P r(C|[o]))

+

(16)

k k vN N πN (P r(¬C|[o])).

Based on the restrictions of value functions of our model, all possible conditions of the value relationship between k k can be obtained, which are shown in Table 4. Hence, we can acquire the cumulative prospect value and viN viP k V (ai |[o])(i = P, B, N, k = 1, 2, . . . , m) associated with taking different actions under each possible condition. For the cumulative prospect value V k (aP |[o]) associated with the action aP , we have: V k (aP |[o]) =

   

vPk P w+ (P r(C|[o])) + vPk N (1 − w+ (P r(C|[o]))), 0 ≤ vPk N < vPk P ,

vPk P w+ (P r(C|[o])) + vPk N w− (P r(¬C|[o])), vPk N < 0 ≤ vPk P ,    v k (1 − w− (P r(¬C|[o]))) + v k w− (P r(¬C|[o])), v k < v k < 0. PP PN PN PP

(17)

For the cumulative prospect value V k (aB |[o]) associated with the action aB , it holds:  k k vBP w+ (P r(C|[o])) + vBN (1 − w+ (P r(C|[o]))),      k + k +   vBP (1 − w (P r(¬C|[o]))) + vBN w (P r(¬C|[o])),     v k (1 − w− (P r(¬C|[o]))) + v k w− (P r(¬C|[o])), BP BN V k (aB |[o]) = k − k  v w (P r(C|[o])) + v (1 − w− (P r(C|[o]))),  BP BN    k k   vBP w+ (P r(C|[o])) + vBN w− (P r(¬C|[o])),     k k vBP w− (P r(C|[o])) + vBN w+ (P r(¬C|[o])),

9

k k 0 ≤ vBN ≤ vBP ,

k k 0 ≤ vBP < vBN ,

k k vBN ≤ vBP < 0,

k k vBP < vBN < 0, k k vBN < 0 ≤ vBP ,

k k vBP < 0 ≤ vBN .

(18)

k and v k Table 4: All possible conditions of the value relationship between viP iN

Case

The decision rule

k k Value relationship between viP and viN

1

(P)

2

(P)

0 ≤ vPk N < vPk P

3

(P)

4

(B)

5

(B)

6

(B)

7

(B)

k k vBP < vBN <0

8

(B)

9

(B)

k k vBN < 0 ≤ vBP

10

(N)

11

(N)

12

(N)

vPk N < 0 ≤ vPk P

vPk N < vPk P < 0

k k 0 ≤ vBN ≤ vBP

k k 0 ≤ vBP < vBN

k k vBN ≤ vBP <0

k k vBP < 0 ≤ vBN

k k 0 ≤ vN P < vN N

k k vN P < vN N < 0

k k vN P < 0 ≤ vN N

For the cumulative prospect value V k (aN |[o]) associated with taking the action aN , it is induced:

 k + k k  v k (1 − w+ (P r(¬C|[o]))) + vN  N w (P r(¬C|[o])), 0 ≤ vN P < vN N ,  NP k − k − k k V k (aN |[o]) = vN P w (P r(C|[o])) + vN N (1 − w (P r(C|[o]))), vN P < vN N < 0,    k − k + k k vN P w (P r(C|[o])) + vN N w (P r(¬C|[o])), vN P < 0 ≤ vN N .

(19)

Cumulative prospect theory suggests that decision-makers prefer the decision option with the maximum cumulative prospect value. Therefore, the maximum-cumulative-prospect-value decision rules can be induced as: (P3) If V k (aP |[o]) ≥ V k (aB |[o]) and V k (aP |[o]) ≥ V k (aN |[o]), decide o ∈ POS(C);

(B3) If V k (aB |[o]) ≥ V k (aP |[o]) and V k (aB |[o]) ≥ V k (aN |[o]), decide o ∈ BND(C);

(N3) If V k (aN |[o]) ≥ V k (aP |[o]) and V k (aN |[o]) ≥ V k (aB |[o]), decide o ∈ NEG(C). Although the new decision rules (P3) − (N3) have been derived based on the principle of the cumulative prospect value maximization, it is still complex for decision-makers to make the immediate decision. In most 3WD researches, the decision rules can be further simplified based on the probability and thresholds. Thus, it is important to make the further analysis on the constructed 3WD model. 4. The analysis of thresholds and simplification of decision rules The thresholds are critical parameters to simplify decision rules in the 3WD theory. As weight functions are the nonlinear functions of probability, the analytic solutions of thresholds cannot be directly calculated. Thus, it is important to further analyze the existence and uniqueness of thresholds. Apparently, if the thresholds of our model exist and are unique, we can also simplify decision rules with the numerical solutions of thresholds. In order to analyze and prove the existence and uniqueness of thresholds, some properties of the weight functions and cumulative prospect value functions should be given. Proposition 3. Let w+ (ph ), w− (ph ) be the weight functions of CPT. Then weight functions w+ (ph ) and w− (ph ) are both monotonically increasing functions. 10

Proof. Let F (ph ) = ln w+ (ph ). Then, we can compute the derivative of F (ph ) as follows. F 0 (ph ) =

σ ph σ−1 − (1 − ph )σ−1 − . ph ph σ + (1 − ph )σ

As σ = 0.61 and ph ∈ [0, 1], we can use MATLAB to graph the function and figure out that F 0 (ph ) > 0. Since F (ph ) is a continuous function when ph ∈ [0, 1], we have F (ph ) is a monotonically increasing function of ph . Thus, we obtain w+ (ph ) is a monotonically increasing function. Similarly, we can prove: w− (ph ) is a monotonically increasing function. Proposition 4. Let V k (aP |[o]) and V k (aN |[o]) be the cumulative prospect values of the decision-maker dk associated with aP and aN . Then, V k (aP |[o]) is a monotonically increasing function of P r(C|[o]). V k (aN |[o]) is a monotonically decreasing function of P r(C|[o]). Proof. Based on formulas (17), (19) and Proposition 3, the proof of Proposition 4 is straightforward. Proposition 5. Let V k (aB |[o]) be the cumulative prospect value of the decision-maker dk associated with aB . k k satisfies the Case 4, Case 6 or Case 8 in Table 4, V k (aB |[o]) is and vBN When the value relationship between vBP k k a monotonically increasing function of P r(C|[o]). When the value relationship between vBP and vBN satisfies the k Case 5, Case 7 or Case 9, V (aB |[o]) is a monotonically decreasing function of P r(C|[o]). Proof. Similar to Proposition 4, the proof of Proposition 5 is straightforward. k = V k (aB |[o]) − V k (aN |[o]) and VPkN = V k (aP |[o]) − V k (aN |[o]). Let VPkB = V k (aP |[o]) − V k (aB |[o]), VBN k and VPkN can be written as follows. Then, VPkB , VBN k k k k VPkB = vPk P πPk (P r(C|[o])) + vPk N πPk (P r(¬C|[o])) − vBP πB (P r(C|[o])) − vBN πB (P r(¬C|[o])),

k k k k k k k k k VBN = vBP πB (P r(C|[o])) + vBN πB (P r(¬C|[o])) − vN P πN (P r(C|[o])) − vN N πN (P r(¬C|[o])),

VPkN

=

vPk P πPk (P r(C|[o]))

+

vPk N πPk (P r(¬C|[o]))

−

k k vN P πN (P r(C|[o]))

−

(20)

k k vN N πN (P r(¬C|[o])).

k Based on formula (20), we can induce that: if P r(C|[o]) = 0, we have VPkB < 0, VBN ≤ 0 and VPkN < 0. If k k k k k k P r(C|[o]) = 1, we have VP B ≥ 0, VBN > 0 and VP N > 0. If VP B , VBN and VP N can be proven to be monotonically functions, we can obviously certify that thresholds of our model exist and are unique. Thus, we have the following propositions.

Proposition 6. Let VPkB = V k (aP |[o]) − V k (aB |[o]). Then, VPkB is a monotonically increasing function of P r(C|[o]). Proof. On the basis of Proposition 4, we obtain that V k (aP |[o]) is a monotonically increasing function of P r(C|[o]). Besides, according to Proposition 5, we have V k (aB |[o]) is a monotonically decreasing function of P r(C|[o]) when k k the value relationship between vBP and vBN satisfies the Case 5, Case 7 or Case 9 in Table 4. Therefore, we can k obviously have VP B is a monotonically increasing function of P r(C|[o]) in these conditions. k k When the value relationship between vBP and vBN satisfies the Case 4, Case 6 or Case 8, we can induce that k V (aB |[o]) is a monotonically increasing function of P r(C|[o]). Hence, the totally nine (3 × 3) conditions of the k k value relationship between vPk P , vPk N , vBP and vBN need to be analyzed to determine the monotonicity of VPkB . k k k k According to Proposition 2, we have vP N < vP P , vBP ≤ vPk P and vPk N < vBN . Then, there are totally five possible k k k k conditions of the value relationship between vP P , vP N , vBP and vBN . k k (1) Case 1 and Case 4: 0 ≤ vPk N < vPk P , 0 ≤ vBN ≤ vBP . For this case, we have: k k VPkB = (vPk P − vBP )w+ (P r(C|[o])) + (vPk N − vBN )(1 − w+ (P r(C|[o]))).

11

Since w+ (P r(C|[o]) is a monotonically increasing function, (1 − w+ (P r(C|[o]))) is a monotonically decreasing k k function and vPk P − vBP ≥ 0, vPk N − vBN < 0, we obtain VPkB is a monotonically increasing function of P r(C|[o]) by this time. k k (2) Case 2 and Case 4: vPk N < 0 ≤ vPk P , 0 ≤ vBN ≤ vBP . In this case, we have: k k VPkB = (vPk P − vBP )w+ (P r(C|[o])) + vPk N w− (P r(¬C|[o])) − vBN (1 − w+ (P r(C|[o]))).

According to Proposition 3, we have w+ (P r(C|[o])) is a monotonically increasing function, w− (P r(¬C|[o])) is a k monotonically decreasing function and (1 − w+ (P r(C|[o]))) is a monotonically decreasing function. As vPk P − vBP ≥ k k k 0, vP N < 0 and vBN ≥ 0, we obtain VP B is a monotonically increasing function of P r(C|[o]) at the moment. k k (3) Case 2 and Case 6: vPk N < 0 ≤ vPk P , vBN ≤ vBP < 0. In this case, we have: k k VPkB = vPk P w+ (P r(C|[o])) − vBP (1 − w− (P r(¬C|[o]))) + (vPk N − vBN )w− (P r(¬C|[o])).

We can obtain that w+ (P r(C|[o])) is a monotonically increasing function, (1 − w− (P r(¬C|[o]))) is a monotonically k k increasing function and w− (P r(¬C|[o])) is a monotonically decreasing function. As vPk P ≥ 0, vBP < 0, vPk N −vBN < k 0, we have VP B is a monotonically increasing function of P r(C|[o]) at this moment. k k (4) Case 2 and Case 8: vPk N < 0 ≤ vPk P , vBN < 0 ≤ vBP . By this time, we have: k k VPkB = (vPk P − vBP )w+ (P r(C|[o])) + (vPk N − vBN )w− (P r(¬C|[o])).

Considering w+ (P r(C|[o])) is a monotonically increasing function and w− (P r(¬C|[o])) is a monotonically decreasing k k function. As vPk P − vBP ≥ 0, vPk N − vBN < 0, we have VPkB is a monotonically increasing function of P r(C|[o]) at the moment. k k (5) Case 3 and Case 6: vPk N < vPk P < 0, vBN ≤ vBP < 0. At this moment, we have: k k VPkB = (vPk P − vBP )(1 − w− (P r(¬C|[o]))) + (vPk N − vBN )w− (P r(¬C|[o])).

Because (1−w− (P r(¬C|[o]))) is a monotonically increasing function, w− (P r(¬C|[o])) is a monotonically decreasing k k function, and vPk P − vBP ≥ 0, vPk N − vBN < 0, we have: VPkB is a monotonically increasing function of P r(C|[o]) at this time. Therefore, the monotonicity of VPkB is proven to be the monotonically increasing function of P r(C|[o]) at all five possible conditions. Apart from the above five conditions, all the other four conditions do not exist. Thus, VPkB is a monotonically increasing function of P r(C|[o]). Example 2. Let xP P = 5, xBP = 0, xN P = −6, xP N = −5, xBN = 1, xN N = 4. Suppose the reference point x ¯2 of the second decision-maker is 0. Based on formula (11), the value functions can be calculated: vP2 P ≈ 4.12, 2 2 2 2 2 2 2 2 2 vBP = 0.00, vN P ≈ −10.89, vP N ≈ −9.27, vBN = 1.00, vN N ≈ 3.39. Since vP N < 0 < vP P , 0 ≤ vBP < vBN , we can compute VP2B based on formulas (14), (15) and (20) as follows: VP2B = 4.12w+ (P r(C|[o])) + (−9.27)w− (P r(¬C|[o])) − w+ (P r(¬C|[o])). According to Proposition 3, because w+ (P r(C|[o])) is a monotonically increasing function, w− (P r(¬C|[o])) is a monotonically decreasing function and w+ (P r(¬C|[o])) is a monotonically decreasing function, we can obtain VP2B is a monotonically increasing function. k k Proposition 7. Let VBN = V k (aB |[o]) − V k (aN |[o]). Then, VBN is a monotonically increasing function of P r(C|[o]).

Proof. In the light of Proposition 4, we obtain V k (aN |[o]) is a monotonically decreasing function of P r(C|[o]). Besides, according to Proposition 5, we have V k (aB |[o]) is a monotonically increasing function of P r(C|[o]) when 12

k k the value relationship between vBP and vBN satisfies the Case 4, Case 6 or Case 8 in Table 4. Therefore, we can k obviously obtain VBN is a monotonically increasing function of P r(C|[o]) at these conditions. k k When the value relationship between vBP and vBN satisfy the Case 5, Case 7 or Case 9, we can induce that k V (aB |[o]) is a monotonically decreasing function of P r(C|[o]). Hence, the totally nine (3 × 3) conditions of the k k k k k value relationship between vBP , vBN , vN P and vN N need to be analyzed to determine the monotonicity of VBN . k k k k k k In accordance with Proposition 2, we have vN P < vBP , vBN ≤ vN N and vN P < vN N . Then, there are totally five k k k k possible positive and negative relationships between vBP , vBN , vN P and vN N . k k k k (1) Case 5 and Case 10: 0 ≤ vBP < vBN , 0 ≤ vN P < vN N . At this moment, we have: k k k + k k + VBN = (vBP − vN P )(1 − w (P r(¬C|[o]))) + (vBN − vN N )w (P r(¬C|[o])).

According to Proposition 3, (1 − w+ (P r(¬C|[o]))) is a monotonically increasing function and w+ (P r(¬C|[o])) is a k k k k k monotonically decreasing function. As vBP − vN P > 0, vBN − vN N ≤ 0, we have: VBN is a monotonically increasing function of P r(C|[o]) by this time. k k k k (2) Case 5 and Case 12: 0 ≤ vBP < vBN , vN P < 0 ≤ vN N . At this moment, we have: k k k − k k + VBN = vBP (1 − w+ (P r(¬C|[o]))) − vN P w (P r(C|[o])) + (vBN − vN N )w (P r(¬C|[o])).

We can obtain that (1 − w+ (P r(¬C|[o]))) is a monotonically increasing function, w− (P r(C|[o])) is a monotonically k k k k increasing function and w+ (P r(¬C|[o])) is a monotonically decreasing function. As vBP ≥ 0, vN P < 0, vBN −vN N ≤ k 0, we have: VBN is a monotonically increasing function of P r(C|[o]) at the moment. k k k k (3) Case 7 and Case 11: vBP < vBN < 0, vN P < vN N < 0. At this time, we have: k k k − k k − VBN = (vBP − vN P )w (P r(C|[o])) + (vBN − vN N )(1 − w (P r(C|[o]))).

Because w− (P r(C|[o])) is a monotonically increasing function, (1 − w− (P r(C|[o]))) is a monotonically decreasing k k k k k function, and vBP − vN P > 0, vBN − vN N ≤ 0, we have: VBN is a monotonically increasing function of P r(C|[o]) in this case. k k k k (4) Case 7 and Case 12: vBP < vBN < 0, vN P < 0 ≤ vN N . At this time, we have: k k k − k − k + VBN = (vBP − vN P )w (P r(C|[o])) + vBN (1 − w (P r(C|[o]))) − vN N w (P r(¬C|[o])).

We obtain that w− (P r(C|[o])) is a monotonically increasing function, (1 − w− (P r(C|[o]))) is a monotonically decreasing function and w+ (P r(¬C|[o])) is a monotonically decreasing function according to Proposition 3. As k k k k k − vN vBP P > 0, vBN < 0, vN N ≥ 0, we have: VBN is a monotonically increasing function of P r(C|[o]) in this case. k k k k (5) Case 9 and Case 12: vBP < 0 ≤ vBN , vN P < 0 ≤ vN N . At this moment, we have: k k k − k k + VBN = (vBP − vN P )w (P r(C|[o])) + (vBN − vN N )w (P r(¬C|[o])).

Since w− (P r(C|[o])) is a monotonically increasing function, w+ (P r(¬C|[o])) is a monotonically decreasing function, k k k k k and vBP − vN P > 0, vBN − vN N ≤ 0, we have: VBN is a monotonically increasing function of P r(C|[o]) in this case. k Therefore, the monotonicity of VBN is proven to be the monotonically increasing function of P r(C|[o]) at all five k possible conditions. Apart from the above five conditions, all the other four conditions do not exist. Thus, VBN is a monotonically increasing function of P r(C|[o]). Proposition 8. Let VPkN = V k (aP |[o]) − V k (aN |[o]). Then, VPkN is a monotonically increasing function of P r(C|[o]). Proof. In the light of Proposition 4, V k (aP |[o]) is a monotonically increasing function of P r(C|[o]) and V k (aN |[o]) is a monotonically decreasing function of P r(C|[o]). Thus, VPkN is definitely the monotonically increasing function 13

of P r(C|[o]). k Proposition 9. Let VPkB = V k (aP |[o]) − V k (aB |[o]), VBN = V k (aB |[o]) − V k (aN |[o]) and VPkN = V k (aP |[o]) − k k k k V (aN |[o]). Then, VP B , VBN and VP N all have only one zero point. k Proof. Let VPkB = V k (aP |[o]) − V k (aB |[o]). When P r(C|[o]) = 0, we can calculate VPkB = vPk N − vBN < 0. When k k k k P r(C|[o]) = 1, VP B = vP P − vBP ≥ 0. As VP B is a monotonically increasing function of P r(C|[o]), we obtain VPkB k has only one zero point. Similarly, we can prove VBN and VPkN both have only one zero point respectively. k After all the monotonic properties of VPkB , VBN and VPkN are attained respectively, the existence and uniqueness of thresholds can be further obtained. Thus, we have the following proposition.

Proposition 10. Let α2k , β2k and γ2k be the thresholds of the decision-maker dk . Then, α2k , β2k and γ2k exist and are unique. Proof. According to Proposition 9, we have V k (aP |[o]), V k (aB |[o]) have only one intersection. V k (aB |[o]), V k (aN |[o]) have only one intersection, and V k (aP |[o]), V k (aN |[o]) have only one intersection. From the 3WD theory, we obtain α2k is defined as the intersection between V k (aP |[o]) and V k (aB |[o]). β2k is defined as the intersection between V k (aB |[o]) and V k (aN |[o]). γ2k is defined as the intersection between V k (aP |[o]) and V k (aN |[o]). Thus, α2k , β2k and γ2k exist and are unique. Example 3. Let xP P = 5, xBP = 0, xN P = −6, xP N = −5, xBN = 1, xN N = 4. Suppose the reference point x ¯3 3 of the third decision-maker is −6. Based on formula (11), the value functions can be calculated as: vP P ≈ 8.25, 3 3 3 3 3 3 3 3 3 ≈ 4.84, vN vBP P = 0.00, vP N = 1.00, vBN ≈ 5.54, vN N ≈ 7.59. Since 0 < vP N < vP P , 0 < vBP < vBN , 3 3 3 3 3 0 ≤ vN P < vN N , we can compute VP B , VBN and VP N based on formulas (14), (15) and (20) as follows: VP3B = 8.25w+ (P r(C|[o])) + (1 − w+ (P r(C|[o]))) − 4.84(1 − w+ (P r(¬C|[o]))) − 5.54w+ (P r(¬C|[o])) = 7.25w+ (P r(C|[o])) − 0.7w+ (P r(¬C|[o])) − 3.84,

3 VBN = 4.84(1 − w+ (P r(¬C|[o]))) + 5.54w+ (P r(¬C|[o])) − 7.59w+ (P r(¬C|[o]))

= 4.84 − 6.89w+ (P r(¬C|[o])),

VP3N = 8.25w+ (P r(C|[o])) + (1 − w+ (P r(C|[o]))) − 7.59w+ (P r(¬C|[o])) = 7.25w+ (P r(C|[o])) − 7.59w+ (P r(¬C|[o])) + 1.

On the basis of Proposition 3, because w+ (P r(C|[o])) is a monotonically increasing function, w+ (P r(¬C|[o])) is a monotonically decreasing function, we can obtain VP3B , VP3B and VP3B are all monotonically increasing functions. 3 When P r(C|[o]) = 0, we can calculate: VP3B = −4.54, VBN = −2.05, VP3N = −6.59. When P r(C|[o]) = 1, we can 3 3 3 calculate: VP B = 3.41, VBN = 4.84, VP N = 8.25. Therefore, α23 , β23 and γ23 exist and are unique. In accordance with above mentioned propositions, because all the thresholds are proven to be unique, the simplified decision rules of the decision-maker dk can be induced as follows: (P4) If P r(C|[o]) ≥ α2k and P r(C|[o]) ≥ γ2k , decide o ∈ POS(C),

(B4) If P r(C|[o]) ≤ α2k and P r(C|[o]) ≥ β2k , decide o ∈ BND(C),

(N4) If P r(C|[o]) ≤ β2k and P r(C|[o]) ≤ γ2k , decide o ∈ NEG(C).

In 3WD, it is important to compare the values of α and β to determine whether it is a three-way decision process. As mentioned above, since the weight functions are the nonlinear transformation of the probability, the proposed model suggests that α2k , β2k and γ2k all do not have the analytic solutions. However, the existence and uniqueness of thresholds have already been proven. Therefore, to attain the simplified decision rules, the numerical solutions of thresholds are necessary. In practical problems and circumstances, we can use the dichotomy to calculate the 14

Algorithm 1 The algorithm for deriving three-way decision rules Input: The outcome matrix. Output: The three-way decision rules for each o ∈ U of every decision-maker. 1: for k ∈ [1, m] do 2: Select the reference point x ¯k according to the decision-making preference; k 3: Compute the value function vij based on formula (11); k k 4: Compare and order the values of viP and viN , and then determine the weight functions associated with each action; 5: Compute the cumulative prospect value V k (ai |[o]) with the aid of value functions and weight functions; 6: Calculate the numerical solutions of thresholds α2k , β2k and γ2k ; 7: if α2k > β2k then 8: for o ∈ U do 9: if P r(C|[o]) ≥ α2k then 10: decide o ∈ POS(C); 11: else if β2k < P r(C|[o]) < α2k then 12: decide o ∈ BND(C); 13: else if P r(C|[o]) ≤ β2k then 14: decide o ∈ NEG(C). 15: end if 16: end for 17: else 18: for o ∈ U do 19: if P r(C|[o]) ≥ γ2k then 20: decide o ∈ POS(C); 21: else if P r(C|[o]) < γ2k then 22: decide o ∈ NEG(C). 23: end if 24: end for 25: end if 26: end for k numerical solutions of thresholds by computing zeros of VPkB , VBN and VPkN . After that, we need to compare the values of α2k and β2k to decide whether it is a three-way decision process or a two-way decision process. If α2k > β2k , the simplified decision rules of the decision-maker dk can be further derived as:

(P5) If P r(C|[o]) ≥ α2k , decide o ∈ POS(C);

(B5) If β2k < P r(C|[o]) < α2k , decide o ∈ BND(C); (N5) If P r(C|[o]) ≤ β2k , decide o ∈ NEG(C).

Otherwise, the simplified decision rules of the decision-maker dk can be induced as: (P6) If P r(C|[o]) ≥ γ2k , decide o ∈ POS(C);

(N6) If P r(C|[o]) < γ2k , decide o ∈ NEG(C).

In general, we sum up six steps to describe the whole decision process for deriving three-way decision rules of the proposed 3WD model. The detailed information of the whole decision process is presented in Algorithm 1. Step 1: Given the outcome matrix. For k ∈ [1, m], select the reference point x ¯k according to the decision preference of the decision-maker dk . k k k Step 2: Compute the value function vij based on formula (11). Compare and order the values of viP and viN , and then determine the weight functions associated with each action.

15

Table 5: The outcome matrix of the emergency decision-making problem

C

¬C

aP

8

aB

−1

−9

aN

−9

−3 7

Step 3: Compute the cumulative prospect value V k (ai |[o]) with the aid of value functions and weight functions. Then, calculate the numerical solutions of thresholds α2k , β2k and γ2k of every decision-maker. Step 4: Compare the values of thresholds α2k and β2k . If α2k > β2k , then go to Step 5. Otherwise, go to Step 6. Step 5: For o ∈ U , derive its corresponding three way decision rules based on the probability P r(C|[o]) and thresholds α2k , β2k . If P r(C|[o]) ≥ α2k , decide o ∈ POS(C); else if β2k < P r(C|[o]) < α2k , decide o ∈ BND(C); else if P r(C|[o]) ≤ β2k , decide o ∈ NEG(C). Step 6: For o ∈ U , derive its corresponding three way decision rules based on the probability P r(C|[o]) and the threshold γ2k . If P r(C|[o]) ≥ γ2k , decide o ∈ POS(C); else if P r(C|[o]) < γ2k , decide o ∈ NEG(C). 5. Illustrative example and comparative analysis 5.1. An illustrative example With the improvement of the level of social and economic development, the process of industrialization and modernization is accelerated. At the same time, various types of social disasters and natural disasters such as safe production, earthquakes and tsunami constantly happened [3]. In recent years, the frequency of emergencies has been increasing, which shows characteristics of suddenness, uncertainty, unpredictability and complexity [49]. The higher frequency of these emergencies, the higher degree of persecution and the wider range of negative impacts are constantly challenging emergency management capabilities of human beings [29]. In order to minimize the negative impact of emergencies, effective management policies are demanded to reduce adverse effects and limit their spread. Thus, how to effectively evaluate the emergency management policy for reducing risks becomes an important research problem. For an emergency decision-making problem, if we reject a good management policy, some reliable choices may be missed. On the contrary, if we accept a bad management policy, there could be losses produced. Apparently, in the face of practical decision problems, decision-makers may hesitate to make up their mind in the case that the information is not sufficient. In this case, the non-commitment can be a credible decision action when it is hard to make an immediate decision. For the emergency decision-making problem, we assume there are two states Ω = {C, ¬C}, which represent that the emergency management policy is positive or negative, respectively. The set of action is given as A = {aP , aB , aN }, in which three actions for the policy denote acceptance, further consideration and rejection, respectively. Suppose there are nine experts evaluating the emergency management policies, which are denoted as D = {d1 , d2 , d3 , d4 , d5 , d6 , d7 , d8 , d9 }. Table 5 presents the outcome matrix of management policies. xP P , xBP and xN P denote the outcomes incurred for taking actions of acceptance, further consideration and rejection respectively, if it is a good policy. Similarly, xP N , xBN and xN N denote the outcomes incurred for taking the same actions, if it is a bad policy. From Table 5, it is obviously seen that xN P < xBP ≤ xP P , xP N < xBN ≤ xN N , xP N < xP P , xN P < xN N . In this example, there are eight different emergency management policies, which are denoted as U = {o1 , o2 , o3 , o4 , o5 , o6 , o7 , o8 }. Table 6 presents the conditional probability P r(C|[ot ])(t = 1, 2, . . . , 8) of eight emergency management policies. Thus, the proposed model is utilized to support the emergency decision-making problem. In our model, since it considers different risk attitudes of decision-makers, it is necessary to acquire the reference points of nine experts, which are given in Table 7. With the help of the reference point of each expert, the outcomes 16

Table 6: The probabilities of management policies

The probability

o1

o2

o3

o4

o5

o6

o7

o8

P r(C|[ot ])

0.30

0.35

0.41

0.43

0.48

0.55

0.59

0.67

Table 7: The reference points of nine experts

The reference point

d1

d2

d3

d4

d5

d6

d7

d8

d9

x ¯k

−11

−9

−7

−5

−3

−1

1

3

5

15

vPP

15

vNP

5 0 −5 −10

−5 −10

−20

−20 2

3

4 5 6 The index of experts

7

8

−25

9

vPN

0

−15

1

vBN

5

−15

−25

vNN

10 The value functions

The value functions

10

vBP

1

2

3

(a)

4 5 6 The index of experts

7

8

9

(b) Figure 3: The value functions of different experts.

can be transformed into value functions by formula (11). As the reference points of nine experts are not the same, the value functions of all the experts are different, which are all shown in Fig. 3. Since xN P < xBP ≤ xP P , k k k k k k k k xP N < xBN ≤ xN N , xP N < xP P , xN P < xN N , we can obtain vN P < vBP ≤ vP P , vP N < vBN ≤ vN N , vP N < vP P , k k k k vN P < vN N . From Fig. 3, we observe that the value function curves of viP and viN are becoming steeper when value functions are smaller than 0. The result indicates that decision-makers are more sensitive on losses than gains. After the value functions are all calculated, the weight functions associated with taking each action need to be obtained by comparing the corresponding value functions. For instance, for the expert d1 , we obtain the value 1 1 1 1 functions satisfy: 0 ≤ vP1 N < vP1 P , 0 ≤ vBN < vBP , 0 ≤ vN P < vN N . Thus, the cumulative prospect value 1 V (ai |[o])(i = P, B, N ) associated with taking different actions in A = {aP , aB , aN } can be written by formulas (17)-(19) as follows: V 1 (aP |[o]) = vP1 P w+ (P r(C|[o])) + vP1 N (1 − w+ (P r(C|[o]))),

1 1 V 1 (aB |[o]) = vBP w+ (P r(C|[o])) + vBN (1 − w+ (P r(C|[o]))), 1

V (aN |[o]) =

1 vN P (1

+

− w (P r(¬C|[o]))) +

(21)

1 + vN N w (P r(¬C|[o])).

Similarly, the cumulative prospect values of all the other eight experts can also be calculated. Due to the limited space, we only present the calculative process of the cumulative prospect values of the first expert. Then, the numerical solutions of all the thresholds of nine experts can be calculated by computing the zeros of VPk B , 17

α

β

γ

0.8

The value

0.7 0.6 0.5 0.4 0.3 0.2 -11

-9

-7

-5 -3 -1 1 The reference points

3

5

Figure 4: The calculated thresholds with the variation of reference points.

k VBN and VPk N with the aid of the dichotomy. Therefore, all the thresholds of nine experts are shown in Fig. 4. With the increase of reference points, the value of the threshold α2k will get larger first and then get smaller. The value of the threshold β2k will get smaller first and then get larger. Moreover, the value of the threshold γ2k will be always between α2k and β2k . From Fig. 4, we can obtain that the thresholds of experts d1 , d2 , d3 , d4 , d5 and d6 all satisfy: α2k > β2k . Therefore, these six experts select the decision rules (P5) − (N5), which means that the decision processes are all three-way decision processes. For the experts d7 , d8 and d9 , we obtain α2k ≤ β2k . Thus, these three experts may select the decision rules (P6) − (N6), which indicates that the decision processes are two-way decision processes. Moreover, since the reference points of nine experts are different, the calculated thresholds and their simplified decision rules are not the same. Given a management policy ot , we can determine its decision action by comparing the conditional probability P r(C[ot ]) with the thresholds (α2k , β2k , γ2k ). In the light of the decision rules (P5) − (N5) and (P6) − (N6), the concrete decisions of emergency management policies of different experts are shown in Table 8. From Table 8, we can observe that the reference points of experts can influence the decision result of each management policy. For example, when x ¯1 = −11, we are able to obtain the decision results of the expert d1 as: deciding o6 , o7 , o8 ∈ POS(C), o5 ∈ BND(C) and o1 , o2 , o3 , o4 ∈ NEG(C). When x ¯2 = −9, we can also obtain the decision results of the expert d2 as: deciding o7 , o8 ∈ POS(C), o4 , o5 , o6 ∈ BND(C) and o1 , o2 , o3 ∈ NEG(C). Moreover, the experts with different reference points may acquire the same decision results for management policies. For instance, for d4 and d5 , the decision results for these two experts are both determined as: deciding o1 , o2 , o3 , o4 , o5 , o6 , o7 , o8 ∈ BND(C). For d8 and d9 , the decision results are both obtained as: deciding o5 , o6 , o7 , o8 ∈ POS(C) and o1 , o2 , o3 , o4 ∈ NEG(C). Furthermore, we can observe the general tendency of the variation of decision results from Table 8. With the increase of the reference point, the number of management policies in the region POS(C) decreases first and then increases. The number of management policies in the region BND(C) increases first and then decreases. These are all in connection with the reference points and thresholds of different experts, and are in line with Fig. 4.

5.2. Comparative analysis In order to illustrate and verify the effectiveness and feasibility of the proposed model, we compare our model with the classical 3WD model [38] and utility theory based 3WD model [45]. The two 3WD models are both applied into the above emergency decision-making problem, and the results are compared with that of the proposed model. For the classical 3WD model [38], since risks are measured by loss functions, the outcome matrix of policies needs to be transformed into the loss function matrix. For all xij ≥ 0, it is suggested that it will not produce any 18

Table 8: The decision results of management policies of our model

d1

POS(C)

BND(C)

NEG(C)

{o6 , o7 , o8 }

{o5 }

{o1 , o2 , o3 , o4 }

{o2 , o3 , o4 , o5 , o6 , o7 }

{o1 }

{o7 , o8 }

d2

{o4 , o5 , o6 }

{o8 }

d3 d4

∅

{o1 , o2 , o3 , o4 , o5 , o6 , o7 , o8 }

{o8 }

{o3 , o4 , o5 , o6 , o7 }

∅

d5 d6

{o5 , o6 , o7 , o8 }

d8

1

∅

{o1 , o2 , o3 , o4 , o5 }

∅

{o1 , o2 , o3 , o4 }

{o1 , o2 , o3 , o4 }

β

1

β

8

9

2

0.8 The value of threholds

The value of thresholds

{o1 , o2 }

1

0.8

0.6

0.4

0.2

0

∅

∅

{o5 , o6 , o7 , o8 }

d9

∅

{o1 , o2 , o3 , o4 , o5 , o6 , o7 , o8 }

{o6 , o7 , o8 }

d7

{o1 , o2 , o3 }

0.6

0.4

0.2 α1 1

α2 2

3

4 5 6 The index of experts

7

8

0

9

(a)

1

2

3

4 5 6 The index of experts

7

(b)

Figure 5: The comparison of thresholds α and β between Yao’s model and our model.

loss, so that the loss function is set as 0. For all xij < 0, the loss function is set as λij = −xij , which is contrary to the outcome. With all the loss functions, the thresholds of the classical 3WD model can be calculated by formulas (2) and (3): α1 = 0.8571, β1 = 0.2727, γ1 = 0.5000. In Fig. 5, we compare the thresholds of the classical 3WD model with those of our model. As the classical 3WD model does not consider different risk attitudes of decision-makers, all the thresholds of nine experts are the same. For our model, since the reference points of nine experts are varied, the thresholds and decision rules will change based on the decision preference of experts. Then, we obtain the thresholds of the classical 3WD model satisfy that α1 > β1 . Thus, all the nine experts will select the decision rules (P1) − (N1), which indicates that the decision processes are all three-way decision processes. Since all the probabilities of policies satisfy: 0.2727 < P r(C|[ot ]) < 0.8571, the decision results of eight management policies are all decided as: deciding ot ∈ BND(C). Compared with the classical 3WD model [38], the proposed model has the following advantages: (1) Different risk attitudes of decision-makers and risk appetites towards gains and losses are taken into account in our model. By selecting the reference point, the outcomes can be transformed into different value functions according to the preference of decision-makers.

19

Table 9: The risk attitude parameters and thresholds of Zhang’s model

x∗ x ∗

0

x

d1

d2

d3

d4

d5

d6

d7

d8

d9

15

45

10

30

15

25

45

50

55

−11

−9

−7

−5

−3

−1

1

3

5

−12

−10

−15

−12

−11

−15

−10

−9

−9

α3

0.5962

0.6824

0.4705

0.5518

0.5160

0.4610

0.5941

0.6944

0.6813

β3

0.4056

0.3344

0.5020

0.4398

0.4669

0.5092

0.4057

0.3196

0.3310

γ3

0.4917

0.4937

0.4878

0.4904

0.4891

0.4874

0.4913

0.4934

0.4931

(2) The cumulative probabilities are weighted rather than single probability. The weight function incorporates the nonlinear transformation of the probability since individuals will assign large weight to small probability and assign small weight to moderate and large probability. (3) The decision rules are determined based on the cumulative prospect value maximization rather than cost minimization, which may describe and reflect the risk attitude and preference of decision-makers better. For the utility theory based 3WD model [45], since the model takes into account risk attitudes of decision-makers, it is necessary to acquire the risk attitude parameters of nine experts. Suppose the risk attitude parameters of nine experts are given in Table 9. In the utility theory based 3WD model, the utility functions are utilized to calculate thresholds from the outcomes with the aid of risk attitude parameters. Suppose the thresholds of the utility theory based 3WD model are defined as α3 , β3 and γ3 . Thus, all the thresholds of nine experts in the utility theory based 3WD model can be computed, which are shown in Table 9. The decision rules of different experts under our model and Zhang’s model are all presented in Fig. 6. In Fig. 6, we use three ranges to individually represent three disjoint regions POS(C), BND(C) and NEG(C). For our model, with the increase of reference points, the region POS(C) is decreasing first and then increasing. The region BND(C) is the opposite. For the experts d7 , d8 and d9 , the decision rules do not have the region BND(C), which implies that the decision processes are two-way decision processes. For Zhang’s model, the regions POS(C), BND(C) and NEG(C) do not change regularly with the variation of reference points since there are three risk attitude parameters. As shown in Fig. 6, we can observe that the decision rules of our model and Zhang’s model are different. As all the probabilities of management policies are given in Table 6, all the decision results of these management policies under Zhang’s model can also be attained, shown in Table 10. According to the above analysis in Section 3, the decision procedure of our model can always determine the maximum-cumulative-prospect-value decision rules. Therefore, after all the decision results of Zhang’s model and our model are determined, the cumulative prospect values of each management policy can be calculated with the corresponding decision actions. All the calculated cumulative prospect values are shown in Fig. 7, which clearly illustrates that the cumulative prospect values of our model are always greater than or equal to those of utility theory based 3WD model. These results demonstrate that the decision results and rules comply with the principle of the cumulative prospect value maximization. Moreover, the cumulative prospect values may be positive or negative, which is accordance with the reference point selected by experts. Compared with the utility theory based 3WD model [45], the proposed model has the following advantages: (1) Although both 3WD models consider risk attitudes of decision-makers, the number of risk attitude parameters of our model is smaller than that of parameters of Zhang’s model. From Table 7 and Table 9, we obtain the values of the reference point x ¯k are equal to that of the parameter x0 . It is shown that the risk attitude parameter of our model is more reasonable.

20

BND(C)

POS(C)

NEG(C)

The value

The value

NEG(C) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1

2

3

4

5

6

7

8

BND(C)

POS(C)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1

9

2

The index of experts

(a) Our model

3

4 5 6 The index of experts

7

8

9

(b) Zhang’s model

Figure 6: The decision rules of our model and Zhang’s model.

Table 10: The decision results of management policies under Zhang’s model

d1 d2 d3 d4 d5 d6 d7 d8 d9

POS(C)

BND(C)

NEG(C)

{o8 }

{o3 , o4 , o5 , o6 , o7 }

{o1 , o2 }

∅

{o1 , o2 , o3 , o4 , o5 }

{o5 }

{o1 , o2 , o3 , o4 }

∅

{o2 , o3 , o4 , o5 , o6 , o7 , o8 }

{o7 , o8 }

{o5 , o6 }

{o6 , o7 , o8 }

∅

{o1 , o2 , o3 , o4 , o5 }

{o2 , o3 , o4 , o5 , o6 , o7 , o8 }

{o1 }

{o6 , o7 , o8 } {o6 , o7 , o8 } {o8 } ∅

∅

{o3 , o4 , o5 , o6 , o7 }

{o2 , o3 , o4 , o5 , o6 , o7 , o8 }

{o1 }

{o1 , o2 , o3 , o4 }

{o1 , o2 } {o1 }

(2) In our model, the influence of both gains and losses are considered under risk and uncertainty. The model expresses that decision-makers are risk aversion towards gains and risk seeking towards losses. Besides, our model takes into account that decision-makers are more sensitive on losses than gains. (3) The proposed model generalizes the probability into the weight function, which indicates that the decisionmaker may underweight large probability and overweight moderate and small probability. Moreover, cumulative probabilities are weighted rather than single probability, such that the weight functions of our model are determined with the aid of the comparison between value functions. (4) The decision rules are deduced based on the cumulative prospect value maximization rather than utility maximization. It is reported that CPT can achieve good results and performance when describing the behavior of decision-makers under uncertainty and risk, and it implies that our model may describe and reflect the risk attitude and preference better. 6. Conclusion In the 3WD theory, how to describe the risk attitude of decision-makers is a challenge. In this paper, we introduce cumulative prospect theory into three-way decision to construct a novel 3WD model. In the light of 21

7.5

7 6.5 6 2 3 4 5 6 7 8 The index of management policies

6

5.5 5 4.5 1

2.5 2 1.5 1 0.5 8 2 3 4 5 6 7 The index of management policies

Zhang's model

Cumulative prospect values

Cumulative prospect values

Our model

0 -1 -2 2 3 4 5 6 7 8 The index of management policies

-4 -5 -6 -7 -8 2 3 4 5 6 7 8 The index of management policies

Zhang's model

Cumulative prospect values

Cumulative prospect values

Our model

-3

(g) d7

2.5 2 8 2 3 4 5 6 7 The index of management policies

1

Zhang's model

-2

-3 -4 -5 1

Our model

-6

-8 -10 -12 2 3 4 5 6 7 8 The index of management policies

(h) d8

2 3 4 5 6 7 8 The index of management policies

(f) d6

-4

1

Our model

-1

(e) d5

-2

1

3

Our model

1

(d) d4 Zhang's model

4

(c) d3

2

1

Our model

3.5

(b) d2

3

1

2 3 4 5 6 7 8 The index of management policies

Zhang's model

4.5

Cumulative prospect values

(a) d1 Zhang's model

Our model

Zhang's model

Cumulative prospect values

1

Zhang's model

6.5

Cumulative prospect values

Our model

Cumulative prospect values

Cumulative prospect values

Zhang's model

8

Our model

-5 -7 -9 -11 -13 -15 -17 1

2 3 4 5 6 7 8 The index of management policies

(i) d9

Figure 7: The comparison of cumulative prospect values under decision results between our model and Zhang’s model.

CPT, different reference points of decision-makers and risk appetites towards gains and losses are described with the value function. The probability is generalized into two weight functions towards gains and losses, which are performed over the cumulative distribution functions. Moreover, the existence and uniqueness of thresholds of our model are analyzed and proven. Then, we simplify the decision rules and give the whole decision process for deriving three-way decision rules. However, there are still some deficiencies in our model. For example, the analytic solutions of thresholds cannot be calculated. The decision rules can only be simplified and derived based on the conditional probability and numerical solutions of thresholds. Our future work will take into account the application and experiment of the proposed model in the information tables. Acknowledgement This work is supported by National Nature Science Foundation (Nos. 61876079, 71671086, and 71732003), and the National Key Research and Development Program of China (Nos. 2016YFD0702100, 2018YFB1402600).

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25

Author Contribution Statement Tianxing Wang: Writing- Original draft preparation, Data curation, Software Huaxiong Li: Conceptualization, Formulation of overarching research goals, Reviewing and Editing, Funding acquisition. Libo Zhang: Methodology, Formal analysis. Xianzhong Zhou: Funding acquisition, supervision Bing Huang: Reviewing and Editing

Dear Editors, We declare that we DO NOT have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Thank you for your consideration. With kind regards, Sincerely yours, Huaxiong Li (Corresponding author, On behalf of all authors)