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# Linear programming problems with sensitivity analysis

Here's the process I want you to use to answer sensitivity questions. For attached problems, please copy and paste the sensitivity report from QM for Windows.

? Determine if the change is within the permitted boundaries. If it is, the answer to your question can be found within the data. You are to reference the data and derive your answer from it.

? If the change exceeds the permitted bounds, then indicate so and resolve the problem. You would state that "a change in the profit from XX to YY exceeds the upper bounds of ZZ and this results in a new optimal solution of X1 = , X2 = , and Z = ".

1) Irwin Textile Mills produces two types of cotton cloth - denim and corduroy. Corduroy is a heavier grade of cotton cloth and, as such, requires 7.5 pounds of raw cotton per yard, whereas denim requires 5 pounds of raw cotton per yard. A yard of corduroy requires 3.2 hours of processing time; a yard of denim requires 3.0 hours. Although the demand for denim is practically unlimited, the maximum demand for corduroy is 510 yards per month. The manufacturer has 6,500 pounds of cotton and 3,000 hours of processing time available each month. The manufacturer makes a profit of \$2.25 per yard of denim and \$3.10 per yard of corduroy.

a. What is the effect on the optimal solution if the profit per yard of denim is increased from \$2.25 to \$3.00? What is the effect if the profit per yard of corduroy is increased from \$3.10 to \$4.00?

b. What will be the effect on the optimal solution if Irwin Mills could obtain only 6,000 pounds of cotton per month?

2) The Bradley family owns 410 acres of farmland on which they grow corn and tobacco. Each acre of corn costs \$105 to plan, cultivate and harvest; each acre of tobacco costs \$210 The Bradley's have a budge of \$52,500 for next year. The government limits the number of acres of tobacco that can be planted to 100. The profit from each acre of corn is \$300; the profit from each acre of tobacco is \$520.

a. What would the profit for corn have to be for the Bradley's to plant only corn?
b. If the Bradleys can obtain an additional 100 acres of land, will the number of acres of corn and tobacco they plan to grow change?
c. If the Bradleys decide not to cultivate a 50-acre section as part of a crop recovery program, how will it affect their crop plans?

When asked to provide the linear programming model, I need it in the original form, not standard form. (In other words, do not show slack variables.) I also want each constraint labeled. Also, your linear programming models must conform to the additive property. This means ratio constraints will have to be converted to linear inequalities. Remember that constraints have decision variables to the left of the inequality and a constant number to the right.

When asked to supply the graphical solution, you are to supply a graph of the feasible region along with the (X1 = , X2 = , Z = ) values for each of the extreme points. Please do this with QM for Windows and produce a graph of the feasible region and the extreme points for linear programming models of two decision variables. You can save the graph as an HTML file and then copy and paste it into your word document. Remember, I need both the graph and the extreme points. You also need to indicate the optimum solution from the extreme points.

3) Julie is considering leasing a food booth outside the Tech stadium at home football games. Tech sells out every home game and Julia knows that everyone eats a lot of food. She has to pay \$1,000 per game for a booth and the booths are not very large. Vendors can sell either food or drinks on Tech property, but not both. She thinks slices of pizza, hot dogs and BBQ sandwiches are the most popular food items.

Most items are sold during the hour before the game stars and during half time; thus it will not be possible for Julia to prepare the food while she is selling it. She must prepare the food ahead of time and then store it in a warming oven. For \$600 she can lease a warming oven for the 6-game home season. The oven has 16 shelves and each shelf is 3 feet by 4 feet. She plans to fill the oven with the three food items before the game and then again before half time.

Julia has negotiated with a local pizza delivery company to deliver 14-inch pizza twice each game?2hrs before game time and right after the opening kickoff. Each pizza will cost her \$6 and will include 8 slices. She estimates it will cost her \$0.45 for each hot dog and \$0.90 for each BBQ sandwich if she makes the BBQ herself the night before. She measured a hot dog and found it took up 16 inches (squared). She plans to sell a slice of pizza and a hot dog for \$1.50 a piece and a BBQ sandwich for \$2.25. She has \$1,500 in cash available to purchase and prepare the food items for the first home game; for the remaining five games she will purchase her ingredients with the money she has already made from the previous games.

Julie has talked to some students and vendors who have sold food and has discovered that she can expect to sell at least as many slices of pizza as hot dogs and BBQ sandwiches combined. She also anticipates that she will probably sell at least twice as many hot gods as BBQ sandwiches. She believes that she will sell everything she can stock and develop a customer base for the season if she follows these general guidelines for demand. If Julia clears at least \$1,000 in profit for each game after paying all her expenses, she believes it will be worth leasing the booth.

A) Formulate and solve a linear programming model for Julia that will help you advise her if she should lease the booth.

B) If Julia were to borrow some more money before the first game to purchase more ingredients, could she increase her profit? If so, how much should she borrow and how much additional profit would she make? What factor constrains her from borrowing even more money than this amount (indicated in your answer to the previous question)?

C) When Julia looked at the solution in (A) she realized that it would be physically difficult for her to prepare all the hot dogs and BBQ sandwiches indicated in this solution. She believes she can hire a friend of hers to help her for \$100 per game. Based on the results in (A) and (B), is this something you think she could reasonably do and should do?

#### Solution Summary

This posting contains solution to following linear programming problems using QM for windows:

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