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?