Optimal Resource Allocation: With Practical Statistical Applications and Theory

CHAPTER 8

Monte Carlo Simulation

8.1 Introductory Remarks

Very often a reliability goal function cannot be expressed in a convenient analytical form and makes even calculation of reliability decrements practically impossible. For instance, such situations arise when system units are mutually dependent or their reliability simultaneously depends on some common environmental factors (temperature, mechanical impacts, etc.). In these cases, the Monte Carlo simulation is usually used for reliability indices calculation. However, the problem arises: how can one use the Monte Carlo simulation for optimization?

Roughly speaking, the idea is in observing the process of the spare unit expenditure (replacement of failed units) until specified restrictions allow one to do so. This may be a simulation process or an observation of the real deployment of the system. After the stopping moment, we start another realization of simulation process or observation of the real data. When the appropriate statistical data are collected, the process of finding the optimal solution starts.

Avoiding a formal description of the algorithm, let us demonstrate it on numerical examples which will make the idea of the method and its specific technique clearer.

8.2 Formulation of Optimal Redundancy Problems in Statistical Terms

Standard methods do not give a solution if the goal function is the mean time to failure:

(8.1)

or if units are dependent, for instance, via a vector of some external factors g (temperature, ...