Question

The Department of Navy has been downsizing and must find cost savings opportunities to meet mandated congressional budget cuts. One suggestion under consideration is to change the makeup of the content of Navy sea rations, the canned food supplies containing certain minimum daily requirements (MDR) of Vitamin A, Vitamin D, iron, and other nutrients, which combat troops carry into battle. The current supplier of Navy sea rations is Texfoods. A two-ounce portion of their product supplies 20% of the required amount of Vitamin A, 25% of the MDR of Vitamin D, and 50% of the required amount of iron. Each two-ounce portion cost the Navy $0.60. Because all minimum standards for nutrient must be met in each serving, the current sea ration container must contain 10 ounces of the Texfoods product (to meet the MDR of Vitamin A). This cost the Navy $3.00 (5 * $0.60 = $3.00) per serving. A new supplier, Calration, is pushing the Navy to switch to their product. A two-ounce portion of Calration cost $0.50, a saving of $0.10 per two-ounce portion. A two-ounce portion of Calration provides 50% of the MDR for Vitamin A, 25% of the requirement for Vitamin D, but only 10% of the MDR for iron. If a switch is made to Calration the sea ration container would have to contain 20 ounces in order to meet the MDR for iron. The cost to the Navy would be $5.00 (10 * $0.50) per serving. In your new position as Chief Procurement Officer, Northern Fleet Command you suggested mixing the two products might meet the nutrient standards at a lower cost than the current $3 per serving. While the Navy has always prided itself on the quality of foods served, taste has not been a worry for Congress; hence, mixing the rations, either by combining them or by packing a portion of each, is an acceptable alternative. You have been given instructions to evaluate the data.

Prepare a report for congress that recommends the number of two-ounce portions of Texfoods and Calrations to include in a single serving of sea rations that will provide the MDR of Vitamins A and D and iron (it is not necessary to know the MDR of these items [Why?]). Congress is interested in the preserving cost of your serving and the Navy might be interested in the total weight of the serving (Why?). Senator Joe Ask Questions will want to know if your proposed mixture supplies an excess of either Vitamin A, D, or iron. Also, Senator Meg Curious will want to know what might happen if Texfoods or Calration changes their pricing; does your proposed mixture remain the same. Address your report to Members of Congress, Naval Food Committee. Be sure to follow the quantitative analysis approach in your report.

Answer 1

This problem can be solved using concept of linear programming problem as follows.

Suppose, 2x ounces of Texfoods and 2y ounces of Calration are to be provided in ration.

So, we have to minimize

Subject to

Using graphical method we can solve as follows.

The feasible region under 3 constraints is shaded by green.

The objective function is drawn in red moving which towards and away of origin we can find different values of z.

Clearly, moving red line we find that minimum value occurs in (1.5, 2.5).

In a ration,

• Amount of Vitamin A with respect to MDR = 1.5*20%+2.5*50% = 155%
• Amount of Vitamin B with respect to MDR = 1.5*25%+2.5*25% = 100%
• Amount of iron with respect to MDR = 1.5*50%+2.5*10% = 100%

Report-

1.5 two-ounce of Texfoods and 2.5 two-ounce of Calration is two be provided n each ration. Therefore, mixing of two types of food is required.

This combination is fulfilling MDR.

Cost per ration is $ (1.5*0.60+2.5*0.50) = $ 2.15

Total weight of each ration = (1.5+2.5)*2 ounces = 8 ounces

Vitamin A is more in the ration than MDR.

If any of prices changes, then new calculation to get proportion is required. But if both the prices change in such a manner that the ratio of prices remain i,e, then this amounts of each type in the mixture remains valid.