Question $3(15\%)$:

Pictures Inc. produces printers. The marketing manager has developed the following forecasts for the last six month of year $2024($in units$)$:

$\backslash$table$[[$Month$,$July,August,September,October,November,December$],[$Forecast Demand,$5,000,3,500,4,000,5,000,4,800,5,200]]$

The company has $130$ full$-$time workers and has two options in case of shortage: One is to employ part$-$time workers and two is to use overtime with the full$-$time workers. Initial inventory at the beginning of July is $2,000$ units and the desired ending inventory at the end of the year is $0.$ Using the following information and the mentioned two options, production manager wants to develop the minimum cost production plan using level output$/$workforce and calculate the total cost:

$\backslash$table$[[$Regular labour cost,$$80$ per unit$],[$Regular production rate by full$-$time workers,$3,250$ units per month for $130$ full$-$time workers$],[$Overtime labour cost,$$100$ per unit$],[$Part time labour cost,$$120$ per unit$],[$Holding cost$,$$$10$ per unit per month on the average inventory$],[$Back$-$order cost,$$150$ per unit$]]$

a$.$ Use overtime $($up to $1,500$ units per month$)$ to avoid shortage in the plan, provide the plan and calculate the total cost.

b$.$ Use part$-$time workers $($up to $1,000$ units per month$)$ to avoid shortage in the plan, provide the plan and calculate the total cost.

c$.$ According to the developed plan in part a and b$,$ which plan should be selected by Pictures Inc.? Why?