Slides by JOHN LOUCKS St. Edward’s University

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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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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
• 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
• 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
• 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
• 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
• 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.

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