Abstract
We extend and improve recent results given by Singh and Watson on using classical bounds on the union of sets in a chance-constrained optimization problem. Specifically, we revisit the so-called Dawson and Sankoff bound that provided one of the best approximations of a chance constraint in the previous analysis. Next, we show that our work is a generalization of the previous work, and in fact the inequality employed previously is a very relaxed approximation with assumptions that do not generally hold. Computational results demonstrate on average over a 43% improvement in the bounds. As a byproduct, we provide an exact reformulation of the floor function in optimization models.
Highlights
1.1 BackgroundIn a recent work, Singh and Watson [13] approximate a two-stage joint chanceconstrained optimization model using approximations based on the union of sets
We extend and improve recent results given by Singh and Watson on using classical bounds on the union of sets in a chance-constrained optimization problem
We provide an exact reformulation of the floor function in optimization models
Summary
Singh and Watson [13] approximate a two-stage joint chanceconstrained optimization model using approximations based on the union of sets. They rewrite a joint chance constraint (JCC) as a union of sets of “failed” scenarios. Using classical Bonferroni-styled approximations of the union, they bound the chance constraint, thereby bounding the optimization model itself. B. Singh mization objective, a lower (upper) bound for a union (or, the JCC) provides an upper (lower) bound for the optimization model. The contribution of Singh and Watson is to utilize classical (upper and lower) approximations of this union within optimization model (1)
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