$4$ Multivariate derivatives and LP optimization

$1.$ Find the derivatives for the following function:

a$.=(,)=2+24$

Find the partial derivates $$

$$ and $$

$$

$2.$ An agricultural supply company is developing a livestock feed mix that will consist of three

ingredients: ingredient A$,$ ingredient B$,$ and ingredient C$.$ The table below displays nutrition and

cost information per ounce of each of these ingredients.

Ingredient Calories Fat grams $\%$RDA

Vitamin A

$\%$RDA

Vitamin D

$\%$RDA

Protein

Cost

A $830435\%5\%35\%$ $$0\mathrm{.}75$

B $600625\%41\%21\%$ $$0\mathrm{.}63$

C $75034\%15\%22\%$ $$0\mathrm{.}20$

The company wants to create a mix that contains no more than $750$ calories per ounce and no more

than $10$ grams of fat. The desired mix should also meet at least $25\%$ of the Recommended Daily

Allowance $($RDA$)$ of each of the following nutrients: Vitamin A$,$ Vitamin D$,$ and Protein. The

company wants to develop the feed mix as cheaply as possible.

A$)$ Following the steps below formulate this problem as a linear programming model, and use the Excel

solver to find the optimal percentages of each ingredient to include in the feed mix. Submit your Excel

workbook together with your answer sheet.

Decision variables.

$($We want to find out the percentage of each one the three ingredients in one ounce of the mix$)$

Objective function.

$($How do we measure the cost of the mix in terms of percentages of the ingredients? What is the sense

of optimization?$)$

Constraints.

The total of the percentages of ingredients has to be equal to $100\%.$

Limits on the following nutrients in the mix.

$($Left hand side: express the amount of nutrient in the mix in terms of percentages of ingredients

Right hand side: limit amount$)$

Calories

Fat

Vitamin A

Vitamin D

Protein

B$)$ Inspect the sensitivity report and find out the change in the optimal cost if we reduce the calorie

limit from $750$ to $700,$ and the fat limit from $10$ to $8.$ Does the cost increase or decrease and by how

much?