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Slides by John Loucks St. Edward’s University

Slides by John Loucks St. Edward’s University. Agenda. Some Review from Last Class Data Envelopment Analysis Revenue Management Game Theory Concepts. Chapter 5 Advanced Linear Programming Applications. Data Envelopment Analysis Compares one unit to similar others

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Slides by John Loucks St. Edward’s University

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  1. Slides by John Loucks St. Edward’s University

  2. Agenda • Some Review from Last Class • Data Envelopment Analysis • Revenue Management • Game Theory Concepts

  3. Chapter 5 Advanced Linear Programming Applications • Data Envelopment Analysis • Compares one unit to similar others • Ie branch of a bank, franchise of a chain • Revenue Management • Maximize revenue with a fixed inventory • Portfolio Models and Asset Allocation • Determine best portfolio composition • Game Theory • Competition with a zero sum

  4. Data Envelopment Analysis • Data envelopment analysis (DEA): used to determine the relative operating efficiency of units with the same goals and objectives. • DEA creates a hypothetical composite • optimal weighted average (W1, W2,…) of existing units. • E – Efficiency Index • Allows comparison between composite and unit • “what the outputs of the composite would be, given the units inputs” • If E < 1, unit is less efficient than the composite unit If E = 1, there is no evidence that unit k is inefficient.

  5. Data Envelopment Analysis • The DEA Model MIN E s.t.OUTPUTS INPUTS Sum of weights = 1 E, weights > 0

  6. Data Envelopment Analysis 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. Outputs: performances on SAT scores, the number of seniors finishing high school the number of students who enter college Inputs number of teachers teaching senior classes the prorated budget for senior instruction number of students in the senior class.

  7. Data Envelopment Analysis • Input Roosevelt1Lincoln2Washington3 Senior Faculty 37 25 23 Budget ($100,000's) 6.4 5.0 4.7 Senior Enrollments 850 700 600

  8. Data Envelopment Analysis • Output Roosevelt1Lincoln2Washington3 Average SAT Score 800 830 900 High School Graduates 450 500 400 College Admissions 140 250 370

  9. Data Envelopment Analysis • Define the Decision Variables E = Fraction of Roosevelt's input resources required by the composite high school w1 = Weight applied to Roosevelt's input/output resources by the composite high school w2 = Weight applied to Lincoln’s input/output resources by the composite high school w3 = Weight applied to Washington's input/output resources by the composite high school

  10. Data Envelopment Analysis • Define the Objective Function Since our objective is to DETECT INEFFICIENCIES, we want to minimize the fraction of Roosevelt High School's input resources required by the composite high school: MIN E

  11. Data Envelopment Analysis • Define the Constraints Sum of the Weights is 1: (1) w1 + w2 + w3 = 1 • Output Constraints • General form for each output: • output for composite >= output for Roosevelt • Output for composite = • (Output for Roosevelt * weight for Roosevelt ) +(output for Lincoln * weight for Lincoln ) + (output for Washington * weight for Washington ) +

  12. Data Envelopment Analysis Output Constraints: Since w1 = 1 is possible, each output of the composite school must be at least as great as that of Roosevelt: (2) 800w1 + 830w2 + 900w3> 800 (SAT Scores) (3) 450w1 + 500w2 + 400w3> 450 (Graduates) (4) 140w1 + 250w2 + 370w3> 140 (College Admissions)

  13. Data Envelopment Analysis • Input Constraints • General Form • Input for composite <= input for Roosevelt * E • Input for composite = • (Input for Roosevelt * Input for Roosevelt ) +(Input for Lincoln * Input for Lincoln ) + (Input for Washington * Input for Washington ) (5) 37w1 + 25w2 + 23w3< 37E (Faculty) (6) 6.4w1 + 5.0w2 + 4.7w3< 6.4E (Budget) (7) 850w1 + 700w2 + 600w3< 850E (Seniors) Nonnegativity: E, w1, w2, w3> 0

  14. Data Envelopment Analysis • MIN E • ST • (1) w1+ w2 + w3 = 1 (2) 800w1+ 830w2 + 900w3> 800 (SAT Scores) (3) 450w1+ 500w2 + 400w3> 450 (Graduates) (4) 140w1+ 250w2 + 370w3> 140 (College Admissions) (5) 37w1+ 25w2 + 23w3< 37E (Faculty) (6) 6.4w1+ 5.0w2 + 4.7w3< 6.4E (Budget) (7) 850w1 + 700w2 + 600w3< 850E (Seniors) (8) E, w1, w2, w3> 0

  15. Data Envelopment Analysis • Computer Solution OBJECTIVE FUNCTION VALUE = 0.765 VARIABLEVALUE REDUCED COSTS E 0.765 0.000 W1 (R) 0.000 0.235 W2 (L) 0.500 0.000 W3 (W) 0.500 0.000 *Composite is 50% Lincoln, 50% Washington *Roosevelt is no more than 76.5% efficient as composite

  16. Data Envelopment Analysis • Computer Solution (continued) CONSTRAINTSLACK/SURPLUSDUAL VALUES 1 0.000 -0.235 2 (SAT) 65.000 0.000 3 (grads) 0.000 -0.001 4 (college) 170.000 0.000 5 (fac) 4.294 0.000 6 (budget) 0.044 0.000 7 (seniors) 0.000 0.001 Zero Slack – Roosevelt is 76.5% efficient in this area (ie grads) Positive slack – Roosevelt is LESS THAN 76.5% efficient (ie SAT) ie SAT scores are 65 points higher in the composite school

  17. Revenue Management • Another LP application is revenue management. • Revenue managementmanaging the short-term demand for a fixed perishable inventory in order to maximize revenue potential. • first used to determine how many airline seats to sell at an early-reservation discount fare and many to sell at a full fare.

  18. Revenue Management • General Form • MAX (revenue per unit * units allocated) • ST • CAPACITY • DEMAND • NONNEGATIVE

  19. Revenue Management LeapFrog Airways provides passenger service for Indianapolis, Baltimore, Memphis, Austin, and Tampa. LeapFrog has two WB828 airplanes, one based in Indianapolis and the other in Baltimore. Each morning the Indianapolis based plane flies to Austin with a stopover in Memphis. The Baltimore based plane flies to Tampa with a stopover in Memphis. Both planes have a coach section with a 120-seat capacity.

  20. Revenue Management LeapFrog uses two fare classes: a discount fare D class and a full fare F class. Leapfrog’s products, each referred to as an origin destination itinerary fare (ODIF), are listed on the next slide with their fares and forecasted demand. LeapFrog wants to determine how many seats it should allocate to each ODIF.

  21. IND BAL Each day a plane Leaves both IND And BAL for AUS and TAM Respectively. Both flights lay over In MEM No return flights (for simplicity) Each plane holds 120 Leg 1 Leg 2 MEM Leg 4 Leg 3 AUS TAM

  22. 8 different origin-destination combinations Plus two different fare classes: Discount and Full Fare 8 Orig-Desination combinations * 2 fare classes = 16 combinations

  23. Revenue Management Fare Class D D D F F F D D D F F F D D F F ODIF Code IMD IAD ITD IMF IAF ITF BMD BAD BTD BMF BAF BTF MAD MTD MAF MTF Fare 175 275 285 395 425 475 185 315 290 385 525 490 190 180 310 295 Demand 44 25 40 15 10 8 26 50 42 12 16 9 58 48 14 11 ODIF 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Origin Indianapolis Indianapolis Indianapolis Indianapolis Indianapolis Indianapolis Baltimore Baltimore Baltimore Baltimore Baltimore Baltimore Memphis Memphis Memphis Memphis Destination Memphis Austin Tampa Memphis Austin Tampa Memphis Austin Tampa Memphis Austin Tampa Austin Tampa Austin Tampa

  24. Revenue Management • Define the Decision Variables There are 16 variables, one for each ODIF: IMD = number of seats allocated to Indianapolis-Memphis- Discount class IAD = number of seats allocated to Indianapolis-Austin- Discount class ITD = number of seats allocated to Indianapolis-Tampa- Discount class IMF = number of seats allocated to Indianapolis-Memphis- Full Fare class IAF = number of seats allocated to Indianapolis-Austin-Full Fare class

  25. Revenue Management • Define the Decision Variables (continued) ITF = number of seats allocated to Indianapolis-Tampa- Full Fare class BMD = number of seats allocated to Baltimore-Memphis- Discount class BAD = number of seats allocated to Baltimore-Austin- Discount class BTD = number of seats allocated to Baltimore-Tampa- Discount class BMF = number of seats allocated to Baltimore-Memphis- Full Fare class BAF = number of seats allocated to Baltimore-Austin- Full Fare class

  26. Revenue Management • Define the Decision Variables (continued) BTF = number of seats allocated to Baltimore-Tampa- Full Fare class MAD = number of seats allocated to Memphis-Austin- Discount class MTD = number of seats allocated to Memphis-Tampa- Discount class MAF = number of seats allocated to Memphis-Austin- Full Fare class MTF = number of seats allocated to Memphis-Tampa- Full Fare class

  27. Revenue Management • Define the Objective Function Maximize total revenue: Max (fare per seat for each ODIF) x (number of seats allocated to the ODIF) Max 175IMD + 275IAD + 285ITD + 395IMF + 425IAF + 475ITF + 185BMD + 315BAD + 290BTD + 385BMF + 525BAF + 490BTF + 190MAD + 180MTD + 310MAF + 295MTF

  28. Revenue Management • Define the Constraints There are 4 capacity constraints, one for each flight leg: Indianapolis-Memphis leg (1)IMD + IAD + ITD + IMF + IAF + ITF < 120 Baltimore-Memphis leg (2)BMD + BAD + BTD + BMF + BAF + BTF < 120 Memphis-Austin leg (3)IAD + IAF + BAD + BAF + MAD + MAF < 120 Memphis-Tampa leg (4)ITD + ITF + BTD + BTF + MTD + MTF < 120

  29. Revenue Management • Define the Constraints (continued) Demand Constraints Limit the amount of seats for each ODIF There are 16 demand constraints, one for each ODIF: (5) IMD < 44 (11) BMD < 26 (17) MAD <58 (6) IAD < 25 (12) BAD < 50 (18) MTD < 48 (7) ITD < 40 (13) BTD < 42 (19) MAF < 14 (8) IMF < 15 (14) BMF < 12 (20) MTF < 11 (9) IAF < 10 (15) BAF < 16 (10) ITF < 8 (16) BTF < 9

  30. Revenue Management Max 175IMD + 275IAD + 285ITD + 395IMF + 425IAF + 475ITF + 185BMD + 315BAD + 290BTD + 385BMF + 525BAF + 490BTF + 190MAD + 180MTD + 310MAF + 295MTF ST: IMD + IAD + ITD + IMF + IAF + ITF <120BMD + BAD + BTD + BMF + BAF + BTF <120IAD + IAF + BAD + BAF + MAD + MAF <120ITD + ITF + BTD + BTF + MTD + MTF <120 IMD <44, BMD <26, MAD <58, IAD <25, BAD <50MTD <48, ITD <40, BTD <42, MAF <14, IMF <15BMF <12, MTF <11, IAF <10, BAF <16, ITF <8BTF <9 IMD, IAD, ITD, IMF, IAF, ITF, BMD, BAD, BTD, BMF, BAF, BTF, MAD, MTD, MAF, MTF > 0

  31. Revenue Management • Computer Solution • Revenue Contribution is $96265

  32. Revenue Management • Computer Solution (continued) • IMD dual value is 90 • IMF dual value is 310

  33. Introduction to Game Theory • In decision analysis, a single decision maker seeks to select an optimal alternative. • In game theory, there are two or more decision makers, called players, who compete as adversaries against each other. • 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. • Each player selects a strategy independently without knowing in advance the strategy of the other player(s). continue

  34. Introduction to Game Theory • The combination of the competing strategies provides the value of the game to the players. • Examples of competing players are teams, armies, companies, political candidates, and contract bidders.

  35. Two-Person Zero-Sum Game • Two-person means there are two competing players in the game. • Zero-sum means the gain (or loss) for one player is equal to the corresponding loss (or gain) for the other player. • The gain and loss balance out so that there is a zero-sum for the game. • What one player wins, the other player loses.

  36. Two-Person Zero-Sum Game Example • 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.

  37. Two-Person Zero-Sum Game Example • Strategy Choices Strategy 1: Offer a cash rebate on a new vehicle. Strategy 2: Offer free optional equipmenton a new vehicle. Strategy 3: Offer a 0% loan on a new vehicle.

  38. Two-Person Zero-Sum Game Example • Payoff Table: Number of Vehicle Sales Gained Per Week by Dealership A (or Lost Per Week by Dealership B) Dealership B Cash Rebate b1 Free Options b2 0% Loan b3 Dealership A Cash Rebate a1 Free Options a2 0% Loan a3 2 2 1 -3 3 -1 3 -2 0

  39. Two-Person Zero-Sum Game • Step 1: Identify the minimum payoff for each row (for Player A). • Step 2: For Player A, select the strategy that provides the maximum of the row minimums (called the maximin).

  40. Two-Person Zero-Sum Game Example • Identifying Maximin and Best Strategy Dealership B Cash Rebate b1 Free Options b2 0% Loan b3 Row Minimum Dealership A Cash Rebate a1 Free Options a2 0% Loan a3 1 -3 -2 2 2 1 -3 3 -1 3 -2 0 Best Strategy For Player A Maximin Payoff

  41. Two-Person Zero-Sum Game • Step 3: Identify the maximum payoff for each column (for Player B). • Step 4: For Player B, select the strategy that provides the minimum of the column maximums (called the minimax).

  42. Two-Person Zero-Sum Game Example • Identifying Minimax and Best Strategy Dealership B Best Strategy For Player B Cash Rebate b1 Free Options b2 0% Loan b3 Dealership A Cash Rebate a1 Free Options a2 0% Loan a3 2 2 1 -3 3 -1 Minimax Payoff 3 -2 0 3 3 1 Column Maximum

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

  44. Pure Strategy Example • Saddle Point and Value of the Game Dealership B Value of the game is 1 Cash Rebate b1 Free Options b2 0% Loan b3 Row Minimum Dealership A Cash Rebate a1 Free Options a2 0% Loan a3 1 -3 -2 2 2 1 -3 3 -1 3 -2 0 3 3 1 Column Maximum Saddle Point

  45. Pure Strategy Example • Pure Strategy Summary • Player A should choose Strategy a1 (offer a cash rebate). • Player A can expect a gain of at least 1 vehicle sale per week. • Player B should choose Strategy b3 (offer a 0% loan). • Player B can expect a loss of no more than 1 vehicle sale per week.

  46. Mixed Strategy • 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. • In this case, a mixed strategy is best. • With a mixed strategy, each player employs more than one strategy. • Each player should use one strategy some of the time and other strategies the rest of the time. • The optimal solution is the relative frequencies with which each player should use his possible strategies.

  47. Mixed Strategy Example • Consider the following two-person zero-sum game. The maximin does not equal the minimax. There is not an optimal pure strategy. Player B Row Minimum b1 b2 Player A Maximin 4 5 a1 a2 4 8 11 5 Column Maximum 11 8 Minimax

  48. Mixed Strategy Example p = the probability Player A selects strategy a1 (1 -p) = the probability Player A selects strategy a2 If Player B selects b1: EV = 4p + 11(1 – p) If Player B selects b2: EV = 8p + 5(1 – p)

  49. Mixed Strategy Example To solve for the optimal probabilities for Player A we set the two expected values equal and solve for the value of p. 4p + 11(1 – p) = 8p + 5(1 – p) 4p + 11 – 11p = 8p + 5 – 5p 11 – 7p = 5 + 3p Hence, (1 - p) = .4 -10p = -6 p = .6 Player A should select: Strategy a1 with a .6 probability and Strategy a2 with a .4 probability.

  50. Mixed Strategy Example q = the probability Player B selects strategy b1 (1 -q) = the probability Player B selects strategy b2 If Player A selects a1: EV = 4q + 8(1 – q) If Player A selects a2: EV = 11q + 5(1 – q)

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