Intuitionistic Multiplicative Set Research Articles

Generalized correlation coefficients of intuitionistic multiplicative sets and their applications to pattern recognition and clustering analysis

ABSTRACT Intuitionistic multiplicative preference relations (IMPRs) and intuitionistic multiplicative sets (IMSs) play a significant role in real-life problems that contain unsymmetrical and nonuniform information. Correlation coefficients are critical tools for evaluating such information, especially in medical areas and clustering analysis, where the relationship between objects in the given data is required. Despite the importance of this subject, there is only one approach in the literature regarding the correlation coefficients of IMSs and existing coefficients have certain disadvantages. In this paper, we propose a parametric generalisation of these correlation coefficients on IMSs and apply them to medical diagnosis, taxonomy, and clustering. To that end, some disadvantages of existing correlation coefficients are listed first. Then, with some theoretical work, we derive a parametric generalisation of these coefficients and their weighted forms. To better illustrate how the parametric generalisation of correlation coefficients improves the results, numerical parametric solutions of existing examples are presented with detailed comparisons. Moreover, a novel algorithm is introduced for clustering using proposed correlation coefficients in IMSs. Finally, three real-life examples are provided to demonstrate the superiority of the proposed generalised correlation coefficients and the clustering algorithm in specific applications.

A bi-direction performance evaluation model for water pollution treatment engineering under the intuitionistic multiplicative linguistic environment

As the increasingly serious water pollution problem affects the sustainable development of the ecological environment, the research of water pollution treatment engineering cannot be delayed. Among them, the performance evaluation of water pollution treatment engineering is a major focus. After reading the existing studies, it is found that most of the existing performance evaluation indicators of water pollution treatment engineering have qualitative indicators and there is an unbalanced preference representation. Intuitionistic multiplicative linguistic sets can be a good representation of the qualitative preference and non-preference of decision-makers in the context of decision-making containing unbalanced phenomena. Therefore, to better solve the problem of water pollution treatment engineering, this paper introduces intuitionistic multiplicative linguistic sets to the problem of water pollution treatment engineering and proposes an effective theory for it. First, considering the multiplicative nature of the intuitionistic multiplicative linguistic set, a new score function and accuracy function are defined, and on this basis, the priority rules of intuitionistic multiplicative linguistic set are given to prepare for the subsequent water pollution treatment engineering performance ranking. And the distance measure of intuitionistic multiplicative linguistic set is introduced and a CRITIC attribute weight determination model under intuitionistic multiplicative linguistic set is obtained on this basis. Secondly, the Choquet integral operator is applied to better represent the correlation between elements. However, the nature of membership degree and non-membership degree shows that it is not reasonable to aggregate the information of intuitionistic multiplicative linguistic sets with a single increasing and decreasing transformation. Therefore, in this paper, we propose the IMLS bi-direction exponent Choquet integral operator, which is inspired by the bi-direction Choquet integral. Lastly, we improve the original preference function of the classical PROMETHEE II method to obtain the bi-directional PROMETHEE II method in intuitionistic multiplicative linguistic information. Finally, a numerical case is also provided to illustrate the scientific and rational application of the bi-directional PROMETHEE II method in intuitionistic multiplicative linguistic information for the performance evaluation of water pollution treatment engineering.

A new inclusion measure-based clustering method and its application to product classification

Appropriate product classification can not only promote the sale of products but also highly improve the profits of manufacturers and retailers. To derive a desirable clustering of products, we need to consider qualitative (e.g., customers’ evaluation) and quantitative (e.g., product sales) information simultaneously. However, traditional methods such as K-means clustering, hierarchical clustering, and density-based clustering fail to depict the fuzzy cognitions of decision makers (DMs) in real cases. This article investigates clustering methods to aid DMs in their classification management of products. To do so, an intuitionistic multiplicative set (IMS), a new form of information expression, is applied to express the DMs’ evaluations of products due to its strong ability to handle unbalanced or asymmetric information. This study includes an in-depth analysis of inclusion measures of IMSs to measure the inclusion relationships between products under the given evaluation criteria. We propose two types of clustering techniques—the intuitionistic multiplicative (IM) transitive closure clustering method and the IM netting clustering method—to tackle the clustering problem with IMS information. Lastly, a case study of product classification management is presented to illustrate the applicability of the proposed methods. Furthermore, the validity and superiority of these clustering methods are verified by theoretical and practical comparisons.

INTUITIONISTIC MULTIPLICATIVE SET APPROACH FOR GREEN SUPPLIER SELECTION PROBLEM USING TODIM METHOD

Production and supply chain can be considered as the fundamentals of economic cycle for all countries. As a result of climate crisis in the world and the disasters it brought, the use of green products has become inevitable. With the restrictions imposed by some governments, suppliers have started to canalize their productions to environmental goods and the selection of most appropriate green supplier become a strategic decision. However, classical methods remain incapable to making these decisions due to the uncertain information contained in real life problems. Intuitionistic multiplicative information is a good choice dealing with these uncertainties in this kind of problems. Therefore, in this work, green supplier selection problem is discussed using TODIM method with intuitionistic multiplicative sets.

Multi-attribute group decision-making process based on possibility degree and operators for intuitionistic multiplicative set

This paper aims to present a novel multiple attribute group decision-making process under the intuitionistic multiplicative preference set environment. In it, Saaty’s 1/9-9 scale is used to express the imprecise information which is asymmetrical distribution about 1. To achieve it, the present work is divided into three folds. First, a concept of connection number-based intuitionistic multiplicative set (CN-IMS) is formulated by considering three degrees namely “identity”, “contrary”, and “discrepancy” of the set and study their features. Second, to rank the given number, we define a novel possibility degree measure which compute the degree of possibility within the given objects. Finally, several aggregation operators on the pairs of the given numbers are designed and investigated their fundamental inequalities and relations. To explain the presented measures and operators, a group decision-making approach is promoted to solve the problems with uncertain information and illustrated with several examples. The advantages, comparative, as well as perfection analysis of the proposed framework are furnished to confirm the approach.

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. 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.

Exponential operational laws and new aggregation operators for intuitionistic multiplicative set in multiple-attribute group decision making process

The intuitionistic multiplicative preference set is one of the replacements to the intuitionistic fuzzy preference set, where the preferences related to the object is asymmetrical distribution about 1. In it, Saaty’s 1–9 scale has been used to represent the uncertain and imprecise information. Meanwhile, an aggregation operator by using general operational laws for some fuzzy sets is an important task to aggregate the different numbers. Motivated by these primary characteristics, it is interesting to present the concept of exponential operational laws, which differs from the traditional laws by the way, in which bases are real numbers while exponents are the intuitionistic multiplicative numbers. In this paper, we develop a methodto solve the Multiple Attribute Group Decision Making (MAGDM) problem under the Intuitionistic Multiplicative Sets (IMS) environment. To do it, firstly, we define some new exponential operational laws and a score function for IMS and studied their properties. Secondly, based on this, we develop some averaging and geometric aggregation operators and characterize their various properties. Thirdly, a novel approach is promoted to solve MAGDM problems with IMS information. Finally, some numerical illustrations are given with a comparative study to verify the approach.

A simplified neutrosophic multiplicative set‐based TODIM using water‐filling algorithm for the determination of weights

AbstractDue to the technological developments around the world, the amount of information that researchers work on increases continuously. This information density contains incomplete and uncertain data that cannot be fully expressed with crisp numbers. Fuzzy sets, intuitionistic fuzzy sets, and neutrosophic sets are useful tools to manage such information, but these concepts use a symmetrical and uniform scale to express data, whereas real‐life problems contain nonsymmetrical and non‐uniform information. Intuitionistic multiplicative sets (IMSs) are effective tools for dealing with these real‐life problems. However, IMSs cannot handle real‐life problems completely because indeterminate information depends on membership and non‐membership information of IMSs, which is a restriction for decision makers and also for decision problems. To overcome this limitation, this paper generalizes the IMSs by using simplified neutrosophic set and introduces a novel approach that is called simplified neutrosophic multiplicative sets (SNMSs). Firstly, we define SNMS, show their set‐based operations, and then give a description of simplified neutrosophic multiplicative numbers (SNMNs). Based on SNMNs, we develop two simplified neutrosophic multiplicative aggregation operators on SNMNs that are called simplified neutrosophic multiplicative weighted arithmetic average operator and simplified neutrosophic multiplicative weighted geometric average operator. Furthermore, we define some simplified neutrosophic multiplicative distance measures. Finally, using a model based on water‐filling algorithm for determining criteria weights, we give a numerical example to demonstrate the effectiveness of the introduced concept with the proposed simplified neutrosophic multiplicative simplified neutrosophic multiplicative‐TODIM method.

A priority-based intuitionistic multiplicative UTASTAR method and its application in low-carbon tourism destination selection

The intuitionistic multiplicative preference relation extends Saaty’s multiplicative preference relation by membership and non-membership functions to describe decision-makers’ preferences on objects (alternatives or criteria), and the product of pairwise membership and non-membership degrees is less than or equal to one. In recent years, the intuitionistic multiplicative preference relation has attracted increasing attention and many methods have been proposed to handle multiple criteria decision making (MCDM) problems. However, few of the existing methods can be used to tackle the MCDM problems with both quantitative and qualitative criteria. To overcome this challenge, in this paper, a novel MCDM method called the priority-based intuitionistic multiplicative UTASTAR method is proposed, which combines a new intuitionistic multiplicative prioritization method and the UTASTAR method within the context of intuitionistic multiplicative sets. The consistency checking and improving processes are considered in the weight-determining method. A case study regarding the low-carbon tourism destination selection is adopted to illustrate the applicability of the proposed method. The effectiveness and superiority of the proposed method are further verified by comparative analyses and discussions.

Distance‐based intuitionistic multiplicative multiple criteria decision‐making methods for healthcare management in West China Hospital

AbstractIntuitionistic multiplicative sets use an asymmetric, unbalanced scale to express information from positive, negative, and indeterminate information. They have been found capable of comprehensively and objectively representing a person's intuitive understanding and hence have attracted much attention. Distance techniques are widely used to measure the degree to which arguments deviate from one another. Several fuzzy set extensions have been developed, but little research has been conducted on measures of distance between intuitionistic multiplicative sets. In this paper, we start by presenting a variety of measures of the distance between intuitionistic multiplicative sets, including Hausdorff distance measures, weighted distance measures, ordered weighted distance measures, and continuous weighted distance measures. We then develop a distance‐based intuitionistic multiplicative‐technique for order preference by similarity to ideal solution method and a distance‐based intuitionistic multiplicative‐Vlsekriterijumska Optimizacija I Kompromisno Resenje method for handling multiple criteria decision‐making problems with intuitionistic multiplicative evaluation information. To demonstrate the practical application of these distance measures and the proposed methods, we provide a case study of hospital management of inpatient admission. The paper ends with comparative analyses of the two methods and some concluding remarks.

Distance-based intuitionistic multiplicative MULTIMOORA method integrating a novel weight-determining method for multiple criteria group decision making

Intuitionistic multiplicative set is a powerful technique to depict asymmetric or unbalanced evaluation information. Given that the MULTIMOORA method is a multiple criteria decision making method with strong robustness, this paper aims to propose a distance-based intuitionistic multiplicative MULTIMOORA method integrating a novel weight-determining method. To do so, firstly, new distance measures, including the intuitionistic multiplicative Jaccard distance, the intuitionistic multiplicative Dice distance, and the intuitionistic multiplicative generalized distance measure, are defined. Subsequently, an interesting weight determining method motivated by the grey relation analysis theory is presented to determine the weights of criteria. Based on these research findings, we then develop an improved intuitionistic multiplicative MULTIMOORA method to address the multiple criteria decision making problems with intuitionistic multiplicative evaluation information. A case study concerning medical equipment selection is presented to illustrate the applicability of the proposed method. The effectiveness and superiority of the distance-based intuitionistic multiplicative MULTIMOORA method are verified by some comparative analyses.

A multiple attribute group decision making method based on two novel intuitionistic multiplicative distance measures

Distance measure is used to measure the deviation degree between different arguments. Since the existing distance measures between intuitionistic multiplicative sets cannot represent the angle of two intuitionistic multiplicative sets, this paper introduces some new distance measures between intuitionistic multiplicative sets, which include the projection-based distance measure and the psychological distance measures. After that, a multiple attribute group decision making method is proposed to handle the problem in which the weights of experts and attributes are unknown or partially known. A case study concerning the drug supplier selection in hospital management is provided to demonstrate the calculation process of the proposed method. Finally, some comparative analyses are given to illustrate the efficiency and applicability of the proposed method.

Novel correlation coefficients under the intuitionistic multiplicative environment and their applications to decision-making process

The objective of this work is to present novel correlation coefficients under the intuitionistic multiplicative preference relation (IMPR), for measuring the relationship between the two intuitionistic multiplicative sets, instead of intuitionistic fuzzy preference relation (IFPR). As IFPR deals under the conditions that the attribute values grades are symmetrical and uniformly distributed. But in our day-to-day life, these conditions do not fulfill the decision maker requirement and hence IFPR theory is not applicable in that domain. Thus, for handling this, an intuitionistic multiplicative set theory has been utilized where grades are distributed asymmetrical around 1. Further, under this environment, a decision making method based on the proposed novel correlation coefficients has been presented. Pairs of membership and non-membership degree are considered to be a vector representation during formulation. Three numerical examples have been taken to demonstrate the efficiency of the proposed approach.

A Robust Ranking Method for Intuitionistic Multiplicative Sets Under Crisp, Interval Environments and Its Applications

In this paper, a new generalized improved score function for the intuitionistic multiplicative set (IMS) has been presented. IMS uses the unbalanced distributed scale instead of balanced one to describe the membership information. Under this environment and by integrating the idea of hesitation degree, a score function has been proposed for interval-valued as well as for crisp numbers. Lastly, a multicriteria decision-making approach, based on proposed function, has been presented and illustrated with a numerical example for showing their superiority in decision making. Sensitivity analysis of the various parameters has also been done for showing their impact on ranking the alternative.

Methods for ranking intuitionistic multiplicative numbers by distance measures in decision making

Intuitionistic multiplicative number (IMN) is the basic component of intuitionistic multiplicative set (IMS) and intuitionistic multiplicative preference relation, which is suitable for describing the preference information in the unbalance distribution. In this paper, we propose two methods of ranking IMNs by distance measures for applications in decision making. To do it, we first develop the normalized and weighted normalized Manhattan distances between IMSs by analyzing the correlations of IMSs and intuitionistic fuzzy sets quantitatively. As a specific case, we also present the normalized Manhattan distance between IMNs, based on which, we introduce the distance and accuracy functions of IMNs and give a comparison law for them. After that, we propose a method for ranking IMNs in decision making. Furthermore, considering the characteristics of the distance measures, we extend the distance and accuracy functions into IMSs and give the extended distance and accuracy functions of IMSs. Then we apply them to the decision making, especially group decision making, and propose another ranking method for IMNs based on an extended comparison law. Our methods can not only overcome the shortages of the existing method that uses the score and accuracy functions for ranking IMNs which may occur contradiction with our intuition, but also reduce the amount and time of computation.

Indefinite integrals of generalized intuitionistic multiplicative functions

By using the unsymmetrical scale and the non-discrete values, the generalized intuitionistic multiplicative sets (GIMSs) can reflect our intuition more objectively. The GIMFs, which are defined on ...

Point operators for intuitionistic multiplicative information

Intuitionistic multiplicative set uses the unbalanced distributed scale instead of the balanced one to describe the membership and non-membership information. It can avoid some disadvantages in calculating process. We develop a series of point operators to reduce the uncertain information in intuitionistic multiplicative set. The proposed operator can redistribute the uncertain information according to the difference preference of decision makers.

Interval-Valued Intuitionistic Multiplicative Sets

A generalization of the notion of intuitionistic multiplicative set (IMS) is given in this paper, which is called interval-valued intuitionistic multiplicative set (IVIMS). The IVIMS uses an unsymmetrical scale (Saaty's 1-9 scale) instead of the ordinary symmetrical scale in an intuitionistic fuzzy set or an interval-valued intuitionistic fuzzy set. IVIMSs can reflect our intuition more objectively in some special situations. The basic component of an IVIMS is interval-valued intuitionistic multiplicative number (IVIMN). To rank IVIMNs, the comparison laws are developed. Based on the pseudo-multiplication, we explore some interval-valued intuitionistic multiplicative operations and obtain several specific results when some specific forms are assigned. In addition, some distance measures and weighted distance measures between IVIMSs are developed to describe the difference of any two IVIMSs. By analyzing the range of the uncertain degree, a generalized interval-valued intuitionistic multiplicative set is presented, which is a generalization of the notions of IMS and IVIMS. Finally, a numerical example is given to illustrate our results.