Question 5:
Ani Abu, campus dietitian for a small community college, is responsible for formulating a
nutritious meal plan for students. For an evening meal, she feels that the following five meal
content requirements should be met: (1) between 900 and 1,500 calories; (2) at least 4
milligrams of iron; (3) no more than 50 grams of fat; (4) at least 26 grams of protein; and
(5) no more than 50 grams of carbohydrates. On a particular day, Ani’s food stock includes
seven items that can be prepared and served for supper to meet these requirements. The cost per
pound for each food item and the contribution to each of the five nutritional requirements are
given in the table below.
TABLE OF FOOD VALUES AND COSTS
FOOD ITEM
CALORIES/
LB
IRON
(MG/LB)
FAT
(GM/LB)
PROTEIN CARBOHYDRATES
(GM/LB)
(GM/LB)
COST/
LB ($)
Milk (x1)
295
0.2
16
16
22
0.60
Ground Meat (x2)
1216
0.2
96
81
0
2.35
Chicken (x3)
394
4.3
9
74
0
1.15
Fish (x4)
358
3.2
0.5
83
0
2.25
Beans (x5)
128
3.2
0.8
7
28
0.58
Spinach (x6)
118
14.1
1.4
14
19
1.17
Potatoes (x7)
279
2.2
0.5
8
63
0.33
What combination and amounts of food items will provide the nutrition Ani requires at the least
total food cost?
(a) Formulate as an LP problem.
7 items need to be prepared and served for supper to meet the requirement:
Cost = 0.60X1 + 2.35X2 + 1.15X3 + 2.25X4 + 0.58X5 + 1.17X6 + 0.33X7
Constraint:
1. 295x1 + 1216x2 + 394x3 + 358x4 + 128x5 + 118x6 + 279x7 ≥900 (calories)
2. 295x1 + 1216x2 + 394x3 + 358x4 + 128x5 + 118x6 + 279x7 ≤ 1500 (calories)
3. 0.2x1 + 0.2x2 + 4.3x3 + 3.2x4 + 3.2x5 + 14.1x6 + 2.2x7 ≥4 (iron)
1
4.16x1 + 96x2 + 9x3 + 0.5x4 + 0.8x5 + 1.4x6 + 0.5x7 ≤50 (fat)
5. 16x1 + 81x2 + 74x3 + 83x4 + 7x5 + 14x6 + 8x7 ≥26 (protein)
6. 22x1 +0x2 + 0x3 + 0x4 + 28x5 + 19x6 + 63x7 ≤50 (carbohydrates)
The cost per pound for each food item cannot be negative, X1, X2, X3, X4, X5, X6, X7
≥0. non-negative constraints.
The cost to formulate nutritious meal should be minimal. Therefore, it is
minimization of an objective function.
Min Z = 0.60X1 + 2.35X2 + 1.15X3 + 2.25X4 + 0.58X5 + 1.17X6 + 0.33X7
(b) What is the cost per meal?
Because the variables are more than 2, therefore excel solver was used to solve the
problems.
2
Based on excel solver result, cost per meal is $ 1.75.
(c) Is this a well-balanced diet?
Yes, based on the data solver, we can see that all nutrients are well balanced diet because
it fulfils all the requirement of nutrients needed within the identified constraint using the
minimum cost at $ 1.75.
Nutrition
C1 (calories)
C2 (calories)
C3 (iron)
C4 (fat)
C5 (protein)
C6 (carbohydrates)
Final Value
900
900
4
50
60.78
50
≥
<
≥
<
≥
<
3
RHS
900
1500
4
50
26
50