Problem:
The optimal quantity of bookcase
Work-force planning
All-Basa, Inc., produces two models of bookcases, for which relevant data are summarized as follows:
Bookcase 1
- Selling Price $15
- Labor Required 0.75 hour/unit
- Bottleneck machine time required 1.5 hours/unit
- Raw Material Required 2 bf/unit
Bookcase 2
- Selling Price $8
- Labor Required 0.5 hour/unit
- Bottleneck machine time required 0.8 hours/unit
- Raw Material Required 1 bf/unit
Each week, up to 400 board feet (bf) or raw material (RM) is available at a cost $1.50/bf. The company employs four workers, who work 40 hours per week for a total regular-time labor supply of 160 hours per week. They work regardless of production volumes, so that their salaries are treated as fixed costs. There are 320 hours per week available on the bottleneck machine.
In the absence of advertising, 50 units per week of bookcase 1 and 60 units of bookcase 2 will be demanded. Advertising can be used to stimulate the demand for each product. Experience shows that each dollar spent on advertising bookcase 1 increases demand for bookcase 1 by 10 units, while each dollar spent on advertising bookcase 2 increases demand for bookcase 2 by 15 units. At most $100 per week can be spent on advertising.
Formulate a LP program (not integer programming) to maximize the profit and determine how much of each product (P1 and P2) to produce each week, how much raw material (RM) to buy, how much overtime to use (OT), and how much advertising to buy for each product (A1 and A2). You must use the notation given in the parenthesis for each variable.
a) What is the maximum profit value?
b) If overtime costs only $4/hour (and all other parameters unchanged), how much overtime should All-Bala use?
c) If each unit of bookcase 1 sold for $15.50 (and all other parameters unchanged), what will the optimal profit per week be?
d) What is the most All-Bala should be willing to pay for another unit of raw material?
e) If each worker were required to work 45 hours per week (and all other parameters unchanged), what would the company's profit be?
f) If each unit of bookcase 2 sold for $10 (and all other parameters unchanged), what would be the optimal quantity of bookcase 2 to produce.