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11SlideSlide 2005 Thomson/South 2005 Thomson/South--WesternWestern

Slides Prepared bySlides Prepared by

JOHN S. LOUCKSJOHN S. LOUCKSST. EDWARDS UNIVERSITYST. EDWARDS UNIVERSITY

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Chapter 4Chapter 4Linear Programming ApplicationsLinear Programming Applications

Blending ProblemBlending Problem

Portfolio Planning ProblemPortfolio Planning Problem

Product Mix ProblemProduct Mix Problem

Transportation ProblemTransportation Problem

Data Envelopment AnalysisData Envelopment Analysis

Revenue ManagementRevenue Management

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Ferdinand Feed Company receives four rawFerdinand Feed Company receives four rawgrains from which it blends its dry pet food. The petgrains from which it blends its dry pet food. The pet

food advertises that each 8food advertises that each 8--ounce packetounce packet

meets the minimum daily requirementsmeets the minimum daily requirements

for vitamin C, protein and iron. Thefor vitamin C, protein and iron. Thecost of each raw grain as well as thecost of each raw grain as well as the

vitamin C, protein, and iron units pervitamin C, protein, and iron units per

pound of each grain are summarized onpound of each grain are summarized on

the next slide.the next slide.

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Vitamin C Protein IronVitamin C Protein IronGrain Units/lb Units/lb Units/lb Cost/lbGrain Units/lb Units/lb Units/lb Cost/lb

1 91 9 1212 0 .750 .75

2 162 16 1010 14 .9014 .903 83 8 1010 15 .8015 .80

4 104 10 88 7 .707 .70

Ferdinand is interested in producing the 8Ferdinand is interested in producing the 8--ounceounce

mixture at minimum cost while meeting the minimummixture at minimum cost while meeting the minimum

daily requirements of 6 units of vitamin C, 5 units ofdaily requirements of 6 units of vitamin C, 5 units of

protein, and 5 units of iron.protein, and 5 units of iron.

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Define the decision variablesDefine the decision variables

xxjj = the pounds of grain= the pounds of grainjj ((jj = 1,2,3,4)= 1,2,3,4)

used in the 8used in the 8--ounce mixtureounce mixture

Define the objective functionDefine the objective function

Minimize the total cost for an 8Minimize the total cost for an 8--ounce mixture:ounce mixture:

MIN .75MIN .75xx11 + .90+ .90xx22 + .80+ .80xx33 + .70+ .70xx44

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Define the constraintsDefine the constraintsTotal weight of the mix is 8Total weight of the mix is 8--ounces (.5 pounds):ounces (.5 pounds):

(1)(1) xx11 ++ xx22 ++ xx33 ++ xx44 = .5= .5

Total amount of Vitamin C in the mix is at least 6Total amount of Vitamin C in the mix is at least 6units:units:

(2) 9(2) 9xx11 + 16+ 16xx22 + 8+ 8xx33 + 10+ 10xx44 > 6> 6

Total amount of protein in the mix is at least 5 units:Total amount of protein in the mix is at least 5 units:

(3) 12(3) 12xx11 + 10+ 10xx22 + 10+ 10xx33 + 8+ 8xx44 > 5> 5

Total amount of iron in the mix is at least 5 units:Total amount of iron in the mix is at least 5 units:

(4) 14(4) 14xx22 + 15+ 15xx33 + 7+ 7xx44 > 5> 5Nonnegativity of variables:Nonnegativity of variables: xxjj >> 0 for all0 for alljj

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The Management ScientistThe Management Scientist OutputOutput

OBJECTIVE FUNCTION VALUE = 0.406OBJECTIVE FUNCTION VALUE = 0.406

VARIABLEVARIABLE VALUEVALUE REDUCED COSTSREDUCED COSTS

X1X1 0.0990.099 0.0000.000X2X2 0.2130.213 0.0000.000X3X3 0.0880.088 0.0000.000X4X4 0.0990.099 0.0000.000

Thus, the optimal blend is about .10 lb. of grain 1, .21 lb.Thus, the optimal blend is about .10 lb. of grain 1, .21 lb.

of grain 2, .09 lb. of grain 3, and .10 lb. of grain 4. Theof grain 2, .09 lb. of grain 3, and .10 lb. of grain 4. The

mixture costs Fredericks 40.6 cents.mixture costs Fredericks 40.6 cents.


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Winslow Savings has $20 million availableWinslow Savings has $20 million availablefor investment. It wishes to investfor investment. It wishes to invest

over the next four months in suchover the next four months in such

a way that it will maximize thea way that it will maximize the

total interest earned over the fourtotal interest earned over the fourmonth period as well as have at leastmonth period as well as have at least

$10 million available at the start of the fifth month for$10 million available at the start of the fifth month for

a high rise building venture in which it will bea high rise building venture in which it will be

participating.participating.

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For the time being, Winslow wishes to investFor the time being, Winslow wishes to investonly in 2only in 2--month government bonds (earning 2% overmonth government bonds (earning 2% over

the 2the 2--month period) and 3month period) and 3--month construction loansmonth construction loans

(earning 6% over the 3(earning 6% over the 3--month period). Each of thesemonth period). Each of these

is available each month for investment. Funds notis available each month for investment. Funds notinvested in these two investments are liquid and earninvested in these two investments are liquid and earn

3/4 of 1% per month when invested locally.3/4 of 1% per month when invested locally.

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Formulate a linear program that will helpFormulate a linear program that will helpWinslow Savings determine how to invest over theWinslow Savings determine how to invest over the

next four months if at no time does it wish to havenext four months if at no time does it wish to have

more than $8 million in either government bonds ormore than $8 million in either government bonds or

construction loans.construction loans.

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Define the decision variablesDefine the decision variables

ggjj = amount of new investment in= amount of new investment in

government bonds in monthgovernment bonds in month jj

ccjj=

amount of new investment in=

amount of new investment inconstruction loans in monthconstruction loans in month jj

lljj = amount invested locally in month= amount invested locally in month jj,,

wherewhere jj = 1,2,3,4= 1,2,3,4

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Define the objective functionDefine the objective functionMaximize total interest earned over the 4Maximize total interest earned over the 4--month period.month period.

MAX (interest rate on investment)(amount invested)MAX (interest rate on investment)(amount invested)

MAX .02MAX .02gg11 + .02+ .02gg22 + .02+ .02gg33 + .02+ .02gg44+ .06+ .06cc11 + .06+ .06cc22 + .06+ .06cc33 + .06+ .06cc44

+ .0075+ .0075ll11 + .0075+ .0075ll22 + .0075+ .0075ll33 + .0075+ .0075ll44

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Define the constraintsDefine the constraintsMonth 1's total investment limited to $20 million:Month 1's total investment limited to $20 million:

(1)(1) gg11 ++ cc11 ++ ll11 = 20,000,000= 20,000,000

Month 2's total investment limited to principle andMonth 2's total investment limited to principle andinterest invested locally in Month 1:interest invested locally in Month 1:

(2)(2) gg22 ++ cc22 ++ ll22 = 1.0075= 1.0075ll11oror gg22 ++ cc22 -- 1.00751.0075ll11 ++ ll22 = 0= 0

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Define the constraints (continued)Define the constraints (continued)Month 3's total investment amount limited toMonth 3's total investment amount limited toprinciple and interest invested in government bondsprinciple and interest invested in government bondsin Month 1 and locally invested in Month 2:in Month 1 and locally invested in Month 2:

(3)(3) gg33 ++ cc33 ++ ll33 = 1.02= 1.02gg11 + 1.0075+ 1.0075ll22oror -- 1.021.02gg11 ++gg33 ++ cc33 -- 1.00751.0075ll22 ++ ll33 = 0= 0

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Define the constraints (continued)Define the constraints (continued)Month 4's total investment limited to principle andMonth 4's total investment limited to principle andinterest invested in construction loans in Month 1,interest invested in construction loans in Month 1,goverment bonds in Month 2, and locally invested ingoverment bonds in Month 2, and locally invested in

Month 3:Month 3:(4)(4) gg44 ++ cc44 ++ ll44 = 1.06= 1.06cc11 + 1.02+ 1.02gg22 + 1.0075+ 1.0075ll33oror -- 1.021.02gg22 ++gg44 -- 1.061.06cc11 ++ cc44 -- 1.00751.0075ll33 ++ ll44 = 0= 0

$10 million must be available at start of Month 5:$10 million must be available at start of Month 5:(5) 1.06(5) 1.06cc22 + 1.02+ 1.02gg33 + 1.0075+ 1.0075ll44 >> 10,000,00010,000,000

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Define the constraints (continued)Define the constraints (continued)No more than $8 million in government bonds at anyNo more than $8 million in government bonds at anytime:time:

(6)(6) gg11

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Define the constraints (continued)Define the constraints (continued)No more than $8 million in construction loans at anyNo more than $8 million in construction loans at anytime:time:

(10)(10) cc11

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Floataway Tours has $420,000 that can be usedFloataway Tours has $420,000 that can be usedto purchase new rental boats for hire during theto purchase new rental boats for hire during the

summer. The boats cansummer. The boats can

be purchased from twobe purchased from two

different manufacturers.different manufacturers.Floataway Tours wouldFloataway Tours would

like to purchase at least 50 boats and would like tolike to purchase at least 50 boats and would like to

purchase the same number from Sleekboat as frompurchase the same number from Sleekboat as from

Racer to maintain goodwill. At the same time,Racer to maintain goodwill. At the same time,Floataway Tours wishes to have a total seatingFloataway Tours wishes to have a total seating

capacity of at least 200.capacity of at least 200.

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Formulate this problem as a linear program.Formulate this problem as a linear program.

Maximum ExpectedMaximum Expected

Boat Builder Cost Seating Daily ProfitBoat Builder Cost Seating Daily Profit

Speedhawk Sleekboat $60003

$ 70Speedhawk Sleekboat $60003

$ 70Silverbird Sleekboat $7000 5 $ 80Silverbird Sleekboat $7000 5 $ 80

Catman Racer $5000 2 $ 50Catman Racer $5000 2 $ 50

Classy Racer $9000 6 $110Classy Racer $9000 6 $110


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Define the decision variablesDefine the decision variablesxx11 = number of Speedhawks ordered= number of Speedhawks ordered

xx22 = number of Silverbirds ordered= number of Silverbirds ordered

xx33= number of Catmans ordered= number of Catmans ordered

xx44

= number of Classys ordered= number of Classys ordered

Define the objective functionDefine the objective function

Maximize total expected daily profit:Maximize total expected daily profit:

Max: (Expected daily profit per unit)Max: (Expected daily profit per unit)

x (Number of units)x (Number of units)Max: 70Max: 70xx11 + 80+ 80xx22 + 50+ 50xx33 + 110+ 110xx44


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Define the constraintsDefine the constraints(1) Spend no more than $420,000:(1) Spend no more than $420,000:

60006000xx11 + 7000+ 7000xx22 + 5000+ 5000xx33 + 9000+ 9000xx44 > 5050(3) Number of boats from Sleekboat equals number(3) Number of boats from Sleekboat equals number

of boats from Racer:of boats from Racer:

xx11 ++ xx22 ==xx33 ++ xx44 oror xx11 ++ xx22 -- xx33 -- xx44 = 0= 0


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Define the constraints (continued)Define the constraints (continued)(4) Capacity at least 200:(4) Capacity at least 200:

33xx11 + 5+ 5xx22 + 2+ 2xx33 + 6+ 6xx44 >> 200200

Nonnegativity of variables:Nonnegativity of variables:

xxjj >> 0, for0, forjj = 1,2,3,4= 1,2,3,4


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Complete FormulationComplete Formulation

Max 70Max 70xx11 + 80+ 80xx22 + 50+ 50xx33 + 110+ 110xx44s.t.s.t.

60006000xx11

+ 7000+ 7000xx22

+ 5000+ 5000xx33

+ 9000+ 9000xx44

> 5050

xx11 ++ xx22 -- xx33 -- xx44 = 0= 0

33xx11 + 5+ 5xx22 + 2+ 2xx33 + 6+ 6xx44 >> 200200

xx11,, xx22,, xx33,, xx44 >> 00


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Partial Spreadsheet Showing Problem DataPartial Spreadsheet Showing Problem DataA B C D E F

1

2 Constr. X1 X2 X3 X4 RHS

3 #1 6 7 5 9 420

4 #2 1 1 1 1 50

5 #3 1 1 -1 -1 0

6 #4 3 5 2 6 200

7 Object. 70 80 50 110

LHS Coefficients


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Partial Spreadsheet Showing SolutionPartial Spreadsheet Showing SolutionA B C D E F

9

10 X1 X2 X3 X4

11 28 0 0 28

12

13 5040

14

15 LHS RHS

16 420.0 = 50

18 0.0 = 0

19 252.0 >= 200Min. Seating

Decision Variable Values

No. of Boats

Maximum Total Profit

Constraints

Spending Max.

Min. # Boats

Equal Sourcing


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The Management ScienceO

utputThe Management ScienceO

utput


VariableVariable ValueValue Reduced CostReduced Costxx11 28.000 0.00028.000 0.000

xx22 0.000 2.0000.000 2.000xx33 0.000 12.0000.000 12.000xx44 28.000 0.00028.000 0.000

ConstraintConstraint Slack/SurplusSlack/Surplus Dual PriceDual Price1 0.000 0.0121 0.000 0.0122 6.000 0.0002 6.000 0.0003 0.0003 0.000 --2.0002.0004 52.000 0.0004 52.000 0.000


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Solution SummarySolution Summary Purchase 28 Speedhawks from Sleekboat.Purchase 28 Speedhawks from Sleekboat. Purchase 28 Classys from Racer.Purchase 28 Classys from Racer. Total expected daily profit is $5,040.00.Total expected daily profit is $5,040.00.

The minimum number of boats was exceeded by 6The minimum number of boats was exceeded by 6(surplus for constraint #2).(surplus for constraint #2). The minimum seating capacity was exceeded by 52The minimum seating capacity was exceeded by 52

(surplus for constraint #4).(surplus for constraint #4).


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Sensitivity ReportSensitivity Report

Adjustable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$D$12 X1 28 0 70 45 1.875

$E$12 X2 0 -2 80 2 1E+30

$F$12 X3 0 -12 50 12 1E+30

$G$12 X4 28 0 110 1E+30 16.36363636


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Sensitivity ReportSensitivity Report

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease

$E$17 #1 420.0 12.0 420 1E+30 45$E$18 #2 56.0 0.0 50 6 1E+30

$E$19 #3 0.0 -2.0 0 70 30

$E$20 #4 252.0 0.0 200 52 1E+30


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The Navy has 9,000 pounds of material in Albany,The Navy has 9,000 pounds of material in Albany,Georgia that it wishes to ship to three installations:Georgia that it wishes to ship to three installations:

San Diego, Norfolk, and Pensacola. TheySan Diego, Norfolk, and Pensacola. They

require 4,000, 2,500, and 2,500 pounds,require 4,000, 2,500, and 2,500 pounds,

respectively. Government regulationsrespectively. Government regulationsrequire equal distribution of shippingrequire equal distribution of shipping

among the three carriers.among the three carriers.

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The shipping costs per pound forThe shipping costs per pound for truck, railroad,truck, railroad,and airplane transit are shown on the next slide.and airplane transit are shown on the next slide.

Formulate and solve a linear program toFormulate and solve a linear program to

determine the shipping arrangementsdetermine the shipping arrangements

(mode, destination, and quantity) that(mode, destination, and quantity) thatwill minimize the total shipping cost.will minimize the total shipping cost.

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DestinationDestination

ModeMode San Diego Norfolk PensacolaSan Diego Norfolk Pensacola

TruckTruck $12 $ 6 $ 5$12 $ 6 $ 5

RailroadRailroad 20 11 920 11 9

AirplaneAirplane 30 26 2830 26 28


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Define the Decision VariablesDefine the Decision VariablesWe want to determine the pounds of material,We want to determine the pounds of material, xxijij,,to be shipped by modeto be shipped by mode ii to destinationto destinationjj. The. Thefollowing table summarizes the decision variables:following table summarizes the decision variables:

San Diego Norfolk PensacolaSan Diego Norfolk Pensacola

TruckTruck xx1111 xx1212 xx1313RailroadRailroad xx2121 xx2222 xx2323AirplaneAirplane xx3

13

1xx3

23

2xx3333


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Define theO

bjective FunctionDefine theO

bjective FunctionMinimize the total shipping cost.Minimize the total shipping cost.

Min: (shipping cost per pound for each mode perMin: (shipping cost per pound for each mode perdestination pairing) x (number of pounds shippeddestination pairing) x (number of pounds shipped

by mode per destination pairing).by mode per destination pairing).

Min: 12Min: 12xx1111 + 6+ 6xx1212 + 5+ 5xx1313 + 20+ 20xx2121 + 11+ 11xx2222 + 9+ 9xx2323+ 30+ 30xx3131 + 26+ 26xx3232 + 28+ 28xx3333


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Define the ConstraintsDefine the ConstraintsEqual use of transportation modes:Equal use of transportation modes:

(1)(1) xx1111 ++ xx1212 ++ xx1313 =3000=3000

(2)(2) xx2121 ++ xx2222 ++ xx2323 =3000=3000

(3

)(3

) xx3131 ++ xx3232 ++ xx3333=

3

000=

3

000Destination material requirements:Destination material requirements:

(4)(4) xx1111 ++ xx2121 ++ xx3131 = 4000= 4000

(5)(5) xx1212 ++ xx2222 ++ xx3232 = 2500= 2500

(6)(6) xx1313 ++ xx2323 ++ xx3333 = 2500= 2500Nonnegativity of variables:Nonnegativity of variables:

xxijij >> 0,0, ii = 1,2,3 and= 1,2,3 and jj = 1,2,3= 1,2,3


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Partial Spreadsheet Showing Problem DataPartial Spreadsheet Showing Problem DataA B C D E F G H I J K

1

2 Con. X11 X12 X13 X21 X22 X23 X31 X32 X33 RHS

3 #1 1 1 1 3000

4 #2 1 1 1 3000

5 #3 1 1 1 3000

6 #4 1 1 1 4000

7 #5 1 1 1 2500

8 #61 1 1

2500

9 Obj. 12 6 5 20 11 9 30 26 28

LHS Coefficients


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Partial Spreadsheet Showing SolutionPartial Spreadsheet Showing SolutionA B C D E F G H I J K

12 X11 X12 X13 X21 X22 X23 X31 X32 X33

13 1000 2000 0 0 500 2500 3000 0 0

14

15

16 LHS RHS

17 3000 = 3000

18 3000 = 3000

19 3000 = 3000

20 4000 = 4000

21 2500 = 2500

22 2500 = 2500

Constraints

Truc

Rail

Minimized Total Shipping Cost 142000

Nor

Pen

San

Air


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The Management Scientist OutputThe Management Scientist Output


VariableVariable ValueValue Reduced CostReduced Costxx1111 1000.000 0.0001000.000 0.000

xx1212 2000.000 0.0002000.000 0.000xx1313 0.000 1.0000.000 1.000xx2121 0.000 3.0000.000 3.000xx2222 500.000 0.000500.000 0.000xx2323 2500.000 0.0002500.000 0.000

xx3131 3000.000 0.0003000.000 0.000xx3232 0.000 2.0000.000 2.000xx3333 0.000 6.0000.000 6.000


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Solution SummarySolution Summary San Diego will receive 1000 lbs. by truckSan Diego will receive 1000 lbs. by truckand 3000 lbs. by airplane.and 3000 lbs. by airplane.

Norfolk will receive 2000 lbs. by truckNorfolk will receive 2000 lbs. by truck

and 500 lbs. by railroad.and 500 lbs. by railroad. Pensacola will receive 2500 lbs. by railroad.Pensacola will receive 2500 lbs. by railroad. The total shipping cost will be $142,000.The total shipping cost will be $142,000.


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Data envelopment analysisData envelopment analysis (DEA) is an LP application(DEA) is an LP applicationused to determine the relative operating efficiency ofused to determine the relative operating efficiency ofunits with the same goals and objectives.units with the same goals and objectives.

DEA creates aDEA creates a fictitious composite unitfictitious composite unit made up of anmade up of anoptimal weighted average (optimal weighted average (WW11,, WW22,) of existing units.,) of existing units.

An individual unit,An individual unit, kk, can be compared by determining, can be compared by determiningEE, the fraction of unit, the fraction of unit kks input resources required bys input resources required bythe optimal composite unit.the optimal composite unit.

IfIf EE < 1, unit< 1, unit kk is less efficient than the composite unitis less efficient than the composite unit

and be deemed relatively inefficient.and be deemed relatively inefficient. IfIf EE = 1, there is no evidence that unit= 1, there is no evidence that unit kk is inefficient, butis inefficient, but

one cannot conclude thatone cannot conclude that kk is absolutely efficient.is absolutely efficient.

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The DEA ModelThe DEA Model

MINMIN EE

s.t.s.t. Weighted outputsWeighted outputs >> UnitUnit kks outputs output

(for each measured output)(for each measured output)Weighted inputsWeighted inputs > 00

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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.

The district is evaluating

performances on SAT scores, the

number of seniors finishing high

school, and the number of students

who enter college as a function of the

number of teachers teaching seniorclasses, the prorated budget for senior instruction,

and the number of students in the senior class.


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Input

Roosevelt Lincoln Washington

Senior Faculty 37 25 23

Budget ($100,000's) 6.4 5.0 4.7

Senior Enrollments 850 700 600


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Output

Roosevelt Lincoln Washington

Average SAT Score 800 830 900

High School Graduates 450 500 400

College Admissions 140 250 370


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Decision VariablesDecision Variables

E = Fraction of Roosevelt's input resources required bythe composite high school

w1 = Weight applied to Roosevelt's input/output

resources by the composite high schoolw2 = 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


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Objective FunctionObjective Function

Minimize the fraction of Roosevelt High School's inputresources required by the composite high school:

MIN E


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ConstraintsConstraints

Sum of the Weights is 1:

(1) w1 + w2 + w3 = 1

Output Constraints:

Since w1 = 1 is possible, each output of the compositeschool 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)


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ConstraintsConstraints

Input Constraints:

The input resources available to the composite school isa fractional multiple, E, of the resources available toRoosevelt. Since the composite high school cannot use

more input than that available to it, the inputconstraints are:

(5) 37w1 + 25w2 + 23w3 < 37E (Faculty)

(6) 6.4w1 + 5.0w2 + 4.7w3 < 6.4E (Budget)

(7) 850w

1 + 700w

2 + 600w3 < 850

E(Seniors)

Nonnegativity of variables:

E, w1, w2, w3 > 0


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OBJECTIVE FUNCTION VALUE = 0.765

VARIABLE VALUE REDUCED COSTS

E 0.765 0.000W1 0.000 0.235

W2 0.500 0.000

W3 0.500 0.000


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CONSTRAINT SLACK/SURPLUS DUAL PRICES

1 0.000 -0.235

2 65.000 0.000

3 0.000 -0.0014 170.000 0.000

5 4.294 0.000

6 0.044 0.000

7 0.000 0.001


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ConclusionConclusion

The output shows that the composite school ismade up of equal weights of Lincoln and Washington.Roosevelt is 76.5% efficient compared to this compositeschool when measured by college admissions (because

of the 0 slack on this constraint (#4)). It is less than76.5% efficient when using measures of SAT scores andhigh school graduates (there is positive slack inconstraints 2 and 3.)


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Revenue ManagementRevenue Management

Another LP application is revenue management.Another LP application is revenue management. Revenue managementRevenue management involves managing the shortinvolves managing the short--

term demand for a fixed perishable inventory interm demand for a fixed perishable inventory inorder to maximize revenue potential.order to maximize revenue potential.

The methodology was first used to determine howThe methodology was first used to determine howmany airline seats to sell at an earlymany airline seats to sell at an early--reservationreservationdiscount fare and many to sell at a full fare.discount fare and many to sell at a full fare.

Application areas now include hotels, apartmentApplication areas now include hotels, apartmentrentals, car rentals, cruise lines, and golf courses.rentals, car rentals, cruise lines, and golf courses.

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5353SlideSlide 2005 Thomson/South 2005 Thomson/South WesternWestern

End of Chapter 4End of Chapter 4

ms11 04 (1)

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