$5\mathrm{.}8$ GRT $($Pty$)$ Ltd manufactures two products, and the raw materials go through two processes in the production process. The one product is called the Basic, and the other the Superior. The information available is given in the table below.

$\backslash$table$[[,$Basic,Superior$],[$Maximum sales potential,$12000,20000],[$Selling price,R$80,$R$100],[$Machine time:,,$],[$Process $1,0\mathrm{.}5,0\mathrm{.}3],[$Process $2,0\mathrm{.}4,0\mathrm{.}6]]$

Process $1$ and Process $2$ have maximum available hours of $6800$ and $7680$ respectively. The processes are used for the sole purpose of manufacturing the two products mentioned above.

The maximum available material is $15600kg.$ Each product requires $1kg$ of material. The material is purchased at R$12$ per kg$.$ Variable machine overheads are estimated at R$15$ per machine hour for Process $1,$ and R$18$ per machine hour for Process $2.$

All units are sold in the year in which they are produced.

There is no limit to the number of units of each product that can be sold.

Required:

$($a$)$ Use a linear programming graph and simultaneous equations to determine the optimal production plan for the two products, the Basic and the Superior:

$($b$)$ Determine the shadow price for one additional kilogram of material.

Linear programming