Dynamic control in multi-item production/inventory systems

Abstract

We consider a production/inventory system consisting of one production line and multiple products. Finished goods are kept in stock to serve stochastic demand. Demand is fulfilled immediately if there is an item of the requested product in stock and otherwise it is backordered and fulfilled later. The production line is modeled as a non-preemptive single server and the objective is to minimize the sum of the average inventory holding costs and backordering costs. We investigate the structure of the optimal production policy, propose a new scheduling policy, and develop a method for calculating base stock levels under an arbitrary but given scheduling policy. The performance of the various production policies is evaluated in extensive numerical experiments.

Calculation of E(P|w)

Let P represent an exponentially distributed time variable with mean $1/\mu$. Consider a queuing system where customers arrive according to a Poisson distribution with constant rate $\lambda$. In this appendix, we show that E(P|w) , the expectation of a time interval given that during this time interval w customer arrivals occur, is equal to $(w+1)/(\lambda + \mu )$. Using the definition of conditional expectation and the property that the number of Poisson arrivals during an exponentially distributed time interval with mean $1/\mu$ has a geometric distribution with success probability $\frac{1}{1 + \frac{\lambda }{\mu }}$, we obtain the following expression for E(P|w) :

$$\begin{aligned} E(P|w)&= \frac{\int \limits _{0}^{\infty } t \quad \mu \mathrm {e}^{-\mu \, t} \quad \frac{(\lambda \, t)^w}{w!} \quad \mathrm {e}^{-\lambda \, t} \quad \mathrm {d}t}{(1 - \frac{1}{1 + \frac{\lambda }{\mu }})^w \quad (\frac{1}{1 + \frac{\lambda }{\mu }})} \end{aligned}$$

(13a)

Rearranging terms, and using that $\frac{(\lambda + \mu )^{w+2} \quad t^{w+1}}{(w+1)!} \quad \mathrm {e}^{-(\lambda + \mu ) \, t}$ is the probability density function of the Erlang distribution with parameters $(w + 2)$ and $(\lambda + \mu )$, we can rewrite expression (13a) as follows:

$$\begin{aligned} E(P|w)= & {} \frac{\frac{\mu \, \lambda ^w \, (w+1)}{(\lambda + \mu )^{w+2}} \int \limits _{0}^{\infty } \frac{(\lambda + \mu )^{w+2} \quad t^{w+1}}{(w+1)!} \quad \mathrm {e}^{-(\lambda + \mu ) \, t}}{\left( 1 - \frac{1}{1 + \frac{\lambda }{\mu }}\right) ^w \quad \left( \frac{1}{1 + \frac{\lambda }{\mu }}\right) } \end{aligned}$$

(13b)

$$\begin{aligned}= & {} \frac{\frac{\mu \, \lambda ^w \, (w+1)}{(\lambda + \mu )^{w+2}} }{\left( 1 - \frac{1}{1 + \frac{\lambda }{\mu }}\right) ^w \quad \left( \frac{1}{1 + \frac{\lambda }{\mu }}\right) } \end{aligned}$$

(13c)

Finally, we obtain the desired result $E(P|w) = (w+1)/(\lambda + \mu )$ from (13c) via straightforward algebraic manipulations.