9,752 research outputs found
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
Different optimum notions for fuzzy functions and optimality conditions associated
Fuzzy numbers have been applied on decision and optimization problems
in uncertain or imprecise environments. In these problems, the necessity to define
optimal notions for decision-maker’s preferences as well as to prove necessary and
sufficient optimality conditions for these optima are essential steps in the resolution
process of the problem. The theoretical developments are illustrated and motivated
with several numerical examples.The research in this paper has been supported by MTM2015-66185 (MINECO/FEDER, UE) and
Fondecyt-Chile, Project 1151154
Numerical solution of a fuzzy time-optimal control problem
In this paper, we consider a time-optimal control problem with uncertainties.
Dynamics of controlled object is expressed by crisp linear system of
differential equations with fuzzy initial and final states. We introduce a
notion of fuzzy optimal time and reduce its calculation to two crisp optimal
control problems. We examine the proposed approach on an example.Comment: 11 pages, 3 figure
Improving the quality of the industrial enterprise management based on the network-centric approach
The article examines the network-centric approach to the industrial enterprise management to improve the ef ciency and effectiveness in the implementation of production plans and maximize responsiveness to customers. A network-centric management means the decentralized enterprise group management. A group means a set of enterprise divisions, which should solve by joint efforts a certain case that occurs in the production process. The network-centric management involves more delegation of authority to the lower elements of the enterprise’s organizational structure. The industrial enterprise is considered as a large complex system (production system) functioning and controlled amidst various types of uncertainty: information support uncertainty and goal uncertainty or multicriteria uncertainty. The information support uncertainty occurs because the complex system functioning always takes place in the context of incomplete and fuzzy information. Goal uncertainty or multicriteria uncertainty caused by a great number of goalsestablished for the production system. The network-centric management task de nition by the production system is formulated. The authors offer a mathematical model for optimal planning of consumers’ orders production with the participation of the main enterprise divisions. The methods of formalization of various types of uncertainty in production planning tasks are considered on the basis of the application of the fuzzy sets theory. An enterprise command center is offered as an effective tool for making management decisions by divisions. The article demonstrates that decentralized group management methods can improve the ef ciency and effectiveness of the implementation of production plans through the self-organization mechanisms of enterprise divisions.The work has been prepared with the financial support from the Russian Ministry of Education and Science (Contract No. 02.G25.31.0068 of 23.05.2013 as part of the measure to implement Decision of the Russian Government No. 218)
Advanced theoretical and experimental studies in automatic control and information systems
A series of research projects is briefly summarized which includes investigations in the following areas: (1) mathematical programming problems for large system and infinite-dimensional spaces, (2) bounded-input bounded-output stability, (3) non-parametric approximations, and (4) differential games. A list of reports and papers which were published over the ten year period of research is included
Uncertainty in Soft Temporal Constraint Problems:A General Framework and Controllability Algorithms forThe Fuzzy Case
In real-life temporal scenarios, uncertainty and preferences are often
essential and coexisting aspects. We present a formalism where quantitative
temporal constraints with both preferences and uncertainty can be defined. We
show how three classical notions of controllability (that is, strong, weak, and
dynamic), which have been developed for uncertain temporal problems, can be
generalized to handle preferences as well. After defining this general
framework, we focus on problems where preferences follow the fuzzy approach,
and with properties that assure tractability. For such problems, we propose
algorithms to check the presence of the controllability properties. In
particular, we show that in such a setting dealing simultaneously with
preferences and uncertainty does not increase the complexity of controllability
testing. We also develop a dynamic execution algorithm, of polynomial
complexity, that produces temporal plans under uncertainty that are optimal
with respect to fuzzy preferences
New optimality conditions for multiobjective fuzzy programming problems
In this paper we study fuzzy multiobjective optimization problems de ned for n variables. Based on a new p-dimensional fuzzy stationary-point de nition, necessary e ciency conditions are obtained. And we prove that these conditions are also su cient under new fuzzy generalized convexity notions. Furthermore, the results are obtained under general di erentiability hypothesis.The research in this paper has been supported by Fondecyt-Chile, project 1151154 and by Ministerio de Economía y
Competitividad, Spain, through grant MINECO/FEDER(UE) MTM2015-66185-P
A Better Approach for Solving a Fuzzy Multiobjective Programming Problem by Level Sets
In this paper, we deal with the resolution of a fuzzy multiobjective programming problem
using the level sets optimization. We compare it to other optimization strategies studied until now
and we propose an algorithm to identify possible Pareto efficient optimal solutions
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