Joint redundancy and maintenance optimization for multistate series–parallel systems

Abstract

This paper formulates the joint redundancy and replacement schedule optimization problem generalized to multistate system, where the system and its components have a range of performance levels. Multistate system reliability is defined as the ability to maintain a specified performance level. The system elements are chosen from a list of available products on the market and the number of such elements is determined for each system component. Each element is characterized by its capacity, reliability and cost. The reliability of a system element is characterized by its lifetime distribution with the hazard rate, which increases with time. It is specified as the expected number of failures during different time intervals. The optimal system structure and the number of element replacements during the study period are defined as those which provide the desired level of system reliability with minimal sum of costs of capital investments, maintenance and unsupplied demand caused by failures. A universal generating function technique is applied to evaluate the multistate system reliability. A genetic algorithm is used as an optimization technique. Examples of determination of the optimal system structure and replacement schedule are provided.

Introduction

Redundancy and maintenance are used to provide a required level of system reliability. Engineers typically try to achieve this level with minimal cost by solving the problems of redundancy optimization or maintenance optimization separately.

The well-known problem of redundancy optimization has been addressed in a number of studies, e.g. for binary state systems [1], [2]. Reliability is considered a measure of the system's ability to meet the demand (required performance level) when applied to multistate systems. In power engineering, for example, the ability of the system to provide an adequate supply of electrical energy is used to evaluate its availability [2], [3]. In this case, the outage effect, which depends on consumer demand, will vary for units with different nominal capacities. Therefore, the capacities of system elements should be taken into account, as well as the consumer demand curve. The redundancy optimization problem of a system with different element capacities may be considered a problem of system structure optimization. This problem is addressed in Refs. [3], [4], where algorithms are suggested for multistate series–parallel system structure optimization. (In Ref. [5] the method has been extended to systems with bridge topology.)

For systems containing elements with failure rates increasing in time preventive replacement of the elements can also be used to enhance system reliability. Replacing elements that have a high risk of failure, while reducing the chance of failure, can incur significant expenses, especially in systems with high replacement rates. Minimal repair, the less expensive option, enables the system element to resume its work after failure, but does not affect its hazard rate [6], [7]. Maintenance policies that reach a compromise between preventive replacements and minimal repairs aim at achieving an optimal solution for problems with different criteria. They have been addressed in a number of studies [1], [7], [8], [9], [10]. All of these studies considered the reliability of binary state systems.

It is recognized [11] that obtaining the component lifetime distribution is the bottleneck in implementing existing maintenance optimization approaches. The expected number of element failures during any time interval can be obtained either from mathematical models [10] or from expert opinion [11]. For minimal repairs whose durations are relatively short compared with the time between failures, the expected number of failures is equal to the expected number of repairs during any time interval. Thus, it is possible to obtain the renewal function of each element (expected number of its repairs at time interval (0, t]). The expected number of element failures/repairs f(t_j) can be estimated for different time intervals (0, t_j] between consecutive replacements.

A tradeoff exists between investments into system redundancy and its maintenance (preventive replacements) cost. The optimal reliability design should take both of these factors into account in order to reach a solution that provides the desired system reliability at minimum cost. Because these factors influence each other, the problem cannot be solved in two separate stages, i.e. by finding the minimum cost replacement schedule for a preliminarily determined optimal structure. This paper presents an algorithm that can be used to determine both the optimal configuration for a multistate series–parallel system and the optimal schedule of cyclic replacements of system elements. Each element of this system is characterized by its nominal capacity and renewal function, obtained from mathematical models or elicited from expert opinion. The time and cost of repair and replacement are available for each system element. The cost of inclusion of the element within the system is also given. In general, this cost is not equal to the cost of element replacement because the inclusion of the element can incur additional expenses, such as investments into the corresponding infrastructure (communication, foundation, etc.). The objective is to provide the desired system availability with minimal sum of the costs of system structure, maintenance and penalties caused by system mission losses (unsupplied demand).

Incorporating the performance levels (capacities) of elements into the optimization problem makes it difficult for traditional optimization procedures because of “dimension damnation”. A genetic algorithm (GA) is used to solve this complicated combinatorial optimization problem. The technique for encoding the solution is adapted to represent replacement policies. A solution quality index includes both reliability and cost estimations. The effective procedure, which is based on the universal generating function, is used to evaluate the multistate system availability.

An illustrative example is presented in which the optimal structure and replacement schedule is found for the series–parallel system.