# Math 160: Study Guide - Chapter 5

1. A feasible region is sketched. Place a ≤ or ≥ into each inequality so that the shaded region is the solution to the system of linear inequalities. Given an objective function, determine the maximum and minimum values.
2. A feasible region is sketched. Find the point where the maximum occurs for the given objective functions. Look at problems 5.2.1-8.
3. Construct a mathematical model in the form of a linear programming problem. Be sure to define your variables. Do not set up the system or solve the problem. Look at problems 5.2.31-46.
4. Given a system of linear equations, complete the table of intersection points. Look at problems 5.3.7-8.
5. 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 and the point where it occurs. Complete the original problem by writing ≤ or ≥ in each problem constraint. Look at problems 5.3.5-6.
6. 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.
7. You are given a tableau (not the preliminary) for a non-standard maximization problem. Identify which variables are basic and which are non-basic and give their values. Determine where the next pivot should occur.
8. 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, and decide the next step (optimal solution has been found, an additional pivot is required, or the problem has no optimal solution). Look at problems 5.4.1-8.
9. 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.
10. You are given the preliminary tableau for a non-standard maximization problem. Write the initial 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.
11. You are given a tableau (not the initial) from a standard maximization problem. 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.
12. Take the standard maximization problem and write the initial system by adding the slack variables. Then write the initial tableau. Look at problems 5.4.9-12.
13. Form the dual problem from the minimization problem. Do not solve the dual problem, only set it up. Look at problems 5.5.13-20.
14. Identify whether each problem is a standard maximization problem, is a standard minimization problem, can be changed into a standard maximization problem (explain how), can be changed into a standard minimization problem (explain how), or is a non-standard problem that must be worked using the Big-M method. In the cases where the problem can be changed into a standard problem, indicate how. For example, you might write "Can be changed into a standard minimization problem by multiplying the second constraint by -1." Look at problems 5.6.25-32.
15. 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 and maybe to solve a system of linear equations for question 4.
• There is a take home portion of the exam worth 10 points. It is due the day of the in-class exam. The problems on the take home exam are listed below.

## Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Take Home Total 5 3 5 6 7 6 6 7 6 6 10 6 5 5 7 10 100

## Take Home Exam Problems

Work the following even problems from the textbook. You are free to use the simplex and/or pivot programs once the tableaus are set up. You should write the initial and final tableaus. You should write the intermediate tableaus if you are using the pivot program but not if you are using the simplex program. Each question is worth 5 points.

1. 5.6.28. Use the pivot program and show all of the the intermediate tableaus.
2. 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 ).