The Monte Carlo Method and optimization of spare parts in complex realistic scenarios

Optimizing resources, spare parts, and repair teams, in a complex logistic system is a major problem. It is a frequent problem in almost every organization and industry and it involves a large capital investment. In this paper we address this problem in a systematic manner and suggest a methodology that leads to a satisfactory and viable solution to the problem in a very general sense; that is, for almost any situation that may arise in reality. We first discuss the problem of calculating the performance of a general system in all its complexity and propose that the Monte Carlo method is the main tool and method to achieve that goal. We then formally define the optimization problem in terms of the performance function and the cost of resources. We next show that even though the Monte Carlo method can yield the performance function for realistic systems it is not a practical tool for optimization because of the prohibitively high calculation time involved. We then suggest an analytic approximation for the "surface" of the performance function in the resources space. This approximation is based on parameters "learned" from a small number of Monte Carlo calculations. The optimization process is then performed analytically using marginal analysis over an approximate analytic surface. After a certain number of analytical steps the obtained set of resources is checked with a new Monte Carlo calculation. This calculation serves both to correct any deviations in the analytic performance and improves the approximated surface by updating the parameters used to create it; thus, creating a hybrid optimization method that combines the speed of analytic calculations with the modeling power of the Monte Carlo method. It is then explained how the method could be extended to multiple fields and echelons and a number of realistic case studies are presented.