$1.$ The manufacturing department of the Thomas Lee Company is planning

its production level over the next twelve months. Based on orders received

and sales forecasts provided by the marketing department, the estimate of

monthly demand forecast for the twelve$-$month period is as follows:

Month $123456$

Forecasted Demand $(000)161910161817$

The monthly unit holding cost is $$4,$ the cost of hiring a worker is $$3000$; the

cost of laying off a worker is $$2000.$ Regular employee works $40$ hours per

week with monthly salary of $$3200$ and the cost of utilizing overtime is $1\mathrm{.}5$

times the regular time labor cost. Each worker can produce $50$ units per

month during regular time and each worker can work no more than $10$ hours

of overtime per week. Current starting inventory is $500$ units, the initial

workforce level is $150$ and the maximum physical production capacity is

$20000$ units per month. We may assume that there are $4$ weeks in a month.

Thomas Lee would like to find the optimal $($i$.$e$.,$ least cost$)$ production plan.

$($a$)$ Formulate a linear program to solve this problem. Define

your decision variables and explain the constraints.

$($b$)$Solve the linear program.