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.

我们扩展和改进了Singh和Watson给出的关于在机会受限的优化问题中对集合的并集使用经典界的最新结果。具体来说，我们将回顾所谓的Dawson和Sankoff边界，该边界在先前的分析中提供了机会约束的最佳近似值之一。接下来，我们表明我们的工作是对先前工作的概括，实际上，先前采用的不等式是一个非常宽松的近似，其假设通常不成立。计算结果表明，边界平均提高了43％以上。作为副产品，我们在优化模型中提供了下限函数的精确表述。