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Part $1$

The following is a linear programming model for analyzing the product mix of$$ Maxine's Hat$$ Company, which produces three hat$$ types:

$$Maximize: Z$=$$$7$X Subscript $1+$$$8$X Subscript $2+$$$6$X Subscript $3$

Subject$$ to:$2$X Subscript $1+4$X Subscript $2+2$X Subscript $3$ Subscript less than or equals$120($machine A$$ time$)$

$5$X Subscript $1+3$X Subscript $2+2$X Subscript $3$ Subscript less than or equals$400($machine B$$ time$)$

$2$X Subscript $1+2$X Subscript $2+4$X Subscript $3$ Subscript less than or equals$110($machine C$$ time$)$

$$X Subscript $1$ greater than or equals $0,$X Subscript $2$ greater than or equals $0,$ and X Subscript $3$ greater than or equals Subscript $0$

The POM for Windows printout below shows the optimal solution to the problem.

$($a$)$ Results Screen

X Subscript $1$

X Subscript $2$

X Subscript $3$

RHS

Shadow Price

Maximize

$7$

$8$

$6$

Machine A

$2$

$4$

$2$

$<=$

$120$

$0\mathrm{.}5$

Machine B

$0\mathrm{.}20$

$0\mathrm{.}05$

$0\mathrm{.}10$

$<=$

$400$

$0$

Machine C

$0\mathrm{.}15$

$0\mathrm{.}10$

$0\mathrm{.}10$

$<=$

$110$

$3$

Solution

$50$

$5$

$0$

$390$

$($b$)$ Ranging screen

Variable

Value

Reduced Cost

Original Val

Lower Bound

Upper Bound

X$1$

$50$

$0$

$7$

$4\mathrm{.}6667$

$8$

X$2$

$5$

$0$

$8$

$7$

$14$

X$3$

$0$

$7$

$6$

$-$Infinity

$13$

Constraint

Shadow Price

$$Slack$/$Surplus

Original Val

Lower Bound

Upper Bound

Machine A

$\mathrm{.}5$

$0$

$120$

$110$

$220$

Machine B

$0$

$135$

$400$

$265$

Infinity

Machine C

$3$

$0$

$110$

$60$

$120$

Consider each of the following statements$$ independently, and state whether it is true or false. Explain each answer.

Part $2$

a$.$ If the price of hat $3$ were increased to $$$$11\mathrm{.}75,$ it would be part of the optimal product mix. $($Hint$$: Hat $3$ is represented by X Subscript $3$ and its optimal value currently is$0,$ which means that it is not to be produced$($and not part of the optimal product$$ mix$).)$

A$.$

$$False$,$ because the optimal value of Hat $3$ would remain at its current value until its price increases above$$ $$13\mathrm{.}00.$

B$.$

$$True$,$ because the price of Hat $3$ increased more than$50\%$ of its current price.

C$.$

$$Unknown$,$ because the optimal value of Hat $3$ should be recalculated with the new selling price.

D$.$

$$True$,$ because the reduced cost of Hat $3$ is$$ $$7\mathrm{.}00.$