Flexible solutions to linear programs under uncertainty: Equality constraints

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Abstract
This paper presents a framework which allows uncertainties in the matrix elements of an equality constrained linear program to be taken into account without requiring detailed knowledge of the statical characteristics of these uncertainties. The results are derived using the model of the linear program with flexibility previously introduced for the inequality constrained case. However, because a feasible region common to all perturbed constraint sets does not exist in the equality constrained case, a flexibility set which intersects all perturbed sets individually rather than jointly is defined. The flexibility set is constructed by identifying a finite subset of all perturbed constraint sets which need to be investigated. Three cases for the equality constrained problem are considered: independent variations in the array elements, column dependent variations, and row dependent variations. In each case the problem is solved using a possibly large but decomposable linear program. In the first two cases, this program needs to be solved only once; while in the row dependent case an iterative but finite solution procedure is required.

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