Given: (This is the information located above the graph) The contribution to pro

Question

Given: (This is the information located above the graph)

The contribution to profit from the private label Brand X is $100 per case compared to only $12.50 per case from the company’s nationally advertised Brand Y.

The recipe for a case of X is 1 unit of Nutrient, 3 units of Color, 15 units of Flavor and 10 units of Grain.

The recipe for case Y is 2 units of Nutrient, 1 unit of Color, 1 unit of Flavor and 1 unit of Grain.

In order to meet government health regulations the manufacturer must use at least 250 units of Flavor.

The other ingredients are in short supply, so you must use no more than 400 units of Nutrient, no more than 500 units of Color, and no more than 600 units of Grain.

B. Write the four constraints that are described in the scenario given in the graph above

1. Describe why each of the four written constraints from part B is a minimum or a maximum constraint.

C. Analyze the optimum production yielding the greatest amount of profit by doing the following:

1. Determine the number of cases of Brand X that should be produced, showing all of your work.

2. Determine the number of cases of Brand Y that should be produced, showing all of your work.

3. Discuss how the feasible region was used to arrive at your calculations for parts C1 and C2.

4. Determine the total profit that would be generated by the production levels determined in parts C1 and C2, showing all of your work.

Explanation / Answer

Answer:

B.1) Let X represent the number of cases of Brand X product that are produced and Y represent the number of cases of Brand Y product that are produced.

Since the contribution to profit for a case of Brand X product is $100, and the contribution due to a case of Brand Y product is $12,50, the total profit is:

P = 100X + 12.50Y

This is the objective function.

The constraints are:

1) Nutrient Constraint

Each case of Brand X product requires 1 unit of nutrient, while each case of Brand Y product requires 2 units of nutrient. The total nutrient used is then:

Nutrient = X + 2Y

Since each batch must use no more than 400 units, this is a maximum constraint. The constraint is:

X + 2Y 400


2) Color Constraint

Each case of Brand X product requires 3 units of color, while each case of Brand Y product requires 1 unit of nutrient. The total color used is then:

Color = 3X + Y

Since each batch can use no more than 500 units of color, this constraint is a maximum constraint. The constraint is:

3X + Y 500

3) Flavor Constraint

Each case of Brand X product requires 15 units of flavor, while each case of Brand Y product requires 1 unit of flavor. The total flavor used is then:

Flavor = 15X + Y

Since each batch can use no more than 250 units of flavour, this constraint is a maximum constraint. The constraint is:

15X + Y 250


4) Grain Constraint

Each case of Brand X product requires 10 units of grain, while each case of Brand Y product requires 1 unit of grain. The total flavor used is then:

Grain = 10X + Y

Since each batch can use no more than 600 units of grain, this constraint is a maximum constraint. The constraint is:

10X + Y 600

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