$2.$ A DIET PROBLEM

The Feed $\text{'}$N Ship feedlot fattens cattle for local farmers and ships them to meat markets in Kansas City and Omaha. The owners of the feedlot seek to determine the amounts of cattle feed to buy to satisfy minimum nutritional standards and, at the same time, minimize total feed costs. Each grain stock contains different amounts of four nutritional ingredients: A$,$ B$,$ C$,$ and D$.$ Here are the ingredient contents of each grain, in ounces per pound of grain.

The cost per pound of grains $\backslash (\backslash$mathrm$\{$X$\},\backslash$mathrm$\{$Y$\}\backslash ),$ and Z is $\backslash (\backslash$$ $0\mathrm{.}02,\backslash$$ $0\mathrm{.}04\backslash ),$ and $\backslash (\backslash$$ $0\mathrm{.}025\backslash ),$ respectively. The minimum requirement per cow per month is $64$ ounces of ingredient $\backslash ($ A$,80\backslash )$ ounces of ingredient $\backslash ($ B $\backslash ),16$ ounces of ingredient C $,$ and $128$ ounces of ingredient D $.$

The feedlot faces one additional restriction$-$it can obtain only $500$ pounds of stock Z per month from the feed supplier, regardless of its need. Because there are usually $100$ cows at the Feed $\text{'}$N Ship feedlot at any given time, this constraint limits the amount of stock Z for use in the feed of each cow to no more than $5$ pounds, or $80$ ounces, per month.