# Linear Programming - not looking for answer just some help

• June 29th 2013, 06:35 AM
CMoSeattle
Linear Programming - not looking for answer just some help
I am stuck on the following question for my task, any help or comments are appreciated.

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.

I am stuck on the following question:
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.

I have the determined the following corners but don't know if I am right or at least on the right track

(0,6)
(2.5,5)
(5,3.5) I rounded up to 5 because it's in half case increments.
(6,0)

I can do the calculations once I have determined the corners.
Graph is attached thank you in advance for your help
• June 30th 2013, 08:35 AM
Soroban
Re: Linear Programming - not looking for answer just some help
Hello, CMoSeattle!

The problem has too many words . . .

Quote:

A company produces and sells two brands of pet food: brand X and brand Y.

A case of brand X requires 4 units of nutrient C, 12 units of flavor additive and 8 units of color additive.
A case of brand Y requires 4 units of nutrient C, 6 units of flavor additive and 15 units of color additive.

The supply is limited to: 30 units of nutrient C, 72 units of flavor additive, and 90 units of color additive.

The profit is: $40 per case of brand X,$30 per case of brand Y.

How much of each brand must be produced to maximize profit?

Let $x$ = number of cases of brand X. . $(x \ge 0)$
Let $y$ = number of cases of brand Y. . $(y \ge 0)$

We have these facts:

. . $\begin{array}{c|c|c|c|} & \text{Nutr. C} & \text{Flavor} & \text{Color} \\ \hline \text{Brand X} & 4x & 12x & 6x \\ \text{Brand Y} & 4y & 6y & 15y \\ \hline \text{Total:} & 30 & 72 & 90 \\ \hline \end{array}$

We have these inequalities:

. . $\begin{Bmatrix}4x + 4y & \le & 30 && \Rightarrow && 2x + 2y &\le & 15 \\ 12x + 6y &\le& 72 && \Rightarrow && 2x + \;y & \le & 12 \\ 6x + 15y & \le & 90 && \Rightarrow && 2x + 5y &\le&30 \end{Bmatrix}$

Graph and shade the inequalities.
You should get a pentagon with vertices:
. . $(0,0),\;(6,0),\;(4.5,3),\;(2.5,5),\;(0,6)$

Substitute into the Profit Function, $P \:=\:40x+30y$
. . and determine which point produces maximum profit.
• June 30th 2013, 03:12 PM
CMoSeattle
Re: Linear Programming - not looking for answer just some help
Thanks for your help, so you determined those point from the constraints? I was looking for intersecting lines to establish the vertices. Based off your recommendation (4.5,3) is the optimal production.