1. 1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2. 2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 5 Advanced Linear Programming Applications n Data Envelopment Analysis n Game Theory – Part I

3. 3 3 Slide © 2008 Thomson South-Western. All Rights Reserved Data Envelopment Analysis n Data envelopment analysis (DEA) is an LP application used to determine the relative operating efficiency of units with the same goals and objectives. n DEA creates a fictitious composite unit made up of an optimal weighted average ( W 1, W 2,…) of existing units. n An individual unit, k, can be compared by determining E, the fraction of unit k ’s input resources required by the optimal composite unit. n If E < 1, unit k is less efficient than the composite unit and be deemed relatively inefficient. n If E = 1, there is no evidence that unit k is inefficient, but one cannot conclude that k is absolutely efficient.

4. 4 4 Slide © 2008 Thomson South-Western. All Rights Reserved Data Envelopment Analysis n The DEA Model MIN E s.t.Weighted outputs > Unit k ’s output (for each measured output) Weighted inputs < E [Unit k ’s input] (for each measured input) Sum of weights = 1 E, weights > 0

5. 5 5 Slide © 2008 Thomson South-Western. All Rights Reserved The Langley County School District is trying to determine the relative efficiency of its three high schools. In particular, it wants to evaluate Roosevelt High. The district is evaluating performances on SAT scores, the number of seniors finishing high school, and the number of students who enter college as a function of the number of teachers teaching senior classes, the prorated budget for senior instruction, and the number of students in the senior class. Data Envelopment Analysis

6. 6 6 Slide © 2008 Thomson South-Western. All Rights Reserved n n Input Roosevelt Lincoln Washington Roosevelt Lincoln Washington Senior Faculty 37 25 23 Budget ($100,000's) 6.4 5.0 4.7 Senior Enrollments 850 700 600 Data Envelopment Analysis

7. 7 7 Slide © 2008 Thomson South-Western. All Rights Reserved n n Output Roosevelt Lincoln Washington Roosevelt Lincoln Washington Average SAT Score 800 830 900 High School Graduates 450 500 400 High School Graduates 450 500 400 College Admissions 140 250 370 College Admissions 140 250 370 Data Envelopment Analysis

8. 8 8 Slide © 2008 Thomson South-Western. All Rights Reserved n Define the Decision Variables E = Fraction of Roosevelt's input resources required by the composite high school E = Fraction of Roosevelt's input resources required by the composite high school w 1 = Weight applied to Roosevelt's input/output resources by the composite high school w 2 = Weight applied to Lincoln’s input/output resources by the composite high school w 3 = Weight applied to Washington's input/output resources by the composite high school Data Envelopment Analysis

9. 9 9 Slide © 2008 Thomson South-Western. All Rights Reserved n Define the Objective Function Minimize the fraction of Roosevelt High School's input resources required by the composite high school: MIN E Data Envelopment Analysis

10. 10 Slide © 2008 Thomson South-Western. All Rights Reserved n Define the Constraints Sum of the Weights is 1: (1) w 1 + w 2 + w 3 = 1 (1) w 1 + w 2 + w 3 = 1 Output Constraints: Output Constraints: Since w 1 = 1 is possible, each output of the composite school must be at least as great as that of Roosevelt: Since w 1 = 1 is possible, each output of the composite school must be at least as great as that of Roosevelt: (2) 800 w 1 + 830 w 2 + 900 w 3 > 800 (SAT Scores) (3) 450 w 1 + 500 w 2 + 400 w 3 > 450 (Graduates) (3) 450 w 1 + 500 w 2 + 400 w 3 > 450 (Graduates) (4) 140 w 1 + 250 w 2 + 370 w 3 > 140 (College Admissions) (4) 140 w 1 + 250 w 2 + 370 w 3 > 140 (College Admissions) Data Envelopment Analysis

11. 11 Slide © 2008 Thomson South-Western. All Rights Reserved n Define the Constraints (continued) Input Constraints: The input resources available to the composite school is a fractional multiple, E, of the resources available to Roosevelt. Since the composite high school cannot use more input than that available to it, the input constraints are: The input resources available to the composite school is a fractional multiple, E, of the resources available to Roosevelt. Since the composite high school cannot use more input than that available to it, the input constraints are: (5) 37 w 1 + 25 w 2 + 23 w 3 < 37 E (Faculty) (6) 6.4 w 1 + 5.0 w 2 + 4.7 w 3 < 6.4 E (Budget) (6) 6.4 w 1 + 5.0 w 2 + 4.7 w 3 < 6.4 E (Budget) (7) 850 w 1 + 700 w 2 + 600 w 3 < 850 E (Seniors) (7) 850 w 1 + 700 w 2 + 600 w 3 < 850 E (Seniors) Nonnegativity of variables: Nonnegativity of variables: E, w 1, w 2, w 3 > 0 E, w 1, w 2, w 3 > 0 Data Envelopment Analysis

12. 12 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution OBJECTIVE FUNCTION VALUE = 0.765 VARIABLE VALUE REDUCED COSTS VARIABLE VALUE REDUCED COSTS E 0.765 0.000 E 0.765 0.000 W1 0.000 0.235 W1 0.000 0.235 W2 0.500 0.000 W2 0.500 0.000 W3 0.500 0.000 W3 0.500 0.000 Data Envelopment Analysis

13. 13 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Solution (continued) CONSTRAINT SLACK/SURPLUS DUAL PRICES 1 0.000 -0.235 1 0.000 -0.235 2 65.000 0.000 2 65.000 0.000 3 0.000 -0.001 3 0.000 -0.001 4 170.000 0.000 4 170.000 0.000 5 4.294 0.000 5 4.294 0.000 6 0.044 0.000 6 0.044 0.000 7 0.000 0.001 7 0.000 0.001 Data Envelopment Analysis

14. 14 Slide © 2008 Thomson South-Western. All Rights Reserved n Conclusion The output shows that the composite school is made up of equal weights of Lincoln and Washington. Roosevelt is 76.5% efficient compared to this composite school when measured by college admissions (because of the 0 slack on this constraint (#4)). It is less than 76.5% efficient when using measures of SAT scores and high school graduates (there is positive slack in constraints 2 and 3.) Data Envelopment Analysis

15. 15 Slide © 2008 Thomson South-Western. All Rights Reserved Introduction to Game Theory n In decision analysis, a single decision maker seeks to select an optimal alternative. n In game theory, there are two or more decision makers, called players, who compete as adversaries against each other. n It is assumed that each player has the same information and will select the strategy that provides the best possible outcome from his point of view. n Each player selects a strategy independently without knowing in advance the strategy of the other player(s). continue

16. 16 Slide © 2008 Thomson South-Western. All Rights Reserved Introduction to Game Theory n The combination of the competing strategies provides the value of the game to the players. n Examples of competing players are teams, armies, companies, political candidates, and contract bidders.

17. 17 Slide © 2008 Thomson South-Western. All Rights Reserved n Two-person means there are two competing players in the game. n Zero-sum means the gain (or loss) for one player is equal to the corresponding loss (or gain) for the other player. n The gain and loss balance out so that there is a zero- sum for the game. n What one player wins, the other player loses. Two-Person Zero-Sum Game

18. 18 Slide © 2008 Thomson South-Western. All Rights Reserved n Competing for Vehicle Sales Suppose that there are only two vehicle dealer- ships in a small city. Each dealership is considering three strategies that are designed to take sales of new vehicles from the other dealership over a four-month period. The strategies, assumed to be the same for both dealerships, are on the next slide. Two-Person Zero-Sum Game Example

19. 19 Slide © 2008 Thomson South-Western. All Rights Reserved n Strategy Choices Strategy 1: Offer a cash rebate Strategy 1: Offer a cash rebate on a new vehicle. on a new vehicle. Strategy 2: Offer free optional Strategy 2: Offer free optional equipment on a equipment on a new vehicle. new vehicle. Strategy 3: Offer a 0% loan Strategy 3: Offer a 0% loan on a new vehicle. on a new vehicle. Two-Person Zero-Sum Game Example

20. 20 Slide © 2008 Thomson South-Western. All Rights Reserved 2 2 1 2 2 1 CashRebate b 1 0%Loan b 3 FreeOptions b 2 Dealership B n Payoff Table: Number of Vehicle Sales Gained Per Week by Dealership A Gained Per Week by Dealership A (or Lost Per Week by Dealership B) (or Lost Per Week by Dealership B) -3 3 -1 3 -2 0 3 -2 0 Cash Rebate a 1 Free Options a 2 0% Loan a 3 Dealership A Two-Person Zero-Sum Game Example

21. 21 Slide © 2008 Thomson South-Western. All Rights Reserved n Step 1: Identify the minimum payoff for each row (for Player A). row (for Player A). n Step 2: For Player A, select the strategy that provides the maximum of the row minimums (called the maximum of the row minimums (called the maximin). the maximin). Two-Person Zero-Sum Game

22. 22 Slide © 2008 Thomson South-Western. All Rights Reserved n Identifying Maximin and Best Strategy RowMinimum 1-3-2 2 2 1 2 2 1 CashRebate b 1 0%Loan b 3 FreeOptions b 2 Dealership B -3 3 -1 3 -2 0 3 -2 0 Cash Rebate a 1 Free Options a 2 0% Loan a 3 Dealership A Best Strategy For Player A MaximinPayoff Two-Person Zero-Sum Game Example

23. 23 Slide © 2008 Thomson South-Western. All Rights Reserved n Step 3: Identify the maximum payoff for each column (for Player B). (for Player B). n Step 4: For Player B, select the strategy that provides the minimum of the column maximums the minimum of the column maximums (called the minimax). (called the minimax). Two-Person Zero-Sum Game

24. 24 Slide © 2008 Thomson South-Western. All Rights Reserved n Identifying Minimax and Best Strategy 2 2 1 2 2 1 CashRebate b 1 0%Loan b 3 FreeOptions b 2 Dealership B -3 3 -1 3 -2 0 3 -2 0 Cash Rebate a 1 Free Options a 2 0% Loan a 3 Dealership A Column Maximum 3 3 1 3 3 1 Best Strategy For Player B MinimaxPayoff Two-Person Zero-Sum Game Example

25. 25 Slide © 2008 Thomson South-Western. All Rights Reserved Pure Strategy n Whenever an optimal pure strategy exists: n the maximum of the row minimums equals the minimum of the column maximums (Player A’s maximin equals Player B’s minimax) n the game is said to have a saddle point (the intersection of the optimal strategies) n the value of the saddle point is the value of the game n neither player can improve his/her outcome by changing strategies even if he/she learns in advance the opponent’s strategy

26. 26 Slide © 2008 Thomson South-Western. All Rights Reserved RowMinimum 1-3-2 CashRebate b 1 0%Loan b 3 FreeOptions b 2 Dealership B -3 3 -1 3 -2 0 3 -2 0 Cash Rebate a 1 Free Options a 2 0% Loan a 3 Dealership A Column Maximum 3 3 1 3 3 1 Pure Strategy Example n Saddle Point and Value of the Game 2 2 1 2 2 1 SaddlePoint Value of the game is 1

27. 27 Slide © 2008 Thomson South-Western. All Rights Reserved Pure Strategy Example n Pure Strategy Summary n Player A should choose Strategy a 1 (offer a cash rebate). n Player A can expect a gain of at least 1 vehicle sale per week. n Player B should choose Strategy b 3 (offer a 0% loan). n Player B can expect a loss of no more than 1 vehicle sale per week.

28. 28 Slide © 2008 Thomson South-Western. All Rights Reserved Mixed Strategy n If the maximin value for Player A does not equal the minimax value for Player B, then a pure strategy is not optimal for the game. n In this case, a mixed strategy is best. n With a mixed strategy, each player employs more than one strategy. n Each player should use one strategy some of the time and other strategies the rest of the time. n The optimal solution is the relative frequencies with which each player should use his possible strategies.

29. 29 Slide © 2008 Thomson South-Western. All Rights Reserved Mixed Strategy Example b1b1b1b1 b2b2b2b2 Player B 11 5 a1a1a2a2a1a1a2a2 Player A 4 8 4 8 n Consider the following two-person zero-sum game. The maximin does not equal the minimax. There is not an optimal pure strategy. ColumnMaximum 11 8 11 8 RowMinimum 4 5 Maximin Minimax

30. 30 Slide © 2008 Thomson South-Western. All Rights Reserved Mixed Strategy Example p = the probability Player A selects strategy a 1 (1  p ) = the probability Player A selects strategy a 2 If Player B selects b 1 : EV = 4 p + 11(1 – p ) If Player B selects b 2 : EV = 8 p + 5(1 – p )

31. 31 Slide © 2008 Thomson South-Western. All Rights Reserved Mixed Strategy Example 4 p + 11(1 – p ) = 8 p + 5(1 – p ) To solve for the optimal probabilities for Player A we set the two expected values equal and solve for the value of p. 4 p + 11 – 11 p = 8 p + 5 – 5 p 11 – 7 p = 5 + 3 p -10 p = -6 p =.6 Player A should select: Strategy a 1 with a.6 probability and Strategy a 1 with a.6 probability and Strategy a 2 with a.4 probability. Strategy a 2 with a.4 probability. Hence, (1  p ) =.4

32. 32 Slide © 2008 Thomson South-Western. All Rights Reserved Mixed Strategy Example q = the probability Player B selects strategy b 1 (1  q ) = the probability Player B selects strategy b 2 If Player A selects a 1 : EV = 4 q + 8(1 – q ) If Player A selects a 2 : EV = 11 q + 5(1 – q )

33. 33 Slide © 2008 Thomson South-Western. All Rights Reserved Mixed Strategy Example n Value of the Game For Player A: EV = 4 p + 11(1 – p ) = 4(.6) + 11(.4) = 6.8 For Player B: EV = 4 q + 8(1 – q ) = 4(.3) + 8(.7) = 6.8 Expected gain per game for Player A Expected loss per game for Player B

34. 34 Slide © 2008 Thomson South-Western. All Rights Reserved End of Chapter 5