Linear Programming : Writing Constraints and Maximizing Profit

The Chop is the manufacturer of the store brand of hatchets and axes sold by various home hardware supply stores. Each item consists of a hickory handle produced in the local processing facility and a steel blade forged and polished in the local machine shop. These two items are then transported to an assembly area where the blade is attached to the handle and the item is packaged for shipment to destination stores. Because of its exclusive contracts with various stores, the shop can sell any amount of product it produces.

Skilled labor is required to produce the handles and the blades. Cheaper unskilled labor is used to assemble and package the final products. There is sufficient storage area of store the finished product prior to shipping. Assume all raw material is on-hand at the start of the quarter and there is no way to obtain additional raw material or to schedule additional labor hours.

The following provides data on a per item, per product basis:

Hatchet Ax
Selling price to stores $16.00 $29.50
Hickory required (ft/unit) 1.6 3.1
Steel required (lbs/unit) 1.8 4.2
Skilled labor required per unit produced
Handle production (min/unit) 5.0 2.6
Blade production (min/unit) 11.0 9.4
Unskilled labor required per unit produced
Assembly (min/unit) 4.0 5.5
Packaging (min/unit) 2.0 2.5

The following data summarizes important information for the next three months (the next quarter):

June July August
Contracted requirements
Minimum Hatchet production 200 200 225
Minimum Ax production 200 250 300
Costs
Skilled labor ($ per hour) $12.00 $12.00 $12.84
Unskilled labor ($ per hour) $7.00 $7.00 $7.35
Hickory ($ per linear foot) $1.00 $1.20 $1.05
Steel ($ per pound) $1.18 $1.06 $0.98

The following table provides and summary of the resource availability over the next three months.

Availability
Resource June July August
Skilled Labor (total hours) 220 250 200
Unskilled Labor (total hours) 120 100 140
Hickory (total linear feet) 2000 2200 1500
Steel (pounds) 2000 2000 2000

Tasking:

Management needs to know the production schedule for June, July and August that maximizes profit. Given this production schedule, provide a 1) summary of resources used by month, by product and 2) profits by month, by product.