Single-machine lot-sizing with uniform yields and rigid demands: Robustness of the optimal solution

We consider Ihe lot-sizing discrete problem of a single machine whose yield is uniformly distributed. We assume that the customers’ demand is rigid, i.e. all demand must be satisfied. The costs involved are a setup cost paid each time a run is initiated plus a unit variable cost per unit produced. No salvage cost is associated with extra or defective units. We prove that there exists an optimal sequence of lot-sizes that is strictly increasing in the demand levels and therefore an optimal lot-size that is at least as large as the demand level. These properties have been proved in the literature only for binomial yields. We also provide an extremely simple algorithm to compute the optimal lot-sizes. Moreover, we show that the cost function is strongly robust in the lot-size: for any e > 0 we develop a procedure that generates a (usually large) class of policies whose relative error is bounded by e.