The Munchies Cereal Company makes a cereal from several ingredients. Two of the

Question

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 vitamin A and 2 milligrams of vitamin B. An ounce of oats costs $0.05, and an ounce of rice costs $0.03. Formulate a linear programming model for this problem and solve using the simplex method.

Explanation / Answer

a)

Let the ounces of oats be x

Let the ounces of rice be y

Vitamin A Required: 8x + 6y > = 48

Vitamin B Required: x + 2y >= 12

Cost Function Z = 0.05x + 0.03y (Minimization Function)

b)

Solving using simplex method

Tableau #1
x      y      s1     s2     -p          
8      6      -1     0      0      40   
1      2      0      -1     0      10   
0.05   0.03   0      0      1      0    

Tableau #2
x       y       s1      s2      -p            
1       0.75    -0.125 0       0       5     
0       1.25    0.125   -1      0       5     
0       -0.0075 0.00625 0       1       -0.25

Tableau #3
x      y      s1     s2     -p          
1      0      -0.2   0.6    0      2    
0      1      0.1    -0.8   0      4    
0      0      0.007 -0.006 1      -0.22

Tableau #4
x         y         s1        s2        -p                
1.66667   0         -0.333333 1         0         3.33333
1.33333   1         -0.166667 0         0         6.66667
0.01      0         0.005     0         1         -0.2    

Hence the optimal cost will be 8 units of rice

The Cost to be integer = 8 * 0.03 = 0.24$

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