Capacity expansion for random exponential demand growth with lead times

The combination of demand uncertainty and a lead time for adding capacity creates the risk of capacity shortage during the lead time. We formulate a model of capacity expansion for uncertain exponential demand growth and deterministic expansion lead times when there is an obligation to provide a specified level of service. The service level, defined in terms of the ratio of expected lead-time shortage to installed capacity, is guaranteed by timing each expansion to begin when demand reaches a fixed proportion of the capacity position. Under this timing rule, the optimal facilities to install can be determined by solving an equivalent deterministic problem without lead times. Numerical results show the effects of the demand parameters and lead-time length on the expansion timing. The interaction of timing with expansion size is explored for the case when continuous facility sizes are available with economies of scale.