Abstract
A technique is developed for finding a closed form expression for the cumulative distribution function of the maximum value of the objective function in a stochastic linear programming problem, where either the objective function coefficients or the right hand side coefficients are continuous random vectors with known probability distributions. This is the “wait and see” problem of stochastic linear programming. Explicit results for the distribution problem are extremely difficult to obtain; indeed, previous results are known only if the right hand side coefficients have an exponential distribution [1]. To date, no explicit results have been obtained for stochastic c, and no new results of any form have appeared since the 1970’s. In this paper, we obtain the first results for stochastic c, and new explicit results if b an c are stochastic vectors with an exponential, gamma, uniform, or triangle distribution. A transformation is utilized that greatly reduces computational time.
Highlights
Consider the linear programming problem, Max z ( x) = cx (1)s.t : ( A, I ) x = b (2) x≥0 (3)where c is an 1× (m + n) vector whose jth component is cj (where, c j = 0, for ( ) j > n ) and b is an m ×1 vector whose ith component is bi, A = aij is an m × n matrix, I is an m × m identity matrix and x is an (m + n)×1 vector
No explicit results have been obtained for stochastic c, and no new results of any form have appeared since the 1970’s
We obtain the first results for stochastic c, and new explicit results if b an c are stochastic vectors with an exponential, gamma, uniform, or triangle distribution
Summary
A technique is developed for finding a closed form expression for the cumulative distribution function of the maximum value of the objective function in a stochastic linear programming problem, where either the objective function coefficients or the right hand side coefficients are continuous random vectors with known probability distributions. We obtain the first results for stochastic c, and new explicit results if b an c are stochastic vectors with an exponential, gamma, uniform, or triangle distribution. Consider the linear programming problem, Max z ( x) = cx s.t : ( A, I ) x = b where c is an 1× (m + n) vector whose jth component is cj
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