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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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Blending ProblemBlending Problem
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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Blending ProblemBlending Problem
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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Blending ProblemBlending Problem
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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Blending ProblemBlending Problem
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.
Blending ProblemBlending Problem
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Portfolio Planning ProblemPortfolio Planning Problem
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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Portfolio Planning ProblemPortfolio Planning Problem
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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Portfolio Planning ProblemPortfolio Planning Problem
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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Portfolio Planning ProblemPortfolio Planning Problem
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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Portfolio Planning ProblemPortfolio Planning Problem
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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Portfolio Planning ProblemPortfolio Planning Problem
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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Portfolio Planning ProblemPortfolio Planning Problem
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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Portfolio Planning ProblemPortfolio Planning Problem
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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Portfolio Planning ProblemPortfolio Planning Problem
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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Portfolio Planning ProblemPortfolio Planning Problem
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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Product Mix ProblemProduct Mix Problem
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
Product Mix ProblemProduct Mix Problem
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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
Product Mix ProblemProduct Mix Problem
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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
Product Mix ProblemProduct Mix Problem
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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
Product Mix ProblemProduct Mix Problem
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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
Product Mix ProblemProduct Mix Problem
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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
Product Mix ProblemProduct Mix Problem
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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
Product Mix ProblemProduct Mix Problem
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The Management ScienceO
utputThe Management ScienceO
utput
OBJECTIVE FUNCTION VALUE = 5040.000OBJECTIVE FUNCTION VALUE = 5040.000
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
Product Mix ProblemProduct Mix Problem
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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).
Product Mix ProblemProduct Mix Problem
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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
Product Mix ProblemProduct Mix Problem
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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
Product Mix ProblemProduct Mix Problem
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Transportation ProblemTransportation Problem
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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Transportation ProblemTransportation Problem
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
Transportation ProblemTransportation Problem
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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
Transportation ProblemTransportation Problem
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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
Transportation ProblemTransportation Problem
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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
Transportation ProblemTransportation Problem
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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
Transportation ProblemTransportation Problem
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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
Transportation ProblemTransportation Problem
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The Management Scientist OutputThe Management Scientist Output
OBJECTIVE FUNCTION VALUE = 142000.000OBJECTIVE FUNCTION VALUE = 142000.000
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
Transportation ProblemTransportation Problem
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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.
Transportation ProblemTransportation Problem
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Data Envelopment AnalysisData Envelopment Analysis
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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Data Envelopment AnalysisData Envelopment Analysis
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.
Data Envelopment AnalysisData Envelopment Analysis
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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
Data Envelopment AnalysisData Envelopment Analysis
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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
Data Envelopment AnalysisData Envelopment Analysis
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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
Data Envelopment AnalysisData Envelopment Analysis
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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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The Management ScientistThe Management Scientist OutputOutput
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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The Management ScientistThe Management Scientist OutputOutput
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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End of Chapter 4End of Chapter 4