Control policies for single-stage production systems with perishable inventory and customer impatience

Abstract

We consider problems of inventory and admission control for make-to-stock production systems with perishable inventory and impatient customers. Customers may balk upon arrival (refuse to place orders) and renege while waiting (withdraw delayed orders) during stockouts. Item lifetimes and customer patience times are random variables with general distributions. Processing, setup, and customer inter-arrival times are however assumed to be exponential random variables. In particular, the paper studies two models. In the first model, the system suspends its production when its stock reaches a safety level and can resume later without incurring any setup delay or cost. In the second model, the system incurs setup delays and setup costs; during stockouts, all arriving customers are informed about anticipated delays and either balk or place their orders but cannot withdraw them later. Using results from the queueing literature, we derive expressions for the system steady-state probabilities and performance measures, such as profit from sales and costs of inventory, setups, and delays in filling customer orders. We use these expressions to find optimal inventory and admission policies, and investigate the impact of product lifetimes and customer patience times on system performance.

Appendix: Solution of Model B

Consider the Markov chain shown in Fig. 3. For α=N, the Chapman-Kolmogorov equations of the Markov chain (α,n) are

These equations can easily be solved; the probabilities P(N,s),…,P(N,−c) on the left sides can be expressed as functions of P(W,s−1).

For states (W,n),n>σ, the Chapman-Kolmogorov equations are

In these equations, the probabilities P(W,s−2),…,P(W,n−1),…,P(W,σ) on the right sides are also expressed as functions of P(W,s−1).

Finally, for (W,n),n≤σ, we have

As previously, the probabilities P(W,n−1) on the right sides are expressed as functions of P(W,s−1). The last equation is redundant. To find P(W,s−1) we use the normalization equation, $\sum_{n=-c}^{s}{[P(W,n)+P(N,n)]}=1$.