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.     

Optimizing fuzzy p-hub center problem with generalized value-at-risk criterion

In this study, we reduce the uncertainty embedded in secondary possibility distribution of a type-2 fuzzy variable by fuzzy integral, and apply the proposed reduction method to p-hub center problem, which is a nonlinear optimization problem due to the existence of integer decision variables. In order to optimize p-hub center problem, this paper develops a robust optimization method to describe travel times by employing parametric possibility distributions. We first derive the parametric possibility distributions of reduced fuzzy variables. After that, we apply the reduction methods to p-hub center problem and develop a new generalized value-at-risk (VaR) p-hub center problem, in which the travel times are characterized by parametric possibility distributions. Under mild assumptions, we turn the original fuzzy p-hub center problem into its equivalent parametric mixed-integer programming problems. So, we can solve the equivalent parametric mixed-integer programming problems by general-purpose optimization software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the efficiency of the proposed solution methods.     

Asymptotic structural characteristics of fuzzy measure and their applications

In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary.     

On the nearest parametric approximation of a fuzzy number

Many nearest parametric approximation methods of fuzzy sets are proposed in the literature. It is clear that the specific approximations may lead to the loss of information about fuzziness. To overcome this problem, most of these methods rely on the minimization of the distance between the original fuzzy set and its approximation. But these approximations mostly are not flexible to the decision maker's choice. Hence, in this paper, we offer a parametric fuzzy approximation method based on the decision maker's strategy as an extension of trapezoidal approximation of a fuzzy number. This method comprises the selection of the form of the parametric membership function and its evaluation.     

Basic concepts for a theory of evaluation: Hierarchical aggregation via autodistributive connectives in fuzzy set theory

The ‘aggregative operator’ introduced by Combi (1982) in his analysis of ‘basic concepts for a theory of evaluation’ in this journal is based on the axioms of associativity, continuity, and monotonicity. In this paper, an alternative approach is given where associativity is replaced by another axiom, viz. autodistributivity, implying a kind of hierarchical aggregation instead of the former horizontal one. The new set of properties yields a general class of operators for connecting fuzzy sets which are formally similar to the aggregative operators, but are purely compensative (in a strong sense). Moreover, this modified approach generalizes a recent approach based on generalized means as compensative connectives. It appears to be similar to expected utility approaches in decision making under uncertainty, and seems to be a very promising tool to handle representation problems as they grow from other fields of human decision making, e.g. of multicriteria analysis and of welfare theory.     

Attribute reduction based on generalized fuzzy evidence theory in fuzzy decision systems

Attribute reduction is viewed as an important issue in data mining and knowledge representation. This paper studies attribute reduction in fuzzy decision systems based on generalized fuzzy evidence theory. The definitions of several kinds of attribute reducts are introduced. The relationships among these reducts are then investigated. In a fuzzy decision system, it is proved that the concepts of fuzzy positive region reduct, lower approximation reduct and generalized fuzzy belief reduct are all equivalent, the concepts of fuzzy upper approximation reduct and generalized fuzzy plausibility reduct are equivalent, and a generalized fuzzy plausibility consistent set must be a generalized fuzzy belief consistent set. In a consistent fuzzy decision system, an attribute set is a generalized fuzzy belief reduct if and only if it is a generalized fuzzy plausibility reduct. But in an inconsistent fuzzy decision system, a generalized fuzzy belief reduct is not a generalized fuzzy plausibility reduct in general.     

Some aspects of intuitionistic fuzzy sets

We first discuss the significant role that duality plays in many aggregation operations involving intuitionistic fuzzy subsets. We then consider the extension to intuitionistic fuzzy subsets of a number of ideas from standard fuzzy subsets. In particular we look at the measure of specificity. We also look at the problem of alternative selection when decision criteria satisfaction is expressed using intuitionistic fuzzy subsets. We introduce a decision paradigm called the method of least commitment. We briefly look at the problem of defuzzification of intuitionistic fuzzy subsets.     

New models for shortest path problem with fuzzy arc lengths

This paper considers the shortest path problem with fuzzy arc lengths. According to different decision criteria, the concepts of expected shortest path, α-shortest path and the most shortest path in fuzzy environment are originally proposed, and three types of models are formulated. In order to solve these models, a hybrid intelligent algorithm integrating simulation and genetic algorithm is provided and some numerous examples are given to illustrate its effectiveness.     

Exact solution to the periodic boundary value problem for a first-order linear fuzzy differential equation with impulses

Using the expression of the exact solution to a periodic boundary value problem for an impulsive first-order linear differential equation, we consider an extension to the fuzzy case and prove the existence and uniqueness of solution for a first-order linear fuzzy differential equation with impulses subject to boundary value conditions. We obtain the explicit solution by calculating the solutions on each level set and justify that the parametric functions obtained define a proper fuzzy function. Our results prove that the solution of the fuzzy differential equation of interest is determined, under the appropriate conditions, by the same Green’s function obtained for the real case. Thus, the results proved extend some theorems given for ordinary differential equations.     

Fuzzy decision trees

We propose here to extend the decision trees method to the case when the involved data (probabilities, cost, profits, losses) appear as words belonging to the common language whose semantic representations are fuzzy sets. First we discuss the reasons why such an extension is to be aimed at. Then in the fuzzy case we carry out a reformalization of the basic concepts of probability and utility theory. Finally we show how these reformalized concepts can be applied to fuzzy decision trees.     

Decomposition of a complex fuzzy controller for the truck-and-trailer reverse parking problem

The use of fuzzy logic has, in the last twenty years, become standard practice in the field of control. The reason lies in the fuzzy logic’s ability to relatively quickly transfer uncertain experience and knowledge about the observed object’s behaviour into the process of decision making. Nevertheless, one of the biggest problems that arises when using a fuzzy approach is the large number of fuzzy rules that have to be processed in order to produce one decision (i.e. one control output). The number of rules in a fuzzy controller primarily originates from the number of input variables that are entering the decision process and one possible solution for decreasing it is to use the method of decomposition. Its main goal is to implement the equivalent control functionality with a hierarchy of simpler fuzzy controllers. Their main characteristic is a lower number of input variables, which as a consequence leads to a smaller number of fuzzy rules. In our paper we apply the decomposition approach to the classical complex control case of the Truck-and-Trailer (T&T) reverse parking control problem. In such cases the implementation of control using only one fuzzy controller is very complex and the existing solutions, in some details, even deviate from the classical fuzzy approach. Our solution is, on the other hand, based only on the uncertain knowledge about the behaviour of the T&T driver and the results achieved are even better than those achieved by using the existing solutions.     

Solving generalized fuzzy data envelopment analysis model: a parametric approach

Data envelopment analysis (DEA) is a non-parametric technique to assess the performance of a set of homogeneous decision making units (DMUs) with common crisp inputs and outputs. Regarding the problems that are modelled out of the real world, the data cannot constantly be precise and sometimes they are vague or fluctuating. So in the modelling of such data, one of the best approaches is using the fuzzy numbers. Substituting the fuzzy numbers for the crisp numbers in DEA, the traditional DEA problem transforms into a fuzzy data envelopment analysis (FDEA) problem. Different methods have been suggested to compute the efficiency of DMUs in FDEA models so far but the most of them have limitations such as complexity in calculation, non-contribution of decision maker in decision making process, utilizable for a specific model of FDEA and using specific group of fuzzy numbers. In the present paper, to overcome the mentioned limitations, a new approach is proposed. In this approach, the generalized FDEA problem is transformed into a parametric programming, in which, parameter selection depends on the decision maker’s ideas. Two numerical examples are used to illustrate the approach and to compare it with some other approaches.     

多重模糊蕴涵与生成模糊蕴涵的新方法

提出一种生成模糊蕴涵的新方法,即通过一个或多个选定的模糊蕴涵作多重迭代生成新的模糊蕴涵,称为多重模糊蕴涵.从选定的模糊蕴涵出发,根据不同的迭代次数,分别证明迭代结果仍为模糊蕴涵.进一步,分析当选定的模糊蕴涵具有某种性质时,所生成的多重模糊蕴涵是否仍保持这些性质.还对选定的模糊蕴涵分别为(S,N)-蕴涵和R-蕴涵这两种重要情形,进行更深入的分析.这些结果将为模糊蕴涵在控制及决策等领域的应用提供支持.     

Using fuzzy numbers in knapsack problems

This paper investigates knapsack problems in which all of the weight coefficients are fuzzy numbers. This work is based on the assumption that each weight coefficient is imprecise due to the use of decimal truncation or rough estimation of the coefficients by the decision-maker. To deal with this kind of imprecise data, fuzzy sets provide a powerful tool to model and solve this problem. Our work intends to extend the original knapsack problem into a more generalized problem that would be useful in practical situations. As a result, our study shows that the fuzzy knapsack problem is an extension of the crisp knapsack problem, and that the crisp knapsack problem is a special case of the fuzzy knapsack problem.     

Production and payment policies for an imperfect manufacturing system with discount cash flows analysis in fuzzy random environments

ABSTRACT

This paper considers an imperfect manufacturing system with credit policies in fuzzy random environments. The supplier simultaneously offers the retailer either a permissible delay in payments or a cash discount and retailer in turn provides its customer a permissible delay period. We used an alternate approach – discount cash flow analysis to establish an inventory problem. It is assumed that the elapsed time until the machine shifts from ‘in-control’ state to ‘out-of-control’ state is characterized as a fuzzy random variable. As a function of this parameter, the profit function is also a random fuzzy variable. Based on the credibility measure of fuzzy event, the model with fuzzy random elapsed time can be transformed into a crisp model . We establish several theoretical results to obtain the solution that provides the largest present value of all future cash flows. Finally, numerical example is given to illustrate the results and obtain some managerial insights.     

Multiple criteria decision problems with fuzzy domination structures

The concepts of domination structures and nondominated solutions in multiple criteria decision problems, which were introduced by Yu, enable us to tackle general situations in which there exists information concerning the decision maker's preferences.In many of the multiple criteria decision problems the underlying domination structures are not known precisely but only fuzzily determined. Yu primarily works with the case where the domination structure at each point is a convex cone. As a result, there exists a sharp borderline dividing all solutions into nondominated solutions and the others.This paper fuzzifies the concepts of domination structures and nondominated solutions to allow them to be applied to a larger class of the multiple criteria decision problems mentioned above. Introducing the concepts of fuzzy convex cones and fuzzy polar cones, it is shown how some of the main results obtained by Yu are extended.     

Transportation policies for single and multi-objective transportation problem using fuzzy logic

In this paper, single and multi-objective transportation models are formulated with fuzzy relations under the fuzzy logic. In the single-objective model, objective is to minimize the transportation cost. In this case, the amount of quantities transported from an origin to a destination depends on the corresponding transportation cost and this relation is verbally expressed in an imprecise sense i.e., by the words ‘low’, ‘medium’, ‘high’. For the multi-objective model, objectives are minimization of (i) total transportation cost and (ii) total time for transportation required for the system. Here, also the transported quantity from a source to a destination is determined on the basis of minimum total transportation cost as well as minimum transportation time. These relations are imprecise and stated by verbal words such as ‘very high’, ‘high’, ‘medium’, ‘low’ and ‘very low’. Both single objective and multi-objective problems using Real coded Genetic Algorithms (GA and MOGA) are developed and used to solve the single level and bi-level logical relations respectively. The models are illustrated with numerical data and optimum results are presented.     

Correspondence between fuzzy measures and classical measures

In this paper we study the question whether, given a fuzzy measure (as defined in [3] and [4]). there exists a classical measure such that the fuzzy measure of a measurable fuzzy set μ equals the classical measure of the area below the membership function of μ. The results are that in the case of finite additivity there is a one-to-one correspondence between classical measures and fuzzy measures, whereas in the case of countable additivity this result only holds for generated fuzzy σ-algebras. Finally, some connections of that problem with the existence of an extension of a fuzzy measure defined on an arbitrary fuzzy σ-algebra σ to the generated fuzzy σ-algebra $\text{σ}$ are discussed.     

A rough set approach to intuitionistic fuzzy soft set based decision making

The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. Since its appearance, there has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making. The intuitionistic fuzzy soft set is a combination of an intuitionistic fuzzy set and a soft set. The rough set theory is a powerful tool for dealing with uncertainty, granuality and incompleteness of knowledge in information systems. Using rough set theory, this paper proposes a novel approach to intuitionistic fuzzy soft set based decision making problems. Firstly, by employing an intuitionistic fuzzy relation and a threshold value pair, we define a new rough set model and examine some fundamental properties of this rough set model. Then the concepts of approximate precision and rough degree are given and some basic properties are discussed. Furthermore, we investigate the relationship between intuitionistic fuzzy soft sets and intuitionistic fuzzy relations and present a rough set approach to intuitionistic fuzzy soft set based decision making. Finally, an illustrative example is employed to show the validity of this rough set approach in intuitionistic fuzzy soft set based decision making problems.     

A parametric total order on fuzzy numbers and a fuzzy shortest route problem

A parametric total order relation is introduced in the form of a certain modification of the "fuzzy max" order on the class of fuzzy numbers generated by a shape function. By the parametric relation. fuzzy numbers unordered with respect of the fuzzy max order can be ordered according either to their value of center or the size of ambiguity. A fuzzy shortest route problem in which are distances are given by fuzzy numbers is discussed under the criterion of the parametric total order, and solved by the dynamic programming approach. A method is proposed to find all of fuzzy routes mimmal in the sense of the fuzzy max order.