## 3 Bi-Level Programming Model for MEIFM
### Model Formulation
#### 3.2.1 Notations  
This paper uses the following notations in the development of the mathematical model for the proposed MEIFM problem, as shown in Table 1.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & & **Meaning** \\ \hline \multirow{3}{*}{Optimization objectives} & \(C_{\max}\) & \multicolumn{1}{c}{Makespan} \\  & _TEC_ & Total energy consumption \\  & _MHQ_ & Material handling quantity \\ \hline \multirow{3}{*}{Index} & \(i\) & Index of job, \(i=1,2,\cdots,n\) \\  & \(j\) & Index of operation for job \(i,j=1,2,\cdots,o_{i}\) \\  & \(k,l\) & Index of machine, \(k,l=1,2,\cdots,m\) \\  & \(r\) & Index of machine row number, \(r=1,2,\cdots,g\) \\ \hline \multirow{6}{*}{Intermediate variables} & \(O_{ij}\) & \(j\)-th operation of the job \(i\) \\  & \(ct_{ijk}\) & Completion time of \(O_{ij}\) on machine \(k\) \\  & \(ct_{j/k}\) & Completion time of immediate operation of \(O_{ij}\) on machine \(k\) \\  & \(st_{ijk}\) & Start time of \(O_{ij}\) on machine \(k\) \\  & \(it_{k}\) & Idle time of machine \(k\) \\  & \(tt_{lk}\) & Transportation time from machine \(l\) to machine \(k\) \\  & \(dt_{ijk}\) & Delay time of \(O_{ij}\) on machine \(k\) due to machine resource constraints \\  & \(d_{lk}\) & Distance between machine \(l\) to machine \(k\) \\  & \(f_{lk}\) & Material handling frequency from machine \(l\) to machine \(k\) \\  & _PE_ & Processing energy consumption of all machines \\  & _IE_ & Idle energy consumption of all machines \\  & _TE_ & Transportation energy consumption of all jobs in workshop \\  & \(x_{l}\) & Horizontal coordinate of machine \(l\) in workshop \\  & \(y_{l}\) & Vertical coordinate of machine \(l\) in workshop \\ \hline \multirow{6}{*}{Input variables} & \(pt_{ijk}\) & Processing time of \(O_{ij}\) on machine \(k\) \\  & \(v\) & Transportation speed of transporter in workshop \\  & \(pe_{k}\) & Processing energy consumption per unit time of machine \(k\) \\  & \(ie_{k}\) & Idle energy consumption per unit time of machine \(k\) \\  & \(te\) & Transportation energy consumption per unit time of transporter \\ \cline{2-4}  & \(e_{lk}\) & Minimal distance between machine \(l\) and machine \(k\) that must be maintained in horizontal direction \\ \cline{2-4}  & \(\Delta_{lk}\) & Net distance between machine \(l\) and machine \(k\) in horizontal direction \\  & \(e_{l}\) & Length of machine \(l\) in horizontal direction \\  & \(w_{l}\) & Width of machine \(l\) in vertical direction \\  & \(s\) & Center distance of two adjacent rows \\  & \(E\) & Length of workshop in horizontal direction \\  & \(W\) & Width of workshop in vertical direction \\ \hline \multirow{6}{*}{Decision variables} & \(x_{ijk}\) & Binary variable, if \(O_{ij}\) is processed on machine \(k\), then \(x_{ijk}=1\); \\  & & otherwise, \(x_{ijk}=0\) \\ \cline{2-4}  & \(x_{ilk}\) & Binary variable, if job \(i\) is transported from machine \(l\) to machine \(k\), then \(x_{ilk}=1\); otherwise, \(x_{ilk}=0\) \\ \cline{2-4}  & \(z_{lr}\) & Binary variable, if machine \(l\) is located on \(r\)-th row in the workshop, then \(z_{lr}=1\); otherwise \(z_{lr}=0\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The notations used in this paper.