# LINEAR SYSTEMS

Given:

Company A produces and sells a popular pet food product packaged under two brand names, with formulas that contain different proportions of the same ingredients. Company A made this decision so that their national branded product would be differentiated from the private label product. Some product is sold under the company’s nationally advertised brand (Brand Y), while the re-proportioned formula is packaged under a private label (Brand X) and is sold to chain stores.

Because of volume discounts and other stipulations in the sales agreements, the contribution to profit from the Brand Y product sold to distributors under the company’s national brand is only \$12.50 per case compared to \$100 per case for private label product Brand X. There are four ingredients involved in this problem. The recipes specifying the use of each ingredient in the two product brands are given in the template. Also note, an ingredient may either be in limited supply or may have government regulations requiring a minimum or maximum amount of an ingredient.

A.  Set up the system of four constraints that are plotted as shown on the graph in the attached Excel Spreadsheet, showing all work necessary to arrive at the equations.
1.  Explain why each of the four identified constraints is a minimum or a maximum constraint.
2.  Identify the objective function.

B.  Determine the total profit to be made if the company produces a combination of cases of Brand X and Brand Y that lies on the black-dashed objective function line (profit line) as shown on the graph in the attached Excel Spreadsheet.

C.  Determine the optimum production (number of cases of Brand X and Y) that yields the greatest amount of profit by doing the following:
1.  Determine how many cases of Brand X should be produced during each production period, showing all your work.
2.  Determine how many cases of Brand Y should be produced during each production period, showing all your work.
3.  Explain how the feasible region of the provided graph was used to arrive at your calculations for parts C1 and C2.

Note: Partial cases are allowed as part of the solution.

D.  Determine the total contribution to profit that would be generated by the production levels you gave in parts C1 and C2, showing all of your work.
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