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# Operations Research and Linear Programming : Optimizing Product Mix

The Androgynous Bicycle Company (ABC) has the hottest new products on the upscale toy market -- boys' and girls' bicycles in bright fashion colors, with oversized hubs and axles, shell design safety tires, a strong padded frame, chrome-plated chains, brackets and valves, and a non-slip handlebar. Due to the seller's market for high-quality toys, ABC can sell all the bicycles it manufactures at the following prices: boys' bikes -- \$220, girls' bikes -- \$175. This is the price payable to ABC at its Orlando plant.

The firm's accountant has determined that direct labor costs will be 45% of the price ABC receives for the boys' model and 40% of the price received for the girls' model. Production costs other than labor, but excluding painting and packaging, are \$44 per boys' bicycle and \$30 per girls' bicycle. Painting and packaging are \$20 per bike, regardless of model.

The Orlando plant's overall production capacity is 390 bicycles per day. Each boy's bike requires 2.5 labor hours while each girl's model takes 2.4 hours to complete. ABC currently employs 120 workers, who each work an 8-hour day. The firm has no desire to hire or fire employees to affect labor availability, believing its stable work force is one of its biggest assets.

a. Formulate a linear programming model that can be used to develop a product mix for the company that yields an optimal profit contribution.
b. Determine the optimal product mix using the Management Scientist software.
c. Explain the impact (if any) on the optimal solution resulting from the stable workforce policy adopted by the firm.

#### Solution Preview

Suppose each day the company will produce x boy's bicycle and y girl's bicycle.
We begin to analyze the conditions.

1. The Orlando plant's overall production capacity is 390 bicycles per day. It means x+y<=390.
2. Each boy's bike requires 2.5 labor hours while each girl's model takes 2.4 hours to complete. So totally 2.5x+2.4y labor hours are required.
3. ABC currently employs 120 workers, who each work an 8-hour day. So the total labor per day is 120*8=960. Thus ...

#### Solution Summary

LP is used to optimize product mix. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

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