Linear Programming Problem - The EnergyBoat Company

The Background

The EnergyBoat, Inc. produces and distributes high quality motor boat with highly seasonal demand. To handle this seasonality, the company has several options for its supply chain aggregated planning purposes: hiring an additional workers during peak seasons, outsourcing part of the demand to a third party manufacturer, building up inventory during the slow months, or backorder the demand which will be delivered in the following month.

To determine how to best use these options, the vice president of production and distribution started with the demand forecast over the next six months, as shown in the following table:

Month Demand forecast
May 2,000
June 3,200
July 4,000
August 5,200
September 2,500
October 2,500

EnergyBoat sells each product to retailer for $900. The company has a starting inventory at the beginning of May of 1,200 units. At the beginning of May the company has a workforce of 200 employees. The plant has a total of 20 working days in each month, and each employee earns $30 per regular hour. Each worker works 8 regular hours a day, if they work overtime, their hourly wage is $45/hour. The capacity of the plant is determined by the total labor hours worked. Therefore, machine capacities and raw materials do not limit the capacity of the production operation. Due to the labor rules, no employee will work more than 8 hours of overtime per month. The various costs are shown in the following table:

Item Cost
Material cost $120/unit
Inventory Holding cost $20/unit/month
Marginal cost of backorder $60/unit/month
Hiring & Training $2000/worker
Layoff cost $5000/worker
Labor hours required 15 hours/unit
Regular labor cost $30/hour
Overtime labor cost $45/hour
Cost of subcontracting $760/unit

EnergyBoat has no limits on subcontracting, inventory storage and backorder. All backorders are supplied from the following months' production. Inventory costs are incurred based on the ending inventory of each month. The supply chain manager's goal is to obtain the optimal aggregate plan that allows EnergyBoat to end October with at least 1800 units in inventory.

The Questions

1. The optimal aggregate plan is the one that results in the highest profits over the six-month planning period. Given EnergyBoat's desire to maintain high customer service, all demands must be satisfied. Formulate the optimization problem and solve it.

2. While the optimal production plan can be obtained, the vice president is concerned about several issues. One issue is the over-time labor cost, the union is considering raising the over-time wages from $45 to $47 in the coming contract. Given this becomes true, the vice president is wondering its impact on the optimal plan and the optimal profit, will the original optimal plan be still optimal? How much does the optimal profit change?

3. The second issue is demand forecast which is generated by actual sales of the past several years. From previous experience, the forecasts are subject to errors, the vice president wants to know the impact of the forecast errors on the optimal profits, i.e., suppose the actual sales is slightly higher (lower) than the forecast, how will the optimal profit change? And in what range of the forecast error the rates of changes are valid?