# Big M method

• February 14th 2013, 01:31 PM
Yoodle15
Big M method
I cannot understand only a small bit of how this is solved. Please see below. I have written about what I cannot understand in red.

"A company produces two brands of trail mix, regular and deluxe, by mixing dried fruit, nuts and cereal. The recipes for the mixes are given in the table. The company has 1,200 pounds of dried fruits, 750 pounds of nuts and 1,500 pounds of cereal to be used in producing the mixes. The company makes a profit of \$0.40 on each pound of regular mix and \$0.60 on each pound of deluxe mix. How many pounds of each ingredient should be used in each mix in order to maximize the company's profit?

Type of Mix | Ingredients
Regular | At least 20% nuts
| At most 40% cereal
Deluxe | At least 30% nuts
| At most 25% cereal

Note: State the mathematical model (objective function and all constraints); no solution is required.

Then I identified the variables
Variables:
X1 = Nuts in Regular mix
X2 = Nuts in Deluxe Mix
X3 = Cereal in Regular Mix
X4 = Cereal in Deluxe Mix
X5 = Fruit in Regular Mix
X6 = Fruit in Deluxe Mix

Constraints:

First we need to limit total supplies:

http://www.sosmath.com/CBB/latexrend...b85a16a0f9.png
http://www.sosmath.com/CBB/latexrend...d262ac0fc3.png
http://www.sosmath.com/CBB/latexrend...061b89c0f5.png

Than we limit content:

http://www.sosmath.com/CBB/latexrend...f9dc1bd5d0.png This one just means that at least 20% of total regular mix is the nuts in the regular. We can rewrite this one as:
http://www.sosmath.com/CBB/latexrend...2c6a5791c0.png How does 0.2(X1+X3+X5) become -4X1+X3+X5<=0?

Than we follow the same idea to guarantee the nut content of the deluxe and limit the cereal content of both:

http://www.sosmath.com/CBB/latexrend...5226a3d615.png
http://www.sosmath.com/CBB/latexrend...88a3a35687.png How are those three also rewritten into what they are right now?
http://www.sosmath.com/CBB/latexrend...7622c19218.png

Finally, the objective function is to maximize:
http://www.sosmath.com/CBB/latexrend...b8b5d45504.png "
• February 14th 2013, 01:52 PM
ILikeSerena
Re: Big M method
Quote:

Originally Posted by Yoodle15
I cannot understand only a small bit of how this is solved. Please see below. I have written about what I cannot understand in red.

"A company produces two brands of trail mix, regular and deluxe, by mixing dried fruit, nuts and cereal. The recipes for the mixes are given in the table. The company has 1,200 pounds of dried fruits, 750 pounds of nuts and 1,500 pounds of cereal to be used in producing the mixes. The company makes a profit of \$0.40 on each pound of regular mix and \$0.60 on each pound of deluxe mix. How many pounds of each ingredient should be used in each mix in order to maximize the company's profit?

Type of Mix | Ingredients
Regular | At least 20% nuts
| At most 40% cereal
Deluxe | At least 30% nuts
| At most 25% cereal

Note: State the mathematical model (objective function and all constraints); no solution is required.

Then I identified the variables
Variables:
X1 = Nuts in Regular mix
X2 = Nuts in Deluxe Mix
X3 = Cereal in Regular Mix
X4 = Cereal in Deluxe Mix
X5 = Fruit in Regular Mix
X6 = Fruit in Deluxe Mix

Constraints:

First we need to limit total supplies:

http://www.sosmath.com/CBB/latexrend...b85a16a0f9.png
http://www.sosmath.com/CBB/latexrend...d262ac0fc3.png
http://www.sosmath.com/CBB/latexrend...061b89c0f5.png

Than we limit content:

http://www.sosmath.com/CBB/latexrend...f9dc1bd5d0.png This one just means that at least 20% of total regular mix is the nuts in the regular. We can rewrite this one as:
http://www.sosmath.com/CBB/latexrend...2c6a5791c0.png How does 0.2(X1+X3+X5) become -4X1+X3+X5<=0?

Than we follow the same idea to guarantee the nut content of the deluxe and limit the cereal content of both:

http://www.sosmath.com/CBB/latexrend...5226a3d615.png
http://www.sosmath.com/CBB/latexrend...88a3a35687.png How are those three also rewritten into what they are right now?
http://www.sosmath.com/CBB/latexrend...7622c19218.png

Finally, the objective function is to maximize:
http://www.sosmath.com/CBB/latexrend...b8b5d45504.png "

Hi Yoodle15! :)

Let's take a closer look at the 20% nuts in the regular mix.

It means that the amount of nuts (x1) is at least 20% of the total (x1+x3+x5).
In a formula: $x_1 \ge 0.2(x_1+x_3+x_5)$.

Let's simplify that:

$x_1 \ge 0.2(x_1+x_3+x_5)$

$5 x_1 \ge x_1+x_3+x_5$

$0 \ge -4 x_1+x_3+x_5$

$-4 x_1+x_3+x_5 \le 0$

Does that make sense to you?
How would the others look then?
• February 14th 2013, 10:53 PM
Yoodle15
Re: Big M method
Thank you! It does! :)