# An Application of an Unequal-Area Facilities Layout Problem with Fixed-Shape Facilities
## 2 Mathematical Model  
### Mixed Integer Linear Programming Formulation  
The indices, parameters, and variables for the UA-FLP are defined as follows:  
_Indices_:  
\(i,j=1,\ldots,N\) where \(N\) is the number of facilities.  
Parameters:  
\(f_{ij}=f_{ij}+f_{ji}=\) units of materials which flow between facility \(i\) and facility \(j\) (consider only upper triangular matrix)  
\(a_{i}=\) shorter side length of facility \(i\)  
\(b_{i}=\) longer side length of facility \(i\)  
\(L=\) length of the plant floor space available for layout  
\(W=\) width of the plant floor space available for layout  
_Variables_:  
\((x_{i},y_{i})=\) the centroid (or location) of facility \(i\)  
\(l_{i}\), \(w_{i}=\) the length and width of facility \(i\)  
\(dx_{ij}\), \(dy_{ij}=\) horizontal and vertical distance between the centers of facilities \(i\) and \(j\)  
\(o_{i}=1\) if facility \(i\) is vertically oriented and zero otherwise  
\(lept_{ij}=1\) if facility \(i\) is to the left of facility \(j\) and zero otherwise  
\(below_{ij}=1\) if facility \(i\) is below facility \(j\) and zero otherwise  
Next, a mixed integer linear programming (MILP) formulation is presented for the UA-FLP:  
\[\text{Minimize z}=\sum_{i=1}^{N-1}\sum_{j>i}^{N}f_{ij}\big{(}dx_{ij}+dy_{ij} \big{)} \tag{1}\]  
subject to:  
\[x_{i}\geq 0.5l_{i}\;\forall\;i \tag{2}\]  
\[x_{i}\leq L-0.5l_{i}\;\forall i \tag{3}\]  
\[y_{i}\geq 0.5w_{i}\;\forall i \tag{4}\]  
\[y_{i}\leq W-0.5w_{i}\;\forall i \tag{5}\]  
\[x_{i}+0.5l_{i}-x_{j}-0.5l_{j}\leq L\left(1-lept_{ij}\right)\forall i,j\
eq i \tag{6}\]  
\[y_{i}+0.5w_{i}-y_{j}-0.5w_{j}\leq W\left(1-below_{ij}\right)\forall i,j\
eq i \tag{7}\]  
\[lept_{ij}+lept_{ji}+below_{ij}+below_{ji}=1\;\forall i,j\
eq i \tag{8}\]  
\[l_{i}=a_{i}o_{i}+b_{i}\left(1-o_{i}\right)\forall i \tag{9}\]  
\[w_{i}=b_{i}o_{i}+a_{i}\left(1-o_{i}\right)\forall i \tag{10}\]  
\[dx_{ij}\geq x_{i}-x_{j}\;\forall i,j>i \tag{11}\]  
\[dx_{ij}\geq x_{j}-x_{i}\;\forall i,j>i \tag{12}\]  
\[dy_{ij}\geq y_{i}-y_{j}\;\forall i,j>i \tag{13}\]  
\[dy_{ij}\geq y_{j}-y_{i}\;\forall i,j>i \tag{14}\]  
\[x_{i},y_{i},l_{i},w_{i},dx_{ij},dy_{ij}\geq 0\;\forall i,j\
eq i \tag{15}\]  
\[o_{i},lept_{ij},below_{ij}=0\;\text{or}\;1\;\forall i,j\
eq i \tag{16}\]  
Objective function (1) minimizes the sum of the weighted distances. Constraints (2)-(5) ensure that the facilities are within the boundary (area) of the plant floor space available for layout. Constraints (6)-(8) are used to ensure that facilities do not overlap. Constraints (9) and (10) control the orientation of the facilities. For example, if facility \(i\) is horizontally (vertically) oriented, set \(o_{i}=0\) (\(o_{i}=1\)). If \(o_{i}\) is not set to \(0\) or \(1\), the model will determine the orientation of facility \(i\) such that total weighted distance is minimized. Constraints (11)-(14) are used to obtain the rectilinear distances between the centers offacilities. Last, the restrictions on the variables are given in Constraints (15) and (16). Since the MILP model presented above can only be used to solve small-size problems in reasonable time, a heuristic technique is presented next for the UA-FLP, but first a small problem instance is solved using the model.