# Optimal replacement policy for multicomponent systems: An application to a dairy herd

## Abstract

We consider the problem of finding an optimal replacement policy for a system which has many components. The main difficulty in this problem is that there is an interaction among the items in the system. Thus, the optimal replacement decision for each item depends not only on its state, but also on those of the other items in the systems. This interaction is due to the the fact that the both the stock size and the supply of replacement items is limited. (Instead of an unlimited supply of standard replacement items as is implicitly assumed by most of the replacement models). In our application to dairy herd managenent the problem is further complicated by the fact that this limited supply is not exogenous to the process but is actually generated by it. This is due to the fact that almost all the replacement young cows are home grown. The traits of these young cows have genetic dependence on those of their parents. The dairy herd management problem is actually a special case of the joint replacement and inventory problem, where the groups of cows are the stock of replacement items. At each point of time the decision problem is to find the optimal composition of items from the available population of items. An exact derivation of the optimal replacement policy for such problems is very complicated because the optimal decisions for each period depends on the state of the whole stock, and of all the available replacement components. This leads to a dynamic programming problem with a very large number of state variables which is not feasible to solve numerically due to the great amount of computer time involved. This paper presents a practical method for obtaining an approximate solution for the above described problem. The computational difficulty caused by the tremendously large dimensionality of the state variable is overcome by means of an iterative method which combines simulation and Dynamic Programming approach to compute successive linear approximations of the value function. © 1986.