$($a$)$ Develop a layout for the matrix in Figure below using the $2-$opt algorithm. $($b$)$ Identify a maximal PAG. $($c$)$ Draw the dual of the maximal PAG identified in $($a$).($d$)$ Based on the dual and using the department lengths provided in the example, develop a detailed layout by trial and error. $($e$)$ How does it compare with the $2-$opt solution provided in $($a$)?$ Explain.