The last two rows in this spreadsheet indicate the minimum and$/$or maximum nutrients required per week for men between the ages of $20$ and $50.($Maximum values of $9,999$ indicate that no maximum value applies to that particular nutritional requirement.$)$

The USDA uses this information to design a diet $($or weekly consumption plan$)$ that meets the indicated nutritional requirements. The last two columns in Figure $7\mathrm{.}20$ represent two different objectives that can be pursued in creating a diet. First, we may want to identify the diet that meets the nutritional requirements at a minimum cost. Although such a diet might be very economical, it might also be very unsatisfactory to the tastes of the people who are expected to eat it$.$ To help address this issue, the USDA conducted a survey to assess people$$s preferences for different food groups. The last column in Figure $7\mathrm{.}20$ summarizes these preference ratings, with higher scores indicating more desirable foods, and lower scores indicating less desirable foods. Thus, another objective that could be pursued would be that of determining the diet that meets the nutritional requirements and produces the highest total preference rating. However, this solution is likely to be quite expensive. Assume that the USDA has asked you to help them analyze this situation using MOLP.

Find the weekly diet that meets the nutritional requirements in the least costly manner. What is the lowest possible minimum cost? What preference rating does this solution have?

Find the weekly diet that meets the nutritional requirements with the highest preference rating. What preference rating does this solution have? What cost is associated with this solution?

Find the solution that minimizes the maximum percentage deviation from the optimum values for each individual objective. What cost and preference rating is associated with this solution?

Suppose that deviations from the optimal cost value are weighted twice as heavily as those from the optimal preference value. Find the solution that minimizes the maximum weighted percentage deviations. What cost and preference rating is associated with this solution?

What other factors or constraints might you want to include in this analysis if you had to eat the resulting diet?