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Chapter 4 Chapter 4 Linear Programming ApplicationsLinear Programming Applications
Portfolio Planning ProblemPortfolio Planning Problem Product Mix ProblemProduct Mix Problem Blending ProblemBlending Problem Data Envelopment AnalysisData Envelopment Analysis Revenue ManagementRevenue Management
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Portfolio Planning ProblemPortfolio Planning Problem
Winslow Savings has $20 million availableWinslow Savings has $20 million available
for 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 four
month period as well as have at leastmonth period as well as have at least
$10 million available at the start of the fifth $10 million available at the start of the fifth month formonth 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 invest
only in 2-month government bonds (earning 2% only in 2-month government bonds (earning 2% overover
the 2-month period) and 3-month construction the 2-month period) and 3-month construction loansloans
(earning 6% over the 3-month period). Each of (earning 6% over the 3-month period). Each of thesethese
is available each month for investment. Funds notis available each month for investment. Funds not
invested in these two investments are liquid and invested in these two investments are liquid and earnearn
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 help
Winslow Savings determine how to invest over Winslow Savings determine how to invest over thethe
next four months if at no time does it wish to next four months if at no time does it wish to havehave
more than $8 million in either government more than $8 million in either government bonds orbonds 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 j j
ccjj = amount of new investment in = amount of new investment in construction loans in month construction loans in month jj
lljj = amount invested locally in month = amount invested locally in month j j, ,
wherewhere j j = 1,2,3,4 = 1,2,3,4
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Portfolio Planning ProblemPortfolio Planning Problem
Define the objective functionDefine the objective function
Maximize total interest earned over the 4-month Maximize total interest earned over the 4-month period.period.
MAX (interest rate on investment)(amount MAX (interest rate on investment)(amount invested)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 constraints
Month 1's total investment limited to $20 Month 1's total investment limited to $20 million:million:
(1) (1) gg11 + + cc11 + + ll11 = 20,000,000 = 20,000,000
Month 2's total investment limited to principle Month 2's total investment limited to principle and interest invested locally in Month 1:and interest invested locally in Month 1:
(2) (2) gg22 + + cc22 + + ll22 = 1.0075 = 1.0075ll11
or or gg22 + + cc22 - 1.0075 - 1.0075ll11 + + 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 to Month 3's total investment amount limited to principle and interest invested in government principle and interest invested in government bonds in Month 1 and locally invested in Month bonds in Month 1 and locally invested in Month 2:2:
(3) (3) gg33 + + cc33 + + ll33 = 1.02 = 1.02gg11 + 1.0075 + 1.0075ll22
or - 1.02or - 1.02gg11 + + gg33 + + cc33 - 1.0075 - 1.0075ll22 + + 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 and Month 4's total investment limited to principle and interest invested in construction loans in Month 1, interest invested in construction loans in Month 1, goverment bonds in Month 2, and locally invested goverment bonds in Month 2, and locally invested in Month 3:in Month 3:
(4) (4) gg44 + + cc44 + + ll44 = 1.06 = 1.06cc11 + 1.02 + 1.02gg22 + 1.0075 + 1.0075ll33
or - 1.02or - 1.02gg22 + + gg44 - 1.06 - 1.06cc11 + + cc44 - 1.0075 - 1.0075ll33 + + 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 >> 10,000,000 10,000,000
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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 No more than $8 million in government bonds at any time:at any time:
(6) (6) gg11 << 8,000,000 8,000,000
(7) (7) gg11 + + gg22 << 8,000,000 8,000,000
(8) (8) gg22 + + gg33 << 8,000,000 8,000,000
(9) (9) gg33 + + gg44 << 8,000,000 8,000,000
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Portfolio Planning ProblemPortfolio Planning Problem
Define the constraints (continued)Define the constraints (continued)
No more than $8 million in construction loans No more than $8 million in construction loans at any time:at any time:
(10) (10) cc11 << 8,000,000 8,000,000
(11) (11) cc11 + + cc22 << 8,000,000 8,000,000
(12) (12) cc11 + + cc22 + + cc33 << 8,000,000 8,000,000
(13) (13) cc22 + + cc33 + + cc44 << 8,000,000 8,000,000
Nonnegativity: Nonnegativity: ggjj, , ccjj, , lljj >> 0 for 0 for jj = 1,2,3,4 = 1,2,3,4
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Product Mix ProblemProduct Mix Problem
Floataway Tours has $420,000 that can be Floataway Tours has $420,000 that can be usedused
to 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 purchase the same number from Sleekboat as fromfrom
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 Maximum Expected Expected
Boat Builder Cost Seating Boat Builder Cost Seating Daily ProfitDaily Profit
Speedhawk Sleekboat $6000 3 Speedhawk Sleekboat $6000 3 $ 70$ 70
Silverbird Sleekboat $7000 5 Silverbird Sleekboat $7000 5 $ 80$ 80
Catman Racer $5000 2 Catman Racer $5000 2 $ 50 $ 50
Classy Racer $9000 6 Classy Racer $9000 6 $110 $110
Product Mix ProblemProduct Mix Problem
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Define the decision variablesDefine the decision variables
xx11 = 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 Max: (Expected daily profit per
unit) 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 << 420,000420,000
(2) Purchase at least 50 boats: (2) Purchase at least 50 boats:
xx11 + + xx22 + + xx33 + + xx44 >> 50 50
(3) Number of boats from Sleekboat equals (3) Number of boats from Sleekboat equals number number of boats from Racer:of boats from Racer:
xx11 + + xx22 = = xx33 + + xx44 or or 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 >> 200 200
Nonnegativity of variables: Nonnegativity of variables:
xxjj >> 0, for 0, for jj = 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 + 110xx44
s.t.s.t.
60006000xx11 + 7000 + 7000xx22 + 5000 + 5000xx33 + 9000 + 9000xx44 << 420,000 420,000
xx11 + + xx22 + + xx33 + + xx44 >> 50 50
xx11 + + xx22 - - xx33 - - xx44 = 0 = 0
33xx11 + 5 + 5xx22 + 2 + 2xx33 + 6 + 6xx44 >> 200200
xx11, , xx22, , xx33, , xx44 >> 0 0
Product Mix ProblemProduct Mix Problem
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The Management Science OutputThe Management Science Output
OBJECTIVE FUNCTION VALUE = 5040.000OBJECTIVE FUNCTION VALUE = 5040.000
VariableVariable ValueValue Reduced CostReduced Cost xx11 28.000 0.000 28.000 0.000 xx22 0.000 2.000 0.000 2.000 xx33 0.000 12.000 0.000 12.000 xx44 28.000 0.000 28.000 0.000
ConstraintConstraint Slack/SurplusSlack/Surplus Dual PriceDual Price 1 0.000 0.012 1 0.000 0.012 2 6.000 0.000 2 6.000 0.000 3 0.000 -2.000 3 0.000 -2.000 4 52.000 0.000 4 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 Classy’s from Racer.Purchase 28 Classy’s from Racer.
• Total expected daily profit is $5,040.00.Total expected daily profit is $5,040.00.
• The minimum number of boats was exceeded The minimum number of boats was exceeded by 6 (surplus for constraint #2).by 6 (surplus for constraint #2).
• The minimum seating capacity was exceeded The minimum seating capacity was exceeded by 52 (surplus for constraint #4).by 52 (surplus for constraint #4).
Product Mix ProblemProduct Mix Problem
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Sensitivity ReportSensitivity Report
Adjustable CellsFinal Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease$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$12 X4 28 0 110 1E+30 16.36363636
Product Mix ProblemProduct Mix Problem
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Sensitivity ReportSensitivity Report
ConstraintsFinal Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease$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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Blending ProblemBlending Problem
Ferdinand Feed Company receives four Ferdinand Feed Company receives four rawraw
grains from which it blends its dry pet food. grains from which it blends its dry pet food. The petThe pet
food advertises that each 8-ounce packetfood advertises that each 8-ounce packet
meets the minimum daily requirementsmeets the minimum daily requirements
for vitamin C, protein and iron. Thefor vitamin C, protein and iron. The
cost 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 Iron Vitamin C Protein Iron
Grain Units/lb Units/lb Units/lb Cost/lbGrain Units/lb Units/lb Units/lb Cost/lb
1 9 1 9 12 12 0 .75 0 .75
2 16 2 16 10 10 14 .9014 .90
3 83 8 10 10 15 .8015 .80
4 10 4 10 8 8 7 .70 7 .70
Ferdinand is interested in producing the 8-ounceFerdinand is interested in producing the 8-ounce
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 grain jj ( (jj = = 1,2,3,4) 1,2,3,4)
used in the 8-ounce mixtureused in the 8-ounce mixture
Define the objective functionDefine the objective function
Minimize the total cost for an 8-ounce Minimize the total cost for an 8-ounce mixture: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 8-ounces (.5 pounds):Total weight of the mix is 8-ounces (.5 pounds):
(1) (1) xx11 + + xx22 + + xx33 + + xx44 = .5 = .5Total amount of Vitamin C in the mix is at least 6 Total amount of Vitamin C in the mix is at least 6 units: units:
(2) 9(2) 9xx11 + 16 + 16xx22 + 8 + 8xx33 + 10 + 10xx44 > 6 > 6Total amount of protein in the mix is at least 5 Total amount of protein in the mix is at least 5 units:units:
(3) 12(3) 12xx11 + 10 + 10xx22 + 10 + 10xx33 + 8 + 8xx44 > 5 > 5Total 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 > 5
Nonnegativity of variables: Nonnegativity of variables: xxjj >> 0 for all 0 for all jj
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The Management ScientistThe Management Scientist Output Output
OBJECTIVE FUNCTION VALUE = 0.406OBJECTIVE FUNCTION VALUE = 0.406
VARIABLEVARIABLE VALUEVALUE REDUCED COSTSREDUCED COSTS X1 X1 0.099 0.099 0.0000.000 X2 X2 0.213 0.213 0.0000.000 X3 X3 0.088 0.088 0.0000.000 X4 X4 0.099 0.099 0.0000.000
Thus, the optimal blend is about .10 lb. of grain Thus, the optimal blend is about .10 lb. of grain 1, .21 lb.1, .21 lb.
of grain 2, .09 lb. of grain 3, and .10 lb. of grain 4. of grain 2, .09 lb. of grain 3, and .10 lb. of grain 4. TheThe
mixture costs Frederick’s 40.6 cents.mixture costs Frederick’s 40.6 cents.
Blending ProblemBlending Problem
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Data Envelopment AnalysisData Envelopment Analysis
Data envelopment analysisData envelopment analysis (DEA) is an LP application (DEA) is an LP application used to determine the relative operating efficiency of used to determine the relative operating efficiency of units with the same goals and objectives.units with the same goals and objectives.
DEA creates a DEA creates a fictitious composite unitfictitious composite unit made up of an made up of an optimal 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 determining EE, the fraction of unit , the fraction of unit kk’s input resources required by ’s input resources required by the optimal composite unit.the optimal composite unit.
If If EE < 1, unit < 1, unit kk is less efficient than the composite unit is less efficient than the composite unit and be deemed relatively inefficient.and be deemed relatively inefficient.
If If EE = 1, there is no evidence that unit = 1, there is no evidence that unit kk is inefficient, is inefficient, but one cannot conclude that but one cannot conclude that kk is absolutely efficient. is absolutely efficient.
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Data Envelopment AnalysisData Envelopment Analysis
The DEA ModelThe DEA Model
MIN MIN EE
s.t.s.t. Weighted outputs Weighted outputs >> Unit Unit kk’s output ’s output (for each measured output)(for each measured output)
Weighted inputs Weighted inputs << E E [Unit [Unit kk’s input]’s input](for each measured input)(for each measured input)
Sum of weights = 1Sum of weights = 1
EE, weights , weights >> 0 0
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The Langley County School District is trying todetermine the relative efficiency ofits three high schools. In particular,it wants to evaluate Roosevelt High.
The district is evaluating performances on SAT scores, thenumber of seniors finishing highschool, and the number of studentswho enter college as a function of thenumber 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 23Budget ($100,000's) 6.4 5.0 4.7Senior 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 by the composite high schoolw1 = Weight applied to Roosevelt's input/output
resources by the composite high schoolw2 = Weight applied to Lincoln’s input/output
resources by the composite high schoolw3 = 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 input resources required by the composite high school:
MIN E
Data Envelopment AnalysisData Envelopment Analysis
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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
composite school 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)
Data Envelopment AnalysisData Envelopment Analysis
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ConstraintsConstraints
Input Constraints: The input resources available to the composite
school is a fractional multiple, E, of the resources available to Roosevelt. Since the composite high school cannot use more input than that available to it, the input constraints are:
(5) 37w1 + 25w2 + 23w3 < 37E (Faculty)
(6) 6.4w1 + 5.0w2 + 4.7w3 < 6.4E (Budget)
(7) 850w1 + 700w2 + 600w3 < 850E (Seniors)
Nonnegativity of variables: E, w1, w2, w3 > 0
Data Envelopment AnalysisData Envelopment Analysis
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The Management ScientistThe Management Scientist Output Output
OBJECTIVE FUNCTION VALUE = 0.765
VARIABLE VALUE REDUCED COSTS
E 0.765 0.000 W1 0.000
0.235 W2 0.500
0.000 W3 0.500
0.000
Data Envelopment AnalysisData Envelopment Analysis
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The Management ScientistThe Management Scientist Output Output
CONSTRAINT SLACK/SURPLUS DUAL PRICES 1 0.000 -0.235
2 65.000 0.000 3 0.000 -0.001 4 170.000 0.000 5 4.294 0.000 6 0.044 0.000 7 0.000 0.001
Data Envelopment AnalysisData Envelopment Analysis
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ConclusionConclusionThe output shows that the composite
school is made up of equal weights of Lincoln and Washington. Roosevelt is 76.5% efficient compared to this composite school when measured by college admissions (because of the 0 slack on this constraint (#4)). It is less than 76.5% efficient when using measures of SAT scores and high school graduates (there is positive slack in constraints 2 and 3.)
Data Envelopment AnalysisData Envelopment Analysis
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Revenue ManagementRevenue Management
Another LP application is revenue Another LP application is revenue management.management.
Revenue managementRevenue management involves managing the involves managing the short-term demand for a fixed perishable short-term demand for a fixed perishable inventory in order to maximize revenue inventory in order to maximize revenue potential.potential.
The methodology was first used to determine The methodology was first used to determine how many airline seats to sell at an early-how many airline seats to sell at an early-reservation discount fare and many to sell at a reservation discount fare and many to sell at a full fare.full fare.
Application areas now include hotels, Application areas now include hotels, apartment rentals, car rentals, cruise lines, apartment rentals, car rentals, cruise lines, and golf courses.and golf courses.