Capacity selection under uncertainty with ratio objectives

Capacity choice or expansion, whether organic or via mergers and acquisitions, creates firms of widely varying scales. The ex-post profitability of such a transformed firm relative to its original size will typically be evaluated on ratio (rate) measures like earnings per share or profits to asset ratio, as such are the only meaningful ways to compare profitability of firms of substantially different sizes. It thus seems desirable that ex-ante capacity selection decisions will also be guided by a ratio objective. We explore capacity choice decisions under demand uncertainty through the ratio measures: (1) Expected “newsvendor” (i.e., single period) costs per unit capacity. (2) Expected (newsvendor) profit per unit of capacity. (3) Expected profit per costs of acquiring capacity. (4) A weighted average of a ratio and non-ratio objectives. We allow the capacity-acquisition costs and capacity's production capability to be general non-linear functions. Cost and profit objectives cease to be equivalent when ratio objectives are involved. We show that cost ratio optimizers will select a larger capacity than absolute cost optimizers, while profit ratio optimizers will select smaller capacities than absolute counterpart. We provide examples which show that the difference in optimal capacities, and the associated difference in objective value can be substantial. We also perform comparative statics analyses of the effect of change in item's shortage cost and of stochastic shifts in demand.