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Linear Programming

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Red Brand Canners

On Monday, September 13, 1999, Mitchell Gordon, Vice president of operations at Red Brand Canners, asked the controller, the sales manager, and the production manager to meet with him to discuss the amount of tomato products to pack that season. The tomato crop, which had been purchases at planting, was beginning to arrive at the cannery, and packing operations would have to be started by the following Monday. Red Brand Canners is a medium-sized company that cans and distributes a variety of fruit and vegetable products under private brands in the western states.

William Cooper, the controller, and Charles Myer, the sales manager, were the first to arrive in Gordon's office. Dan Tucker, the production manager came in a few minutes later and said that he had picked up Produce Inspection's latest estimate of the quality of the incoming tomatoes. According to the report, about 20% of the crop was grade A quality and the remaining portion of the 3 million pound crop was grade B.

Gordon asked Myers about the demand for the tomato products for the coming year. Myers replied that they could sell all of the whole canned tomatoes they could produce. The expected demand for the tomato juice and tomato paste, on the other hand, was limited. The sales manager then passed around the latest demand forecast, which is showing in Table 8.6 (below) He reminded the group that the selling prices had been set in light of the long term marketing strategy of the company and that the potential sales had been forecast at these prices.

Bill Cooper, after looking at Myers's estimates of demand, said that it looked like the company, "should do quite well (on the tomato crop) this year." With the new accounting system that had been set up, he had been able to compute the contribution for each product, and according to his analysis, the incremental profit on whole tomatoes was greater than the incremental profit on any other tomato product. In May, after Red Brand had signed contracts agreeing to purchase the grower's production at an average delivered price of 6 cents per pound, Cooper had computed the tomato products' contributions. Table 8.7 (below)

Table 8.6
Product Selling Price/Case Demand Forecast (cases)
24-2 ½ Whole tomatoes 4.00 800,000
24-2 ½ peach halves 5.40 10,000
24-2 ½ peach nectar 4.60 5,000
24-2 ½ tomato juice 4.50 50,000
24-2 ½ cooking apples 4.90 15,000
24-2 ½ tomato paste 3.80 80,000

Table 8.7
Product 24-2 ½ whole tomatos 24-2 ½ peach halves 24-2 ½ peach nectar 24-2 ½ tomato juice 24-2 ½ cooking apples 24-2 ½ tomato paste
Selling Price 4.00 5.40 4.60 4.50 4.90 3.80
Variable Cost (direct labor) 1.18 1.40 1.27 1.32 0.70 0.54
Variable Overhead 0.24 0.32 0.23 0.36 0.22 0.26
Variable Selling 0.40 0.30 0.40 0.85 0.28 0.38
Packaging Material 0.70 0.56 0.60 0.65 0.70 0.77
Fruit* 1.08 1.80 1.70 1.20 0.90 1.50
Total Variable Costs 3.60 4.38 4.20 4.38 2.80 3.45

Contribution 0.40 1.02 0.40 0.12 1.10 0.35
Less allocated overhead 0.28 0.70 0.52 0.21 0.75 0.23
Net Profit 0.12 0.32 -0.12 -0.09 0.35 0.12

* Product Usate is as follows:
Product Pounds Per Case
whole tomatos 18
peach halves 18
peach nectar 17
tomato juice 20
cooking apples 27
tomato paste 25

The discussion questions are shown on page 3 of the attached file.

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Red Brand Canners
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This is an Integer Programming problem.
We want to maximize the net profit of tomato canning, while trying to meet the demand with limited tomato resource.

Let decision variable be x1, x2, and x3, representing the ...

Solution Summary

The setup and solution of Linear Programming are provided.

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