James E. Ward

Using Block Norms for Location Modeling

In formulating a continuous location model with facilities represented as points in Rn e.g., typically in the plane, one must characterize the distance between two points as a function of their coordinates. Two criteria in selecting a distance function are 1 to obtain good approximations of actual distances, and 2 to obtain a mathematical model of the location problem that is easy to solve. In this paper, we show how a class of norms with polygonal contours, called block norms, can yield attractive choices as distance functions with respect to these criteria. In particular, we consider the following relevant properties of block norms: they generalize the concepts of rectilinear or city-block travel; they are dense in the set of all norms; they have interesting travel interpretations; in the plane, they can be expressed as a sum of the absolute values of linear functions; they often give better approximations to actual highway distances than the most frequently used family of norms, the lp norms; and, finally, they yield linear programming formulations of certain facility location problems i.e., the Weber problem and the Rawls problem.

Integrated Facility Location and Vehicle Routing Models: Recent Work and Future Prospects

SYNOPTIC ABSTRACTFacility location and vehicle routing are two widely used and studied management science resource planning models. According to the 1986 Statistical Abstract of the United States, freight transportation outlays for local trucking totaled 80.9 billion dollars in 1983. Thus, location/routing decisions have considerable economic importance in domains such as distribution systems planning. Since location and routing decisions are closely related, integrated models that consider the two aspects simultaneously offer the promise of more effective and economical decisions. However, such integrated models are complex and their design poses challenges in combining the short-term operational considerations of vehicle routing with the medium/long-term strategic issues of facility location. This paper discusses recent research related to various modeling approaches for location/routing problems, including comprehensive mathematical programming formulations, analytical approximations, and modified faci...

Approaches to sensitivity analysis in linear programming

A continuing priority in sensitivity and parametric analysis is to develop approaches that provide useful information, that are easy for a decision-maker to use, and that are computationally practical. Herein we review approaches to sensitivity analysis in linear programming and discuss how they meet the above needs. Special emphasis is given to sensitivity analysis of the objective function coefficients.

Some Properties of Location Problems with Block and Round Norms

The point-objective problem and the Weber problem are two well-known formulations for locating a new facility with respect to a set of fixed facilities. When locations are represented as points on a plane, the point-objective problem is a multiple objective formulation of minimizing the distance from a variable point to each of the fixed points. Similarly, the Weber problem is a single objective formulation of minimizing the sum of transportation costs between the variable point and the fixed points, where transportation cost is a function of distance. Generalizing solution properties for these problems from distance measures given by the Euclidean, rectilinear, Ip, and one-infinity norms; this paper develops solution properties under the broad classes of distance measures given by block and round norms. For the point-objective problem, we show that i the efficient set for all round norms is the convex hull of the set of fixed points and ii the efficient set under a block norm tends to the convex hull for a sequence of block norms approaching a round norm. For the Weber problem, we prove that i an optimal location for any block norm may be found in a finite set of intersection points belonging to the convex hull and ii this set tends to the convex hull for a sequence of block norms approaching a round norm. Finally, we use these results to propose a synthesis of some of the main properties in continuous and network location theory.

On the location of a tree-shaped facility

This paper considers the problem of locating a central facility on a tree network. The central facility takes the form of a subtree of the network, and provides service to several demand points located at nodes of the network. Two types of costs are involved in evaluating a given facility. The setup cost represents the cost of establishing the facility and is taken to be directly proportional to the total length of the facility. The transportation cost is the cost associated with the travel of customers to the facility. The objective function is to select a tree-shaped facility that will minimize the sum of the setup cost and the total transportation cost. We introduce a general model and a simple {open_quotes}bottom-up{close_quotes} dynamic programming algorithm for its solution. We then focus on the important class of covering problems, where the transportation cost of a customer to the serving facility is zero if the distance between the two is below a certain threshhold, and is equal to some penalty term otherwise. We improve upon the performance of existing methods, and present algorithms with subquadratic complexity.

Technical Note—A New Norm for Measuring Distance Which Yields Linear Location Problems

We propose a new norm, called the one-infinity norm, for characterizing distance in facility location problems. This new norm, which is a hybrid version of the rectilinear and Tchebycheff norms, not only gives a good characterization of distance (as compared, for example, to results by Love and Morris) but also has two alternate interpretations of travel. Furthermore, it yields linear programming formulations of location problems.

Balancing Retailer Inventories

This paper examines the optimality of inventory “balancing” in a one-warehouse N-retailer distribution system facing stochastic demand for a single product over T successive time intervals. In particular, we consider the division of predetermined quantities of warehouse stock among retailers in each interval. Balancing attempts to bring the retailer-inventories to the same (normalized with respect to first-interval demand) net inventory level. When the demand distribution over the T periods is symmetric with respect to all pairs of retailers, and retailer cost over the T-intervals is a convex function of the T shipments, we show that balancing divisions are optimal. The required convexity condition is shown to be satisfied for some familiar cost functions when shortages are backordered and lost. However for other cost functions, the convexity condition is satisfied under backordering, but not under lost sales. We also consider the nonidentical-retailer (demands are not symmetric) case and provide examples...

A parts selection model with one-way substitution

Abstract This paper develops a parts selection model with one-way substitution. The problem horizon is assumed to be infinite and cost and demand parameters are assumed to be stationary over the problem horizon. The objective is to select a subset of parts from the available set of parts to minimize the average per-period cost. In addition to the purchase and inventory related costs, we assume that there is an overhead cost to manage the stock of parts which is a function of the number of parts kept in stock. The purchase price function used in the paper covers all the commonly used quantity discount models. Several analytical results are developed for the problem and an efficient dynamic programming algorithm is developed to find an optimal solution.

A minimum-length covering subtree of a tree

One shows that the problem of finding a minimum-length covering subtree of a tree can be solved by appropriately modifying a known algorithm for finding a minimum cardinality point cover. Optimality of the unique covering subtree is established by the use of duality theory: a weak duality theorem and a strong duality theorem. The solution provided by the algorithm is shown to solve certain other optimization problems as well

A minimum length covering subgraph of a network

This paper concerns the problem of locating a central facility on a connected networkN. The network,N, could be representative of a transport system, while the central facility takes the form of a connected subgraph ofN. The problem is to locate a central facility of minimum length so that each of several demand points onN is covered by the central facility: a demand point atvi inN is covered by the central facility if the shortest path distance betweenvi and the closest point in the central facility does not exceed a parameterri. This location problem is NP-hard, but for certain special cases, efficient solution methods are available.