OPTIMIZATION OF RELIABILITY OF A SERIES SYSTEM

Abstract

The present work deals with the optimal design of a system by using structural redundancy. A basic consideration in the design of a complex system is the reliability which should be very high. Generally, the reliability of the constituent components is not sufficient to meet the system reliability requirement. One way of enhancing the system reliability is to curtail the complexity of the system which may result in poor stability and transient response of the system and degradation in the quality of product. The other practical way is to introduce structural redundancy at the subsystem level. The amount of redundancies to be employed depend on the resources available which are usually limited and pose a problem to the system Designers. Therefore, in the opti mal design of a system, the problem of optimal allocation of redundancies to optimize reliability subject to the multiple constraints such as cost, weight, power consumption etc./ arises. An attempt has been made to solve this problem in the present work. In the interest of generality, any parti cular system is not considered in this stucy. This thesis embodies the mathematical modelling of the optimal design problem of a system having active or dynamic redundancy. The active redundancy includes parallel, series, series parallel, majority voting and multiple-line redun dancy while dynamic redundancy comprises standby and hybrid.

redundancy. Generalized expressions are derived for the models suggested. The effect of switch failures, i.e. false switching, gradual failure and failure to operate, and dor mancy in the dynamic redundancy are considered in the mathe matical modelling. The systems having standby redundancy with spare and repair facilities are also considered. These models result only in partial optimization of the design problem. A true optimal design requires optimal allocation of reliability as well as redundancy in a system. Considering this fact, reliability problem is formulated. It takes the final form of nonlinear mixed integer programming problem. These nonlinear integer programming problems are linear ized by using the bivalent variables. The linearized relia bility problem has same feasible solution region as the origi nal one but the number of variables are increased. The nonlinear integer programming reliability problem is converted into the Geometric Programming formulation by assuming variables to be continuous which leads it to a system of nonlinear simultaneous equations with variables one less than the number of constraints. When the system has only one constraint, expressions are derived to get optimal number of redundant components required in terms of resources available. These expressions are very useful to the system designer. An algorithm is devised for solving reliability problem by using SUMT formulation. The constrained problem is solved by s^tcep&st "escent and tree search method. This algorithm is effective when system is subjected to multiple constraints and provides an exact solution.

The use of nonbinary tree search based on graphy theory is made to solve the linearized reliability problem. The method is conputationally efficient than the other available zero-one programming methods as it requires only few branch ing and less computer sorage. The same method is modified to avoid the calculation of external stable set to find upper bound on the objective function. The linearized reliability problem is solved by the flexi' ble enumeration scheme which allows a great deal of flexibi lity in the backtracking process and thus improving the effi ciency of the search procedure. This method requires simple algebraic computation and provides accurate results. The multiple constraints linearized reliability problem is converted into an equivalent knapsack type problem having a single constraint by aggregating the constraints. This equivalent problem is easier to solve than the original prob lem. A Branch and Bound method is brought out to solve the equivalent problem. A very efficient method is developed to solve nonlinear integer programming reliability problem. The method is based on-the fact that for maximizing the system reliability one component must be added sequentially to that particular stage which has lowest reliability. As the method needs only simple calculations and very little memory, it can be used to solve large systems. The optimal allocation of reliability and redundancy problem is solved by using SUMT formulation with discretiz ation penalty function.

The computer programs are developed and have been applied to solve various problems with success. To illustrate the methods of attack, numerical examples are incorporated. These methods can be used for the reliability-based design of the system such as control system, digital system At the end, the various methods discussed in this thesis are compared so that a system designer may know their limitations and advantages. Future avenues of research are also discussed. In short, the mathematical models have been presented for the optimal reliability design problem. Various types of redundancies are considered and methods to solve the reliability problem are discussed.