slides by john loucks st. edward’s university

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© 2008 Thomson South-Western. All Rights Reserved© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Slides byJOHN

LOUCKSSt. Edward’sUniversity

INTRODUCTION TO MANAGEMENT SCIENCE, 13e

AndersonSweeneyWilliams

Martin

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Chapter 5 Advanced Linear Programming

Applications Data Envelopment Analysis Revenue Management Portfolio Models and Asset Allocation Game Theory

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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.

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Data Envelopment Analysis The DEA Model

MIN Es.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?

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Input

Output

Data Envelopment Analysis

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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 = 1Full time physician : 48.14wg + 34.62wu + 36.72wc+ 33.16ws >= 36.72Medicare patients 285.2wg + 162.3wu + 275.7wc + 210.4ws <= 275.7E


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Formulaiton


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OutputVariable 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


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이해하기Data Envelopment AnalysisGeneral University County State0.212266 0.260447 0 0.527287

Medicare 48.14 34.62 36.72 33.16NonMedi 43.1 27.11 45.98 56.46Nurses 253 148 175 160Interns 41 27 23 84

10.21849 9.016682 0 17.48483 36.729.148673 7.060724 0 29.77061 45.9853.70335 38.54619 0 84.36587 176.61548.702914 7.032074 0 44.29208 60.02707

General University County State

FTE 285.2 162.3 275.7 210.4 249.5741Supply 123.8 128.7 348.5 154.1 315.4754Beddays 106.72 64.21 104.1 104.04 94.23527

60.53832 42.27058 0 110.9411 213.75 0.77529926.27856 33.51955 0 81.25488 141.053 0.40474322.65305 16.72331 0 54.85891 94.23527 0.905238

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General HospitalMin = 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 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

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Output 값이 최고든지 input 값이 최소이면 E=1Data Envelopment Analysis

General University County State0 1 0 0

Medicare 48.14 34.62 36.72 33.16NonMedi 43.1 27.11 45.98 56.46Nurses 253 148 175 160Interns 41 27 23 84

0 34.62 0 0 34.620 27.11 0 0 27.110 148 0 0 1480 27 0 0 27

General University County State

FTE 285.2 162.3 275.7 210.4 162.3Supply 123.8 128.7 348.5 154.1 128.7Beddays 106.72 64.21 104.1 104.04 64.21

0 162.3 0 0 162.3 10 128.7 0 0 128.7 10 64.21 0 0 64.21 1

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문제점Inefficient 한 unit 을 찾아낼 수는 있는데 Efficient unit은 찾기가 어렵다 .output 이든 input 이든 무엇 하나라도 제일 잘하면 (output measure 가 최대이거나 input measure 가 최소 ) 설사 다른 부분에서 매우 Inefficient 해도 나타나지 않는다 .


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Fleight legs

Fleight Reservation

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Fares and Demand forcasts

Fleight Reservation

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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

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Constraints

Fleight Reservation

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Output

Fleight Reservation

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Output

Fleight Reservation

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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)

Fleight Reservation

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

Year5)

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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

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End of Chapter 5

slides by john loucks st. edward’s university

Documents

ewg wu wc ws

ws unit ks output

decision variables wg

cengage learning

accessible website

input e

general hospital

state hospitals e