Tom M. Cavalier

A heuristic for the multiple tour maximum collection problem

Abstract The multiple tour maximum collection problem (MTMCP) consists of determining the m optimal time constrained tours which visit a subset of weighted nodes in a graph such that the total weight collected from the subset of nodes is maximized. In this paper, a heuristic for the MTMCP is developed. The results of computational testing reveal that the heuristic has three very attractive features. First, the heuristic will always produce a feasible solution if one exists. Second, the heuristic produces very good solutions. In most cases where the exact solution is known, it produces the optimal solution. And finally, the heuristic has the ability to handle large problems in a very short amount of computation time.

Modeling and integer programming techniques applied to propositional calculus

Abstract This paper discusses alternative methods for constructing a 0–1 integer programming problem from a propositional calculus problem and the use of the resulting mathematical program to solve the related logic problem. This paper also identifies some special structures associated with the constraint sets and discusses several fundamental results concerning methods of preprocessing the logical inferences into constraints.

Virtual manufacturing cells : exploiting layout design and intercell flows for the machine sharing problem

Existing approaches to cell formation concentrate on machine grouping and part family formation. Since their input data usually do not consider flow directions and volumes, they neglect layout and handling strategies that can simplify the machine sharing problem. Hence, these approaches fail to relate the intracell and intercell layout decisions to the machine grouping and sharing decisions. This paper introduces a new approach for cell formation which integrates machine grouping and layout design, neglecting part family formation. The concepts of a hybrid cellular layout and virtual manufacturing cells are related. It is shown that a combination of overlapping GT cells, functional layout and handling reduces the need for machine duplication among cells. This approach questions the traditional emphasis on machine duplication to create independent cells that is suggested by the standard machine-part matrix clustering methods. The steps in the method are demonstrated by using two illustrative examples obtai...

An efficient algorithm for facility location in the presence of forbidden regions

Abstract This paper investigates a constrained form of the classical Weber problem. Specifically, we consider the problem of locating a new facility in the presence of convex polygonal forbidden regions such that the sum of the weighted distances from the new facility to n existing facilities is minimized. It is assumed that a forbidden region is an area in the plane where travel and facility location are not permitted and that distance is measured using the Euclidean-distance metric. A solution procedure for this nonconvex programming problem is presented. It is shown that by iteratively solving a series of unconstrained problems, this procedure terminates at a local optimum to the original constrained problem. Numerical examples are presented.

Discriminant analysis via mathematical programming: certain problems and their causes

Abstract In the past several years there has been some interest shown in the use of mathematical programming (i.e. specifically either linear programming or linear goal programming) for use in discriminant analysis—an area previously approached only by the more conventional methods of statistical analysis. However, those who have actually attempted to apply the mathematical programming approach have, in some instances, encountered a number of problems which raise questions about the usefulness of the methodology in such an application. In this paper, we describe these problems, identify their causes and suggest ways in which such problems may be avoided. This work represents an ongoing effort on the part of the authors in the development of an efficient revised approach to discriminant analysis via mathematical programming.

Constrained location of competitive facilities in the plane

This paper examines a competitive facility location problem occurring in the plane. A new gravity-based utility model is developed, in which the capacity of a facility serves as its measure of attractiveness. A new problem formulation is given, having elastic gravity-based demand, along with capacity, forbidden region, and budget constraints. Two solution algorithms are presented, one based on the big square small square method, and the second based on a penalty function formulation using fixed-point iteration. Computational testing is presented, comparing these two algorithms along with a general-purpose nonlinear solver.

A new algorithm for the minimal-area convex enclosure problem

Abstract The problem of cutting parts from a piece of material occurs in a number of settings. When the pieces to be cut are non-rectangular, the problem is called the irregular pattern layout problem (or nesting problem). This problem occurs in sheet metal and apparel industries, for example. One possible approach is to combine individual pieces into low-waste ‘modules’ which are then arranged by a heuristic technique. In this spirit, the Minimal-Area Convex Enclosure Problem is solved in this paper. Given two simple polygons, the algorithm finds the relative position of one with respect to the other such that the area of their convex enclosure is minimized. The technique searches along the envelope (or no-fit-polygon), dynamically maintaining the convex enclosure vertices as well as an analytic representation of the area. The algorithm runs quickly and computational experience is included.

Scheduling for machining and assembly in a job-shop environment

The problem of scheduling n assembly jobs in a job-shop environment is addressed. Each job is characterized by multiple parts that must themselves be scheduled for processing through the shop. The individual parts combine at assembly stations to form sub-assemblies or final assemblies. Sub-assemblies combine with other subassemblies to form final assemblies. The objective pursued in the developed model is to maximize the machine utilization subject to satisfying job due date requirements. Essentially, one must consider machine availability, the amount of work on each machine for each operation, the precedence constraints, and the dispatching criteria to perform scheduling in this kind of environment. A heuristic algorithm is developed to solve the problem. The application of the algorithm is demonstrated with an example problem.

A heuristic algorithm for minimax sensor location in the plane

This paper addresses the problem of locating a finite number of sensors to detect an event in a given planar region. The objective is to minimize the maximum probability of non-detection where the underlying region consists of a convex polygon. The sensor location problem has a multitude of applications, including the location of aircraft detection sensors, the placement of sentries along a border to detect enemy penetration, the detection of nuclear tests, and the detection of natural and hazardous man-made events. The problem is a difficult nonlinear nonconvex programming problem even in the case of two sensors. A fast heuristic based on Voronoi polygons is developed in this paper. The algorithm can quickly generate high-quality solutions. Computational experience is provided.

Facility location in the presence of congested regions with the rectilinear distance metric

This paper considers the planar p-median problem in the presence of congested regions, where distances are measured with the rectilinear distance metric. We define a congested region as a convex polygonal area of the plane in which a new facility cannot be located but through which travel is permitted at an additional cost per unit distance. Based on these assumptions, we show that this constrained form of the planar P-median problem can be transformed into an equivalent unconstrained P-median problem on a network. Hence, this constrained form of the P-median problem is reduced to a combinatorial search where an optimal set of new facilities is chosen from a finite set of candidate points.