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Question:
A cereal company makes cereals from several ingredients (oats and rice). These ingredients have Vitamins A and B in them. Each box of cereal needs to have 48 miligrams of Vitamin A and 12 miligrams of Vitamin B, while minimizing cost. An ounce of oat contributes 8 mg of Vitamin A and 1 mg of Vitamin B whereas an ounce of rice contributes 6 mg of Vitamin A and 2 mg of Vitamin B. The sodium level in the cereal box must be less than 6 mg and amonut of sodium in oats and rice are 0.4 mg and 0.5 mg respectively. The last constraint is that the box of cereals must contain at least 10 ounces of oats. An ounce of oats cost $0.05 whereas an onuce of rice costs $0.03.
For the optimal solution, what are the slack&surplus variables for Vitamin A, Vitamin B, Sodium and Oats?
Hint: You'll have to calculate the optimal solution first, then move on from there.
Its very easy if you think about it a little. Good luck to all!
Update: Congrats to Batman534.
The solution is:
Vitamin A: 38 surplus
Vitamin B: 0 surplus
Sodium: 1.5 slack
Oats: 0 surplus
As a lot of you asked about slack&surplus, here:
Slack means resources that weren't used for the optimal solutions, surplus means resources exceeded (more than required).
Your objective function is, minimizing cost.
Thus,
Z= cost
x1 =oats
x2=rice
Minimize Z = 0.05 x1 + 0.03x2
If you graph all the constraints and the objective function, you can mark corner points of the feasible region and plug in those cordinates to the objective function. The lowest value will be your optimal solution as you are trying to minimize cost (Z).
To find surplus for Vitamin A, you use the Vitamin A constraint; 8x1 + 6x2 - s1 = 48
8(10) + 6(1) - s1 = 48
s1 = 38
To find surplus for Vitamin B, you use the Vitamin B constraint; x1 + 2x2 - s2 = 12
(10) + 2(1) - s2 = 12
s2 = 0
To find slack for Sodium, you use the Sodium constraint; 0.4x1 + 0.5x2 + s3 = 6
0.4(10) + 0.5(1) + s3 = 6
s3 = 1.5
To find surplus for Oats, you use the Oats constraint; x1 - s4 = 10
(10) - s4 = 10
s4 = 10