# 4 Questions - Linear Programming and Optimization Problems

A cook is puzzling over the number of pounds of food he should purchase in order to minimize his cost. He has always brought his food from a small health food store in town. The store sells two types of mixtures. Both of these mixtures contain the three ingredients needed, but the cook needs his own special ratio of these ingredients to meet the requirements of a certain diet. The chart below shows that the diet is to be made up of at least 9 grams of Vitamins B-12, at least 30 grams of calcium, and at least 24 grams of iron. As the chart also shows the ratios for each of the three ingredients in the two mixtures the store sells is not the same as what the cook needs. The cook wants to minimize his costs, and have the minimize requirements met once he combines the two mixtures and cook up the entire blend.

Dietary Constraints

Vitamin B-12 Calcium Iron

Minimum Requirements 9 Grams 30 Grams 24 Grams

Farmer's Mixture A =$3.00 per pound I g per pound 6 g per pound 8 g per pound

Farmer's Mixture B =$3.00 3 g per pound 5 g per pound 3 g per pound

Question 1

First set up an equation for the total cost the book will have to pay for the two mixtures he will buy. This will be called the cost function.

Let x = the number of pounds of mixture A, and Y = the number of pounds of mixture B.

Question 2

Next, develop three inequalities utilizing the cook’s dietary constraints.

Question 3

Draw a graph for the solution of the three inequalities (this does not need to be turned in). Shading will indicate the solution to each inequality. In this case, since the wording of the inequalities was “at least” 9 grams of Vitamin B-12, etc., the inequalities will utilize the = symbol. The shaded area that satisfies all three inequalities will form a solution area bounded by the segments of straight lines.

Next, calculate the coordinates of each point in the solution area where any two of the boundary lines intersect, as well as where any of the boundary lines cross the axes

a. 1st point of intersection is:

b. 2ndpoint of intersection is:

c. x-intercept is:

d. y-intercept is:3

Explain how to obtain these.

Question 4

Calculate the cost at each of the points of intersection, including the intercepts. Then compare these values and pick the one that is the least.

a. The value of the cost function at the 1st point of intersection is:

b. The value of the cost function at the 2nd point of intersection is:

c. The value of the cost function at the x intercept is:

d. The value of the cost function at the y intercept is:

e. The cost is the least at:

https://brainmass.com/math/optimization/4-questions-linear-programming-and-optimization-problems-2577

#### Solution Summary

This shows how to minimize costs while still meeting nutritional requirements for a given mixture problem in an attached Word document. The solution are is graphed for (2) in an attached .jpg file.

Linear Optimization Problems

I need help with these problems. Problems 8-4, 8-6 MUST use Excel's Solver to complete the problems. Note, the template is also attached for these 2 problems and must be solved using this template.

8-4 (Animal feed mix problem) The Battery Park Stable feeds and houses the horses used to pull tourist-filled

carriages through the streets of Charleston's historic waterfront area. The stable owner, an ex-racehorse

trainer, recognizes the need to set a nutritional diet for the horses in his care. At the same time, he would like

to keep the overall daily cost of feed to a minimum. The feed mixes available for the horses' diet are

an oat product, a highly enriched grain, and a mineral product. Each of these mixes contains a certain

amount of five ingredients needed daily to keep the average horse healthy. The table on this page shows

these minimum requirements, units of each ingredient per pound of feed mix, and costs for the three mixes.

In addition, the stable owner is aware that an overfed horse is a sluggish worker. Consequently, he

determines that 6 pounds of feed per day are the most that any horse needs to function properly. Formulate

this problem and solve for the optimal daily mix of the three feeds. Please see attached excel spead sheet for original data from problem.

8-6 Eddie Kelly is running for re-election as mayor of a small town in Alabama. Jessica Martinez, Kelly's

campaign manager during this election, is planning the marketing campaign, and there is some stiff

competition. Martinez has selected four ways to advertise: television ads, radio ads, billboards, and

newspaper ads. The costs of these, the audience reached by each type of ad, and the maximum number

of each is shown in the following table:

See attached spreadsheet for table

In addition, Martinez has decided that there should be at least six ads on TV or radio or some

combination of those two. The amount spent on billboards and newspapers together must not exceed the

amount spent on TV ads. While fundraising is still continuing, the monthly budget for advertising has

been set at $15,000. How many ads of each type should be placed to maximize the total number of

people reached?