Optimal lot sizes with geometric production yield and rigid demand

A job shop has to deliver a given number of custom-order items. Production is performed in "lots" that require costly setup, and output quality is stochastic. The size of each lot must be set before output quality is observed. We study production processes whose yield is distributed according to a generalized truncated geometric distribution: Production can randomly go "out of control", in which case all subsequent output of that lot is defective. The "generalization" allows both hazard rates and marginal production costs to vary as production progresses. Our results characterize the optimal lot sizes. In particular, we show that for small demands the optimal lot size equals the outstanding demand, and for larger demands it is less than the outstanding demand, with all lot sizes uniformly bounded. For sufficiently large demands, we identify conditions under which the optimal lots are precisely those that minimize the ratio of production cost to the expected number of good items. A tighter characterization is given for the standard case, where hazard rates and marginal costs are constant.