Introduction:
Graphical models enable a manager to visualize the objective function (profit line), constraints, and possible solutions to a given problem, and to make more informed decisions based on that information.
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 X), while the re-proportioned formula is packaged under a private label (Brand Y) and is sold to chain stores.
Because of volume discounts and other stipulations in the sales agreements, the contribution to profit is $40 per case for product sold to distributors under the company’s Brand X national brand compared to $30 per case for the Brand Y private label product.
An ample supply is available of most of the pet food ingredients; however, three additives are in limited supply. The tight supply of nutrient C (one of several nutrient additives), a flavor additive, and a color additive all limit production of both Brand X and Brand Y.
The formula for a case of Brand X calls for 4 units of nutrient C, 12 units of flavor additive, and 6 units of color additive. The Brand Y formula per case requires 4 units of nutrient C, 6 units of flavor additive, and 15 units of color additive. The supply of the three ingredients for each production period is limited to 30 units of nutrient C, 72 units of flavor additive, and 90 units of color additive.
Task:
A. Determine the system for the three constraints as shown on the attached “Graph 1,” showing all work necessary to arrive at the equations.
1. Identify each constraint by labeling them as a minimum or a maximum constraint.
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 purple dashed objective function (profit line) as shown on the attached “Graph 1.”
C. Determine the optimum production levels for Brand X and Brand Y (which yields the greatest amount of profit). Your response should include the number of cases of Brand X and Brand Y to be produced during each production period, showing all of your work.
D. Determine the total contribution to profit that would be generated by the production level you recommend in part C, showing all of your work.
