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Article Excerpt
We consider the notoriously difficult discrete-time inventory model with stochastic demands, a constant lead time, and lost sales. We show that the effective state space is a relatively manageable compact set. Then, we test various plausible heuristics. We find that several perform reasonably well, although none is perfect. However, the standard base-stock policy (a direct analogue of the optimal policy for a backlog system) performs badly. We also show that the optimal cost is increasing in the lead time.
Subject classifications: inventory; lost sales; dynamic programming.
Area of review: Manufacturing, Service, and Supply Chain Operations.
History: Received May 2006; revisions receive October 2006, March 2007; accepted April 2007. Published online in Articles in Advance March 31,2008.
1.Introduction
We have a fairly good understanding of inventory systems where stockouts are backlogged. We know good policies for these systems, and we understand their behavour. For example. for a basic system with stationary data constant lead time, linear order cost, and independent random demands, we know that the optimal policy has a simple form, abase-stock policy, and we can readily compute the optimal one. Also, we know that the optimal policy and the over-all cost depend on the physical system's parameters roughly through the standard deviation of lead-time demand-the none-period demand's standard deviation times the square root of the lead time. Finally, we know how to extend these results in several directions- a fixed as well as a linear order cost, stochastic lead times, correlated and/or nonstationary demands, etc. (see, e.g., Porteus 2002 and Zipkin 2000).
Many real systems have lost sales, is much weaker. The problem was formulated a half century ago by Karlin and Scarf (1958). They and Morton (1969) established some basic structural results. Still, it is much harder to compute and optimal policy, even for the simplest systems (with stationarly data, etc.). The state space grows rapidly as the lead time increases. Partly for this reason, plausible heuristics have been tested only on a limited range of systems. Moreover, we know almost nothing about how good policies and their costs behave with respect to the system's parameters, Extensions to more complex systems are few (see, e.g., Nahmias 1979, Cohen et al. 1988, Johansen and Thorstenson 1993, and Melchiors et al. 2000).
We do know that a lost-sales system is fundamentally different from one with backorders, For example, provided the mean demand is finite, the optimal cost in the lost-sales system is bounded as a function of the lead time and the demand variance. (To see this, just consider the policy of never ordering. Its average const is the lost-sales penalty cost times the mean demand.) The backorders model, in contrast, has unbounded cost-that is one implication of the square-root relation mentioned above.
We show that, under any pluasible policy, the effective state space is a relatively manageable compact set. It still grows rapidly in the lead time, but less so than it would otherwise. Armed with this technique, plus modern computer technology, we are able to solve models with longer lead times than any reported previously. We then evaluate various plausible heuristics. Some of these were proposed decades ago, but others quite recently. We find that several perform reasonably well, although none is perfect. However, the standard base-stock policy (a direct analogue of the optimal policy for a backlog system) performs poorly.
We also show that the optimal cost is increasing in the lead time. This fact is not surprising, but it is useful to know. (This monotonicity result applies to any system with lagged control, not just the lost-sales system.)
This paper is organized as follows. Section 2 presents the formulation. Section 3 describes the heuristics. Section 4 derives the state-space result. Section 5 presents the lead-time monotonicity result. Section 6 reports numerical results.
2. Formulation
Consider the standard, single-item inventory system in discrete time with lost sales. The demands and sate variables can be either integers or continuous. Denote
L = order lead time, L >
t = time index, t=1, ...,T
[d.sub.t] = demand in period t.
[w.sub.t] = inventory at time t, before the order due at t arrives.
[z.sub.t] = order at time t.
[y.sub.t] = inventory at time t, after the order due at t arrives
= [w.sub.t] + [z.sub.t-L]
[u.sub.t] = [y.sub.t] - [d.sub.t].
[x.sub.0t] = [y.sub.t].
[x.sub.1t] = [z.sub.t+1-L]
...
[x.sub.L-1,t] = [z.sub.t-1].
[x.sub.t] = ([[x.sub.0t], x.sub.1t], ..., [x.sub.L-1,t]]).
Assume that the [d.sub.t] are independent and nonnegative. The state of the system is the L-vector [x.sub.t]. The dynamics are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[X.sub.t+1] = ([[x.sub.0t] - [d.sub.t]].sup.+] + [x.sub.1], [x.sub.2t], ..., [x.sub.L-1,t, [z.sub.t]).
For simplicity, assume that the data (cost factors and demand distributions) are stationary, and the order...
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