Richard M. Soland

A Closed Queueing Network Model for Multi-Echelon Repairable Item Provisioning

Abstract A model of a two-echelon (two levels of repair, one level of supply) repairable-item provisioning system is presented. It is desired to find the capacities of the base and depot repair facilities as well as the spares level which together guarantee a specified system service level at minimum cost. Closed queueing network theory is used to model the stochastic process, and an implicit enumeration algorithm is used to solve the optimization problem.

A multicriteria approach to the location of public facilities

Abstract We argue that practical problems involving the location of public facilities are really multicriteria problems, and ought to be modeled as much. The general criteria are those of cost and service, but there exist several distinct criteria in each of those two categories. For the first category, fixed investment cost, fixed operating cost, variable operating cost, total operating cost, and total discounted cost are all reasonable criteria to consider. In terms of service, both demand served and response time (or distance traveled) are appropriate criteria, either agglomerated or considered on the basis of the individual clients. In this paper we treat such multicriteria questions in the framework of a model for selecting a subset of M sites at which to establish public facilities in order to serve client groups located at N distinct points. We show that for some combinations of specific criteria, parametric solutions of a generalized assignment problem (GAP) will yield all efficient solution. In most other cases the efficient solutions can be found through parametric solution of a GAP with additional constraints of a type which can be incorporated into an existing algorithm for the GAP. Rather than attempting to find all efficient solutions, however, we advocate an interactive approach to the resolution of multicriteria location problems and elaborate on a specific interactive algorithm for multicriteria optimization which for the present model solves a finite sequence of GAPs or GAP-type problems. Finally, some similar aspects of private sector location problems are discussed.

An interactive computer system for multicriteria facility location

Abstract A computer system developed to assist a decision maker in finding his most preferred efficient solution to a multicriteria location model is described. The system requires interaction between the decision maker and optimization software to conduct a heuristic search of the set of efficient solutions to the location model. A command language developed to give the user control over the system and an optimization algorithm developed for finding efficient solutions are presented.

Optimal terminal defense tactics when several sequential engagements are possible

In this paper we treat a situation in which a terminal defense wishes to minimize the expected fraction of a target destroyed. Attacking reentry vehicles RVs arrive simultaneously, and therefore the defense knows the attack size. Several sequential engagements of the RVs by defensive interceptors are possible, and the defense has the capability to shoot-look-shoot between these engagements. Using techniques of stochastic dominance and stochastic dynamic programming, we determine optimal defensive tactics as functions of the numbers of RVs, interceptors, and engagements remaining; the various single-shot kill probabilities are parameters. We examine extensions of the basic model that determine both the expected number of interceptors remaining after the attack and the minimum number of interceptors needed to provide a desired level of protection of the target.

A Renewal Theoretic Approach to the Estimation of Future Demand for Replacement Parts

This expository paper presents a renewal theoretic approach to the determination of the probability distribution of the demand for replacement parts in a specified future time interval. Ordinary, modified, and equilibrium renewal processes are discussed as models for physical processes in which parts are replaced as they fail. Some results from renewal theory are stated and then used to estimate the demand for replacement parts under a number of different conditions. Practical difficulties, and ways to circumvent them, are then discussed. The paper concludes with an illustrative example.

On Some Common Interests Among Reliability, Inventory, and Queuing

Queuing networks can be used to model maintained systems. Under many conditions, closed-network queuing theory can be applied to ascertain the availability of such systems. Multi-echelon repairable-item inventory systems are one such class. Problems of common interest to the reliability, queuing, and inventory communities are highlighted, and solution techniques for these problems are presented.

Iterative solution methods for obtaining the steady-state probability distributions of Markovian multi-echelon repairable item inventory systems

Abstract The Jacobi, Gauss-Seidel and bi-conjugate gradient methods are used to compute the steady-state probability distributions for finite state-space continuous time Markov processes (closed queueing networks) that arise in the modeling of two- and three-echelon repairable item inventory systems. Alternative systems of linear equations that express the steady-state conditions are examined, generation and storage of the transition rate matrix are discussed briefly and various initializations and stopping criteria are tested. Numerical results are given for problems with up to one million states. Good results are obtained with a two-phase algorithm that uses the Gauss-Seidel method first and the bi-conjugate gradient method subsequently.

Optimal Provisioning for Single-Echelon Repairable Item Inventory Control in a Time-Varying Environment

Abstract This paper presents an algorithm to find a least-cost mix of spares and repair channels, satisfying minimum performance measures, in a single-echelon repairable item inventory system. The monotonicity of the performance measures is the basis for using implicit enumeration in the optimization algorithm. The system analyzed is in a transient environment and is modeled as a birthdeath process with finite state space. The systems transition rates are allowed to vary over time. Transient probabilities are computed using randomization. Illustrative examples are presented.

Optimal Allocation of Missiles Against Area and Point Defenses

This paper considers the problem of allocating a given number of reliable offensive missiles of one type against reliable subtractive area and point defenses of known size and the cities protected by these defenses so as to maximize the total damage done to the cities. An integer nonlinear programming formulation of the problem is developed and a branch-and-bound algorithm for its solution is offered. Computational considerations are discussed and numerical results are given.

Letter to the Editor-Availability of Renewal Functions for Gamma and Weibull Distributions with Increasing Hazard Rate

This note draws attention to the availability of tables of the principal renewal functions for cases in which the underlying lifetime distribution of the renewal process is gamma or Weibull with increasing hazard rate. The mathematical formulation is given, along with a description of the method of computation and an example illustrating the use of the tables.