## 2 Related Work
### Ua-Flp  
The UA-FLP was originally formulated by [PERSON] and [PERSON] [37]. It considered on the assumptions that there is a given rectangular region, and there are a set of facilities needed to be placed into the region. The fixed dimensions of the rectangular region are \(W(width)\times H(height)\), and each of the rectangular facilities occupies a specified area (\(B_{i}\)). The constraints of the problem include the sum of the facility areas must be less than or equal to the fixed rectangular region (see Equation (1)) and the facilities cannot overlap. The objective function, or the fitness function in our study, is on basis of total material handling cost (MFC) between facilities [38], which is presented in Equation (2). \[\sum_{i=1}^{p}B_{i}\leq W\times H \tag{1}\]  
\[\text{min}MFC=\sum_{i=1}^{p}\sum_{j=1,j\
eq j}^{p}f(i,j)d(i,j)+(D_{\text{inf}}) ^{k}(V_{feas}-V_{all})\;\;i,j=1,2,\ldots,p, \tag{2}\]  
where:  
\(p=\) the number of facilities;  
\(f(i,j)=\) the total flow of moving materials between facility \(i\) and \(j\), where \(i,j=1,2,\ldots,p\);  
\(d(i,j)=\) the weighted rectilinear distance between facilities \(i\) and \(j\), where \(i,j=1,2,\ldots,p\);  
\(D_{\text{inf}}\) = the number of infeasible departments;  
\(V_{feas}\) = the best feasible objective function value yet found;  
\(V_{all}\) = the best overall objective function value yet found;  
\(k\) = the parameter for adjusting the \"severity\", which was set to \(k=3\)[39]. The calculation of distance between facilities can be Euclidean (Equation (3)) or rectilinear (Equation (4)), where the point defined by \(x\) and \(y\) is the center of the facility  
\[d(i,j)=\sqrt{(x_{j}-x_{i})^{2}+(y_{i}-y_{j})^{2}} \tag{3}\]  
\[d(i,j)=\abs{x_{i}-x_{j}}+\abs{y_{i}-y_{j}} \tag{4}\]  
The FBS in our UA-FLP consists of two parts: the facility sequence codes and the bay division codes. The former encodes the order of \(p\) facilities represented by integer numbers which will be placed into the fixed rectangular area. The order of the facilities, named 'facility sequence', is organized by by bay from top to bottom and from left to right. The latter encodes the bay divisions which have \((p-1)\) binary elements in the area, each facility is associated with one element except the first one. A value of 1 indicates that the corresponding facility is the last facility in the present bay, while 0 means the facility and its previous one are in the same bay. We take an example to detail the encoding method of FSB. Assume there are seven facilities in a UA-FLP. The facility layout problem is to minimize the overall cost by determining how many bays are required, which facilities are assigned to which bays, what kind of ordering is adopted for placing the facilities in each bay. For instance, one solution is illustrated in Figure 1.  
There are four bays in this example: Bay 1, Bay 2, Bay 3, and Bay 4. Each bay contains one or more facilities. For instance, Bay 1 contains two facilities, F and C, while Bay 2 contains only one facility, A. To represent the layout shown in Figure 1 by using FSB encoding, two arrays are required, as shown in Figure 2. The first array, facility sequence, describes the ordering of the facilities from top to bottom and from left to right in the layout. The second array, bay divisions, indicates whether the corresponding facility is the last one in one bay or not. Since the top left facility is F, it is the first one placed into the two arrays, as shown the first step in Figure 2. Moreover, because it is not the last one in the first bay, the corresponding element is set to 0. That is, the first element in the array of bay divisions is set to 0. Next, the second facility, C, is inserted into the arrays because it is in Bay 1 and below Facility F, as shown in Step 2 in Figure 2. Since it is the last one in Bay 1, its value in the second element in the array of bay division is set to 1. At Step 3, we move to Facility A in Bay 2. Because Facility A is also the last one in Bay 2, the third element in Bay Divisions is set to 1.