Math 160: Study Guide - Chapters 5&6

1. 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.
2. A feasible region is sketched and the slope of each edge is given. Determine where the maximum (or minimum) 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. Five parts.
3. 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.
4. 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. This is an explanation of the how and why, not just "pick the column with the most negative number in the bottom row." That much is the answer to the "how", but then you need to explain "why" you pick that column. The how do you pick the pivot row would involve explaining how to find the proper ratios, not just "pick the smallest ratio".
5. 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.
6. 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).
7. 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).
8. 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.
9. 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.
10. A preliminary tableau from a non-standard maximization (or minimization) 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. It also involves being being able to interpret the objective function and might involve knowing how to handle a non-standard minimization problem. Remember that there are no slack, surplus, or artificial variables in the problem. The problem contains only decision variables and, of course, the objective function.
11. 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.
12. Form the dual problem, but do not solve it.
13. 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 "Standard minimization after multiplying the second constraint by -1." Six parts.
14. 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.
15. 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 25 points. See below for those problems.
• The in-class exam is worth 100 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.
• 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.
• Know the difference between a problem, system, and tableau. Also know the difference between a preliminary tableau and an initial tableau for a non-standard problem. Problems ask for a specific type of answer.
• Always make sure that the cleared column has a 1 as its sole number before trying to interpret the values in a tableau. For example, if the coefficient in the x₂ column is 3 and the right hand side is 12, then the value of x₂ is 4, not 12.

Points per problem

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.

1. 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.
2. 6.3.50. Write the initial problem and then form the dual problem. Use the simplex program. Be sure you answer the original question.
3. 6.4.28. Use the pivot program and show all of the the intermediate tableaus.
4. 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 ).
5. Take the problem below and work it like it was a problem in section 6.1 Set up a system of equations like those in section 6.1 where you add a slack variable for each ≤ constraint. Be sure to include the non-negativity constraints when you write your system. Find the following:
   1. The number of decision variables.
   2. The number of slack variables.
   3. The number of basic variables.
   4. The number of non-basic variables.
   5. Set up a table, like we did in section 6.1, with a column for each variable, a column labeled "feasible?", and a column for the objective function P.
      1. List all of possible combinations that can be found by setting the non-basic variables equal to zero.
      2. Find the values of the basic variables for each point. If the values are "nice" decimals, you can leave them in decimal form, otherwise write them as fractions.
      3. Identify whether or not each point is feasible.
      4. Find the value of the objective function for each of the feasible solutions.
   6. What is the maximum value of the objective function and where does it occur?

Linear Programming Problem to use for #5

The take home exam is due the day of the in-class exam.

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Last updated June 29, 2010 8:26 PM