Parametric integer programming with application in reliability optimization problems

Abstract

The reliable performance of a system for a mission under various conditions is of utmost importance in many situations. Hence, the reliability allocation models and reliability optimization in design have been extensively studied8,9. Traditionally redundancy allocation has been used in improving the reliability of a multi-stage system. That is to use additional components or subassemblies above the minimum required for an existing system to operate. The optimal design depends on maximizing the reliability of the system within the imposed constraint bounds. A series system with active component redundancy can be formulated as a nonlinear integer programming problem. When a given problem has two constraints, Fyffe, Himes and Lee3 provided dynamic programming approach using the Lagrange multiplier, Misra5 and Sharma7 provided heuristic methods, and Nakagawa and Miyazaki5 presented surrogate constraints method. However, finding an "optimal" solution to the reliability optimization model is not the only requirement. Decision makers may want to know what happens if a certain change is made in the right-hand side of the constraints. Some right-hand sides may not be known with certainty (usually, right-hand sides are estimated values), and hence the decision maker must know how the optimal solution behaves as these parameters are varied in the model. In general, problems of this type can be formulated as parametric nonlinear integer programming. A review of postoptimal analysis and parametric analysis appeared in Geoffrion and Nauss4, which treats integer linear programming. As for nonlinear programming, Cooper1 provided postoptimal right-hand side analysis which discusses right-hand sides with discrete scales. In this paper, we modify Cooper's algorithm so that we can simultaneously find optimal solutions for a family of problems of the type described above which differ only in the right-hand side vector of the constraints.