Solution of minisum and minimax location–allocation problems with Euclidean distances

A new method for the solution of minimax and minisum location–allocation problems with Euclidean distances is suggested. The method is based on providing differentiable approximations to the objective functions. Thus, if we would like to locate m service facilities with respect to n given demand points, we have to minimize a nonlinear unconstrained function in the 2m variables x₁,y₁, ?,x_m,y_m. This has been done very efficiently using a quasi-Newton method. Since both the original problems and their approximations are neither convex nor concave, the solutions attained may be only local minima. Quite surprisingly, for small problems of locating two or three service points, the global minimum was reached even when the initial position was far from the final result. In both the minisum and minimax cases, large problems of locating 10 service facilities among 100 demand points have been solved. The minima reached in these problems are only local, which is seen by having different solutions for different initial guesses. For practical purposes, one can take different initial positions and choose the final result with best values of the objective function. The likelihood of the best results obtained for these large problems to be close to the global minimum is discussed. We also discuss the possibility of extending the method to cases in which the costs are not necessarily proportional to the Euclidean distances but may be more general functions of the demand and service points coordinates. The method also can be extended easily to similar three-dimensional problems.     

New heuristic methods for the capacitated multi‐facility Weber problem

In this paper we consider the capacitated multi‐facility Weber problem with the Euclidean, squared Euclidean, and ?_p‐distances. This problem is concerned with locating m capacitated facilities in the Euclidean plane to satisfy the demand of n customers with the minimum total transportation cost. The demand and location of each customer are known a priori and the transportation cost between customers and facilities is proportional to the distance between them. We first present a mixed integer linear programming approximation of the problem. We then propose new heuristic solution methods based on this approximation. Computational results on benchmark instances indicate that the new methods are both accurate and efficient. © 2006 Wiley Periodicals, Inc. Naval Research Logistics 2006     

Minimax and maximin facility location problems on a sphere

The problem dealt with in this article is as follows. There are n “demand points” on a sphere. Each demand point has a weight which is a positive constant. A facility must be located so that the maximum of the weighted distances (distances are the shortest arcs on the surface of the sphere) is minimized; this is called the minimax problem. Alternatively, in the maximin problem, the minimum weighted distance is maximized. A setup cost associated with each demand point may be added for generality. It is shown that any maximin problem can be reparametrized into a minimax problem. A method for finding local minimax points is described and conditions under which these are global are derived. Finally, an efficient algorithm for finding the global minimax point is constructed.     

The gradual covering problem

In this paper we investigate the gradual covering problem. Within a certain distance from the facility the demand point is fully covered, and beyond another specified distance the demand point is not covered. Between these two given distances the coverage is linear in the distance from the facility. This formulation can be converted to the Weber problem by imposing a special structure on its cost function. The cost is zero (negligible) up to a certain minimum distance, and it is a constant beyond a certain maximum distance. Between these two extreme distances the cost is linear in the distance. The problem is analyzed and a branch and bound procedure is proposed for its solution. Computational results are presented. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Relaxation method for the solution of the minimax location-allocation problem in euclidean space

A method previously devised for the solution of the p-center problem on a network has now been extended to solve the analogous minimax location-allocation problem in continuous space. The essence of the method is that we choose a subset of the n points to be served and consider the circles based on one, two, or three points. Using a set-covering algorithm we find a set of p such circles which cover the points in the relaxed problem (the one with m < n points). If this is possible, we check whether the n original points are covered by the solution; if so, we have a feasible solution to the problem. We now delete the largest circle with radius r_p (which is currently an upper limit to the optimal solution) and try to find a better feasible solution. If we have a feasible solution to the relaxed problem which is not feasible to the original, we augment the relaxed problem by adding a point, preferably the one which is farthest from its nearest center. If we have a feasible solution to the original problem and we delete the largest circle and find that the relaxed problem cannot be covered by p circles, we conclude that the latest feasible solution to the original problem is optimal. An example of the solution of a problem with ten demand points and two and three service points is given in some detail. Computational data for problems of 30 demand points and 1–30 service points, and 100, 200, and 300 demand points and 1–3 service points are reported.     

Multi-item service constrained (s,S) policies for spare parts logistics systems

Many organizations providing service support for products or families of products must allocate inventory investment among the parts (or, identically, items) that make up those products or families. The allocation decision is crucial in today's competitive environment in which rapid response and low levels of inventory are both required for providing competitive levels of customer service in marketing a firm's products. This is particularly important in high-tech industries, such as computers, military equipment, and consumer appliances. Such rapid response typically implies regional and local distribution points for final products and for spare parts for repairs. In this article we fix attention on a given product or product family at a single location. This single-location problem is the basic building block of multi-echelon inventory systems based on level-by-level decomposition, and our modeling approach is developed with this application in mind. The product consists of field-replaceable units (i.e., parts), which are to be stocked as spares for field service repair. We assume that each part will be stocked at each location according to an (s, S) stocking policy. Moreover, we distinguish two classes of demand at each location: customer (or emergency) demand and normal replenishment demand from lower levels in the multiechelon system. The basic problem of interest is to determine the appropriate policies (s_i S_i) for each part i in the product under consideration. We formulate an approximate cost function and service level constraint, and we present a greedy heuristic algorithm for solving the resulting approximate constrained optimization problem. We present experimental results showing that the heuristics developed have good cost performance relative to optimal. We also discuss extensions to the multiproduct component commonality problem.     

The conditional p-center problem in the plane

An algorithm is given for the conditional p-center problem, namely, the optimal location of one or more additional facilities in a region with given demand points and one or more preexisting facilities. The solution dealt with here involves the minimax criterion and Euclidean distances in two-dimensional space. The method used is a generalization to the present conditional case of a relaxation method previously developed for the unconditional p-center problems. Interestingly, its worst-case complexity is identical to that of the unconditional version, and in practice, the conditional algorithm is more efficient. Some test problems with up to 200 demand points have been solved. © 1993 John Wiley & Sons, Inc.     

Solving quadratic assignment problems with rectangular distances and integer programming

The problem considered involves the assignment of n facilities to n specified locations. Each facility has a given nonnegative flow from each of the other facilities. The objective is to minimize the sum of transportation costs. Assume these n locations are given as points on a two-dimensional plane and transportation costs are proportional to weighted rectangular distances. Then the problem is formulated as a binary mixed integer program. The number of integer variables (all binary) involved equals the number of facilities squared. Without increasing the number of integer variables, the formulation is extended to include “site costs” Computational results of the formulation are presented.     

Polynomial algorithms for center location on spheres

When locating facilities over the earth or in space, a planar location model is no longer valid and we must use a spherical surface. In this article, we consider the one-and two-center problems on a sphere that contains n demand points. The problem is to locate facilities to minimize the maximum distance from any demand point to the closest facility. We present an O(n) algorithm for the one-center problem when a hemisphere contains all demand points and also give an O(n) algorithm for determining whether or not the hemisphere property holds. We present an O(n³ log n) algorithm for the two-center problem for arbitrarily located demand points. Finally, we show that for general p, the p center on a sphere problem is NP-hard. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 341–352, 1997     

Duality in constrained multi‐facility location models

We consider the ??_p‐norm multi‐facility minisum location problem with linear and distance constraints, and develop the Lagrangian dual formulation for this problem. The model that we consider represents the most general location model in which the dual formulation is not found in the literature. We find that, because of its linear objective function and less number of variables, the Lagrangian dual is more useful. Additionally, the dual formulation eliminates the differentiability problem in the primal formulation. We also provide the Lagrangian dual formulation of the multi‐facility minisum location problem with the ??_pb‐norm. Finally, we provide a numerical example for solving the Lagrangian dual formulation and obtaining the optimum facility locations from the solution of the dual formulation. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 410–421, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10010     

Estimating travel distances by the weighted lp norm

Empirical distance functions are used to estimate actual travel distances in a transportation network, to verify the accuracy of road mileage data, and to formulate continuous location models. In this article we consider the problem of fitting the weighted l_p norm to a given network. Mathematical properties are derived for two fitting criteria found in the literature. These properties are used to develop an accurate and efficient methodology to solve for the best-fitting parameter values. The directional bias of the l_p norm is analyzed for its effect on the range of search for the optimal p value. Concepts and methodology are applied to a case study of the road system in Southern Ontario. In conclusion, a general framework for other types of distance functions is briefly discussed.     

Location of regional facilities

In this paper we consider the single-facility and multifacility problems of the minisum type of locating facilities on the plane. Both demand locations and the facilities to be located are assumed to have circular shapes, and demand and service is assumed to have a uniform probability density inside each shape. The expected distance between two facilities is calculated. Euclidean and squared-Euclidean distances are discussed.     

Computing demand properties at the wholesale warehouse level

Computational formulas are given for the mean, variance, and autocorrelation function of the demand process at an upper-echelon facility (warehouse). The demand process at the warehouse is induced by the aggregated inventory replenishment processes of N independently operated lower-echelon facilities (stores) in parallel. Each store, we assume, employs an (s,S) inventory replenishment policy with complete backlogging to satisfy its own random, independently and identically distributed demand. The formulas result from an analysis of the stochastic replenishment process at a single store. Examples of the properties of the demand process at the upper-echelon facility are presented for several lower-echelon environments.     

Facility location when demand is time dependent

In this article we investigate the problem of locating a facility among a given set of demand points when the weights associated with each demand point change in time in a known way. It is assumed that the location of the facility can be changed one or more times during the time horizon. We need to find the time “breaks” when the location of the facility is to be changed, and the location of the facility during each time segment between breaks. We investigate the minisum Weber problem and also minimax facility location. For the former we show how to calculate the objective function for given time breaks and optimally solve the rectilinear distance problem with one time break and linear change of weights over time. Location of multiple time breaks is also discussed. For minimax location problems we devise two algorithms that solve the problem optimally for any number of time breaks and any distance metric. These algorithms are also applicable to network location problems.     

Note: Facility location with closest rectangular distances

This paper considers the problem of locating one or more new facilities on a continuous plane, where the destinations or customers, and even the facilities, may be represented by areas and not points. The objective is to locate the facilities in order to minimize a sum of transportation costs. What is new in this study is that the relevant distances are the distances from the closest point in the facility to the closest point in the demand areas. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 77–84, 2000     

On the rectangular p-center problem

The p-center problem involves finding the best locations for p facilities such that the furthest among n points is as close as possible to one of the facilities. Rectangular (sometimes called rectilinear, Manhattan, or l₁) distances are considered. An O(n) algorithm for the 1-center problem, an O(n) algorithm for the 2-center problem, and an O(n logn) algorithm for the 3-center problem are given. Generalizations to general p-center problems are also discussed.     

A computation procedure for the exact solution of location-allocation problems with rectangular distances

A method is presented to locate and allocate p new facilities in relation to n existing facilities. Each of the n existing facilities has a requirement flow which must be supplied by the new facilities. Rectangular distances are assumed to exist between all facilities. The algorithm proceeds in two stages. In the first stage a set of all possible optimal new facility locations is determined by a set reduction algorithm. The resultant problem is shown to be equivalent to finding the p-median of a weighted connected graph. In the second stage the optimal locations and allocations are obtained by using a technique for solving the p-median problem.     

Performance analysis of queue length monitoring of M/G/1 systems

This study investigates the statistical process control application for monitoring queue length data in M/G/1 systems. Specifically, we studied the average run length (ARL) characteristics of two different control charts for detecting changes in system utilization. First, the nL chart monitors the sums of successive queue length samples by subgrouping individual observations with sample size n. Next is the individual chart with a warning zone whose control scheme is specified by two pairs of parameters, (upper control limit, d_u) and (lower control limit, d_l), as proposed by Bhat and Rao (Oper Res 20 (1972) 955–966). We will present approaches to calculate ARL for the two types of control charts using the Markov chain formulation and also investigate the effects of parameters of the control charts to provide useful design guidelines for better performance. Extensive numerical results are included for illustration. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

One-facility location with rectilinear tour distances

This article concerns the location of a facility among n points where the points are serviced by “tours” taken from the facility. Tours include m points at a time and each group of m points may become active (may need a tour) with some known probability. Distances are assumed to be rectilinear. For m ≤ 3, it is proved that the objective function is separable in each dimension and an exact solution method is given that involves finding the median of numbers appropriately generated from the problem data. It is shown that the objective function becomes multimodal when some tours pass through four or more points. A bounded heuristic procedure is suggested for this latter case. This heuristic involves solving an auxiliary three-point tour location problem.     

Evaluating cost and reliability integrated performance of stochastic logistics systems

The significance of integrating reliability into logistics performance has been established [The Logistics Performance Index and Its Indicators, World Bank International Trade and Transport Departments, (2010)]. Hence, as a response to the work by the World Bank, the present article aims to evaluate the performance index R_b,d of logistics systems as the probability that a specified demand d can be distributed successfully through multistate arc capacities from the source to the destination under the constraint that the total distribution cost should not exceed the cost limitation b. This article provides a pioneering approach for a straightforward computation of the performance index R_b,d. The proposed algorithm is a hybrid between the polynomial time capacity‐scaling algorithm, which was presented by Edmonds and Karp [JACM 19 (1972)], and the decomposition algorithm, which was presented by Jane and Laih [IEEE (2008)]. Currently, the proposed approach is the only algorithm that can directly compute R_b,d. An illustration of the proposed algorithm is presented. The results of the computational experiments indicate that the presented algorithm outperforms existing algorithms. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012