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# The production manager for the Classic Boat Corporation must determine

how many units of the Classic 21 model to produce over the next four quarters. The company has a beginning inventory of 100 Classic 21 boats, and demand for the four quarters is 2000 units in quarter 1, 4000 units in quarter 2, 3000 units in quarter 3, and 1500 units in quarter 4. The firm has limited production capacity in each quarter. That is, up to 4000 units can be produced in quarter 1, 3000 units in quarter 2, 2000 units in quarter 3, and 4000 units in quarter 4. Each boat held in inventory in quarters 1 and 2 incurs an inventory holding cost of \$250 per unit; the holding cost for quarters 3 and 4 is \$300 per unit. The production costs for the first quarter are \$10,000 per unit; these costs are expected to increase by 10% each quarter because of increases in labor and material costs. Management specified that the ending inventory for quarter 4 must be at least 500 boats.
1.    Formulate a linear programming model that can be used to determine the production schedule that will minimize the total cost of meeting demand in each quarter subject to the production capacities in each quarter and also to the required ending inventory in quarter 4.

2.    Solve the linear program formulated in question 1. Then develop a table that will show for each quarter the number of units to manufacture, the ending inventory, and the costs incurred.

3.    Interpret each of the shadow prices corresponding to the constraints developed to meet demand in each quarter. Based on these shadow prices, what advice would you give the production manager?

The shadow prices tell us how much it would cost if demand were to increase by one additional unit. For example, in Quarter 2 the shadow price is \$12,760; thus, demand for one more boat in Quarter 2 will increase costs by \$12,760.
4.    Interpret each of the shadow prices corresponding to the production capacity in each quarter. Based on each of these shadow prices, what advice would you give the production manager?

The shadow price of 0 for Quarter 4 tells us we have excess capacity in Quarter 4. The negative shadow prices in Quarters 1-3 tell us how much increasing the production capacity will decrease costs. For example, the shadow price of −\$2510 for Quarter 1 tells us that if capacity were increased by 1 unit for this quarter, costs would go down \$2510.

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