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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.

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slide1

Slides by

JOHN

LOUCKS

St. Edward’s

University

INTRODUCTION TO MANAGEMENT SCIENCE, 13e

Anderson

Sweeney

Williams

Martin

chapter 5 advanced linear programming applications
Chapter 5 Advanced Linear Programming Applications
  • Data Envelopment Analysis
  • Revenue Management
  • Portfolio Models and Asset Allocation
  • Game Theory
data envelopment analysis
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 analysis1
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 analysis3
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 analysis5
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 analysis7
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 analysis8
Data Envelopment Analysis
  • Output 값이 최고든지 input값이 최소이면 E=1
data envelopment analysis9
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 reservation1
Fleight Reservation
  • Fares and Demand forcasts
fleight reservation2
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 reservation6
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)

slide20

Portfolio Model (p.233)

  • Five scenarios (5 previous returns, Year1, . . ., Year5)
slide21

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