A Multicriteria Interval-Valued Intuitionistic Fuzzy Set TOPSIS Decision-Making Approach Based on the Improved Score Function

Wei-wei Li; Chong Wu

doi:10.1515/jisys-2015-0096

Article Open Access

A Multicriteria Interval-Valued Intuitionistic Fuzzy Set TOPSIS Decision-Making Approach Based on the Improved Score Function

Wei-wei Li and Chong Wu

Published/Copyright: November 6, 2015

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From the journal Journal of Intelligent Systems Volume 25 Issue 2

Abstract

In this paper, an improved technique for order preference by similarity to an ideal solution (TOPSIS) method was proposed based on completely unknown attribute weight information, as well as taking multiple criteria decision making of interval-valued intuitionistic fuzzy number as the evaluation information. First of all, a cumulative interval score function considering the influence of hesitation as well as a cumulative score function containing the risk preference of the decision maker were constructed. Then, a new information entropy function was constructed by using the newly defined score function, which presents a new method that directly utilizes evaluation information to obtain criterion weight. Next, all schemes were sequenced by virtue of relative closeness and the criterion weight of each alternative and ideal scheme. Finally, the effectiveness of the proposed method was illustrated by comparison with examples.

Keywords: Interval-valued intuitionistic fuzzy number (IVIFN); TOPSIS; entropy weight method; cumulative interval-valued score function (SCIV); dot operator of the cumulative interval-valued score function (SDOCIV)

1 Introduction

With the rapid development of society, economy, and technology, especially network technology and information technology, the situation now is more complex and varying than that in the past. In such a situation, decision makers are always coming up with fuzzy concepts [18]. Only based on the traditional fuzzy set theory, even with a known membership degree of one element in set A, the hesitancy degree cannot be determined. Atanassov [1], a scholar from the Republic of Bulgaria, proposed the theory of intuitionistic fuzzy sets (IFSs), which describes unknown information from the aspect of membership degree, non-membership degree, and hesitancy degree. This theory is suitable for dealing with unknown information and can be used to describe fuzzy concepts. Atanassov and Gargov [2] also extended the theory of IFSs. To be specific, they illustrated the concept of interval-valued IFSs (IVIFSs) and defined their basic algorithm. IVIFSs are more flexible and practical in dealing with fuzzy and uncertain problems than IFSs. Therefore, scholars have conducted extensive research on them from the angles of distance metric [4], score function [6, 22], entropy [9], attribute weight [14], prospect theory [7], algorithm [25, 27, 30], aggregation operators [3, 12, 21, 25, 27, 30, 35], and dynamic decision making [16, 29]. In recent years, IVIFSs have been widely applied in the field of project decision [36], risk investment [6], logic programming [36], and pattern identification [20].

The technique for order preference by similarity to an ideal solution (TOPSIS) [4] is a priority method for multiattribute decision-making problems. By this method, the closeness degree between the project to be evaluated and the ideal project is determined. It is judged based on similarity degree [9], distance function [4], probability [6], and similarity coefficient [8, 34]. In the process of judgment, it is necessary to determine attribute weight, as attribute varies in importance. Here, the key problem is how to use matrix information to solve score function and to weigh attribute on the condition that the attribute weight is unknown. It is a combination of two problems: one is the choice of the weighting method, and the other is the selection of the score function. As to the choice of the weighting method, most scholars figure out the weight by constructing distortion function and establishing a goal-programming problem [6, 25, 33]. This weighting method is too complex [17]. Wu and Wan [23] worked out the weight by directly using the information on the score function matrix to construct the information entropy function. This method is relatively simple and effective. The weighting function constructed in Wu and Wan [23] has two disadvantages. One is that the score function is chosen on the condition that hesitancy is not considered. The other is that when the score function is normalized by the weighting function, S(α) may be negative. A negative S(α) cannot be substituted into the mean information entropy function Hs(Xj) = − ∑i = 1nS˜(αij)lnS˜(αij)lnn. It is possible to figure out the logarithm of a negative number with e as the base. When S(α)≤0, the problem cannot be further solved after S(α) is substituted into the weighted function. As to the selection of the score function, Chen et al. proposed an IVIFS sorting method based on score function [5]. Liu and Wang have proposed several score functions [10, 11, 19, 32]. However, the result obtained by using the functions above sometimes appears to be inconsistent with the real results [6]. Scholars generally hold the view that there are two reasons for the defects in the present score functions: one is that the interval number is directly converted into a certain number, and the other is that the effect of hesitancy degree on the decision influence is not considered [6]. In the previous literature, the effect of hesitancy degree on score function is analyzed based on subjective judgment made by decision makers. There are three score functions proposed to analyze the effect: S_N(α)=u_α−v_α+(p−q)π_α [36], S_Ln(α)=u_α+pπ_α/(p+q) [29], and S_L(α)=u_α–v_α+ρπ_α [31], where p, q, and ρ are the parameters of decision makers’ subjective judgment. However, in practice, p, q, and ρ are not fixed within a certain range. Scholars define the new score function of hesitancy [22, 29]. In fact, the selection of the score function is related to the specific situation of the actual problem. However, few scholars realize that they should choose the kind of score function according to specific circumstances.

With regard to the above questions, an IVIFS TOPSIS decision-making model based on cumulative interval-valued score function is proposed. For this model, breakthrough is made in three aspects:

In this paper, a new cumulative score function of IFSs and a new cumulative interval-valued score function are defined based on the premise that hesitated people have a herd mentality. As a result, the two problems of the existing score function are overcome: subjective consideration of hesitancy degree and applicable scope. In the final part, the applicable scope is specified and analyzed.
On the basis of decision makers’ preference of risk, a dot operator equation of the cumulative interval-valued score function is defined. The interval-valued intuitionistic fuzzy information integration method based on the risk preference factor is proposed. By this method, the project sequence can adequately reflect decision makers’ preference of risk; therefore, a more reasonable decision making is given. The application of interval-valued intuitionistic fuzzy theory is also widened.
The weighting function is improved in two aspects. First, the weighting equation of IVIFSs based on information entropy in Wu and Wan [23] is improved. Second, the dot operator equation of the cumulative interval-valued score function defined in this paper is used. The improvement makes possible the modification of the risk factor according to the dot operator equation of the cumulative interval-valued score function. The decision sequence is given for decision makers with different risk preferences.

2 Multicriteria Group Decision Environment Based on IVIFSs

Considering that sometimes the membership degree and non-membership degree are difficult to precisely numerically determine, Atanassov and Gargov extended IFSs and presented the notion of IVIFSs.

Definition 1 [2]: Let X be a non-empty set and D [0, 1] be the set of all closed subintervals of [0, 1]. An IVIFS A in X is an expression defined by

A˜ = {〈x, MA˜˜(x), NA˜˜(x)〉|x ∈ X} = {〈x, [MA˜L(x), MA˜U(x)], [NA˜L(x), NA˜U(x)]〉|x ∈ X},

where MA˜˜(x):X → D[0, 1] and NA˜˜(x):X → D[0, 1] denote the membership degree and the non-membership degree for any x∈X, respectively. MA˜˜(x) and NA˜˜(x) are closed intervals rather than real number, and their lower and upper boundaries are denoted by MA˜L(x), MA˜U(x), 0 ≤ NA˜L(x) + NA˜˜(x) ≤ 1 and NA˜U(x), respectively. The expression is subject to the condition 0 ≤ MA˜˜(x) + NA˜˜(x) ≤ 1.

Definition 2 [2]: For each element x, the hesitancy degree of an intuitionistic fuzzy interval of x∈X in A˜ is defined as follows:

πA˜˜(x) = 1 − MA˜˜(x) − NA˜˜(x) = [1 − MA˜U˜(x) − NA˜U˜(x), 1 − MA˜L˜(x) − NA˜L˜(x)] = [πA˜L˜(x), πA˜U˜(x)].

Especially, if MA˜U˜(x) = MA˜L˜(x) and NA˜U˜(x) = NA˜L˜(x), then the IVIFS is reduced to an ordinary IFS.

Definition 3 [19]: For any interval-valued intuitionistic fuzzy number (IVIFN) α=(u_A(x), v_A(x)), its score function is S(α) = [S,ijl S]iju = [μα− − vα+, μα+ − vα−], where S(α)∈[–1, 1].

According to Definition 3, the interval-valued intuitionistic fuzzy decision matrix D=(d_ij)_m×n can be converted into interval-valued score function matrix S=(S_ij)_m×n, where Sij = [S,ijl S]iju (interval numbers).

Definition 4 [19]: Assume Sij = [S,ijl S]iju is an interval number; Gα(Sij) = S ijl+ Siju2 + αS ijl− Siju2 is defined as the dot operator of Sij = [S,ijl S]iju, where α is the risk factor; and α∈[–1, 1]. When α=0, the decision maker neither likes nor dislikes risk; when α>0, the decision maker likes risk; and when α<0, the decision maker dislikes risk.

Definition 5 [32]: Assume α = ([uα−, uα+], [vα−, vα+]) and β = ([uβ−, uβ+], [vβ−, vβ+]) are two interval-valued IFNs, d(α, β) = 14(|uα− − uβ−| + |uα+ − uβ+| + |vα− − vβ−| + |vα+ − vβ+|).

3 The Cumulative Interval-Valued Score Function that Takes the Hesitant Population into Consideration

In this section, the cumulative score function of IFNs is first analyzed, and then the cumulative score function of interval-valued IFNs.

3.1 The Cumulative Score Function of IFNs

In the previous references, the effect of hesitancy degree on score function is analyzed based on the subjective judgment made by decision makers. There are three score functions proposed to analyze the effect: S_N(α)=u_α−v_α+(p−q)π_α [36], S_Ln(α)=u_α+pπ_α/(p+q) [29], and S_L(α)=u_α−v_α+ρπ_α [31] are the parameters of the decision makers’ subjective judgment. However, in practice, p, q, and ρ are not fixed within a certain range. In Gao et al. [7], based on the herd mentality individuals have, the effect of hesitancy degree on score function is analyzed. However, there is a little deviation about the analysis. The cumulative score function of IFN proposed in the references above is S(α) = μ − v1 − (μ + v)π. The cumulative score function of IFNs and dot aggregators have been investigated by many authors [15, 13, 24, 26, 28]; based on these references, the cumulative score function of intuitionistic fuzzy number (IFN) is analyzed in details as follows.

In actual life, when people are hesitant in making a choice or a decision, they are susceptible to others’ influence, and do as others do. For the IFN α=(μ, v), the hesitation part is π on the first round. On the second round, the hesitation part of the first round π is divided, according to voter turnout of the first round, into three parts: part 1, part 2, and part 3. These three parts represent those who are susceptible to supporting people, those who are susceptible to objecting to people, and those who are not susceptible to both, respectively. For part 1, those who are susceptible to supporting people are μπ; for part 2, those who are susceptible to objecting to people are vπ; and for part 3, those who are susceptible to neither are π². On the third round, the neutral part of the second round π² is redivided into three parts: the supporting part, the objecting part, and the neutral part. These three parts are μπ², vπ², and π³. All in all, after the first round of ballot, the neutral part is π, the supporting part is u, and the objecting part is v. The neutral part vetoes again. After the second round of ballot, the neutral part is π²; the supporting part is u+uπ, as uπ is added; and the objecting part is v+vπ, as vπ is added. After the nth round of ballot, the neutral part is πⁿ; to the supporting part, uπⁿ⁻¹ is added; and to the objecting part, vπⁿ⁻¹ is added. For these n rounds of ballot (do n−1 times of analysis of the hesitation part), the cumulative membership and non-membership degrees are illustrated as follows:

μn(α) = u + uπ + uπ2 + ⋯uπn−1,vn(α) = v + vπ + vπ2 + ⋯vπn−1.

After several ballots, find the limit to obtain the cumulative membership and the cumulative non-membership:

limn→∞μn(α) = uu + v, limn→∞vn(α) = vu + v

Thus, the new score function of IFNs are as follows:

Definition 6: Assume α=(μ, v) is an IFN, then SC(α) = μ − vμ + v is its cumulative score function.

3.2 The Cumulative Interval-Valued Score Function of IVIFSs

For any IVIFN α=(u_A(x), v_A(x)), the hesitancy degree is considered. For the left endpoint and the right endpoint Sijl = μα− − vα+ and Siju = μα+ − vα− of the interval-valued score function S(α) = [S,ijl S]iju of Definition 3, the hesitancy degree is also considered. The hesitancy degree of the lower limit Sijl = μα− − vα+ is 1 − (μα− + vα+), while that of the upper limit Siju = μα+ − vα− is 1 − (μα+ + vα−). For the upper and lower limit Sijl and Siju, the hesitancy degree is considered using the method of Definition 6. The score function is improved. After n ballots of the hesitated people, the lower limit of the score function is

μn−(α) = u− + u−π + u−π2 + ⋯u−πn, limn→∞μn−(α) = uα−uα− + vα+,vn+(α) = v+ + v+π + v+π2 + ⋯v+πn, limn→∞vn−(α) = vα+uα− + vα+

Thus, the lower limit of the improved cumulative interval-valued score function is SCijl = μα− − vα+μα− + vα+; in a similar way, it can be concluded that the upper limit of the improved cumulative interval-valued score function is SCiju = μα+ − vα−μα+ + vα−.

Thus, the new interval-valued score function of IVIFNs are defined as follows:

Definition 7: For an IVIFN α=(u_A(x), v_A(x)), its cumulative interval-valued score function is SCIV(α) = [μα− − vα+μα− + vα+, μα+ − vα−μα+ + vα−].

3.3 Theorem of the Cumulative Interval-Valued Score Function

Theorem 1:With regard to an IVIFNα = ([uα−, uα+], [vα−, vα+]),if uαi ∈ (uα−, uα+), and vαi ∈ [vα−, vα+] for any (u_αi, v_αi), then SCIV(αi) = uαi − vαiuαi + vαi ∈ [μα− − vα+μα− + vα+, μα+ − vα−μα+ + vα−].

Proof: First, let A = μα+ − vα−μα+ + vα− − uαi − vαiuαi + vαi. To prove μα+ − vα−μα+ + vα− ≥ uαi − vαiuαi + vαi is to prove A≥0; because the denominator of A is >0, to prove A≥0 is to prove B = (μα+ − vα−)(uαi + vαi) − (uαi − vαi)(μα+ + vα−) ≥ 0, B = 2(μα+vα+ − vαi−μαi−). According to the definition of the IVIFN, μα+ ≥ μα−, vα+ ≥ vα−. Obviously, B≥0, so uαi − vαiuαi + vαi ≤ μα+ − vα−μα+ + vα− is correct.

In the similar way, it can be proved that uαi − vαiuαi + vαi ≥ μα− − vα+μα− + vα+ is correct. Therefore, for any (u_αi, v_αi), if uαi ∈ (uα−, uα+) and vαi ∈ [vα−, vα+], then SCIV(αi) = uαi − vαiuαi + vαi ∈ [μα− − vα+μα− + vα+, μα+ − vα−μα+ + vα−].

According to Theorem 1, if hesitated people do as others do, the cumulative interval-valued score function has the largest value SCIViju = μα+ − vα−μα+ + vα− when the number of the supporting people is μα+ (the largest number) and the number of the opposing people is vα− (the smallest number). In contrast, the cumulative interval-valued score function has the smallest value SCIVijl = μα− − vα+μα− + vα+ when the number of the supporting people is μα− (the smallest number) and the number of the opposing people is vα+ (the largest number). □

Theorem 2:The upper and lower limitSijlandSijuof the cumulative interval-valued score functionSCIV(α) = [μα− − vα+μα− + vα+, μα+ − vα−μα+ + vα−]of an IVIFNα=(u_A(x), v_A(x)) are both the strictly monotonically increasing function of u_A(x) and the strictly monotonically decreasing function of v_A(x).

Proof:Sijl = μα− − vα+μα− + vα+ = 1 − 2vα+μα− + vα+, ∂Sijl∂μα− = ∂SCIV(μα−, vα+)∂μα− = 2vα+(μα− + vα+)2,

∂Sijl∂vα+ = ∂SCIV(μα−, vα+)∂vα+ = 2vα+(μα− + vα+)2 − 2μα− + vα+ = 2vα+ − 2(μα−+vα+)(μα− + vα+)2 = −2μα−(μα− + vα+)2.

Because μα− ≥ 0, vα+ ≥ 0, ∂Sijl∂μα− ≥ 0, and only when vα+ = 0, this equation is equal to 0; ∂Sijl∂vα+ ≤ 0, and only when μα− = 0, this equation is equal to 0. Therefore, the lower limit Sijl of the cumulative interval-valued score function SCIV(α) = [μα− − vα+μα− + vα+, μα+ − vα−μα+ + vα−] is the strictly monotonically increasing function of u_A(x) and the strictly monotonically decreasing function of v_A(x).

In a similar way, it can be demonstrated that Siju is the strictly monotonically increasing function of u_A(x) and the strictly monotonically decreasing function of v_A(x).

According to Theorem 2, it can be drawn that the cumulative interval-valued score function increases with the increase of supporting degree, while it decreases with the increase of opposing degree. □

Theorem 3:For an IVIFN α=(u_A(x), v_A(x)), if u_A(x)=vA(x), that is to say,μα+ = vα+, μα− = vα−,the upper and lower limit Sijl and Sijuof the cumulative interval-valued score functionSCIV(α) = [μα− − vα+μα− + vα+, μα+ − vα−μα+ + vα−]are symmetrical about the origin.

Proof:μα+ = vα+, μα− = vα−, Sijl = μα− − vα+μα− + vα+ = vα− − μα+vα− + μα+ = − μα+ − vα−μα+ + vα− = − S.iju

According to Theorem 3, when the number of supporting people is equal to that of opposing people, the cumulative interval-valued score function proposed in this paper is symmetrical about the origin. □

3.4 Dot Operator of the Cumulative Interval-Valued Score Function

In Ye [32], the dot aggregator method of the interval-valued score function is proposed (Definition 4). Xia and Xu [24] also proposed generalized point operators for aggregating intuitionistic fuzzy information. On the basis of these references, a new dot operator method of the cumulative interval-valued score function is proposed in this paper.

Definition 8: For any IVIFN α=(u_A(x), v_A(x)), the dot operator of its cumulative interval-valued score function SCIV(α) = [μα− − vα+μα− + vα+, μα+ − vα−μα+ + vα−] is GDOCIV(α) = μα− − vα+μα− + vα+ + μα+ − vα−μα+ + vα−2 + βμα− − vα+μα− + vα+ − μα+ − vα−μα+ + vα−2, where β is the risk factor and β∈[–1, 1]. When β=0, the decision maker neither likes nor dislikes risk; when β>0, the decision maker likes risk; and when β<0, the decision maker dislikes risk.

4 Improved TOPSIS Based on Grey Relation – New Entropy Weight Method

4.1 Grey Relation Degree Evaluation Method for IVIFNs

As for the calculation of IVIFNs’ distance, a Euclidean distance formula is used in Wu and Wan [23], while the Hamming distance formula (Definition 5) based on Wu and Wan [23] was used in this paper. The reason for using this formula is to take into consideration the difference between uα−, uα+, vα−, vα+ and uβ−, uβ+, vβ−, vβ+.

After that, the correlation coefficient matrix of the IVIFN’s positive ideal solution and negative ideal solution are calculated, and the formula is as follows:

(1)ξij+ = miniminjDi(αij − A+) + ρmaximaxkDi(αij − A+)Di(αij − A+) + ρmaximaxkDi(αij − A+). (1)

(2)ξij− = miniminjDi(αij − A−) + ρmaximaxiDi(αij − A−)Di(αij − A−) + ρmaximaxiDi(αij − A−). (2)

Among them, the distance calculation formula of the attribute value of each scheme and the ideal point is defined by Definition 5.

4.2 New Entropy Weight Method with Cumulative Interval-Valued Score Function of Interval-Valued Intuitionistic Numbers

For the interval-valued intuitionistic fuzzy decision-making matrix F=[α_ij]_n×m, α_ij=(u_ij, v_ij). For any proposal A_i(i=1, 2, …, n), the score function value of the information characteristics of A_i in the aspect of attribute X_j is first determined and then processed with a normalization method. In Wu and Wan [23], the formula used is S˜(αij) = S(αij)/∑i = 1nS(αij). The mean information entropy of attribute X_j is determined: Hs(Xj) = − 1lnn∑i = 1nS˜(αij)lnS˜(αij). The condition where S(α_ij) is below 0 is not taken into consideration.

In this paper, Eqs. (3) and (4) are modified in the following aspects. First, for the calculation of S˜(αij), the effect of hesitant population and the risk preference of decision makers are considered, and the cumulative score function of IVIFNs redefined in this paper is used (Definition 8: GDOCIV(α) = (μα− − vα+μα− + vα+ + μα+ − vα−μα+ + vα−)/2 + β(μα− − vα+μα− + vα+ − μα+ − vα−μα+ + vα−)/2). Second, when calculating S˜(αij), according to Li et al. [9], the absolute value of S(α_ij) in the denominator of S˜(αij) is obtained and then summed, considering S(α_ij) can be positive or negative. The absolute value of S˜(αij) is no more than 1. Third, the modification of H_s(X_j) is as follows: the logarithm of S˜(αij) is obtained and then its absolute value. This makes it possible to calculate H_s(X_j) even when S(α_ij)≤0.

(3)S˜(αij) = GDOCIV(α)∑i=1n|GDOCIV(α)|. (3)

Utilize Eq. (4) to calculate the entropy weight of X_j:

(4)Hs(Xj) = − 1lnn∑i=1nS˜(αij)ln|S˜(αij)|, (4)

when D+ = (dij+)6×5 = [0.12500.4250.0250.1750.1250.1250.350.3500.2250.37500.050.050.02500.30.1500.2750.3500.350.0500.37510.5250.075], let

(4)D− = (dij−)6×5 = [0.30.42500.40.250.2750.2750.050.050.40.1500.3750.3250.3250.2750.300.150.30.10.0250.3750.0250.3250.5250.151.12500.45]. (4)

Utilize Eq. (5) to calculate the weight:

(5)ωj = 1 − Hs(Xj)∑k=1m(1 − Hs(Xk)). (5)

4.3 TOPSIS of IVIFSs Based on the New Information Entropy Function Method

On the basis of the idea of the TOPSIS method, the calculation steps of improved TOPSIS method based on cumulative interval-valued score function (S_CIV) of IVIFN are given as follows:

Utilize Eqs. (6) and (7). Determine the positive ideal solution and negative ideal solution:
(6)A+ = {〈[u1+−, u1++], [v1+−, v1++]〉, …, 〈[un+−, un++], [vn+−, vn++]〉}, (6)
(7)A− = {〈[u1−−, u1−+], [v1−−, v1−+]〉, …, 〈[un−−, un−+], [vn−−, vn−+]〉}, (7)
where, ∀j=1, 2, …, n
〈[uj+−, uj++], [vj+−, vj++]〉 = 〈[maxiuij−, maxiuij+], [minivij−, minivij+]〉,
〈[uj−−, uj−+], [vj−−, vj−+]〉 = 〈[miniuij−, miniuij+], [maxivij−, maxivij+]〉.
Utilize Eqs. (1) and (2) to calculate the correlation coefficient matrix of each alternative’s properties with the IVIFNs’ ideal solution and negative ideal solution.
Utilize Eq. (3) to normalize the score function values of each attribute, and obtain the final score function matrix.
Utilize Eqs. (4) and (5) to calculate the weight coefficient of each attribute.
Utilize Eqs. (8) and (9) to calculate the correlation coefficients of each scheme to interval-valued intuitionistic fuzzy positive and negative ideal points:
(8)Pi+ = ∑j=1mωj × Pij+(j = 1, 2, …, m), (8)
(9)Pi− = ∑j=1mωj × Pij−(j = 1, 2, …, m). (9)
Utilize Eq. (10) to calculate the R(i), which is the relative correlation of each scheme to the positive ideal point:
(10)R(i) = Pi+Pi+ + Pi−. (10)
According to the relative closeness degree, the programs are sorted by large to small order. The bigger the R(i), the better the scheme.

5 Case Analysis

In this section, we apply the above method to the problem of vendor selection (adapted from Wu and Wan [23]). Different from Wu and Wan [23], this paper is based on the cumulative score function and the newly established weighted function. In addition, the risk preference of investors is considered in this paper.

When choosing products supplier, a company first establishes six evaluation indices: price (X₁), product shelf life (X₂), quality (X₃), technical level (X₄), after-sales service (X₅), and future cooperation possible (X₆). Experts use these indicators to evaluate five suppliers A_i(i=1, 2, 3, 4, 5). After statistical processing, each supplier under various indicators of evaluation information can be represented by an IVIFN, as shown in Table 1.

Table 1

Interval-Valued Intuitionistic Fuzzy Decision Matrix.

	A₁	A₂	A₃	A₄	A₅
X₁	([0.4, 0.5], [0.2, 0.3])	([0.5, 0.7], [0.1, 0.2])	([0.2, 0.3], [0.6, 0.7])	([0.5, 0.6], [0.1, 0.2])	([0.4, 0.5], [0.3, 0.4])
X₂	([0.6, 0.8], [0.1, 0.2])	([0.6, 0.8], [0.1, 0.2])	([0.4, 0.5], [0.3, 0.4])	([0.3, 0.4], [0.2, 0.3])	([0.8, 0.9], [0.0, 0.1])
X₃	([0.4, 0.5], [0.2, 0.4])	([0.3, 0.4], [0.4, 0.6])	([0.7, 0.8], [0.1, 0.2])	([0.5, 0.8], [0.1, 0.2])	([0.5, 0.8], [0.1, 0.2])
X₄	([0.8, 0.9], [0.1, 0.1])	([0.8, 0.9], [0.0, 0.1])	([0.2, 0.5], [0.1, 0.2])	([0.6, 0.7], [0.1, 0.2])	([0.8, 0.9], [0.0, 0.1])
X₅	([0.2, 0.6], [0.2, 0.3])	([0.2, 0.5], [0.3, 0.4])	([0.7, 0.8], [0.0, 0.1])	([0.3, 0.4], [0.3, 0.4])	([0.7, 0.8], [0.1, 0.2])
X₆	([0.5, 0.7], [0.1, 0.2])	([0.1, 0.2], [0.4, 0.5])	([0.5, 0.6], [0.2, 0.4])	([0.1, 0.2], [0.7, 0.8])	([0.3, 0.6], [0.1, 0.2])

Step 1. Determine the positive ideal solution and negative ideal solution according to Eqs. (6) and (7).

A+ = {([0.5, 0.7], [0.1, 0.2]), ([0.8, 0.9], [0.0, 0.1]), ([0.7, 0.8], [0.1, 0.2]),([0.8, 0.9], [0.0, 0.1]), ([0.7, 0.8], [0.0, 0.1]), ([0.5, 0.7], [0.1, 0.2])},

A− = {([0.2, 0.3], [0.6, 0.7]), ([0.3, 0.4], [0.3, 0.4]), ([0.3, 0.4], [0.4, 0.6]),([0.2, 0.5], [0.1, 0.2]), ([0.2, 0.4], [0.3, 0.4]), ([0.1, 0.2], [0.7, 0.8])}.

Step 2. Calculate scheme attribute weight and their distance with the corresponding positive and negative ideal point:

D+ = (dij+)6×5 = [0.12500.4250.0250.1750.1250.1250.350.3500.2250.37500.050.050.02500.30.1500.2750.3500.350.0500.37510.5250.075],

D− = (dij−)6×5 = [0.30.42500.40.250.2750.2750.050.050.40.1500.3750.3250.3250.2750.300.150.30.10.0250.3750.0250.3250.5250.151.12500.45].

Each solution is obtained, and the positive and negative ideal point grey correlation coefficient Pij+, Pij−:

P+ = (Pij+)6×5 = [0.6310.330.890.550.580.580.330.3310.450.3310.790.790.8610.330.510.390.3310.330.7810.570.330.490.87],

P− = (Pij−)6×5 = [0.410.3310.350.460.530.53110.420.5610.330.370.370.350.3310.50.330.7410.3810.410.520.790.3310.56].

Step 3. Calculate the scoring function matrix of different schemes according to the type of each attribute as follows (the risk coefficient β=0):

S(αij) = [0.1290.29−0.2580.2580.0650.2620.2620.0480.0480.3810.103−0.1030.4140.3450.3450.2460.2620.0660.1640.2620.10300.48300.4140.237−0.1580.132−0.3160.158].

Step 4. Calculate the weight of each attribute by Eqs. (4) and (5):

w = ( 0.22, 0.07, 0.14, 0.02, 0.18, 0.37).

Steps 5 and 6. According to Eqs. (8)–(10), calculate the correlation Pi+ of scheme A_i and A⁺, Pi− of A_i and A^–, and R(i). The results are shown as follows:

i	Pi+	Pi−	R(i)
1	0.702	0.536	0.567
2	0.601	0.727	0.453
3	0.545	0.549	0.498
4	0.581	0.757	0.434
5	0.783	0.468	0.626

Step 7. Utilize R(i) (i=1, 2, …, n) to sort the scheme by large-to-small order. The sorting result is A₅>A₁>A₃>A₂>A₄. Therefore, the optimal solution is A₅.

In Wu and Wan [23], the attribute weights are w=(0.2, 0.1, 0.25, 0.1, 0.15, 0.2). The final sorting is A₅>A₁>A₃>A₄>A₂. Although this paper’s relative weight of each attribute are different from that in Wu and Wan [23], the sorting results are the same. In this paper, by using the method that can make the weight and decision expert assessment information consistent, more objective and reliable sorting results were obtained. The method proposed in this paper can enable decision makers, according to their own risk preference, choose appropriate risk factors (given in Wu and Wan [23], and equal to the special circumstances of β=0). Here are two kinds of special situations:

When risk factor β=1 (pursue maximum risk), w=(0.2, 0.1, 0.25, 0.1, 0.15, 0.2). The sorting result is A₅>A₁>A₃>A₄>A₂. The sequencing results are exactly the same with the results in [23].
When risk factor β=–1 (pursue minimal risk), w=(0.17, 0.09, 0.1, 0.04, 0.34, 0.26). The sorting result is A₅>A₃>A₁>A₂>A₄.

Evidently, when decision makers have different risk preferences, although the optimal solution is the same, the sorting result is not consistent; the ranking results in Wu and Wan [23] are the same as those for decision makers who pursue the biggest risk in this paper.

6 Conclusion

In this paper, we did further research on the multiple-attribute decision-making problem of IVIFNs, and put forward an integrated decision method based on the cumulative interval-valued score function (S_CIV) and improved the TOPSIS method. The method used in this paper can enable decision makers to make use of evaluation information in making a decision, and consider the different risk preferences of decision makers, thus making the decision more comprehensive and objective.

It should be noted that the conclusion made in this paper is based on the premise that hesitant people would do as others do. The conclusion is only suitable for cases where the neutral part π is small. Only in such cases is it highly likely to happen that the neutral part is redivided into the supporting part, the objecting part, and the neutral part at a ratio of μ:ν:π on the second round of ballot. For cases where the neutral part π is large, take α=(0.1, 0.01) for an example. The precision degree h(α₂) is only 0.11, while the hesitancy degree π is up to 0.89, μ=0.1, ν_α=0.01. In such cases, it is highly unlikely to happen that the neutral part is redivided into the supporting part, the objecting part, and the neutral part at a ratio of μ:ν:π on the second round of ballot. Therefore, for cases with a large hesitancy part, before the score function is constructed, hesitancy should be analyzed with other methods, such as the prospect theory [7].

Acknowledgments

The paper was supported by the National Natural Science Foundation of China (grant no. 71271070).

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Received: 2015-9-9

Published Online: 2015-11-6

Published in Print: 2016-4-1

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Articles in the same Issue

https://doi.org/10.1515/jisys-2015-0096

Keywords for this article

Interval-valued intuitionistic fuzzy number (IVIFN); TOPSIS; entropy weight method; cumulative interval-valued score function (SCIV); dot operator of the cumulative interval-valued score function (SDOCIV)

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BY-NC-ND 3.0