Polygonal approximation of flows

The analysis of the qualitative behavior of flows generated by ordinary differential equations often requires quantitative information beyond numerical simulation which can be difficult to obtain analytically. In this paper we present a computational scheme designed to capture qualitative information using ideas from the Conley index theory. Specifically we design an combinatorial multivalued approximation from a simplicial decomposition of the phase space, which can be used to extract isolating blocks for isolated invariant sets. These isolating blocks can be computed rigorously to provide computer-assisted proofs. We also obtain local conditions on the underlying simplicial approximation that guarantees that the chain recurrent set can be well-approximated.     

Robust strategies for facility location under uncertainty

This paper considers a stochastic facility location problem in which multiple capacitated facilities serve customers with a single product, and a stockout probabilistic requirement is stated as a chance constraint. Customer demand is assumed to be uncertain and to follow either a normal or an ambiguous distribution. We study robust approximations to the problem in order to incorporate information about the random demand distribution in the best possible, computationally tractable way. We also discuss how a decision maker’s risk preferences can be incorporated in the problem through robust optimization. Finally, we present numerical experiments that illustrate the performance of the different robust formulations. Robust optimization strategies for facility location appear to have better worst-case performance than nonrobust strategies. They also outperform nonrobust strategies in terms of realized average total cost when the actual demand distributions have higher expected values than the expected values used as input to the optimization models.     

Group decision-making based on concepts of ideal and anti-ideal points in a fuzzy environment

Decision-making problems (location selection) often involve a complex decision-making process in which multiple requirements and uncertain conditions have to be taken into consideration simultaneously. In evaluating the suitability of alternatives, quantitative/qualitative assessments are often required to deal with uncertainty, subjectiveness and imprecise data, which are best represented with fuzzy data. This paper presents a new method of analysis of multicriteria based on the incorporated efficient fuzzy model and concepts of positive ideal and negative ideal points to solve decision-making problems with multi-judges and multicriteria in real-life situations. As a result, effective decisions can be made on the basis of consistent evaluation results. Finally, this paper uses a numerical example of location selection to demonstrate the applicability of this method, with its simplicity in both concept and computation. The results show that this method can be implemented as an effective decision aid in selecting location or decision-making problems.     

Fuzzy random regression based multi-attribute evaluation and its application to oil palm fruit grading

Multi-attribute decision-making is usually concerned with weighting alternatives, thereby requiring weight information for decision attributes from a decision maker. However, the assignment of an attribute’s weight is sometimes difficult, and may vary from one decision maker to another. Additionally, imprecision and vagueness may affect each judgment in the decision-making process. That is, in a real application, various statistical data may be imprecise or linguistically as well as numerically vague. Given this coexistence of random and fuzzy information, the data cannot be adequately treated by simply using the formalism of random variables. To address this problem, fuzzy random variables are introduced as an integral component of regression models. Thus, in this paper, we proposed a fuzzy random multi-attribute evaluation model with confidence intervals using expectations and variances of fuzzy random variables. The proposed model is applied to oil palm fruit grading, as the quality inspection process for fruits requires a method to ensure product quality. We include simulation results and highlight the advantage of the proposed method in handling the existence of fuzzy random information.     

Quantifying vagueness in decision models

When modelling specific decision situations the decision maker often feels overstrained when he is asked for precise numerical quotations concerning the objectives or the constraints, whereas qualitative statements are easily given. In the recent past the theory of fuzzy sets has proven to be very useful for representing this type of information. Though it is quite advanced formally, the practical determination of its core elements, i.e. membership functions and operators, has only been explored to a very limited extent. This paper presents results of empirical research which focused on the problem of modelling vagueness, i.e. determining membership functions of fuzzy sets which are considered as quantitative representations of vague concepts such as ‘young man’, ‘long sticks’, ‘high profits’, etc.     

Multi-criteria analysis for dredger dispatching under uncertainty

Dredger dispatching is a complex decision making process in which multiple requirements and uncertain site conditions have to be taken into consideration simultaneously, for a specific dredging task. In evaluating the suitability of dredgers, besides quantitative assessments, qualitative assessments are often required to deal with uncertainty, subjectiveness and imprecision, which are best represented with fuzzy data. This paper formulates dredger dispatching as a fuzzy multi-criteria analysis model, and presents an effective algorithm for handling both crisp and fuzzy data in a straightforward manner. As a result, effective decisions can be made based on consistent evaluation results. An empirical study of a real case in China is conducted to demonstrate the applicability of the model. With its simplicity in both concept and computation, the model can be implemented as an effective decision aid in selecting dredgers for specific dredging tasks.     

Simulation for lattice-valued doubly labeled transition systems

During the last decades, a large amount of multi-valued transition systems, whose transitions or states are labeled with specific weights, have been proposed to analyze quantitative behaviors of reactive systems. To set up a unified framework to model and analyze systems with quantitative information, in this paper, we present an extension of doubly labeled transition systems in the framework of residuated lattices, which we will refer to as lattice-valued doubly labeled transition systems (LDLTSs). Our model can be specialized to fuzzy automata over complete residuated lattices, fuzzy transition systems, and multi-valued Kripke structures. In contrast to the traditional yes/no approach to similarity, we then introduce lattice-valued similarity between LDLTSs to measure the degree of closeness of two systems, which is a value from a residuated lattice. Further, we explore the properties of robustness and compositionality of the lattice-valued similarity. Finally, we extend the Hennessy–Milner logic to the residuate lattice-valued setting and show that the obtained logic is adequate and expressive with lattice-valued similarity.     

A genetic design of linguistic terms for fuzzy rule based classifiers

The determination of fuzzy information granules including the estimation of their membership functions play a significant role in fuzzy system design as well as in the design of fuzzy rule based classifiers (FRBCSs). However, although linguistic terms are fundamental elements in the process of elucidating expert’s knowledge, the problem of linguistic term design along with their fuzzy-set-based semantics has not been fully addressed, since term-sets of attributes have not been interpreted as a formalized structure. Thus, the essential relationship between linguistic terms, as syntax, and the constructed fuzzy sets, as their quantitative semantics, or in other words, the problem of the natural semantics of terms behind the linguistic literal has not been addressed. In this paper, we introduce the problem of the design of optimal linguistic terms and propose a method of the design of FRBCSs which may incorporate with the design of linguistic terms to ensure that the presence of linguistic literals are supported not only by data but also by their natural semantics. It is shown that this problem plays a primordial role in enhancing the performance and the interpretability of the designed FRBCSs and helps striking a better balance between the generality and the specificity of the desired fuzzy rule bases for fuzzy classification problems. A series of experiments concerning 17 Machine Learning datasets is reported.     

Fuzzy temporal fault tree analysis of dynamic systems

Fault tree analysis (FTA) is a powerful technique that is widely used for evaluating system safety and reliability. It can be used to assess the effects of combinations of failures on system behaviour but is unable to capture sequence dependent dynamic behaviour. A number of extensions to fault trees have been proposed to overcome this limitation. Pandora, one such extension, introduces temporal gates and temporal laws to allow dynamic analysis of temporal fault trees (TFTs). It can be easily integrated in model-based design and analysis techniques. The quantitative evaluation of failure probability in Pandora TFTs is performed using exact probabilistic data about component failures. However, exact data can often be difficult to obtain. In this paper, we propose a method that combines expert elicitation and fuzzy set theory with Pandora TFTs to enable dynamic analysis of complex systems with limited or absent exact quantitative data. This gives Pandora the ability to perform quantitative analysis under uncertainty, which increases further its potential utility in the emerging field of model-based design and dependability analysis. The method has been demonstrated by applying it to a fault tolerant fuel distribution system of a ship, and the results are compared with the results obtained by other existing techniques.     

Solution of a possibilistic multiobjective linear programming problem

The estimate of the parameters which define a conventional multiobjective decision making model is a difficult task. Normally they are either given by the Decision Maker who has imprecise information and/or expresses his considerations subjectively, or by statistical inference from the past data and their stability is doubtful. Therefore, it is reasonable to construct a model reflecting imprecise data or ambiguity in terms of fuzzy sets and several fuzzy approaches to multiobjective programming have been developed 1, 9, 10, 11. The fuzziness of the parameters gives rise to a problem whose solution will also be fuzzy, see 2, 3, and which is defined by its possibility distribution. Once the possibility distribution of the solution has been obtained, if the decision maker wants more precise information with respect to the decision vector, then we can pose and solve a new problem. In this case we try to find a decision vector, which approximates as much as possible the fuzzy objectives to the fuzzy solution previously obtained. In order to solve this problem we shall develop two different models from the initial solution and based on Goal Programming: an Interval Goal Programming Problem if we define the relation “as accurate as possible” based on the expected intervals of fuzzy numbers, as we showed in [4], and an ordinary Goal Programming based on the expected values of the fuzzy numbers that defined the goals. Finally, we construct algorithms that implement the above mentioned solution method. Our approach will be illustrated by means of a numerical example.     

An Approach to Fuzzy Noncooperative Nash Games

Systems that involve more than one decision maker are often optimized using the theory of games. In the traditional game theory, it is assumed that each player has a well-defined quantitative utility function over a set of the player decision space. Each player attempts to maximize/minimize his/her own expected utility and each is assumed to know the extensive game in full. At present, it cannot be claimed that the first assumption has been shown to be true in a wide variety of situations involving complex problems in economics, engineering, social and political sciences due to the difficulty inherent in defining an adequate utility function for each player in these types of problems. On the other hand, in many of such complex problems, each player has a heuristic knowledge of the desires of the other players and a heuristic knowledge of the control choices that they will make in order to meet their ends.In this paper, we utilize fuzzy set theory in order to incorporate the players' heuristic knowledge of decision making into the framework of conventional game theory or ordinal game theory. We define a new approach to N-person static fuzzy noncooperative games and develop a solution concept such as Nash for these types of games. We show that this general formulation of fuzzy noncooperative games can be applied to solve multidecision-making problems where no objective function is specified. The computational procedure is illustrated via application to a multiagent optimization problem dealing with the design and operation of future military operations.     

Integration of particle swarm optimization-based fuzzy neural network and artificial neural network for supplier selection

This study is intended to develop an intelligent supplier decision support system which is able to consider both the quantitative and qualitative factors. It is composed of (1) the collection of quantitative data such as profit and productivity, (2) a particle swarm optimization (PSO)-based fuzzy neural network (FNN) to derive the rules for qualitative data, and (3) a decision integration model for integrating both the quantitative data and fuzzy knowledge decision to achieve the optimal decision. The results show that the decision support system developed in this study make more precise and favorable judgments in selecting suppliers after taking into account both qualitative and quantitative factors.     

A fuzzy MCDM approach for stock selection

A fuzzy MCDM approach is applied to the stock selection problem, where the proposed approach can deal with qualitative information in addition to quantitative information. A hierarchy of major–sub criteria is then established to reduce the dependence between criteria. The ratings of alternatives versus qualitative sub-criteria and the weights of major- and sub-criteria are assessed in linguistic terms represented by fuzzy numbers. Each sub-criterion is in a benefit, cost, or balanced nature. New standardization methods for fuzzy numbers in the cost and balanced nature are presented. The algorithms of membership functions of the final aggregation are completely developed instead of approximation. The final aggregations in fuzzy numbers are then defuzzified to crisp values in order to rank the performance of alternatives. Moreover, the ratio of market price to performance (PP) is suggested to filter the over/under-pricing of alternatives. A set of buying/selling strategies are recommended according to the performance and PP. An empirical example then demonstrates the processing of the proposed approach.     

基于混合fuzzy entropy-TOPSIS法的供应链范式选择

针对精益、敏捷、精敏供应链范式选择决策涉及到多种定性指标和定量指标这一特点,提出一种基于模糊集理论、熵和TOPSIS的混合型多属性决策模型。为了避免主观赋权法无法反映客观条件变化,或客观赋权法无法反映专家经验的双重弊端,在对定量指标赋权时,采用主观赋权的三角模糊数法和客观赋权的熵权法对定量指标进行组合赋权。对于定性指标仅依靠三角模糊数法确定其权重。然后,运用fuzzy TOPSIS法选择出最佳的供应链范式。最后通过实例对模型进行验证以及对评价指标权重进行敏感性分析,证实了该模型能为企业选择适用自身运作的供应链范式提供高效方法。     

含语言评价信息的一种模糊多属性群决策方法

对于多属性群决策问题的处理,有时需要采用先决策、后综合的处理方法,而含有语言评价信息的多属性群决策问题,定性目标一般用语言评价信息描述,由决策人给出定性目标和权系数的语言变量评价,用梯形模糊数表示,对定量目标进行无量纲化处理;将决策人对于单一目标的评价指标聚合成多个目标的评价模糊数,采用Bass-Kw akernaak模糊数排序方法对方案进行排序;群体的评价通过Borda函数来集结方案集的群体排序.     

The fuzzy p-median problem: A global analysis of the solutions

We apply fuzzy techniques to incorporate external data into p-median problems. So we can detect certain solutions that would be discarded by usual crisp and fuzzy algorithms but that contrasted with this additional information can be advantageous. This usually reveals a pathology of the model and hence our methods provide some fuzzy validation criteria for p-median models.     

Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis

Hesitant fuzzy sets (HFSs), which allow the membership degree of an element to a set represented by several possible values, can be considered as a powerful tool to express uncertain information in the process of group decision making. We derive some correlation coefficient formulas for HFSs and apply them to clustering analysis under hesitant fuzzy environments. Two real world examples, i.e. software evaluation and classification as well as the assessment of business failure risk, are employed to illustrate the actual need of the clustering algorithm based on HFSs, which can incorporate the difference of evaluation information provided by different experts in clustering processes. In order to extend the application domain of the clustering algorithm in the framework of HFSs, we develop the interval-valued HFSs and the corresponding correlation coefficient formulas, and then demonstrate their application in clustering with interval-valued hesitant fuzzy information through a specific numerical example.     

The construction of a comprehensive model for production strategy evaluation

To acquire reasonable profits and to be competitive in the globalize market, more and more firms today have exploited various kinds of production strategies, such as outsourcing, joint venture, or some kind of strategic alliance. Since every production strategy has its benefits and costs and may bring a firm different opportunities and risks, which kind of production strategy is the most suitable for a firm to carry out is a difficult and complicated decision with a high degree of uncertainty. Therefore, this research proposes a comprehensive production strategy evaluation model that can facilitate such a decision making for a firm. The factors that should be considered for devising a production strategy are listed for the benefits, opportunities, costs and risks (BOCR) merits first, and fuzzy Delphi method is applied next to select the most important factors under each merit. A network with BOCR merits is constructed to incorporate the selected factors, and fuzzy analytic network process is then applied to consider the interdependence and feedback inside the network. The proposed model can effectively aid decision making on which kind of production strategy should be adopted by a firm. A case of a printed circuit board manufacturer is presented for the illustration of the proposed model.     

Modeling fuzzy capacitated p-hub center problem and a genetic algorithm solution

Hub and spoke networks are used to switch and transfer commodities between terminal nodes in distribution systems at minimum cost and/or time. The p-hub center allocation problem is to minimize maximum travel time in networks by locating p hubs from a set of candidate hub locations and allocating demand and supply nodes to hubs. The capacities of the hubs are given. In previous studies, authors usually considered only quantitative parameters such as cost and time to find the optimum location. But it seems not to be sufficient and often the critical role of qualitative parameters like quality of service, zone traffic, environmental issues, capability for development in the future and etc. that are critical for decision makers (DMs), have not been incorporated into models. In many real world situations qualitative parameters are as much important as quantitative ones. We present a hybrid approach to the p-hub center problem in which the location of hub facilities is determined by both parameters simultaneously. Dealing with qualitative and uncertain data, Fuzzy systems are used to cope with these conditions and they are used as the basis of this work. We use fuzzy VIKOR to model a hybrid solution to the hub location problem. Results are used by a genetic algorithm solution to successfully solve a number of problem instances. Furthermore, this method can be used to take into account more desired quantitative variables other than cost and time, like future market and potential customers easily.     

Mining performance data through nonlinear PCA with optimal scaling

Performance data are usually collected in order to build well‐defined performance indicators. Since such data may conceal additional information, which can be revealed by secondary analysis, we believe that mining of performance data may be fruitful. We also note that performance databases usually contain both qualitative and quantitative variables for which it may be inappropriate to assume some specific (multivariate) underlying distribution. Thus, a suitable technique to deal with these issues should be adopted. In this work, we consider nonlinear principal component analysis (PCA) with optimal scaling, a method developed to incorporate all types of variables, and to discover and handle nonlinear relationships. The reader is offered a case study in which a student opinion database is mined. Though generally gathered to provide evidence of teaching ability, they are exploited here to provide a more general performance evaluation tool for those in charge of managing universities. We show how nonlinear PCA with optimal scaling applied to student opinion data enables users to point out some strengths and weaknesses of educational programs and services within a university. Copyright © 2009 John Wiley & Sons, Ltd.