A 30-year-old investment analyst has been experimenting with Optimum Eating, a new iPhone app. The app shows you how to divide the proportion of your weekly meals among the four categories listed in the following table. The table's bottom two rows list the health attributes of the meal categories. The row labeled BMI (standing for Body Mass Index) rates the categories in terms of maintaining a trim and healthy body weight. Healthy meals deliver a maximum BMI score (100); fast- food meals deliver the worst score (20). The Heart row uses a similar scale to rate the meals as to their impact on cardiovascular factors such as blood pressure and cholesterol. Both scales have been developed by the Office of the U.S. Surgeon General. The last column lists the Surgeon General's recommended goals for a healthy weekly diet, 70 for BMI and 80 for heart.
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Portions
|
Healthy 0.25
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Standard0.25
|
Fast Food0.25
|
Restaurant0.25
|
Diet Score
|
|
|
Value
|
$3.00
|
$8.00
|
$12.00
|
$28.00
|
$12.75
|
Goals
|
|
Cost
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$6.00
|
$5.00
|
$8.00
|
$20.00
|
$9.75
|
$10.00
|
|
BMI
|
100
|
70
|
20
|
50
|
60.00
|
70
|
|
Heart
|
100
|
70
|
40
|
70
|
70.00
|
80
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The Cost row lists the average cost per meal (from a database of cities across the United States. (The costs in the table are those for Milwaukee, Wisconsin.) Finally, the values in the first row are purely personal and have been entered by the financial analyst himself. (Notice that the analyst's taste buds are quite averse to healthy meals. He rates a $6.00 healthy meal as worth only $3.00 in value.)
At the outset, the app sets the meal proportions by default at .25 across the board. (You might think of the individual as eating 6 or 7 meals per week in each category.) For these default proportions, the diet column calculates the average cost, BMI score, and heart score per meal. (These scores are computed by multiplying each meal category proportion by its score and then adding these products over the four categories.) Note that the default diet falls short of meeting both the BMI and heart goals listed in the last column.
a. On a spreadsheet, re-create this app in the form of a linear program.
b. Use your spreadsheet's optimizer to find the meal proportions that meet the BMI and heart goals at minimum cost. Which category is not a part of the least-cost weekly meal plan?
c. Instead, the analyst wants to adopt a plan that maximizes his value per meal while meeting both health goals and spending no more than $10 per meal on average. Use the optimizer to determine this meal plan. How does the plan change if the analyst is able to budget $15 per meal?