A$2$: Diet Problem

A prison is trying to decide what to feed its prisoners. They would like to offer some combination of milk, beans and oranges in the next meal. Their goal is to minimize cost while satisfying a set of minimum nutritional requirements. The cost and nutritional content of each food$-$stuff is given in the table along with minimum nutritional guidelines.

$\backslash$table$[[,$Food stuff,$],[,$Milk,Beans,Oranges,MDR$],[$Nutrient$,($gallons$),($cups$),($large$),($micrograms$)],[$Niacin $(mg),3\mathrm{.}20,4\mathrm{.}90,0\mathrm{.}80,13\mathrm{.}00],[$Thiamin $(mg),1\mathrm{.}12,1\mathrm{.}30,0\mathrm{.}19,1\mathrm{.}50],[$Vitamin C $(mg),32\mathrm{.}00,0\mathrm{.}00,93\mathrm{.}0,45\mathrm{.}00],[$Cost $($$$),2\mathrm{.}00,0\mathrm{.}20,0\mathrm{.}25,]]$

Suppose we want to ensure that the cost of Milk and Beans should not account for more than $60\%$ of the total cost. Can we also model this restriction?

Wi Win

The LP formulation of this problem is given below.

Decision Variables:

$x1$ : the amount of milk offered

X$2$: the amount of beans offered

X$3$: the amount of oranges offered

Objective Function:

Minimize $Z=21+0\mathrm{.}22+0\mathrm{.}253$

Constraints:

$3\mathrm{.}21+4\mathrm{.}92+0\mathrm{.}8313$

$1\mathrm{.}121+1\mathrm{.}32+0\mathrm{.}1931\mathrm{.}5$

$321+93345$