| We say product A is a partial substitute for product B if a fraction of the customers who prefer B are willing to accept A when B is out of stock. When demand is uncertain, it is intuitive and true that a larger "willing to substitute" fraction implies larger expected profits. A higher "willing to substitute" fraction allows one to pool the risk of individual products. It may also be intuitive that a larger "willing to substitute" fraction might result in lower optimal total inventory. For the full substitution structure, in which all customers are willing to accept an alternative when their first choice is not available, a few researchers have shown that this latter intuition is not always true. We show that the latter intuition is not true for a number of realistic situations of partial substitution with commonly used demand distributions such as normal, exponential, Poisson, and uniform.; Even though multi-product inventory problems have been intensively studied since at least 1965, there is effectively no analytic solution to multi-product models. Thus, we resort to Monte Carlo simulation to answer various questions about such systems. We develop a new form of quasi-random sampling, called Stratified Latin Hypercube (SLH) sampling, to obtain very accurate results in simulation based numerical analysis for the total inventory level of partial substitution case with Poisson and Normal demands. SLH sampling combines Latin square sampling with multidimensional stratified sampling. We compare this approach with other well-known variance reduction methods when applied to a variety of simulation problems in the operations management area, ranging from multi-product inventory management with demand substitution, stochastic PERT networks, petroleum reserve evaluation, international supply chains under exchange rate uncertainty, option pricing, and stochastic programming. We consider three settings: (1) simple evaluation of a policy, (2) optimization under uncertainty, and (3) quantile estimation as occurs in Value at Risk analysis. In the optimization setting we analyze estimation bias. For all the above, we show that for practical purposes, SLH sampling is never worse than other methods, and in many applications it is significantly better. |
