Proceedings of the 14th International Middle East Power Systems Conference (MEPCON’10), Cairo University, Egypt, December 19-21, 2010, Paper ID 306.
Fuzzy Geometric Programming Optimization using New Arithmetic Fuzzy Logic-based Representation Sameh Farid Saad Eid
Ahmed Mohammed A. M. Kamel
Hassen Taher Dorrah
Automatic Control and System Engineering Group, Dept. of Electric Power and Machines Engineering Faculty of Engineering, Cairo University, Giza, Egypt
[email protected] [email protected] [email protected] The advantages of applying fuzzy optimization model to solving practical problems as follows: on the one hand, it can avoid rigidity and stiffness arising from conventional optimization model deal with some practical optimization problems; on the other hand, it can also efficiently reduce information loss arising from the conventional optimization model operation to data [2].
Abstract - The paper is directed towards solving the unconstrained geometric programming problem in fully fuzzy environment. The approach is based on using the arithmetic fuzzy logic based representation. In this representation, each parameter has a pair of value and fuzzy level. The fuzzy level is considered as an indication of the uncertainty in the corresponding parameter. Using the new representation, the conventional geometric programming algorithm is modified to take into consideration the effect of the uncertainty (the fuzzy level). This modification is achieved by making the calculations of each step of the modified algorithm in pairs. Consequently, the proposed fuzzy geometric programming optimization algorithm gives the optimal solution in a form of an optimal value and its corresponding fuzzy level. The fuzzy level of the optimal solution gives the uncertainty in its value due to the uncertainties of the problem input parameters. Two examples are given to demonstrate the efficacy and reliability of the suggested technique. It is shown that the provided technique is powerful to handle the large scale fuzzy problem in a systematic way, and possesses the properties of linearity and reversibility.
Deterministic Heuristic Optimization
State Space Search
Branch and Bound
Probabilistic
Algebraic Geometry
Artificial Intelligence (AI)
Monte Carlo Algorithms
Expert Systems
Soft Computing Computational Intelligence (CI)
Stochastic Dynamic Programming
Index Terms – Geometric Programming, Fuzzy systems, Fuzzy Optimization, Arithmetic Fuzzy Logic-based Representation.
Evolutionary Computation (EC) Memetic Algorithms
Mathematical Programming Random Optimization
I. INTRODUCTION
Possibilistic Programming
O
ptimization 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 find a best possible option or solution. Examples of various optimization techniques are shown in Figure 1.
Simulated Annealing (SA)
Genetic Algorithms (GA) (LAS) Learning Classifier System
Swarm Intelligence (SI) Ant Colony Optimization (ACO) Particle Swarm Optimization (PSO)
Tabu Search (TS)
Evolutionary Programming
Parallel Tempering
Evolution Strategy (ES)
Differential Evolution (DE)
(GP) Genetic Programming
Standard Genetic Programming
Stochastic Tunneling
Real applications of optimization often contain information and data that is imperfect. Thus, attempts have been made since the early days to develop optimization models for handling such cases. As the first, natural approaches in this respect one can mention value intervals and probability distributions as representations of uncertain data. They have led to the development of various interval and stochastic optimization models.
Harmonic Search (HS)
Evolutionary Algorithms (EA)
Other Operations Research Techniques
Linear Genetic Programming Grammar Guided Genetic Prog.
Fig. 1 The Taxonomy of Global Optimization Algorithms.
Geometric programming (GP) is a relatively new method of solving a class of nonlinear programming problems. It is used to minimize functions that are in the form of posynomials subject to constraints of the same type. It differs from other optimization techniques in the emphasis it places on the relative magnitudes of the terms of the objective function rather than the variables. Instead of finding optimal values of the design variables first, geometric programming first finds the optimal value of the objective function. This feature is
Fuzzy sets theory has provided conceptually powerful and constructive tools and techniques to handle another aspect of imperfect information related to vagueness and imprecision. This has resulted in the emergence of a new field, called fuzzy optimization (and its related fuzzy mathematical programming) [1].
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especially advantageous in situations where the optimal value of the objective function may be all that is of interest. In such cases, calculation of the optimum design vectors can be omitted. Another advantage of geometric programming is that it often reduces a complicated optimization problem to one involving a set of simultaneous linear algebraic equations. The major disadvantage of the method is that it requires the objective function and the constraints in the form of posynomials [3].
Furthermore, any uncertainty in the problem parameters requires resolution of the problem using the modified parameter values. In order to overcome these difficulties the new arithmetic fuzzy logic-based representation developed by Gabr and Dorrah [6-11] is used for solving geometric programming in fully fuzzy environment. In this representation, each parameter is expressed as a pair of parentheses, the first is the actual value and the second is the corresponding fuzzy level, (Value, Fuzzy Level)
Geometric Programming can be considered to be an innovative modus operandi to solve a nonlinear problem in comparison with other nonlinear techniques. It was originally developed to design engineering problems. It has become a very popular technique since its inception in solving nonlinear problems [4].
II. BRIEF DESCRIPTION of ARITHMETIC FUZZY LOGICBASED REPRESENTATION The newly suggested Arithmetic fuzzy logic-based Representation approach developed by Gabr and Dorrah [611] is based on representing each parameter by two components is the deterministic equivalence, and is the fuzzy equivalence representing a small uncertainty or value tolerance in the parameter . The term is modeled by the formula: where is the relative unit fuzziness (usually a certain small percentage), and is the corresponding fuzzy level. For the sake of simplicity is omitted in the representation and the parameter is expressed by the following pair:
Fuzzy logic optimization is an extension of global optimization techniques operating in fuzzy environment. A short survey of the common fuzzy logic optimization techniques is elucidated in Fig.2. In general, these methods represent an extension of global optimization techniques in fuzzy environment [5].
Type 2 Fuzzy Modeling
Linguistic Modeling
Fuzzy Cognitive Mapping
Fuzzy Matrices
Fuzzy Modeling and Optimization Techniques
Fuzzy Modeling Techniques
Fuzzy Optimization Techniques
Fuzzy Monte Carto Methods
(1) Fuzzy BiMatrices
where the first term in the pair is the equivalent deterministic component, and is an integer value indicating the corresponding fuzzy level of X . The scaled or normalized fuzzy term is , such that .
Fuzzy Interval Matrices
Fuzzy Super Matrices Heuristic Optimization
Fuzzy Mathematical Programming
In general, the fuzzy level could be extended from level to where and are integers. If the levels are selected from to , the value of the relative fuzziness is restricted such as and (preferably will be
