Flexible service capacity: Optimal investment and the impact of demand correlation

We consider a firm that provides multiple services using both specialized and flexible capacity, Pie problem is formulated as a two-stage, single-period stochastic program. The firm invests in capacity before the actual demand is known and optimally assigns capacity to customers when demand is realized. Sample applications include a car rental company's use of mid-sized cars to satisfy unexpectedly high demand for compact cars and an airline's use of business-class seats to satisfy economy-class demand. We obtain an analytical solution for a particular case, when services may be upgraded by one class. The simple form of the solution allows us to compare the optimal capacities explicitly with a solution that does not anticipate flexibility. Given that demand follows a multivariate normal distribution, we analytically characterize the effects of increasing demand correlation on the optimal solution. For the case with two customer classes, the effects of demand correlation are intuitive: Increasing correlation induces a shift from flexible to dedicated capacity. When there are three or more classes, there are also adjustments to the resources not directly affected by the correlation change. As correlation rises, these changes follow an alternating pattern (for example, if the optimal capacity of one resource rises, then the optimal capacity of the adjacent resource falls). These results make precise conjectures based on numerical experiments that have existed in the literature for some time.