# Double Integral-Based Method for Ranking Intuitionistic Multiplicative Sets and Its Application in Selecting Logistics Transfer Station

Authors: Zhiyong Tian; Jian Lin
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Intuitionistic multiplicative sets can be applied in many practical situations, most of which are based on ranking of intuitionistic multiplicative numbers. This study develops an integral method for ranking intuitionistic multiplicative numbers based on the new definitions of multiplicative score function and accuracy function. The ranking method considers both the risk preference and infinitely many possible values in feasible region. Some reasonable properties of multiplicative score function and accuracy function are studied, respectively. We construct a total order relation on the set of intuitionistic multiplicative numbers. The multiplicative score function and accuracy function are utilized to select the optimal logistics transfer station. A comparison example is developed to highlight the advantage of the risk preference-based ranking method.


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## Body

## 1. Introduction

Intuitionistic fuzzy set [1] contains three aspects of preference information, and it can represent the characteristics of things more comprehensively in reality. Since the concept of the intuitionistic fuzzy set was put forward, the related theory and application have become the research hotspots among various fields, such as intelligent computing [2–4], cluster analysis [5–7], decision-making [8–12], green supplier selection [13–15], and so on.Representing the symmetric and uniform preferences of decision makers only, the intuitionistic fuzzy set is often inconsistent with human intuition in actual life. To overcome this in constructing intuitionistic preference relation, Xia et al. [16] defined the intuitionistic multiplicative set based on the 1/9-9 scale. The intuitionistic multiplicative set can effectively express asymmetric and uneven preference information that appears in many practical decision-making problems. For example, the intuitionistic multiplicative set is very suitable to describe the law of diminishing marginal utility in economics [17–19]. So far, a series of important results have been achieved in the theory and application of intuitionistic multiplicative sets. Xu [20] proposed the expected intuitionistic multiplicative preference relation based on intuitionistic multiplicative numbers. The priority weight intervals were derived by the geometric aggregation operator and the error propagation formula. Xu and Xia [21] defined the intuitionistic multiplicative preference relation by considering two parts of information describing the intensity degrees. Choquet integral was used to aggregate the intuitionistic multiplicative intuitionistic multiplicative numbers. Jiang and Xu [22] proposed two kinds of methods to ranking alternatives; meanwhile, transformation mechanism and aggregation operators were developed. Yu and Fang [23] fused the intuitionistic multiplicative information by defining the concepts of two aggregation operators. Yu and Xu [24] extended the intuitionistic multiplicative set to the intuitionistic multiplicative triangular fuzzy set. The operational laws and desirable properties are studied. Jiang et al. [25] proposed the interval-valued intuitionistic multiplicative set based on an unsymmetrical scale. The comparison laws of interval-valued intuitionistic multiplicative numbers were given. Jiang et al. [19] investigated an approach for group decision-making based on incomplete intuitionistic multiplicative preference relation. Ren et al. [26] applied the intuitionistic multiplicative numbers to the analytic hierarchy process. Qian and Niu [27] defined some effective operational laws of intuitionistic multiplicative numbers. Moreover, two useful aggregation operators were proposed. Jiang et al. [28] studied some universal distances based on the classical Minkowski distance. Garg [29] defined some distance measures between two or more intuitionistic multiplicative preference relations. Zhang and Pedrycz [30] checked the consistency of intuitionistic multiplicative preference relation. The generating weights were derived based on the consistent preference relation. Zhang and Guo [31] analyzed the consistency definition of intuitionistic multiplicative preference relation and established a linear programming-based algorithm to improve the flaws. Ma and Xu [32] utilized the parameterized hyperbolic scale to describe the preference values. Liao et al. [33] investigated some novel distance measures between intuitionistic multiplicative sets. Jin et al. [34] derived the normalized intuitionistic multiplicative weights from consistent intuitionistic multiplicative preference relation. Zhang and Pedrycz [35] presented the concept of intuitionistic multiplicative preference to deal with multicriteria group decision-making problems. Mamata [36] estimated initial values for all missing entries based on incomplete interval-valued intuitionistic multiplicative preference relation. Zhang and Chen [37] studied decision-making with incomplete intuitionistic multiplicative preference relations. A reasonable consistency of incomplete intuitionistic multiplicative preference relation was introduced.Many practical applications of intuitionistic multiplicative sets were based on the ranking of intuitionistic multiplicative numbers. Xia et al. [16] defined the score and accuracy functions of intuitionistic multiplicative numbers. To construct a total order, some comparison laws were introduced in detail. Jiang et al. [38] developed two approaches for ranking intuitionistic multiplicative numbers based on distance measures. In order to consider the decision maker’s personal preference in the process of ranking, Chen [39] proposed a ranking formula of intuitionistic fuzzy numbers with preference parameters. However, the existing method for ranking intuitionistic multiplicative numbers do not take into account the preference information of decision makers. The above two ranking methods only consider the single point value and ignore the infinite number of possible values, which makes that the ranking result may be distorted and invalid in some cases. To overcome the shortages of existing ranking methods, this study studies a novel method for ranking intuitionistic multiplicative numbers based on the risk preference of decision makers. The proposed ranking method takes all the potential possible values in feasible region into account. Therefore, the ranking results obtained by the new method are more reliable and reasonable. Moreover, the risk preference-based ranking method is applied to select the optimal logistics transfer station with intuitionistic multiplicative information.The rest of this study is structured as follows. Section2 reviews some related concepts on the intuitionistic multiplicative set. In Section 3, the accuracy function and score function of intuitionistic multiplicative numbers are defined based on risk preference of the decision maker. Section 4 proposes the total order among intuitionistic multiplicative numbers. Numerical example and comparison are calculated in Section 5. Conclusion and further study are stated in Section 6.


## 2. Preliminaries

For the convenience of the follow-up discussion, intuitionistic multiplicative number and its order relation are reviewed as follows.Definition 1 (See [16]).
Let setX be fixed. An intuitionistic multiplicative set M over X is defined as(1)M˜=<x,ρMx,σMx>|x∈X,where ρMx and σMx are the membership information and nonmembership information of x with respect to intuitionistic multiplicative set M, respectively, such that ρMx,σMx∈1/9,9, and 0<ρMxσMx≤1. The hesitation or uncertain information of x corresponding to M is denoted by τMx=1/ρMxσMx.
Clearly,∀x∈X, and 1≤τMx≤92. An intuitionistic multiplicative number (IMN) can be represented by α˜=ρα,σα, such that ρα,σα∈1/9,9, 0<ρασα≤1, and τα=τα˜=1/ρασα. IM denotes the set of all IMNs.Definition 2 (See [16]).
Letα˜=ρα,σα be an intuitionistic multiplicative number. The score function and accuracy function of α˜ are expressed by(2)SXiaα˜=ρασα,(3)AXiaα˜=ρασα.Definition 3 (Reference [16]).
Letα˜1=ρ1,σ1 and α˜2=ρ2,σ2 be two intuitionistic multiplicative numbers. The order relation between α˜1 and α˜2 is defined as follows:(1)
IfSXiaα˜1>SXiaα˜2, then α˜1≻Xiaα˜2(2)
IfSXiaα˜1=SXiaα˜2, then(a)
IfAXiaα˜1=AXiaα˜2, then α˜1=α˜2(b)
IfAXiaα˜1>AXiaα˜2, then α˜1≻Xiaα˜2


## 3. The Multiplicative Accuracy and Score Functions of Intuitionistic Multiplicative Numbers

In the following, we will define the multiplicative accuracy function and score function with considering the risk preference of the decision maker.


### 3.1. The Multiplicative Accuracy Function Based on Risk Preference

Definition 4.
Letα˜=ρα,σα be an intuitionistic multiplicative number. The feasible region after transformation of α˜ is denoted by(4)Rα=x,y|ρ¯α≤x≤−σ¯α,σ¯α≤y≤−x,where ρ¯α=lnρα, σ¯α=lnσα, and τ¯α=lnτα, respectively.
Sinceρασατα=1, we have lnρα+lnσα+lnτα=0. It follows that ρ¯α+σ¯α+τ¯α=0. Obviously, (4) can be further written as(5)Rα=x,y|lnρα≤x≤−lnσα,lnσα≤y≤−x,where Rα represents the mapping feasible region with respect to intuitionistic multiplicative numbers α˜. The mapping feasible region Rα is shown in Figure 1.Figure 1
The mapping feasible region with respect to.α˜.Definition 5.
Letα˜=ρα,σα be an intuitionistic multiplicative number. The multiplicative accuracy function of α˜ based on risk preference is expressed as(6)Aλα˜=e∬Rαλx+1−λydxdy/MRα,where Rα is the mapping feasible region of intuitionistic multiplicative number α˜, and MRα is the area of mapping feasible region Rα. The parameter λ reflects the risk tendency of the decision maker. When λ∈0.5,1, the decision maker is a risk lover. When λ∈0,0.5, the decision maker is averse to risk. Especially, the attitude of the decision maker is neutral to the risk when λ=0.5.
Aλα˜ represents the geometric average of accuracy function over all feasible values. From (5), we have(7)MRα=12τ¯α2=12ρ¯α+σ¯α2.
According to the operational properties of double integral, we have(8)∬Rαλx+1−λydxdy=∫ρ¯α−σ¯α∫σ¯α−xλx+1−λydydx=∫ρ¯α−σ¯α1−3λ2x2−λσ¯αx+λ−1σ¯α22dx=2−3λ6σ¯α3−1−3λ6ρ¯α3+λσ¯α2ρ¯α2−λ−12σ¯α2ρ¯α.
Based on (6), it follows that(9)Aλα˜=e∬Rαλx+1−λydxdy/MRα=e2−3λ/6σ¯α3−1−3λ/6ρ¯α3+λσ¯α/2ρ¯α2−λ−1/2σ¯α2ρ¯α/1/2ρ¯α+σ¯α2=e2σ¯α3−ρ¯α3+3σ¯α2ρ¯α+3λρ¯α3−σ¯α3+ρ¯α2σ¯α−σ¯α2ρ¯α/3ρ¯α+σ¯α2=e2σ¯α−ρ¯ασ¯α+ρ¯α2+3λρ¯α−σ¯ασ¯α+ρ¯α2/3ρ¯α+σ¯α2=e1/32σ¯α−ρ¯α+λρ¯α−σ¯α=eλ2ρ¯α−σ¯α/3+1−λ2σ¯α−ρ¯α/3.
Sinceρ¯α=lnρα and σ¯α=lnσα, we have(10)Aλα˜=eλ2lnρα−lnσα/3+1−λ2lnσα−lnρα/3=eλ/3lnρα2/σα+1−λ/3lnσα2/ρα.
The expression of multiplicative accuracy function ofα˜ can be further written as(11)Aλα˜=ρα2/3σα1/3λσα2/3ρα1/31−λ=ρα2σα3λσα2ρα31−λ,which can be equivalently written as(12)Aλα˜=ρασαλ⋅σα2/3ρα1/3=ρασαλ⋅σα2ρα3.
The multiplicative accuracy function based on risk preference satisfies some desirable properties as follows.Property 1.
For any two intuitionistic multiplicative numbersα˜1=ρ1,σ1 and α˜2=ρ2,σ2, AXiaα˜1>AXiaα˜2 if and only if A0.5α˜1>A0.5α˜2.Proof.
IfAXiaα˜1>AXiaα˜2, by equation (3), we have ρ1σ1>ρ2σ2.
Whenλ=0.5, from (12), we have A0.5α˜i=ρiσi1/6, i=1,2.
Fori=1,2, it follows that AXiaα˜i=A0.5α˜i6.
Therefore,AXiaα˜1>AXiaα˜2 is equivalently written as(13)A0.5α˜1=AXiaα˜11/6>AXiaα˜21/6=A0.5α˜2.
The proof of Property1 is completed.Property 2.
Letα˜=ρα,σα be an intuitionistic multiplicative number. ∀λ∈0,1, we have(14)19≤minρα2σα3,σα2ρα3≤Aλα˜≤maxρα2σα3,σα2ρα3≤9.Proof.
We prove Property1 in the following two cases:(1)
Ifρα≥σα, then we have σα2/ρα3≤ρα2/σα3. ∀λ∈0,1, we have(15)σα2ρα3≤ρα2σα3λσα2ρα31−λ≤ρα2σα3.Sinceρα,σα∈1/9,9, we have(16)19≤σα2ρα3≤ρα2σα3λσα2ρα31−λ≤ρα2σα3≤9.(2)
Ifρα<σα, then we have ρα2/σα3≤σα2/ρα3. Since ρα,σα∈1/9,9, for all λ∈0,1, we have(17)19≤ρα2σα3≤ρα2σα3λσα2ρα31−λ≤σα2ρα3≤9.Considering both cases (1) and (2), we have(18)19≤minρα2σα3,σα2ρα3≤Aλα˜≤maxρα2σα3,σα2ρα3≤9.Therefore, Property2 is proved.Property 3.
Letα˜1=ρ1,σ1 and α˜2=ρ2,σ2 be two intuitionistic multiplicative numbers. ∀λ∈0,1; if σ1/σ2≤ρ1/ρ2≤σ1/σ22, then Aλα˜1≥Aλα˜2.Proof.
Based on equation (7), we have(19)Aλα˜1=ρ1σ1λ⋅σ12/3ρ11/3,Aλα˜2=ρ2σ2λ⋅σ22/3ρ21/3.
Aλα˜1 divided by Aλα˜2, we have(20)Aλα˜1Aλα˜2=ρ1σ2ρ2σ1λ⋅σ1/σ22/3ρ1/ρ21/3.
The above expression can be equivalently written as
lnAλα˜1−lnAλα˜2=λlnρ1−lnρ2+lnσ2−lnσ1+1/32lnσ1−2lnσ2−lnρ1+lnρ2. Since σ1/σ2≤ρ1/ρ2≤σ1/σ22, we have lnσ1−lnσ2≤lnρ1−lnρ2≤2lnσ1−lnσ2. Accordingly, for all λ∈0,1, lnAλα˜1−lnAλα˜2≥0 holds. Namely, we have Aλα˜1≥Aλα˜2. Therefore, the proof of Property 3 is completed.Property 4.
LetΔλ be a perturbation of parameter λ with λ+Δλ∈0,1. Given two intuitionistic multiplicative numbers α˜1=ρ1,σ1 and α˜2=ρ2,σ2. If Aλα˜1≤Aλα˜2, then Aλ+Δλα˜1≤Aλ+Δλα˜2 holds if and only if(21)−λ≤Δλ≤1−λ,ρ1σ2=ρ2σ1;max−λ,lnAλα˜2−lnAλα˜1lnρ1−lnρ2+lnσ2−lnσ1≤Δλ≤1−λ,ρ1σ2<ρ2σ1,−λ≤Δλ≤min1−λ,lnAλα˜2−lnAλα˜1lnρ1−lnρ2+lnσ2−lnσ1,ρ1σ2>ρ2σ1.Proof.
Since0≤λ+Δλ≤1, we have −λ≤Δλ≤1−λ. By equation (12), it holds that(22)Aλ+Δλα˜1=Aλα˜1⋅ρ1σ1Δλ,Aλ+Δλα˜2=Aλα˜2⋅ρ2σ2Δλ.
Accordingly, the following two inequalities are equivalent:(23)Aλ+Δλα˜1≤Aλ+Δλα˜2⟺Aλα˜1Aλα˜2≤ρ2σ1ρ1σ2Δλ.
In the following, we analyze (23) in three cases:(1)
Ifρ1σ2=ρ2σ1, then inequalities Aλα˜1≤Aλα˜2 and Aλ+Δλα˜1≤Aλ+Δλα˜2 are equivalent. Thus, ∀Δλ∈−λ,1−λ, and relational expression (12) is true.(2)
Ifρ1σ2<ρ2σ1, then inequality Aλ+Δλα˜1≤Aλ+Δλα˜2 is equivalent to Δλ≥lnAλα˜2−lnAλα˜1/lnρ1−lnρ2+lnσ2−lnσ1.Considering−λ≤Δλ≤1−λ, we have(24)Aλ+Δλα˜1≤Aλ+Δλα˜2⟺max−λ,lnAλα˜2−lnAλα˜1lnρ1−lnρ2+lnσ2−lnσ1≤Δλ≤1−λ.(3)
Ifρ1σ2>ρ2σ1, then inequality Aλ+Δλα˜1≤Aλ+Δλα˜2 is equivalent to Δλ≤lnAλα˜2−lnAλα˜1/lnρ1−lnρ2+lnσ2−lnσ1.Since−λ≤Δλ≤1−λ, we have(25)Aλ+Δλα˜1≤Aλ+Δλα˜2⟺−λ≤Δλ≤min1−λ,lnAλα˜2−lnAλα˜1lnρ1−lnρ2+lnσ2−lnσ1.In sum, the proof of Property4 is completed.


### 3.2. The Multiplicative Score Function Based on Risk Preference

Definition 6.
Letα˜=ρα,σα be an intuitionistic multiplicative number. The multiplicative score function of α˜ based on risk preference is defined by(26)Sλα˜=e∬Rα1−λx−λydxdy/MRα,where Rα is the mapping feasible region of α˜, and MRα is the area of mapping feasible region Rα. The parameter λ reflects the risk tendency of the decision maker.
It clear thatSλα˜ represents the geometric average of score functions over all feasible values. According to the operational properties of double integral, we have(27)∬Rα1−λx−λydxdy=∫ρ¯α−σ¯α∫σ¯α−x1−λx−λydydx=∫ρ¯α−σ¯α−2−λ2x2−1−λσ¯αx+λσ¯α22dx=−λ+16σ¯α3+2−λ6ρ¯α3+1−λσ¯α2ρ¯α2−λ2σ¯α2ρ¯α.
Based on (26), it follows that(28)Sλα˜=e∬Rα1−λx−λydxdy/MRα=e−λ+1/6σ¯α3+2−λ/6ρ¯α3+1−λσ¯α/2ρ¯α2−λ/2σ¯α2ρ¯α/1/2ρ¯α+σ¯α2=e−2σ¯α3+ρ¯α3−3σ¯α2ρ¯α+1−λρ¯α3+σ¯α3+3ρ¯α2σ¯α+3σ¯α2ρ¯α/3ρ¯α+σ¯α2=eρ¯α−2σ¯ασ¯α+ρ¯α2+1−λσ¯α+ρ¯α3/3ρ¯α+σ¯α2=e1/3ρ¯α−2σ¯α+1−λ/3ρ¯α+σ¯α=e1−λ2ρ¯α−σ¯α/3+λρ¯α−2σ¯α/3.
Sinceρ¯α=lnρα and σ¯α=lnσα, we have(29)Sλα˜=e1−λ2lnρα−lnσα/3+λlnρα−2lnσα/3=e1−λ/3lnρα2/σα+λ/3lnρα/σα2.
The expression of multiplicative score function ofα˜ can be further written as(30)Sλα˜=ρα2/3σα1/31−λρα1/3σα2/3λ=ρασα23λρα2σα31−λ,which can be equivalently written as(31)Sλα˜=1ρασα3λ⋅ρα2σα3.
The multiplicative score function based on risk preference satisfies some desirable properties as follows.Property 5.
Letα˜=ρα,σα be an intuitionistic multiplicative number. ∀λ1,λ2∈0,1; if λ1≤λ2, then Sλ1α˜≤Sλ2α˜.Proof.
Based on equation (31), we have(32)Sλ1α˜=1ρασα3λ1⋅ρα2σα3,Sλ2α˜=1ρασα3λ2⋅ρα2σα3.
It follows that(33)Sλ1α˜Sλ2α˜=1/ρασα3λ1⋅ρα2/σα31/ρασα3λ2⋅ρα2/σα3=ρασα3λ2−λ1.
Since0<ρασα≤1 and λ2−λ1≥0, we have Sλ1α˜≤Sλ2α˜.
The proof of Property5 is completed.Property 6.
For any two intuitionistic multiplicative numbersα˜1=ρ1,σ1 and α˜2=ρ2,σ2, SXiaα˜1>SXiaα˜2 if and only if S0.5α˜1>S0.5α˜2.Proof:.
IfSXiaα˜1>SXiaα˜2, by equation (2), we have ρ1/σ1>ρ2/σ2.
Whenλ=0.5, from (31), we have S0.5α˜i=ρi/σi1/2, i=1,2.
Fori=1,2, it holds that SXiaα˜i=S0.5α˜i2.
Therefore,SXiaα˜1>SXiaα˜2 is equivalently written as(34)S0.5α˜1=SXiaα˜11/2>SXiaα˜21/2=S0.5α˜2.
The proof of Property6 is completed.Property 7.
Letα˜=ρα,σα be an intuitionistic multiplicative number. ∀λ∈0,1, we have 1/9≤ρα2/σα3≤Sλα˜≤ρα/σα23≤9.Proof.
From equation (31), we have S0α˜=ρα2/σα3 and S1α˜=ρα/σα23. By Property 5, for all λ∈0,1, we have S0α˜≤Sλα˜≤S1α˜. Since ραx,σαx∈1/9,9, we have(35)Sλα˜≥S0α˜=ρα2σα3≥19,Sλα˜≤S1α˜=ρασα23≤9.
Therefore, Property7 is proved.Property 8.
LetΔλ be a perturbation of parameter λ with λ+Δλ∈0,1. Given two intuitionistic multiplicative numbers α˜1=ρ1,σ1 and α˜2=ρ2,σ2. If Sλα˜1≤Sλα˜2, then Sλ+Δλα˜1≤Sλ+Δλα˜2 holds if and only if(36)−λ≤Δλ≤1−λ,ρ1σ1=ρ2σ2,max−λ,3lnSλα˜2−3lnSλα˜1lnρ2−lnρ1+lnσ2−lnσ1≤Δλ≤1−λ,ρ1σ1>ρ2σ2,−λ≤Δλ≤min1−λ,3lnSλα˜2−3lnSλα˜1lnρ2−lnρ1+lnσ2−lnσ1,ρ1σ1<ρ2σ2.Proof.
Since0≤λ+Δλ≤1, we have −λ≤Δλ≤1−λ. By equation (31), it holds that(37)Sλ+Δλα˜1=Sλα˜1⋅1ρ1σ13Δλ,Sλ+Δλα˜2=Sλα˜2⋅1ρ2σ23Δλ.
Accordingly, the following two inequalities are equivalent:(38)Sλ+Δλα˜1≤Sλ+Δλα˜2⟺Sλα˜2Sλα˜1≥ρ2σ2ρ1σ13Δλ.
In the following, we analyze (38) in three cases:(1)
Ifρ1σ1=ρ2σ2, then inequalities Sλα˜1≤Sλα˜2 and Sλ+Δλα˜1≤Sλ+Δλα˜2 are equivalent. Thus, ∀Δλ∈−λ,1−λ, and the relational expression (38) is true.(2)
Ifρ1σ1>ρ2σ2, then inequality Sλ+Δλα˜1≤Sλ+Δλα˜2 is equivalent to Δλ≥3lnSλα˜2−3lnSλα˜1/lnρ2−lnρ1+lnσ2−lnσ1.Considering−λ≤Δλ≤1−λ, we have(39)Sλ+Δλα˜1≤Sλ+Δλα˜2⟺max−λ,3lnSλα˜2−3lnSλα˜1lnρ2−lnρ1+lnσ2−lnσ1≤Δλ≤1−λ.(3)
Ifρ1σ1<ρ2σ2, then inequality Sλ+Δλα˜1≤Sλ+Δλα˜2 is equivalent to Δλ≤3lnSλα˜2−3lnSλα˜1/lnρ2−lnρ1+lnσ2−lnσ1.Since−λ≤Δλ≤1−λ, we have(40)Sλ+Δλα˜1≤Sλ+Δλα˜2⟺−λ≤Δλ≤min1−λ,3lnSλα˜2−3lnSλα˜1lnρ2−lnρ1+lnσ2−lnσ1.In sum, the proof of Property8 is completed.Property 9.
Letα˜1=ρ1,σ1 and α˜2=ρ2,σ2 be two intuitionistic multiplicative numbers. If ρ1≥ρ2 and σ1≤σ2, then for all λ∈0,1, we haveSλα˜1≥Sλα˜2.Proof.
From equation (31), we have(41)Sλα˜1=1ρ1σ13λ⋅ρ12σ13,Sλα˜2=1ρ2σ23λ⋅ρ22σ23.
It follows that(42)Sλα˜1Sλα˜2=1/ρ1σ13λ⋅ρ12/σ131/ρ2σ23λ⋅ρ22/σ23=σ22σ12ρ1ρ23λρ12ρ22σ2σ131−λ.
Sinceρ1≥ρ2 and σ1≤σ2, we have ρ1/ρ2≥1 and σ2/σ1≥1. Considering 0≤λ≤1, we have Sλα˜1≥Sλα˜2.
Therefore, the proof of Property9 is completed.


## 3.1. The Multiplicative Accuracy Function Based on Risk Preference

Definition 4.
Letα˜=ρα,σα be an intuitionistic multiplicative number. The feasible region after transformation of α˜ is denoted by(4)Rα=x,y|ρ¯α≤x≤−σ¯α,σ¯α≤y≤−x,where ρ¯α=lnρα, σ¯α=lnσα, and τ¯α=lnτα, respectively.
Sinceρασατα=1, we have lnρα+lnσα+lnτα=0. It follows that ρ¯α+σ¯α+τ¯α=0. Obviously, (4) can be further written as(5)Rα=x,y|lnρα≤x≤−lnσα,lnσα≤y≤−x,where Rα represents the mapping feasible region with respect to intuitionistic multiplicative numbers α˜. The mapping feasible region Rα is shown in Figure 1.Figure 1
The mapping feasible region with respect to.α˜.Definition 5.
Letα˜=ρα,σα be an intuitionistic multiplicative number. The multiplicative accuracy function of α˜ based on risk preference is expressed as(6)Aλα˜=e∬Rαλx+1−λydxdy/MRα,where Rα is the mapping feasible region of intuitionistic multiplicative number α˜, and MRα is the area of mapping feasible region Rα. The parameter λ reflects the risk tendency of the decision maker. When λ∈0.5,1, the decision maker is a risk lover. When λ∈0,0.5, the decision maker is averse to risk. Especially, the attitude of the decision maker is neutral to the risk when λ=0.5.
Aλα˜ represents the geometric average of accuracy function over all feasible values. From (5), we have(7)MRα=12τ¯α2=12ρ¯α+σ¯α2.
According to the operational properties of double integral, we have(8)∬Rαλx+1−λydxdy=∫ρ¯α−σ¯α∫σ¯α−xλx+1−λydydx=∫ρ¯α−σ¯α1−3λ2x2−λσ¯αx+λ−1σ¯α22dx=2−3λ6σ¯α3−1−3λ6ρ¯α3+λσ¯α2ρ¯α2−λ−12σ¯α2ρ¯α.
Based on (6), it follows that(9)Aλα˜=e∬Rαλx+1−λydxdy/MRα=e2−3λ/6σ¯α3−1−3λ/6ρ¯α3+λσ¯α/2ρ¯α2−λ−1/2σ¯α2ρ¯α/1/2ρ¯α+σ¯α2=e2σ¯α3−ρ¯α3+3σ¯α2ρ¯α+3λρ¯α3−σ¯α3+ρ¯α2σ¯α−σ¯α2ρ¯α/3ρ¯α+σ¯α2=e2σ¯α−ρ¯ασ¯α+ρ¯α2+3λρ¯α−σ¯ασ¯α+ρ¯α2/3ρ¯α+σ¯α2=e1/32σ¯α−ρ¯α+λρ¯α−σ¯α=eλ2ρ¯α−σ¯α/3+1−λ2σ¯α−ρ¯α/3.
Sinceρ¯α=lnρα and σ¯α=lnσα, we have(10)Aλα˜=eλ2lnρα−lnσα/3+1−λ2lnσα−lnρα/3=eλ/3lnρα2/σα+1−λ/3lnσα2/ρα.
The expression of multiplicative accuracy function ofα˜ can be further written as(11)Aλα˜=ρα2/3σα1/3λσα2/3ρα1/31−λ=ρα2σα3λσα2ρα31−λ,which can be equivalently written as(12)Aλα˜=ρασαλ⋅σα2/3ρα1/3=ρασαλ⋅σα2ρα3.
The multiplicative accuracy function based on risk preference satisfies some desirable properties as follows.Property 1.
For any two intuitionistic multiplicative numbersα˜1=ρ1,σ1 and α˜2=ρ2,σ2, AXiaα˜1>AXiaα˜2 if and only if A0.5α˜1>A0.5α˜2.Proof.
IfAXiaα˜1>AXiaα˜2, by equation (3), we have ρ1σ1>ρ2σ2.
Whenλ=0.5, from (12), we have A0.5α˜i=ρiσi1/6, i=1,2.
Fori=1,2, it follows that AXiaα˜i=A0.5α˜i6.
Therefore,AXiaα˜1>AXiaα˜2 is equivalently written as(13)A0.5α˜1=AXiaα˜11/6>AXiaα˜21/6=A0.5α˜2.
The proof of Property1 is completed.Property 2.
Letα˜=ρα,σα be an intuitionistic multiplicative number. ∀λ∈0,1, we have(14)19≤minρα2σα3,σα2ρα3≤Aλα˜≤maxρα2σα3,σα2ρα3≤9.Proof.
We prove Property1 in the following two cases:(1)
Ifρα≥σα, then we have σα2/ρα3≤ρα2/σα3. ∀λ∈0,1, we have(15)σα2ρα3≤ρα2σα3λσα2ρα31−λ≤ρα2σα3.Sinceρα,σα∈1/9,9, we have(16)19≤σα2ρα3≤ρα2σα3λσα2ρα31−λ≤ρα2σα3≤9.(2)
Ifρα<σα, then we have ρα2/σα3≤σα2/ρα3. Since ρα,σα∈1/9,9, for all λ∈0,1, we have(17)19≤ρα2σα3≤ρα2σα3λσα2ρα31−λ≤σα2ρα3≤9.Considering both cases (1) and (2), we have(18)19≤minρα2σα3,σα2ρα3≤Aλα˜≤maxρα2σα3,σα2ρα3≤9.Therefore, Property2 is proved.Property 3.
Letα˜1=ρ1,σ1 and α˜2=ρ2,σ2 be two intuitionistic multiplicative numbers. ∀λ∈0,1; if σ1/σ2≤ρ1/ρ2≤σ1/σ22, then Aλα˜1≥Aλα˜2.Proof.
Based on equation (7), we have(19)Aλα˜1=ρ1σ1λ⋅σ12/3ρ11/3,Aλα˜2=ρ2σ2λ⋅σ22/3ρ21/3.
Aλα˜1 divided by Aλα˜2, we have(20)Aλα˜1Aλα˜2=ρ1σ2ρ2σ1λ⋅σ1/σ22/3ρ1/ρ21/3.
The above expression can be equivalently written as
lnAλα˜1−lnAλα˜2=λlnρ1−lnρ2+lnσ2−lnσ1+1/32lnσ1−2lnσ2−lnρ1+lnρ2. Since σ1/σ2≤ρ1/ρ2≤σ1/σ22, we have lnσ1−lnσ2≤lnρ1−lnρ2≤2lnσ1−lnσ2. Accordingly, for all λ∈0,1, lnAλα˜1−lnAλα˜2≥0 holds. Namely, we have Aλα˜1≥Aλα˜2. Therefore, the proof of Property 3 is completed.Property 4.
LetΔλ be a perturbation of parameter λ with λ+Δλ∈0,1. Given two intuitionistic multiplicative numbers α˜1=ρ1,σ1 and α˜2=ρ2,σ2. If Aλα˜1≤Aλα˜2, then Aλ+Δλα˜1≤Aλ+Δλα˜2 holds if and only if(21)−λ≤Δλ≤1−λ,ρ1σ2=ρ2σ1;max−λ,lnAλα˜2−lnAλα˜1lnρ1−lnρ2+lnσ2−lnσ1≤Δλ≤1−λ,ρ1σ2<ρ2σ1,−λ≤Δλ≤min1−λ,lnAλα˜2−lnAλα˜1lnρ1−lnρ2+lnσ2−lnσ1,ρ1σ2>ρ2σ1.Proof.
Since0≤λ+Δλ≤1, we have −λ≤Δλ≤1−λ. By equation (12), it holds that(22)Aλ+Δλα˜1=Aλα˜1⋅ρ1σ1Δλ,Aλ+Δλα˜2=Aλα˜2⋅ρ2σ2Δλ.
Accordingly, the following two inequalities are equivalent:(23)Aλ+Δλα˜1≤Aλ+Δλα˜2⟺Aλα˜1Aλα˜2≤ρ2σ1ρ1σ2Δλ.
In the following, we analyze (23) in three cases:(1)
Ifρ1σ2=ρ2σ1, then inequalities Aλα˜1≤Aλα˜2 and Aλ+Δλα˜1≤Aλ+Δλα˜2 are equivalent. Thus, ∀Δλ∈−λ,1−λ, and relational expression (12) is true.(2)
Ifρ1σ2<ρ2σ1, then inequality Aλ+Δλα˜1≤Aλ+Δλα˜2 is equivalent to Δλ≥lnAλα˜2−lnAλα˜1/lnρ1−lnρ2+lnσ2−lnσ1.Considering−λ≤Δλ≤1−λ, we have(24)Aλ+Δλα˜1≤Aλ+Δλα˜2⟺max−λ,lnAλα˜2−lnAλα˜1lnρ1−lnρ2+lnσ2−lnσ1≤Δλ≤1−λ.(3)
Ifρ1σ2>ρ2σ1, then inequality Aλ+Δλα˜1≤Aλ+Δλα˜2 is equivalent to Δλ≤lnAλα˜2−lnAλα˜1/lnρ1−lnρ2+lnσ2−lnσ1.Since−λ≤Δλ≤1−λ, we have(25)Aλ+Δλα˜1≤Aλ+Δλα˜2⟺−λ≤Δλ≤min1−λ,lnAλα˜2−lnAλα˜1lnρ1−lnρ2+lnσ2−lnσ1.In sum, the proof of Property4 is completed.


## 3.2. The Multiplicative Score Function Based on Risk Preference

Definition 6.
Letα˜=ρα,σα be an intuitionistic multiplicative number. The multiplicative score function of α˜ based on risk preference is defined by(26)Sλα˜=e∬Rα1−λx−λydxdy/MRα,where Rα is the mapping feasible region of α˜, and MRα is the area of mapping feasible region Rα. The parameter λ reflects the risk tendency of the decision maker.
It clear thatSλα˜ represents the geometric average of score functions over all feasible values. According to the operational properties of double integral, we have(27)∬Rα1−λx−λydxdy=∫ρ¯α−σ¯α∫σ¯α−x1−λx−λydydx=∫ρ¯α−σ¯α−2−λ2x2−1−λσ¯αx+λσ¯α22dx=−λ+16σ¯α3+2−λ6ρ¯α3+1−λσ¯α2ρ¯α2−λ2σ¯α2ρ¯α.
Based on (26), it follows that(28)Sλα˜=e∬Rα1−λx−λydxdy/MRα=e−λ+1/6σ¯α3+2−λ/6ρ¯α3+1−λσ¯α/2ρ¯α2−λ/2σ¯α2ρ¯α/1/2ρ¯α+σ¯α2=e−2σ¯α3+ρ¯α3−3σ¯α2ρ¯α+1−λρ¯α3+σ¯α3+3ρ¯α2σ¯α+3σ¯α2ρ¯α/3ρ¯α+σ¯α2=eρ¯α−2σ¯ασ¯α+ρ¯α2+1−λσ¯α+ρ¯α3/3ρ¯α+σ¯α2=e1/3ρ¯α−2σ¯α+1−λ/3ρ¯α+σ¯α=e1−λ2ρ¯α−σ¯α/3+λρ¯α−2σ¯α/3.
Sinceρ¯α=lnρα and σ¯α=lnσα, we have(29)Sλα˜=e1−λ2lnρα−lnσα/3+λlnρα−2lnσα/3=e1−λ/3lnρα2/σα+λ/3lnρα/σα2.
The expression of multiplicative score function ofα˜ can be further written as(30)Sλα˜=ρα2/3σα1/31−λρα1/3σα2/3λ=ρασα23λρα2σα31−λ,which can be equivalently written as(31)Sλα˜=1ρασα3λ⋅ρα2σα3.
The multiplicative score function based on risk preference satisfies some desirable properties as follows.Property 5.
Letα˜=ρα,σα be an intuitionistic multiplicative number. ∀λ1,λ2∈0,1; if λ1≤λ2, then Sλ1α˜≤Sλ2α˜.Proof.
Based on equation (31), we have(32)Sλ1α˜=1ρασα3λ1⋅ρα2σα3,Sλ2α˜=1ρασα3λ2⋅ρα2σα3.
It follows that(33)Sλ1α˜Sλ2α˜=1/ρασα3λ1⋅ρα2/σα31/ρασα3λ2⋅ρα2/σα3=ρασα3λ2−λ1.
Since0<ρασα≤1 and λ2−λ1≥0, we have Sλ1α˜≤Sλ2α˜.
The proof of Property5 is completed.Property 6.
For any two intuitionistic multiplicative numbersα˜1=ρ1,σ1 and α˜2=ρ2,σ2, SXiaα˜1>SXiaα˜2 if and only if S0.5α˜1>S0.5α˜2.Proof:.
IfSXiaα˜1>SXiaα˜2, by equation (2), we have ρ1/σ1>ρ2/σ2.
Whenλ=0.5, from (31), we have S0.5α˜i=ρi/σi1/2, i=1,2.
Fori=1,2, it holds that SXiaα˜i=S0.5α˜i2.
Therefore,SXiaα˜1>SXiaα˜2 is equivalently written as(34)S0.5α˜1=SXiaα˜11/2>SXiaα˜21/2=S0.5α˜2.
The proof of Property6 is completed.Property 7.
Letα˜=ρα,σα be an intuitionistic multiplicative number. ∀λ∈0,1, we have 1/9≤ρα2/σα3≤Sλα˜≤ρα/σα23≤9.Proof.
From equation (31), we have S0α˜=ρα2/σα3 and S1α˜=ρα/σα23. By Property 5, for all λ∈0,1, we have S0α˜≤Sλα˜≤S1α˜. Since ραx,σαx∈1/9,9, we have(35)Sλα˜≥S0α˜=ρα2σα3≥19,Sλα˜≤S1α˜=ρασα23≤9.
Therefore, Property7 is proved.Property 8.
LetΔλ be a perturbation of parameter λ with λ+Δλ∈0,1. Given two intuitionistic multiplicative numbers α˜1=ρ1,σ1 and α˜2=ρ2,σ2. If Sλα˜1≤Sλα˜2, then Sλ+Δλα˜1≤Sλ+Δλα˜2 holds if and only if(36)−λ≤Δλ≤1−λ,ρ1σ1=ρ2σ2,max−λ,3lnSλα˜2−3lnSλα˜1lnρ2−lnρ1+lnσ2−lnσ1≤Δλ≤1−λ,ρ1σ1>ρ2σ2,−λ≤Δλ≤min1−λ,3lnSλα˜2−3lnSλα˜1lnρ2−lnρ1+lnσ2−lnσ1,ρ1σ1<ρ2σ2.Proof.
Since0≤λ+Δλ≤1, we have −λ≤Δλ≤1−λ. By equation (31), it holds that(37)Sλ+Δλα˜1=Sλα˜1⋅1ρ1σ13Δλ,Sλ+Δλα˜2=Sλα˜2⋅1ρ2σ23Δλ.
Accordingly, the following two inequalities are equivalent:(38)Sλ+Δλα˜1≤Sλ+Δλα˜2⟺Sλα˜2Sλα˜1≥ρ2σ2ρ1σ13Δλ.
In the following, we analyze (38) in three cases:(1)
Ifρ1σ1=ρ2σ2, then inequalities Sλα˜1≤Sλα˜2 and Sλ+Δλα˜1≤Sλ+Δλα˜2 are equivalent. Thus, ∀Δλ∈−λ,1−λ, and the relational expression (38) is true.(2)
Ifρ1σ1>ρ2σ2, then inequality Sλ+Δλα˜1≤Sλ+Δλα˜2 is equivalent to Δλ≥3lnSλα˜2−3lnSλα˜1/lnρ2−lnρ1+lnσ2−lnσ1.Considering−λ≤Δλ≤1−λ, we have(39)Sλ+Δλα˜1≤Sλ+Δλα˜2⟺max−λ,3lnSλα˜2−3lnSλα˜1lnρ2−lnρ1+lnσ2−lnσ1≤Δλ≤1−λ.(3)
Ifρ1σ1<ρ2σ2, then inequality Sλ+Δλα˜1≤Sλ+Δλα˜2 is equivalent to Δλ≤3lnSλα˜2−3lnSλα˜1/lnρ2−lnρ1+lnσ2−lnσ1.Since−λ≤Δλ≤1−λ, we have(40)Sλ+Δλα˜1≤Sλ+Δλα˜2⟺−λ≤Δλ≤min1−λ,3lnSλα˜2−3lnSλα˜1lnρ2−lnρ1+lnσ2−lnσ1.In sum, the proof of Property8 is completed.Property 9.
Letα˜1=ρ1,σ1 and α˜2=ρ2,σ2 be two intuitionistic multiplicative numbers. If ρ1≥ρ2 and σ1≤σ2, then for all λ∈0,1, we haveSλα˜1≥Sλα˜2.Proof.
From equation (31), we have(41)Sλα˜1=1ρ1σ13λ⋅ρ12σ13,Sλα˜2=1ρ2σ23λ⋅ρ22σ23.
It follows that(42)Sλα˜1Sλα˜2=1/ρ1σ13λ⋅ρ12/σ131/ρ2σ23λ⋅ρ22/σ23=σ22σ12ρ1ρ23λρ12ρ22σ2σ131−λ.
Sinceρ1≥ρ2 and σ1≤σ2, we have ρ1/ρ2≥1 and σ2/σ1≥1. Considering 0≤λ≤1, we have Sλα˜1≥Sλα˜2.
Therefore, the proof of Property9 is completed.


## 4. The Order Relation between Intuitionistic Multiplicative Numbers

In the following, the order relation between any two intuitionistic multiplicative numbers is defined based on the multiplicative score function and accuracy function.Property 10.
Letα˜1=ρ1,σ1 and α˜2=ρ2,σ2 be two intuitionistic multiplicative numbers. For each λ∈0,1, α˜1=α˜2 if and only if Aλα˜1=Aλα˜2 and Sλα˜1=Sλα˜2 hold.Proof.
(1)
Ifα˜1=α˜2, then we have ρ1=ρ2 and σ1=σ2. Accordingly, Sλα˜1=Sλα˜2 and Aλα˜1=Aλα˜2 hold.(2)
For allλ∈0,1, if Sλα˜1=Sλα˜2 and Aλα˜1=Aλα˜2, we have(43)ρ1σ1−λ/3⋅ρ12/3σ11/3=ρ2σ2−λ/3⋅ρ22/3σ21/3,(44)ρ1σ1λ⋅σ12/3ρ11/3=ρ2σ2λ⋅σ22/3ρ21/3.From (43), we have(45)2−λlnρ1−lnρ2=1+λlnσ1−lnσ2.By (44), we have(46)3λ−1lnρ1−lnρ2=3λ−2lnσ1−lnσ2,(45) can be further written as(47)lnρ1−lnρ2=1+λ2−λ⋅lnσ1−lnσ2.Applying equation (47) to (46), it follows that(48)3λ−11+λlnσ1−lnσ2=2−λ3λ−2lnσ1−lnσ2.Obviously, we can derive from (48) that 2λ2−2λ+1lnσ1−lnσ2=0Since2λ2−2λ+1>0 always holds, we have σ1=σ2. Based on (47), we have ρ1=ρ2. Therefore, it concludes that α˜1=α˜2.In sum, the proof of Property10 is completedIn the following, the order relation between any two intuitionistic multiplicative numbers is definedDefinition 7.
Letα˜1=ρ1,σ1 and α˜2=ρ2,σ2 be two intuitionistic multiplicative numbers. For each λ∈0,1, if one of the following conditions is true,(1)
Sλα˜1>Sλα˜2(2)
Sλα˜1=Sλα˜2 and Aλα˜1>Aλα˜2then α˜1 is said to be superior to α˜2, denoted by α˜1≻λα˜2.Property 11.
Letα˜1, α˜2, andα˜3 be three intuitionistic multiplicative numbers. The order relation ≻λ in Definition 7 satisfies the following conditions:(1)
Ifα˜2≻λα˜1, then α˜1≻λα˜2 does not hold(2)
For each intuitionistic multiplicative number,α˜1, α˜1≻λα˜1 does not hold(3)
Ifα˜1≻λα˜2 and α˜2≻λα˜3, then α˜1≻λα˜3Proof.
(1)
Sinceα˜2≻λα˜1, from Definition 7, the results can be analyzed in the following two cases. When Sλα˜2>Sλα˜1, then α˜1≻λα˜2 does not hold. WhenSλα˜1=Sλα˜2 and Aλα˜2>Aλα˜1, then α˜1≻λα˜2 still holds.(2)
Ifα˜1≻λα˜1 holds, then from Definition 7, we haveSλα˜1>Sλα˜1 or Aλα˜1>Aλα˜1, which lead to the illogical results. Thus, for each intuitionistic multiplicative number,α˜1, α˜1≻λα˜1 does not hold.(3)
Sinceα˜1≻λα˜2, the results can be considered in two cases:(a)
Sλα˜1>Sλα˜2. Because α˜2≻λα˜3, from Definition 7, we have Sλα˜2≥Sλα˜3. Thus, we get Sλα˜1>Sλα˜3. It follows that α˜1≻λα˜3.(b)
Sλα˜1=Sλα˜2 and Aλα˜1>Aλα˜2. Because α˜2≻λα˜3, we have(i)
IfSλα˜2>Sλα˜3, then Sλα˜1>Sλα˜3. Namely, we have α˜1≻λα˜3.(ii)
IfSλα˜2=Sλα˜3 and Aλα˜2>Aλα˜3, then Sλα˜1=Sλα˜3 and Aλα˜1>Aλα˜3. From Definition 7, we have α˜1≻λα˜3.
In sum, the proof of Property11 is complete.It is clear that the order relation≻λ on the set of intuitionistic multiplicative numbers is a strict order. The frame diagram of the proposed ranking method is illustrated in Figure 2.Accordingly, the proposed multiplicative accuracy and score functions can be used to compare any two intuitionistic multiplicative numbers. By equations (2), (3), (12), and (31), we have S0.5α˜=ρασα1/2=SXiaα˜1/2 and A0.5α˜=ρασα1/6=AXiaα˜1/6. It means that the total order relation defined in Definition 7 is equivalent to Xia’s method [16] when λ=0.5. Clearly, Xia’s ranking method is a special case of the proposed method based on multiplicative accuracy and score functions.Figure 2
The frame diagram of the proposed ranking method.


## 5. Numerical Example and Comparison

In the following, we apply the proposed score and accuracy functions to select the optimal logistics transfer station. In addition, a comparison example is developed to highlight the advantage and effectiveness of the proposed ranking method.


### 5.1. Application of the Multiplicative Score and Accuracy Functions in Selecting Logistics Transfer Station

With the increasingly frequent interaction of materials, the pressure of logistics enterprises is becoming more and more serious. It is necessary to setup some logistics transfer stations to make the material circulation process smoother. The optimal location of the transfer station can maximize the transportation efficiency of logistics enterprises. Suppose there is a panel with three logistics transfer stations:TS1 (transfer station I), TS2 (transfer station II), and TS3 (transfer station III). Logistics enterprise expects to select the optimal location of logistics transfer station according to four attributes: B1 (traffic conditions), B2 (built-up area), B3 (number of surrounding residents), and B4 (local economic developing level). The weighting vector corresponding to attributes is ω=ω1,ω2,ω3,ω4=0.1,0.4,0.3,0.2.Step 1.
Assume that three logistics transfer stationsTS1, TS2, and TS3 are evaluated by intuitionistic multiplicative numbers under four attributes B1B2, B3, and B4. Accordingly, the intuitionistic multiplicative decision matrix D˜=d˜ij3×4 is obtained as(49)B1B2B3B4D˜=TS1TS2TS314,31,33,512,115,21,73,916,212,11,21,314,3.Step 2.
By equation (31), the multiplicative score matrix SλD˜=Sλd˜ij3×4 with respect to parameter λ is derived as follows:(50)SλD˜=0.275⋅1.101λ0.693⋅0.693λ1.216⋅0.405λ0.630⋅1.260λ0.271⋅1.357λ0.523⋅0.523λ1.000⋅0.333λ0.240⋅1.442λ0.630⋅1.260λ0.794⋅0.794λ0.693⋅0.693λ0.275⋅1.101λ.Step 3.
The overall multiplicative score values of alternativesTS1, TS2, and TS3 are calculated as follows: (51)OSTS1,λ=∏j=14Sλd˜1jωj=0.734⋅0.696λ,OSTS2,λ=∏j=14Sλd˜2jωj=0.509⋅0.615λ,OSTS3,λ=∏j=14Sλd˜3jωj=0.602⋅0.852λ.
For parameterλ∈0,1, the curves corresponding to three overall multiplicative score values are illustrated in Figure 3.Step 4.
It is clear thatOSTS1,λ>OSTS2,λ and OSTS3,λ>OSTS2,λ hold for all λ∈0,1. Since OSTS1,λ/OSTS3,λ=1.219⋅0.817λ, we consider three cases as follows:(1)
Ifλ∈0,0.98, then we have OSTS1,λ>OSTS3,λ, i.e., TS1>TS3(2)
Ifλ∈0.98,1, then we have OSTS3,λ>OSTS1,λ, i.e., TS3>TS1(3)
Ifλ=0.98, then OSTS1,λ=OSTS3,λ holds. In this case, the following multiplicative accuracy values should be considered:(52)A0.98d˜11=0.289,A0.98d˜12=0.709,A0.98d˜13=1.229,A0.98d˜14=0.639,A0.98d˜31=0.639,A0.98d˜32=0.805,A0.98d˜33=0.709,A0.98d˜34=0.289.Step 5.
Accordingly, the overall multiplicative accuracy values of alternativesTS1 and TS3 are calculated as (53)OATS1,0.98=∏j=14A0.98d˜1jωj=0.749,OATS3,0.98=∏j=14A0.98d˜3jωj=0.617.
It is apparent thatOATS1,0.98>OATS3,0.98. Based on Definition 7, we have TS1>TS3. In sum, the ranking order of alternatives TS1, TS2, and TS3 is shown as follows: (1) if λ∈0,0.98, then TS1>TS3>TS2; (2) if λ∈0.98,1, then TS3>TS1>TS2.Figure 3
The overall multiplicative score values of three alternatives.


### 5.2. Comparison with Xia’s Method

Suppose we need to rank three intuitionistic multiplicative numbersα˜1=3,5, α˜2=1,3, and α˜3=2,6 in descending order. Using (31), the multiplicative score values of α˜1, α˜2, and α˜3 are obtained as(54)Sλα˜1=1.216⋅0.405λ,Sλα˜2=0.693⋅0.693λ,Sλα˜3=0.874⋅0.437λ.For allλ∈0,1, we have Sλα˜1>Sλα˜2 and Sλα˜1>Sλα˜3. From Definition 7, it follows that α˜1≻λα˜2 and α˜1≻λα˜3. Furthermore, if λ∈0,0.5, then Sλα˜3>Sλα˜2. Accordingly, we have α˜3≻λα˜2. If λ∈0.5,1, then Sλα˜2>Sλα˜3. Therefore, we have α˜2≻λα˜3. Considering S0.5α˜2=S0.5α˜3, we calculate the multiplicative accuracy values A0.5α˜2=1.201 and A0.5α˜3=1.513. Based on Definition 7, we have α˜3≻0.5α˜2.In sum, the ranking order ofα˜1, α˜2, and α˜3 can be concluded in two cases: (1) if λ∈0,0.5, then α˜1≻α˜3≻α˜2; (2) if λ∈0.5,1, then α˜1≻α˜2≻α˜3.For comparing with the proposed ranking method, Xia’s method [16] is employed to rank intuitionistic multiplicative numbers α˜1, α˜2, and α˜3. By (2), the multiplicative score values of α˜1, α˜2, and α˜3 are calculated as SXiaα˜1=3/5, SXiaα˜2=1/3, and SXiaα˜3=1/3. Moreover, from (3), we have AXiaα˜2=3 and AXiaα˜3=12. According to Definition 3, the ranking order is determined as α˜1≻α˜3≻α˜2. The comparative results of two ranking methods are illustrated in Table 1.Table 1
Comparative results of two ranking methods.
Ranking methodParameterRanking orderXia’s methodNullα˜1≻α˜3≻α˜2Proposed methodλ∈0,0.5α˜1≻α˜3≻α˜2Proposed methodλ∈0.5,1α˜1≻α˜2≻α˜3From Table1, it is obvious that the ranking result of our method is the same as Xia’s method when λ∈0,0.5. On the other hand, the ranking results of the two methods are different when λ∈0.5,1. An effective ranking method should consider the risk preference of decision makers. It should be pointed out that Xia’s ranking method is a special case of the proposed ranking method. Xia’s method only considers the single point value and ignores the infinite number of potential possible values in feasible region Rα. As a result, the ranking result of Xia’s method may be distorted and invalid in some cases. The proposed ranking method in this study considers the risk preference of decision makers in ranking results and traverses all potential possible values. Therefore, the ranking results of our method are more reasonable and effective.


## 5.1. Application of the Multiplicative Score and Accuracy Functions in Selecting Logistics Transfer Station

With the increasingly frequent interaction of materials, the pressure of logistics enterprises is becoming more and more serious. It is necessary to setup some logistics transfer stations to make the material circulation process smoother. The optimal location of the transfer station can maximize the transportation efficiency of logistics enterprises. Suppose there is a panel with three logistics transfer stations:TS1 (transfer station I), TS2 (transfer station II), and TS3 (transfer station III). Logistics enterprise expects to select the optimal location of logistics transfer station according to four attributes: B1 (traffic conditions), B2 (built-up area), B3 (number of surrounding residents), and B4 (local economic developing level). The weighting vector corresponding to attributes is ω=ω1,ω2,ω3,ω4=0.1,0.4,0.3,0.2.Step 1.
Assume that three logistics transfer stationsTS1, TS2, and TS3 are evaluated by intuitionistic multiplicative numbers under four attributes B1B2, B3, and B4. Accordingly, the intuitionistic multiplicative decision matrix D˜=d˜ij3×4 is obtained as(49)B1B2B3B4D˜=TS1TS2TS314,31,33,512,115,21,73,916,212,11,21,314,3.Step 2.
By equation (31), the multiplicative score matrix SλD˜=Sλd˜ij3×4 with respect to parameter λ is derived as follows:(50)SλD˜=0.275⋅1.101λ0.693⋅0.693λ1.216⋅0.405λ0.630⋅1.260λ0.271⋅1.357λ0.523⋅0.523λ1.000⋅0.333λ0.240⋅1.442λ0.630⋅1.260λ0.794⋅0.794λ0.693⋅0.693λ0.275⋅1.101λ.Step 3.
The overall multiplicative score values of alternativesTS1, TS2, and TS3 are calculated as follows: (51)OSTS1,λ=∏j=14Sλd˜1jωj=0.734⋅0.696λ,OSTS2,λ=∏j=14Sλd˜2jωj=0.509⋅0.615λ,OSTS3,λ=∏j=14Sλd˜3jωj=0.602⋅0.852λ.
For parameterλ∈0,1, the curves corresponding to three overall multiplicative score values are illustrated in Figure 3.Step 4.
It is clear thatOSTS1,λ>OSTS2,λ and OSTS3,λ>OSTS2,λ hold for all λ∈0,1. Since OSTS1,λ/OSTS3,λ=1.219⋅0.817λ, we consider three cases as follows:(1)
Ifλ∈0,0.98, then we have OSTS1,λ>OSTS3,λ, i.e., TS1>TS3(2)
Ifλ∈0.98,1, then we have OSTS3,λ>OSTS1,λ, i.e., TS3>TS1(3)
Ifλ=0.98, then OSTS1,λ=OSTS3,λ holds. In this case, the following multiplicative accuracy values should be considered:(52)A0.98d˜11=0.289,A0.98d˜12=0.709,A0.98d˜13=1.229,A0.98d˜14=0.639,A0.98d˜31=0.639,A0.98d˜32=0.805,A0.98d˜33=0.709,A0.98d˜34=0.289.Step 5.
Accordingly, the overall multiplicative accuracy values of alternativesTS1 and TS3 are calculated as (53)OATS1,0.98=∏j=14A0.98d˜1jωj=0.749,OATS3,0.98=∏j=14A0.98d˜3jωj=0.617.
It is apparent thatOATS1,0.98>OATS3,0.98. Based on Definition 7, we have TS1>TS3. In sum, the ranking order of alternatives TS1, TS2, and TS3 is shown as follows: (1) if λ∈0,0.98, then TS1>TS3>TS2; (2) if λ∈0.98,1, then TS3>TS1>TS2.Figure 3
The overall multiplicative score values of three alternatives.


## 5.2. Comparison with Xia’s Method

Suppose we need to rank three intuitionistic multiplicative numbersα˜1=3,5, α˜2=1,3, and α˜3=2,6 in descending order. Using (31), the multiplicative score values of α˜1, α˜2, and α˜3 are obtained as(54)Sλα˜1=1.216⋅0.405λ,Sλα˜2=0.693⋅0.693λ,Sλα˜3=0.874⋅0.437λ.For allλ∈0,1, we have Sλα˜1>Sλα˜2 and Sλα˜1>Sλα˜3. From Definition 7, it follows that α˜1≻λα˜2 and α˜1≻λα˜3. Furthermore, if λ∈0,0.5, then Sλα˜3>Sλα˜2. Accordingly, we have α˜3≻λα˜2. If λ∈0.5,1, then Sλα˜2>Sλα˜3. Therefore, we have α˜2≻λα˜3. Considering S0.5α˜2=S0.5α˜3, we calculate the multiplicative accuracy values A0.5α˜2=1.201 and A0.5α˜3=1.513. Based on Definition 7, we have α˜3≻0.5α˜2.In sum, the ranking order ofα˜1, α˜2, and α˜3 can be concluded in two cases: (1) if λ∈0,0.5, then α˜1≻α˜3≻α˜2; (2) if λ∈0.5,1, then α˜1≻α˜2≻α˜3.For comparing with the proposed ranking method, Xia’s method [16] is employed to rank intuitionistic multiplicative numbers α˜1, α˜2, and α˜3. By (2), the multiplicative score values of α˜1, α˜2, and α˜3 are calculated as SXiaα˜1=3/5, SXiaα˜2=1/3, and SXiaα˜3=1/3. Moreover, from (3), we have AXiaα˜2=3 and AXiaα˜3=12. According to Definition 3, the ranking order is determined as α˜1≻α˜3≻α˜2. The comparative results of two ranking methods are illustrated in Table 1.Table 1
Comparative results of two ranking methods.
Ranking methodParameterRanking orderXia’s methodNullα˜1≻α˜3≻α˜2Proposed methodλ∈0,0.5α˜1≻α˜3≻α˜2Proposed methodλ∈0.5,1α˜1≻α˜2≻α˜3From Table1, it is obvious that the ranking result of our method is the same as Xia’s method when λ∈0,0.5. On the other hand, the ranking results of the two methods are different when λ∈0.5,1. An effective ranking method should consider the risk preference of decision makers. It should be pointed out that Xia’s ranking method is a special case of the proposed ranking method. Xia’s method only considers the single point value and ignores the infinite number of potential possible values in feasible region Rα. As a result, the ranking result of Xia’s method may be distorted and invalid in some cases. The proposed ranking method in this study considers the risk preference of decision makers in ranking results and traverses all potential possible values. Therefore, the ranking results of our method are more reasonable and effective.


## 6. Conclusion and Further Study

In multiattribute decision-making problems under intuitionistic multiplicative environment, the ranking of fuzzy numbers is directly related to the priority of alternatives. Due to the limitation of resources, knowledge level, and behavior habits, decision makers tend to have different preferences for risk. In this study, a novel and effective ranking method of intuitionistic multiplicative numbers is proposed by defining the multiplicative score function and accuracy function, which integrates the risk preference information of the decision maker. We verify the necessary properties of multiplicative score function and accuracy function and then establish the total order relation of intuitionistic multiplicative numbers. The main advantages and innovations of the proposed ranking method are stated as follows:(1)
The proposed ranking method takes all the potential possible values in feasible region into account. Therefore, the ranking results obtained by our method are more reliable and reasonable. Xia’s method only considers the single point value and ignores infinite number of possible values. The ranking result of Xia’s method may be distorted and invalid in some cases.(2)
The ranking method proposed in this study integrates the risk preference of decision makers. The ranking results are more flexible and effective than Xia’s method. We prove that Xia’s ranking method is a special case of the proposed ranking method.(3)
We establish the total order relation of intuitionistic multiplicative numbers with risk preference information of decision makers. The necessary axiomatic properties (asymmetry, irreflexivity, and transitivity) of total order relation are verified.In addition, the proposed ranking method is applied to select the optimal logistics transfer station with intuitionistic multiplicative information. A comparison example is developed to highlight the advantage and effectiveness of the proposed ranking method. In the future, we consider extending the ranking method to nonequilibrium multiplicative environment. We will try to apply the series ranking method to deal with practical problems such as pollution control, system assessment, and energy management.



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