Inventory management under highly uncertain demand
Section snippets
Problem statement
We are concerned with inventory management situations where there is significant demand uncertainty as measured by the coefficient of variation, . In semiconductor manufacturing, for example, it is common to find parts with , and it is not unusual to see parts with . Inventory managers often use base-stock levels of the form , where is the mean, is the standard deviation and z is a safety factor based on the normal distribution. Typical values of z are and
Cost minimization
Most of the issues concerning large 's are more serious for products with short life cycles. To simplify the presentation of the relevant issues, we model the case where there is a single opportunity to order and demand occurs over a single period. We assume that demand D is a non-negative random variable with mean and variance . To simplify the exposition we will also assume that we start with zero inventories. Extensions to positive initial inventories and multi-period models with
Behavior of base-stock levels
Here, we show that optimal base-stock levels converge to zero as demand variability gets large for the Lognormal, the Gamma and the Negative Binomial distributions. For the Pareto distribution, we show that base-stock levels converge to zero relative to the mean as the mean becomes large.
Distribution-free bounds on base-stock levels
We provide a tight, distribution-free, upper bound on base-stock levels for non-negative random variable with mean and variance ; see also Agrawal and Seshadri [2] and Gallego [4] for distribution-free bounds for policies. Let D be a non-negative random variables with mean and variance and let be the smallest base-stock level y such that . Let Theorem 5 Proof We investigate the behavior of for fixed for a non-negative
Profit comparison
We now compare the performance of base-stock levels set by the normal, maximal and the Lognormal formulas in terms of the expected profit. Managers who use the normal model to compute base-stock levels intend to use the resulting base-stock levels against a non-negative distribution with the same mean and variance. For this reason, we test these base-stock levels against the worst case, the Lognormal and the Gamma distributions. The shape and scale parameters of the Gamma distributions are
Inventory models with service constraints
Managers often use service levels instead of a cost model to determine base-stock policies by selecting a desired service level . The normal prescribes a base-stock level to satisfy demand with probability . The Lognormal model prescribes a base-stock level . Notice that as . It is easy to see that is maximized when , so .
It is instructive to see how the base-stock levels for the normal and the Lognormal
Managerial implications
Practical wisdom suggests that larger safety stocks are needed to protect against demand uncertainty when the standard deviation of demand increases. A key message of this paper is that this is not always true. Negative safety stocks, i.e., base-stock levels below the mean, can result in high service levels. When demand variability is very high, it may be enormously expensive and unnecessary to insist on a base-stock level of the form as suggested by the normal distribution. In addition,
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