基于改进向量相似度的区间数动态多指标决策模型及应用

针对决策信息为区间数的不确定性动态决策问题,在属性权重和时间权重未知的情况下,基于改进向量相似度的方法,构建一种兼顾决策信息和决策偏好的动态多指标决策模型.利用区间型决策信息的相对相似性和属性重要度,构造相对相似度最小规划模型以确定指标权重;在综合考虑决策信息时间价值、决策者偏好的基础上,构建极大熵模型以确定时间权重;结合向量相似度计算存在的缺陷,提出一种基于向量投影思想的向量综合相似度测度方法,从而建立不确性动态决策模型,并通过实例分析检验该模型的合理性和有效性.     

MAGDM Linear-Programming Models With Distinct Uncertain Preference Structures

Group decision making with preference information on alternatives is an interesting and important research topic which has been receiving more and more attention in recent years. The purpose of this paper is to investigate multiple-attribute group decision-making (MAGDM) problems with distinct uncertain preference structures. We develop some linear-programming models for dealing with the MAGDM problems, where the information about attribute weights is incomplete, and the decision makers have their preferences on alternatives. The provided preference information can be represented in the following three distinct uncertain preference structures: 1) interval utility values; 2) interval fuzzy preference relations; and 3) interval multiplicative preference relations. We first establish some linear-programming models based on decision matrix and each of the distinct uncertain preference structures and, then, develop some linear-programming models to integrate all three structures of subjective uncertain preference information provided by the decision makers and the objective information depicted in the decision matrix. Furthermore, we propose a simple and straightforward approach in ranking and selecting the given alternatives. It is worth pointing out that the developed models can also be used to deal with the situations where the three distinct uncertain preference structures are reduced to the traditional ones, i.e., utility values, fuzzy preference relations, and multiplicative preference relations. Finally, we use a practical example to illustrate in detail the calculation process of the developed approach.      

An approach to group decision making with heterogeneous incomplete uncertain preference relations

For practical group decision making problems, decision makers tend to provide heterogeneous uncertain preference relations due to the uncertainty of the decision environment and the difference of cultures and education backgrounds. Sometimes, decision makers may not have an in-depth knowledge of the problem to be solved and provide incomplete preference relations. In this paper, we focus on group decision making (GDM) problems with heterogeneous incomplete uncertain preference relations, including uncertain multiplicative preference relations, uncertain fuzzy preference relations, uncertain linguistic preference relations and intuitionistic fuzzy preference relations. To deal with such GDM problems, a decision analysis method is proposed. Based on the multiplicative consistency of uncertain preference relations, a bi-objective optimization model which aims to maximize both the group consensus and the individual consistency of each decision maker is established. By solving the optimization model, the priority weights of alternatives can be obtained. Finally, some illustrative examples are used to show the feasibility and effectiveness of the proposed method.     

An integrated probabilistic linguistic projection method for MCGDM based on ELECTRE III and the weighted convex median voting rule

In the multi-criteria group decision-making (MCGDM) problems with great uncertainty, making full use of participants' evaluation information could help improve the accuracy and reliability of decision results. Probabilistic linguistic term set (PLTS) is an effective tool to represent qualitative data and can fully express the hesitation and preference of decision makers. Therefore, this paper aims to propose an MCGDM method based on PLTSs. In the proposed method, the projection of PLTSs is explored to measure the distance and angle differences between two objects, and Bayesian best–worst method (Bayesian BWM) is used to determine the aggregated final weights of criteria. Besides, the elimination and choice translating reality III (ELECTRE III) method combined with distillation algorithm deals with the projection of PLTSs to obtain the alternatives' ranking of each decision maker. Then, the weighted convex median voting rule is developed to integrate the rankings results regarding all decision makers, which can solve the conflict of ranking results among experts and ensure that the comprehensive ranking results are reasonable and practical. Finally, a case study of health-care waste management is designed and comparative analyses are implemented to show the effectiveness and advantages of the proposed method.     

A new method of obtaining the priority weights from an interval fuzzy preference relation

In analyzing a multiple criteria decision-making problem, the decision maker may express her/his opinions as an interval fuzzy or multiplicative preference relation. Then it is an interesting and important issue to investigate the consistency of the preference relations and obtain the reliable priority weights. In this paper, a new consistent interval fuzzy preference relation is defined, and the corresponding properties are derived. The transformation formulae between interval fuzzy and multiplicative preference relations are further given, which show that two preference relations, consistent interval fuzzy and multiplicative preference relations, can be transformed into each other. Based on the transformation formula, the definition of acceptably consistent interval fuzzy preference relation is given. Furthermore a new algorithm for obtaining the priority weights from consistent or inconsistent interval fuzzy preference relations is presented. Finally, three numerical examples are carried out to compare the results using the proposed method with those using other existing procedures. The numerical results show that the given procedure is feasible, effective and not requisite to solve any mathematical programing.     

A PROMETHEE-based outranking method for multiple criteria decision analysis with interval type-2 fuzzy sets

This paper develops new methods based on the preference ranking organization method for enrichment evaluations (PROMETHEE) that use a signed distance-based approach within the environment of interval type-2 fuzzy sets for multiple criteria decision analysis. The theory of interval type-2 fuzzy sets provides an intuitive and computationally feasible way of addressing uncertain and ambiguous information in decision-making fields. Many studies have developed multiple criteria decision analysis methods in the context of interval type-2 fuzzy sets; most of these methods can be characterized as scoring or compromising models. Nevertheless, the extended versions of outranking methods have not been thoroughly investigated. This paper establishes interval type-2 fuzzy PROMETHEE methods for ranking alternative actions among multiple criteria based on the concepts of signed distance-based generalized criteria and comprehensive preference indices. We develop interval type-2 fuzzy PROMETHEE I and interval type-2 fuzzy PROMETHEE II procedures for partial and complete ranking, respectively, of the alternatives. Finally, the feasibility and applicability of the proposed methods are illustrated by a practical problem of landfill site selection. A comparative analysis is also performed with ordinary fuzzy PROMETHEE methods to validate the effectiveness of the proposed methodology.     

A simulated annealing-based permutation method and experimental analysis for multiple criteria decision analysis with interval type-2 fuzzy sets

The aim of this paper is to develop a simulated annealing-based permutation method for multiple criteria decision analysis within the environment of interval type-2 fuzzy sets. The outranking methodology constitutes one of the most fruitful approaches in multiple criteria decision making and has been applied in numerous real-world problems. The permutation method is a classical outranking model, which generalizes Jacquet–Lagreze's permutation method and is based on a pairwise criterion comparison of the alternatives. Because modeling of the uncertainty in the decision-making process becomes increasingly important, an extension to the interval type-2 fuzzy environment is a useful generalization of the permutation method and is appropriate for handling uncertain and imprecise information in practical decision-making situations. This paper produces a signed-distance-based comparison among the comprehensive rankings of alternatives for concordance and discordance analyses. An integrated nonlinear programming model is constructed for estimation of the criterion weights and the optimal ranking order of the alternatives under incomplete preference information. To enhance the implementation efficiency, a simulated annealing-based permutation method and its meta-heuristic algorithm are developed to produce a polynomial time solution for the total completion time problem. Furthermore, computational experiments with notably large amounts of simulation data are conducted to test the solution approach and validate the correctness of the approximate solution compared with the optimal all-permutation-based result.     

Distance‐based multicriteria group decision‐making approach with probabilistic linguistic term sets

Probabilistic linguistic term sets (PLTSs) are an important expression for hesitant linguistic preference information under group decision‐making circumstances. This study investigates problems of multicriteria group decision making (MCGDM) with PLTSs. A novel and rational comparison method is first proposed, and two distance measures for PLTSs are defined. The weight of each criterion is then obtained via maximum deviation method. Subsequently, extended Techniques for Order Preference by Similarity to Ideal Solution (TOPSIS) ‐ VIseKriterijumska Optimizacija I Kompromisno Resenje a Serbian name (VIKOR) and TODIM (an acronym in Portuguese of interactive and multiple attribute decision making) methods are developed as decision support models to handle MCGDM problems. An illustrative example is also analysed to demonstrate the rationality and feasibility of the proposed methods.     

Decider: A fuzzy multi-criteria group decision support system

Multi-criteria group decision making (MCGDM) aims to support preference-based decision over the available alternatives that are characterized by multiple criteria in a group. To increase the level of overall satisfaction for the final decision across the group and deal with uncertainty in decision process, a fuzzy MCGDM process (FMP) model is established in this study. This FMP model can also aggregate both subjective and objective information under multi-level hierarchies of criteria and evaluators. Based on the FMP model, a fuzzy MCGDM decision support system (called Decider) is developed, which can handle information expressed in linguistic terms, boolean values, as well as numeric values to assess and rank a set of alternatives within a group of decision makers. Real applications indicate that the presented FMP model and the Decider  software are able to effectively handle fuzziness in both subjective and objective information and support group decision-making under multi-level criteria with a higher level of satisfaction by decision makers.     

Consistency and consensus models for group decision-making with uncertain 2-tuple linguistic preference relations

Due to the uncertainty of the decision environment and the lack of knowledge, decision-makers may use uncertain linguistic preference relations to express their preferences over alternatives and criteria. For group decision-making problems with preference relations, it is important to consider the individual consistency and the group consensus before aggregating the preference information. In this paper, consistency and consensus models for group decision-making with uncertain 2-tuple linguistic preference relations (U2TLPRs) are investigated. First of all, a formula which can construct a consistent U2TLPR from the original preference relation is presented. Based on the consistent preference relation, the individual consistency index for a U2TLPR is defined. An iterative algorithm is then developed to improve the individual consistency of a U2TLPR. To help decision-makers reach consensus in group decision-making under uncertain linguistic environment, the individual consensus and group consensus indices for group decision-making with U2TLPRs are defined. Based on the two indices, an algorithm for consensus reaching in group decision-making with U2TLPRs is also developed. Finally, two examples are provided to illustrate the effectiveness of the proposed algorithms.     

Group decision making based on multiple types of linguistic preference relations

In this paper, we investigate group decision making problems with multiple types of linguistic preference relations. The paper has two parts with similar structures. In the first part, we transform the uncertain additive linguistic preference relations into the expected additive linguistic preference relations, and present a procedure for group decision making based on multiple types of additive linguistic preference relations. By using the deviation measures between additive linguistic preference relations, we give some straightforward formulas to determine the weights of decision makers, and propose a method to reach consensus among the individual preferences and the group’s opinion. In the second part, we extend the above results to group decision making based on multiple types of multiplicative linguistic preference relations, and finally, a practical example is given to illustrate the application of the results.     

Multi-criteria group decision making based on trapezoidal intuitionistic fuzzy information

With respect to multi-criteria group decision making (MCGDM) problems under trapezoidal intuitionistic fuzzy environment, a new MCGDM method is investigated. The proposed method can effectively avoid the failure caused by the use of inconsistent decision information and provides a decision-making idea for the case of “the truth be held in minority”. It consists of three interrelated modules: weight determining mechanism, group consistency analysis, and ranking and selection procedure. For the first module, distance measures, expected values and arithmetic averaging operator for trapezoidal intuitionistic fuzzy numbers are used to determine the weight values of criteria and decision makers. For the second module, a consistency analysis and correction procedure based on trapezoidal intuitionistic fuzzy weighted averaging operator and OWA operator is developed to reduce the influence of conflicting opinions prior to the ranking process. For the third module, a trapezoidal intuitionistic fuzzy TOPSIS is used for ranking and selection. Then a procedure for the proposed MCGDM method is developed. Finally, a numerical example further illustrates the practicality and efficiency of the proposed method.     

Multiple-attribute group decision making with different formats of preference information on attributes.

Interval utility values, interval fuzzy preference relations, and interval multiplicative preference relations are three common uncertain-preference formats used by decision-makers to provide their preference information in the process of decision making under fuzziness. This paper is devoted in investigating multiple-attribute group-decision-making problems where the attribute values are not precisely known but the value ranges can be obtained, and the decision-makers provide their preference information over attributes by three different uncertain-preference formats i.e., 1) interval utility values; 2) interval fuzzy preference relations; and 3) interval multiplicative preference relations. We first utilize some functions to normalize the uncertain decision matrix and then transform it into an expected decision matrix. We establish a goal-programming model to integrate the expected decision matrix and all three different uncertain-preference formats from which the attribute weights and the overall attribute values of alternatives can be obtained. Then, we use the derived overall attribute values to get the ranking of the given alternatives and to select the best one(s). The model not only can reflect both the subjective considerations of all decision-makers and the objective information but also can avoid losing and distorting the given objective and subjective decision information in the process of information integration. Furthermore, we establish some models to solve the multiple-attribute group-decision-making problems with three different preference formats: 1) utility values; 2) fuzzy preference relations; and 3) multiplicative preference relations. Finally, we illustrate the applicability and effectiveness of the developed models with two practical examples.     

Interval-valued intuitionistic fuzzy mathematical programming method for hybrid multi-criteria group decision making with interval-valued intuitionistic fuzzy truth degrees

As an important component of group decision making, the hybrid multi-criteria group decision making (MCGDM) is very complex and interesting in real applications. The purpose of this paper is to develop a novel interval-valued intuitionistic fuzzy (IVIF) mathematical programming method for hybrid MCGDM considering alternative comparisons with hesitancy degrees. The subjective preference relations between alternatives given by each decision maker (DM) are formulated as an IVIF set (IVIFS). The IVIFSs, intuitionistic fuzzy sets (IFSs), trapezoidal fuzzy numbers (TrFNs), linguistic variables, intervals and real numbers are used to represent the multiple types of criteria values. The information of criteria weights is incomplete. The IVIFS-type consistency and inconsistency indices are defined through considering the fuzzy positive and negative ideal solutions simultaneously. To determine the criteria weights, we construct a novel bi-objective IVIF mathematical programming of minimizing the inconsistency index and meanwhile maximizing the consistency index, which is solved by the technically developed linear goal programming approach. The individual ranking order of alternatives furnished by each DM is subsequently obtained according to the comprehensive relative closeness degrees of alternatives to the fuzzy positive ideal solution. The collective ranking order of alternatives is derived through establishing a new multi-objective assignment model. A real example of critical infrastructure evaluation is provided to demonstrate the applicability and effectiveness of this method.     

犹豫不确定离散语言Z-numbers及其基于离散T模的多准则群决策问题的应用

介绍基于Z-numbers和语言模型的犹豫不确定离散语言Z-numbers(HUDLZNs).HUDLZNs能够有效地描述决策信息的复杂性和不确定性,并能较好地反映出决策者的犹豫性.在此基础上,提出一种基于离散T模融合和正理想方案的多准则群决策方法.首先,借助语言尺度函数来处理语言信息,并定义HUDLZNs间的距离、$\lambda$ 截集和$C^λ截集;其次,提出基于HUDLZNs的离散T模融合;再次,结合离散T模融合和语言尺度函数的优点提出一种HUDLZNs的多准则群决策方法;最后,用ERP系统选型的实例进行阐明,并通过灵敏度分析和已有方法的比较进一步表明所提出方法的有效性和可行性.     

Pythagorean Fuzzy LINMAP Method Based on the Entropy Theory for Railway Project Investment Decision Making

The uncertainty and complexity of the decision‐making environment and the subjectivity of the decision makers will lead to the inevitable errors of the decision‐making data. A poor decision will be produced with those errors, whereas the linear programming technique for multidimensional analysis of preference (LINMAP) method can adjust such errors through constructing an optimal programming model based on the consistency of the decision‐making information, and it has been applied widely in multiple attribute group decision making (MAGDM). Moreover, Pythagorean fuzzy information is useful to simulate the ambiguous and uncertain decision‐making environment. Therefore, the LINMAP method under the Pythagorean fuzzy circumstance will be proposed in this paper to solve MAGDM problems. To measure the fuzziness and uncertainty of Pythagorean fuzzy set (PFS) and interval‐valued PFS, Pythagorean fuzzy entropy (PFE) and interval‐valued PFE (IVPFE) grounded on the similarity and hesitancy parts have been defined, respectively. Then, Pythagorean fuzzy LINMAP (PF LINMAP) methods are constructed on the basis of the PFE and IVPFE correspondingly. Under the given preference relations, the maximum consistency and the amount of knowledge can be realized by the proposed methods. After investigating the relevant indicator system, the decision‐making problem concerning railway project investment has been solved through the proposed PF LINMAP method with PFE. Finally, the practicability and effectiveness of the PF LINMAP method has been verified via the comparative analysis with the existing methods.     

Consistency improvement under a personalized individual semantics context in distributed linguistic group decision making

Consistency measurement is a significant issue in linguistic decision making when preferences are expressed via linguistic preference relations. However, the extant literatures on linguistic consistency generally overlook the fact that even the same word can have diverse meanings for different people, which indicates that people usually possess personalized individual semantics (PISs) over words. Furthermore, with the complexity of the practical decision-making problem increases, decision makers become more likely to be uncertain and hesitant to make their preferences due to the lack of knowledge, therefore, their linguistic preferences may be represented through distributed linguistic representations. However, there are few consistency improving studies on distributed linguistic representations. Therefore, in this study we devise a novel consistency improving approach for distribution linguistic preference relations under a PISs context. Furthermore, the usability and effectiveness of the PISs based consistency improvement method are verified through the detail numerical analysis and comparative study.     

Interval‐valued probabilistic linguistic term sets in multi‐criteria group decision making

The theory of probabilistic linguistic term sets (PLTSs) is very useful in objectively dealing with the multi‐criteria group decision making (MCGDM) problems in which there is hesitancy in providing linguistic assessments; and PLTSs allow experts to express their preferences on one linguistic term over another. In order to reflect the uncertainty and inconsistency of decision‐makers and handle incomplete linguistic information, we propose a new PLTS called interval‐valued probabilistic linguistic term set (IVPLTS). In addition, the existing approaches associated with PLTSs are limited or highly complex in real applications. Therefore, new operations, comparison laws, and aggregation operators are developed for IVPLTS. Furthermore, we establish an efficient framework for MCGDM problems based on the proposed comparison method and the fuzzy preference relation. Then we apply it to a real‐life case under linguistic environment. The extended TOPSIS methods combined with PLTSs by using different operational laws are also included for comparison. The final results demonstrate the efficiency and practicality of the new framework.     

犹豫模糊环境下准则具有偏好关系的TODIM方法

针对智慧制造评估时专家的决策信息具有犹豫模糊不确定性问题，提出了一种关于准则具有犹豫模糊偏好关系的改进交互式多准则决策（TODIM）方法。首先提出了准则间的犹豫模糊偏好关系概念，并证明了其基本性质。在TODIM方法优势度的计算过程中，将准则权重犹豫模糊偏好关系替代原有的精确值权重，使信息的准确性最大化。将该方法用于智能制造的评估上，实例分析结果表明所提方法是可行和有效的。     

Design concept evaluation considering information reliability,uncertainty, and subjectivity: An integrated rough-Z-number-enhanced MCGDM methodology

Design concept evaluation plays a vital role in new product development due to its critical impact on successive design activities as well as the novelty and competence of the final product. Lots of decision-making techniques have been developed for the design concept evaluation under various contexts. However, existing methods mainly focus on the evaluation under precise environments or fuzzy set scenarios. Whereas in reality, most risk assessments are derived from experts’ subjective judgments. The reliability of the evaluation is rarely taken into consideration. To bridge this gap, this study proposes a rough-Z-number-enhanced MCGDM (multi-criteria group decision-making) to resolve the design concept evaluation considering information reliability, uncertainty, and subjectivity. A rough-Z-number is introduced to incorporate into the MCGDM framework to represent the individual evaluation and aggregate group risk assessments. A rough-Z-number-based AHP (analytic hierarchy process) is presented to determine the criteria weights. A rough-Z-number-based MABAC (multi-attributive border approximation area comparison) is proposed to rank the design concepts. Experiments and comparative analyses demonstrate the advantage of the rough-Z-number extended MCGDM in complex and uncertain decision-making environments. With the rough-Z-number-based enhancement, the information reliability, uncertainty, and subjectivity in design concept evaluation are well characterized.