Multiplicative Preference Relations Research Articles

Performance measurement of decision making units through interval efficiency with slacks-based measure: an application to tourist hotels in Taipei

Data envelopment analysis (DEA) is widely used to evaluate the performance of a group of homogeneous decision making units (DMUs). Considering the uncertainty, interval DEA has been introduced to fit into more situations. In this paper, an interval efficiency method based on slacks-based measure is proposed to solve the uncertain problems in DEA. Firstly, the maximum and minimum efficiency values of the evaluated DMU are calculated by the furthest and closest distance from the evaluated DMU to the projection points on the Pareto-efficient frontier, respectively. Then, the AHP method is used for the full ranking of DMUs. The paper uses the pairwise comparison relationship between each pair of DMUs to construct the interval multiplicative preference relations (IMPRs) matrix. If the matrix does not meet the consistency condition, a method to obtain consistency IMPRs is introduced. According to the consistency judgment matrix, the full ranking of DMUs can be obtained. Finally, we apply our method to the performance evaluation of 12 tourist hotels in Taipei in 2019.

Consensus measure for interval multiplicative preference relations from a probability distribution perspective

Interval multiplicative preference relations are an extension of multiplicative preference relations, which are commonly used by experts to express their uncertain preference information about objects in group decision making. This paper proposes a new consensus measure of interval multiplicative preference relations in the group decision making from a probability distribution perspective. First, the collective interval multiplicative preference relation is obtained by assuming that the interval obeys the lognormal distribution. Then, a new consensus measure of interval multiplicative preference relations is proposed. Finally, a numerical example is given to illustrate the proposed approach.

SVNMPR: A new single‐valued neutrosophic multiplicative preference relation and their application to decision‐making process

The aim of the paper is to present the concept of a multiplicative preference relation (MPR) with the features of the single-valued neutrosophic (SVN) set and named as SVN multiplicative preference relation (SVNMPR). The SVN set (SVNS) handles uncertainties more broadly than the intuitionistic fuzzy set by considering the three independent degree systems with a scale rating of 0–1 and is symmetrical about 0.5. However, for asymmetrical distribution, Saaty discusses the 1–9 scale to display the information. Driven by the advantages of the scale of 1–9 and the independence nature of degrees in SVNS, in this paper, we introduce the concept of SVNMPR with ratings values as SVN numbers with range 1 ∕ q − q with q > 1. The main advantages of the proposed relation are the asymmetric and nonuniform distributions of about 1. To complete address this, we divide the stated work into three parts. First, an idea of SVNMPR is added wherein pairwise evaluation values are represented the usage of SVN numbers over a scale of 1 ∕ q − q with q > 1. Also, we outline the idea of the SVN multiplicative preference set (SVNMPS). Second, to rank the proposed SVNMPS, we define a new ranking method with improved score functions by adding the expert's attitude to the analysis. The relationship between the proposed and the existing functions is also derived. Third, we are defining some new operation laws for SVNMPS to preserve the closeness property. On the basis of these laws, we define a number of operators to aggregate the different pairs of the SVNMPSs and to derive a number of relations and inequalities. Finally, a group decision-making approach is established and the practicality of it is demonstrated with a numerical example and compared with a number of existing approaches.

Eigenproblem driven triangular fuzzy analytic hierarchy process

The eigenproblem plays a crucial role on checking acceptability of decision-makers’ pairwise comparison results and deriving priorities from preference relation matrices in the analytic hierarchy process (AHP). This paper analyzes two recent fuzzy eigenvector methods and illustrates their deficiencies. Two frameworks of additively normalized triangular fuzzy priorities (ANTFPs) are introduced to characterize a cluster of equivalent fuzzy priority vectors. Based on the approximation relationship between triangular fuzzy multiplicative preference relation matrices and their most appropriate ANTFPs, three eigenproblems with positive real matrices are established and an eigenvector based linear program is developed to obtain support interval based ANTFPs from fuzzy multiplicative preference relation matrices. The largest eigenvalue weighting consistency index and consistency ratio are defined and an acceptability checking method is then proposed by both examining acceptable consistency of fuzzy multiplicative preference relation matrices and acceptable vagueness of eigenvector based ANTFPs. Afterwards, an eigenproblem driven triangular fuzzy AHP is devised in detail. A numerical illustration including four fuzzy multiplicative preference relation matrices is provided and a comparative study is carried out to demonstrate the superiority and effectiveness of the presented models. Meanwhile, a multi-criteria graduate job selection problem is used to show the application of the proposed triangular fuzzy AHP.

A decision making model based on intuitionistic multiplicative preference relations with approximate consistency

A decision making model based on intuitionistic multiplicative preference relations with approximate consistency

Probabilistic Linguistic Group Decision-Making Method Based on Attribute Decision and Multiplicative Preference Relations

Probabilistic Linguistic Group Decision-Making Method Based on Attribute Decision and Multiplicative Preference Relations

A decision-making framework for evaluating appropriate business blockchain platforms using multiple preference formats and VIKOR

• Blockchain platform selection problem is evaluated with a hybrid approach. • Group decision making with multiple preference relations is used. • VIKOR method is combined in the evaluation process. • A business application is provided to validate the proposed approach. • It is novel in the blockchain research area with its criteria and method combination. The blockchain is a shared, reliable, public transaction ledger that can be programmed for virtually anything valuable. Blockchains in terms of permissions and transactions with private ledgers are relatively new. The goal of this study is to present a decision-making process to guide organizations in the evaluation of blockchains as an enterprise platform service. Enterprise platforms can be selected by decision makers who voice their opinions in diverse forms by considering a range of criteria. Multiple preference formats offer suitable methods for evaluating such criteria. In this study, the VIKOR (Vlsekriterijumska Optimizacija I Kompromisno Resenje) technique is employed for evaluating the alternatives and solving the selection problem. It contributes to the blockchain literature by combining multiple preference formats with VIKOR for the first time for the blockchain platform selection problem. The findings suggest that the proposed evaluation framework is powerful for selecting the most suitable enterprise blockchain platform among alternatives.

Decision making with incomplete interval multiplicative preference relations based on stochastic program and interval category

• The risk attitude of DM is introduced to define a new consistency index of IMPR. • The most pessimistic and optimistic acceptably consistent IMPRs are obtained. • A stochastic programming model is built to generate the priority weights. • A 0–1 mixed integer program is established to determine the DMs’ weights. • Derive the adjusted DMs’ weights with the group consensus and category information. Interval multiplicative preference relations (IMPRs) have been widely used in decision making for their ability to efficiently express the uncertainty of information. This paper investigates the decision making with incomplete IMPRs. First, a new consistency index of IMPR is defined. By minimizing the consistency index, the missing values in an incomplete IMPR can be estimated. Subsequently, considering the risk attitude of decision-makers (DMs), two optimization models are constructed to obtain the most pessimistic and optimistic acceptably multiplicative consistent IMPRs, respectively. Based on the triangular distribution of intervals, a stochastic programming model is built to derive the priority weights from an acceptably multiplicative consistent IMPR. Thus, a new method is put forward for individual decision making with an incomplete IMPR. To reach maximum group support degree, a 0–1 mixed integer programming model is established to determine DMs’ weights for group decision making with IMPRs. Considering the category information of the individual intervals and collective intervals, the adjusted DMs’ weights are defined. The individual IMPRs are integrated into the collective IMPR. The ranking of alternatives is generated by the collective IMPR. Two real-life examples are demonstrated to validate the proposed methods. A decision support system based on the proposed methods is designed.

A geometrical method for consensus building in GDM with incomplete heterogeneous preference information

In real-life group decision-making (GDM) problems, the preferences given by decision-makers(DMs) are often incomplete, because the complexity of decision-making problems and the limitation of knowledge of DM make it difficult for DMs to take a determined evaluation of alternatives. In addition, preference relations provided by DMs are often heterogeneous because they always have different decision habits and hobbies. However, the consensus method for GDM under incomplete heterogeneous preference relations is rarely studied. For four common preference relations: utility values, preference orderings, and (incomplete) multiplicative preference relations and (incomplete) fuzzy preference relations, this paper proposes a geometrical method for consensus building in GDM. Specifically, we integrate incomplete heterogeneous preference structures using a similarity-based optimization model and set a corresponding geometrical consensus measurement. Then, preference modification and weighting processes are proposed to improve consensus degree. Finally, we conduct a comparison analysis based on a qualitative analysis and algorithm complexity analysis of existing consensus reaching methods. Numerical analyses and convergence tests show that our method can promote the improvement of the consensus degree in GDM, and has less time complexity than the previous methods. The proposed geometrical method is a more explainable model due to operability and simplicity.

Multimethod to prioritize projects evaluated in different formats

The prioritization of Research, Development & Innovation Projects is an essential step in the innovation management process. As a rule, it is carried out applying methods that allow one to process experts' preferences concerning each project according to established criteria. However, there are different preference formats which experts can utilize: Ordering of Alternatives, Utility Values, Multiplicative Preference Relations, Fuzzy Estimates, Fuzzy Preference Relations, etc. Wherein, each prioritization method usually handles only one of these formats. Thus, the following question arises: how do we prioritize projects taken from portfolios evaluated in different formats? The proposed methodology presents a way to overcome this gap by achieving three main objectives. First, develop techniques that make it possible to crossover between preference formats and prioritization methods. Second, merge two portfolios of projects built applying different prioritization methods. Third, prioritize projects evaluated using different formats. The results of this study are universal and can be applied to replace any method of prioritization. In the specific case, the Mapping method is replaced by the Analytic Hierarchy Process and, then, by the Interactive Multicriteria Decision Making method (so called TODIM method). Techniques are also proposed to ensure compatibility between different preference formats and prioritization methods.•Prioritization of projects evaluated in different formats using the Mapping, AHP, and TODIM methods.•Providing fully consistent evaluation matrices.•Application of techniques to make different preference formats and prioritization methods compatible.

Expected consistency-based model and multiplicative DEA cross-efficiency for group decision-making with incomplete distribution linguistic preference relations

Faced with complex decision problems, distribution linguistic preference relation (DLPR) is an effective way for decision-makers (DMs) to express preference information. However, due to the complexity of the decision-making environment, DMs may not be able to provide complete linguistic distribution for all linguistic terms in DLPRs, which results in incomplete DLPRs. Therefore, in order to solve group decision-making (GDM) with incomplete DLPRs, this paper proposes expected consistency-based model and multiplicative DEA cross-efficiency. For a given incomplete DLPRs, we first propose an optimization model to obtain complete DLPR. This optimization model can evaluate the missing linguistic distribution and ensure that the obtained DLPR has a high consistency level. And then, we develop a transformation function that can transform DLPRs into multiplicative preference relations (MPRs). Furthermore, we design an improved multiplicative DEA model to obtain the priority vector of MPR for ranking all alternatives. Finally, a numerical example is provided to show the rationality and applicability of the proposed GDM method.

A group decision making method considering both the consistency and consensus of intuitionistic multiplicative preference relations

This paper aims to develop a new group decision making (GDM) approach with intuitionistic multiplicative preference relations (IMPRs) by considering the consistency and consensus. Using the distance between a given IMPR and its corresponding underlying consistent IMPR, the concept of acceptably consistent IMPR is introduced, then an automatic algorithm is designed to repair the inconsistent IMPR to be of acceptable consistency. Meanwhile, each decision maker's consensus level is evaluated by the deviation between his/her individual IMPR and the group IMPR, and another algorithm for reaching acceptable level of consensus is provided. Moreover, the consensus improving process can guarantee that the modified IMPRs still be acceptably consistent, then the normalized intuitionistic multiplicative priority weight vector can be obtained from a mathematical programming model. A step-by-step algorithm based on the consistency and consensus of IMPRs is offered. Finally, two examples and the corresponding comparative analyses are presented to demonstrate the effectiveness of the proposed method.

Geometric consistency of triangular fuzzy multiplicative preference relation and its application to group decision making

The triangular fuzzy multiplicative preference relation (TFMPR) has attracted the attention of many scholars. This paper investigates the geometric consistency of TFMPR and applies it to group decision making (GDM). Firstly, by introducing two parameters, a triangular fuzzy number is transformed into an interval. According to the geometric consistency of interval multiplicative preference relation (IMPR), the geometric consistency of TFMPR is defined. Then, two corresponding IMPRs are extracted from the TFMPR by programming models in the majority case and minority case, respectively. Using the constructed linear programming models, two interval priority weight vectors are obtained from the two extracted IMPRs, respectively. Combining two interval priority weight vectors, a linear programming model is established to derive the triangular fuzzy priority weights. Subsequently, the closeness degrees of alternatives by experts are defined to obtain the group utility indices and individual regret indices of alternatives. Then, the compromise indices of alternatives are calculated considering experts’ compromise attitude. By minimizing the compromise indices of alternatives, a multi-objective programming model is constructed to obtain experts’ weights. By aggregating the individual TFMPRs, the collective TFMPR is obtained to derive the triangular fuzzy priority weights. Using the arithmetic mean values, the ranking order of alternatives is generated. Therefore, a method is proposed to solve GDM with TFMPRs. Finally, a performance evaluation example of precise poverty alleviation is provided to illustrate the advantage of the proposed method.

A novel method to derive the intuitionistic multiplicative priority vector for the intuitionistic multiplicative preference relation

A novel method to derive the intuitionistic multiplicative priority vector for the intuitionistic multiplicative preference relation

L-R geometric consistency definition of triangular multiplicative preference relation in group decision making

Triangular multiplicative preference relation (TMPR) is an efficient technology which can make experts to comfortably express their paired comparisons. Existing consistency definitions of TMPR overlook experts' trust levels on their judgments. Considering experts' trust levels, this paper proposes a left and right (L-R) geometric consistency definition of TMPR and develops a new group decision-making (GDM) method. The L-R geometric means of triangular fuzzy number (TFN) are defined. Then, L-R geometric consistency of TMPR is given involving experts' trust levels about their judgments. To improve the consistency, a programming model is constructed to derive an acceptably consistent TMPR from an unacceptably consistent one. Using the difference degree between two TMPRs, a new approach is presented to extract experts' weights. Two algorithms are established for the methods of individual decision-making (IDM) with a TMPR and GDM with TMPRs. A simulation algorithm is devised to verify the superiority of the proposed IDM with a TMPR. Simulation results reveal that the proposed IDM method outperforms existing methods in logarithmic Hamming distance and difference degree. L-R geometric consistency makes contribution to TMPR since it considers experts' trust levels and has reciprocity, invariance and robustness. Several real-life examples are furnished to explain the validity of two proposed algorithms.

A new consistency definition of interval multiplicative preference relation

This paper presents a new consistency definition of interval multiplicative preference relation (IMPR). Several desirable properties of this new consistency are fully discussed, and the sufficient and necessary conditions of the new consistency definition are also provided. It is proven that this new consistency of IMPR has robustness and invariance with respect to permutation of the compared objects' labels. Moreover, comparative analyses are conducted to reveal the relationships among five existing consistency definitions and this new consistency definition of IMPR. Based on the new consistency definition of IMPR, a new consistency index is introduced to measure the consistency degree of IMPR. An iterative algorithm is proposed to improve the consistency level of an inconsistent IMPR. Subsequently, a goal programming model is built to derive an interval priority vector from an acceptably consistent IMPR. Eventually, a new individual decision making method with an IMPR is put forward. Numerical examples and simulation analyses are conducted to illustrate the superiority and validity of the proposed individual decision making method.

Generalized hesitant multiplicative preference relations and the analytic risk-network process

The hesitant fuzzy preference relation (HFPR) and the hesitant multiplicative preference relation (HMPR) are emerging techniques in decision-making research field. Clearly, an intuitionistic idea to improve the above preference relations is to synthesize the HFPR and the HMPR, which is also a primary contribution of this paper. Based on which, we propose the generalized HMPR (GHMPR) with full desired properties using transformation formulas that combine HFPR with HMPR and convert the HMPR into the GHMPR. Then, the GHMPR not only considers different risk appetites of a decision maker, but also preserves the advantages of both. Moreover, we design the corresponding consistency check algorithms, weight calculation methods and convergence algorithms for the GHMPR. Based on these new methods, we further propose a qualitative decision-making method under the GHMPR environment named as the analytic risk-network process (ARNP). Saaty’s well-known analytic network process case is then utilized as an illustrated example to verify the feasibility of the proposed methods via comparisons. The simulation results show our proposed methods are effective and useful.

A Novel Triangular Fuzzy Analytic Hierarchy Process

Triangular fuzzy multiplicative preference relation (TFMPR) is a widely used preference representation framework in the fuzzy analytic hierarchy process (FAHP). In this article, we develop a triangular fuzzy multiplication-based equation to characterize transitivity among original fuzzy assessments in a consistent TFMPR. A geometric consistency index is then proposed to measure the inconsistency of TFMPRs. By capturing row fuzziness proportionalities and the difference between the row increasing and decreasing part fuzziness indices, this article establishes two logarithmic least square models to find normalized fuzzy weights from two kinds of TFMPRs. The two logarithmic least square models are further integrated into one whose analytic solution is found by the method of Lagrange multipliers. A novel method is put forward to check the acceptability of TFMPRs by both examining acceptable consistency and acceptable fuzziness. This article devises a parametric defuzzification approach to obtain the real-valued weights of criteria for aggregating the local normalized fuzzy weights into the global fuzzy weights in the triangular FAHP. A comparative analysis of the proposed model with existing fuzzy eigenvector methods and fuzzy average methods is carried out by a numerical example with four TFMPRs to clarify its validity and merits. An outstanding teacher award selection problem is provided to show the practicality of the proposed triangular FAHP.

Optimal group selection model for large-scale group decision making

Research on large-scale group decision making (LSGDM) has generally focused on clustering and reaching a consensus. Often, participants (or decision makers) are unwilling to share their consensus or revise their preferences, which leads to the loss of time and money and even reduces the efficiency of decision making. Accordingly, in LSGDM, an optimal group derived from a large number of participants could improve the decision-making efficiency and reduce time consumption. In this paper, we investigate a model to select the optimal group for LSGDM with multiplicative preference relations. Notably, for individual and group preference relations, we define an individual logarithmic square compatibility measure and a group logarithmic square compatibility measure, respectively. Some properties associated with the individual logarithmic square compatibility measure and group logarithmic square compatibility measure are addressed. For a large number of preferences in LSGDM, the redundant preferences and optimal group are distinguished based on an optimal group selection model using the optimal participant weights obtained by minimizing the group logarithmic square compatibility model. Then, we demonstrate that the model is a convex quadratic programming problem. Moreover, the conditions for the existence of an optimal solution and the conditions for redundant preference relations in the group logarithmic square compatibility model are provided. In this context, the proposed model is verified to effectively identify the optimal group.

A comparative study for consistency-based decision making with interval multiplicative preference relations

ABSTRACTTo ensure the reasonable application and perfect the theory of decision making with interval multiplicative preference relations (IMPRs), this paper continues to discuss decision making with IMPRs. After reviewing previous consistency concepts for IMPRs, we find that Krejčí’s consistency concept is more flexible and natural than others. However, it is insufficient to address IMPRs only using this concept. Considering this fact, this paper researches inconsistent and incomplete IMPRs that are usually encountered. First, programming models for addressing inconsistent and incomplete IMPRs are constructed. Then, this paper studies the consensus of individual IMPRs and defines a consensus index using the defined correlation coefficient. When the consensus requirement does not satisfy requirement, a programming model for improving consensus level is built, which can ensure the consistency. Subsequently, a procedure for group decision making with IMPRs is offered, and associated examples are provided to specifically show the application of main theoretical results.