# Calibration of GA Parameters for Layout Design Optimization Problems Using Design of Experiments
## 4 Mathematical Model of the Cell-Formation Problem  
According to the review of the literature, the minimization of intercell flows and the total cell load variation can be considered the essential objectives in the manufacturing cell-formation research. Thus, the bi-objective fitness function used to evaluate the solution incorporates intercell flows and cell load variation. The mathematical model is given as:  
Intercell flow fitness function:  
\[\text{Minimize}f_{1}=\sum_{i}\sum_{j}\sum_{k}a_{ij}|X_{ik}-Y_{ik}| \tag{1}\]  
Cell load variation fitness function:  
Minimize  
\[f_{2}=\sum_{i}\sum_{k}X_{ik}\sum_{j}\bigl{(}\omega_{ij}-m_{ik}\bigr{)}^{2} \tag{2}\]  
Variable:  
\[m_{ik}=\sum_{i}X_{ik}\times Y_{jk}\times\omega_{ij}/\sum_{i}X_{ik} \tag{3}\]  
Decision variables:  
\[\omega_{ij}=t_{ij};\text{ if part }j\text{ is processed on machine }i \tag{4}\]  
0; otherwise  
\[X_{ik}=1;\text{ if machine }i\text{ is in cell }k \tag{5}\]  
0; otherwise  
\[Y_{jk}=1;\text{ if part }j\text{ is in cell }k \tag{6}\]  
0; otherwise  
\[a_{ij}=1;\text{ if part }j\text{ needs to be processed on machine }i \tag{7}\]  
0; otherwise  
Constraints:  
\[\sum_{k}^{c}X_{ik}=1\;\forall\in\{1,2,\dots,m\} \tag{8}\]  
Figure 1: Representation of a chromosome using real coding.  
\[\sum_{k}Y_{jk}=1\;\forall\;\in\{1,2,\ldots,p\} \tag{9}\]  
\[\sum X_{ik}\geq 1\;\forall\;k\;\in\{1,2,\ldots,c\} \tag{10}\]  
\[\sum Y_{jk}\geq 1\;\forall\;k\;\in\{1,2,\ldots,c\} \tag{11}\]  
Bi-objective fitness function:  
\[Z(t)=\alpha\cdot f_{1}+(1-\alpha)\cdot f_{2} \tag{12}\]  
The given model of cell-formation problem works with processing time. According to this model, we search to obtain the number of cells and the number of machines and parts within each cell. Equation (1) shows the calculation of the intercell flows, and Equation (2) shows the cell load variation. Equation (3) shows the average intercell processing time for the \(j\)-th part and \(k\)-th cell. Equation (4) shows the decision variables that assign the time \(t_{ij}\) of the \(i\)-th machine required to process the \(j\)-th part. Equations (5)-(7) are decision variables that state that \(x_{ik},y_{jk}\) and \(a_{jj}\) are 0-1 binary numbers. Equations (8) and (9) ensure that each machine and part is attached to only one cell. Equations (10) and (11) ensures that, in each cell, there must be allocated at least one machine and one part, respectively. Equation (12) is the bi-objective fitness function for a non-binary cell-formation problem balanced by weight factor \(\alpha\).  
A fitness function value is computed for each chromosome in the population, and the objective is to find a chromosome with the maximum fitness function value. Due to objective of minimizing both the total cell load variation and the exceptional elements, it is necessary to map it inversely and then maximize the result. [PERSON] [50] suggested a mapping function given as:  
\[F(t)=Z_{max}-Z(t) \tag{13}\]  
The symbol \(F(t)\) stands for the fitness function of the \(t\)-th chromosome, and \(Z_{max}\) is the \(max[Z(t)]\) of all chromosomes (\(t\)). The advantage is that the worst chromosomes obtain a zero-fitness function value, so they are not going to be reproduced into the next generation.  
The following procedure given by [PERSON] and [PERSON] [51] is used to assign parts into the machine cells:  
\[P_{kj}=\left(\frac{f_{kj}}{f_{k}}\right)\cdot\left(\frac{f_{kj}}{f_{j}}\right) \cdot\left(\frac{T_{kj}}{T_{j}}\right) \tag{14}\]  
where, \(P_{kj}\) is the membership index of the \(j\)-th part belonging to the \(k\)-th cell; \(f_{kj}\) is the number of machines in the \(k\)-th cell required by the \(j\)-th part; \(f_{k}\) is the total number of machines in the \(k\)-th cell; \(f_{j}\) is total number of machines required by the \(j\)-th part; \(T_{kj}\) is the processing time of the \(j\)-th part in the \(k\)-th cell; and \(T_{j}\) is the total processing time required by the \(j\)-th part.