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2010 | Buch

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Fuzzy Optimization

Recent Advances and Applications

herausgegeben von: Weldon A. Lodwick, Janusz Kacprzyk

Verlag: Springer Berlin Heidelberg

Buchreihe : Studies in Fuzziness and Soft Computing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Optimization is an extremely important area in science and technology which provides powerful and useful tools and techniques for the formulation and solution of a multitude of problems in which we wish, or need, to to find a best possible option or solution. The volume is divided into a coupe of parts which present various aspects of fuzzy optimization, some related more general issues, and applications.

Inhaltsverzeichnis

Frontmatter

Introductory Sections

Frontmatter

Fuzziness, Rationality, Optimality and Equilibrium in Decision and Economic Theories

Abstract

This essay presents structural categories of theories of optimization. It begins with the classical system leading to the establishment of an entry point of fuzzy optimization from the logic of the classical optimization. Four categories of optimization problems are identified. They are exact (non-fuzzy) and nonstochastic, exact (non-fuzzy) and stochastic categories that are associated with classical laws of thought and mathematics. The other categories are fuzzy and non-stochastic, and fuzzy-stochastic problems that are associated with fuzzy laws of thought and fuzzy mathematics. The canonical structures of the problems and their solutions are presented.

From these structures, similarities and differences in the problem structures and corresponding solutions are abstracted and discussed. The similarities and differences in the problem-solution structures of different categories are attributed to properties of exactness and completeness about information-knowledge structures in which the optimization problems are formulated and solved. The assumed degrees of exactness and completeness establish defective informationknowledge structure that generates uncertainties and produces inter-category differences in the optimization problem. The specific differences of intra-category algorithms are attributed to differences in the assumed functional relationships of the variables that establish the objective and constraint sets. The essay is concluded with taxonomy of solution structures and discussions on future research directions.

Kofi Kissi Dompere

Introduction to Fuzzy and Possibilistic Optimization

Introduction

Deterministic optimization is a normative process which extracts the best from a set of options, usually under constraints. It is arguably true that optimization is one of the most used areas of mathematical applications. It is the thesis of this book that applied mathematical programming problems should be solved predominantly by using a fuzzy and possibilistic approaches. Rommelfanger ([42], p. 295), states that the only operations research methods that is widely applied is linear programming. He goes on to state that even though this is true, of the 167 production (linear) programming systems investigated and surveyed by Fandel [18], only 13 of these were ”purely” (my interpretation) linear programming systems. Thus, Rommelfanger concludes that even with this most highly used and applied operations research method, there is a discrepancy between classical linear programming and what is applied. Deterministic and stochastic optimization models require well-defined input parameters (coefficients, right-hand side values), relationships (inequalities, equalities), and preferences (real valued functions to maximize, minimize) either as real numbers or real valued distribution functions. Any large scale model requires significant data gathering efforts. If the model has projections of future values, it is clear that real numbers and real valued distributions are inadequate representations of parameters, even assuming that the model correctly captures the underlying system. It is also known from mathematical programming theory that only a few of the variables and constraints are necessary to describe an optimal solution (basic variables and active constraints), assuming a correct deterministic normative criterion (objective function). The ratio of variables that are basic and constraints that are active compared to the total becomes smaller, in general, as the model increases in size since in general large-scale models tend to become more sparse. Thus, only a few parameters need to be obtained precisely. Of course the problem is that it is not known a priori which variables will be basic and which constraints will be active.

Weldon A. Lodwick, Elizabeth Untiedt

Basic Issues

Frontmatter

Aggregation Operators for Evaluating Alternatives

Abstract

This chapter reviews the use of aggregation functions and operators in the field of decision making. We first present an overview of main decision making problems, and, then, we show that aggregation operators are in common use for solving them. Once having presented their interest, we describe the major aggregation operators, their properties and their major differences.

Vicenç Torra

Optimization and Aggregation Functions

Introduction

In this work we will look at connections between aggregation functions and optimization. There are two such connections: 1) aggregation functions are used to transform a multiobjective optimization problem into a single objective problem by aggregating several criteria into one, and 2) construction of aggregation functions often involves an optimization problem.

Gleb Beliakov

Chebyshev Approximation of Inconsistent Fuzzy Relational Equations with Max-T Composition

Abstract

This paper considers resolving the inconsistency of a system of fuzzy relational equations with max-T composition by simultaneously modifying the coefficient matrix and the right hand side vector. We show that resolving the inconsistency of fuzzy relational equations with max-T composition by means of Chebyshev approximation is closely related to the generalized solvability of interval-valued fuzzy relational equations with max-T composition. An efficient procedure is proposed to obtain a consistent system with the smallest perturbation in the sense of Chebyshev distance.

Pingke Li, Shu-Cherng Fang

Various Types of Fuzzy Optimization and Fuzzy Mathematical Programming Models

Frontmatter

A Survey of Fuzzy Convex Programming Models

Abstract

Optimization is a procedure of finding and comparing feasible solutions until no better solution can be found. It can be divided into several fields, one of which is the Convex Optimization. It is characterized by a convex objective function and convex constraint functions over a convex set which is the set of the decision variables. This can be viewed, on the one hand, as a particular case of nonlinear programming and, on the other hand, as a general case of linear programming. Convex optimization has applications in a wide range of real-world applications, whose data often cannot be formulate precisely. Hence it makes perfect sense to apply fuzzy set theory as a way to mathematically describe this vagueness. In this paper we review the theory about this topic and describe some flexible and possibilistic programming models to solve fuzzy convex programming problems. Flexible programming uses fuzzy sets to represent the vagueness of the decision maker’s aspirations and constraints, while possibilistic programming models imprecise or ambiguous data by possibility distributions.

Ricardo C. Silva, Carlos Cruz, José L. Verdegay, Akebo Yamakami

Approaches to Linear Programming Problems with Interactive Fuzzy Numbers

Abstract

Fuzzy programming has been developed mainly under the assumption of non-interaction among fuzzy coefficients. However, this assumption is not always suitable in the treatment of real world problems. Several approaches have been proposed to treat the interaction among fuzzy coefficients. In this paper, we review treatments of interaction among fuzzy coefficients in fuzzy linear programming problems. Using a necessity fractile model of a simple linear program with fuzzy coefficients, we will see the differences between non-interactive and interactive problems. We review the five approaches to interactive fuzzy numbers, i.e., weak independent fuzzy numbers, a fuzzy vector with a quadratic membership function, scenario decomposed fuzzy numbers, an oblique fuzzy vector, and a fuzzy polytope.

Masahiro Inuiguchi

Possibilistic Optimization Tasks with Mutually T-Related Parameters: Solution Methods and Comparative Analysis

Abstract

The problems of possibilistic linear programming are studied in the article. Unlike in other known related publications, t-norms are used to describe the interaction (relatedness) of fuzzy parameters. Solution methods are proposed, models of possibilistic optimization are compared for different t-norms.

Alexander Yazenin, Ilia Soldatenko

A Parametrized Model for Optimization with Mixed Fuzzy and Possibilistic Uncertainty

Abstract

Fuzzy and possibilistic uncertainty are very closely related, and sometimes coexist in optimization under uncertainty problems. Fuzzy uncertainty in mathematical programming problems typically represents flexibility on the part of the decision maker. On the other hand, possibilistic uncertainty generally expresses a lack of information about the values the parameters will assume.

Several models for mixed fuzzy and possibilistic programming problems have previously been published. The semantic interpretation of these models, however, is of questionable value. The mixed models in the literature find solutions in which the fuzzy uncertainty (or flexibility) and the possibilistic uncertainty (or lack of confidence in the outcome) are held to the same levels.

This chapter proposes a new mixed model which allows a trade-off between fuzzy and possibilistic uncertainty. This trade-off corresponds to a semantic interpretations consistent with human decision-making. The new model shares characteristics with multi-objective programming and Markowitz models. Model structure, semantic justification, and solution approaches are covered.

Elizabeth Untiedt

On Solving Optimization Problems with Ordered Average Criteria and Constraints

Abstract

The problem of aggregating multiple numerical attributes to form overall measure is of considerable importance in many disciplines. The ordered weighted averaging (OWA) aggregation, introduced by Yager, uses the weights assigned to the ordered values rather than to the specific attributes. This allows one to model various aggregation preferences, preserving simultaneously the impartiality (neutrality) with respect to the individual attributes. However, importance weighted averaging is a central task in multiattribute decision problems of many kinds. It can be achieved with theWeighted OWA (WOWA) aggregation though the importanceweightsmake the WOWA concept much more complicated than the original OWA. In this paper we analyze solution procedures for optimization problems with the ordered average objective functions or constraints. We show that the WOWA aggregation with monotonic preferential weights can be reformulated in a way allowing to introduce linear programming optimization models, similar to the optimization models we developed earlier for the OWA aggregation. Computational efficiency of the proposed models is demonstrated.

Włodzimierz Ogryczak, Tomasz Śliwiński

Fuzzy Dynamic Programming Problem for Extremal Fuzzy Dynamic System

Abstract

This work deals with the problems of the Expremal Fuzzy Continuous Dynamic System (EFCDS) optimization problems and briefly discuss the results developed by G. Sirbiladze [31]–[38]. The basic properties of extended extremal fuzzy measure are considered and several variants of their representation are given. In considering extremal fuzzy measures, several transformation theorems are represented for extended lower and upper Sugeno integrals. Values of extended extremal conditional fuzzy measures are defined as a levels of an expert knowledge reflections of EFCDS states in the fuzzy time intervals. The notions of extremal fuzzy time moments and intervals are introduced and their monotone algebraic structures that form the most important part of the fuzzy instrument of modeling extremal fuzzy dynamic systems are discussed. New approaches in modeling of EFCDS are developed. Applying the results of [31] and [32], fuzzy processes with possibilistic uncertainty, the source of which is extremal fuzzy time intervals, are constructed. The dynamics of EFCDS’s is described. Questions of the ergodicity of EFCDS’s are considered. Fuzzy-integral representations of controllable extremal fuzzy processes are given. Sufficient and necessary conditions are presented for the existence of an extremal fuzzy optimal control processes, for which we use R. Bellman’s optimality principle and take into consideration the gain-loss fuzzy process. A separate consideration is given to the case where an extremal fuzzy control process acting on the EFCDS does not depend on an EFCDS state. Applying Bellman’s optimality principle and assuming that the gain-loss process exists for the EFCDS, a variant of the fuzzy integral representation of an optimal control is given for the EFCDS. This variant employs the instrument of extended extremal fuzzy composition measures constructed in [32]. An example of constructing of the EFCDS optimal control is presented.

Gia Sirbiladze

Vaguely Motivated Cooperation

Abstract

The transferable utility (TU) cooperative games are used as an effective mathematical representation of cooperation and coalitions forming. This contribution deals with a modified form of such games in which the expected pay-offs of coalitions are known only vaguely, where the vagueness is modelled by means of fuzzy quantities and some other fuzzy set theoretical concepts. Such games were investigated in [8] and in some other papers. Their cores and Shapley values were analyzed and some of their basic properties were shown. This contribution is to extend that analysis, namely from the point of view of the motivation of players to cooperate in coalitions, as well as the relation between the willingness to cooperate and the ability to find the conditions under that the cooperation can be percepted as fair.

Milan Mareš

Fuzzy Network and Combinatorial Optimization

Frontmatter

Computing Min-Max Regret Solutions in Possibilistic Combinatorial Optimization Problems

Abstract

In this chapter we discuss a wide class of combinatorial optimization problems with a linear sum and a bottleneck cost function. We first investigate the case when the weights in the problem are modeled as closed intervals. We show how the notion of optimality can be extended by using a concept of a deviation interval. In order to choose a solution we adopt a robust approach. We seek a solution that minimizes the maximal regret, that is the maximal deviation from optimum over all weight realizations, called scenarios, which may occur. We then explore the case in which the weights are specified as fuzzy intervals. We show that under fuzzy weights the problem has an interpretation consistent with possibility theory. Namely, fuzzy weights induce a possibility distribution over the scenario set and the possibility and necessity measures can be used to extend the optimality evaluation and the min-max regret approach.

Adam Kasperski, Paweł Zieliński

Stochastic Bottleneck Spanning Tree Problem on a Fuzzy Network

Abstract

This paper considers a fuzzy network version of the stochastic bottleneck spanning tree problem. Existence of each edge is not necessary certain and it is given by a certain value between 0 and 1. 1 means that it exists certainly and 0 means it does not exist. For intermediate numbers, a higher value corresponds to a higher possibility of existence. Furthermore each edge has a random cost independent to other edges. The probability that the maximum burden among these selected edges is not greater than the capacity should be not less than the fixed probability. Under the above setting, we seek a spanning tree minimizing the capacity and maximizing the minimal existence possibility among these selected edges. Since usually there is no spanning tree optimizing two objectives at a time, we derive an efficient solution procedure to obtain a set of some non-dominated spanning tree after defining non-domination of spanning trees. Finally we discuss the further research problems.

Yue Ge, Hiroaki Ishii

The Use of Fuzzy Numbers in Practical Project Planning and Control

Abstract

The paper proposes how to use fuzzy numbers in project planning and control in such a way that it would have a chance to be used in practice. The method is destined for all the projects, but especially for those where in the initial phase the knowledge about the project is very incomplete and is made stepwise more precise during the project execution, also for those in which initial assumptions about the project execution are due to later changes.

Dorota Kuchta

Applications

Frontmatter

Ant Feature Selection Using Fuzzy Decision Functions

Abstract

One of the most important stages in data preprocessing for data mining is feature selection. Real-world data analysis, data mining, classification and modeling problems usually involve a large number of candidate inputs or features. Less relevant or highly correlated features decrease in general the classification accuracy, and enlarge the complexity of the classifier. Feature selection is a multi-criteria optimization problem with contradictory objectives, which are difficult to properly describe by conventional cost functions. This chapter proposes the use of fuzzy optimization to improve the performance of this type of system, since it allows for an easier and more transparent description of the criteria used in the feature selection process. In our previous work, an ant colony optimization algorithm for feature selection was proposed, which minimized two objectives: number of features and classification error. In this chapter, a fuzzy objective function is proposed to cope with the difficulty of weighting the different criteria involved in the optimization algorithm. The application of fuzzy feature selection to two benchmark problems show the usefulness of the proposed approach.

Susana M. Vieira, João M. C. Sousa, Uzay Kaymak

Application of Fuzzy Theory to the Investment Decision Process

Abstract

In the present paper, we propose a new approach to portfolio optimization that allows portfolio managers to construct portfolios that reflect their views about risk assets by applying fuzzy theory. The proposed approach to the investment decision process is based on the mean-variance approach proposed by Markowitz (1952,1959) and uses the concept of asset market equilibrium proposed by Sharpe (1964). For portfolio managers, it is very meaningful to use the equilibrium expected excess returns associated with the capital market as a reference. The proposed approach enables a new method for incorporating the views of portfolio managers to aid in the investment decision process. Moreover, in order to estimate the distribution of an unknown true membership function of the views of portfolio managers concerning risk assets, we propose a fuzzy information criterion to evaluate the fitness of the distribution between an unknown true membership function and a hypothetical membership function. In particular, the proposed approach enables a group of portfolio managers of pension funds to obtain an important solution that realizes optimal expected excess returns of risk assets by specifying the vague views of portfolio managers as a fuzzy number.

Hiroshi Tsuda, Seiji Saito

Decision Making Techniques in Political Management

Abstract

In this paper, we develop a new decision making model and apply it in political management. We use a framework based on the use of ideals in the decision process and several similarity measures such as the Hamming distance, the adequacy coefficient and the index of maximum and minimum level. For each similarity measure, we use different types of aggregation operators such as the simple average, the weighted average, the ordered weighted averaging (OWA) operator and the generalized OWA (GOWA) operator. This new approach considers several attributes and different scenarios that may occur in the uncertain environment. We see that depending on the particular type of aggregation operator used the results may lead to different decisions.

Anna M. Gil-Lafuente, José M. Merigó

Mathematical Approaches for Fuzzy Portfolio Selection Problems with Normal Mixture Distributions

Abstract

This chapter considers some versatile portfolio selection models with general normal mixture distributions and fuzzy or interval numbers. Then, these mathematical approaches to obtain the optimal portfolio are developed. Furthermore, in order to compare our proposed models with standard models and represent the advantage of our proposed models, a numerical example is provided.

Takashi Hasuike, Hiroaki Ishii

Fuzzy Random Redundancy Allocation Problems

Abstract

Due to subjective judgement, imprecise human knowledge and perception in capturing statistical data, the real data of lifetimes in many systems are both random and fuzzy in nature. Based on the fuzzy random variables that are used to characterize the lifetimes, this paper studies the redundancy allocation problems to a fuzzy random parallel-series system.

Two fuzzy random redundancy allocation models (FR-RAM) are developed through reliability maximization and cost minimization, respectively. Some properties of the FR-RAM are obtained, where an analytical formula of reliability with convex lifetimes is derived and the sensitivity of the reliability is discussed. To solve the FR-RAMs, we first address the computation of reliability. A random simulation method based on the derived analytical formula is proposed to compute the reliability with convex lifetimes. As for the reliability with nonconvex lifetimes, the technique of fuzzy random simulation together with the discretization method of fuzzy random variable is employed to compute the reliability, and a convergence theorem of the fuzzy random simulation is proved. Subsequently, we integrate the computation approaches of the reliability and genetic algorithm (GA) to search for the approximately optimal redundancy allocation of the models. Finally, some numerical examples are provided to illustrate the feasibility of the solution algorithm and quantify its effectiveness.

Shuming Wang, Junzo Watada

Reliable Biological Circuit Design Including Uncertain Kinetic Parameters

Abstract

In the context of possibilistic decision making, this work deals with biological design problems particularly important in the near future when it will be possible to produce biological entities and synthetic organisms for pharmacological and medical usage. The biological systems is investigated in terms of performances or main key features of the system. The analysis of the biological system is based on the idea that the set of parameters involved in the model can be classified into two different typologies: the uncertain kinetic parameters and the control design parameters. In order to design a robust and reliable biological system with respect to a target performance, the design parameter values are set up to balance the uncertainty of the kinetic parameters. To take into account these uncertainties arising from the estimations of the kinetic parameters, the function representing the feedback of the system is fuzzified and a measure of failure of the designed biological circuit is minimized to reach the required performance. An application of this methodology is illustrated on a case study of an autonomously oscillatory system: the Drosophila Period Protein which is a central component of the Drosophila circadian clocks. Finally, the results of the fuzzy methodology are compared with a deterministic method.

Eva Sciacca, Salvatore Spinella

Fuzzy Optimal Algorithms for Multiple Target Convergence

Abstract

The proposed fuzzy application is the use of fuzzy algorithms for a networked swarm of autonomous vehicles, such as those used in planet exploration, and to be used in target location determination and convergence. In particular, an algorithm of this type could be used in an Autonomous Stratospheric Aircraft (ASA), thus having the possibility of being used for the exploration of a planet as well as many other applications. Upon finding an unknown location of a specified target, the algorithm would then swarm and eventually converge upon the location. There are two similar, but fundamentally different algorithms proposed in this presentation. These algorithms are capable of locating and converging upon multiple targeted locations simultaneously. This project is inspired by the current thought of NASA in the search of life on Mars, which is ”Follow the Water” [17] where the targeted location would be a water source. These algorithms make use of combining a modified Particle Swarm Optimization algorithm combined with fuzzy variables for added intelligence.

Zach Richards

Fuzzy Linear Programming in Practice: An Application to the Spanish Football League

Abstract

FLP problems are perhaps one of the most and best studied topics of Soft Computing, and are among the most fruitful in applications and in theoretical and practical results. Areas of application of FLP problems are many and varied and in fact suppose an extraordinary example of technology transfer in action. In this paper, Fuzzy Linear Programming models are applied to the European football game in which the inherent uncertainty of the parameters relating to the football teams in the Spanish Football League serve to establish these models and so optimize the returns of the investments made to maintain a high quality competition. In this context, fuzzy DEA models are established which provide teams predictions as to their efficiency score. At the end of the study we offer some experiments using data from the Spanish Football League 2006/07.

J. M. Cadenas, V. Liern, R. Sala, J. L. Verdegay

Backmatter

Titel: Fuzzy Optimization
herausgegeben von: Weldon A. Lodwick
Janusz Kacprzyk
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-13935-2
Print ISBN: 978-3-642-13934-5
DOI: https://doi.org/10.1007/978-3-642-13935-2

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