Chapter 5

Chapter 5

Slides by John Loucks St. Edwards University 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 1 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Agenda Some Review from Last Class Data Envelopment Analysis Revenue Management Game Theory Concepts 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 2 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide

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 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 3 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 4 thator unit inefficient. or duplicated, posted k to is a publicly accessible website, in whole or in part. Slide Data Envelopment Analysis

The DEA Model MIN E s.t. OUTPUTS INPUTS Sum of weights = 1 E, weights > 0 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 5 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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 theMay senior class. copied 2011 Cengage Learning. All Rights Reserved. not be scanned, 6 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Data Envelopment Analysis Input Roosevelt1 Lincoln2 Washington3 Senior Faculty 23

Budget ($100,000's) 4.7 Senior Enrollments 600 37 6.4 850 25 5.0 700 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 7 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Data Envelopment Analysis Output Roosevelt1 Lincoln2 Washington3 Average SAT Score 800 830 900

High School Graduates 450 500 400 College Admissions 140 250 370 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 8 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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 Lincolns input/output resources by the composite high school w3 = Weight applied to Washington's input/output

resources by the composite high school 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 9 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 10 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 11 Washington ) + or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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) 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 12 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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) 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 13

(6) 6.4w1 + 5.0w 4.7w < 6.4E 2 + or duplicated, or posted to a publicly accessible website, in 3whole or in part. Slide 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

Cengage Learning. All Rights Reserved. May not be scanned, copied 14 (Seniors) 2011 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Data Envelopment Analysis Computer Solution OBJECTIVE FUNCTION VALUE = VARIABLE VALUE E 0.765 W1 (R) 0.000 W2 (L) 0.500 W3 (W) 0.500 0.765 REDUCED COSTS

0.000 0.235 0.000 0.000 *Composite is 50% Lincoln, 50% Washington *Roosevelt is no more than 76.5% efficient as composite 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 15 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Data Envelopment Analysis Computer Solution (continued) CONSTRAINT SLACK/SURPLUS DUAL 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 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 16 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Revenue Management

Another LP application is revenue management. Revenue management managing the shortterm 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. 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 17 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Revenue Management General Form MAX (revenue per unit * units allocated) ST

CAPACITY DEMAND NONNEGATIVE 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 18 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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. 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 19 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Revenue Management

LeapFrog uses two fare classes: a discount fare D class and a full fare F class. Leapfrogs 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. 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 20 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide IND BAL Leg 1 Leg 2 MEM Leg 3 AUS

Each day a plane Leaves both IND And BAL for AUS and TAM Respectively. Both flights lay over In MEM Leg 4 TAM No return flights (for simplicity) Each plane holds 120 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 21 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 8 different origin-destination combinations Orig Dest

IND MEM IND AUS IND TAM BAL MEM BAL AUS BAL TAM MEM

AUS MEM TAM Plus two different fare classes: Discount and Full Fare 8 Orig-Desination combinations * 2 fare classes = 16 combinations 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 22 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Revenue Management Fare ODIF ODIF Origin Destination Class Code Fare Deman D IMD 175 d 1 Indianapoli Memphis D IAD 275 44

2 s Austin D ITD 285 25 3 Indianapoli Tampa F IMF 395 40 4 s Memphis F IAF 425 15 5 Indianapoli Austin F ITF

475 10 6 s Tampa D BMD 185 8 7 Indianapoli Memphis D BAD 315 26 8 s Austin D BTD 290 50 9 Indianapoli Tampa F BMF 385 42 10

s Memphis F BAF 525 12 11 Indianapoli Austin F BTF 490 16 12 s Tampa D MAD 190 9 13 Baltimore Austin D MTD 180 58 14 Baltimore Tampa F

MAF 310 48 15 Baltimore Austin F MTF 295 14 16 Baltimore Tampa 11 Baltimore 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 23 Baltimore or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Memphis Revenue Management Define the Decision Variables There are 16 variables, one for each ODIF: IMD = number of seats allocated to IndianapolisMemphisDiscount class

IAD = number of seats allocated to IndianapolisAustin- Discount class ITD = number of seats allocated to IndianapolisTampa- Discount class IMF = number of seats allocated to IndianapolisMemphis- Full Fare class IAF = number of seats allocated to IndianapolisAustin-Full Fare class 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 24 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Revenue Management Define the Decision Variables (continued) ITF = number of seats allocated to IndianapolisTampaFull Fare class BMD = number of seats allocated to BaltimoreMemphisDiscount class BAD = number of seats allocated to BaltimoreAustinDiscount class BTD = number of seats allocated to BaltimoreTampaDiscount class BMF = number of seats allocated to BaltimoreMemphis 2011 CengageFull Learning. All class Rights Reserved. May not be scanned, copied 25 Fare or duplicated, or posted to a publicly accessible website, in whole or in part. Slide

Revenue Management Define the Decision Variables (continued) BTF = number of seats allocated to BaltimoreTampaFull Fare class MAD = number of seats allocated to MemphisAustinDiscount class MTD = number of seats allocated to MemphisTampaDiscount class MAF = number of seats allocated to MemphisAustinFull Fare class MTF = number of seats allocated to MemphisTampa 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 26 Full Fare class or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 27 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 28 or duplicated, or posted to a publicly accessible website, in whole or in part.

Slide 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 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 29 or duplicated, posted a publicly accessible (9) orIAF < to10(15) BAF

16 in whole or in part.Slide 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 < 120 BMD + BAD + BTD + BMF + BAF + BTF < 120 IAD + IAF + BAD + BAF + MAD + MAF < 120 ITD + ITF + BTD + BTF + MTD + MTF < 120 IMD < 44, BMD < 26, MAD < 58, IAD < 25, BAD < 50 2011 Cengage All Rights Reserved. not beMAF scanned, MTD

< copied 14, IMF 30 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Revenue Management Computer Solution Revenue Contribution is $96265 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 31 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Revenue Management Computer Solution (continued) - IMD dual value is

90 - IMF dual value is 310 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 32 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 33 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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. 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 34 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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. 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 35 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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. 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 36 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Two-Person Zero-Sum Game Example Strategy Choices Strategy 1: Offer a cash rebate on a new vehicle. Strategy 2: Offer free optional equipment on a new vehicle. Strategy 3: Offer a 0% loan on a new vehicle. 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 37

or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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 Free 0% Rebate Options Loan b1 b2 b3 Dealership A Cash Rebate Free Options 0% Loan a1

a2 a3 2 -3 3 1 -1 0 2 3 -2 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 38 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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). 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 39 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Two-Person Zero-Sum Game Example Identifying Maximin and Best Strategy Dealership B Cash Free 0% Rebate Options Loan Row b1 b2 b3 Minimum

Dealership A Cash Rebate Free Options 0% Loan a1 a2 a3 2 -3 3 1 -1 Best 0 Strategy For Player A 2 3 -2 1 -3

-2 Maximin Payoff 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 40 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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). 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 41 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide

Two-Person Zero-Sum Game Example Identifying Minimax and Best Strategy Dealership B Cash Free 0% Rebate Options Loan b1 b2 b3 Dealership A Cash Rebate Free Options a1 a2 0% Loan a3 Column Maximum

2 -3 3 3 1 -1 0 2 3 -2 3 Best Strategy For Player B Minimax Payoff 1 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 42 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide

Pure Strategy Whenever an optimal pure strategy exists: the maximum of the row minimums equals the minimum of the column maximums (Player As maximin equals Player Bs 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 opponents strategy 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 43 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Pure Strategy Example Saddle Point and Value of the Game Dealership B

Cash Free 0% Rebate Options Loan Row b1 b2 b3 Minimum Dealership A Cash Rebate Free Options Value of the game is 1 a1 a2 0% Loan a3 Column Maximum 2

-3 3 3 1 -1 0 1 2 3 -2 3 1 -3 -2 Saddle Point 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 44 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Pure Strategy Example

Pure Strategy Summary Player A should choose Strategy a (offer a 1 cash rebate). Player A can expect a gain of at least 1 vehicle sale per week. Player B should choose Strategy b (offer a 3 0% loan). Player B can expect a loss of no more than 1 vehicle sale per week. 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 45 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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. 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 46 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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

Player A a1 a2 Column Maximum b1 4 11 8 5 11 8 b2 Row Minimum 4 Maximin 5 Minimax 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 47

or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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) 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 48 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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 Hence, 11 7p = 5 + 3p (1 - p) -10p = -6 = .4

p = .6 Player A should select: Strategy a1 with a .6 probability and Strategy a2 with a .4 probability. 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 49 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 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) 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 50 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide Mixed Strategy Example Value of the Game

Expected gain per game for Player A For Player A: EV = 4p + 11(1 p) = 4(.6) + 11(.4) = 6.8 For Player B: EV = 4q + 8(1 q) = 4(.3) + 8(.7) = Expected loss per game 6.8for Player B 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied 51 or duplicated, or posted to a publicly accessible website, in whole or in part. Slide

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