RELIABILITY:SYSTEM RELIABILITY

By -
SYSTEM RELIABILITY

The structure function of a system containing n components is a binary random variable that takes on the value 1 or 0. Furthermore, the reliability of this system can be expressed as2

INTRODUCTION TO OPERATIONS RESEARCH-0608

1It is evident that the loss of the front or rear brakes will affect the braking capability of the automobile, but the definition of “perform successfully” may allow for either set working.

2The time t is now suppressed in the notation. Recall that the time is implicitly included in determining whether or not the ith component performs satisfactorily.

In general, such conditional probabilities require careful analysis. For example, P{X2 = 1½X1 = 1} is the probability that component 2 will perform successfully, given that component 1 performs successfully. Consider a system where the heat from component 1 affects the temperature of component 2 and thereby its probability of success. The performance of these components is then dependent, and the evaluation of the conditional probability is extremely difficult. If, on the other hand, the performance characteristics of these components do not interact, e.g., the temperature of one component does not affect the performance of the other component, then the components can be said to be independent. The expression for the reliability then simplifies and becomes

INTRODUCTION TO OPERATIONS RESEARCH-0609

INTRODUCTION TO OPERATIONS RESEARCH-0610

This result is analogous to, and dependent upon, the assumption that the structure function of the system is coherent. The implication of this intuitive result is that the reliability of the automobile will improve if the reliability of one or more components is improved.

Comments

Post a Comment

Popular posts from this blog

DUALITY THEORY:THE ESSENCE OF DUALITY THEORY

By -
Image
THE ESSENCE OF DUALITY THEORY Given our standard form for the primal problem at the left (perhaps after conversion from another form), its dual problem has the form shown to the right. Thus, with the primal problem in maximizatio n form, the dual problem is in minimization form instead. Furthermore, the dual problem uses exactly the same pa r amete r s as the primal problem, but in different locations, as summarized below. 1. The coefficients in the objective function of the primal problem are the right-hand sides of the functional constraints in the dual problem. 2. The right-hand sides of the functional constraints in the primal problem are the coef- ficients in the objective function of the dual problem. 3. The coefficients of a variable in the functional constraints of the primal problem are the coefficients in a functional constraint of the dual problem. To highlight the comparison, now look at these same two problems in matrix notation (as introduced at the beginn
Read more

NETWORK OPTIMIZATION MODELS:THE MINIMUM SPANNING TREE PROBLEM

By -
Image
THE MINIMUM SPANNING TREE PROBLEM The minimum spanning tree problem bears some similarities to the main version of the shortest-path problem presented in the preceding section. In both cases, an undi r ecte d and connected network is being considered, where the given information includes some mea- sure of the positive length (distance, cost, time, etc.) associated with each link. Both problems also involve choosing a set of links that have the shortest total length among all sets of links that satisfy a certain property. For the shortest-path problem, this property is that the chosen links must provide a path between the origin and the destination. For the minimum spanning tree problem, the required property is that the chosen links must provide a path between ea c h pair of nodes. The minimum spanning tree problem can be summarized as follows: 1. You are given the nodes of a network but not the links. Instead, you are given the po- tential links and the positive length for each
Read more

NETWORK OPTIMIZATION MODELS:THE SHORTEST-PATH PROBLEM

By -
Image
THE SHORTEST-PATH PROBLEM Although several other versions of the shortest-path problem (including some for directed networks) are mentioned at the end of the section, we shall focus on the following simple version. Consider an undi r ecte d and connected network with two special nodes called the origin and the destination. Associated with each of the links (undirected arcs) is a non- negative distance. The objective is to find the shortest path (the path with the minimum total distance) from the origin to the destination. A relatively straightforward algorithm is available for this problem. The essence of this procedure is that it fans out from the origin, successively identifying the shortest path to each of the nodes of the network in the ascending order of their (shortest) distances from the origin, thereby solving the problem when the destination node is reached. We shall first outline the method and then illustrate it by solving the shortest-path problem encountered by the See
Read more