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Optimization under uncertainty: Stateoftheart and opportunities
 Computers and Chemical Engineering
, 2004
"... A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemi ..."
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A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemicals. A key difficulty in optimization under uncertainty is in dealing with an uncertainty space that is huge and frequently leads to very largescale optimization models. Decisionmaking under uncertainty is often further complicated by the presence of integer decision variables to model logical and other discrete decisions in a multiperiod or multistage setting. This paper reviews theory and methodology that have been developed to cope with the complexity of optimization problems under uncertainty. We discuss and contrast the classical recoursebased stochastic programming, robust stochastic programming, probabilistic (chanceconstraint) programming, fuzzy programming, and stochastic dynamic programming. The advantages and shortcomings of these models are reviewed and illustrated through examples. Applications and the stateoftheart in computations are also reviewed. Finally, we discuss several main areas for future development in this field. These include development of polynomialtime approximation schemes for multistage stochastic programs and the application of global optimization algorithms to twostage and chanceconstraint formulations.
c ○ TÜB˙ITAK Solving Fuzzy Linear Programming Problems with Linear Membership Functions
"... In this paper, we concentrate on two kinds of fuzzy linear programming problems: linear programming problems with only fuzzy technological coefficients and linear programming problems in which both the righthand side and the technological coefficients are fuzzy numbers. We consider here only the ca ..."
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In this paper, we concentrate on two kinds of fuzzy linear programming problems: linear programming problems with only fuzzy technological coefficients and linear programming problems in which both the righthand side and the technological coefficients are fuzzy numbers. We consider here only the case of fuzzy numbers with linear membership functions. The symmetric method of Bellman and Zadeh [2] is used for a defuzzification of these problems. The crisp problems obtained after the defuzzification are nonlinear and even nonconvex in general. We propose here the “modified subgradient method ” and use it for solving these problems. We also compare the new proposed method with well known “fuzzy decisive set method”. Finally, we give illustrative examples and their numerical solutions. Key Words: Fuzzy linear programming; fuzzy number; modified subgradient method; fuzzy decisive set method.
Handling Integrated Quantitative and Qualitative Search Space
 in a Real World Optimization Problem”, Congress on Evolutionary Computation (CEC
, 2003
"... problems can be both quantitative (Q T) and qualitative (Q L) in nature, combining both types of information can result in more realistic solution for real world optimisation problems. However, most of the approaches reported in literature are incapable of conducting optimisation search in such mixe ..."
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problems can be both quantitative (Q T) and qualitative (Q L) in nature, combining both types of information can result in more realistic solution for real world optimisation problems. However, most of the approaches reported in literature are incapable of conducting optimisation search in such mixed environment. Therefore this paper proposes a mathematically proven methodology for handling integrated Q T and Q L search space in real world optimisation problems. The paper begins by presenting the definition of these optimisation problems an analysis of the challenges that they pose for existing optimisation strategies and related research. The paper then presents the proposed solution strategy and the mathmatical proof. Furthermore, a case study on rod rolling problem is presented to validate the effectiveness of the proposed metholodology. The paper concludes with a brief outline of limitations and future research activities. 1.
Modeling Uncertainty Using Probabilistic Based Possibility Theory With Applications To Optimization
, 1998
"... It is shown that possibility distributions can be formulated within the context of probability theory and that membership values of fuzzy set theory can be interpreted as cumulative probabilities. The basic functions and operations of possibility theory are interpreted within this setting. The proba ..."
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It is shown that possibility distributions can be formulated within the context of probability theory and that membership values of fuzzy set theory can be interpreted as cumulative probabilities. The basic functions and operations of possibility theory are interpreted within this setting. The probabilistic information that can be derived from possibility distributions is examined. This leads to two functionals that provide estimates for the expected value of a random variable, the expected average of a single possibility distribution and the estimated expectation that requires two special possibility distributions to compute. Secondly, the space of fuzzy numbers is examined. It is shown that this space can be partitioned into a vector space and that the expected average functional motivates a norm on this space. It is shown that for most applications, Cauchy sequences converge in this space. Thirdly, applications of this theory to problems in optimization are examined. The concept of ...
Control and Cybernetics
"... A generalized varyingdomain optimization method for fuzzy goal programming with priorities based on a genetic algorithm by ..."
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A generalized varyingdomain optimization method for fuzzy goal programming with priorities based on a genetic algorithm by
Fuzzy Variable Linear Programming with Fuzzy Technical Coefficients
"... Fuzzy linear programming is an application of fuzzy set theory in linear decision making problems and most of these problems are related to linear programming (LP) with fuzzy variables. In this paper, an approximate but convenient method for solving these problems with fuzzy nonnegative technical c ..."
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Fuzzy linear programming is an application of fuzzy set theory in linear decision making problems and most of these problems are related to linear programming (LP) with fuzzy variables. In this paper, an approximate but convenient method for solving these problems with fuzzy nonnegative technical coefficient and without using the ranking functions, is proposed. With the help of numerical examples, the method is illustrated. Keyword:Fuzzy linear programming, Fuzzy variable linear programming, Fuzzy number, Optimal solution, Decomposition method.
1 NESTED INTERVALS AND SETS: CONCEPTS, RELATIONS TO FUZZY SETS, AND APPLICATIONS
"... In data processing, we often encounter the following problem: Suppose that we have processed the measurement results ˜x1,..., ˜xn, and, from this processing, have obtained an estimate ˜y = f(˜x1,..., ˜xn) for a quantity y = f(x1,..., xn); we know the intervals xi of possible values of xi, and we wan ..."
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In data processing, we often encounter the following problem: Suppose that we have processed the measurement results ˜x1,..., ˜xn, and, from this processing, have obtained an estimate ˜y = f(˜x1,..., ˜xn) for a quantity y = f(x1,..., xn); we know the intervals xi of possible values of xi, and we want to find the interval y of possible values of y. Interval computations are one of the main techniques for solving this problem. In some cases, for each i, in addition to the guaranteed interval xi of possible values, we have a smaller interval that an expert believes to contain xi. There may be several such nested intervals. In these cases, in addition to the guaranteed interval y, it is desirable to know the possible intervals of y that correspond to the opinions of different experts. Techniques of such nested interval computations and reallife applications of these techniques are described in this paper. ∗ This work was partially carried out while Hung T. Nguyen was on sabbatical leave
A METHOD FOR SOLVING FULLY FUZZY QUADRATIC PROGRAMMING PROBLEMS
"... Abstract. In this paper we focus on a kind of quadratic programming with fuzzy numbers and variables. First by using a fuzzy ranking and arithmetic operations, we transform these problems to crisp model with nonlinear objective and linear constraints, then by solving this problem we obtain a fuzzy ..."
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Abstract. In this paper we focus on a kind of quadratic programming with fuzzy numbers and variables. First by using a fuzzy ranking and arithmetic operations, we transform these problems to crisp model with nonlinear objective and linear constraints, then by solving this problem we obtain a fuzzy optimal solution. Finally, we give an illustrative example. 2000 Mathematics Subject Classification: 90C70, 90C20. 1.
Evolutionary Techniques for Fuzzy Optimization Problems
"... In this paper we deal with mathematical programming problems with fuzzy constraints. Fuzzy solutions are obtained by means of a parametric approach in conjuntion with evolutionary techniques. Some important characteristics of the evolutionary algorithm are a natural representation of solution ..."
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In this paper we deal with mathematical programming problems with fuzzy constraints. Fuzzy solutions are obtained by means of a parametric approach in conjuntion with evolutionary techniques. Some important characteristics of the evolutionary algorithm are a natural representation of solutions, a problemindependent technique for constraint satisfaction, tournament selection, complete generational replacement, and elitism strategy. A numerical example is shown for the sake of ilustration. Keywords: Fuzzy mathematical programming, evolutionary algorithms. 1 Introduction A nonlinear programming problem can be stated as Min f(x) s:t: : g j (x) 0; j = 1; : : : ; m x 2 X (1) where f(x), g j (x), j = 1; : : : ; m are defined on ! n , X is a subset of ! n , and x is a vector of n components x 1 ; : : : ; x n . This problem must be solved for the values x 1 ; : : : ; x n that satisfy the constraints and minimize the function f . This is to be meant as one needs to fi...