# Math 160: Study Guide - Chapter 5

1. Write the requirements for a problem to be a standard maximization or standard minimization problem.
2. Indicate how to convert a non-standard problem into the initial system. Know what to do with a ≤, ≥, and = constraints. Know how to handle a minimization problem.
3. You are given a linear programming problem and asked to shade the feasible region. The graph is drawn, you just need to shade the appropriate region. Then add slack variables and/or subtract surplus variables to create a system that could be used in the geometric approach (section 5.3) to create a table of decision, slack, and/or surplus variables and determine if they're feasible or not.
4. Given a table of intersection points, determine which points are feasible and which aren't. Find the solution to the problem (maximum or minimum depending on the problem). The solution includes the values of the variables and the objective function, not just the point where it occurs.
5. Given a feasible region and a constant-profit (isoprofit) line, determine where the maximum value occurs. Explain how you arrived at your answer.
6. You are given a tableau (not the initial) for a standard maximization problem. Identify the number of slack variables and decision variables. Label the columns of the tableau with the appropriate variable names, identify which variables are basic and non-basic, give the value of each of the variables, identify a function that could be the objective function (multiple choice), and state whether or not the optimal solution has been found.
7. You are given the final tableau for a dual problem. Identify which variables are basic and non-basic and give their values for both the dual (maximization) and original (minimization) problems.
8. You are given the initial tableau for a non-standard maximization problem. Write the initial problem. This will require you to know all of the material in problem 2 and be able to reverse it to obtain the problem. The columns (variables) will not be named - you will need to be able to determine which are decision, slack, surplus, artificial, and objective based on the tableau. Remember that the problem has the inequality constraints in it; it does not contain slack, surplus, or artificial variables.
9. You are given a tableau (not the initial). Identify which variables are basic and non-basic and give their values. Identify which variable is basic in each row, indicate the pivot column, find the appropriate ratios, indicate the pivot row, circle the pivot element, identify the entering variable, identify the exiting variable, identify how much the objective function will increase after the pivot.
10. Explain the pivot procedure. Know why the pivot column is chosen. Know why the pivot row is chosen. The answer is more than "pick the column with the most negative number in the objective function" and "pick the row with the smallest non-negative ratio." You should also explain what the values in the objective function represent and how the ratios are formed and what they represent.
11. Form the dual problem from the minimization problem. Do not solve the dual problem, only set it up.
12. You are given a geometric representation of a system. Identify which variables are basic and non-basic for the indicated points. A tableau (not the initial) is given; identify which point corresponds to the tableau, give the value of the objective function at that point, and indicate where you would pivot to move to a specified point.

## Notes:

• The test is individual. There are no groups allowed.
• Wherever possible, I have tried to compose the test so that the graphing calculator is useless and to test your actual understanding of the material, not your ability to run the simplex or pivot programs. In fact, the only place you really need the calculator on the test is to do division.
• There is a take home portion of the exam worth 30 points. It is due the day of the in-class exam. The problems on the take home exam are listed below.
• To balance points properly, the total below is 100 points. That total will be multiplied by 0.70 and then the take-home point total added to that to get the final score.

## Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 Total 6 8 8 8 5 12 8 8 15 6 6 10 100

## Take Home Exam Problems

Work the following even problems from the textbook. You are free to use the simplex and pivot programs once the tableaus are set up. You should write the initial and final tableaus, but you do not need to write the intermediate tableaus unless you're using them to determine where to pivot (sect 5.6).

1. 5.4.28
2. 5.4.46 part a
3. 5.5.24
4. 5.5.46 part a
5. 5.6.28
6. 5.6.42, go ahead and solve. Hint: When properly setup, this problem can be turned into a standard maximization problem and then the simplex program ran. That's why the "go ahead and solve" instructions are there, the difficult part is setting it up properly. Another hint: remember that the amount loaned doesn't have to be all \$3 million, so when it says things like "50% of the amount loaned", don't use 50% of \$3 million, use 0.50( x1+x2+x3+x4 ).