Location Set Covering Problem Research Articles

Integers in the location set-covering problem

Pastor's discovery of a historical example of a noninferior fractional solution to the location set-covering problem is not unexpected. We argue here that a method or observation that is generally good (or true) has merit and should not be dismissed as lacking in utility because of occasional counterexamples. A further report concerning fractions/integers in set-covering problems is made and the specific example cited by Pastor is analyzed.

Fractions in the location set-covering problem

Inferior and noninferior solutions to the location set-covering problem arc differentiated and computational experience is presented which strongly suggests that noninferior solutions, from the relaxed linear programming formulation, will be fully integral with a high degree of reliability. The authors have not yet encountered a noninferior fractional solution. An explanation of this phenomenon is presented.

Covering-Location Models for Emergency Situations That Require Multiple Response Units

In this paper we propose a new criterion for coverage which is suitable for two kinds of applications: (i) location of fire trucks in a geographical area in which some demands require multiple fire trucks to be within an acceptable distance standard to achieve coverage; and (ii) location of ambulances in an environment in which a large demand volume often leads to the unavailability of the most desirable response unit. The set covering location problem and the maximal covering location problem are examined in the proposed modeling framework. Properties for these models are developed, and limited computational experience is reported.

A simple model for predicting the number of servers necessary to cover a region

A simple regression model is developed to estimate the number of servers necessary to cover a region. Data for the model were obtained within a research design that included five replications of 180 experimental conditions. A total of 300 rectangular regions were generated, and a set covering location problem was solved for each region using three distance metrics. Statistical results established coverage radius, compactness, and the number of nodes in the region to be significant factors. The resulting regression model was applied to a South Carolina county. Its estimates are seen to be very close to the number of units needed to cover. The model is shown to be a convenient technique for quickly estimating the number of servers while exhibiting an error distribution that is conservatively biased towards overestimation. This last property is desirable for public emergency services.

Capacitated Covering Models

Because of their widespread applicability, the set covering location problem and the maximal covering location problem have received considerable attention in the facility-location literature. There have been many extensions and modifications to these problems as they have been applied to various planning scenarios. A basic underlying assumption of the location-covering models formulated to date is that the facilities being sited are uncapacitated. Although this assumption is valid in many location-planning settings, there certainly exist situations in which this assumption severely limits the application of covering models. Capacitated versions of the set covering location problem and the maximal covering location problem have thus been formulated. In addition, the theoretical links between these models and the capacitated plant location problem, the capacitated p-median problem and, the generalized assignment problem are shown. By exploiting these links, planners can solve small and moderately sized real-world problems with existing solution methods. It is expected that these theoretical links will also give insight into developing new heuristics for large-sized capacitated covering problems.

Concepts and Applications of Backup Coverage

Backup coverage, the second coverage of a demand node, is suggested as a decision criterion in modelling the location of emergency services on a network. The efficient handling of stochastic demand by vehicles which can respond to only one call at a time may require backup coverage in areas of high demand as a means to maintain a more uniform level of service. This new criterion is applied in the context of the classic covering models, the Location Set Covering Problem and the Maximal Covering Location Problem. First coverage as defined in these models is traded off against backup coverage in the present work. Other efforts which incorporate additional levels of coverage are reviewed. We also show how to extend these models to third as well as subsequent coverage. Based on an example problem we show that significant levels of backup coverage may possibly be provided within a system without substantial loss of first coverage.

Location Modeling Utilizing Maximum Service Distance Criteria

The location set‐covering problem (LSCP) and the maximal covering location problem (MCLP) have been the subject of considerable interest. As originally defined, both problems allowed facility placement only at nodes. This paper deals with both problems for the case when facility placement is allowed anywhere on the network. Two theorems are presented that show that when facility placement is unrestricted, for either the LSCP or MCLP at least one optimal solution exists that is composed entirely of points belonging to a finite set of points called the network intersect point set (NIPS). Optimal solution approaches to the unrestricted site LSCP and MCLP problems that utilize the NIPS and previously developed solution methodologies are presented. Example solutions show that considerable improvement in the amount of coverage or the number of facilities needed to insure total coverage can be achieved by allowing facility placement along arcs of the network. In addition, extensions to the arc‐covering model and the ambulance‐hospital model of ReVelle, Toregas, and Falkson are developed and solved.