A concept for the mathematical modeling of multiple repairable equipment and logistic systems (MREAL systems) is developed; These systems consist of multiple populations of repairable equipment, and their associated design, procurement, maintenance, and supply support. MREAL systems present management and design problems which parallel the·management and design of multiple, consumable item inventory systems. However, the MREAL system is more complex since it has a repair component.

The MREAL system concept is described in a classification hierarchy which attempts to categorize the components of such systems. A specific mathematical model (MREAL1) is developed for a subset of these components. Included in MREAL1 are representations of the equipment reliability and maintainability design problem, the maintenance capacity problem, the retirement age problem, and the population size problem, for each of the multiple populations. MREAL1 models the steady state stochastic behavior of the equipment repair facilities using an approximation which is based upon the finite source, multiple server queuing system. System performance measures included in MREAL1 are: the expected MREAL total system life cycle cost (including a shortage cost penalty); the steady state expected number of shortages; the probability of catastrophic failure in each equipment population; and two budget based measures of effectiveness.

Two optimization methods are described for a test problem developed for MREAL1. The first method computes values of the objective function and the constraints for a specified subset of the solution space. The best feasible solution found is recorded. This method can also examine all possible solutions, or can be used in a manual search. The second optimization method performs an exhaustive enumeration. of the combinatorial programming portion of MREAL1, which represents equipment design. For each enumerated design combination, an attempt is made to find the optimal solution to the remaining nonlinear discrete programming problem. A sequential unconstrained minimization technique is used which is based on an augmented Lagrangian penalty function adapted to the integer nature of MREAL1. The unconstrained minimization is performed by a combination of Rosenbrock's search technique, the steepest descent method, and Fibonacci line searches, adapted to the integer nature of the search. Since the model contains many discrete local minima, the sequential unconstrained minimization is repeated from different starting solutions, based upon a heuristic selection procedure. A gradient projection method provides the termination criteria for each unconstrained minimization.