Review the school lunch diet problem scenario below. Formulate and solve the appropriate linear programming problem to solve the diet problem facing the county.

The Diet Problem Case Study

The Stigler diet was named after George Stigler, a Nobel Laureate in economics, who posed the following diet problem decades ago: "For a moderately active man weighing 154 pounds, how much of each of the 77 foods should be eaten on a daily basis so that a man's intake of nine nutrients will be at least equal to the recommended dietary allowances (RDAs) suggested by the National Research Council, with the cost of the diet being minimal?" The nutrient RDAs required to be met in Stigler's experiment were calories, protein, calcium, iron, vitamin A, thiamine, riboflavin, niacin, and ascorbic acid.

The result was an annual budget allocated to foods such as evaporated milk, cabbage, dried navy beans, and beef liver at a cost of approximately $0.11 a day in 1939 U.S. dollars. That was then!

Rosanne McBoman, is the Montgomery County, PA, public schools director responsible for the lunch plan for county students. Ms. McBoman is concerned about providing nutritious meals for the students at a reasonable cost. After consultations with other experts, it was agreed that the lunch plan for a meal should contain the right amount of the five following required nutrient items: calories, protein, fat, carbohydrates, and iron. The meal plan was code named Rosanne's Diet.

The county kitchen stock includes seven food items that can be prepared and served for lunch to meet the requirements of Rosanne's Diet. The cost per pound for each food item and the dietary contribution to each of the five nutritional components required for Rosanne's Diet (according to Linear Programming Assessment Table 1 below) are given in the Linear Programming Assessment Table 2, shown below.

Linear Programming Assessment Table 1

Nutrient	Required Intake
Calories	Between 900 and 1550
Carbohydrates	Less than 50 grams
Fat	Less than 50 grams
Proteins	At least 25 grams
Iron	At least 4 milligrams

 

Deliverable

Find the optimal combination and amounts of food items that will meet Rosanne's Dietary requirement at the least total cost.

Linear Programming Assessment Table 2

Food Item	Calories /LB	Protein (gm/lb)	Fat (gm/lb)	Carbs (gm/lb)	Iron (mg/lb)	Cost ($)/lb
Beans	130	8	0.9	30	3.5	0.76
Chicken	400	75	11	0	4.6	1.24
Fish	360	86	0.7	0	3.4	2.31
Red meat	1200	92	98	0	0.3	2.44
Milk	300	15	19	21	0.3	0.58
Potatoes/Bread	280	9	0.7	67	2.5	0.41
Spinach	115	16	1.3	21	15.7	1.19

 

Here are things to consider when developing the analysis and recommendation.

1. Formulate as a linear programming (LP) problem.
2. Write the appropriate objective function and constraint sets. 
   
   • What is the cost per meal?
   • What combination and amount of food items will provide the minimum cost? To minimize cost, use software to find the least cost based on your objective function and constraint sets.
3. Explain whether you consider Rosanne's Diet to be well balanced. It is important to note that ensuring that the meal is balanced is sensitive to the cost of the items. Therefore, you need to conduct a sensitivity analysis.