Given the projected demands for the next six months, prepare an initial aggregate production plan that uses only regular time. Then develop an alternative aggregate production plan that also shows overtime production is utilised to fulfill backorders.

Regular time is $150$ units per month. Overtime is a maximum of $20$ units per month. Overtime cost is $$30$ per unit, backorder cost is $$20$ per unit, inventory holding cost is $$5$ per unit, regular time cost of $$20$ per unit, and beginning inventory is zero.