In this work, a make-to-stock production system with two production rates, normal and emergency, was investigated. Such a production system is not uncommon in practice. Usually, production occurs at the normal production rate. However, when the inventory level is low and anticipated demands cannot be effectively satisfied, the emergency production rate is employed to prevent costly stock-outs. In real life, the emergency production rate can be achieved in many ways such as by purchasing products from outside, installing new production equipment, hiring temporary workers, or simply working overtime. The normal production rate incurs a lower unit production cost but with lower throughput while the emergency production rate increases throughput at the expense of a higher unit production cost.

The most important production decision in this system, which significantly affects the total cost, is to determine the optimal production rate employed for a given inventory level. Such decisions must be carefully made in order to minimise the total cost.

With the assumptions of exponential production times and Poisson demands, it was found that the optimal control policy for the production system with two production rates is characterised by two critical inventory levels. One level can be considered as the base-stock level where production stops when the inventory level reaches the first level. The second critical inventory level determines which production rate should be employed. A normal production rate is employed if the inventory level is above the second level, and an emergency production rate is employed when it falls below the second level. This optimal control policy is straightforward and can be easily implemented in practice.

In addition, an optimal control policy for the production system with N production rates was also developed, which is characterised by M critical inventory levels (M < N). Though there are N production rates available in the production system, only M production rates are employed under the optimal control policy. The other N – M production rates will never be employed because this increased production capacity cannot justify the additional production costs.

Numerical studies were performed to compare the expected total cost per unit time of two production systems, one with a single (normal) production rate, and another with two (normal and emergency) production rates. Each of the production systems operates under its optimal control policy. The results show that significant cost reduction can be achieved by employing the emergency production rate in most cases studied. The magnitude of this cost reduction depends on how often the emergency production rate is employed. The average cost reduction over our experimental problems is greater than 60%. Figures 1-2 show effects of utilisation of the production system and the production rate ratio on the cost reduction.

Figure 1: Effect of utilisation on cost reduction.	Figure 2 : Effect of production rate ratio on cost reduction.