Fuzzy Bilevel Optimization

In the dissertation the solution approaches for different fuzzy optimization problems are presented. The single-level optimization problem with fuzzy objective is solved by its reformulation into a biobjective optimization problem. A special attention is given to the computation of the membership function of the fuzzy solution of the fuzzy optimization problem in the linear case. Necessary and sufficient optimality conditions of the the convex nonlinear fuzzy optimization problem are derived in differentiable and nondifferentiable cases. A fuzzy optimization problem with both fuzzy objectives and constraints is also investigated in the thesis in the linear case. These solution approaches are applied to fuzzy bilevel optimization problems. In the case of bilevel optimization problem with fuzzy objective functions, two algorithms are presented and compared using an illustrative example. For the case of fuzzy linear bilevel optimization problem with both fuzzy objectives and constraints k-th best algorithm is adopted.:1 Introduction 1 1.1 Why optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Fuzziness as a concept . . . . . . . . . . . . . . . . . . . . .. . . . . . . 2 1.3 Bilevel problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Preliminaries 11 2.1 Fuzzy sets and fuzzy numbers . . . . . . . . . . . . . . . . . . . . . 11 2.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Fuzzy order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Fuzzy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 3 Optimization problem with fuzzy objective 19 3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Local optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 Existence of an optimal solution . . . . . . . . . . . . . . . . . . . . 25 4 Linear optimization with fuzzy objective 27 4.1 Main approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 Membership function value . . . . . . . . . . . . . . . . . . . . . . . . 34 4.4.1 Special case of triangular fuzzy numbers . . . . . . . . . . . . 36 4.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 5 Optimality conditions 47 5.1 Differentiable fuzzy optimization problem . . . . . . . . . . .. . . . 48 5.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.1.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . .. 49 5.1.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Nondifferentiable fuzzy optimization problem . . . . . . . . . . . . 51 5.2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . . 52 5.2.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 54 5.2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6 Fuzzy linear optimization problem over fuzzy polytope 59 6.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2 The fuzzy polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 6.3 Formulation and solution method . . . . . . . . . . . . . . . . . . .. . 65 6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7 Bilevel optimization with fuzzy objectives 73 7.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7.2 Solution approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74 7.3 Yager index approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.4 Algorithm I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 7.5 Membership function approach . . . . . . . . . . . . . . . . . . . . . . .78 7.6 Algorithm II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80 7.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8 Linear fuzzy bilevel optimization (with fuzzy objectives and constraints) 87 8.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.2 Solution approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 8.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 9 Conclusions 95 Bibliography 9

Evolutionary multi-objective worst-case robust optimisation

Many real-world problems are subject to uncertainty, and often solutions should not only be good, but also robust against environmental disturbances or deviations from the decision variables. While most papers dealing with robustness aim at finding solutions with a high expected performance given a distribution of the uncertainty, we examine the trade-off between the allowed deviations from the decision variables (tolerance level), and the worst case performance given the allowed deviations. In this research work, we suggest two multi-objective evolutionary algorithms to compute the available trade-offs between allowed tolerance level and worst-case quality of the solutions, and the tolerance level is defined as robustness which could also be the variations from parameters. Both algorithms are 2-level nested algorithms. While the first algorithm is point-based in the sense that the lower level computes a point of worst case for each upper level solution, the second algorithm is envelope-based, in the sense that the lower level computes a whole trade-off curve between worst-case fitness and tolerance level for each upper level solution. Our problem can be considered as a special case of bi-level optimisation, which is computationally expensive, because each upper level solution is evaluated by calling a lower level optimiser. We propose and compare several strategies to improve the efficiency of both algorithms. Later, we also suggest surrogate-assisted algorithms to accelerate both algorithms

Game theoretic optimization for product line evolution

Product line planning aims at optimal planning of product variety. In addition, the traditional product line planning problem develops new product lines based on product attributes without considering existing product lines. However, in reality, almost all new product lines evolve from existing product lines, which leads to the product line evolution problem. Product line evolution involves trade-offs between the marketing perspective and engineering perspective. The marketing concern focuses on maximizing utility for customers; the engineering concern focuses on minimizing engineering cost. Utility represents satisfaction experienced by the customers of a product. Engineering cost is the total cost involved in the process of the development of a product line. These two goals are in conflict since the high utility requires high-end product attributes which could increase the engineering cost and vice versa. Rather than aggregating both problems as one single level optimization problem, the marketing and engineering concerns entail a non-collaborative game per se. This research investigates a game-theoretic approach to the product line evolution problem. A leader-follower joint optimization model is developed to leverage conflicting goals of marketing and engineering concerns within a coherent framework of game theoretic optimization. To solve the joint optimization model efficiently, a bi-level nested genetic algorithm is developed. A case study of smart watch product line evolution is reported to illustrate the feasibility and potential of the proposed approach.M.S

Soft Computing

Soft computing is used where a complex problem is not adequately specified for the use of conventional math and computer techniques. Soft computing has numerous real-world applications in domestic, commercial and industrial situations. This book elaborates on the most recent applications in various fields of engineering

Soft Computing

Soft computing is used where a complex problem is not adequately specified for the use of conventional math and computer techniques. Soft computing has numerous real-world applications in domestic, commercial and industrial situations. This book elaborates on the most recent applications in various fields of engineering

Continual improvement: A bibliography with indexes, 1992-1993

This bibliography lists 606 references to reports and journal articles entered into the NASA Scientific and Technical Information Database during 1992 to 1993. Topics cover the philosophy and history of Continual Improvement (CI), basic approaches and strategies for implementation, and lessons learned from public and private sector models. Entries are arranged according to the following categories: Leadership for Quality, Information and Analysis, Strategic Planning for CI, Human Resources Utilization, Management of Process Quality, Supplier Quality, Assessing Results, Customer Focus and Satisfaction, TQM Tools and Philosophies, and Applications. Indexes include subject, personal author, corporate source, contract number, report number, and accession number

Bilevel programming: reformulations, regularity, and stationarity

We have considered the bilevel programming problem in the case where the lower-level problem admits more than one optimal solution. It is well-known in the literature that in such a situation, the problem is ill-posed from the view point of scalar objective optimization. Thus the optimistic and pessimistic approaches have been suggested earlier in the literature to deal with it in this case. In the thesis, we have developed a unified approach to derive necessary optimality conditions for both the optimistic and pessimistic bilevel programs, which is based on advanced tools from variational analysis. We have obtained various constraint qualifications and stationarity conditions depending on some constructive representations of the solution set-valued mapping of the follower’s problem. In the auxiliary developments, we have provided rules for the generalized differentiation and robust Lipschitzian properties for the lower-level solution setvalued map, which are of a fundamental interest for other areas of nonlinear and nonsmooth optimization. Some of the results of the aforementioned theory have then been applied to derive stationarity conditions for some well-known transportation problems having the bilevel structure

An adaptive multi-agent system for self-organizing continuous optimization

Cette thèse présente une nouvelle approche pour la distribution de processus d'optimisation continue dans un réseau d'agents coopératifs. Dans le but de résoudre de tels problèmes, le domaine de l'optimisation multidisciplinaire a été proposé. Les méthodes d'optimisation multidisciplinaire proposent de distribuer le processus d'optimisation, généralement en reformulant le problème original d'une manière qui réduit les interconnexions entre les disciplines. Cependant, ces méthodes présentent des désavantages en ce qui concerne la difficulté de les appliquer correctement, ainsi que leur manque de flexibilité. En se basant sur la théorie des AMAS (Adaptive Multi-Agent Systems), nous proposent une représentation générique à base d'agents des problèmes d'optimisation continue. A partir de cette représentation, nous proposons un comportement nominal pour les agents afin d'exécuter le processus d'optimisation. Nous identifions ensuite certaines configurations spécifiques qui pourraient perturber le processus, et présentons un ensemble de comportements coopératifs pour les agents afin d'identifier et de résoudre ces configurations problématiques. Enfin, nous utilisons les mécanismes de coopération que nous avons introduit comme base à des patterns de résolution coopérative de problèmes. Ces patterns sont des recommandations de haut niveau pour identifier et résoudre des configurations potentiellement problématiques qui peuvent survenir au sein de systèmes de résolution collective de problèmes. Ils fournissent chacun un mécanisme de résolution coopérative pour les agents, en utilisant des indicateurs abstraits qui doivent être instanciés pour le problème en cours.In an effort to tackle such complex problems, the field of multidisciplinary optimization methods was proposed. Multidisciplinary optimization methods propose to distribute the optimization process, often by reformulating the original problem is a way that reduce the interconnections between the disciplines. However these methods present several drawbacks regarding the difficulty to correctly apply them, as well as their lack of flexibility. Based on the AMAS (Adaptive Multi-Agent Systems) theory, we propose a general agent-based representation of continuous optimization problems. From this representation we propose a nominal behavior for the agents in order to do the optimization process. We then identify some specific configurations which would disturb this nominal optimization process, and present a set of cooperative behaviors for the agents to identify and solve these problematic configurations. At last, we use the cooperation mechanisms we introduced as the basis for more general Collective Problem Solving Patterns. These patterns are high-level guideline to identify and solve potential problematic configurations which can arise in distributed problem solving systems. They provide a specific cooperative mechanism for the agents, using abstract indicators that are to be instantiated on the problem at hand

Toll competition in highway transportation networks

Within a highway transportation network, the social welfare implications of two different groups of agents setting tolls in competition for revenues are studied. The first group comprises private sector toll road operators aiming to maximise revenues. The second group comprises local governments or jurisdictions who may engage in tax exporting. Extending insights from the public economics literature, jurisdictions tax export because when setting tolls to maximise welfare for their electorate, they simultaneously benefit from revenues from extra-jurisdictional users. Hence the tolls levied by both groups will be higher than those intended solely to internalise congestion, which then results in welfare losses. Therefore the overarching question investigated is the extent of welfare losses stemming from such competition for toll revenues. While these groups of agents are separately studied, the interactions between agents in each group in competition can be modelled within the common framework of Equilibrium Problems with Equilibrium Constraints. Several solution algorithms, adapting methodologies from microeconomics as well as evolutionary computation, are proposed to identify Nash Equilibrium toll levels. These are demonstrated on realistic transportation networks. As an alternative paradigm to competition, the possibilities for co-operation between agents in each group are also explored. In the case of toll road operators, the welfare consequences of competition could be positive or adverse depending on the interrelationships between the toll roads in competition. The results therefore generalise those previously obtained to a more realistic setting investigated here. In the case of competition between jurisdictions, it is shown that the fiscal externality of tax exporting resulting from their toll setting decisions can substantially reduce the welfare gains from internalising congestion. The ability of regulation, co-operation and bilateral bargaining to reduce the welfare losses are assessed. The research thus contributes to informing debates regarding the appropriate level of institutional governance for toll pricing policies