# Linear programming

#### mitchfel

##### New member
The Bitz-Karan Corporation faces a blending decision in developing a new cat food called Yum-Mix. Two basic ingredients have been combined and tested, and the firm has determined that to each can of Yum-Mix at least 30 units of protein and at least 80 units of riboflavin must be added. These two nutrients are available in two competing brands of animal food supplements. The cost per kilogram of the brand A supplement is $9, and the cost per kilogram of brand B supplement is$15. A kilogram of brand A added to each production batch of YumMix provides a supplement of 1 unit of protein and 1 unit of riboflavin to each can. A kilogram of brand B provides 2 units of protein and 4 units of riboflavin in each can. Bitz-Karan must satisfy these minimum nutrient standards while keeping costs of supplements to a minimum.
(a) Formulate this problem to find the best combination of the two supplements to meet the minimum requirements at the least cost.
(b) Solve for the optimal solution by the simplex method.

#### Jomo

##### Elite Member
This is a math help forum and not a homework service site. That is we do not solve problems for students--mainly because that will not help them.

#### nasi112

##### Full Member
As I have always said, lengthy questions makes me sleepy. I have slept while reading the second line.

#### ISTER_REG

##### New member
First of all, I would like to note that I did not ask the question, but I find the problem interesting, so I would like to discuss my solution or rather my approach with you here....

The required info from the text in a nutshell are:
...at least 30 units of protein and at least 80 units of riboflavin must be added, cost per kilogram of the brand A is $9, and the cost per kilogram of brand B is$15, A kilogram of brand A provides 1 unit of protein and 1 unit of riboflavin, brand B provides 2 units of protein and 4 units of riboflavin

Now I introduce the variables $$\displaystyle x$$ (product A) and $$\displaystyle y$$ (product B) and enter the information into a table.

 Product A Product B Amount in mixture protein 1 2 30 riboflavin 1 4 80 price 9 15

My objective function now would be the cost minimization, so something like $$\displaystyle z(x,y) = 9x+15y \quad\text{(min)}$$.

Now it is clear from the text that at least as many proportions of protein and riboflavin must be included, so I would further formulate the constraints with:

$$\displaystyle 1x+2y \geq 30$$
$$\displaystyle 1x+4y \geq 80$$

In addition we accept only positive values for $$\displaystyle x$$ and $$\displaystyle y$$, so:

$$\displaystyle x \geq 0$$
$$\displaystyle y \geq 0$$

I had it all plotted once and saw that the intersection of the products is in the negative range ($$\displaystyle S=(-20|25)$$), this confuses me (or I don't know how to interpret this). What do you say to the approach, any suggestions?

#### ISTER_REG

##### New member
Now that a bit of time has passed, I'm still interested in discussing my calculation with you

#### ISTER_REG

##### New member
Hello,
I am still interested in discussing my solution that I proposed in #4. Unfortunately, the thread creator has not spoken again, too bad....