$(45$ points$)$ Deckard Electric produces old fashioned chandeliers, and ceiling fans. Both chandeliers and fans require a two$-$step production process involving wiring and assembly. It takes about $2$ hours to wire a chandelier and $3$ hours to wire a fan. Final assembly of the chandelier and fan require $6$ and $5$ hours respectively. The production capability is such that only $12$ hours of wiring and $30$ hours of assembly time are available. Each chandelier produced nets the firm $$7$ and each fan $$6.$ The firm's management wants to achieve the following goals with the given priorities:

$G_1$ : Reach a profit of at least $$30$

$G_2$ : Fully use wiring department hours available.

$G_3$ : Avoid assembly department overtime.

a$)$ Formulate the problem as a goal programming problem.

b$)$ Suppose that unused labor in the wiring department incurs a penalty of $$2$ while overtime in assembly department costs an additional $$3,$ what would be the objective function using the weights method?

c$)$ Assuming $G_1>G_2>G_3$ determine the optimum solution using the pre$-$emptive method.