Problem Modeling
First we decide on decision variables. Let us label the foods 1, 2, ::: 6, and let xi represent the
number of servings of food i in the diet.
Our objective is to minimize cost, which can be written
3x1 + 24x2 + 13x3 + 9x4 + 20x5 + 19x6
We have constraints for energy, protein, calcuim, and for each serving/day limit. This gives the
formulation:
Minimize 3x1 + 24x2 + 13x3 + 9x4 + 20x5 + 19x6
Sub ject to
110x1 + 205x2 + 160x3 + 160x4 + 420x5 + 260x6 2000
4x1 + 32x2 + 13x3 + 8x4 + 4x5 + 14x6 55
2x1 + 12x2 + 54x3 + 285x4 + 22x5 + 80x6 800
x1 4
x2 3
x3 2
x4 8
x5 2
x6 2
xi 0 (for all i)
Discussion
The creation of optimal diets was one of the rst uses of linear programming. Some diculties
with linear programming include diculties in formulating palatability" requirements, and issues
of divisibility (no one wants to eat half a green bean). These linear programming models do give
an idea on how much these palatability requirements are costing.