GFUZ : An interactive program for solving a multicriteria decision making problem

In this paper, an interactive approach for dealing with a linear programing model under multiple and conflicting objectives is presented. The technique used can be regarded as fuzzy mathematical programming. A compromise solution is determined by eliciting from a single decision maker, a curve of preference for given aspiration levels of the objectives involved. Software was developed for a personal computer and can be helpful in providing support to practitioners utilizing mathematical modelling to justify their decisions.     

knowCube: a visual and interactive support for multicriteria decision making

In this paper, we present a novel multicriteria decision support system (MCDSS), called knowCube, consisting of components for knowledge organization, generation, and navigation. Knowledge organization rests upon a database for managing qualitative and quantitative criteria, together with add-on information. Knowledge generation serves filling the database via, e.g. identification, optimization, classification or simulation. For “finding needles in haycocks”, the knowledge navigation component supports graphical database retrieval and interactive, goal-oriented problem solving. Navigation “helpers” are, for instance, cascading criteria aggregations, modifiable metrics, ergonomic interfaces, and customizable visualizations. Examples from real-life projects, e.g. in industrial engineering and in the life sciences, illustrate the application of our MCDSS.     

An outranking method for multicriteria decision making with probabilistic hesitant information

Defects of hesitant fuzzy set (HFS) manifest in actual decision-making process, so adding probabilities to the values in HFS is necessary. The probabilistic HFS (PHFS) is a useful tool to describe the uncertainty of elements in HFS by introducing occurrence probabilities. However, some important issues in PHFS utilization remain to be addressed. In this study, an outranking method for multicriteria decision making (MCDM) with probabilistic hesitant information is presented. First, the binary relations between two probabilistic hesitant fuzzy elements (PHFEs) are defined on the basis of the elimination and choice translating reality method. Some outranking relations between the alternatives are then introduced. Second, we provide a Hausdorff distance between two PHFEs. The main characteristic of the proposed Hausdorff distance is that it does not require the same length and arrangement of the PHFEs. Third, a maximizing Hausdorff distance deviation method is developed to obtain the evaluation criteria weights under a probabilistic hesitant fuzzy environment. Finally, an illustrative example in conjunction with comparative analysis is used to demonstrate that the proposed method is feasible for practical MCDM problems.     

An interactive rectangle elimination method for biobjective decision making

This paper presents a new man-machine interactive method for biobjective decision making. It is specifically designed to cope with both the ill-defined nature of the decision problem and the high cost of computation points in the tradeoff (Pareto optimal) set. With this method, the decision maker may efficiently approximate the tradeoff set and/or estimate his preferred objective value. First, the notion of a rectangle representation of the tradeoff set by a set of points, called experiments, and a set of rectangles, defined by the experiments, is introduced. Next, a special class of decision makers is considered. For a decision maker in this special class, the rectangle representation of the tradeoff set defines a rectangle of uncertainty that contains the decision maker's preferred objective value. A measure of the worst ease uncertainity is formulated and minimized to yield an optimal strategy for interactively selecting experiments. Finally, this strategy is employed in a general interactive algorithm that works under minimal assumptions on the tradeoff set and on the decision maker.     

An interactive method for fuzzy multiple attribute group decision making

In this paper, we develop an interactive method for multiple attribute group decision making under fuzzy environment. The method can be used in situations where the information about attribute weights is partly known, the weights of decision makers are expressed in exact numerical values or triangular fuzzy numbers, and the attribute values are triangular fuzzy numbers. The method transforms fuzzy decision matrices into their expected decision matrices, constructs the corresponding normalized expected decision matrices by two simple formulas, and then aggregates these normalized expected decision matrices into a complex decision matrix. Moreover, the decision makers are asked to provide their preferences gradually in the course of interactions. By solving linear programming models, the method diminishes the given alternative set gradually, and finally finds the most preferred alternative. By using the method, the decision makers can provide and modify their preference information gradually in the process of decision making so as to make the decision result more reasonable. The method can not only reflect the importance of the given arguments and the ordered positions of the arguments, but also relieve the influence of unfair arguments on the decision result. Finally, a practical problem is used to illustrate the developed method.     

An interactive method for dynamic intuitionistic fuzzy multi-attribute group decision making

This paper investigates the dynamic intuitionistic fuzzy multi-attribute group decision making (DIF-MAGDM) problems, in which all the attribute values provided by multiple decision makers (DMs) at different periods take the form of intuitionistic fuzzy numbers (IFNs), and develops an interactive method to solve the DIF-MAGDM problems. The developed method first aggregates the individual intuitionistic fuzzy decision matrices at different periods into an individual collective intuitionistic fuzzy decision matrix for each decision maker by using the dynamic intuitionistic fuzzy weighted averaging (DIFWA) operator, and then employs intuitionistic fuzzy TOPSIS method to calculate the individual relative closeness coefficient of each alternative for each decision maker and obtain the individual ranking of alternatives. After doing so, the method utilizes the hybrid weighted averaging (HWA) operator to aggregate all the individual relative closeness coefficients into the collective relative closeness coefficient of each alternative and obtain the aggregate ranking of alternatives, by which the optimal alternative can be selected. In addition, the spearman correlation coefficient for both the aggregate ranking and individual ranking of alternatives is calculated to measure the consensus level of the group preferences. Finally, a numerical example is used to illustrate the developed method.     

An interactive computerized algorithm for multicriteria vehicle routing problems

The Multicriteria vehicle routing problem (VRP) is a VRP which allows several relevant objectives to be achieved. The use of an interactive computer program combined with a powerful vehicle routing algorithm can be a valuable tool in the hands of a scheduler who can apply his own skills and knowledge to full effect in conjunction with the speed and flexibility of the computer program.

An interactive computerized algorithm is developed that utilizes a heuristic algorithm to determine the most favorable vehicle routes for multicriteria VRPs. This algorithm takes into account that the problem being addressed may in fact be complicated by several conflicting objectives. Although many different objectives could be considered, the specific model presented here consists of the following three relevant objectives: minimization of the travel distance of vehicles, minimization of total deterioration of goods during transportation, and maximization of the total fulfillment of emergent services and conditional dependencies of stations.

The interactive algorithm successfully performs the trade-off between the achievement levels of the objectives through changes of the decision maker's goal priority structure, the target values of the objectives, and the subsets formation of stations.     

Consensus-based decision support for multicriteria group decision making

Consensus decision making is complex and challenging in multicriteria group decision making due to the involvement of several decision makers, the presence of multiple, and often conflicting criteria, and the existence of subjectiveness and imprecision in the decision making process. To ensure effective decisions being made, the interest of all the decision makers usually represented by the degree of consensus in the decision making process has to be adequately considered. This paper presents a consensus-based approach for effectively solving the multicriteria group decision making problem. The subjectiveness and imprecision of the decision making process is adequately handled by using intuitionistic fuzzy numbers. An interactive algorithm is developed for consensus building in the group decision making process. A decision support system framework is presented for improving the effectiveness of the consensus building process. An example is presented for demonstrating the applicability of the proposed approach for solving the multicriteria group decision making problem in real world situations.     

An interactive system for supporting multiobjective decision making

The purpose of this paper is to develop an interactive system for supporting the decision making process under multiple objectives and to empirically evaluate its performance. An interactive algorithm underlying the system is proposed with emphasis on the psychological aspects of the decision maker (DM). A choice process model is developed, based on pairwise comparison judgments of alternatives, because the judgments are basic and easy for a DM. A corresponding interactive algorithm is implemented and compared with other existing algorithms. Two kinds of comparative experiments, numerical and subject experiments, are conducted to verify the validity of the choice model as well as the practical effectiveness and the convergence of the algorithm.     

Modeling prioritized multicriteria decision making

We consider the problem of multicriteria decision making (MCDM) in the situation in which there exists a prioritization of criteria. A good example of prioritization among criteria occurs in the case of air travel, where concerns about passenger safety have a higher priority then economic concerns. Tradeoffs between saving on gasoline usage and jeopardizing passenger safety are unacceptable. We show how this prioritization of criteria can be modeled by using importance weights in which the weights associated with the lower priority criteria are related to the satisfaction of the higher priority criteria. We provide some models that allow for the formalization of these prioritized MCDM problems using both the Bellman-Zadeh paradigm for MCDM and the ordered weighted averaging (OWA) operator method.     

A new method selection approach for fuzzy group multicriteria decision making

Fuzzy multicriteria decision making (MCDM) has been widely used in ranking a finite number of decision alternatives characterized by fuzzy assessments with respect to multiple criteria. In group decision settings, different fuzzy group MCDM methods often produce inconsistent ranking outcomes for the same problem. To address the ranking inconsistency problem in fuzzy group MCDM, this paper develops a new method selection approach for selecting a fuzzy group MCDM method that produces the most preferred group ranking outcome for a given problem. Based on two group averaging methods, three aggregation procedures and three defuzzification methods, 18 fuzzy group MCDM methods are developed as an illustration to solve the general fuzzy MCDM problem that requires cardinal ranking of the decision alternatives. The approach selects the group ranking outcome of a fuzzy MCDM method which has the highest consistency degree with its corresponding ranking outcomes of individual decision makers. An empirical study on the green bus fuel technology selection problem is used to illustrate how the approach works. The approach is applicable to large-scale group multicriteria decision problems where inconsistent ranking outcomes often exist between different fuzzy MCDM methods.     

Soft information set for multicriteria decision making

An information source value is perceived differently by different agents. In this paper, we present a new knowledge representation structure, termed as soft information set (SIS), to provide a parameterized representation of the information values, as perceived by an agent. The properties of SIS are investigated and the notion of relations in SIS is devised. The significance of the proposed SIS in aiding multi criteria decision making is illustrated through real case-studies.     

Pythagorean fuzzy multicriteria group decision making through similarity measure based on point operators

In this paper, a series of similarity measures based on point operators for Pythagorean fuzzy sets are proposed. Using the proposed similarity measures, two new aggregation operators, viz., Pythagorean fuzzy‐dependent averaging operator and Pythagorean fuzzy‐dependent geometric operator, are developed. The advantage of using these operators is that the influence of unfair arguments of aggregated results could be eliminated, since the associated weights are taken from the aggregated Pythagorean fuzzy arguments. Also, the proposed operators have the capability to adjust the degree of aggregated arguments with the controlling parameters. To establish the application potentiality of those operators, a methodology for solving multicriteria group decision‐making problems having Pythagorean fuzzy arguments is developed. A numerical example is provided to demonstrate the proficiency of the proposed method. The achieved results are compared with the results of other existing technique.     

Pythagorean模糊环境下基于交叉熵和TOPSIS的多准则决策方法

考虑到Pythagorean模糊集（Pythagorean Fuzzy Set,PFS）具有的优势,提出了一个Pythagorean模糊环境下解决多准则决策（Multicriteria Decision Making,MCDM）问题的新方法。根据TOPSIS理论计算Pythagorean模糊环境下的正、负理想解,同时提出两个Pythagorean模糊集之间的交叉熵定义,并对其性质给予证明。计算每个方案各自和正、负理想解之间的交叉熵,再根据相对贴近度对所有方案进行排序。通过一个在绿色环境下的供应商选择的算例验证了有效性和实用性。     

Relating decision under uncertainty and multicriteria decision making models

This short overview paper points out the striking similarity between decision under uncertainty and multicriteria decision making problems, two areas which have been developed in an almost completely independent way until now. This pertains both to additive and non‐additive (including qualitative) approaches existing for the two decision paradigms. This leads to an emphasis on the remarkable formal equivalence between postulates underlying these approaches (like between the “sure‐thing principle” and mutual preferential independence of criteria). This analogy is exploited by surveying classical results as well as very recent advances. This unified view should be fruitful for a better understanding of the postulates underlying the approaches, for cross‐fertilization, and for adapting artificial intelligence uncertainty representation frameworks to preference modelling. © 2000 John Wiley & Sons, Inc.     

不确定多属性Vague集决策方法

针对指标权重信息未知的Vague集多指标决策问题,通过计算各方案与理想方案的相离度来建立最优化决策模型,求解该模型获得指标的权重,并引入了方案集的最优点和最劣点的概念,给出了Vague集多指标决策的新方法。该方法根据模糊值矩阵得到方案的最优点和最劣点,通过计算各方案到最优点和最劣点的距离对方案进行排序。最后用实例进行了验证。     

Risk and confidence analysis for fuzzy multicriteria decision making

Recent research has recognised that multicriteria decision making (MCDM) should take account of uncertainty, risk and confidence. This paper takes this research forward by using linguistic variables and triangular fuzzy numbers to model the decision maker’s (DM) risk and confidence attitudes in order to define a more complete MCDM solution. To illustrate the computation process and demonstrate the feasibility of the results we use a travel problem that has been used previously to assess MCDM techniques. The results show that the method is useful for tackling imprecision and subjectivity in complex, ill-defined and human-oriented decision problems.     

Hesitant fuzzy Hamacher aggregation operators for multicriteria decision making

As a fuzzy set extension, the hesitant set is effectively used to model situations where it is allowable to determine several possible membership degrees of an element to a set due to the ambiguity between different values. We first introduce some new operational rules of hesitant fuzzy sets based on the Hamacher t-norm and t-conorm, in which a family of hesitant fuzzy Hamacher operators is proposed for aggregating hesitant fuzzy information. Some basic properties of these proposed operators are given, and the relationships between them are shown in detail. We further discuss the interrelations between the proposed aggregation operators and the existing hesitant fuzzy aggregation operators. Applying the proposed hesitant fuzzy operators, we develop a new technique for hesitant fuzzy multicriteria decision making problems. Finally, the effectiveness of the proposed technique is illustrated by mean of a practical example.     

An interactive evolutionary multi-objective optimization and decision making procedure

With the advent of efficient techniques for multi-objective evolutionary optimization (EMO), real-world search and optimization problems are being increasingly solved for multiple conflicting objectives. During the past decade of research and application, most emphasis has been spent on finding the complete Pareto-optimal set, although EMO researchers were always aware of the importance of procedures which would help choose one particular solution from the Pareto-optimal set for implementation. This is also one of the main issues on which the classical and EMO philosophies are divided on. In this paper, we address this long-standing issue and suggest an interactive EMO procedure which will involve a decision-maker in the evolutionary optimization process and help choose a single solution at the end. This study uses many year’s of research on EMO and would hopefully encourage both practitioners and researchers to pay more attention in viewing the multi-objective optimization as a aggregate task of optimization and decision-making.