Capacity expansion under a service level constraint for uncertain demand with lead times
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For a service provider, stochastic demand growth along with expansion lead times and economies of scale may complicate a capacity planning problem. We consider a service provider who has to maintain certain minimum level of service and is interested in knowing the optimal timings and sizes of the future capacity expansions. This service level is defined in terms of unsatisfied demand over an expansion cycle. Under this service level constraint, the service provider wants to minimize the infinite time horizon cost of expansion. We assume a stationary policy where the timing and the sizes of the expansions are determined as fixed proportions of the capacity position, where the capacity position is the capacity that will be available when the current expansion is completed. We assume that the demand for the capacity follows a geometric Brownian motion (GBM) process. We discuss a method to check the GBM process fit for any data series representing the demand values and find that the data for electric utility consumption in the US, and the airline passenger enplanement data over a period of 15 years satisfy the assumptions of a GBM process. Using properties of the demand process, we can use financial option pricing theory to express the service level in terms of the decision variables. Particularly, we use the Up-and-Out partial barrier call option price expression to formulate the service level constraint. We use cutting plane algorithm to solve the optimization problem. Numerical optimization shows that it could be optimal to accumulate initial shortage before initiating the next capacity expansions for a low growth, low volatility demand and also when the expansion lead times are shorter. However, when the demand grows at a high rate or is more volatile, it is optimal to start the next expansion project before the demand reaches the current capacity position.