## 7 Numerical Illustration
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Machine Type**} & \multirow{2}{*}{**Available Time**} & \multicolumn{4}{c}{**Machine Time**} \\ \cline{3-5}  & & **Product (1)** & **Product (2)** & **Product (3)** \\ \hline \multirow{5}{*}{Band saw} & [1075,1275] & \multirow{5}{*}{[9.5,11.5]} & \multirow{5}{*}{[9.5,11.5]} & [4,6] \\  & or & & or & or \\  & [1275,1475] & or & & [11.5,13.5] & [8,10] \\  & or & & & or \\  & [1475,1675] & & & & [10,12] \\ \hline \multirow{5}{*}{Profit} & \multirow{5}{*}{[50,60]} & [40,50] & [90,100] & \multirow{5}{*}{[16.5,17.5]} \\  & & or & or & & \\  & & [50,60] & [100,110] & or & \\  & & or & or & & [17.5,18.5] \\  & & [60,70] & [110,120] & & \\ \hline \multirow{5}{*}{Product liability} & \multirow{5}{*}{[0.70,0.72]} & \multirow{5}{*}{[0.81,0.85]} & [0.75,0.78] \\  & & or & [0.81,0.85] & [0.78,0.81] \\  & & or & & & \\  & & or & [0.85,0.89] & [0.81,0.84] \\  & & [0.74,0.76] & & & \\  & & & & [0.84,0.87] \\ \hline \multirow{5}{*}{Quality} & \multirow{5}{*}{[82,92]} & [65,75] & \multirow{5}{*}{or} & [40,50] \\  & & or & & [40,50] \\ \cline{1-1}  & & or & & [50,60] \\ \cline{1-1}  & & [92,102] & [85,95] & or & \\ \cline{1-1}  & & or & & [60,70] \\ \hline \multirow{5}{*}{Workers’ satisfaction} & \multirow{5}{*}{[15,25]} & [90,100] & \multirow{5}{*}{[65,75]} \\  & & or & & \\ \cline{1-1}  & & or & & \\ \cline{1-1}  & & [25,35] & [110,120] & [75,85] \\ \cline{1-1}  & & or & & \\ \cline{1-1}  & & & [120,130] & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._The graphical presentation of the results are given in the Figure 1:  
To validate the comparison, we now calculate the trace value of all the cases for the industrial planning problem to know which scalarization technique provides the best efficient solution. The trace value of all the case I, II, III, and IV are given in the Table 4 and Figure 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cases** & **FGP** & **Bi-Level FGP** \\ \hline \multirow{4}{*}{**I**} & \multicolumn{2}{c}{**Ist Level**} & **IInd Level** \\ \cline{2-3}  & \(\begin{array}{c}\text{Z}_{1}=8878.5,\text{Z}_{2}=0.84536,\\ \text{Z}_{3}=10786,\text{Z}_{4}=11185\\ \text{x}_{1}=28,\text{x}_{2}=56,\text{x}_{3}=41\end{array}\) & \(\begin{array}{c}\text{Z}_{1}=6079.50,\text{Z}_{2}=0.77886,\\ \text{x}_{1}=86,\text{x}_{2}=7,\text{x}_{3}=47\end{array}\) & \(\begin{array}{c}\text{Z}_{1}=4929.5,\text{Z}_{2}=0.801329,\\ \text{Z}_{3}=9516,\text{Z}_{4}=8510\end{array}\) \\  & \(\begin{array}{c}\text{x}_{1}=53,\text{x}_{2}=8,\text{x}_{3}=67\end{array}\) \\ \hline \multirow{4}{*}{**II**} & \(\begin{array}{c}\text{Z}_{1}=10,792,\text{Z}_{2}=0.82057,\\ \text{Z}_{3}=10,634,\text{Z}_{4}=12350\\ \text{x}_{1}=17,\text{x}_{2}=75,\text{x}_{3}=49\end{array}\) & \(\begin{array}{c}\text{Z}_{1}=5762.5,\text{Z}_{2}=0.77532,\\ \text{x}_{1}=79,\text{x}_{2}=0,\text{x}_{3}=81\end{array}\) & \(\begin{array}{c}\text{Z}_{1}=5772.5,\text{Z}_{2}=0.77611,\\ \text{Z}_{3}=11,624,\text{Z}_{4}=8110\end{array}\) \\  & \(\begin{array}{c}\text{x}_{1}=77,\text{x}_{2}=2,\text{x}_{3}=81\end{array}\) \\ \hline \multirow{4}{*}{**III**} & \(\begin{array}{c}\text{Z}_{1}=9575,\text{Z}_{2}=0.80512,\\ \text{Z}_{3}=10,256,\text{Z}_{4}=9335\\ \text{x}_{1}=43,\text{x}_{2}=56,\text{x}_{3}=28\end{array}\) & \(\begin{array}{c}\text{Z}_{1}=7437.5,\text{Z}_{2}=0.76719,\\ \text{x}_{1}=110,\text{x}_{2}=5,\text{x}_{3}=45\end{array}\) & \(\begin{array}{c}\text{Z}_{1}=7187.5,\text{Z}_{2}=0.76943,\\ \text{x}_{1}=12,\text{x}_{2}=6,\text{x}_{3}=49\end{array}\) \\ \hline \multirow{4}{*}{**IV**} & \(\begin{array}{c}\text{Z}_{1}=11,403,\text{Z}_{2}=0.83419,\\ \text{Z}_{3}=10,929.99,\\ \text{Z}_{4}=10,102.83\\ \text{x}_{1}=48,\text{x}_{2}=61,\text{x}_{3}=58\end{array}\) & \(\begin{array}{c}\text{Z}_{1}=6645.50,\text{Z}_{2}=0.79027,\\ \text{x}_{1}=97,\text{x}_{2}=2,\text{x}_{3}=83\end{array}\) & \(\begin{array}{c}\text{Z}_{1}=5697.5,\text{Z}_{2}=0.80734,\\ \text{x}_{1}=71,\text{x}_{2}=3,\text{x}_{3}=95\end{array}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: A compromise solution to the problem.