Multiplicative Preference Relations Research Articles

Group decision making with incomplete triangular fuzzy multiplicative preference relations for evaluating third-party reverse logistics providers

The strategic management of reverse logistics (RL) is essential for enterprises to enhance their operational efficiency, customer satisfaction, and sustainability performance in today's competitive marketplace. Many manufacturing firms have to cooperate with professional RL providers to overcome resource constraints and technological limitations, ultimately driving business success. Thus, it is critical for every enterprise to select the most suitable third-party RL provider (3PRLP). This paper provides a novel group decision making (GDM) method with incomplete triangular fuzzy multiplicative preference relations (TFMPRs) to cope with the selection of the most optimal 3PRLP. Firstly, a definition of acceptable incomplete TFMPRs is given. Then, the sufficient and necessary condition of an acceptable incomplete TFMPR is proposed. By analyzing the properties of consistent TFMPRs, a graph-based algorithm is designed to estimate the unknown elements in incomplete TFMPRs. Based on the proposed acceptable consistency definition of TFMPRs, an optimization model is set up to improve the consistency degree of inconsistent TFMPRs. The optimal normalized triangular fuzzy multiplicative weight vector (Tri-MWV) is obtained by computing two analytic expressions and solving a linear programming model. To measure the closeness degree of two TFMPRs, the concept of logarithmic correlation coefficient (LCC) between two TFMPRs is proposed. Combining the incomplete preference information in incomplete TFMPRs with the LCCs of any two TFMPRs, an algorithm of computing experts’ weights is displayed. Subsequently, a novel method of GDM with incomplete TFMPRs is presented. Lastly, a practical example of evaluating 3PRLPs is conducted to illustrate the effectiveness of the proposed GDM method with incomplete TFMPRs.

The strategy of consensus and consistency improving considering bounded confidence for group interval-valued intuitionistic multiplicative best-worst method

To significantly and reasonably enhance the individual consistency and group consensus in group best-worst method (GBWM) under interval-valued intuitionistic multiplicative environment, this study proposes a strategy of consensus and consistency improving considering bounded confidence. Firstly, based on the consistency definition of interval-valued intuitionistic multiplicative preference relations (IVIMPRs), the consistency definition for interval-valued intuitionistic multiplicative reference comparison relations (IVIMRCRs) is proposed. Subsequently, a two-stage consistency improving optimization model is constructed. Then, an interactively iterative consistency improving algorithm considering bounded confidence is designed. For GBWM with IVIMRCRs, the concept of consensus measure is first defined to assess the consensus level among individual IVIMRCRs, and then a two-stage consensus improving optimization model considering bounded confidence is constructed. Additionally, based on the above theories and models, an interactively iterative consensus improving algorithm considering bounded confidence and consistency management is designed for GBWM with IVIMRCRs. Finally, the practicality and effectiveness of the proposed GBWM are verified through an actual application case and simulation analyses. Meanwhile, the advantages of this method are explained through comparative analyses with the existing BWMs and group decision making (GDM) with IVIMPRs.

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 two-objective-optimization-driven group decision making model under the bipolarity of decision information

When building consensus in group decision making (GDM) under uncertainty, an important yet rarely studied issue is to find the Pareto solutions of multi-objective optimization model. This paper reports a two-objective (2Ob) optimization driven consensus model in GDM by describing the bipolarity of judgements through intuitionistic multiplicative preference relations (IMPRs). First, it is realized that the inherent property of IMPRs is the hesitancy degree. A novel inconsistency index of IMPRs is proposed by combining the effects of hesitancy degree and inconsistency of boundary matrices. Second, the compatibility measure between two IMPRs is utilized to quantify the consensus level (CL) of decision makers. The threshold of acceptable group CL is found to decrease with the order of IMPRs for the first time. A 2Ob optimization model is constructed by minimizing group inconsistency degree and group CL, respectively. Third, the method of equipping two flexibility degrees to each expert is proposed for optimizing individual IMPRs. It is interesting to find that the constructed granularity matrix is different from interval-valued IMPRs. A multi-objective particle swarm optimization algorithm is adopted to obtain a set of Pareto solutions to GDM problems. Case studies are carried out to illustrate the proposed consensus reaching model. The results help to identify how to provide flexible decisions in GDM under some complexity and uncertainty of a practical problem.

Consistency and consensus enhancing in group decision making with interval-valued intuitionistic multiplicative preference relations based on bounded confidence

Consistency and consensus enhancing in group decision making with interval-valued intuitionistic multiplicative preference relations based on bounded confidence

Stochastic group preference acceptability analysis for interval-valued multiplicative preference relations based on TODIM method

Among many decision problems, interval-valued multiplicative preference relation (IMPR) is widely utilized due to its ability to express uncertain information. A new approach to solve group decision making (GDM) with IMPRs is proposed in this paper, named stochastic group preference acceptability analysis with TODIM (SGPAA-TODIM) method, by combining TODIM method (an acronym in Portuguese of Interactive and multi-criteria Decision Making) with stochastic multi-criteria acceptability analysis (SMAA-2). It effectively circumvents information loss and considers the weight and risk preferences of experts. Firstly, the stochastic multiplicative preference relation is defined through stochastic simulation employing a certain density function, and its priority weight vector is determined using the logarithmic least squares method (LLSM). Then, the priority preference comprehensive matrix is proposed by extracting information of priority vectors. Moreover, the novel SGPAA-TODIM method is developed to analyze the stochastic parameter spaces, with the optimal rank determined based on the analysis of acceptability degrees associated with dominance rank. Finally, to demonstrate the validity and applicability of the proposed method, numerical examples are given.

Neutrosophic multiplicative preference relations based on consensus analysis and additive consistency in group decision making: A goal programming approach

Multiplicative preference relations can be expanded into Neutrosophic multiplicative preference relations (NMPR) and Interval Neutrosophic Multiplicative Preference Relations (INMPR). They are appropriate for capturing the experts' assessments' uncertainty, ambiguity, and indeterminacy. This work aims to provide a consistency and consensus-based approach for dealing with group decision-making using NMPRs and INMPRs, as well as many goal programming models to manage the consistency and consensus of NMPRs and INMPRs. To define and measure acceptable consistency for NMPRs and INMPRs, the study first offers a consistency index. Several consistency-based programming approaches are designed to address the inconsistency and provide an appropriate consistent NMPR and INMPR for an NMPR and INMPR that are not consistent enough. We provide a consistency-based approach to NMPR and INMPR decision-making. Then, considering the consensus in GDM, a consensus index is suggested for determining the level of agreement between specific NMPRs and INMPRs. Thereafter, a group NMPR and INMPR are created by combining individual NMPRs and INMPRs using an aggregation operator that ensures the consistency of the group NMPRs and INMPRs. With a collection of NMPRs and INMPRs, a consistency- and consensus-based GDM approach is built on single-valued neutrosophic sets (SVNS) and interval-valued neutrosophic sets (IVNS). Finally, two real-world numerical examples are shown, along with a comparison. The proposed method checked the individual consistency level and the group consensus, which is less than the existing method. The ranking of the alternatives was given, which was more convincing than the existing methods. It is also clear that it is much simpler than the previous methods.

An indirect multicriteria group decision-making method with heterogeneous preference relations and reliabilities of decision-makers

This study proposes a new indirect method to solve multicriteria group decision-making (MCGDM) problems with heterogeneous preference relations and decision-makers’ reliabilities. To unify the heterogeneous preference relations, the transformation of multiplicative preference relations (MPRs), fuzzy preference relations (FPRs), intuitionistic fuzzy preference relations (IFPRs), and linguistic preference relations (LPRs) into distributed preference relations (DPRs) is analyzed. As consistency among DPRs is constructed under the assumption that decision-makers are bounded rationality, the abstract transformation of MPR, FPR, IFPR, and LPR into DPR is defined, and its reality depends on the preferences of decision-makers. Two important properties of abstract transformation, namely, internal consistency and restricted max-max transitivity, were theoretically verified. The reliabilities of decision-makers are involved in MCGDM by estimating the reliability of each criterion to improve solution quality. Based on the four types of abstract transformations, a process for analyzing MCGDM problems with heterogeneous preference relations and decision-makers’ reliabilities is developed. The proposed method is used to analyze a new business selection problem for an enterprise located in Changzhou, Jiangsu, China to demonstrate its applicability and validity.

Consensus reaching with minimum adjustment and consistency management in group decision making with intuitionistic multiplicative preference relations

Intuitionistic multiplicative preference relation (IMPR) is an efficient tool for describing the paired comparisons between objectives in group decision making (GDM). Existing GDM methods with IMPRs overlook the hesitancy of IMPRs and lack a feedback mechanism of adjustment in the consensus reaching process (CRP). Thus, to ensure that the adjusted IMPRs can provide reliable information for decision making, the hesitancy measure for IMPRs is first incorporated into the process of enhancing consistency and consensus. A new consistency index for IMPRs is defined based on distance measure. Some goal programming models are constructed for enhancing consistency of IMPR. For a collection of IMPRs with unacceptable consistency and consensus, based on the optimization model and the identification rule and direction rule (IR-DR), an interactive iterative algorithm is designed for enhancing consistency and consensus, which can greatly save the completion time of consensus reaching and maintain primitive information to the maximum. Subsequently, a distance-based aggregation model is constructed to derive the group IMPR by considering different experts’ decision-making principles. Thereby, a new GDM method with IMPRs considering minimum adjustment is proposed. Moreover, the rationality and effectiveness of the proposed method are verified through a numerical example and some related simulation experiments. Meanwhile its merits are illustrated by comparison analyses.

Group decision-making with multiplicative probabilistic linguistic preference relations based on consistency improvement and upgraded multiplicative DEA cross-efficiency

Multiplicative probabilistic linguistic preference relation (MPLPR) has been widely used by decision-makers (DMs) to tackle group decision-making (GDM) problems. However, due to the complexity of the decision-making circumstance and individual subjectivity of DMs, they often provide inconsistent MPLPRs which often lead to unreasonable decision results. To solve this problem, this paper investigates a novel approach to GDM with MPLPRs based on consistency improvement and upgraded multiplicative data envelopment analysis (DEA) cross-efficiency. First, the concept of sequential consistency of MPLPR is defined. Then, a consistency improvement algorithm is proposed, which can convert any unacceptable consistent MPLPR into an acceptable one. Furthermore, we use geometric averages to transform MPLPR into multiplicative preference relation (MPR). Meanwhile, considering the conservative psychology of DMs, an upgraded multiplicative DEA cross-efficiency model based on the pessimistic criterion is constructed, which can derive the priority vector of MPLPR. Therefore, we can obtain the rational ranking results for all alternatives. Finally, a case analysis of emergency logistics under COVID-19 is provided to illustrate the validity and applicability of the proposed approach.

Revisiting the consistency improvement and consensus reaching processes of intuitionistic multiplicative preference relations

Revisiting the consistency improvement and consensus reaching processes of intuitionistic multiplicative preference relations

Group decision-making with interval multiplicative preference relations

Group decision-making with interval multiplicative preference relations

Consensus Reaching with Minimum Adjustment and Consistency Management in Group Decision Making with Intuitionistic Multiplicative Preference Relations

Consensus Reaching with Minimum Adjustment and Consistency Management in Group Decision Making with Intuitionistic Multiplicative Preference Relations

Consistency and Consensus Enhancing in Group Decision Making with Interval-Valued Intuitionistic Multiplicative Preference Relations Based on Bounded Confidence

Consistency and Consensus Enhancing in Group Decision Making with Interval-Valued Intuitionistic Multiplicative Preference Relations Based on Bounded Confidence

Multiplicative Consistency Ascertaining, Inconsistency Repairing, and Weights Derivation of Hesitant Multiplicative Preference Relations

This article investigates multiplicative consistency ascertaining, inconsistency repairing, and weights derivation for hesitant multiplicative preference relations (HMPRs). First, the completely multiplicative consistency and weakly multiplicative consistency of HMPRs are defined. Based on them, 0–1 mixed programming models and simple algebraic operations are proposed to ascertain the multiplicative consistency of HMPRs. Then, some goal programming models are developed to generate the weights from consistent HMPRs and to revise inconsistent HMPRs. An integrated procedure to manage the multiplicative consistencies of HMPRs is designed. The proposed methods are also extended to accommodate incomplete HMPRs, and to estimate missing values. Finally, some numerical examples, a comparative analysis with existent approaches, and a simulation analysis are included to illustrate the practicality and effectiveness of the developed models.

An Interactive Consensus Model in Group Decision Making with Heterogeneous Hesitant Preference Relations

This paper proposes an interactive consensus reaching model in the group decision making for heterogeneous hesitant preference relations (i.e., hesitant fuzzy preference relations, hesitant multiplicative preference relations, hesitant fuzzy linguistic preference relations). First, the consistencies of three hesitant preference relations are defined, respectively. Then, based on their definitions, three optimization models are constructed to obtain the weight vector of alternatives, based on which an interactive consensus adjustment algorithm is established based on the direct consensus framework. This framework adopts feedback mechanism to facilitate the information correction of decision makers. After several rounds of adjustment, the decision results with satisfactory consensus level are achieved. Finally, the practicability and effectiveness of the model are illustrated through a case study of mine accident emergency decision making.

A hesitant fuzzy multiplicative Base-criterion multi-criteria group decision making method

Hesitant fuzzy sets have a unique characteristic that its basic element could manifest the assessment values of different decision makers on the same option under a certain criterion. Base-criterion method is a very significant tool for calculating the weights of the criteria in multiple criteria decision-making. In this paper, we developed a novel approach hesitant fuzzy BCM based on hesitant fuzzy multiplicative preference relation for multiple criteria group decision making. The base-comparison of the preferential criterion relative to other criteria is expressed as linguistic terms, which might be converted into hesitant multiplicative elements (HMEs). HMEs are extended along the same length according to the attitude of the decision makers. Then normalized optimal hesitant fuzzy weights are calculated. The normalized optimal hesitant fuzzy weights of criteria may be transferred to crisp values by employing score function. To illustrate the applicability and suitability of hesitant fuzzy BCM, we analyse the optimal transportation mode selection problem and car selection problem under hesitant fuzzy environment. The outcomes of the proposed model indicate that the hesitant fuzzy BCM is highly consistent and can yield appreciable preference ranking of criteria and alternatives.

Two different approaches for consistency of intuitionistic multiplicative preference relation using directed graph

Consistency is an important issue that causes wide public concern of decision makers in the decision-making process. The lack of consistency in preference relations results in a vague solution. The main goal of this paper is to achieve the consistent intuitionistic multiplicative preference relation using a graphical approach. We have proposed two different approaches of the consistency for intuitionistic multiplicative preference relation (IMPR). In the first approach, we propose an algorithm to achieve the consistency of IMPR by using the cycles of various lengths in a directed graph. The second approach proves isomorphism between IMPRs and asymmetric multiplicative preference relations. The result explores the methodologies developed for asymmetric multiplicative preference relations in the case of IMPRs and achieves the consistency of asymmetric multiplicative preference relations using a directed graph. Sometimes the decision maker may not be able to provide the complete reference. So the above-said method is applied for an incomplete IMPR; consistency plays an important role here. The examples are provided to illustrate both the ways in all cases.

Deriving 2-ranks from multiplicative preference relations in a group context: An investigation with axiomatic design

Multiplicative preference relations (MPRs) are broadly employed in the Analytic Hierarchy Process (AHP) to support group decision making (GDM). In AHP-GDM, aggregation of individual judgments (AIJ) and aggregation of individual priorities (AIP) are frequently used to acquire a full ranking of alternatives. However, in practical AHP-GDM, alternatives sometimes need to be assigned to two preference-ordered classes. In this paper, we call it the 2-rank GDM problem with MPRs and investigate it through axiomatic design. We start by proposing six axioms: Anonymity for alternatives (AA), Anonymity for individuals (AI), Positive responsiveness (PR), Non-dictatorship (ND), Independence of irrelevant alternatives (IIA), and Power invariance (PI) to identify a “perfect” 2-rank approach. Next, an impossibility theorem is presented stating that no 2-rank approach satisfies six axioms concurrently due to the inherent conflict among them. Based on AIJ and AIP, five 2-rank approaches M I, AIJ-M II, AIP-M II, AIP-M III, and AIP-M IV are developed. We then analyze and compare them based on the proposed axioms and prove that M I satisfies all axioms except for IIA, while other approaches only satisfy AA, AI, PR, and ND. Finally, simulation experiments are designed to further explore and compare the properties of these five 2-rank approaches. From the axiomatic and simulation analyses, it is found that M I is the best approach for the 2-rank GDM problem with MPRs.

A new group decision-making method with incomplete intuitionistic multiplicative preference relations

PurposeConsistency and consensus are two important research issues in group decision-making (GDM). Considering some drawbacks associated with these two issues in existing GDM methods with intuitionistic multiplicative preference relations (IMPRs), a new GDM method with complete IMPRs (CIMPRs) and incomplete IMPRs (ICIMPRs) is proposed in this paper.Design/methodology/approachA mathematically programming model is constructed to judge the consistency of CIMPRs. For the unacceptably consistent CIMPRs, a consistency-driven optimization model is constructed to improve the consistency level. Meanwhile, a consistency-driven optimization model is constructed to supplement the missing values and improve the consistency level of the ICIMPRs. As to GDM with CIMPRs, first, a mathematically programming model is built to obtain the experts' weights, after that a consensus-driven optimization model is constructed to improve the consensus level of CIMPRs, and finally, the group priority weights of alternatives are obtained by an intuitionistic fuzzy programming model.FindingsThe case analysis of the international exchange doctoral student selection problem shows the effectiveness and applicability of this GDM method with CIMPRs and ICIMPRs.Originality/valueFirst, a novel consistency definition of CIMPRs is presented. Then, a consistency-driven optimization model is constructed, which supplements the missing values and improves the consistency level of ICIMPRs simultaneously. Therefore, this model greatly improves the efficiency of consistency improving. Experts' weights determination method considering the subjective and objective information is proposed. The priority weights of alternatives are determined by an intuitionistic fuzzy (IF) programming model considering the risk preference of experts, so the method determining priority weights is more flexible and agile. Based on the above theoretical basis, a new GDM method with CIMPRs and ICIMPRs is proposed in this paper.