Cost Functions , Time Horizon , Decision - Dependent

We define what is a replacement policy for a system that fails randomly in time and its main characteristics. There are several parameters that drive the structure of the optimal policy. We provide a detailed discussion of the time horizon, objective functions, and failure time distributions as part of any replacement policy. 1 Definitions and main characteristics Our daily lives involve the observation, use, construction, and destruction of many systems. We replace the light balls when they burn out, take our cars to the repair shop when the breaks need to be replaced or the oil is due to be changed. The situations requiring a replacement are usually connected to wear-out, ageing, deterioration, or failure of the item/system involved. In this presentation we will assume that all of these processes are random. Definition: A replacement policy π is a decision making rule that defines the time and type of replacement of an item or a system such that an objective defined by the decision maker is optimized. The main characteristics of any replacement policy are: • Objective function A common criteria in deciding when to perform a replacement is economically justified, i.e. a cost function (called objective function), G(x), is optimized with respect to a set of parameters. Examples are the smallest average replacement cost per unit time or the total discounted cost. Alternative objectives are the maximum reliability, minimum net present cost, internal rate of return, or any general utility function defined by the decision maker. • Time of replacement A replacement can be performed: as soon as the item fails (also known as corrective maintenance), before failure (preventive maintenance), or in a certain amount of time after failure. This will be one of the objective function’s parameters, call it T . We will try to find the value of T that optimizes the objective function. • Type of replacement The item can be replaced with a new one or with an old (used) one. This is another one of the objective function parameters, denote it by a. As before, we would like to find the age of the system with which we will replace the ”old” one that optimizes the objective function. • Failure time The failure occurrences are assumed to be random following a general counting stochastic process {N(t), t ≥ 0}, see Rausand and Høyland (2004, page 231). The ”failure time” will be the time between two consecutive failures. We can formulate the general stochastic optimization problem of finding the optimal replacement policy π = (T ∗, a∗) as min T,a∈R+ E[G(T, a, t)] (1) where E stands for the expectation operator taken with respect to a filtration up to time t defined by the counting process N(t). The decision variables are T time of replacement and a age of the replacing item. Note that if we want to find the maximum of the objective function, it will be equivalent to finding the minimum of its negative value. Here are some common replacement policies one can obtain for specific values of the time of replacement, T : • Age replacement policy: the item/system is replaced upon failure or at age Y , whichever comes first (see eqr111).