$($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.

Q$\mathrm{.}2.$ There are five work centres in one section of a manufacturing company in the northeast heat treatment $(HT),$ painting $(P),$ welding $(W),$ general machining area $($MA$),$ and cooling $($C$).$ The interaction between the following workstation pairs$($MA$,$ W$),($C$,$ W$)-$is best characterized by an E relationship, between $($MA$,$ P$)$ and $(C,P)$ by an I relationship, and between $(MA,C)$ by an $O$ relationship. All other pairs except $(HT,C)$ and $(P,W)$ have a $U$ relationship. Whereas $HT$ and $C$ work centres must be adjacent due to technological considerations, P and $W$ must not be adjacent due to a relationship chart for the five work centres. Construct a relationship chart for the five work centres. Use the numeric values of $64,32,16,8,4,$ and $2$ for $A,E,$I,$O,U,$ and $x$ and the department areas shown in Table to find a solution to the problem using the PAG heuristic.

$\backslash$table$[[$Work Center,Area $($in Square Meters$)],[$HT$,6000],[$P$,2000],[$W$,2000],[$MA$,4000],[$C$,4000]]$

Q$\mathrm{.}3.$ A recirculating conveyor has a total length of $700ft$ and a speed of $90f\frac{t}{m}in.$ Spacing of part carriers $=14ft.$ Each carrier can hold one part. Automatic machines load and unload the conveyor at the load and unload stations. Time to load a part is $0\mathrm{.}10$min and unload time is the same. To satisfy production requirements, the loading and unloading rates are each $2\mathrm{.}0$ parts per min. Evaluate the conveyor system design with respect to the three principles developed by Kwo.

The assignment is due on $13$ th of May at $19\mathrm{.}00.$

PS$\mathrm{.}1.$ Submit your report as a PDF file ONLY. The other file format is not accepted and will not be marked.

PS$\mathrm{.}2.$ Use your first and last name as a file name, for example, asliakbas.pdf$,$ any other file name and format is not acceptable at all.

PS$\mathrm{.}3.$ Submit your report through itslearning before the deadline. E$-$mail submission is not acceptable and if you do so$,$ your paper will not be marked. However, if there is any strong mitigating circumstance with the evidence, you would send your report by email to my mail address before the deadline.

PS$\mathrm{.}4.$ Plagiarism is a very serious academic misconduct, and the consequence of such offense will be having a zero mark from the project and I am sure you will be sensitive on this and be extra vigilant on the issue.