Slides by JOHN LOUCKS St. Edward’s University

Slides by JOHN LOUCKS St. Edward’s University INTRODUCTION TO MANAGEMENT SCIENCE, 13e Anderson Sweeney Williams Martin

Chapter 5 Advanced Linear Programming Applications • Data Envelopment Analysis • Revenue Management • Portfolio Models and Asset Allocation • Game Theory

Data Envelopment Analysis • Data envelopment analysis (DEA) is an LP application used to determine the relative operating efficiency of units with the same goals and objectives. • DEA creates a fictitious composite unit made up of an optimal weighted average (W1, W2,…) of existing units. • An individual unit, k, can be compared by determining E, the fraction of unit k’s input resources required by the optimal composite unit. • If E < 1, unit k is less efficient than the composite unit and be deemed relatively inefficient. • If E = 1, there is no evidence that unit k is inefficient, but one cannot conclude that k is absolutely efficient.

Data Envelopment Analysis • The DEA Model MIN E s.t.Sum of weights = 1 Weighted composite outputs >Unit k’s output (for each measured output) Weighted inputs <E [Unit k’s input] (for each measured input) E, weights >0 Question : Can we find a combination of units whose output is as much as k unit , but can reduce the input?

Data Envelopment Analysis • Input • Output

Data Envelopment Analysis • About which Hospital? • Maximizing or minimizing? • Constraints? How many? • Decision variables wg, wu, wc, ws : weights for General, University, County, and State hospitals E : Efficient measure for County hospital wg + wu + wc + ws = 1 Full time physician : 48.14wg + 34.62wu + 36.72wc+ 33.16ws >= 36.72 Medicare patients 285.2wg + 162.3wu + 275.7wc + 210.4ws <= 275.7E

Data Envelopment Analysis • Formulaiton

Data Envelopment Analysis • Output Variable Value Reduced Cost E 0.9052379 0.000000 WG 0.2122662 0.000000 WU 0.2604472 0.000000 WC 0.000000 0.9476212E-01 WS 0.5272867 0.000000 Row Slack or Surplus Dual Price 1 0.9052379 -1.000000 2 0.000000 0.2388859 3 0.000000 -0.1396455E-01 4 0.000000 -0.1373087E-01 5 1.615387 0.000000 6 37.02707 0.000000 7 35.82408 0.000000 8 174.4224 0.000000 9 0.000000 0.9606148E-02

Data Envelopment Analysis • 이해하기

Data Envelopment Analysis • General Hospital Variable Value Reduced Cost E 1.000000 0.000000 WG 1.000000 0.000000 WU 0.000000 0.4148155 WC 0.000000 1.784315 WS 0.000000 0.000000 Row Slack or Surplus Dual Price 1 1.000000 -1.000000 2 0.000000 0.000000 3 0.000000 0.000000 4 0.000000 -0.2096828E-01 5 0.000000 -0.3805019E-03 6 0.000000 0.000000 7 0.000000 0.000000 8 0.000000 0.8077544E-02 9 0.000000 0.000000 • General Hospital Min = E; wg + wu + wc + ws = 1; 48.14*wg + 34.62*wu + 36.72*wc + 33.16*ws >= 48.14; 43.10*wg + 27.11*wu + 45.98*wc + 56.46*ws >= 43.10; 253*wg + 148*wu + 175*wc + 160*ws >= 253; 41*wg + 27*wu + 23*wc + 84*ws >= 41; 285.2*wg + 162.3*wu + 275.7*wc + 210.4*ws - 285.2*E <= 0; 123.8*wg + 128.7*wu + 348.5*wc + 154.1*ws - 123.8*E <= 0; 106.72*wg+ 64.21*wu + 104.1*wc + 104.04*ws- 106.72*E <= 0;

Data Envelopment Analysis • Output 값이 최고든지 input값이 최소이면 E=1

Data Envelopment Analysis • 문제점 Inefficient 한 unit을 찾아낼 수는 있는데 Efficient unit은 찾기가 어렵다. output이든 input이든 무엇 하나라도 제일 잘하면 (output measure가 최대이거나 input measure가 최소) 설사 다른 부분에서 매우 Inefficient해도 나타나지 않는다.

Fleight Reservation • Fleight legs

Fleight Reservation • Fares and Demand forcasts

Fleight Reservation • Maximizing or Minimizing? • Constraints? How many? • Decision Variables Pittsburg, Newark, Charlotte, Orlando, Myrtle Beach ODIF code : PCQ, PMQ, POQ, PCY, PMY, . . . • Objective function

Fleight Reservation • Constraints

Fleight Reservation • Output

Fleight Reservation • What is the soluion? • How much is the optimal revenue? • Two weeks earlier than the departure, PMQ( from Pittsburg to Myrtle Beach) reservation is 44.Can you reserve one more seat for PMQ when a customer wants to reserve ? dual prices for 1 & 4 are 4 and 179, it costs 183, but revenue increase is 228. Thus, 228 – 179 = 85. Yes. (read the last paragraph on p.231 about bid price)

Portfolio Model (p.233) • Five scenarios (5 previous returns, Year1, . . ., Year5)

Game theory (p.241) • Two-person, zero-sum game : 2 parties.gain of one party means the loss of the other. • Pay-off table gain of one party depending upon the strategies that two parties take. Pay-off table is known to both party. • Maximin strategy • Minmax regret strategy

End of Chapter 5

Slides by JOHN LOUCKS St. Edward’s University