# An Application of an Unequal-Area Facilities Layout Problem with Fixed-Shape Facilities
## 1 Introduction
### Unequal-Area Facility Layout Problem  
As stated earlier, the UA-FLP considered in this paper is defined as the problem of locating rectangular facilities on a rectangular floor space such that facilities do not overlap while minimizing the total distance materials travel between facilities (i.e., minimizing total weighted distance). The rectangular floor space considered is the machining department space within the transmission production area of the TMMWV plant. The facilities are defined as the manufacturing cells, inspection stations, and pieces of equipment used to support production (e.g., washers). The manufacturing cells consist of groups of machines with their material handling systems (e.g., conveyor system). Recall, the model A machines discussed earlier will be reused, and the model B machines will be removed from the existing layout. Since the model B machines form separate cells from the model A machines, the model A-machine cells will remain in the layout, and the model B-machine cells will be removed. In other words, the cells remaining in the machining department will not change (i.e., the internal layout of the cells will not change) and their dimensions are known. Since the dimensions of the cells and other pieces of equipment are known, the facilities have fixed dimensions.  
As a result, the assumptions for the UA-FLP considered in this paper are as follows:  
1. Facilities may have unequal-areas and are rectangular in shape.
2. Floor space available for the facilities is rectangular in shape. Also, it is continuous and constrained (i.e., available floor space has fixed dimensions).
3. The dimensions of the facilities are fixed and known.
4. Facilities may have free orientations.
5. The objective of the UA-FLP is to obtain a layout such that total weighted distance is minimized.
6. The input and output points of the facilities are at the center of the facilities.
7. The rectilinear distance measure is used to obtain the distances between the centroids of two facilities.  
Note in assumption (2) that the floor space dimensions are fixed (i.e., constrained). Oftentimes this is the case, since most FLPs require re-layout of an already existing department or production area with fixed dimensions (i.e., fixed length and width of layout area), as in the TMMWV layout problem. However, the FLP becomes more complex, since heuristics may oftentimes produce infeasible layouts, especially when the layout area is tightly constrained (i.e., percentage of free space available is low). Many papers in the literature consider an unconstrained rectangular floor space where infeasible layouts are not an issue.  
Assumption (3) added with assumption (2) adds even more complexity, since the facility's dimensions are fixed, and are not allowed to vary as in many of the UA-FLPs in the literature. Therefore, for UA-FLPs with tightly constrained floor space, it becomes extremely difficult to find good feasible solutions. As stated previously, TMMWV wanted to keep the internal layouts of the cells (i.e., some of the facilities). As a result, the dimensions of these cells (facilities) are fixed. Also, they know the dimensions of the additional machines (i.e., other facilities) purchased. It is important to note that fixed facilitiesdimensions are very uncommon in the UA-FLP literature. Oftentimes, researchers consider fixed area facilities but allow the dimensions of the facilities to vary. In this case, the lengths and widths of the facilities are controlled using special constraints so that facilities are not too narrow.  
Assumption (4) states that facilities may have free orientations. That is, facilities may be either horizontally or vertically oriented. If the longer side of the facility is parallel to the \(x\)-axis, the facility is considered horizontally oriented, thermwise, it is deemed vertically oriented.