Importance measure on finite time horizon and application to Markovian multistate production systems

The sensitivity analysis or the reliability importance analysis of complex industrial systems aims to identify, in a multiunit structure, which components contribute the most to a variation of the performance criterion. In this paper an importance measure, called the multidirectional sensitivity measure, is considered; it is defined as the derivative of the performance in the direction of one parameter in the direction of a group of parameters (failure and repair rates of components, for example), or in any direction of the transition rates of a Markovian system. It is really a directional derivative in the direction of a vector in the appropriate space. There is a homotopy on the matrices that acts on the parameter space. This contrasts with the approach through components, and is less dependent on the structure or interaction of components than classical Birnbaum importance factors. This importance measure proposed for sensitivity analysis of steady state reliability is developed herein for the transient state. It is also extended and applied to the study of the production capacity of multistate production systems such as manufacturing, production lines, and power generation, which exhibit performances that can settle on different levels depending on the operative conditions of the constitutive components. A simple numerical example is introduced to show why this measure provides an efficient tool to investigate not only the importance of a given component, but also the importance of a class of components, the importance of the maintenance and, more generally, the effect of the simultaneous change of several design parameters.