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

Tian, Zhiyong; Lin, Jian

doi:https://doi.org/10.1155/2021/6664439

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Numerical Example Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Applications of Fuzzy Sets and their Extensions in Engineering

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Research Article | Open Access

Volume 2021 | Article ID 6664439 | https://doi.org/10.1155/2021/6664439

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

Zhiyong Tian¹and Jian Lin^2,3

Academic Editor: Mingwei Lin

Received28 Oct 2020

Revised15 Dec 2020

Accepted26 Dec 2020

Published13 Jan 2021

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.

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. 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.

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 set be fixed. An intuitionistic multiplicative set over is defined aswhere and are the membership information and nonmembership information of with respect to intuitionistic multiplicative set , respectively, such that , and . The hesitation or uncertain information of corresponding to is denoted by .
Clearly, , and . An intuitionistic multiplicative number (IMN) can be represented by , such that , , and . denotes the set of all IMNs.

Definition 2 (See [16]). Let be an intuitionistic multiplicative number. The score function and accuracy function of are expressed by

Definition 3 (Reference [16]). Let and be two intuitionistic multiplicative numbers. The order relation between and is defined as follows:(1)If , then (2)If , then(a)If , then (b)If , then

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. Let be an intuitionistic multiplicative number. The feasible region after transformation of is denoted bywhere , , and , respectively.
Since , we have . It follows that . Obviously, (4) can be further written aswhere represents the mapping feasible region with respect to intuitionistic multiplicative numbers . The mapping feasible region is shown in Figure 1.

Definition 5. Let be an intuitionistic multiplicative number. The multiplicative accuracy function of based on risk preference is expressed aswhere is the mapping feasible region of intuitionistic multiplicative number , and is the area of mapping feasible region . The parameter reflects the risk tendency of the decision maker. When , the decision maker is a risk lover. When , the decision maker is averse to risk. Especially, the attitude of the decision maker is neutral to the risk when .
represents the geometric average of accuracy function over all feasible values. From (5), we haveAccording to the operational properties of double integral, we haveBased on (6), it follows thatSince and , we haveThe expression of multiplicative accuracy function of can be further written aswhich can be equivalently written asThe multiplicative accuracy function based on risk preference satisfies some desirable properties as follows.

Property 1. For any two intuitionistic multiplicative numbers and , if and only if .

Proof. If , by equation (3), we have .
When , from (12), we have , .
For , it follows that .
Therefore, is equivalently written asThe proof of Property 1 is completed.

Property 2. Let be an intuitionistic multiplicative number. , we have

Proof. We prove Property 1 in the following two cases:(1)If , then we have . , we have Since , we have(2)If , then we have . Since , for all , we have Considering both cases (1) and (2), we have Therefore, Property 2 is proved.

Property 3. Let and be two intuitionistic multiplicative numbers. ; if , then .

Proof. Based on equation (7), we have divided by , we haveThe above expression can be equivalently written as
. Since , we have . Accordingly, for all , holds. Namely, we have . Therefore, the proof of Property 3 is completed.

Property 4. Let be a perturbation of parameter with . Given two intuitionistic multiplicative numbers and . If , then holds if and only if

Proof. Since , we have . By equation (12), it holds thatAccordingly, the following two inequalities are equivalent:In the following, we analyze (23) in three cases:(1)If , then inequalities and are equivalent. Thus, , and relational expression (12) is true.(2)If , then inequality is equivalent to . Considering , we have(3)If , then inequality is equivalent to . Since , we have In sum, the proof of Property 4 is completed.

3.2. The Multiplicative Score Function Based on Risk Preference

Definition 6. Let be an intuitionistic multiplicative number. The multiplicative score function of based on risk preference is defined bywhere is the mapping feasible region of , and is the area of mapping feasible region . The parameter reflects the risk tendency of the decision maker.
It clear that represents the geometric average of score functions over all feasible values. According to the operational properties of double integral, we haveBased on (26), it follows thatSince and , we haveThe expression of multiplicative score function of can be further written aswhich can be equivalently written asThe multiplicative score function based on risk preference satisfies some desirable properties as follows.

Property 5. Let be an intuitionistic multiplicative number. ; if , then .

Proof. Based on equation (31), we haveIt follows thatSince and , we have .
The proof of Property 5 is completed.

Property 6. For any two intuitionistic multiplicative numbers and , if and only if .

Proof:. If , by equation (2), we have .
When , from (31), we have , .
For , it holds that .
Therefore, is equivalently written asThe proof of Property 6 is completed.

Property 7. Let be an intuitionistic multiplicative number. , we have .

Proof. From equation (31), we have and . By Property 5, for all , we have . Since , we haveTherefore, Property 7 is proved.

Property 8. Let be a perturbation of parameter with . Given two intuitionistic multiplicative numbers and . If , then holds if and only if

Proof. Since , we have . By equation (31), it holds thatAccordingly, the following two inequalities are equivalent:In the following, we analyze (38) in three cases:(1)If , then inequalities and are equivalent. Thus, , and the relational expression (38) is true.(2)If , then inequality is equivalent to . Considering , we have(3)If , then inequality is equivalent to . Since , we have In sum, the proof of Property 8 is completed.

Property 9. Let and be two intuitionistic multiplicative numbers. If and , then for all , we have.

Proof. From equation (31), we haveIt follows thatSince and , we have and . Considering , we have .
Therefore, the proof of Property 9 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. Let and be two intuitionistic multiplicative numbers. For each , if and only if and hold.

Proof. (1)If , then we have and . Accordingly, and hold.(2)For all , if and , we have From (43), we have By (44), we have (45) can be further written as Applying equation (47) to (46), it follows that Obviously, we can derive from (48) that Since always holds, we have . Based on (47), we have . Therefore, it concludes that . In sum, the proof of Property 10 is completed In the following, the order relation between any two intuitionistic multiplicative numbers is defined

Definition 7. Let and be two intuitionistic multiplicative numbers. For each , if one of the following conditions is true,(1)(2) and then is said to be superior to , denoted by .

Property 11. Let , , and

be three intuitionistic multiplicative numbers. The order relation in Definition 7 satisfies the following conditions:(1)If , then does not hold(2)For each intuitionistic multiplicative number, , does not hold(3)If and , then

Proof. (1)Since , from Definition 7, the results can be analyzed in the following two cases. When , then does not hold. When and , then still holds.(2)If holds, then from Definition 7, we have or , which lead to the illogical results. Thus, for each intuitionistic multiplicative number, , does not hold.(3)Since , the results can be considered in two cases:(a). Because , from Definition 7, we have . Thus, we get . It follows that .(b) and . Because , we have(i) If , then . Namely, we have .(ii) If and , then and . From Definition 7, we have .In sum, the proof of Property 11 is complete.

It is clear that the order relation 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 and . It means that the total order relation defined in Definition 7 is equivalent to Xia’s method [16] when . Clearly, Xia’s ranking method is a special case of the proposed method based on multiplicative accuracy and score functions.

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: (transfer station I), (transfer station II), and (transfer station III). Logistics enterprise expects to select the optimal location of logistics transfer station according to four attributes: (traffic conditions), (built-up area), (number of surrounding residents), and (local economic developing level). The weighting vector corresponding to attributes is .

Step 1. Assume that three logistics transfer stations , , and are evaluated by intuitionistic multiplicative numbers under four attributes , , and . Accordingly, the intuitionistic multiplicative decision matrix is obtained as

Step 2. By equation (31), the multiplicative score matrix with respect to parameter is derived as follows:

Step 3. The overall multiplicative score values of alternatives , , and are calculated as follows: For parameter , the curves corresponding to three overall multiplicative score values are illustrated in Figure 3.

Step 4. It is clear that and hold for all . Since , we consider three cases as follows:(1)If , then we have , i.e., (2)If , then we have , i.e., (3)If , then holds. In this case, the following multiplicative accuracy values should be considered:

Step 5. Accordingly, the overall multiplicative accuracy values of alternatives and are calculated as It is apparent that . Based on Definition 7, we have . In sum, the ranking order of alternatives , , and is shown as follows: (1) if , then ; (2) if , then .

5.2. Comparison with Xia’s Method

Suppose we need to rank three intuitionistic multiplicative numbers , , and in descending order. Using (31), the multiplicative score values of , , and are obtained as

For all , we have and . From Definition 7, it follows that and . Furthermore, if , then . Accordingly, we have . If , then . Therefore, we have . Considering , we calculate the multiplicative accuracy values and . Based on Definition 7, we have .

In sum, the ranking order of , , and can be concluded in two cases: (1) if , then ; (2) if , then .

For comparing with the proposed ranking method, Xia’s method [16] is employed to rank intuitionistic multiplicative numbers , , and . By (2), the multiplicative score values of , , and are calculated as , , and . Moreover, from (3), we have and . According to Definition 3, the ranking order is determined as . The 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 . On the other hand, the ranking results of the two methods are different when . 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 . 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.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 71601049), the Humanities and Social Sciences Fund of the Ministry of Education (No. 16YJC630064), the Natural Science Foundation of Fujian Province (No. 2020J01576), the Foundation of Beijing Intelligent Logistics System Collaborative Innovation Center (No. BILSCIC-2019KF-16), China Postdoctoral Science Foundation (No. 2019M660242), the Construction Fund for Digital Fujian Big Data for Agriculture and Forestry (No. KJG18019A), and the University Training Program in Scientific Research for Outstanding Young Talents of Fujian Province: COST sharing strategy of incomplete cooperative games and its application in water pollution control (No. K80SCC53A).

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Copyright © 2021 Zhiyong Tian and Jian Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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