# Rough-Number-Based Multiple-Criteria Group Decision-Making Method by Combining the BWM and Prospect Theory

Authors: Fan Jia; Xingyuan Wang
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

Multicriteria group decision-making (MCGDM) problems have been a research hotspot in recent years, and prospect theory is introduced to cope with the risk and imprecision in the process of decision-making. To guarantee the effectiveness of information aggregation and extend the feasibility of prospect theory, this paper proposes a novel decision-making approach based on rough numbers and prospect theory to solve risky and uncertain MCGDM problems. Firstly by combining rough numbers and the best-worst method (BWM), we construct a linear programming model to calculate rough criteria weights, which are defined by lower limitations and upper limitations. Then for the imprecision of value function and weighting function in prospect theory, we propose a novel method with the aid of combining rough numbers and prospect theory to handle the risk in decision-making problems. Finally, a numerical example involving investment is introduced to illustrate the application and validity of the proposed method.


---

## Body

## 1. Introduction

A multiple-criteria decision-making (MCDM) problem, as one of the most significant problems in the fields of management, economics, and engineering, is the process of selecting the optimal option in all possible alternatives according to diverse criteria. Since the complexity and variety of decision-making environments determine that opinions of different decision-makers should be taken into account, multiple-criteria group decision-making (MCGDM) becomes research hotspot and is widely used in practical problems [1], such as supplier selection [2, 3], emergency management [4, 5], and product development [6, 7]. Methods for solving MCDM problems, e.g., AHP [8], TOPSIS [9], VIKOR [10], and PROMETHEE [11], have been improved and applied to group decision-making problems.The increase of imprecision and complexity in real-world problems leads to the fact that decision-makers might be unable to express personal preferences with numerical values, so some theories dealing with imprecision are introduced into MCGDM problems, especially the theory of fuzzy sets developed by Zadeh [12]. Classical methods have been extended to solve uncertain MCGDM problems based on fuzzy sets as well as their generations, such as fuzzy TOPSIS [13], triangular fuzzy AHP [14], intuitionistic fuzzy VIKOR [15], fuzzy prospect theory [16], intuitionistic fuzzy ELECTRE [17], and Pythagorean fuzzy PROMETHEE [18]. There are two defects which make it difficult to overcome these fuzzy methods: one is that the process of group information aggregation such as weighted average is mechanized, leading to the neglect of interaction among decision-makers, and another one is that the inherent subjectivity of membership function can easily result in the decision-making bias. Aiming at these defects, the rough-number method is proposed [19], which is based on the basic notion of approximates in rough set theory, an effective method to handle imprecision information developed by Pawlak [20]. A rough number can characterize imprecise information by means of rough boundary intervals bounded by the upper and lower limits, which can be directly computed from the raw data without any subjective adjustments, assumptions, or membership functions. Therefore, many researches combined traditional decision-making methods with rough numbers, resulting in extended models like rough AHP [7], rough VIKOR [21], rough DEMATEL [22], and rough MABAC [23].As a classical MCDM method, the analytic hierarchy process (AHP) has been used widely for calculating the weights of criteria [7, 8, 14]. The AHP requires to compare the relative importance of each two criteria and obtain a comparison matrix, but due to the complexity of comparison procedures of the AHP, as well as the limitation of human cognition, the results obtained by the AHP always lack consistency in the pairwise comparison matrix; therefore, to improve the traditional AHP, Rezaei introduced a novel pairwise comparison idea and proposed the best-worst method (BWM) [24]. In the process of the BWM, decision-makers only need to compare each criterion with the best criterion and the worst criterion, rather than the comparisons between all the criteria. Therefore, the BWM yields two comparison vectors, and then the weights of criteria can be obtained by solving a mathematical programming model [24]. The BWM has been used widely in many areas, such as water scarcity management [25], supplier evaluation and selection [26, 27], quality assessment of scientific output [28], and sustainable architecture [29]. Some researchers have combined rough numbers with the BWM to handle the MCDM problems: Željko et al. proposed a rough BWM-SAW model to select wagons for the internal transport [30], a rough BWM-WASPAS model to determine the location selection for roundabout construction [31], and then a rough BWM-SERVQUAL model for quality assessment of scientific conferences [32]; Pamučar et al. integrated rough numbers and fuzzy sets, proposed interval-valued fuzzy-rough numbers (IVFRNs) to aggregate fuzzy evaluating values of the decision group, and presented an IVFRN-based BWM to obtain the weights of criteria [33]; and then Pamučar et al. proposed a BWM-WASPAS-MABAC model based on interval rough numbers to evaluate the third-party logistics provider [34]. All the models based on the rough BWM are based on the original BWM [24], which is nonlinear and may not obtain the unique solution of a mathematical programming model, resulting in a decision failure. Then, Rezaei modified the model and proposed a linear BWM, which is based on the same philosophy as the original model but yields a unique solution [35]. Therefore, in this paper, we intend to construct a linear rough BWM to obtain the weights of criteria.The study for risk attitudes of decision-makers is another crucial aspect of decision-making problems, and many researchers introduced prospect theory to MCDM models. Prospect theory developed by Kahneman and Tversky [36] is a descriptive model of individual decision-making under condition of risk. Later, Tversky and Kahneman [37] developed the cumulative prospect theory, which captures psychological aspects of decision-making under risk. In the prospect theory, the outcomes are expressed by means of gains and losses from a reference alternative. The value function in prospect theory assumes an S-shape concave above the reference alternative, which reflects the aversion of risk in face of gains, and the convex part below the reference alternative reflects the propensity to risk in case of losses. Prospect theory has been an arisen behavioral model of decision-making under risk, and in order for the application in an uncertain environment, some research works have begun to explore the combination of prospect theory and imprecise information, such as prospect theory under the fuzzy environment [16], linguistic environment [38], interval type 2 fuzzy environment [39], and rough environment [40]. Unfortunately, the process is still at the primary stage: the imprecise information involved only includes fuzzy numbers and interval numbers, and although Fang et al. referred to the rough environment in [40], they did not explore the combination of rough numbers and prospect theory; proposed methods only concentrate on the imprecision of value function in prospect theory, while they ignore the imprecision of weighting function, and almost all the combined methods pay no attention to group decision-making problems. So it is essential to extend prospect theory to imprecise MCGDM problems.In this paper, we introduce rough numbers to MCGDM models and combine the linear BWM and prospect theory to handle the risk and uncertain MCGDM problems. The rest of this paper is arranged as follows: In Section2, we shortly describe some knowledge on methods and theories involved in this paper. In Section 3, we propose the rough-number-based MCGDM method based on the BWM and prospect theory, including processes of criteria weighting and alternative ranking. In Section 4, we present a practical example to illustrate the application and verify the feasibility and validity of this new method. In Section 5, some conclusions and directions for the future work are proposed.


## 2. Preliminaries

This section is composed of three subsections to review some preliminaries about the rough number, best-worst method, and prospect theory.


### 2.1. Rough Number

Inspired by rough set theory, rough number is first proposed by Zhai et al. [41] in order to handle subjective preferences of customers in quality function deployment. Similar to the notion of approximates in rough sets, a rough number is constructed by lower and upper limits, which determine a rough boundary interval to characterize imprecise information. While the rough number merely depends on original data without any prior knowledge, it can capture the experts’ real perception effectively and aggregate every individual’s preference into an objective and consistent group judgement. In this section, we review some basic definitions of rough number.Definition 1.
(see [41]). Suppose U is the universe containing all the objects and there are n classes expressed as R=C1,C2,…,Cn. If they are ordered as C1<C2<⋯<Cn, then for ∀Y∈U, ∀Ci∈R, the lower approximation (Apr¯Ci), upper approximation (Apr¯Ci), and boundary region (BndCi) of Ci can be defined as(1)Apr¯Ci=∪Y∈URY≤Ci,(2)Apr¯Ci=∪Y∈URY≥Ci,(3)BndCi=∪Y∈URY≠Ci=Y∈URY>Ci∪Y∈URY<Ci.Definition 2.
(see [41]). Ci can be expressed by a rough number RNCi, which is determined by its lower limit RN¯Ci and upper limit RN¯Ci, which are expressed as(4)RN¯Ci=1ML∑RYY∈Apr¯Ci,(5)RN¯Ci=1MU∑RYY∈Apr¯Ci,(6)RBndCi=RN¯Ci−RN¯Ci,(7)RNCi=Ci=RN¯Ci,RN¯Ci,where ML and MU are the number of objects contained in Apr¯Ci and Apr¯Ci, respectively.
For a convenient expression,RN¯Ci, RN¯Ci, and RNCi can be denoted as Ci¯, Ci¯, and Ci for short, respectively.
According to the definition, the rough number is similar to the interval number in form, so with the aid of arithmetic operations of interval analysis [42], Zhai et al. proposed the operations of rough numbers [43].Definition 3.
(see [43]). Suppose a=a¯,a¯ and b=b¯,b¯ are two rough numbers and α is a real number, then the arithmetic operations of rough numbers can be expressed as(8)a×α=α×a=α×a¯,α×a¯,forα≥0,α×a¯,α×a¯,forα<0,(9)a+b=a¯+b¯,a¯+b¯,(10)a−b=a¯−b¯,a¯−b¯,(11)a×b=mina¯b¯,a¯b¯,a¯b¯,a¯b¯,maxa¯b¯,a¯b¯,a¯b¯,a¯b¯,(12)ab=a¯,a¯×1b¯,1b¯,0∉b¯,b¯.
Specifically, ifa>0 and b>0, which means a¯, a¯, b¯, and b¯ are all greater than 0, then the multiplication operation (11) can be simplified as a×b=a¯b¯,a¯b¯ and the division operation (12) can be simplified as a/b=a¯/b¯,a¯/b¯.
To compare the values of different rough numbers, Zhai et al. proposed the ranking rules [41]. For any two rough numbers RN1=RN1¯,RN¯1 and RN2=RN2¯,RN2¯, there are five possible cases, which are shown in Figure 1. Denote M1 and M2 as the medians of RN1 and RN2, and the ranking rules can be easily explained as follows [41]:(a)
WhenM1=M2,(i)
IfRN1¯=RN2¯ and RN1¯=RN2¯, then RN1=RN2 (see Figure 1(a))(ii)
IfRN1¯<RN2¯ and RN1¯>RN2¯, then RN1>RN2(see Figure 1(b))(b)
WhenM1≠M2,(i)
IfM1<M2, then RN1<RN2 (see Figure 1(c))(ii)
IfM1>M2, then RN1>RN2(see Figure 1(d) and (e))Figure 1
Ranking rules for rough numbers.


### 2.2. Best-Worst Method

Pairwise comparison method, like the AHP, has been used widely in MCDM problems [44]. It shows the relative preferences between each two criteria, constructs a preference matrix, and provides a way to find the weights of criteria. As the complexity of comparison procedures and the limitation of human cognition, the pairwise comparison method faces an inevitable defect in practice, which is the lack of consistency of the pairwise comparison matrixes. Rezaei proposed a vector-based method called the best-worst method (BWM), deriving the weights of criteria based on pairwise comparisons in a different way. The steps of the BWM are described as follows [24]:Step 1: the best (e.g., most important) and worst (e.g., least important) criteria are chosen among the criteria setc1,c2,…,cn, denoted as cB and cW, respectively.Step 2: the preference of the best criterion over all the other criteria is determined using a number between 1 and 9, and the best-to-others vector can be expressed asAB=aB1,aB2,...,aBn, where aBi indicates the preference of the best criterion cB over the criterion cj.Step 3: the preference of all the other criteria over the worst criterion is determined using a number between 1 and 9, and the others-to-worst vector can be expressed asAW=a1W,a2W,…,anWT, where aiW indicates the preference of the criterion cj over the best criterion cB.Step 4: a nonlinear min-max mathematical programming problem is constructed as model (13), which can be transferred to the model (14), and the optimal weights w1∗,w2∗,…,wn∗ and consistency index ξ are found by solving the model:(13)min‐maxjwBwj−aBj,wjwW−ajW,s.t.∑jwj=1,wj≥0,forallj,(14)minξ,s.t.wBwj−aBj≤ξ,forallj,wjwW−ajW≤ξ,forallj,∑jwj=1,wj≥0,forallj.Compared with standard pairwise comparison methods such as the AHP, the BWM uses only integer numbers in describing preferences, reduces the times of comparisons, and most importantly provides more consistent and reliable results [24]. Due to the inconsistency of evaluation vectors and the nonlinearity of programming model (13), there may be multiple optimal solutions in some cases [35]. In order to obtain a unique solution, Rezaei improved the mathematical programming by transforming the objective function to the set wB−aBjwj,wj−ajWwW. The programming model can be formulated as follows [35]:(15)min‐maxjwB−aBjwj,wj−ajWwW,s.t.∑jwj=1,wj≥0,forallj.Model (15) is equivalent to the following programming problem:(16)minξLs.t.wB−aBjwj≤ξL,forallj,wj−ajWwW≤ξL,forallj,∑jwj=1,wj≥0,forallj.Apparently, programming model (16) is a linear problem, so we can obtain a unique solution w1∗,w2∗,…,wn∗ and the consistency of comparisons ξL by solving problem (16).


### 2.3. Prospect Theory

Prospect theory, which was initially established by Kahneman and Tversky in 1979, can describe the actual decision behavior of decision-makers under risk and uncertainty [36]. Two core concepts of prospect theory are the value function and decision weighting function. Value function, reflecting the relationship between the decision-maker’s subjective utility and the expected results, can be expressed as(17)vx=Δxα,Δx≥0,−λ−Δxβ,Δx≤0,where Δx is the gain or loss of the outcome relative to the reference point: Δx>0 for a gain, while Δx<0 for a loss; α and β are adjustable coefficients determining the concavity and convexity of the value function, respectively, satisfying 0<αandβ<1; and λ is a parameter describing loss aversion and λ>1. Regarding the adjustable coefficients α and β, the values are larger and the decision-maker is more prone to risk: when α=β=1, the decision-maker shows no change of risk preference for the gain and loss, the value function degenerates to utility function, and the utility vx is linear to the variable Δx, which is depicted as the solid line in Figure 2; in contrast, when α<1 and β<1, the decision-maker is sensitive to the gain and loss and the utility vx is nonlinear to the variable Δx, which can be illustrated as the dashed line in Figure 2.Figure 2
Value function of prospect theory with differentα and β.Tversky and Kahneman considered that decision weights are subjective judgements about the likelihood of occurrence, and they described the form of weighting function as [37](18)πp=pγpγ+1−pγ1/γ,x≥0,pδpδ+1−pδ1/δ,x≤0,where γ and δ are the risk parameters for gains and losses, respectively, satisfying 0<γandδ<1.Many empirical researches have explored the value of parameters in equations (17) and (18). According to Tversky and Kahneman [37], results are consistent with empirical data when α=β=0.88, 2.0≤λ≤2.5, γ=0.61, and δ=0.72. Abdellaoui [45] suggested that parameters in equation (17) should be α=0.89, β=0.92, and λ=2.25, and Gonzalez and Wu [46] considered that γ=δ=0.74 in equation (18).Alternatives can be ranked by the prospect value resulting from the value function and weighting function, whose form is defined as(19)V=∑vΔxπp.


## 2.1. Rough Number

Inspired by rough set theory, rough number is first proposed by Zhai et al. [41] in order to handle subjective preferences of customers in quality function deployment. Similar to the notion of approximates in rough sets, a rough number is constructed by lower and upper limits, which determine a rough boundary interval to characterize imprecise information. While the rough number merely depends on original data without any prior knowledge, it can capture the experts’ real perception effectively and aggregate every individual’s preference into an objective and consistent group judgement. In this section, we review some basic definitions of rough number.Definition 1.
(see [41]). Suppose U is the universe containing all the objects and there are n classes expressed as R=C1,C2,…,Cn. If they are ordered as C1<C2<⋯<Cn, then for ∀Y∈U, ∀Ci∈R, the lower approximation (Apr¯Ci), upper approximation (Apr¯Ci), and boundary region (BndCi) of Ci can be defined as(1)Apr¯Ci=∪Y∈URY≤Ci,(2)Apr¯Ci=∪Y∈URY≥Ci,(3)BndCi=∪Y∈URY≠Ci=Y∈URY>Ci∪Y∈URY<Ci.Definition 2.
(see [41]). Ci can be expressed by a rough number RNCi, which is determined by its lower limit RN¯Ci and upper limit RN¯Ci, which are expressed as(4)RN¯Ci=1ML∑RYY∈Apr¯Ci,(5)RN¯Ci=1MU∑RYY∈Apr¯Ci,(6)RBndCi=RN¯Ci−RN¯Ci,(7)RNCi=Ci=RN¯Ci,RN¯Ci,where ML and MU are the number of objects contained in Apr¯Ci and Apr¯Ci, respectively.
For a convenient expression,RN¯Ci, RN¯Ci, and RNCi can be denoted as Ci¯, Ci¯, and Ci for short, respectively.
According to the definition, the rough number is similar to the interval number in form, so with the aid of arithmetic operations of interval analysis [42], Zhai et al. proposed the operations of rough numbers [43].Definition 3.
(see [43]). Suppose a=a¯,a¯ and b=b¯,b¯ are two rough numbers and α is a real number, then the arithmetic operations of rough numbers can be expressed as(8)a×α=α×a=α×a¯,α×a¯,forα≥0,α×a¯,α×a¯,forα<0,(9)a+b=a¯+b¯,a¯+b¯,(10)a−b=a¯−b¯,a¯−b¯,(11)a×b=mina¯b¯,a¯b¯,a¯b¯,a¯b¯,maxa¯b¯,a¯b¯,a¯b¯,a¯b¯,(12)ab=a¯,a¯×1b¯,1b¯,0∉b¯,b¯.
Specifically, ifa>0 and b>0, which means a¯, a¯, b¯, and b¯ are all greater than 0, then the multiplication operation (11) can be simplified as a×b=a¯b¯,a¯b¯ and the division operation (12) can be simplified as a/b=a¯/b¯,a¯/b¯.
To compare the values of different rough numbers, Zhai et al. proposed the ranking rules [41]. For any two rough numbers RN1=RN1¯,RN¯1 and RN2=RN2¯,RN2¯, there are five possible cases, which are shown in Figure 1. Denote M1 and M2 as the medians of RN1 and RN2, and the ranking rules can be easily explained as follows [41]:(a)
WhenM1=M2,(i)
IfRN1¯=RN2¯ and RN1¯=RN2¯, then RN1=RN2 (see Figure 1(a))(ii)
IfRN1¯<RN2¯ and RN1¯>RN2¯, then RN1>RN2(see Figure 1(b))(b)
WhenM1≠M2,(i)
IfM1<M2, then RN1<RN2 (see Figure 1(c))(ii)
IfM1>M2, then RN1>RN2(see Figure 1(d) and (e))Figure 1
Ranking rules for rough numbers.


## 2.2. Best-Worst Method

Pairwise comparison method, like the AHP, has been used widely in MCDM problems [44]. It shows the relative preferences between each two criteria, constructs a preference matrix, and provides a way to find the weights of criteria. As the complexity of comparison procedures and the limitation of human cognition, the pairwise comparison method faces an inevitable defect in practice, which is the lack of consistency of the pairwise comparison matrixes. Rezaei proposed a vector-based method called the best-worst method (BWM), deriving the weights of criteria based on pairwise comparisons in a different way. The steps of the BWM are described as follows [24]:Step 1: the best (e.g., most important) and worst (e.g., least important) criteria are chosen among the criteria setc1,c2,…,cn, denoted as cB and cW, respectively.Step 2: the preference of the best criterion over all the other criteria is determined using a number between 1 and 9, and the best-to-others vector can be expressed asAB=aB1,aB2,...,aBn, where aBi indicates the preference of the best criterion cB over the criterion cj.Step 3: the preference of all the other criteria over the worst criterion is determined using a number between 1 and 9, and the others-to-worst vector can be expressed asAW=a1W,a2W,…,anWT, where aiW indicates the preference of the criterion cj over the best criterion cB.Step 4: a nonlinear min-max mathematical programming problem is constructed as model (13), which can be transferred to the model (14), and the optimal weights w1∗,w2∗,…,wn∗ and consistency index ξ are found by solving the model:(13)min‐maxjwBwj−aBj,wjwW−ajW,s.t.∑jwj=1,wj≥0,forallj,(14)minξ,s.t.wBwj−aBj≤ξ,forallj,wjwW−ajW≤ξ,forallj,∑jwj=1,wj≥0,forallj.Compared with standard pairwise comparison methods such as the AHP, the BWM uses only integer numbers in describing preferences, reduces the times of comparisons, and most importantly provides more consistent and reliable results [24]. Due to the inconsistency of evaluation vectors and the nonlinearity of programming model (13), there may be multiple optimal solutions in some cases [35]. In order to obtain a unique solution, Rezaei improved the mathematical programming by transforming the objective function to the set wB−aBjwj,wj−ajWwW. The programming model can be formulated as follows [35]:(15)min‐maxjwB−aBjwj,wj−ajWwW,s.t.∑jwj=1,wj≥0,forallj.Model (15) is equivalent to the following programming problem:(16)minξLs.t.wB−aBjwj≤ξL,forallj,wj−ajWwW≤ξL,forallj,∑jwj=1,wj≥0,forallj.Apparently, programming model (16) is a linear problem, so we can obtain a unique solution w1∗,w2∗,…,wn∗ and the consistency of comparisons ξL by solving problem (16).


## 2.3. Prospect Theory

Prospect theory, which was initially established by Kahneman and Tversky in 1979, can describe the actual decision behavior of decision-makers under risk and uncertainty [36]. Two core concepts of prospect theory are the value function and decision weighting function. Value function, reflecting the relationship between the decision-maker’s subjective utility and the expected results, can be expressed as(17)vx=Δxα,Δx≥0,−λ−Δxβ,Δx≤0,where Δx is the gain or loss of the outcome relative to the reference point: Δx>0 for a gain, while Δx<0 for a loss; α and β are adjustable coefficients determining the concavity and convexity of the value function, respectively, satisfying 0<αandβ<1; and λ is a parameter describing loss aversion and λ>1. Regarding the adjustable coefficients α and β, the values are larger and the decision-maker is more prone to risk: when α=β=1, the decision-maker shows no change of risk preference for the gain and loss, the value function degenerates to utility function, and the utility vx is linear to the variable Δx, which is depicted as the solid line in Figure 2; in contrast, when α<1 and β<1, the decision-maker is sensitive to the gain and loss and the utility vx is nonlinear to the variable Δx, which can be illustrated as the dashed line in Figure 2.Figure 2
Value function of prospect theory with differentα and β.Tversky and Kahneman considered that decision weights are subjective judgements about the likelihood of occurrence, and they described the form of weighting function as [37](18)πp=pγpγ+1−pγ1/γ,x≥0,pδpδ+1−pδ1/δ,x≤0,where γ and δ are the risk parameters for gains and losses, respectively, satisfying 0<γandδ<1.Many empirical researches have explored the value of parameters in equations (17) and (18). According to Tversky and Kahneman [37], results are consistent with empirical data when α=β=0.88, 2.0≤λ≤2.5, γ=0.61, and δ=0.72. Abdellaoui [45] suggested that parameters in equation (17) should be α=0.89, β=0.92, and λ=2.25, and Gonzalez and Wu [46] considered that γ=δ=0.74 in equation (18).Alternatives can be ranked by the prospect value resulting from the value function and weighting function, whose form is defined as(19)V=∑vΔxπp.


## 3. The Proposed Decision Model

This section proposes a new method for MCDGM problems. Firstly, we describe the MCDGM problem under risk and imprecision and then develop rough-number-based methods for criteria weighting and alternative ranking, respectively.


### 3.1. Framework of the Proposed Method

Suppose in an MCGDM problem thatA=A1,A2,…,Am is the set of all the alternatives and C=c1,c2,…,cn is the set of criteria; w=w1,w2,…,wn represents the weights of criteria, where wj is the weight of cj; the decision group is composed by h decision-makers, expressed as E=e1,e2,…,eh; there are l states of events in the problem, which can be represented as S=S1,S2,…,Sl, with an occurrence probability ptk of the state St estimated by the decision-maker ek; each decision-maker has an expectation for every criterion as the reference point in prospect theory, and the expectation vector of the decision-maker ek under the state St can be denoted as Rkt=r1kt,r2kt,…,rnkt. So the score table made by ek can be expressed as in Table 1.Table 1
Score table of the decision-makerek.
Criteria
c
1
c
2
…
c
n
States
S
1
S
2
⋯
S
l
S
1
S
2
…
S
l
…
S
1
S
2
…
S
l
Alternatives
A
1
x
11
k
1
x
11
k
2
⋯
x
11
k
l
x
12
k
1
x
12
k
2
…
x
12
k
l
…
x
1
n
k
1
x
1
n
k
2
…
x
1
n
k
l
A
2
x
21
k
1
x
21
k
2
⋯
x
21
k
l
x
22
k
1
x
22
k
2
…
x
22
k
l
…
x
2
n
k
1
x
2
n
k
2
…
x
2
n
k
l
⋮
⋮
⋮
⋱
⋮
A
m
x
m
1
k
1
x
m
1
k
2
⋯
x
m
1
k
l
x
m
1
k
1
x
m
2
k
2
…
x
m
2
k
l
…
x
m
n
k
1
x
m
n
k
2
…
x
m
n
k
l
Expectation
r
1
k
1
r
1
k
2
⋯
r
1
k
l
r
2
k
1
r
2
k
2
…
r
2
k
l
…
r
n
k
1
r
n
k
2
…
r
n
k
lThe framework of the proposed method is depicted in Figure3.Figure 3
Framework of the proposed method.


### 3.2. Rough BWM for Criteria Weighting

As an advanced and efficient pairwise comparison method, the BWM has a special superiority to handle MCDM problems. For the subjectivity and imprecision in the criteria weighting procedure, this section proposes a new criteria weighting method by combining the rough-number method and BWM. The procedure of the rough BWM is described as follows:Step 1: the best criterioncB and the worst criterion cW are determined by the decision group. Based on common rational cognition of individuals in the decision group, each could reach a consensus in choosing the best and worst criteria. If not, an extra criterion c0 can be introduced as the best (or worst) criterion, which makes no difference in the results.Step 2: comparison vectors of each decision-maker are determined.ek can get the best-to-others vector expressed as ABk=aB1k,aB2k,…,aBnk and the others-to-worst vector as AWk=a1Wk,a2Wk,…,anWk using a number between 1 and 9.Step 3: the integrated comparison vectors are constructed. The integrated best-to-others vector and integrated others-to-worst vector can be expressed as(20)AB=aB1,aB2,…,aBn,AW=a1W,a2W,…,anW,whereaBj=aBj1,aBj2,…,aBjs and ajW=ajW1,ajW2,…,ajWs, in which aBj denotes the collection of preferences of cB over cj made by all the decision-makers and ajW denotes the collection of preferences of cj over cW made by all the decision-makers.Step 4: rough comparison vectors are constructed based on the rough-number method. According to equations (1)–(7), the preferences of each decision-maker ek can be transformed to a rough number:(21)RNaBjk=aBjk¯,aBjk¯,RNajWk=ajWk¯,ajWk¯,and then the rough sequences are formed as(22)RNa˜Bj=aBj1¯,aBj1¯,aBj2¯,aBj2¯,…,aBjs¯,aBjs¯,RNa˜jW=ajW1¯,ajW1¯,ajW2¯,ajW2¯,…,ajWs¯,ajWs¯.So the integrated comparison vectors can be expressed as(23)RAB=aB1¯,aB1¯,aB2¯,aB2¯,…,aBn¯,aBn¯,RAW=a1W¯,a1W¯,a2W¯,a2W¯,…,anW¯,anW¯,whereaBj¯,aBj¯=1/h∑k=1saBjk¯,aBjk¯ and ajW¯,ajW¯=1/h∑k=1sajWk¯,ajWk¯.Step 5: the rough weight of each criterion is calculated. Similar to the analysis in the BWM, the optimal rough weight for criteria is the one where, for each pair ofwB/wj and wj/wW, there are wB/wj=aBj and wj/wW=ajW, which can be rewritten as wB¯/wj¯,wB¯/wj¯=aBj¯,aBj¯ and wj¯/wW¯,wj¯/wW¯=ajW¯,ajW¯, respectively. To satisfy these conditions for all j, we should find a solution where the maximum absolute differences wB¯/wj¯−aBj¯, wB¯/wj¯−aBj¯, wj¯/wW¯−ajW¯, and wj¯/wW¯−ajW¯ for all j are minimized. To obtain a unique solution of the model, Rezaei improved the original BWM [35], where the conditions can be transferred to wB¯−aBj¯wj¯, wB¯−aBj¯wj¯, wj¯−ajW¯wW¯, and wj¯−ajW¯wW¯, respectively. So we can construct the programming problem as(24)min‐maxjwB¯−aBj¯wj¯,wB¯−aBj¯wj¯,wj¯−ajW¯wW¯,wj¯−ajW¯wW¯,s.t.12∑jwj¯+wj¯=1,wj¯>wj¯≥0,forallj,where 1/2∑jwj¯+wj¯=1 is the normalization condition of rough-number vectors. Problem (24) can be transferred to the following problem:(25)minζ,s.t.wB¯−aBj¯wj¯≤ζ,forallj,wB¯−aBj¯wj¯≤ζ,forallj,wj¯−ajW¯wW¯≤ζ,forallj,wj¯−ajW¯wW¯≤ζ,forallj,12∑jwj¯+wj¯=1,wj¯>wj¯>0,forallj.Solving problem (25), the optimal rough weight w1¯,w1¯,w2¯,w2¯,…,wn¯,wn¯ is obtained.In pairwise comparison methods, the consistency ratio plays a significant role to illustrate the consistency of evaluating values of criteria. Rezaei proposed the definition of consistency of criteria made by one decision-maker and built the consistency index (CI), as shown in Table2, for the evaluating values between 1 and 9 [24]. To demonstrate the validity of integrated group information in the rough BWM, we define the consistency of integrated comparison as follows.Table 2
Consistency index (CI) table.
a
B
W
1
2
3
4
5
6
7
8
9
CI
0.00
0.44
1.00
1.63
2.30
3.00
3.73
4.47
5.23Definition 4.
The integrated comparison is fully consistent ifaBj×ajW=aBW for all j, where aBj, ajW, and aBW are, respectively, the integrated preference of the best criterion over the criterion j, the integrated preference of the criterion j over the worst criterion, and the integrated preference of the best criterion over the worst criterion.
However, there is usually a situation that some pairs of criteria are not completely consistent in practical decision-making problems. Therefore to indicate how consistent an integrated comparison is, we discuss the consistency ratio of an integrated comparison as follows.
According to the discussion above, the maximum comparison value ofaBW identified by each decision-maker is 9, so the highest integrated rough value is aBW=9,9. Consistency decreases when there exists a difference between aBj×ajW and aBW, which means aBj×ajW≠aBW, and it is clear that the biggest difference occurs when aBj and ajW have the maximum value which is equal to aBW, leading to the value of ξ. The consistent condition can be rewritten as wB/wj×wj/wW=wB/wW, and as the biggest difference occurs when assigning the maximum value to aBj and ajW, we should subtract the value ξ from aBj and ajW and add it to aBW. So we can obtain the equation as(26)aBj−ξ×ajW−ξ=aBW+ξ.
As for the minimum consistencyaBj=ajW=aBW, we obtain(27)aBW−ξ×aBW−ξ=aBW+ξ⟹ξ2−1+2aBWξ+aBW2−aBW=0.
a
B
W is the integrated preference of the best criterion over the worst criterion, and in this paper, it is a rough number, which means aBW=aBW¯,aBW¯. According to aBW¯≤aBW¯, we can conclude that the integrated preference of the best criterion over the worst criterion cannot be greater than aBW¯. It is easy to find that the value of ξ is increasing in a∈1,9 for the function ξ2−1+2aξ+a2−a=0, so we choose the upper limit aBW¯ to calculate the value of CI, which ensures the consistency ratio (CR) satisfying CR∈0,1. So equation (27) can be transformed as(28)ξ2−1+2aBW¯ξ+aBW¯2−aBW¯=0.
Solving equation (28) for different values of aBW¯, we can obtain the maximum possible values of ξ, which compose the consistency index of the rough BWM. As aBW is obtained by aggregating the evaluating information of decision-makers, aBW¯ is varied according to different values, so we cannot predefine the values of ξ. If there is an agreement among all the decision-makers about their preference for the best criterion over the worst, aBW is a crisp number (e.g., aBW¯=aBW¯) and belongs to 1,2,…,9 and then the values of ξ can be determined from the data in Table 2. Based on the value of ξ∗ obtained by model (25) and values of CI, we can calculate the consistency ratio (CR) as(29)CR=ξ∗CI.


### 3.3. Rough Prospect Theory for Alternative Evaluation

It is important to consider decision-makers’ expectations for alternatives in the evaluation process. In prospect theory, expectations of decision-makers can be seen as references, relative to which gains and losses are obtained. This section combines the rough number and prospect theory to handle criteria values, criteria expectations, and probabilities of states and proposes a new method for alternative ranking and selection. The procedure is described as follows.


#### 3.3.1. Step 1: Construction of the Group Rough Evaluation Matrix and Expectation Vector

The evaluation matrixDkt and expectation vector Rkt from the decision-maker ek under the state St can be expressed as(30)Dkt=x11ktx12kt⋯x1nktx21ktx22kt⋅⋅⋅x2nkt⋮⋮⋱⋮xm1ktxm2kt⋯xmnkt,(31)Rkt=r1ktr2kt…rnkt.Evaluation matrixes and expectation vectors are aggregated, and equations (1)–(7) are used to obtain the group rough evaluation matrix and expectation vector under the state St as(32)Dt=x11t¯,x11t¯x12t¯,x12t¯⋯x1nt¯,x1nt¯x21t¯,x21t¯x22t¯,x22t¯⋯x2nt¯,x2nt¯⋮⋮⋱⋮xm1t¯,xm1t¯xm2t¯,xm2t¯⋯xmnt¯,xmnt¯,(33)Rt=r1t¯,r1t¯r2t¯,r2t¯…rnt¯,rnt¯.


#### 3.3.2. Step 2: Calculation of Group Gain and Loss Matrixes

Group evaluation values and expectations are both rough numbers, which are in the form of interval numbers, so we can calculate gains and losses by means of the relationship between interval numbers shown in Table3. The group gain matrix Gt and group loss matrix Lt under the state St can be expressed as(34)Gt=G11tG12t⋯G1ntG21tG22t⋯G2nt⋮⋮⋱⋮Gm1tGm2t⋯Gmnt,Lt=L11tL12t⋯L1ntL21tL22t⋯L2nt⋮⋮⋱⋮Lm1tLm2t⋯Lmnt.Table 3
Gains and losses for all possible cases.
Types
Gains
Losses
a
:
r
j
¯
<
x
i
j
¯
0.5
x
i
j
¯
+
x
i
j
¯
−
r
j
¯
0
b
:
x
i
j
¯
<
r
j
¯
0
0.5
x
i
j
¯
+
x
i
j
¯
−
r
j
¯
c
:
r
j
¯
<
x
i
j
¯
<
r
j
¯
<
x
i
j
¯
0.5
x
i
j
¯
−
r
j
¯
0
d
:
x
i
j
¯
<
r
j
¯
<
x
i
j
¯
<
r
j
¯
0
0.5
x
i
j
¯
−
r
j
¯
e
:
r
i
j
¯
<
x
j
¯
<
x
i
j
¯
<
r
j
¯
0
0
f
:
x
i
j
¯
<
r
j
¯
<
r
i
j
¯
<
x
j
¯
0.5
x
i
j
¯
−
r
j
¯
0.5
x
i
j
¯
−
r
j
¯


#### 3.3.3. Step 3: Calculation of Group Rough Decision Weights

The probability vector of states given by the decision-makerek can be expressed as p1k,p2k,…,plk, where ptk>0 and ∑tptk=1. According to the rough-number method, all the probability vectors from the decision group can be transformed to a rough probability vector p1¯,p1¯,p2¯,p2¯,…,pl¯,pl¯. Referring to the definition of consistency of interval probability initiated by Yager and Kreinovich [47], the following theorem proves that rough probability satisfies the property of consistency.Theorem 1.
The rough probability vectorp1¯,p1¯,p2¯,p2¯,…,pl¯,pl¯ is consistent; in other words, ∑tpt¯≤1 and ∑tpt¯≥1.Proof.
Denotept1,pt2,…,pth as the probability vector of the state St given by h decision-makers, and it can be transformed to a rough number pt¯,pt¯. For an easy and convenient expression, suppose pt1≤pt2≤⋯≤pth. Let 1/h∑k=1hptk=mt, and according to equations (4) and (5), pt1¯=pth¯=mt. So ptk¯≤pth¯ and ptk¯≥pt1¯ for ∀ptk, which lead to pt¯=1/h∑k=1hptk¯≤mt and pt¯=1/h∑k=1hptk¯≥mt. Therefore, we can get(35)∑t=1lpt¯=∑t=1l1h∑k=1hptk¯≤1h∑k=1h∑t=1lptk=1,∑t=1lpt¯=∑t=1l1h∑k=1hptk¯≥1h∑k=1h∑t=1lptk=1.
The theorem is proven.
The group rough decision weights of gainπt+=πt+¯,πt+¯ can be expressed as(36)πt+¯=pt¯γpt¯γ+1−pt¯γ1/γ,πt+¯=pt¯γpt¯γ+1−pt¯γ1/γ.
And the group rough decision weights of lossπt−=πt−¯,πt−¯ can be expressed as(37)πt−¯=pt¯δpt¯δ+1−pt¯δ1/δ,πt−¯=pt¯δpt¯δ+1−pt¯δ1/δ.


#### 3.3.4. Step 4: Calculation of the Group Prospect Matrix

According to equation (34), the value matrixes for gain and loss under the state St are expressed as(38)Vt+=Gijtαm×n,Vt−=−λ−Lijtβm×n,where α, β, and λ are parameters discussed in classical prospect theory.Denote the prospect matrix byV=Vijm×n, where Vij is the prospect value of the alternative Ai at the criterion cj. The calculation formula of Vij is expressed as(39)Vij=∑t=1lVijt+πt++∑t=1lVijt−πt−.


#### 3.3.5. Step 5: Calculation of Prospect Values and Arranging the Alternatives

Standardizing the prospect matrix to eliminate the effect of diverse dimensions, we can obtain the standardized prospect matrixV′=Vij′m×n, where(40)Vij′=VijVj∗,Vj∗=maxiVij¯,Vij¯.According to the rough weight obtained by the rough BWM, the overall prospect valueUi of the alternative Ai is calculated by(41)Ui=∑j=1nwj×Vij′.By the ranking rules for rough numbers shown in Figure1, alternatives are arranged according to their overall prospect values and the maximal one is selected as the optimal alternative.


## 3.1. Framework of the Proposed Method

Suppose in an MCGDM problem thatA=A1,A2,…,Am is the set of all the alternatives and C=c1,c2,…,cn is the set of criteria; w=w1,w2,…,wn represents the weights of criteria, where wj is the weight of cj; the decision group is composed by h decision-makers, expressed as E=e1,e2,…,eh; there are l states of events in the problem, which can be represented as S=S1,S2,…,Sl, with an occurrence probability ptk of the state St estimated by the decision-maker ek; each decision-maker has an expectation for every criterion as the reference point in prospect theory, and the expectation vector of the decision-maker ek under the state St can be denoted as Rkt=r1kt,r2kt,…,rnkt. So the score table made by ek can be expressed as in Table 1.Table 1
Score table of the decision-makerek.
Criteria
c
1
c
2
…
c
n
States
S
1
S
2
⋯
S
l
S
1
S
2
…
S
l
…
S
1
S
2
…
S
l
Alternatives
A
1
x
11
k
1
x
11
k
2
⋯
x
11
k
l
x
12
k
1
x
12
k
2
…
x
12
k
l
…
x
1
n
k
1
x
1
n
k
2
…
x
1
n
k
l
A
2
x
21
k
1
x
21
k
2
⋯
x
21
k
l
x
22
k
1
x
22
k
2
…
x
22
k
l
…
x
2
n
k
1
x
2
n
k
2
…
x
2
n
k
l
⋮
⋮
⋮
⋱
⋮
A
m
x
m
1
k
1
x
m
1
k
2
⋯
x
m
1
k
l
x
m
1
k
1
x
m
2
k
2
…
x
m
2
k
l
…
x
m
n
k
1
x
m
n
k
2
…
x
m
n
k
l
Expectation
r
1
k
1
r
1
k
2
⋯
r
1
k
l
r
2
k
1
r
2
k
2
…
r
2
k
l
…
r
n
k
1
r
n
k
2
…
r
n
k
lThe framework of the proposed method is depicted in Figure3.Figure 3
Framework of the proposed method.


## 3.2. Rough BWM for Criteria Weighting

As an advanced and efficient pairwise comparison method, the BWM has a special superiority to handle MCDM problems. For the subjectivity and imprecision in the criteria weighting procedure, this section proposes a new criteria weighting method by combining the rough-number method and BWM. The procedure of the rough BWM is described as follows:Step 1: the best criterioncB and the worst criterion cW are determined by the decision group. Based on common rational cognition of individuals in the decision group, each could reach a consensus in choosing the best and worst criteria. If not, an extra criterion c0 can be introduced as the best (or worst) criterion, which makes no difference in the results.Step 2: comparison vectors of each decision-maker are determined.ek can get the best-to-others vector expressed as ABk=aB1k,aB2k,…,aBnk and the others-to-worst vector as AWk=a1Wk,a2Wk,…,anWk using a number between 1 and 9.Step 3: the integrated comparison vectors are constructed. The integrated best-to-others vector and integrated others-to-worst vector can be expressed as(20)AB=aB1,aB2,…,aBn,AW=a1W,a2W,…,anW,whereaBj=aBj1,aBj2,…,aBjs and ajW=ajW1,ajW2,…,ajWs, in which aBj denotes the collection of preferences of cB over cj made by all the decision-makers and ajW denotes the collection of preferences of cj over cW made by all the decision-makers.Step 4: rough comparison vectors are constructed based on the rough-number method. According to equations (1)–(7), the preferences of each decision-maker ek can be transformed to a rough number:(21)RNaBjk=aBjk¯,aBjk¯,RNajWk=ajWk¯,ajWk¯,and then the rough sequences are formed as(22)RNa˜Bj=aBj1¯,aBj1¯,aBj2¯,aBj2¯,…,aBjs¯,aBjs¯,RNa˜jW=ajW1¯,ajW1¯,ajW2¯,ajW2¯,…,ajWs¯,ajWs¯.So the integrated comparison vectors can be expressed as(23)RAB=aB1¯,aB1¯,aB2¯,aB2¯,…,aBn¯,aBn¯,RAW=a1W¯,a1W¯,a2W¯,a2W¯,…,anW¯,anW¯,whereaBj¯,aBj¯=1/h∑k=1saBjk¯,aBjk¯ and ajW¯,ajW¯=1/h∑k=1sajWk¯,ajWk¯.Step 5: the rough weight of each criterion is calculated. Similar to the analysis in the BWM, the optimal rough weight for criteria is the one where, for each pair ofwB/wj and wj/wW, there are wB/wj=aBj and wj/wW=ajW, which can be rewritten as wB¯/wj¯,wB¯/wj¯=aBj¯,aBj¯ and wj¯/wW¯,wj¯/wW¯=ajW¯,ajW¯, respectively. To satisfy these conditions for all j, we should find a solution where the maximum absolute differences wB¯/wj¯−aBj¯, wB¯/wj¯−aBj¯, wj¯/wW¯−ajW¯, and wj¯/wW¯−ajW¯ for all j are minimized. To obtain a unique solution of the model, Rezaei improved the original BWM [35], where the conditions can be transferred to wB¯−aBj¯wj¯, wB¯−aBj¯wj¯, wj¯−ajW¯wW¯, and wj¯−ajW¯wW¯, respectively. So we can construct the programming problem as(24)min‐maxjwB¯−aBj¯wj¯,wB¯−aBj¯wj¯,wj¯−ajW¯wW¯,wj¯−ajW¯wW¯,s.t.12∑jwj¯+wj¯=1,wj¯>wj¯≥0,forallj,where 1/2∑jwj¯+wj¯=1 is the normalization condition of rough-number vectors. Problem (24) can be transferred to the following problem:(25)minζ,s.t.wB¯−aBj¯wj¯≤ζ,forallj,wB¯−aBj¯wj¯≤ζ,forallj,wj¯−ajW¯wW¯≤ζ,forallj,wj¯−ajW¯wW¯≤ζ,forallj,12∑jwj¯+wj¯=1,wj¯>wj¯>0,forallj.Solving problem (25), the optimal rough weight w1¯,w1¯,w2¯,w2¯,…,wn¯,wn¯ is obtained.In pairwise comparison methods, the consistency ratio plays a significant role to illustrate the consistency of evaluating values of criteria. Rezaei proposed the definition of consistency of criteria made by one decision-maker and built the consistency index (CI), as shown in Table2, for the evaluating values between 1 and 9 [24]. To demonstrate the validity of integrated group information in the rough BWM, we define the consistency of integrated comparison as follows.Table 2
Consistency index (CI) table.
a
B
W
1
2
3
4
5
6
7
8
9
CI
0.00
0.44
1.00
1.63
2.30
3.00
3.73
4.47
5.23Definition 4.
The integrated comparison is fully consistent ifaBj×ajW=aBW for all j, where aBj, ajW, and aBW are, respectively, the integrated preference of the best criterion over the criterion j, the integrated preference of the criterion j over the worst criterion, and the integrated preference of the best criterion over the worst criterion.
However, there is usually a situation that some pairs of criteria are not completely consistent in practical decision-making problems. Therefore to indicate how consistent an integrated comparison is, we discuss the consistency ratio of an integrated comparison as follows.
According to the discussion above, the maximum comparison value ofaBW identified by each decision-maker is 9, so the highest integrated rough value is aBW=9,9. Consistency decreases when there exists a difference between aBj×ajW and aBW, which means aBj×ajW≠aBW, and it is clear that the biggest difference occurs when aBj and ajW have the maximum value which is equal to aBW, leading to the value of ξ. The consistent condition can be rewritten as wB/wj×wj/wW=wB/wW, and as the biggest difference occurs when assigning the maximum value to aBj and ajW, we should subtract the value ξ from aBj and ajW and add it to aBW. So we can obtain the equation as(26)aBj−ξ×ajW−ξ=aBW+ξ.
As for the minimum consistencyaBj=ajW=aBW, we obtain(27)aBW−ξ×aBW−ξ=aBW+ξ⟹ξ2−1+2aBWξ+aBW2−aBW=0.
a
B
W is the integrated preference of the best criterion over the worst criterion, and in this paper, it is a rough number, which means aBW=aBW¯,aBW¯. According to aBW¯≤aBW¯, we can conclude that the integrated preference of the best criterion over the worst criterion cannot be greater than aBW¯. It is easy to find that the value of ξ is increasing in a∈1,9 for the function ξ2−1+2aξ+a2−a=0, so we choose the upper limit aBW¯ to calculate the value of CI, which ensures the consistency ratio (CR) satisfying CR∈0,1. So equation (27) can be transformed as(28)ξ2−1+2aBW¯ξ+aBW¯2−aBW¯=0.
Solving equation (28) for different values of aBW¯, we can obtain the maximum possible values of ξ, which compose the consistency index of the rough BWM. As aBW is obtained by aggregating the evaluating information of decision-makers, aBW¯ is varied according to different values, so we cannot predefine the values of ξ. If there is an agreement among all the decision-makers about their preference for the best criterion over the worst, aBW is a crisp number (e.g., aBW¯=aBW¯) and belongs to 1,2,…,9 and then the values of ξ can be determined from the data in Table 2. Based on the value of ξ∗ obtained by model (25) and values of CI, we can calculate the consistency ratio (CR) as(29)CR=ξ∗CI.


## 3.3. Rough Prospect Theory for Alternative Evaluation

It is important to consider decision-makers’ expectations for alternatives in the evaluation process. In prospect theory, expectations of decision-makers can be seen as references, relative to which gains and losses are obtained. This section combines the rough number and prospect theory to handle criteria values, criteria expectations, and probabilities of states and proposes a new method for alternative ranking and selection. The procedure is described as follows.


### 3.3.1. Step 1: Construction of the Group Rough Evaluation Matrix and Expectation Vector

The evaluation matrixDkt and expectation vector Rkt from the decision-maker ek under the state St can be expressed as(30)Dkt=x11ktx12kt⋯x1nktx21ktx22kt⋅⋅⋅x2nkt⋮⋮⋱⋮xm1ktxm2kt⋯xmnkt,(31)Rkt=r1ktr2kt…rnkt.Evaluation matrixes and expectation vectors are aggregated, and equations (1)–(7) are used to obtain the group rough evaluation matrix and expectation vector under the state St as(32)Dt=x11t¯,x11t¯x12t¯,x12t¯⋯x1nt¯,x1nt¯x21t¯,x21t¯x22t¯,x22t¯⋯x2nt¯,x2nt¯⋮⋮⋱⋮xm1t¯,xm1t¯xm2t¯,xm2t¯⋯xmnt¯,xmnt¯,(33)Rt=r1t¯,r1t¯r2t¯,r2t¯…rnt¯,rnt¯.


### 3.3.2. Step 2: Calculation of Group Gain and Loss Matrixes

Group evaluation values and expectations are both rough numbers, which are in the form of interval numbers, so we can calculate gains and losses by means of the relationship between interval numbers shown in Table3. The group gain matrix Gt and group loss matrix Lt under the state St can be expressed as(34)Gt=G11tG12t⋯G1ntG21tG22t⋯G2nt⋮⋮⋱⋮Gm1tGm2t⋯Gmnt,Lt=L11tL12t⋯L1ntL21tL22t⋯L2nt⋮⋮⋱⋮Lm1tLm2t⋯Lmnt.Table 3
Gains and losses for all possible cases.
Types
Gains
Losses
a
:
r
j
¯
<
x
i
j
¯
0.5
x
i
j
¯
+
x
i
j
¯
−
r
j
¯
0
b
:
x
i
j
¯
<
r
j
¯
0
0.5
x
i
j
¯
+
x
i
j
¯
−
r
j
¯
c
:
r
j
¯
<
x
i
j
¯
<
r
j
¯
<
x
i
j
¯
0.5
x
i
j
¯
−
r
j
¯
0
d
:
x
i
j
¯
<
r
j
¯
<
x
i
j
¯
<
r
j
¯
0
0.5
x
i
j
¯
−
r
j
¯
e
:
r
i
j
¯
<
x
j
¯
<
x
i
j
¯
<
r
j
¯
0
0
f
:
x
i
j
¯
<
r
j
¯
<
r
i
j
¯
<
x
j
¯
0.5
x
i
j
¯
−
r
j
¯
0.5
x
i
j
¯
−
r
j
¯


### 3.3.3. Step 3: Calculation of Group Rough Decision Weights

The probability vector of states given by the decision-makerek can be expressed as p1k,p2k,…,plk, where ptk>0 and ∑tptk=1. According to the rough-number method, all the probability vectors from the decision group can be transformed to a rough probability vector p1¯,p1¯,p2¯,p2¯,…,pl¯,pl¯. Referring to the definition of consistency of interval probability initiated by Yager and Kreinovich [47], the following theorem proves that rough probability satisfies the property of consistency.Theorem 1.
The rough probability vectorp1¯,p1¯,p2¯,p2¯,…,pl¯,pl¯ is consistent; in other words, ∑tpt¯≤1 and ∑tpt¯≥1.Proof.
Denotept1,pt2,…,pth as the probability vector of the state St given by h decision-makers, and it can be transformed to a rough number pt¯,pt¯. For an easy and convenient expression, suppose pt1≤pt2≤⋯≤pth. Let 1/h∑k=1hptk=mt, and according to equations (4) and (5), pt1¯=pth¯=mt. So ptk¯≤pth¯ and ptk¯≥pt1¯ for ∀ptk, which lead to pt¯=1/h∑k=1hptk¯≤mt and pt¯=1/h∑k=1hptk¯≥mt. Therefore, we can get(35)∑t=1lpt¯=∑t=1l1h∑k=1hptk¯≤1h∑k=1h∑t=1lptk=1,∑t=1lpt¯=∑t=1l1h∑k=1hptk¯≥1h∑k=1h∑t=1lptk=1.
The theorem is proven.
The group rough decision weights of gainπt+=πt+¯,πt+¯ can be expressed as(36)πt+¯=pt¯γpt¯γ+1−pt¯γ1/γ,πt+¯=pt¯γpt¯γ+1−pt¯γ1/γ.
And the group rough decision weights of lossπt−=πt−¯,πt−¯ can be expressed as(37)πt−¯=pt¯δpt¯δ+1−pt¯δ1/δ,πt−¯=pt¯δpt¯δ+1−pt¯δ1/δ.


### 3.3.4. Step 4: Calculation of the Group Prospect Matrix

According to equation (34), the value matrixes for gain and loss under the state St are expressed as(38)Vt+=Gijtαm×n,Vt−=−λ−Lijtβm×n,where α, β, and λ are parameters discussed in classical prospect theory.Denote the prospect matrix byV=Vijm×n, where Vij is the prospect value of the alternative Ai at the criterion cj. The calculation formula of Vij is expressed as(39)Vij=∑t=1lVijt+πt++∑t=1lVijt−πt−.


### 3.3.5. Step 5: Calculation of Prospect Values and Arranging the Alternatives

Standardizing the prospect matrix to eliminate the effect of diverse dimensions, we can obtain the standardized prospect matrixV′=Vij′m×n, where(40)Vij′=VijVj∗,Vj∗=maxiVij¯,Vij¯.According to the rough weight obtained by the rough BWM, the overall prospect valueUi of the alternative Ai is calculated by(41)Ui=∑j=1nwj×Vij′.By the ranking rules for rough numbers shown in Figure1, alternatives are arranged according to their overall prospect values and the maximal one is selected as the optimal alternative.


## 3.3.1. Step 1: Construction of the Group Rough Evaluation Matrix and Expectation Vector

The evaluation matrixDkt and expectation vector Rkt from the decision-maker ek under the state St can be expressed as(30)Dkt=x11ktx12kt⋯x1nktx21ktx22kt⋅⋅⋅x2nkt⋮⋮⋱⋮xm1ktxm2kt⋯xmnkt,(31)Rkt=r1ktr2kt…rnkt.Evaluation matrixes and expectation vectors are aggregated, and equations (1)–(7) are used to obtain the group rough evaluation matrix and expectation vector under the state St as(32)Dt=x11t¯,x11t¯x12t¯,x12t¯⋯x1nt¯,x1nt¯x21t¯,x21t¯x22t¯,x22t¯⋯x2nt¯,x2nt¯⋮⋮⋱⋮xm1t¯,xm1t¯xm2t¯,xm2t¯⋯xmnt¯,xmnt¯,(33)Rt=r1t¯,r1t¯r2t¯,r2t¯…rnt¯,rnt¯.


## 3.3.2. Step 2: Calculation of Group Gain and Loss Matrixes

Group evaluation values and expectations are both rough numbers, which are in the form of interval numbers, so we can calculate gains and losses by means of the relationship between interval numbers shown in Table3. The group gain matrix Gt and group loss matrix Lt under the state St can be expressed as(34)Gt=G11tG12t⋯G1ntG21tG22t⋯G2nt⋮⋮⋱⋮Gm1tGm2t⋯Gmnt,Lt=L11tL12t⋯L1ntL21tL22t⋯L2nt⋮⋮⋱⋮Lm1tLm2t⋯Lmnt.Table 3
Gains and losses for all possible cases.
Types
Gains
Losses
a
:
r
j
¯
<
x
i
j
¯
0.5
x
i
j
¯
+
x
i
j
¯
−
r
j
¯
0
b
:
x
i
j
¯
<
r
j
¯
0
0.5
x
i
j
¯
+
x
i
j
¯
−
r
j
¯
c
:
r
j
¯
<
x
i
j
¯
<
r
j
¯
<
x
i
j
¯
0.5
x
i
j
¯
−
r
j
¯
0
d
:
x
i
j
¯
<
r
j
¯
<
x
i
j
¯
<
r
j
¯
0
0.5
x
i
j
¯
−
r
j
¯
e
:
r
i
j
¯
<
x
j
¯
<
x
i
j
¯
<
r
j
¯
0
0
f
:
x
i
j
¯
<
r
j
¯
<
r
i
j
¯
<
x
j
¯
0.5
x
i
j
¯
−
r
j
¯
0.5
x
i
j
¯
−
r
j
¯


## 3.3.3. Step 3: Calculation of Group Rough Decision Weights

The probability vector of states given by the decision-makerek can be expressed as p1k,p2k,…,plk, where ptk>0 and ∑tptk=1. According to the rough-number method, all the probability vectors from the decision group can be transformed to a rough probability vector p1¯,p1¯,p2¯,p2¯,…,pl¯,pl¯. Referring to the definition of consistency of interval probability initiated by Yager and Kreinovich [47], the following theorem proves that rough probability satisfies the property of consistency.Theorem 1.
The rough probability vectorp1¯,p1¯,p2¯,p2¯,…,pl¯,pl¯ is consistent; in other words, ∑tpt¯≤1 and ∑tpt¯≥1.Proof.
Denotept1,pt2,…,pth as the probability vector of the state St given by h decision-makers, and it can be transformed to a rough number pt¯,pt¯. For an easy and convenient expression, suppose pt1≤pt2≤⋯≤pth. Let 1/h∑k=1hptk=mt, and according to equations (4) and (5), pt1¯=pth¯=mt. So ptk¯≤pth¯ and ptk¯≥pt1¯ for ∀ptk, which lead to pt¯=1/h∑k=1hptk¯≤mt and pt¯=1/h∑k=1hptk¯≥mt. Therefore, we can get(35)∑t=1lpt¯=∑t=1l1h∑k=1hptk¯≤1h∑k=1h∑t=1lptk=1,∑t=1lpt¯=∑t=1l1h∑k=1hptk¯≥1h∑k=1h∑t=1lptk=1.
The theorem is proven.
The group rough decision weights of gainπt+=πt+¯,πt+¯ can be expressed as(36)πt+¯=pt¯γpt¯γ+1−pt¯γ1/γ,πt+¯=pt¯γpt¯γ+1−pt¯γ1/γ.
And the group rough decision weights of lossπt−=πt−¯,πt−¯ can be expressed as(37)πt−¯=pt¯δpt¯δ+1−pt¯δ1/δ,πt−¯=pt¯δpt¯δ+1−pt¯δ1/δ.


## 3.3.4. Step 4: Calculation of the Group Prospect Matrix

According to equation (34), the value matrixes for gain and loss under the state St are expressed as(38)Vt+=Gijtαm×n,Vt−=−λ−Lijtβm×n,where α, β, and λ are parameters discussed in classical prospect theory.Denote the prospect matrix byV=Vijm×n, where Vij is the prospect value of the alternative Ai at the criterion cj. The calculation formula of Vij is expressed as(39)Vij=∑t=1lVijt+πt++∑t=1lVijt−πt−.


## 3.3.5. Step 5: Calculation of Prospect Values and Arranging the Alternatives

Standardizing the prospect matrix to eliminate the effect of diverse dimensions, we can obtain the standardized prospect matrixV′=Vij′m×n, where(40)Vij′=VijVj∗,Vj∗=maxiVij¯,Vij¯.According to the rough weight obtained by the rough BWM, the overall prospect valueUi of the alternative Ai is calculated by(41)Ui=∑j=1nwj×Vij′.By the ranking rules for rough numbers shown in Figure1, alternatives are arranged according to their overall prospect values and the maximal one is selected as the optimal alternative.


## 4. An Illustrative Example

In this section, a practical MCGDM example in a risk environment is given to illustrate the feasibility and validity of the proposed method. The description of this example is as follows.An investment company is planning to select an optimal alternative to invest a sum of money. After the initial screening, there remain six possible alternatives, which are an Internet companyA1, a car company A2, two food companies (A3 and A4), an education company A5, and a chemical company A6. Four criteria, direct benefits c1, indirect benefits c2, social benefits c3, and environmental protection c4, are taken into account in order to evaluate the six alternatives and make the best choice. And there are three natural statuses according to the market forecast, including good S1, fair S2, and poor S3, which influence the evaluating values of alternatives in criteria. To obtain objective and comprehensive results, the investment company chooses five experts from different departments to constitute a decision committee, which is expressed as E=e1,e2,…,e5, and the committee needs to compare the relative importance of each criterion as well as the evaluating values of each alternative according to individual experiences.


### 4.1. Criteria Weighting

According to the discussion of the decision committee,c1 and c4 are determined as the best criterion and the worst criterion, respectively. Five decision-makers determine their comparison vectors by the scoring method in the BWM, and the results are collected in Tables 4 and 5.Table 4
Scores of best-to-others vectors.
c
1
c
2
c
3
c
4
c
1
e
1
1
1
3
5
e
2
1
3
5
7
e
3
1
1
5
7
e
4
1
3
3
5
e
5
1
1
3
7Table 5
Scores of others-to-worst vectors.
c
4
e
1
e
2
e
3
e
4
e
5
c
1
5
7
7
5
7
c
2
3
3
5
3
3
c
3
1
1
3
1
1
c
4
1
1
1
1
1Gathering the comparison vectors, we can obtain the integrated comparison vectors as(42)A1=1,1,1,1,1,1,3,1,3,1,3,5,5,3,3,5,7,7,5,7,A4=5,7,7,5,7,3,3,5,3,3,1,1,3,1,1,1,1,1,1,1.Integrated comparison vectors are transformed to rough comparison vectors as(43)RA1=1,1,1.32,2.28,3.32,4.28,5.72,6.68,RA4=5.72,6.68,3.08,3.72,1.08,1.72,1,1.A programming problem is constructed according to (21) as(44)minζ,s.t.w1¯−1.32w2¯≤ζ,w1¯−2.28w2¯≤ζ,w1¯−3.32w3¯≤ζ,w1¯−4.28w3¯≤ζ,w1¯−5.72w4¯≤ζ,w1¯−6.68w4¯≤ζ,w2¯−3.08w4¯≤ζ,w2¯−3.72w4¯≤ζ,w3¯−1.08w4¯≤ζ,w3¯−1.72w4¯≤ζ,12∑jwj¯+wj¯=1,wj¯>wj¯>0,forj=1,2,3,4.LINGO 16.0 is used to solve this problem, and we can obtain the unique solution asw1¯=0.464, w1¯=0.528, w2¯=0.242, w2¯=0.333, w3¯=0.117, w3¯=0.147, w4¯=0.083, w4¯=0.086, and ξ=0.025.Sincec1 and c4 are the best criterion and the worst criterion, respectively, and a14=5.72,6.68, we use a14¯=6.68 to calculate the value of CI. According to equation (28), we solve ξ2−1+2×6.68ξ+6.682−6.68=0 and obtain CI=3.499 and then CR=0.025/3.499=0.007, which implies a very good consistency.So the optimal rough weights can be denoted as(45)W¯=0.464,0.528,0.242,0.333,0.117,0.147,0.083,0.086.The rough weights of criteria are shown in Figure4, which illustrates that the ranking of weights is c1>c2>c3>c4.Figure 4
Rough weights of criteria.


### 4.2. Alternative Ranking

The evaluation values of alternatives in each criterion are linguistic variables {very poor, poor, fair, good, very good}, corresponding to the numerical variables {1, 3, 5, 7, 9}. Table6 shows the evaluation scores of alternatives by the decision group, and with the aid of transforming them to rough numbers by equations (1)–(7), we can obtain the rough scores shown in Table 7. Then, the gain and loss matrixes under different risk statuses can be calculated according to the results in Table 3.Table 6
Scores and expectations of alternatives.
Criteria
c
1
c
2
c
3
c
4
States
S
1
S
2
S
3
S
1
S
2
S
3
S
1
S
2
S
3
S
1
S
2
S
3
Alternatives
A
1
(7, 9, 7, 5, 7)
(9, 9, 7, 7, 7)
(5, 5, 7, 7, 5)
(5, 7, 7, 5, 7)
(3, 5, 5, 5, 7)
(3, 3, 1, 1, 3)
(7, 7, 9, 7, 7)
(3, 3, 5, 5, 7)
(3, 5, 3, 5, 5)
(5, 5, 7, 7, 5)
(3, 5, 3, 5, 5)
(1, 3, 1, 3, 3)
A
2
(7, 7, 9, 7, 7)
(5, 3, 3, 7, 7)
(3, 3, 5, 7, 7)
(5, 7, 5, 9, 7)
(5, 7, 7, 5, 5)
(3, 1, 5, 5, 3)
(5, 5, 3, 5, 5)
(3, 1, 3, 3, 1)
(5, 7, 5, 7, 5)
(3, 5, 3, 5, 3)
(3, 3, 1, 3, 3)
(1, 1, 3, 5, 1)
A
3
(9, 7, 7, 9, 7)
(7, 9, 5, 5, 9)
(3, 1, 3, 3, 5)
(7, 9, 7, 7, 5)
(5, 5, 7, 5, 5)
(3, 5, 5, 3, 5)
(3, 5, 1, 3, 5)
(1, 3, 1, 3, 1)
(5, 5, 5, 5, 7)
(7, 5, 7, 7, 7)
(7, 5, 5, 7, 5)
(5, 3, 3, 5, 3)
A
4
(5, 7, 9, 9, 7)
(5, 5, 7, 5, 5)
(5, 3, 5, 5, 3)
(5, 5, 5, 7, 7)
(3, 7, 5, 7, 5)
(1, 5, 3, 5, 7)
(9, 7, 7, 5, 7)
(7, 5, 5, 5, 5)
(7, 7, 3, 1, 5)
(7, 7, 9, 7, 9)
(7, 7, 7, 7, 7)
(7, 5, 7, 7, 5)
A
5
(9, 9, 7, 9, 7)
(7, 9, 5, 5, 5)
(5, 5, 3, 3, 7)
(7, 7, 7, 5, 5)
(5, 5, 7, 7, 5)
(3, 3, 5, 5, 3)
(7, 5, 5, 5, 7)
(5, 5, 3, 5, 5)
(3, 5, 3, 5, 5)
(7, 7, 3, 5, 5)
(7, 5, 3, 5, 5)
(5, 3, 3, 3, 5)
A
6
(7, 5, 9, 9, 7)
(7, 7, 9, 7, 5)
(5, 5, 7, 5, 5)
(5, 3, 7, 7, 9)
(3, 1, 5, 7, 7)
(3, 3, 5, 5, 7)
(9, 9, 7, 9, 7)
(7, 7, 7, 7, 5)
(5, 5, 7, 5, 7)
(3, 3, 5, 3, 1)
(3, 1, 3, 1, 3)
(1, 1, 3, 3, 3)
Expectations
(7, 7, 7, 9, 7)
(5, 5, 5, 7, 5)
(5, 5, 5, 3, 3)
(7, 7, 5, 7, 7)
(5, 7, 5, 7, 5)
(3, 5, 3, 5, 3)
(5, 5, 7, 5, 5)
(5, 3, 5, 5, 5)
(5, 3, 5, 3, 3)
(5, 7, 5, 5, 5)
(5, 5, 3, 3, 5)
(3, 3, 1, 1, 3)Table 7
Rough scores and expectations of alternatives.
Criteria
c
1
c
2
c
3
c
4
States
S
1
S
2
S
3
S
1
S
2
S
3
S
1
S
2
S
3
S
1
S
2
S
3
Alternatives
A
1
[6.30, 7.70]
[7.32, 8.28]
[5.32, 6.28]
[5.72, 6.68]
[4.30, 5.70]
[1.72, 2.68]
[7.08, 7.72]
[3.72, 5.50]
[3.72, 4.68]
[5.32, 6.28]
[3.72, 4.68]
[1.72, 2.68]
A
2
[7.08, 7.72]
[3.93, 6.07]
[4.67, 6.07]
[5.72, 7.51]
[5.32, 6.28]
[2.49, 4.28]
[4.28, 4.92]
[1.72, 2.68]
[5.32, 6.28]
[3.32, 4.28]
[2.28, 2.92]
[1.34, 3.12]
A
3
[7.32, 8.28]
[5.93, 8.07]
[2.30, 3.70]
[6.30, 7.70]
[5.08, 5.72]
[3.72, 4.68]
[2.49, 4.28]
[1.32, 2.28]
[5.08, 5.72]
[6.28, 6.92]
[5.32, 6.28]
[3.32, 4.28]
A
4
[6.49, 8.28]
[5.08, 5.72]
[3.72, 4.68]
[5.32, 6.28]
[4.49, 6.28]
[2.84, 5.51]
[6.30, 7.70]
[5.08, 5.72]
[3.04, 6.09]
[7.32, 8.28]
[7.00, 7.00]
[5.72, 6.68]
A
5
[7.72, 8.68]
[5.34, 7.12]
[3.72, 5.51]
[5.72, 6.68]
[5.32, 6.28]
[3.32, 4.28]
[5.32, 6.28]
[4.28, 4.92]
[3.72, 4.68]
[4.49, 6.28]
[4.30, 5.70]
[3.32, 4.28]
A
6
[6.49, 8.28]
[6.30, 7.70]
[5.08, 5.72]
[4.84, 7.51]
[3.04, 6.09]
[3.72, 5.51]
[7.72, 8.68]
[6.28, 6.92]
[5.32, 6.28]
[2.30, 3.70]
[1.72, 2.68]
[1.72, 2.68]
Expectations
[7.08, 7.72]
[5.08, 5.72]
[3.72, 4.68]
[6.28, 6.92]
[5.32, 6.28]
[3.32, 4.28]
[5.08, 5.72]
[4.28, 4.92]
[3.32, 4.28]
[5.08, 5.72]
[3.72, 4.68]
[1.72, 2.68]Gain and loss matrixes under the statusS1 are(46)G1=001.680.2800.295000.280.3900.880.2800.992.080.48000.280.280.2952.480,L1=−0.39−0.28000−0.28−0.48−1.2800−1.6950−0.259−0.48000−0.280−0.295−0.259−0.720−2.08.Gain and loss matrixes under the statusS2 are(47)G2=2.0800.2900.1750001.28001.12000.482.320.70000.511.2801.680,L2=0−0.51−0.280−0.5750−2.08−1.120−0.12−2.4800−0.4150000000−1.140−1.52.Gain and loss matrixes under the statusS3 are(48)G3=1.1200.200.69501.520.2200.21.121.1200.6150.9053.520.41500.21.120.720.6151.540,L3=0−1.12000−0.4150−0.19−0.720000−0.24−0.14000000000.The weight vector of states given by each decision-maker and group rough decision weights derived by equations (31)–(34) are described in Table 8. Here, we take the values of parameters given by Gonzalez and Wu in [46], so the rough decision weights of gain and loss are the same.Table 8
Rough decision weights.
States
S
1
S
2
S
3
Probability vectors
(0.3, 0.4, 0.3, 0.3, 0.4)
(0.4, 0.4, 0.5, 0.4, 0.3)
(0.3, 0.2, 0.2, 0.3, 0.3)
Rough probability
[0.316, 0.364]
[0.365, 0.435]
[0.236, 0.284]
Rough weight vectors
[0.341, 0.375]
[0.376, 0.425]
[0.280, 0.317]Takingα=0.89, β=0.92, and λ=2.25, the value matrixes for gain and loss under different states can be expressed as(49)V1+=001.5870.32200.337000.3220.43300.8930.32200.9911.9190.520000.3220.3220.3372.2440,V1−=−0.946−0.698000−0.698−1.145−2.82400−3.6560−0.649−1.145000−0.6980−0.732−0.649−1.6630−4.414,V2+=1.91900.33200.2120001.246001.106000.5202.1150.728000.5491.24601.5870,V2−=0−1.211−0.6980−1.3520−4.414−2.4970−0.320−5.18900−1.0020000000−2.5380−3.307,V3+=1.10600.23900.72301.4520.26000.2391.1061.10600.6490.9153.0650.45700.2391.1060.7460.6491.4690,V3−=0−2.497000−1.0020−0.488−1.6630000−0.605−0.369000000000.Then, the group prospect matrix is(50)V=0.676,0.844−1.568,−1.3930.436,0.5500.110,0.121−0.292,−0.189−0.453,−0.392−1.899,−1.590−2.202,−1.9560.051,0.1850.079,0.118−3.267,−2.8471.030,1.156−0.134,−0.101−0.865,−0.7310.673,0.7792.308,2.5900.579,0.649−0.262,−0.2380.067,0.0760.351,0.4550.544,0.665−1.406,−1.1891.773,1.982−3.061,−2.749.Also, the standardized prospect matrix is(51)V′=0.207,0.259−0.480,−0.4260.134,0.1680.034,0.037−0.090,−0.058−0.142,−0.120−0.581,−0.487−0.674,−0.5990.016,0.0570.024,0.036−1.000,−0.8720.315,0.354−0.041,−0.031−0.265,−0.2240.206,0.2390.706,0.7930.177,0.199−0.080,−0.0730.020,0.0230.108,0.1390.166,0.204−0.430,−0.3640.543,0.607−0.937,−0.841.Calculating the overall prospect values of each alternative according to rough criteria weights obtained in Section4.1, we obtain(52)U=−0.045,0.062−0.238,−0.163−0.108,−0.029−0.027,0.0350.067,0.103−0.083,0.039.The evaluating values of alternatives are depicted in Figure5. According to the ranking rules of rough numbers shown in Figure 1, it is obvious to find that the ranking order of the alternatives is A5>A1>A4>A6>A3>A2, so the best investment program of this company is A5. The results of ranking order may change when we choose different values of parameters, while there are no regular changes of the final evaluating values in U with the changes of parameters because the evaluating values are comprehensive aggregation of gain and loss matrixes, and the variation tendencies of gain matrix and loss matrixes are offset, which results in an irregular change of the evaluating values. Therefore, it is crucial to determine appropriate values of parameters.Figure 5
Evaluating results of alternatives.


### 4.3. Discussion

In order to discuss the validity of the proposed method, we choose other decision-making techniques with reliable results to calculate the illustrative example and compare the ranks of alternatives. Three MCGDM methods chosen in this section are rough VIKOR [7], prospect theory [36], and fuzzy-number prospect theory [38], respectively. Rough VIKOR takes no account of the expectations of decision-makers, and the synthetic rough evaluating values of alternatives are obtained by the weighted mean model. The crisp values in classical prospect theory can be acquired with the aid of the average values of the decision group, and the fuzzy information in fuzzy-number prospect theory can be obtained by the transforming method from linguistic variables to trapezoidal fuzzy numbers [38]. The ranking results of alternatives derived from different models are shown in Table 9.Table 9
Comparison of the ranks of alternatives by different models.
Alter.
RVIKOR [6]
PT [36]
FPT [38]
RPT
A
1
1
2
2
2
A
2
6
6
6
6
A
3
5
5
5
5
A
4
4
3
3
3
A
5
3
1
1
1
A
6
2
4
4
4It is easy to find that the results derived from methods based on prospect theory are allA5>A1>A4>A6>A3>A2, while the rank by rough VIKOR is A1>A6>A5>A4>A3>A2. The reason giving rise to this difference is that methods based on prospect theory pay attention to the expectations of decision-makers. With no expectations, A1 performs well in each criterion and obtains a high score in comprehensive evaluation, while A5 only holds the third place in the ranking. However, when considering the expectations of decision-makers, A5 becomes the best alternative and A1 falls to the second place. So it is crucial to pay more attention to the expectation, which could change the final result and influence the benefits of decision-makers.The rough-number method based on prospect theory in this paper can acquire the same result as classical prospect theory and fuzzy prospect theory, which declares the validity of the novel model, and there are particular advantages of this new method:(1)
Compared with the traditional MCGDM decision-making methods which acquire group information by the ordinary weighted sum method or interval numbers, the proposed method constructs group preferences with the aid of rough numbers, a new tool to deal with uncertain and subjective information. The aggregated approach by rough numbers considers the relationships between original data from every decision-maker, which brings more accurate group judgements and decision results.(2)
The improvement of the BWM based on the rough number can be effectively applied to group decision-making problems with a high consistency, which reduces the risk for errors derived from tedious calculation and low consistency of traditional pairwise comparison methods such as the AHP. The process of weight calculation is much more accessible and reliable, and the results derived from group preferences are more objective. Besides, as a linear programming model, the proposed rough-number-based BWM in this paper is much easier to calculate and can obtain a unique solution for the weight vector, which reduces the probability of decision failure.(3)
The existing decision-making methods based on prospect theory in dealing with imprecision take advantage of interval numbers, fuzzy numbers, and linguistic variables, and there is no research in developing prospect theory with the aid of rough numbers. The transformations of criteria values, expectations of decision-makers, and even the risk probabilities by rough numbers contain all the group information and bring more objective and reliable decision results than other models. The improvement expands the range of the application of prospect theory.


## 4.1. Criteria Weighting

According to the discussion of the decision committee,c1 and c4 are determined as the best criterion and the worst criterion, respectively. Five decision-makers determine their comparison vectors by the scoring method in the BWM, and the results are collected in Tables 4 and 5.Table 4
Scores of best-to-others vectors.
c
1
c
2
c
3
c
4
c
1
e
1
1
1
3
5
e
2
1
3
5
7
e
3
1
1
5
7
e
4
1
3
3
5
e
5
1
1
3
7Table 5
Scores of others-to-worst vectors.
c
4
e
1
e
2
e
3
e
4
e
5
c
1
5
7
7
5
7
c
2
3
3
5
3
3
c
3
1
1
3
1
1
c
4
1
1
1
1
1Gathering the comparison vectors, we can obtain the integrated comparison vectors as(42)A1=1,1,1,1,1,1,3,1,3,1,3,5,5,3,3,5,7,7,5,7,A4=5,7,7,5,7,3,3,5,3,3,1,1,3,1,1,1,1,1,1,1.Integrated comparison vectors are transformed to rough comparison vectors as(43)RA1=1,1,1.32,2.28,3.32,4.28,5.72,6.68,RA4=5.72,6.68,3.08,3.72,1.08,1.72,1,1.A programming problem is constructed according to (21) as(44)minζ,s.t.w1¯−1.32w2¯≤ζ,w1¯−2.28w2¯≤ζ,w1¯−3.32w3¯≤ζ,w1¯−4.28w3¯≤ζ,w1¯−5.72w4¯≤ζ,w1¯−6.68w4¯≤ζ,w2¯−3.08w4¯≤ζ,w2¯−3.72w4¯≤ζ,w3¯−1.08w4¯≤ζ,w3¯−1.72w4¯≤ζ,12∑jwj¯+wj¯=1,wj¯>wj¯>0,forj=1,2,3,4.LINGO 16.0 is used to solve this problem, and we can obtain the unique solution asw1¯=0.464, w1¯=0.528, w2¯=0.242, w2¯=0.333, w3¯=0.117, w3¯=0.147, w4¯=0.083, w4¯=0.086, and ξ=0.025.Sincec1 and c4 are the best criterion and the worst criterion, respectively, and a14=5.72,6.68, we use a14¯=6.68 to calculate the value of CI. According to equation (28), we solve ξ2−1+2×6.68ξ+6.682−6.68=0 and obtain CI=3.499 and then CR=0.025/3.499=0.007, which implies a very good consistency.So the optimal rough weights can be denoted as(45)W¯=0.464,0.528,0.242,0.333,0.117,0.147,0.083,0.086.The rough weights of criteria are shown in Figure4, which illustrates that the ranking of weights is c1>c2>c3>c4.Figure 4
Rough weights of criteria.


## 4.2. Alternative Ranking

The evaluation values of alternatives in each criterion are linguistic variables {very poor, poor, fair, good, very good}, corresponding to the numerical variables {1, 3, 5, 7, 9}. Table6 shows the evaluation scores of alternatives by the decision group, and with the aid of transforming them to rough numbers by equations (1)–(7), we can obtain the rough scores shown in Table 7. Then, the gain and loss matrixes under different risk statuses can be calculated according to the results in Table 3.Table 6
Scores and expectations of alternatives.
Criteria
c
1
c
2
c
3
c
4
States
S
1
S
2
S
3
S
1
S
2
S
3
S
1
S
2
S
3
S
1
S
2
S
3
Alternatives
A
1
(7, 9, 7, 5, 7)
(9, 9, 7, 7, 7)
(5, 5, 7, 7, 5)
(5, 7, 7, 5, 7)
(3, 5, 5, 5, 7)
(3, 3, 1, 1, 3)
(7, 7, 9, 7, 7)
(3, 3, 5, 5, 7)
(3, 5, 3, 5, 5)
(5, 5, 7, 7, 5)
(3, 5, 3, 5, 5)
(1, 3, 1, 3, 3)
A
2
(7, 7, 9, 7, 7)
(5, 3, 3, 7, 7)
(3, 3, 5, 7, 7)
(5, 7, 5, 9, 7)
(5, 7, 7, 5, 5)
(3, 1, 5, 5, 3)
(5, 5, 3, 5, 5)
(3, 1, 3, 3, 1)
(5, 7, 5, 7, 5)
(3, 5, 3, 5, 3)
(3, 3, 1, 3, 3)
(1, 1, 3, 5, 1)
A
3
(9, 7, 7, 9, 7)
(7, 9, 5, 5, 9)
(3, 1, 3, 3, 5)
(7, 9, 7, 7, 5)
(5, 5, 7, 5, 5)
(3, 5, 5, 3, 5)
(3, 5, 1, 3, 5)
(1, 3, 1, 3, 1)
(5, 5, 5, 5, 7)
(7, 5, 7, 7, 7)
(7, 5, 5, 7, 5)
(5, 3, 3, 5, 3)
A
4
(5, 7, 9, 9, 7)
(5, 5, 7, 5, 5)
(5, 3, 5, 5, 3)
(5, 5, 5, 7, 7)
(3, 7, 5, 7, 5)
(1, 5, 3, 5, 7)
(9, 7, 7, 5, 7)
(7, 5, 5, 5, 5)
(7, 7, 3, 1, 5)
(7, 7, 9, 7, 9)
(7, 7, 7, 7, 7)
(7, 5, 7, 7, 5)
A
5
(9, 9, 7, 9, 7)
(7, 9, 5, 5, 5)
(5, 5, 3, 3, 7)
(7, 7, 7, 5, 5)
(5, 5, 7, 7, 5)
(3, 3, 5, 5, 3)
(7, 5, 5, 5, 7)
(5, 5, 3, 5, 5)
(3, 5, 3, 5, 5)
(7, 7, 3, 5, 5)
(7, 5, 3, 5, 5)
(5, 3, 3, 3, 5)
A
6
(7, 5, 9, 9, 7)
(7, 7, 9, 7, 5)
(5, 5, 7, 5, 5)
(5, 3, 7, 7, 9)
(3, 1, 5, 7, 7)
(3, 3, 5, 5, 7)
(9, 9, 7, 9, 7)
(7, 7, 7, 7, 5)
(5, 5, 7, 5, 7)
(3, 3, 5, 3, 1)
(3, 1, 3, 1, 3)
(1, 1, 3, 3, 3)
Expectations
(7, 7, 7, 9, 7)
(5, 5, 5, 7, 5)
(5, 5, 5, 3, 3)
(7, 7, 5, 7, 7)
(5, 7, 5, 7, 5)
(3, 5, 3, 5, 3)
(5, 5, 7, 5, 5)
(5, 3, 5, 5, 5)
(5, 3, 5, 3, 3)
(5, 7, 5, 5, 5)
(5, 5, 3, 3, 5)
(3, 3, 1, 1, 3)Table 7
Rough scores and expectations of alternatives.
Criteria
c
1
c
2
c
3
c
4
States
S
1
S
2
S
3
S
1
S
2
S
3
S
1
S
2
S
3
S
1
S
2
S
3
Alternatives
A
1
[6.30, 7.70]
[7.32, 8.28]
[5.32, 6.28]
[5.72, 6.68]
[4.30, 5.70]
[1.72, 2.68]
[7.08, 7.72]
[3.72, 5.50]
[3.72, 4.68]
[5.32, 6.28]
[3.72, 4.68]
[1.72, 2.68]
A
2
[7.08, 7.72]
[3.93, 6.07]
[4.67, 6.07]
[5.72, 7.51]
[5.32, 6.28]
[2.49, 4.28]
[4.28, 4.92]
[1.72, 2.68]
[5.32, 6.28]
[3.32, 4.28]
[2.28, 2.92]
[1.34, 3.12]
A
3
[7.32, 8.28]
[5.93, 8.07]
[2.30, 3.70]
[6.30, 7.70]
[5.08, 5.72]
[3.72, 4.68]
[2.49, 4.28]
[1.32, 2.28]
[5.08, 5.72]
[6.28, 6.92]
[5.32, 6.28]
[3.32, 4.28]
A
4
[6.49, 8.28]
[5.08, 5.72]
[3.72, 4.68]
[5.32, 6.28]
[4.49, 6.28]
[2.84, 5.51]
[6.30, 7.70]
[5.08, 5.72]
[3.04, 6.09]
[7.32, 8.28]
[7.00, 7.00]
[5.72, 6.68]
A
5
[7.72, 8.68]
[5.34, 7.12]
[3.72, 5.51]
[5.72, 6.68]
[5.32, 6.28]
[3.32, 4.28]
[5.32, 6.28]
[4.28, 4.92]
[3.72, 4.68]
[4.49, 6.28]
[4.30, 5.70]
[3.32, 4.28]
A
6
[6.49, 8.28]
[6.30, 7.70]
[5.08, 5.72]
[4.84, 7.51]
[3.04, 6.09]
[3.72, 5.51]
[7.72, 8.68]
[6.28, 6.92]
[5.32, 6.28]
[2.30, 3.70]
[1.72, 2.68]
[1.72, 2.68]
Expectations
[7.08, 7.72]
[5.08, 5.72]
[3.72, 4.68]
[6.28, 6.92]
[5.32, 6.28]
[3.32, 4.28]
[5.08, 5.72]
[4.28, 4.92]
[3.32, 4.28]
[5.08, 5.72]
[3.72, 4.68]
[1.72, 2.68]Gain and loss matrixes under the statusS1 are(46)G1=001.680.2800.295000.280.3900.880.2800.992.080.48000.280.280.2952.480,L1=−0.39−0.28000−0.28−0.48−1.2800−1.6950−0.259−0.48000−0.280−0.295−0.259−0.720−2.08.Gain and loss matrixes under the statusS2 are(47)G2=2.0800.2900.1750001.28001.12000.482.320.70000.511.2801.680,L2=0−0.51−0.280−0.5750−2.08−1.120−0.12−2.4800−0.4150000000−1.140−1.52.Gain and loss matrixes under the statusS3 are(48)G3=1.1200.200.69501.520.2200.21.121.1200.6150.9053.520.41500.21.120.720.6151.540,L3=0−1.12000−0.4150−0.19−0.720000−0.24−0.14000000000.The weight vector of states given by each decision-maker and group rough decision weights derived by equations (31)–(34) are described in Table 8. Here, we take the values of parameters given by Gonzalez and Wu in [46], so the rough decision weights of gain and loss are the same.Table 8
Rough decision weights.
States
S
1
S
2
S
3
Probability vectors
(0.3, 0.4, 0.3, 0.3, 0.4)
(0.4, 0.4, 0.5, 0.4, 0.3)
(0.3, 0.2, 0.2, 0.3, 0.3)
Rough probability
[0.316, 0.364]
[0.365, 0.435]
[0.236, 0.284]
Rough weight vectors
[0.341, 0.375]
[0.376, 0.425]
[0.280, 0.317]Takingα=0.89, β=0.92, and λ=2.25, the value matrixes for gain and loss under different states can be expressed as(49)V1+=001.5870.32200.337000.3220.43300.8930.32200.9911.9190.520000.3220.3220.3372.2440,V1−=−0.946−0.698000−0.698−1.145−2.82400−3.6560−0.649−1.145000−0.6980−0.732−0.649−1.6630−4.414,V2+=1.91900.33200.2120001.246001.106000.5202.1150.728000.5491.24601.5870,V2−=0−1.211−0.6980−1.3520−4.414−2.4970−0.320−5.18900−1.0020000000−2.5380−3.307,V3+=1.10600.23900.72301.4520.26000.2391.1061.10600.6490.9153.0650.45700.2391.1060.7460.6491.4690,V3−=0−2.497000−1.0020−0.488−1.6630000−0.605−0.369000000000.Then, the group prospect matrix is(50)V=0.676,0.844−1.568,−1.3930.436,0.5500.110,0.121−0.292,−0.189−0.453,−0.392−1.899,−1.590−2.202,−1.9560.051,0.1850.079,0.118−3.267,−2.8471.030,1.156−0.134,−0.101−0.865,−0.7310.673,0.7792.308,2.5900.579,0.649−0.262,−0.2380.067,0.0760.351,0.4550.544,0.665−1.406,−1.1891.773,1.982−3.061,−2.749.Also, the standardized prospect matrix is(51)V′=0.207,0.259−0.480,−0.4260.134,0.1680.034,0.037−0.090,−0.058−0.142,−0.120−0.581,−0.487−0.674,−0.5990.016,0.0570.024,0.036−1.000,−0.8720.315,0.354−0.041,−0.031−0.265,−0.2240.206,0.2390.706,0.7930.177,0.199−0.080,−0.0730.020,0.0230.108,0.1390.166,0.204−0.430,−0.3640.543,0.607−0.937,−0.841.Calculating the overall prospect values of each alternative according to rough criteria weights obtained in Section4.1, we obtain(52)U=−0.045,0.062−0.238,−0.163−0.108,−0.029−0.027,0.0350.067,0.103−0.083,0.039.The evaluating values of alternatives are depicted in Figure5. According to the ranking rules of rough numbers shown in Figure 1, it is obvious to find that the ranking order of the alternatives is A5>A1>A4>A6>A3>A2, so the best investment program of this company is A5. The results of ranking order may change when we choose different values of parameters, while there are no regular changes of the final evaluating values in U with the changes of parameters because the evaluating values are comprehensive aggregation of gain and loss matrixes, and the variation tendencies of gain matrix and loss matrixes are offset, which results in an irregular change of the evaluating values. Therefore, it is crucial to determine appropriate values of parameters.Figure 5
Evaluating results of alternatives.


## 4.3. Discussion

In order to discuss the validity of the proposed method, we choose other decision-making techniques with reliable results to calculate the illustrative example and compare the ranks of alternatives. Three MCGDM methods chosen in this section are rough VIKOR [7], prospect theory [36], and fuzzy-number prospect theory [38], respectively. Rough VIKOR takes no account of the expectations of decision-makers, and the synthetic rough evaluating values of alternatives are obtained by the weighted mean model. The crisp values in classical prospect theory can be acquired with the aid of the average values of the decision group, and the fuzzy information in fuzzy-number prospect theory can be obtained by the transforming method from linguistic variables to trapezoidal fuzzy numbers [38]. The ranking results of alternatives derived from different models are shown in Table 9.Table 9
Comparison of the ranks of alternatives by different models.
Alter.
RVIKOR [6]
PT [36]
FPT [38]
RPT
A
1
1
2
2
2
A
2
6
6
6
6
A
3
5
5
5
5
A
4
4
3
3
3
A
5
3
1
1
1
A
6
2
4
4
4It is easy to find that the results derived from methods based on prospect theory are allA5>A1>A4>A6>A3>A2, while the rank by rough VIKOR is A1>A6>A5>A4>A3>A2. The reason giving rise to this difference is that methods based on prospect theory pay attention to the expectations of decision-makers. With no expectations, A1 performs well in each criterion and obtains a high score in comprehensive evaluation, while A5 only holds the third place in the ranking. However, when considering the expectations of decision-makers, A5 becomes the best alternative and A1 falls to the second place. So it is crucial to pay more attention to the expectation, which could change the final result and influence the benefits of decision-makers.The rough-number method based on prospect theory in this paper can acquire the same result as classical prospect theory and fuzzy prospect theory, which declares the validity of the novel model, and there are particular advantages of this new method:(1)
Compared with the traditional MCGDM decision-making methods which acquire group information by the ordinary weighted sum method or interval numbers, the proposed method constructs group preferences with the aid of rough numbers, a new tool to deal with uncertain and subjective information. The aggregated approach by rough numbers considers the relationships between original data from every decision-maker, which brings more accurate group judgements and decision results.(2)
The improvement of the BWM based on the rough number can be effectively applied to group decision-making problems with a high consistency, which reduces the risk for errors derived from tedious calculation and low consistency of traditional pairwise comparison methods such as the AHP. The process of weight calculation is much more accessible and reliable, and the results derived from group preferences are more objective. Besides, as a linear programming model, the proposed rough-number-based BWM in this paper is much easier to calculate and can obtain a unique solution for the weight vector, which reduces the probability of decision failure.(3)
The existing decision-making methods based on prospect theory in dealing with imprecision take advantage of interval numbers, fuzzy numbers, and linguistic variables, and there is no research in developing prospect theory with the aid of rough numbers. The transformations of criteria values, expectations of decision-makers, and even the risk probabilities by rough numbers contain all the group information and bring more objective and reliable decision results than other models. The improvement expands the range of the application of prospect theory.


## 5. Conclusion

With the rapid development and changes of the social and the economic environment, decision-makers always face multicriteria group decision-making problems with imprecision, risk, and subjectivity. To handle the realistic decision-making problems in a reasonable and effective way, this paper constructs a risky MCGDM method by combining rough numbers, BWM, and prospect theory, which is able to tackle subjectivity and imprecision. According to the process of decision-making, the proposed method can be divided into two stages. The first stage is to integrate the rough number and BWM to calculate the rough weights of criteria, and the second one is combining the rough number and prospect theory to arrange the alternatives. Then, a case study involving investment is introduced to illustrate the application of this new method, and the results verify the feasibility and validity. The proposed method constructs a linear programming model to obtain the weights of criteria and a psychological decision-making model to handle the subjectivity and risk of problems. It puts forward a new research direction in the MCGDM and can be applied to many practical group decision-making problems with various conditions. In terms of future research, we intend to consider the compensatory and interactive relationship between criteria, and the impact of changes of decision-makers and criteria on the final evaluation results in a dynamic procedure. The expansion of rough numbers to other MCDM methods and realistic problems is also an interesting and significant direction for further research.



---

*Source: 8738327-2020-02-21.xml*