On capital investment

We deal with the problem of making capital investments in machines for manufacturing a product. Opportunities for investment occur over time, every such option consists of a capital cost for a new machine and a resulting productivity gain, i.e., a lower production cost for one unit of product. The goal is that of minimizing the total production costs and capital costs when future demand for the product being produced and investment opportunities are unknown. This can be viewed as a generalization of the ski-rental problem and related to the mortgage problem [3]. If all possible capital investments obey the rule that lower production costs require higher capital investments, then we present an algorithm with constant competitive ratio. If new opportunities may be strictly superior to previous ones (in terms of both capital cost and production cost), then we give an algorithm which is O(min {l + log C, 1 + log log P, 1 + log M}) competitive, where C is the ratio between the highest and the lowest capital costs, P is the ratio between the highest and the lowest production costs, and M is the number of investment opportunities. We also present a lower bound on the competitive ratio of any on-line algorithm for this case, which is Omega (min{log C, log log P/ log log log P, log M/ log log M}). This shows that the competitive ratio of our algorithm is tight (up to constant factors) as a function of C, and not far from the best achievable as a function of P and M.