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# Mathematics - Linear Programming

The Department of the Navy has been downsizing and is looking for cost savings
opportunities to meet mandated congressional budget cuts. One suggestion under
consideration is to change the make up of content of Navy sea rations, the canned food
supplies containing certain minimum quantities of Vitamin A, Vitamin D, iron, and other
nutrients, which combat troops carry into battle.
According to Texfoods, the current supplier of sea rations for the Navy, each two-ounce
portion of its product supplies 20% of the required amount of Vitamin A, 25% of the
required amount of Vitamin D, and 50% of the required amount of iron. Each portion
costs the Navy \$.60. Because all minimum standards must be met in each serving, the
current sea ration container must contain 10 ounces of the Texfoods product (to meet the
minimum amount of Vitamin A). This costs the Navy \$3.00 (=5 * \$.60) per serving.
The Navy is considering switching to another product from a different supplier,
Calration. A two-ounce portion of Calration costs \$.50 and provides 50% of the
minimum requirement of Vitamin A and 25% of the requirement of Vitamin D, but only
10% of the requirement for iron. Substituting the Calration product, the sea ration
container would have to contain 20 ounces in order to meet the minimum iron
requirement, costing the Navy \$5.00 (=10 * \$.50) per serving.
One bright lieutenant has suggested that a mixture of the two products might meet the
overall standards at a lower cost than the current \$3 per serving. The Navy has never
worried about the taste of sea rations; hence, mixing them, either by combining them or
by packing a portion of each, is an acceptable alternative. The lieutenant has been given
permission to evaluate the idea. Considering the problem posed by the lieutenant,
provide an optimal solution that complies with the following three conditions:
1. Determine the amount of two-ounce portions of each sea ration product in the
mix.
2. Minimize the total cost of the sea rations.
3. Meet the minimum requirements for Vitamin A, Vitamin D, and Iron.
Should the cost per 2 oz. portion of the Calration product increase to \$.75,
4. Would the optimal product mix change?
If the cost per 2 oz. portion of the Texfoods product increases to \$1.05 while the cost per
2 oz. portion of the Calration product decreases to \$.30,
5. Would the optimal product mix change?

#### Solution Summary

Neat and step-by-step solution is provided.

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