Question

1. The Munchies Cereal Company makes a cereal from several ingredients. Two of the ingredients, oats and rice, provide Vitamins A and B. The company wants to know how many ounces of oats and rice it should include in each box of cereal to meet the minimum requirements of 48 milligrams of vitamin A and 12 milligrams of vitamin B while minimizing cost. An ounce of oats contributes 8 milligrams of vitamin A and 1 milligram of vitamin B, whereas an ounce of rice contributes 6 milligrams of A and 2 milligrams of B. An ounce of oats costs $0.05, and an ounce of rice costs $0.03.

1. Formulate a linear programming model for this problem.
2. Solve the model using graphical analysis.
3. What would be the effect on the optimal solution in the above problem if the cost of rice increased from $0.03 to $0.06 per ounce?

Answer 1

a)

let x1 be the oats and x2 be rice.

Lp problem

MIN z = 0.05x1 + 0.03x2
subject to
8x1 + 6x2 >= 48
x1 + 2x2 >= 12
and x1,x2 >= 0

b)

1. To draw constraint 8x1+6x2≥48→(1)

Treat it as 8x1+6x2=48

When x1=0 then x2=?

⇒8(0)+6x2=48

⇒6x2=48

⇒x2=486=8

When x2=0 then x1=?

⇒8x1+6(0)=48

⇒8x1=48

⇒x1=488=6

x1	0	6
x2	8	0

2. To draw constraint x1+2x2≥12→(2)

Treat it as x1+2x2=12

When x1=0 then x2=?

⇒(0)+2x2=12

⇒2x2=12

⇒x2=122=6

When x2=0 then x1=?

⇒x1+2(0)=12

⇒x1=12

x1	0	12
x2	6	0

Graphical solution

The minimum value of the objective function z=0.24 occurs at the extreme point (0,8).

Hence, the optimal solution to the given LP problem is : x1=0,x2=8 and min z=0.24.

optimal solution = (2.4 , 4.8)

c)

Hence, the optimal solution to the given LP problem is : x1=2.4,x2=4.8 and min z=0.41.

There is no effect on the optimal solution. Cost will increase.