Discrete OptimizationContour line construction for a new rectangular facility in an existing layout with rectangular departments
Introduction
According to Francis et al. [8] and Bindschedler and Moore [2], a facilities layout problem may arise because of a change in the design of the product, the addition or deletion of a product from a company’s product line, a significant increase or decrease in the demand for a product, changes in the design of the process, etc. Sometimes the layout has to be redesigned to include a new facility such as a single machine, cell or a department. Placement of a new facility in the presence of existing facilities can be considered as a “restricted layout problem” since in a plant layout the existing facilities will act as barriers where travel and new facility placement are not permitted.
There has been significant recent work in the area of planar facility location with barriers. The reader is referred to a recent book by Klamroth [10], a recent chapter by Drezner et al. [6], articles by Dearing et al. [3], Dearing and Segars [4], Frieß, Klamroth and Sprau [9], Dearing et al. [5], and Nandikonda et al. [12]. Recognizing the practical relevance of facility size consideration, Savas et al. [14] consider the optimal placement of a finite-sized facility in the presence of arbitrarily shaped barriers with the median objective and rectilinear distance metric. In a layout context, barriers may be thought of as existing facilities which are usually rectangular. Therefore a special case of their model in which the barriers and the facility are assumed to be rectangular, may be applied to a layout problem where a new rectangular facility has to be optimally placed in the presence of other rectangular facilities. In a layout context, the optimal site may not be always suitable for facility placement. For example the optimal site may pose concerns due to sharp corners, jogs or narrowing of material handling aisles. Hence there is a need to find a nearby location that is usable. Contour lines, that are lines of equal objective function value, help to evaluate the costs of locations other than optimal sites. They help to find the next best solution for an existing layout problem, when the new facility cannot be placed at its intended optimal location. Francis [7] has considered this problem in the context where facilities are points. The finite-area case is more appropriate for facilities layout.
The remainder of this paper is organized as follows. In Section 2, we describe and define the problem, which is defined using the rectangular metric. In Section 3, we briefly visit the grid construction procedure of Larson and Sadiq [11] and the Equal Travel Time Line concept of Batta et al. [1]. We then illustrate some new properties of Equal Travel Time Lines in Section 4. Section 5 illustrates the contour line construction procedure which is followed by a numerical example in Section 6. Section 7 describes the complexity of our solution procedure. Conclusions and directions for future research are presented in Section 8.
Section snippets
Problem definition
We are given a finite number of rectangular existing facilities (EFs) in a 2D plane in which a rectangular new facility (NF) has to be placed. The shape of the NF is given and fixed. We assume that the NF is oriented with its sides parallel to the X and Y axes, and one of the four possible orientations is chosen. The procedure can be repeated for the remaining three orientations. Each EF is characterized by its four corner vertices and has one or more I/O points on its boundary. These EFs are
Background
We divide the plane into regions where the objective function is linear and establish that a contour line is represented by a straight line in each region. The slope of the line potentially changes as we move from one region to the next. This is similar to the situation when EFs and the NF are points (refer [7]). Though the method to find the slope in a region remains similar, the regions are more complex to determine in this case. Like the point case, a grid construction procedure is employed
Additional properties of ETTLs
Like EFs, a rectangular NF can generate ETTLs when the NF interferes with the shortest rectilinear path between a pair of EF I/O points, or between an EF I/O point and the NF I/O point. This is possible only when associated with the affected rectilinear path. Lemma 4.1 Consider an EF–NF rectilinear flow which is interfered by the NF. If an ETTL is generated it is parallel to the affected traversal path and the edges of and the NF which are parallel to the affected traversal path coincide with each
Contour line construction
For the function f(X) the contour line of value z is represented as L(z) where L(z) = {X ∈ F : f(X) = z}. A contour set whose boundary is a contour line is the set of all points having values of f(X) ⩽ z. Francis [7] has shown that for point-sized NF and EFs, the contour line is continuous, with the corresponding contour set being convex. However the finite sizes of the NF and the EFs may present complications as given below:
- 1.
Contour lines may be intercepted by an EF and hence could be disconnected.
- 2.
The
Numerical example
Fig. 15 shows the contour lines constructed for z = 31, 35, 41 and 43 using the above procedure. As can be observed the contour lines for z = 31 and 35 are intercepted by the EFs making them disconnected.
To see the application of contour line construction to a layout context, consider the situation where EF1 and EF2 are existing departments and that a new department indicated by NF needs to be placed. The difficulty is that the NF is a machine that produces significant vibration and has to be set
Solution complexity
Our solution methodology proceeds in two steps. First the number of cells in which objective function value z is potentially present has to be identified using the algorithm of Section 5.1. Each EF generates 2 horizontal and 2 vertical traversal lines from its boundaries. Hence if there are m EFs, 2m horizontal and 2m vertical lines will be produced. Similarly n I/O points can generate at most n horizontal and n vertical lines. Hence the maximum number of cells generated is O((m + n)2).
The
Summary
This paper addresses the contour line construction procedure for a finite-sized rectangular new facility to be placed in a layout having other existing rectangular facilities. Optimal placement of a finite-sized new facility in the presence of other facilities has been studied by Savas et al. [14]. However due to other considerations the optimal site may not be always suitable for placement of the new facility. This necessitates the new facility to be placed at alternate locations and provides
Acknowledgements
This work is supported by the National Science Foundation via grant number DMI–0300370. The authors also wish to acknowledge the help of two anonymous referees, whose comments significantly improved the exposition of this paper.
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