A company produces sofas. The manager wants to prepare a plan for the next six months using the following information:

$123456$

Forecast demand $160150160180170140$

Regular time $$100$

Overtime $$150$

Part$-$time $$120$

Inventory, per month $$10$

Backorder, per month $$50$

There are five workers, each making $30$ sofas a month. The maximum number of sofas produced during overtime is $30$ per month. Part$-$time workers can handle a maximum of $20$ units per month. Beginning inventory is zero. Develop a production plan that minimizes total cost. No ending inventory or back orders are desired at the end of month $6.$