An Asset Assembly Problem

We formulate a finite-horizon nonstationary dynamic single-asset assembly problem, which covers both a liquid-asset assembly problem, based on the work of Rosling published in 1989, where the single asset is the single product being assembled, and a fixed-asset assembly problem, in which the single asset is production capacity. In the latter case, capacity is assembled over time from components and may be used to manufacture many products. In the spirit of Rosling, we provide conditions under which it can be solved by an equivalent analogous serial model in the form developed by Clark and Scarf in 1960, with a separate state variable for the level of assets in each stage of completion, dramatically simplifying the problem and its solution. In the liquid-asset case, we extend Rosling's 1989 work by including nonstationary demands, costs, and revenues in a finite-horizon setting. In the fixed-asset (capacity) expansion case, we show that capacity should be assembled in a balanced way and derive the optimal timing and extent of delays (in previously initiated capacity expansions). Our basic capacity expansion model is deterministic, so under our conditions, it is optimal never to delay a schedule and the state space reduces to a single dimension. However, in the Markov-modulated case, in which cost parameters and customer demand distributions can be influenced by a randomly and exogenously evolving state (of the economy), we illustrate that delays can be optimal.