Machine Servicing Problem

Abstract

The machine servicing problem is shown in Fig. 14.1 and a basic description can be found in almost any operations research book ([1] p. 573(. However, the model in this chapter is considerably different from that in [1]. Items arrive for processing at the first machine M1 and then proceed through the next four machines M2. ... M5 until they reach the “finished] box. All queues in front of each machine are assumed to have unlimited capacity. Queue capacity will not be of importance in this problem so we did not show any queue in Fig. 14.1. Each machine can break down and this will certainly effect throughput, but all other unbroken machines continue to function. The probability that machine Mi breaks down is p_i, 1 ≨ i≨5. For each machine Mi the probability of a break down is modelled as follows: whenever an item arrives at the machine for processing, and goes into queue if the machine is busy, the machine can break down with probability p_i, 1 ≨ i ≨ 5. If an item is being processed at break down it waits until the machine is fixed and then it continues its processing (no processing time is lost). There is a crew of k machine repair workers in the “repair” station in Fig. 14.1. The goal is to find the optimal k to minimize total costs.