Elsevier

Computers & Operations Research

Volume 51, November 2014, Pages 182-189
Computers & Operations Research

Dominance rules for the optimal placement of a finite-size facility in an existing layout

https://doi.org/10.1016/j.cor.2014.05.015Get rights and content

Abstract

In this paper, we consider the problem of optimal placement of a single finite-size, rectangular facility in the presence of other rectangular facilities. There is a non-negative interaction between the new and existing facilities as well as pairs of existing facilities. All interactions are serviced through a finite number of input/output points located on the boundary of each facility. We assume that the travel occurs according to the rectilinear metric and the travel through the facilities is not permitted. It has been established in the previous works that the optimal placement of the new facility belongs to a finite set of candidate points, whose cardinality is polynomially bounded in the number of existing facilities. The optimal placement of the new facility can be found by evaluating the objective function value at each and every candidate point. This explicit enumeration guarantees the optimal solution, however it might become time consuming for a large number of existing facilities. We propose a new procedure based on the lower bounding technique, which can effectively cut down the number of candidate points that need to be evaluated, resulting in significant reduction in the computing time. The procedure was tested on a large number of randomly generated layouts with varying congestion factors (ratio of area occupied by the existing facilities to the total layout area). These extensive numerical tests reveal that, for a moderately congested layout, there is more than 70% reduction in both the number of evaluated candidates and the computing time.

Introduction

Planar facility location problem deals with optimally locating one or more facilities in an existing layout so as to minimize a specific distance function. The most common objective is to minimize the sum of weighted distances between the new and existing facilities, which is also known as median (or minisum) objective. A widely used distance metric is the rectilinear (or Manhattan) metric, in which the travel between any two facilities is assumed to take place along a path parallel to the vertical and horizontal axes.

When the area of the layout is sufficiently large, the size of the facilities can be considered as infinitesimal. A single infinitesimal facility can be located in a layout with existing infinitesimal facilities, using the simple sweep algorithm [5]. The algorithm takes advantage of the fact that the objective function can be separated into functions of x and y coordinates and each sub-function is piecewise linear and convex. Therefore, the optimal location of the new facility can be chosen from a set of points where the slope of the objective function changes from non-increasing to non-decreasing. On the multi-facility location front, Hakimi [6] investigated the problem of locating p new infinitesimal facilities in the presence of other infinitesimal facilities on a network. The author proved the important vertex optimality result which converts the continuous location problem into a discrete search problem. Wesolowsky and Love [13] studied the problem of locating n new infinitesimal facilities in the presence of m point and p area destinations. The authors developed a gradient based algorithm to find the optimal location of the new facilities, under the assumptions that the demand is uniformly distributed over the area destinations and all the distances are rectilinear. Larson and Sadiq [10] studied the problem of locating p new infinitesimal facilities in a layout consisting of arbitrarily shaped barriers with the rectilinear distance metric. The authors established that the feasible candidate points are polynomial in number and the optimal locations can be found by evaluating the objective function value for all the candidates. In the article by Hamacher and Nickel [7], the authors discuss variations of the restricted location problems, in which one or multiple new facilities need to be placed in the presence of existing facilities and forbidden regions. The authors develop the algorithms for solving these problems and they also discuss several practical applications such as circuit board assembly, obnoxious facility planning and location of emergency facilities. Hamacher and Schöbel [8] developed a polynomial procedure for locating a single facility in the presence of existing infinitesimal facilities and forbidden polyhedra, with the center objective and Euclidean distance metric.

As the area of the new facilities is assumed to be infinitesimal, the above procedures cannot be applied to a general problem in which the area of the new facility might be finite with respect to the total area of the layout. For example, if we want to locate a new machine in a manufacturing plant, then the area of the machine cannot be treated as infinitesimal. Drezner [4] studied the problem of locating finite-size facilities in a plane, with median objective, and Euclidean and squared-Euclidean distance metrics. The facilities were assumed to have circular area and the service was assumed to be uniformly distributed over the area. The author found an analytical solution to calculate the effective distance (de) between the various interacting facilities. The effective distance can be multiplied by the respective weights and then added together to obtain the objective function value, which needs to be minimized. Savaş et al. [12] were the first to address the problem of placing a single finite-size facility with arbitrary shape and a known I/O point, in the presence of existing arbitrarily shaped barriers. The authors established that the optimal placement of the new facility belongs to a finite set of candidate points. Sarkar et al. [11] extended the procedure to the placement of a rectangular generalized congested region (GCR), which can be treated as a rectangular facility with finite area.

In the works of Savaş et al. [12] and Sarkar et al. [11], the objective function needs to be evaluated for all the feasible candidate points to find the optimal solution. Since the number of feasible candidate points is in higher order polynomial of the number of existing facilities, this explicit enumeration might require considerable amount of computing effort. For this reason, a better solution procedure needs to be developed which can potentially eliminate some of the non-optimal solutions, before evaluating the objective function value. In this work we propose a new procedure which utilizes some dominance rules to prune the set of feasible candidate locations, resulting in faster convergence to the optimal solution. This procedure can be directly applied to such problems as finding the best location for a new machine in a manufacturing plant, finding the best location for a new building in a campus, etc.

The rest of the paper is organized as follows. In 2 Problem description, 3 Preliminaries, we define our problem and visit some of the important concepts from the previous works. In Section 4, we establish the dominance results and develop the procedure for finding the optimal placement of a new facility. In Section 5, we discuss the algorithms used for implementing the procedure in a computer language and in Section 6, we validate the effectiveness of our procedure for a large set of randomly generated layouts. We conclude the paper with a summary and possible extensions.

Section snippets

Problem statement

The layout under consideration is a rectangular, closed region with finite area. There are a finite number of existing facilities (EFs), with rectangular shapes and edges parallel to the travel axes. A single new facility (NF) having rectangular shape and known dimensions is to be placed in the layout in the presence of the EFs with its edges parallel to the travel axes. Each EF has one or multiple I/O point(s) while the NF has a single I/O point. The I/O points are strictly located on the

Grid construction and cell formation

We first describe the grid construction procedure introduced by Larson and Sadiq [10]. A grid is constructed by passing horizontal and vertical lines through the vertices of each EF and its I/O point(s). Since travel through the EFs is prohibited, the lines will terminate at an EF boundary or at the layout boundary, whichever is encountered first. We will call the resulting lines as gridlines. Let LH and LV denote the sets of horizontal and vertical gridlines respectively and let L=LHLV.

The

Solution procedure

To find and eliminate the dominated candidate points, we will utilize the technique of lower bounding. The main idea behind this is to find a valid relaxation to the original problem and uses its value as a lower bound on the original problem. As the name suggests, lower bound represents the best value that the original problem can possibly achieve, for any solution within a particular solution space. After establishing a lower bound on that solution space, we can solve the original problem to

Specifics of implementation

So far we have discussed the theoretical results and established the procedure for finding the optimal placement of a finite-size NF in a layout. However, to be able to implement the procedure in any computer language, we need the help of some algorithms. In this section we will explain these algorithms and some other practical aspects of implementing the procedure on a computer. Note that the algorithms explained in this section use the simplest and most logical data structures. We acknowledge

Computational results

We conducted extensive computational comparison of the dominance procedure with the explicit enumeration procedure. Both the procedures were coded in Java and executed on Intel® Core™ i7, 2.20 GHz, quad-core processor. The area of the layout in all the problems is 400×400 sq. units. Each layout has four randomly placed EFs, having the same area but different dimensions. Each EF has a single I/O point, randomly located on its boundary. The aspect ratio of each EF is selected from 2U[1,1]

Conclusion

To summarize, we examined the problem of placing a finite-size, rectangular facility with known dimensions, in the presence of other finite-size, rectangular facilities. All the facilities interact with each other through the I/O points and the travel between them occurs according to the rectilinear metric. The objective is to find the optimal placement of the new facility, with the help of some dominance rules, which will reduce the number of feasible placement candidates that need to be

Acknowledgments

The authors would like to thank the anonymous reviewer for his thorough and helpful review comments, which have led to significant improvement in the presentation of this paper.

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