I have a question I need to answer.
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 from the Brand Y private label product is only $30 per case compared to $40 per case for product sold to distributors under the company’s Brand X national brand.
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.
A. Determine the equations for each of the three constraints that are plotted on the attached “Graph 1.”
1. Identify each constraint as a minimum or a maximum constraint.
B. Determine the total contribution to profit that lies on the objective function (profit line) as it is plotted on the graph if the company produces a combination of cases of Brand X and Brand Y.
C. Determine how many cases each of Brand X and of Brand Y you recommend should be produced during each production period for optimum production if Company A wants to generate the greatest amount of profit.
D. Determine the total contribution to profit that would be generated by the production level you recommend in part C.
I have got the answers below:
The constraint for nutrient C = 4x+4y is less or equal than 30 which is the minimum constraint. Therefore y<=-x+7.5 is the maximum constraint.
The constraint for flavor additive = 12x+6y is less or equal than72 which is the minimum constraint.. Therefore y<=-2x+12 is the maximum constraint.
The constraint for color additive = 6x+15y is less or equal than 90 which is the minimum constraint.
Therefore y<=-2x/5+6 is the maximum constraint.
.Objective function (P) is (P) = $40x+$30y
The vertices are:
P(0, 0) = $0 + $0 = $0 profit
P(0, 6) = $0 + $ 180 = $180 profit
P(2.5, 5) = $100 + $150 = $250 profit
P(4.5, 3) = $180 + $90 = $270 profit
P(6, 0) = $240 + $0 = $240 profit
Therefore the company should produce 4.5 cases of brand X and 3 cases of brand Y in each optimum period.
The total contribution to profit that each period would produce is $270.
If someone could let me know if I have got these right I would appreciate it. If there are errors could you please point me in the direction to fixing them.
I have attached a copy of the graph.