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A simple and efficient technique for solving integer-programming problems that normally arise in system-reliability design is introduced. It quickly solves even a very large system problem. It consists of a systematic search near the boundary of constraints and involves functional evaluations only. It can handle system-reliability design problems of any type in which the decision variables are restricted to integer values. Several illustrative examples are given to substantiate these assertions. >

The 15 papers in this special section present new tools, techniques, applications, approaches, and surveys related to various aspects of reliability, and risk analysis and management.

This paper presents a systematic methodology to evaluate usual system effectiveness indexes of large and complex systems using network reduction and transformation techniques. This approach, allows some or all nodes to be imperfect, is efficient and simple, and uses less computer time and memory than existing techniques. Situations where it may not be feasible to use either star-delta or delta-star transformations in its usual form, to reduce a complex network are identified. Five subnetwork configurations are suggested to facilitate simplifying the network. search and subnetwork selection scheme which is adequate for the above purpose, and an efficient array-form representation requiring minimum memory to store the information about the network are offered.

For reliability analysis, the choice between tie-sets and cut-sets is not a matter of convenience but depends on assessment of computational effort. The system configuration alone determines the preference of the approach. The present paper offers a faster algorithm for evaluating system failure frequency using both approaches. An expression for system failure frequency between two specified nodes is derived using either tie-sets or cut-sets. The expression is in terms of failure frequency, availability, and unavailability of the system elements. A heuristic algorithm to obtain the expression can be programmed for a computer. The algorithm is illustrated through an example. The manner in which Abraham's algorithm can be used to determine system probability of failure using cut-sets is suggested. The algorithms are also useful for evaluating sensitivity coefficients.

The reliability literature offers an abundance of methods for the optimal design of systems under some constraints. In most of the papers, the problem considered is: given reliabilities of each constituent component and their constraint-type data, optimize the system reliability. This amounts to the assignment of optimal redundancies to each stage of the system, with each component reliability specified. This is a partial optimization of the system reliability. At the design stage, a designer has many options, e.g., component reliability improvement and use of redundancy. A true optimal system design explores these alternatives explicitly. Our paper demonstrates the feasibility of arriving at an optimal system design using the latter concept. For simplicity, only a cost constraint is used, however, the approach is more general and can be extended to any number of constraints. A particular cost-reliability curve is used to illustrate the approach.

This paper provides a simple algorithm for the solution of the redundancy optimization problem in the presence of multiple linear constraints. Basicaily, the solution to an r-constraint problem is obtained successively from the solution of r-unconstrained problems. At each step, an active constraint is picked out, then the maximum gradient concept is used to find a closer point. The approach requires minimum effort and time for solution.

The redundancy optimization problem is formulated as an integer programming problem of zero-one type variables. The solution is obtained making use of an algorithm due to Lawler and Bell. Objective function and constraints can be any arbitrary functions. Three different variations of the optimization problem are considered. The formulation is easy and the solution is convenient on a digital computer. The size of the problem that can be solved is not restricted by the number of constraints.

An algorithm is presented for evaluation of reliability of any redundant network. It uses the properties of digraphs and is especially suitable for the computer analysis of large complex networks. A method for deriving the reliability expression for any type of network is also described along with a Fortran computer program for the reliability evaluation of series-parallel networks.

A flow graph approach for reliability analysis is applied to redundancy with elements in the simple series-parallel networks and to the more general case with elements in non-series-parallel combinations. The reliability of networks for elements with open or short failures is analyzed with flow graphs. Typical examples are shown.

A mathematical model is formulated for optimizing the reliability of a system subject to given linear constraints; the system has several stages in series; each stage has parallel redundancy to improve the reliability. Part I shows a new way to transform the model of constrained optimization to a saddle point problem by using Lagrange multipliers. Conditions are derived for maximizing the reliability function; Newton's method is used to solve the resulting multidimensional nonlinear algebraic equations. Further modifications are provided to avoid inverting the large Jacobian matrices; therefore this method is practical for large systems. Part II shows how to transform the model of constrained optimization to a multistage decision process and uses the Maximum principle to arrive at the optimal decision. This approach is easy to apply, formulate, and program. The solution can be obtained without fear of nonconvergence (very often experienced with earlier methods) besides providing considerable saving in computer time. Design alternatives can be easily considered.

An algorithm is developed to obtain a simplified reliability expression for a general network. All the success paths of the network are determined; then they are modified to be mutually disjoint. The reliability expression follows directly from the disjoint paths. The algorithm is easy and computationally economical. The reliability expression involves fewer terms and arithmetic operations than any of the existing methods. This is an advantage, considering the size of the system a reliability engineer normally handles and the frequency with which the expression is used for reliability studies.

The method of Lee and the simple application of graph theory seem to give different answers for flow networks. Lee's method is correct. The method whereby graph theory can be extended to give correct results is explained.

An algorithm for the heuristic solution of redundancy optimization problems is presented. There may be any number of constraints which need not be linear. For selecting the stage where redundancy is to be added, a heuristic criterion is introduced which takes into account the relative increment in reliability versus decrement in slacks. The algorithm is simple and requires minimum computation.