Problem $5$: $(20$ points$)$

Creek Pottery employs skilled artisans to produce clay bowls and mugs. The two primary resources used by the company are special pottery clay and skilled labor. Given these limited resources, the company desires to know how many bowls and mugs to produce each day in order to maximize profit. The two products have the following resource requirements for production and profit per item produced $($i$.$e$.,$ the model parameters$)$:

There are $40$ hours of labor and $120$ pounds of clay available each day for production. $1-$Formulate this problem as a linear programming model. $(10$ points$)$

The pottery company has several objectives, listed here in order of importance:

a$.$ To avoid layoffs, the company does not want to use fewer than $40$ hours of labor per day.

b$.$ The company would like to achieve a satisfactory profit level of $\backslash (\backslash$$ $1,600\backslash )$ per day.

c$.$ Because the clay must be stored in a special place so that it does not dry out, the company prefers not to keep more than $120$ pounds on hand each day.

d$.$ Because high overhead costs result when the plant is kept open past normal hours, the company would like to minimize the amount of overtime.

$2-$ Formulate a goal programming model. $(10$ points$)$