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When designing a system, there are two methods that can be used to improve the system's reliability without changing the nature of the system: 1) using more reliable components, and/or 2) providing redundant components within the system. The redundancy allocation problem attempts to find the appropriate mix of components & redundancies within a system in order to either minimize cost subject to a minimum level of reliability, or maximize reliability subject to a maximum cost and weight. Redundancy allocation problems can be classified into two groups; one allows the system to have a mix of components with different characteristics incorporated in the system, while the other only allows one type of each component. The former group has a much larger solution space compared to the latter, and therefore obtaining an exact optimal or even a high quality solution for this problem may be more difficult. Optimization techniques, based on meta-heuristic approaches, have recently been proposed to solve the redundancy allocation problem with a mix of components. However, an exact solution method has not been developed. In this paper, we develop an exact solution method, based on the improved surrogate constraint (ISC) method, and use this method to find optimal solutions to problems previously presented in the literature

A method for solving the problem of optimizing both, redundancy (number of redundant components) and component reliability in each stage of a system under multiple constraints is presented. A mixed-integer nonlinear programming formulation and the surrogate dual method are used. The solution of the surrogate dual problem is not always feasible in the original problem, that is, a 'surrogate gap' exists. Two countermeasures to surrogate gaps are considered: (1) modifying the original problem to tighten the constraints, with the modification being continued until the solution of the surrogate dual problem of the modified problem becomes feasible in the original problem, and (2) decreasing component reliabilities in the vertical direction to the tangential plane of the objective function. The method applies to reliability optimization problems for general systems, enabling complex systems such as communication networks to be treated. Some computational results are shown and compared with other approaches; they show the efficiency of the method. >

The paper presents a method for generating systematically and exactly all possible series-parallel (s-p) structures for several i.i.d. units. The method is modified in order to calculate the failure probabilities and reliabilities of all the possible s-p redundant structures for the i.i.d. units with two failure modes. Included is an example of finding an optimal s-p redundant structure subject to four constraints for reliability (two for failure probabilities and two for fail-safe).

This paper presents a surrogate constraints algorithm for solving nonlinear programming, nonlinear integer programming, and nonlinear mixed integer programming problems. The algorithm contains a new technique for generating a succession of vector values of surrogate multiplier (ie, surrogate problems). By using this technique, a computer can keep a polyhedron, which is a vector space of surrogate multipliers to be considered at a certain time, in its memory. Furthermore it can cut the polyhedron by a given hyperplane, and produce the remaining space as the next polyhedron. Simple examples are included.

Tillman's 1969 treatment for components with several modes of failure and series (or parallel) redundancy using the components is discussed. This note claims that his treatment is wrong. Author replies and further discussion are provided.

The paper presents an efficient method for finding the exact optimal solutions of reliability allocation problems that are formulated as an integer nonlinear programming problem generalized to handle nonlinear constraints and nonseparable problems. The method is based on branch-and-bound and developed by considering separation and relaxation techniques.

The paper presents a method for obtaining an optimal reliability allocation of an n-stage series system. In each stage, redundant comnponents can be added (in parallel, stand-by, or k-out-of-n:G, etc.), or a more reliable component can be used in order to improve the system reliability. The solution is obtained by repeatedly using a more reliable candidate at each stage that has the greatest value of a `weighted sensitivity function'. The balance between the objective unction and the constraints is controlled by a `balancing coefficient'. The overall computational procedure is given and an example is presented. The computations are given for a set of randomly generated test problems in which the optimal parallel redundancy under linear onstraints is determined. The proposed method is then compared with other methods.

The paper considers the problem of determining an optimal design of a reliability system with multiple properties and integer variables. This problem is formulated as a multiobjective nonlinear pure-integer programming problem. The method for solving this problem consists of three techniques: 1) narrowing a given feasible region if the region is too wide, 2) obtaining exactly the set of Pareto-optimal solutions, and 3) selecting the best one for decision makers from the set of Pareto-optimal solutions. An increase in the number of objective functions scarcely affects the size of problem. An example is included.

This paper compares three heuristic methods (Nakagawa-Nakashima, Gopal-Aggarwal-Gupta, Sharma-Venkateswaran) for solving reliability optimization problems with nonlinear constraints. Their effectiveness, measured in terms of computation time, optimality rate, and relative error, is evaluated on several sets of randomly generated test problems with nonlinear constraints for series systems. A FORTRAN program for generating the test problems and best known solutions (some of them are exact optimal) are included.

This note clarifies an interesting paradox concerning s-independent vs mutually exclusive events.