Simulating nonhomogeneous poisson processes with proportional intensities

The purpose of this research is to investigate simulation algorithms for nonhomogeneous Poisson processes with proportional intensities. Two algorithmic approaches are studied: inversion and thinning. Motivated by industrial practices, the covariate vector involved in the simulation is permitted to change after every event (or observation). The algorithms are extended to permit the simulation of general nonhomogeneous Poisson processes with possible discontinuities both in baseline intensity and covariate vector. This latter extension can be used to facilitate a wide range of failure situations that can arise with repairable systems. © 1994 John Wiley & Sons, Inc.     

A penalty for concave minimization derived from the tuy cutting plane

A wide variety of optimization problems have been approached with branch-and-bound methodology, most notably integer programming and continuous nonconvex programming. Penalty calculations provide a means to reduce the number of subproblems solved during the branch-and-bound search. We develop a new penalty based on the Tuy cutting plane for the nonconvex problem of globally minimizing a concave function over linear constraints and continuous variables. Computational testing with a branch-and-bound algorithm for concave minimization indicates that, for the problems solved, the penalty reduces solution time by a factor ranging from 1.2 to 7.2. © 1994 John Wiley & Sons, Inc.     

On the set of (perfect) equilibria of a bimatrix game

This article provides a new approach to the set of (perfect) equilibria. With the help of an equivalence relation on the strategy space of each player. Nash sets and Selten sets are introduced. The number of these sets is finite and each of these sets is a polytope. As a consequence the set of (perfect) equilibria is a finite union of polytopes. © 1994 John Wiley & Sons. Inc.     

Printed circuit board family grouping and component allocation for a multimachine,open-shop assembly cell

This article considers a particular printed circuit board (PCB) assembly system employing surface mount technology. Multiple, identical automatic placement machines, a variety of board types, and a large number of component types characterize the environment studied. The problem addressed is that of minimizing the makespan for assembling a batch of boards with a secondary objective of reducing the mean flow time. The approach adopted is that of grouping boards into production families, allocating component types to placement machines for each family, dividing of families into board groups with similar processing times, and the scheduling of groups. A complete setup is incurred only when changing over between board families. For the environment studied, precedence constraints on the order of component placement do not exist, and placement times are independent of feeder location. Heuristic solution procedures are proposed to create board subfamilies (groups) for which the component mounting times are nearly identical within a subfamily. Assignment of the same component type to multiple machines is avoided. The procedures use results from the theory of open-shop scheduling and parallel processor scheduling to sequence boards on machines. Note that we do not impose an open-shop environment but rather model the problem in the context of an open shop, because the order of component mountings is immaterial. Three procedures are proposed for allocating components to machines and subsequently scheduling boards on the machines. The first two procedures assign components to machines to balance total work load. For scheduling purposes, the first method groups boards into subfamilies to adhere to the assumptions of the open-shop model, and the second procedure assumes that each board is a subfamily and these are scheduled in order of shortest total processing time. The third procedure starts by forming board subfamilies based on total component similarity and then assigns components to validate the open-shop model. We compare the performance of the three procedures using estimated daily, two-day, and weekly production requirements by averaging quarterly production data for an actual cell consisting of five decoupled machines. © 1994 John Wiley & Sons, Inc.     

On the design of a tandem queue with blocking: Modeling,analysis, and gradient estimation

We demonstrate here how recent advances in the study of discrete-event stochastic systems provide fruitful results for the modeling, analysis, and design of manufacturing systems. We consider a multistage make-to-stock system where outputs from the final stage are used to satisfy customer demands. We address the problem of finding the appropriate trade-off between reduced order waiting time and increased process speeds. Using the idea of infinitesimal perturbation analysis (IPA), we establish a simple procedure where sample-path derivatives can be obtained along an arbitrary sample path. Under suitable conditions, we demonstrate that these derivative estimators are unbiased and strongly consistent and can be used in a classical stochastic optimization scheme to solve the problem. The role of continuity and convexity on the validity of the estimator is also addressed. Although the focus of this article is not to solve for the optimal solution, we provide a theoretical justification for such a pursuit. The approach is appealing as it is numerically stable, easy to implement, and can be extended to other system performance measures. © 1994 John Wiley & Sons, Inc.     

Minimax resource allocation problems with ordering constraints

Resource allocation problems consider the allocation of limited resources among numerous competing activities. We address an allocation problem with multiple knapsack resource constraints. The activities are grouped into disjoint sets. Ordering constraints are imposed on the activities within each set, so that the level of one activity cannot exceed the level of another activity in the same set. The objective function is of the minimax type and each performance function is a nonlinear, strictly decreasing and continuous function of a single variable. Applications for such resource allocation problems are found, for example, in high-tech industries confronted with large-scale and complex production planning problems. We present two algorithms to solve the allocation problem with ordering constraints. The first one uses characterization of the optimal decision variables to apply a search method. The second algorithm solves a sequence of problems, each in the format of the original problem without ordering constraints. Whereas the computational effort of the first algorithm depends on the desired degree of accuracy even for linear performance functions, the effort of the latter algorithm is polynomial for certain classes of performance functions. © 1994 John Wiley & Sons, Inc.     

Scheduling about a large common due date with tolerance to minimize mean absolute deviation of completion times

In this article the problem of minimizing the mean absolute deviation (MAD) of job completion times about an unrestricted given common due date with tolerance in the n-job, single-machine scheduling environment is considered. We describe some optimality conditions and show that the unrestricted version of the MAD problem with an arbitrary due date tolerance is polynomial by proposing a polynomial-time algorithm for identifying an optimal schedule. © 1994 John Wiley & Sons, Inc.     

Cost analysis and extension of a simple maintenance-scheduling heuristic

Practitioners of maintenance scheduling sometimes use a simple analytic formula, derived based on inspection scheduling, as a heuristic to determine the length of the preventive maintenance period. The sensitivity of this heuristic solution is analyzed and the cost penalties are calculated compared to the exact solution that utilizes the lifetime distribution in the derivation of the optimal preventive maintenance period. A simple extension of the heuristic is suggested to improve the approximation with a slightly increased computational effort. The sensitivity and cost analysis of the improved heuristic are discussed as well. © 1994 John Wiley & Sons, Inc.     

Setting military reenlistment bonuses

The United States military frequently has difficulty retaining enlisted personnel beyond their initial enlistment. A bonus program within each service, called a Selective Reenlistment Bonus (SRB) program, seeks to enhance reenlistments and thus reduce personnel shortages in critical military occupational specialties (MOSs). The amount of bonus is set by assigning “SRB multipliers” to each MOS. We develop a nonlinear integer program to select multipliers which minimize a function of deviations from desired reenlistment targets. A Lagrangian relaxation of a linearized version of the integer program is used to obtain lower bounds and feasible solutions. The best feasible solution, discovered in a coordinate search of the Lagrangian function, is heuristically improved by apportioning unexpended funds. For large problems a heuristic variable reduction is employed to speed model solution. U.S. Army data and requirements for FY87 yield a 0-1 integer program with 12,992 binary variables and 273 constraints, which is solved within 0.00002% of optimality on an IBM 3033AP in less than 1.7 seconds. More general models with up to 463,000 binary variables are solved, on average, to within 0.009% of optimality in less than 1.8 minutes. The U.S. Marine Corps has used a simpler version of this model since 1986. © 1993 John Wiley & Sons, Inc.