338 ACTIVITY 18: Review for Test

WHY:

Tests help you measure the level of understanding you have gained while learning the material. They can and should be opportunities for learning as well. Preparation for tests helps organize and synthesize the knowledge you have gained during the preceding part of the course.

LEARNING OBJECTIVES:

  1. Discover the general nature of a Math 338 test.
  2. Brainstorm and prioritize questions about topics which confuse one or more team members.
  3. Discover successful approaches to study for the test.

PERFORMANCE CRITERIA:

  1. Quality of the questions identified by the team.
  2. Success in finding an approach to use to study for the test.

RESOURCES:

  1. Sample Test
  2. List of test topics
  3. Activities, quizzes, homework, notes.
  4. 15 minutes

PLAN:

  1. Choose roles if you have not already done so.
  2. Brainstorm and come up with 5 topics which cause confusion in the material covered for the test.
  3. Identify 5 questions whose answers will help clarify the confusion.
  4. Prioritize the questions so the most pressing is asked first.

CRITICAL THINKING QUESTIONS:

  1. Did you have to omit areas of importance in coming up with the answer to Question 1? If so, what were they?

  2. How easy was it to come to consensus on the priority of each question? What were some of the obstacles to this process?

Math 338 Test 2 Topics -- Sections 3.2 - 4.3 (skip parts of 3.6, 3.7, and 4.1)

Note: You may bring in one 2-sided cheat sheet. There will be one proof (choose one out of two). 

Be able to find  A, B, D, B-1, 
CB, CD, 
CBTB-1b, B-1D, b,  
CBTB-1D - CDT 

Slack Variable				     Artificial Variables
Surplus Variable                             2-Phase Simplex Method 
Basic Feasible Solution                      Handling infeasible problems
Feasible Regions F and F-hat 		     Reading SIMPLEX program output
Injection map  Sigma                         Post Optimality analysis
Equivalence of F and F-hat extreme points    Adding a variable
Basic Variable        			     Adding a constraint
Non-Basic Variable      		     Dual of LP problem
Basis Matrix                                 Symmetric Primal
Be able to find Algebraic LP Solutions       Symmetric Dual
Minimum Ratio Rule                           Weak Duality Theorem & Corollary
Non-Degenerate BFS                           Shadow Prices
Simplex Method                               Penalty Costs
Optimality Theorem                           Real problem applications
Handling unbounded problems

Math 338 Sample Test 2

  1. (10 points) Prove one of the following:
    1. Prove that the following equation represents the bottom row of the final tableau for a problem with an optimal solution: . Then prove that the penalty cost corresponding to a non-basic variable is the increase in the value of the objective function for each unit increase in that variable.

    2. Prove that when P designates a primal problem and D designates its dual it follows that D is infeasible whenever P is unbounded. (You may use the Weak Duality theorem and its corollary, but you may not use the Duality theorem to do this proof.)

  2. (5 points) If the constraints of a linear programming problem satisfy AX >= b, define the injection map which maps the feasible region of the original problem into the feasible region for the standard form of the problem.

  3. (5 points) If the standard form of a bounded, feasible linear programming problem has 5 variables and 4 constraints, how many possible basis matrices are there? If the feasible region for the standard form of this problem has 7 extreme points, how many different basic feasible solutions does it have?

  4. (10 points) Given the following problem, determine which variable should enter the basis and which should leave without using the simplex method. Explain your work. 

    
     min -60x1 + 20x2 + 2x3
     subject to:
           6x1 + 2x2 + x3      = 200
           2x1 -  x2    + x4   = 60
            x1            + x5 = 40
              xi >= 0


  5. (30 points) Given the following first and last tableau for the problem in question 4: 
    
        | 6   2  1  0  0 | 200 |	| 0  1  0.2 -0.6  0 |   4 |
	| 2  -1  0  1  0 |  60 |	| 1  0  0.1  0.2  0 |  32 |
	| 1   0  0  0  1 |  40 |	| 0  0 -0.1 -0.2  1 |   8 |
	|----------------------|	|-------------------------|
	|60 -20 -2  0  0 |   0 |	| 0  0 -4   -24   0 |-1840|
    1. Find the final basic and nonbasic variables and compute B, D, CB, CD, CBTB-1b, B-1D, CBTB-1D - CDT, and B-1

    2. Find the optimal solution, the shadow prices and the penalty costs.

    3. Suppose the person posing this problem wishes to make x3 equal to 2. What change will this make in the value of the objective function, presuming that the values of the other variables remain as given in part b?

    4. Suppose the right hand side of the second constraint is changed to 40. Verify that this does not change the set of basic variables. What change does this make in the value of the objective function.

    5. State the dual to the problem in question 4 and give its optimal solution.

  6. (30 points) Given the following problem: Young MBA Erica Cudahy may invest up to $1000. She can invest her money in stocks and loans. Each dollar invested in stocks yields 10 cents profit and each dollar invested in loans yields 15 cents profit. At least 50% of all money invested must be invested in stocks and at least $400 must be invested in loans. She wants to maximize her total profit.

    1. Write this problem in standard form, with the variable for stocks, the variable for loans, the slack variables, the surplus variable, and the artificial variable in that order. Label all your variables.

    2. Complete phase I of the simplex method and write the initial tableau for phase II, keeping the artificial variable column. Put this phase II tableau in a form to which we can apply the simplex method.

    3. Given the following final tableau from phase II with the artificial variable column still in place, identify the basic and non-basic variables, write out the solution to the problem and also compute B, D, CB, CD, B-1D, B-1, and CBTB-1D - CDT. 
      
	| 0  0   0.5    1   1  -1 | 100 |
	| 0  1   0.5    1   0   0 | 500 |
  	| 1  0   0.5   -1   0   0 | 500 |
	|-------------------------------|
	| 0  0 -0.13 -0.05  0 -10 |-125 |


    4. Suppose that Erica received an additional $1000 to invest. How much should she now invest in stocks and loans and what would her profit be?

    5. Suppose that the stock profit increased to 20 cents on the dollar under the original problem conditions. Will this make a difference in the amounts she should invest in stocks and loans? Give a reason for your answer.



  7. (14 points) Given the function f(x) = x2ln x3 - 3x2
    1. Find its domain and derivative.

    2. Find all the critical points for this function

    3. Find all relative max and min points for this function.

    4. Find the absolute max and min over the interval: [ 1, e]


  8. (6 points) Find the gradient of the function x3y2z - xz2

Extra Credit: Find the solution to problem 6(e) above.

Math 338 Activity 18 -- Revised 11/3/98

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