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linear programming model to compute an optimal production policy

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Part I
During the next four months Shoeco must meet, on time, the following demands for pairs of shoes: 300 in month 1; 500 in month 2; 100 in month 3; and 100 in month 4. At the beginning of month 1, 50 pairs of shoes are on hand, and Shoeco has three workers. A worker is paid $1500 per month. Each worker can work up to 160 hours a month before he or she receives overtime. A worker may be required to work up to 20 hours of overtime and is paid $25 per hour for overtime. It takes four hours of labor and $5 of raw material to produce a pair of shoes. At the beginning of each month workers can be hired or fired. Hiring costs are $1600 per worker and firing costs are $2000 per worker. At the end of each month, a holding cost of $30 is charged for each pair left in inventory. Production in a given month can be used to meet that months demand. Use LP to determine an optimal production and labor policy.

Part II
Assuming backlogs are allowed and that it costs $25 for each pair backlogged, adjust the above model to include that aspect.

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The key to this problem is realizing that there are two variables to solve: the number of workers and the number of hours of overtime. Solver accommodates allowing two variables. Here is the set-up in excel. Look at the formulas in the cells to get a grasp of ...

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The solution presents explanations and calculations to arrive at an optimal production policy.

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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.

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