Photon Technologies, Inc., a manufacturer of batteries for mobile phones, signed a contract with a large electronics manufacturer to produce three models of lithium$-$ion battery packs for a new line of phones. The contract calls for the following.

Battery Pack Production Quantity

PT$-100202,000$

PT$-200101,000$

PT$-300149,000$

Photon Technologies can manufacture the battery packs at manufacturing plants located in the Philippines and Mexico. The unit cost of the battery packs differs at the two plants because of differences in production equipment and wage rates. The unit costs for each battery pack at each manufacturing plant are as follows.

Product Philippines Mexico

PT$-100$ $$0\mathrm{.}95$ $$0\mathrm{.}98$

PT$-200$ $$0\mathrm{.}98$ $$1\mathrm{.}06$

PT$-300$ $$1\mathrm{.}34$ $$1\mathrm{.}15$

The PT$-100$ and PT$-200$ battery packs are produced using similar production equipment available at both plants. However, each plant has a limited capacity for the total number of PT$-100$ and PT$-200$ battery packs produced. The combined PT$-100$ and PT$-200$ production capacities are $175,000$ units at the Philippines plant and $160,000$ units at the Mexico plant. The PT$-300$ production capacities are $75,000$ units at the Philippines plant and $100,000$ units at the Mexico plant. The cost of shipping from the Philippines plant is $$0\mathrm{.}18$ per unit, and the cost of shipping from the Mexico plant is $$0\mathrm{.}10$ per unit.

$($a$)$

Develop a linear program that Photon Technologies can use to determine how many units of each battery pack to produce at each plant to minimize the total production and shipping cost $($in dollars$)$ associated with the new contract. $($Assume P$1=$ number of PT$-100$ battery packs produced at the Philippines plant, P$2=$ number of PT$-200$ battery packs produced at the Philippines plant, P$3=$ number of PT$-300$ battery packs produced at the Philippines plant, M$1=$ number of PT$-100$ battery packs produced at the Mexico plant, M$2=$ number of PT$-200$ battery packs produced at the Mexico plant, M$3=$ number of PT$-300$ battery packs produced at the Mexico plant.$)$

Min

Correct: Your answer is correct.

s$.$t$.$

PT$-100$ Production

Correct: Your answer is correct.

PT$-200$ Production

Correct: Your answer is correct.

PT$-300$ Production

Correct: Your answer is correct.

Combined PT$-100$ and PT$-200$ Production Mexico Plant

Correct: Your answer is correct.

Combined PT$-100$ and PT$-200$ Production Philippines Plant

Correct: Your answer is correct.

PT$-300$ Production Mexico Plant

Correct: Your answer is correct.

PT$-300$ Production Philippines Plant

Correct: Your answer is correct.

M$1,$ M$2,$ M$3,$ P$1,$ P$2,$ P$3>=0$

$($b$)$

Solve the linear program developed in part $($a$)$ to determine the optimal production plan.

$($M$1,$ M$2,$ M$3,$ P$1,$ P$2,$ P$3)=$

What is total cost of the production plan $($in dollars$).($Round your answer to the nearest dollar.$)$

$

$($c$)$

Use sensitivity analysis to determine how much the production and$/$or shipping cost per unit $($in dollars per unit$)$ would have to change to produce additional units of the PT$-100$ in the Philippines plant.

It would have to decrease by more than $

per unit.

$($d$)$

Use sensitivity analysis to determine how much the production and$/$or shipping cost per unit $($in dollars per unit$)$ would have to change to produce additional units of the PT$-200$ in the Mexico plant.

It would have to decrease by more than $

per unit.