A Grain Store Has Six Types of Grain, Each Varying

Essay

A grain store has six types of grain, each varying in cost, quality, and nutritional content. Periodically, excess inventory of these grains are consolidated into two local products, Feed-M-All and Supreme-Feed. Feed-M-All sells for $6.50 for a 10-pound bag while Supreme-Feed sells for $8.50 for a 10-pound bag. These feeds are advertised as having the following nutritional content:
$\begin{array}{lccc}\text { Grain } & \text { Minimum Proten } &\text { Minimum Fat }&\text { Maximum Carbohydrates }\\\hline\text { Feed-M-All } & 16 \% & 18 \% & 10 \%\\ \text {Supreme-Feed} &18 \%&18 \% &9 \% \\\end{array}$
The component grains have the following content characteristics:
$$\begin{array} { c c c c c c c } \text { Grain } &\text { Cost/10lbs }& \text { Quality } & \text { Praten } & \text { Fat } & \text { Carbohydrates } & \text { Paunds Ayail. } \\\hline \mathrm { A } & \$ 4.75 & 4 & 15 \% & 10 \% & 10 \% & 90 \\\mathrm {~B} & \$ 4.00 & 2 & 20 \% & 20 \% & 8 \% & 120 \\\mathrm { C } & \$ 3.75 & 1 & 10 \% & 25 \% & 5 \% & 150 \\\mathrm { D } & \$ 4.25 & 3 & 15 \% & 20 \% & 10 \% & 125 \\\mathrm { E } & \$ 4.50 & 3 & 20 \% & 20 \% & 10 \% & 85 \\\mathrm {~F} & \$ 5.00 & 4 & 25 \% & 15 \% & 12 \% & 165\end{array}$$ Targets for Feed-M-All are a cost of $ 4.35 per 10-pound bag, a quality rating of 2.25, along with the minimum percentages of protein and fat, and the maximum percentage of carbohydrates. Similar targets are set for Supreme-Feed with cost set at $ 4.60 and quality at 2.45. There must be at least a 70%-30% mix among these two local feeds.
Formulate an LP model for this product mix problem.