Interval Multiplicative Preference Relations Research Articles

Group decision-making with interval multiplicative preference relations

Group decision-making with interval multiplicative preference relations

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.

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.

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 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.

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.

Multiplicative data envelopment analysis cross-efficiency and stochastic weight space acceptability analysis for group decision making with interval multiplicative preference relations

To deal with group decision making (GDM) with interval multiplicative preference relations (IMPRs), this paper proposes a novel method based on multiplicative data envelopment analysis (DEA) cross-efficiency and stochastic weight space acceptability analysis. We first develop a multiplicative DEA model to evaluate the relative efficiency of all alternatives of a given multiplicative preference relation (MPR). Then, we present a method, free from consistency adjustment, to derive a priority vector using the multiplicative DEA cross-efficiency with respect to the given MPR. For GDM with IMPRs, we consider the decision makers’ weights as a uniform distribution for acceptability analysis. A modified unacceptability index is further defined to measure the unlikeliness for a particular alternative in a particular rank. Finally, we develop an assignment problem model to achieve an optimal ranking by minimizing the total rank unacceptability, and to compute the expected priority vector of all alternatives. Numerical examples are provided to show the applicability and justifications of the proposed GDM method.

Consistency and optimized priority weight analytical solutions of interval multiplicative preference relations

This paper focuses on answering the following question: How to estimate the quality of off-diagonal judgments in an interval multiplicative preference relation (IMPR) and obtain an analytical solution to the optimized interval priority weight vector of an IMPR. By generalizing IMPRs, an interval-multiplication-based transitivity equation is devised to define perfect consistency of generalized IMPRs. A basic interval multiplicative weight (IMW) vector is introduced to characterize different interval weight vectors with equivalency and used as a priority weight benchmark of a generalized IMPR. Two logarithmic least square models are established for determining basic IMWs of a generalized IMPR, and their analytical solutions are found by using the Lagrangian multiplier method. The paper demonstrates that any IMPR with uncertainty has no perfect consistency in terms of interval arithmetic, and constructs a generalized IMPR from an IMPR to introduce consistency of off-diagonal judgments in an IMPR. Two computation formulas are developed to obtain optimized basic IMWs of an IMPR, which can entirely capture off-diagonal judgments if the IMPR is consistent. Numerical examples are offered to illustrate how to apply the proposed models and a comparative study is made to show that off-diagonal judgments in an inconsistent IMPR are properly captured by the analytical solution based IMWs.

Group decision making based on DEA cross-efficiency with intuitionistic fuzzy preference relations

The aim of this paper is to investigate a novel approach to group decision making based on DEA cross-efficiency with intuitionistic fuzzy preference relations, which can avoid information distortion and obtain more credible decision making results. An interval transform function is defined, which can transform an intuitionistic fuzzy preference relation into an interval multiplicative preference relation. Then, an interval transform function based data envelopment analysis model is developed to obtain the ranking vector of consistent intuitionistic fuzzy preference relation, in which each of the alternatives is viewed as a decision making unit. Moreover, for any intuitionistic fuzzy preference relations, we propose two DEA cross-efficiency models to get the cross-efficiency values of all alternatives, and we can calculate the normalized intuitionistic fuzzy priority weight vector of the intuitionistic fuzzy preference relation based on the cross-efficiency values. A goal programming model is investigated to derive the weight vector of decision makers. A step-by-step procedure for group decision making approach based on DEA cross-efficiency with intuitionistic fuzzy preference relations is presented. Finally, numerical examples are given to illustrate the validity and applicability of the proposed method. This is the first attempt of employing the DEA cross-efficiency to the group decision making with intuitionistic fuzzy preference relations.

Multiple Criteria Decision-Making in Heterogeneous Groups of Management Experts

In commercial organizations operations, frequently some dynamic events occur which involve operational, managerial, and valuable information aspects. Then, in order to make a sound decision, the business professional could be supported by a Multi Criteria Decision-Making (MCDM) system for taking an external course of action, as, for instance, forecasting a new market or product, up to an inner decision concerning for instance, the volume of manufacture. Thus, managers need, in a collective manner, to analyze the actual problems, to evaluate various options according to diverse criteria, and finally choose the best solution from a set of various alternatives. Throughout these processes, uncertainty and hesitancy easily arise, when it comes to define and judge criteria or alternatives. Several approaches have been introduced to allow Decision Makers (DMs) to deal with. The Interval Multiplicative Preference Relations (IMPRs) approach is a useful technique and the basis of our proposed methodology to provide reliable consistent and in consensus IMPRs. In this manner, DMs’ choices are implicitly including their uncertainty while maintaining both an acceptable individual consistency, as well as group consensus levels. The present method is based on some recent results and an optimization algorithm to derive reliable consistent and in consensus IMPRs. In order to illustrate our results and compare them with other methodologies, a few examples are addressed and solved.

A Reliable Method for Consistency Improving of Interval Multiplicative Preference Relations Expressed under Uncertainty

When a diverse group of experts share their knowledge to ascertain and solve a particular multiple criteria decision-making (MCDM) problem, uncertainty arises from several sources. In those cases, the interval multiplicative preference relation (IMPR) approach is a useful technique when verifying consistency. As illustrated in this paper, the validation and the improvement of consistency require robust analysis tools and algorithms. The proposed methodology provides reliable and consistent IMPR, which can be quantified in terms of row geometric mean method (RGMM) or the eigenvalue method (EM). In this manner, decision makers’ choices are implicitly including their uncertainty while maintaining acceptable consistency. The present approach is based on the Hadamard’s dissimilarity operator and through an algorithm, the derivation of a reliable and consistent IMPR is synthesized. In order to illustrate our results and compare them appropriately with other methodologies, a few examples are addressed and solved.

Interval cross efficiency for fully ranking decision making units using DEA/AHP approach

Data envelopment analysis (DEA) is a popular technique for measuring the relative efficiency of a set of decision making units (DMUs). Fully ranking DMUs is a traditional and important topic in DEA. In various types of ranking methods, cross efficiency method receives much attention from researchers because it evaluates DMUs by using self and peer evaluation. However, cross efficiency score is usual nonuniqueness. This paper combines the DEA and analytic hierarchy process (AHP) to fully rank the DMUs that considers all possible cross efficiencies of a DMU with respect to all the other DMUs. We firstly measure the interval cross efficiency of each DMU. Based on the interval cross efficiency, relative efficiency pairwise comparison between each pair of DMUs is used to construct interval multiplicative preference relations (IMPRs). To obtain the consistency ranking order, a method to derive consistent IMPRs is developed. After that, the full ranking order of DMUs from completely consistent IMPRs is derived. It is worth noting that our DEA/AHP approach not only avoids overestimation of DMUs’ efficiency by only self-evaluation, but also eliminates the subjectivity of pairwise comparison between DMUs in AHP. Finally, a real example is offered to illustrate the feasibility and practicality of the proposed procedure.

A new consistency concept for interval multiplicative preference relations

Consistency analysis is very important to ensure the reasonable ranking order. However, all previous consistency concepts for interval multiplicative preference relations (IMPRs) are insufficient to address this type of preference relations. This paper introduces a new consistency concept for IMPRs that is a natural extension of the Saaty’s consistency concept for multiplicative preference relations. Several desirable properties are discussed, and the relationship between the new concept and two previous ones is studied. Then, a method to judge the consistency of IMPRs is proposed. Considering the inconsistent case, a 0–1 mixed programming model to derive consistent IMPRs from inconsistent ones is established. To determine missing values in incomplete HFPRs, 0–1 mixed programming models are constructed that can address the situation where ignore objects exist. Meanwhile, illustrative examples are offered to show the feasibility and efficiency of the developed theoretical results, and comparison analysis is provided. Finally, a consistency analysis based algorithm to decision making with IMPRs is developed that can address inconsistent and incomplete cases.

A goal programming model of obtaining the priority weights from an interval preference relation

The main aim of this paper is to investigate the weight vectors of interval fuzzy preference relations and interval multiplicative preference relations. In view of the excellent properties of multiplicative transitivity in modeling the consistency of fuzzy preference relations, based on the multiplicative consistency property, a goal programming model of obtaining the priority weights from an interval fuzzy preference relation is introduced along with some of its desired properties. The priority vector of a fuzzy preference relation can also be derived using this goal programming model, where the optimal value of the objective function is always equal to zero. Numerical examples are provided to illustrate the validity of the proposed model.

Logarithmic least squares approaches to deriving interval weights, rectifying inconsistency and estimating missing values for interval multiplicative preference relations

The aim of this paper is to develop logarithmic least squares prioritization and completion methods for interval multiplicative preference relations. A parameterized transformation formula is proposed to convert a normalized interval weight vector into a consistent interval multiplicative preference relation. A logarithmic least squares model is established to derive a normalized interval weight vector from an interval multiplicative preference relation and construct the optimized consistent interval multiplicative preference relation. Subsequently, a logarithmic least squares model is built to rectify inconsistency for a complete interval multiplicative preference relation without consistency, and a logarithmic least squares completion model is developed to estimate missing values for an incomplete interval multiplicative preference relation. Several numerical examples are examined to illustrate the validity and applicability of the proposed methods, and comparisons with other existing methods are also made.

The optimal group continuous logarithm compatibility measure for interval multiplicative preference relations based on the COWGA operator

The calculation of compatibility measures is an important technique employed in group decision-making with interval multiplicative preference relations. In this paper, a new compatibility measure called the continuous logarithm compatibility, which considers risk attitudes in decision-making based on the continuous ordered weighted geometric averaging (COWGA) operator, is introduced. We also develop a group continuous compatibility model (GCC Model) by minimizing the group continuous logarithm compatibility measure between the synthetic interval multiplicative preference relation and the continuous characteristic preference relation. Furthermore, theoretical foundations are established for the proposed model, such as the sufficient and necessary conditions for the existence of an optimal solution, the conditions for the existence of a superior optimal solution and the conditions for the existence of redundant preference relations. In addition, we investigate certain conditions for which the optimal objective function of the GCC Model guarantees its efficiency as the number of decision-makers increases. Finally, practical illustrative examples are examined to demonstrate the model and compare it with previous methods.

A note on “A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making”

In a recently published paper by Liu et al. [Liu, F., Zhang, W.G., Wang, Z.X. (2012). A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making. European Journal of Operational Research 218, 747–754], two equations are introduced to define consistency of incomplete interval multiplicative preference relations (IMPRs) and employed to develop a goal programming model for estimating missing values. This note illustrates that such consistency definition and estimation model are technically incorrect. New transitivity conditions are proposed to define consistent IMPRs, and a two-stage goal programming approach is devised to estimate missing values for incomplete IMPRs.

A consistency model for group decision making problems with interval multiplicative preference relations

The main aim of this paper is to present a consistency model for interval multiplicative preference relation (IMPR). To measure the consistency level for IMPR, a referenced consistent IMPR of a given IMPR is defined, which has the minimum logarithmic distance from the given IMPR. Based on the referenced consistent IMPR, the consistency level of an IMPR can be measured and an IMPR with unacceptable consistency can be adjusted by a proposed algorithm such that the revised IMPR is of acceptable consistency. A consistency model for group decision making (GDM) problems with IMPRs is proposed to obtain the collective IMPR with highest consistency level. Numerical examples are provided to illustrate the validity of the proposed approaches in decision making.

An approach to incomplete multiplicative preference relations and its application in group decision making

To measure the consistency of multiplicative preference relations, a reasonable concept of consistent multiplicative preference relations is defined, and the multiplicative geometric consistent index (MGCI) is presented. In a simulation method, the average values of the MGCI are studied. Considering incomplete multiplicative preference relations, a consistency-based linear programming model is constructed, which is optimal from the point of view of consistency with respect to the known judgments. In a group decision making context, the group consensus index (GCOI) is given to measure the consensus of individual multiplicative preference relations. Then, the hybrid weighted geometric mean (HWGM) operator is defined to calculate the collective multiplicative preference relation. Based on the consistency and consensus analysis, an approach to group decision making with incomplete multiplicative preference relations is developed. Furthermore, we briefly research the application of the new method to incomplete interval multiplicative preference relations and incomplete fuzzy preference relations. Meanwhile, the associated examples are offered to show the practicability and efficiency of the proposed procedure.