Inventory management under highly uncertain demand

doi:10.1016/j.orl.2006.03.012

Operations Research Letters

Volume 35, Issue 3, May 2007, Pages 281-289

https://doi.org/10.1016/j.orl.2006.03.012 Get rights and content

Abstract

We show that base-stock levels first increase and then decrease as the standard deviation increases for a variety of non-negative random variables with a given mean and provide a distribution-free upper bound for optimal base-stock levels that grows linearly with the standard deviation and then remains constant.

Section snippets

Problem statement

We are concerned with inventory management situations where there is significant demand uncertainty as measured by the coefficient of variation, $cv$ . In semiconductor manufacturing, for example, it is common to find parts with $cv > 2$ , and it is not unusual to see parts with $cv > 3$ . Inventory managers often use base-stock levels of the form $μ + z σ = μ (1 + z * cv)$ , where $μ$ is the mean, $σ$ is the standard deviation and z is a safety factor based on the normal distribution. Typical values of z are $1.28, 1.64$ and

Cost minimization

Most of the issues concerning large $cv$ '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 $μ$ and variance $σ^{2}$ . 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 $μ$ and variance $σ^{2}$ ; see also Agrawal and Seshadri [2] and Gallego [4] for distribution-free bounds for $(Q, r)$ policies. Let D be a non-negative random variables with mean $μ$ and variance $σ^{2}$ and let $y (β)$ be the smallest base-stock level y such that $P (D < y) ⩾ β$ . Let $y^{+} (β) = \{\begin{matrix} μ + \sqrt{\frac{β}{1 - β}} σ & if σ < μ \sqrt{\frac{β}{1 - β}}, \\ \frac{μ}{1 - β} & otherwise . \end{matrix}$

Theorem 5

$y (β) ⩽ y^{+} (β) .$

Proof

We investigate the behavior of $P (D < t)$ for fixed $t > μ$ 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 $β$ . The normal prescribes a base-stock level $y_{n} (β) = μ + σ z_{β}$ to satisfy demand with probability $100 β %$ . The Lognormal model prescribes a base-stock level $y_{\ln} (β) = \exp (ν + τ z_{β})$ . Notice that $y_{\ln} (β) \to 0$ as $σ \to \infty$ . It is easy to see that $y_{\ln} (β)$ is maximized when ${cv}^{2} = \exp (z_{β}^{2}) - 1$ , so $y_{\ln} (β) ⩽ μ \exp (\frac{1}{2} z_{β}^{2})$ .

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 $μ + z σ$ as suggested by the normal distribution. In addition,

References (9)

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Distribution free bounds for fill rate optimization of $(r, Q)$ inventory policies
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Order-statistics calculation, costs and service in an $(s, Q)$ inventory system
Naval Res. Logist.
(1994)
G. Gallego
New bounds and heuristics for $(Q, R)$ policies
Management Sci.
(1998)

There are more references available in the full text version of this article.

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Inventory management under highly uncertain demand

Abstract

Section snippets

Problem statement

Cost minimization

Behavior of base-stock levels

Distribution-free bounds on base-stock levels

Profit comparison

Inventory models with service constraints

Managerial implications

Estimating negative binomial demand for retail inventory management with unobservable lost sales

Naval Res. Logist.

Distribution free bounds for fill rate optimization of (r,Q) inventory policies

Naval Res. Logist.

Order-statistics calculation, costs and service in an (s,Q) inventory system

Naval Res. Logist.

New bounds and heuristics for (Q,R) policies

Management Sci.

Distribution free bounds for fill rate optimization of $(r, Q)$ inventory policies

Order-statistics calculation, costs and service in an $(s, Q)$ inventory system

New bounds and heuristics for $(Q, R)$ policies