توضیحات
ABSTRACT
Fuzzy and possibilistic optimization methods are demonstrated to be effective tools in solving large-scale problems. In particular, an optimization problem in radiation therapy with various orders of complexity from 1000 to 62,250 constraints for fuzzy and possibilistic linear and nonlinear programming implementations possessing (1) fuzzy or soft inequalities, (2) fuzzy right-hand side values, and (3) possibilistic right-hand side is used to demonstrate that fuzzy and possibilistic optimization methods are tractable and useful. We focus on the uncertainty in the right side of constraints which arises, in the context of the radiation therapy problem, from the fact that minimal and maximal radiation tolerances are ranges of values, with preferences within the range whose values are based on research results, empirical findings, and expert knowledge, rather than fixed real numbers. The results indicate that fuzzy/possibilistic optimization is a natural and effective way to model various types of optimization under uncertainty problems and that large fuzzy and possibilistic optimization problems can be solved efficiently.
INTRODUCTION
Many of the hardest optimization problems are those that contain uncertainty because the meanings of inequalities and optima must be defined in the context of the problem in question. Moreover, the complexity of uncertain optimization is formidable. Our research focuses on three approaches to fuzzy and possibilistic uncertainty optimization – (1) fuzzy optimization of Tanaka, Okuda, and Asai (1974) and Zimmermann (1976); (2) the fuzzy optimization based on the surprise functions of Neumaier (2003) and (Lodwick, Neumaier, and Newman (2001)); and (3) the possibilistic optimization of Jamison and Lodwick (2001). Each of these methods is used to solve one of the three uncertainty types that is the focus of our study (fuzzy inequalities, fuzzy right-hand side constraint values, and possibilistic right-hand side values, respectively). We present new results associated with possibilistic optimization which enables a closed form representation and makes possibilistic optimization amenable to algorithms that are fast even for large-scale problems. Our purpose is to show that when fuzzy and possibilistic uncertainty is present in an optimization problem, tractable and efficient numerical methods exist to solve large-scale problems enabling the uncertainty which is present to be explicitly modeled. To do this, an efficient possibilistic method had to be developed (see Section 3, Theorem 1). Moreover, the performance of the surprise function method, which has attractive theoretical properties, needed to be tested on largescale problems to see how it handled fuzzy right-hand side constraint values in practice. Since the Tanaka and Zimmermann approach translates the fuzzy optimization problem into a linear programming problem with one more constraint than its crisp counterpart, its performance and complexity is just that of linear programs, so there is nothing new in what we present except that the application of fuzzy linear programming to a real large- scale problem is novel. The association between the types of uncertainty and solution method is also new.
Year: 2015
Publisher : P.O
By : WELDON A. LODWICK, KATHERINE A. BACHMAN
File Information: English Language/23 Page / size:237 KB
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سال : 2015
ناشر : P.O
کاری از : WELDON A. LODWICK, KATHERINE A. BACHMAN
اطلاعات فایل : زبان انگلیسی / 23 صفحه / حجم :237 KB
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