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Unformatted text preview: 1 Abstract The weak and strong duality theorems in interval-valued linear programming problems are derived. The primal and dual interval-valued linear programming problems are formulated by proposing the concept of scalar (inner) product of closed intervals. We introduce a solution concept that is essentially similar to the notion of nondominated solution in multiobjective programming problems by imposing a partial ordering upon the set of all closed intervals. Under these settings, the weak and strong duality theorems for interval-valued linear programming problems are derived naturally. Keywords: Closed intervals; complementary slackness; scalar (inner) product; solvability; weak duality; strong duality. 2 Introduction and Purpose The methodology for solving optimization problems has widely applied to many research fields. If the coefficients of optimization problems are taken as real numbers, they are categorized as the deterministic optimization problems. However, the coefficients can be taken as uncertain quantities. If the coefficients of optimization problem are assumed as random variables with known distributions, they are categorized as the stochastic optimization problems. The books written by Birge and Louveaux , Kall , Pr´ ekopa , Stancu-Minasian  and Vajda  give the main stream of this topic, and also give many useful techniques for solving the stochastic optimization problems. On the other hand, if the coefficients are taken as closed intervals, they will be categorized as interval-valued optimization problems. As we have known in the stochastic optimization problems, the coefficients are assumed as random variables with known distributions in most of cases. However, the specifications of the distributions are very subjective. For example, many researchers invoke the Gaussian (normal) distributions with different parameters in the stochastic optimization problems. These specifications may not perfectly match the real problems. Therefore, interval-valued optimization problems may provide an alternative choice for considering the uncertainty into the optimization problems. That is to say, the coefficients in the interval-valued optimization problems are assumed as closed intervals. Although the specifications of closed intervals may still be judged as subjective viewpoint, we might 1 argue that the bounds of uncertain data (i.e., determining the closed intervals to bound the possible observed data) are easier to be handled than specifying the Gaussian distributions in stochastic optimization problems. The duality theory for inexact linear programming problem was proposed by Soyster [10, 11, 12] and Thuente . Falk  provided some properties on this problem. However, Pomerol  pointed out some drawbacks of Soyster’s results and also provided some mild conditions to improve Soyster’s results. Rohn  also discussed the duality in interval linear programming problem with real-valued objective function. Now, we shall discuss the duality in interval-valued linear programming problemobjective function....
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This note was uploaded on 11/27/2009 for the course IM MA420 taught by Professor Mar,lee during the Spring '09 term at National Taiwan University.
- Spring '09