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

Intuitionistic multiplicative sets can be applied in many practical situations, most of which are based on ranking of intuitionistic multiplicative numbers. -is study develops an integral method for ranking intuitionistic multiplicative numbers based on the new definitions of multiplicative score function and accuracy function.-e rankingmethod 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. -e 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.

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. e 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][18][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. e 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. e operational laws and desirable properties are studied. Jiang et al. [25] proposed the interval-valued intuitionistic multiplicative set based on an unsymmetrical scale. e 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. e 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. e 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. e proposed ranking method takes all the potential possible values in feasible region into account. erefore, 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. e rest of this study is structured as follows. Section 2 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.

Preliminaries
For the convenience of the follow-up discussion, intuitionistic multiplicative number and its order relation are reviewed as follows.

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.

Definition 5.
Let α � (ρ α , σ α ) be an intuitionistic multiplicative number. e multiplicative accuracy function of α based on risk preference is expressed as where R α is the mapping feasible region of intuitionistic multiplicative number α, and M(R α ) is the area of mapping feasible region R α . e 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. A λ (α) represents the geometric average of accuracy function over all feasible values. From (5), we have According to the operational properties of double integral, we have Based on (6), it follows that Figure 1: e mapping feasible region with respect to.α.

Mathematical Problems in Engineering
Since ρ α � ln ρ α and σ α � ln σ α , we have e expression of multiplicative accuracy function of α can be further written as which can be equivalently written as e multiplicative accuracy function based on risk preference satisfies some desirable properties as follows.

Property 1. For any two intuitionistic multiplicative numbers
When λ � 0.5, from (12), we have is equivalently written as e proof of Property 1 is completed.
Proof. Based on equation (7), we have e above expression can be equivalently written as erefore, the proof of Property 3 is completed.
In sum, the proof of Property 4 is completed. □

e Multiplicative Score Function Based on Risk Preference
Definition 6. Let α � (ρ α , σ α ) be an intuitionistic multiplicative number. e multiplicative score function of α based on risk preference is defined by where R α is the mapping feasible region of α, and M(R α ) is the area of mapping feasible region R α . e parameter λ reflects the risk tendency of the decision maker. It clear that S λ (α) represents the geometric average of score functions over all feasible values. According to the operational properties of double integral, we have

Mathematical Problems in Engineering
Based on (26), it follows that Since ρ α � ln ρ α and σ α � ln σ α , we have e expression of multiplicative score function of α can be further written as which can be equivalently written as e multiplicative score function based on risk preference satisfies some desirable properties as follows.

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. hold.
(3) Since α 1 ≻ λ α 2 , the results can be considered in two cases: In sum, the proof of Property 11 is complete.

□
It is clear that the order relation ≻ λ on the set of intuitionistic multiplicative numbers is a strict order. e frame diagram of the proposed ranking method is illustrated in Figure 2.

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.

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. e optimal location of the transfer station can maximize the transportation efficiency of logistics enterprises. Suppose there is a panel with three logistics transfer stations: TS 1 (transfer station I), TS 2 (transfer station II), and TS 3 (transfer station III). Logistics enterprise expects to select the optimal location of logistics transfer station according to four attri- Step 1. Assume that three logistics transfer stations TS 1 , TS 2 , and TS 3 are evaluated by intuitionistic multiplicative numbers under four attributes B 1 B 2 , B 3 , and B 4 . Accordingly, the intuitionistic multiplicative decision matrix D � (d ij ) 3×4 is obtained as Step 2. By equation (31), the multiplicative score matrix S λ (D) � (S λ (d ij )) 3×4 with respect to parameter λ is derived as follows: Step 3. e overall multiplicative score values of alternatives TS 1 , TS 2 , and TS 3 are calculated as follows: For parameter λ ∈ [0, 1], the curves corresponding to three overall multiplicative score values are illustrated in Figure 3.
e comparative results of two ranking methods are illustrated in Table 1.
From Table 1, 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 R α . As a result, the ranking result of Xia's method may be distorted and invalid in some cases. e proposed ranking method in this study considers the risk preference of decision makers in ranking results and traverses all potential possible values. erefore, the ranking results of our method are more reasonable and effective.

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. e main advantages and innovations of the proposed ranking method are stated as follows: (1) e proposed ranking method takes all the potential possible values in feasible region into account. erefore, 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. e ranking result of Xia's method may be distorted and invalid in some cases.
(2) e ranking method proposed in this study integrates the risk preference of decision makers. e 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. e 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.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.