### MATH 318, Chapter 7 & 8 Quiz

Name:
You may use a calculator on this quiz. You may not use a cell phone or computer. Please read each question carefully, show your work and give justifications for your answers. If you find that you are spending a lot of time on one problem, leave it blank and move on to the next. If you have time left at the end of the quiz please check your work. There are questions on both sides of this quiz paper.
1. The constraint lines for the following linear program are shown on the right below. They are labeled according to which inequality generated them.
 Maximize: 2X + 3Y Subject to: X + 3Y ≤ 6 (1) 5X + 3Y ≤ 15 (2) X,Y ≥ 0

a) (10 points) Use the inequalities provided to shade in the feasible region.

b) (10 points) Approximately what is the optimal solution to this problem? Circle the point corresponding to this solution on the graph above.

b) (10 points) What are the binding constraints in this problem?

c) (10 points) What is the approximate slack associated with each of the other constraints?

2. Par, Inc. is an exclusive manufacturing firm which sells no more than 500 golf bags per year.  The manufacturing times needed to produce their two types of golf bag are shown in the table below, as is the profit contribution for each type of bag.  Their highly trained crafters can provide up to 500 hours of cutting and sewing work and up to 600 hours of finishing and packaging. This year, their board of directors has instructed Par, Inc. to focus on maximizing their profits.

 Product Cutting and Sewing Finishing and Packaging Profit per Bag Standard .6 1 \$27 Deluxe 1 .7 \$25

a) (10 points) Define the decision variables for this problem. Hint: you might want to use the words "how much".

b) (20 points) What is the objective function for this linear programming problem? Is your objective to maximize or minimize the value of this function?

c) (20 points) Write inequalities describing all the constraints for this linear programming problem.

d) (10 points) Par Inc.'s board of directors is considering increasing the number of bags produced per year. Give a rough estimate of the dual price of one golf bag.

Bonus: (5 points) Suppose that in problem 1 the objective was to maximize the value of 2X + 10Y.  Would this change your solution?  If so, how?