Towards a theory of determining size in multi-objective service systems

This paper discusses the determination of the size, in terms of the number of servers, of a simple batch system. The objective of such a system can be, for example, maintenance. The system's optimal size will be determined by maximizing an objective function, which is the expected value per server of a variable, incorporation of two weighted criteria that characterize the system. The first criterion is defined as the percentage of jobs that are completely served in a given time period for the whole system. The second criterion is the rate of non-utilization of the systems as defined by the ratio of unused time in the system to the total time allocated to the system. The rule for determining the size of the system is derived by maximizing the expected weighted sum of the two criteria per server. In this paper we prove a theorem for finding the optimal size of the system by employing the above mentioned objective function. An unexpected result is obtained from the analysis of the sensitivity of the parameters of the system. Contrary to intuition, we find an inverse relationship between the weight of the first criterion and the size of the system. That is, when a greater weight is given to this criterion, the system becomes smaller.