Question $3.(20$ points$)$ Consider the production capabilities of the Springfield Motors factory,

focusing on the Homer. Suppose that Springfield Motors can ramp up production capacity and

currently has the standard production capacity to roll out $200$ Homers per week. The firm can push an extra $30$ sedans out the door each week with overtime efforts. The cost to assemble and inspect

a Homer during regular working hours is $$2,000$ per car. However, producing a Homer in overtime

bumps the cost up to $$3,000$ each. Additionally, it costs $$600$ per week to keep a Homer in

inventory, awaiting future delivery.

Springfield Motors has a clear order schedule for the upcoming six weeks, meaning the demand

for Homers over this period is precisely known: $180,205,185,255,195,$ and $210$ sedans for each

week, respectively. Springfield Motors aims to provide top$-$notch service by fulfilling every order

without any shortages.

The task for Springfield Motors is to devise a production plan that outlines how much standard

and overtime production is needed over the next six weeks, all while keeping costs as low as

possible. It's assumed there are no Homers in inventory at the start of Week $1.$ Also, production is

spread evenly throughout the week, and the demand must be satisfied by the week's end from either

production in that week, or inventory left over from previous weeks.

For instance, if the demand in Week $1$ is $180$ Homers, and Springfield Motors manufactures

exactly $180$ Homers that week, they'll meet the demand spot$-$on$.$

Below are several questions related to creating a linear programming $($LP$)$ model for the

Springfield Motors production challenge. You can solve this LP with the Excel Solver plugin or

any other solver you prefer.

$($a$)$ What is the optimal production schedule? In other words, how many Homers does

Springfield Motors produce per week to meet the demand schedule above? Are there any

cars held in inventory in any of the weeks? If so$,$ which ones?

$($b$)$ Barney, the workers$$ union representative announces that they are willing to increase the

weekly overtime capacity by $10$ cars, to a total of $40$ cars per week. But Barney insists that

the workers will only allow themselves to work extra overtime hours if the firm shows its

love for the workers by sponsoring a day$-$long excursion with a champagne brunch and an

afternoon of playing croquet. This would cost Springfield Motors $$9,000.($Note that

overtime production costs per car would remain at $$3,000/$car $$ this excursion only lifts the cap from $30$ to $40$ cars per week$).$ Should Springfield Motors take the deal?c$)$ Lisa, the production manager, wonders whether $$9,000$ for a champagne brunch and an

afternoon of croquet is really a good idea, and ponders what the most amount the firm

should spend to show its love for its workers to lift the overtime cap from $30$ to $40$ cars per

week. Even more, Lisa wants to make the analysis more awesome by not resolving the

model, but rather, using a sensitivity analysis and good old$-$fashioned mathematics that can

be done with a simple calculator. What is the most that Springfield Motors would be willing

to spend to raise the over cap from $30$ to $40$ cars per week? Can you guess in which weeks

would the extra overtime capacity be used, if any?

$($d$)$ There is now a breaking report by Kent Brockman that suggests the original information

for demand in Weeks $1$ to $6$ is different. What is the optimal production schedule if

Springfield Motors$$ demand for the next six weeks changes to $165,245,145,225,215,$

and $230$ Homers? Like before, assume no inventory on hand at the beginning of Week $1.$