Math 160: Study Guide - Chapters 5&6
- A feasible region is sketched. Make a table of corner points and determine where the maximum and minimum values of the objective function occur. The equations of two of the lines are given without the inequality. Determine whether the inequality needs to be ≤ or ≥ to satisfy the conditions of the problem.
- A feasible region is sketched and the slope of each edge is given. Determine where the maximum value of the objective function will occur. This question is about interpreting the slope of the objective function in relation to the slopes of the edges. Four parts.
- A standard maximization system is given. Set pairs of variables equal to zero and find the values of the other values. Determine whether or not each point is feasible and find the objective function for the feasible points. Determine the maximum of the objective function and where it occurs.
- Explain the simplex process. Know how to pick the pivot column and why that column is picked. Know how to pick the pivot row and why that row is picked.
- The initial tableau for a non-standard maximization problem is given. Identify each variable as being basic or non-basic and give its value. Determine where the next pivot should occur.
- A tableau for a standard maximization problem is given. Label the columns with the variable names. Determine the next step in the simplex procedure (the optimal solution is done, an additional pivot is required, or there is no optimal solution).
- A tableau from a standard maximization problem is given. Identify each variable as being basic or non-basic and give its value. Determine the next step in the simplex procedure (the optimal solution is done, an additional pivot is required, or there is no optimal solution).
- The final tableau from a dual problem is given. Identify which variables
are basic and non-basic and give their values for both the dual (maximization)
and original (minimization) problems.
- Take the standard maximization problem and write the initial system by adding the slack variables. Then write the initial tableau. Do not solve the problem.
- A preliminary tableau from a non-standard maximization problem is given. Label the columns with the appropriate names. Write the original problem. This involves being able to look at a tableau and determine which columns correspond to ≤, ≥, or = constraints.
- A tableau from a standard maximization problem is given. Identify each variable as being basic or non-basic and give its value. Label the tableau with the following items: the variable that is basic in each row, the pivot column, the appropriate ratios, the pivot row, the pivot element, the entering variable, and the exiting variable. Identify the increase in the objective function after the next pivot.
- Form the dual problem, but do not solve it.
- 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." Six parts.
- 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 and indicate where you would pivot
to move to a specified point.
- Construct a mathematical model in the form of a linear programming problem. Be sure to define your variables. Do not write the system or solve the problem.
Notes
- There is a take home exam that is due the day of the in-class exam. It is worth 20 points. See below for those problems.
- The in-class exam is worth 105 points.
- The in-class exam is designed to be worked without the simplex or pivot programs on the calculator. There are no tableaus that need entered into the calculator. The only place you might need a calculator is for solving the system of equations for question 3 or find the ratios for some of the tableaus.
- This will be a one-hour exam. There are no difficult or time consuming questions on the exam. It is a very conceptual exam. Either you know it or you don't. At 6:30, we will continue with lecture material.
Points per problem
| # |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
Take Home |
Total |
| Pts |
7 |
4 |
5 |
6 |
6 |
4 |
5 |
6 |
6 |
6 |
8 |
5 |
6 |
6 |
5 |
20 |
125 |
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 define your variables, write the initial problem, the initial system, the initial and final tableaus, as well as the answer to the question. You should write the intermediate tableaus only if you are using the pivot program but not if you are using the simplex program. Each question is worth 5 points.
- 6.2.46A. Use the simplex program. Read carefully. Make sure the units within each constraint are consistent, but you do not need to use the same units for every constraint.
- 6.3.50. Write the initial problem and then form the dual problem. Use the simplex program. Be sure you answer the original question.
- 6.4.28. Use the pivot program and show all of the the intermediate tableaus.
- 6.4.42, Setup and solve the problem. Hint: When properly setup, this problem can
be turned into a standard maximization problem and then the simplex program
ran. That's why I want you to go ahead and solve the problem, even though the instructions in the book say only to set it up. 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 ).
The take home exam is due the day of the in-class exam.