dolution

Jumbo manufactures bicycles for all ages. The demand forecast for the coming three months is 8k, 12k, and 15k, respectively. Capacity is governed by the number of employees it hires. Employees are paid $10 per hour for regular time and $15 per hour for overtime. Each bicycle requires two hours of work from one employee. The plant works 20 days a month and eight hours a day of regular time. Overtime is restricted to a maximum of 20 hours per employee per month. Jumbo recently has 150 employees and decides not to change that number. Each bicycle uses $35 of material. Carrying a bicycle in inventory from one month to the next costs $4. Jumbo starts with 4000 bicycles in inventory and wants the ending inventory three months later to be equal to 4000 bicycles as well. Backordering customer demand is allowed but only for at most one month (i.e., customer demand in each month must be satisfied either in the current month or in the next month). The per unit backorder cost is $8 for one month. Each month, Jumbo can also buy from a subcontractor at a price of $70 per unit.

Formulate the above sales and operations planning problem as an optimization problem by defining the decisions variables, the objective function, and the constraints.