1 1 Slide © 2005 Thomson/South-Western Chapter 4 Linear Programming Applications n Portfolio Planning Problem n Product Mix Problem n Blending Problem

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© 2005 Thomson/South-Western© 2005 Thomson/South-Western

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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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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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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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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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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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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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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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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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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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


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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


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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


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(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


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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


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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


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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).


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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


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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


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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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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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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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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.


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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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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.


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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


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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 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


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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


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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)


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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


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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


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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


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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.)


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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.

Documents

1 1 Slide © 2005 Thomson/South-Western Chapter 4 Linear Programming Applications n Portfolio Planning Problem n Product Mix Problem n Blending Problem