The diet problem, one of the earliest applications of linear programming, was
originally used by hospitals to determine the most economical diet for patients.
Known in agricultural applications as the feed mix problem, the diet problem
involves specifying a food or feed ingredient combination that satisfies stated
nutritional requirements at a minimum cost level.
The Whole Food Nutrition Center uses three bulk grains to blend a natural cereal
that it sells by the pound. The store advertises that each 2-ounce serving of the
cereal meets an average adult’s minimum requirement for protein, riboflavin,
phosphorus, and magnesium. The cost of each bulk grain and the protein, riboflavin,
phosphorus, and magnesium units per pound are each shown in the table below:
Cost
per Protein Riboflavin Phosphorus Magnesium
Grain Pound (units/lb) (units/lb) (units/lb) (units/lb)
A 33¢ 22 16 8 5
B 47¢ 28 14 7 0
C 38¢ 21 25 9 6
A 2-ounce serving of cereal must have a minimum of 3 units of protein, 2 units of
riboflavin, 1 unit of phosphorus and 0.425 units of magnesium.
Define,
XA = pounds of grain A in one 2-ounce servings of cereal
XB = pounds of grain B in one 2-ounce servings of cereal
XC = pounds of grain C in one 2-ounce servings of cereal
Given the variables are defined as pounds in a 2-ounce servings, the optimal values
for XA, XB, and XC will be fractional, for example, we might find from the computer
solution that X = 0.100, X = 0.020, and X = 0.005