ch04.ppt

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Slides Prepared by

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Chapter 4

Linear Programming Applications

 Blending ProblemBlending Problem

 Portfolio Planning ProblemPortfolio Planning Problem  Product Mix ProblemProduct Mix Problem

 Transportation ProblemTransportation Problem

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Blending Problem

Frederick's Feed Company receives four

raw grains from which it blends its dry pet food.

The pet food advertises that each 8-ounce can

meets the minimum daily requirements for

vitamin C, protein and iron. The cost of each raw

grain as well as the vitamin C, protein, and iron

units per pound of each grain are summarized on

the next slide.

Frederick's is interested in producing the

8-ounce mixture at minimum cost while meeting

the minimum daily requirements of 6 units of

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Blending Problem

Vitamin C Protein Iron Vitamin C Protein Iron

Grain Units/lb Units/lb Units/lb Grain Units/lb Units/lb Units/lb Cost/lb

Cost/lb

1 9 1 9 12 12 0 .75 0 .75

2 16 2 16 10 10 14 14 .90

.90

3 83 8 10 10 15 15 .80

.80

4 10

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Blending Problem

 Define the decision variablesDefine the decision variables

xx_j_j = 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:

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Blending Problem

 Define the constraintsDefine the constraints

Total 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 = .5

Total 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 > 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:

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 The Management ScientistThe Management Scientist Output Output

OBJECTIVE 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 1, .21

lb.

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

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Portfolio Planning Problem

Winslow Savings has $20 million available

for investment. It wishes to invest over the next

four months in such a way that it will maximize

the total interest earned over the four month

period as well as have at least $10 million

available at the start of the fifth month for a high

rise building venture in which it will be

participating.

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Portfolio Planning Problem

For the time being, Winslow wishes to

invest only in 2-month government bonds

(earning 2% over the 2-month period) and

3-month construction loans (earning 6% over the

month construction loans (earning 6% over the

3-month period). Each of these is available each

month for investment. Funds not invested in

these two investments are liquid and earn 3/4 of

1% per month when invested locally.

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Portfolio Planning Problem

Formulate a linear program that will help

Winslow Savings determine how to invest over

the next four months if at no time does it wish to

have more than $8 million in either government

bonds or construction loans.

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Portfolio Planning Problem

gg_j_j = amount of new investment in = amount of new investment in

government bonds in monthgovernment bonds in month j j

cc_j_j = amount of new investment in = amount of new investment in construction loans in month

construction loans in month jj

ll_j_j = amount invested locally in month = amount invested locally in month j j, ,

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Portfolio Planning Problem

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

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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

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Portfolio Planning Problem

 Define the constraints (continued)Define the constraints (continued)

Month 3's total investment amount limited to

principle and interest invested in government

bonds in Month 1 and locally invested in Month

2:

(3) (3) gg₃₃ + + cc₃₃ + + ll₃₃ = 1.02 = 1.02gg₁₁ + 1.0075 + 1.0075ll₂₂ or - 1.02

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Portfolio Planning Problem

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 in goverment 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 + 1.0075ll₃₃

or - 1.02

or - 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:

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No more than $8 million in government bonds 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

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Portfolio Planning Problem

No more than $8 million in construction loans 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:

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Problem: Floataway Tours

Floataway Tours has $420,000 that may be used Floataway Tours has $420,000 that may be used

to purchase new rental boats for hire during the to purchase new rental boats for hire during the summer. The boats can be purchased from two summer. The boats can be purchased from two different manufacturers. Floataway Tours would different manufacturers. Floataway Tours would like to purchase at least 50 boats and would like like to purchase at least 50 boats and would like to purchase the same number from Sleekboat as to purchase the same number from Sleekboat as

from Racer to maintain goodwill. At the same from Racer to maintain goodwill. At the same

time, Floataway Tours wishes to have a total time, Floataway Tours wishes to have a total

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

Pertinent data concerning the boats are Pertinent data concerning the boats are

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Problem: Floataway Tours

 DataData

Maximum Expected Maximum Expected

Boat Builder Cost Seating Boat Builder Cost Seating Daily Profit

Daily 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 $

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Problem: Floataway Tours

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

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

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

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Problem: Floataway Tours

(1) Spend no more than $420,000:

60006000xx₁₁ + 7000 + 7000xx₂₂ + 5000 + 5000xx₃₃ + 9000 + 9000xx₄₄ << 420,000

420,000

(2) Purchase at least 50 boats:

xx₁₁ + + xx₂₂ + + xx₃₃ + + xx₄₄ >> 50 50

(3) Number of boats from Sleekboat equals

number

number of boats from Racer:of boats from Racer:

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Problem: Floataway Tours

 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:

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 Complete FormulationComplete Formulation

Max 70

Max 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

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 Partial Spreadsheet Showing Problem DataPartial Spreadsheet Showing Problem Data

A 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

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Problem: Floataway Tours

 Partial Spreadsheet Showing SolutionPartial Spreadsheet Showing Solution

A 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 <= 420

17 56.0 >= 50

Decision Variable Values

No. of Boats

Maximum Total Profit

Constraints

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Problem: Floataway Tours

 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

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Problem: Floataway Tours

 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).

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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 Adjustable Cells

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Problem: Floataway Tours

 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 Constraints

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Problem: U.S. Navy

The Navy has 9,000 pounds of material in Albany, The Navy has 9,000 pounds of material in Albany,

Georgia which it wishes to ship to three Georgia which it wishes to ship to three

installations: San Diego, Norfolk, and Pensacola. installations: San Diego, Norfolk, and Pensacola.

They require 4,000, 2,500, and 2,500 pounds, They require 4,000, 2,500, and 2,500 pounds, respectively. Government regulations require respectively. Government regulations require equal distribution of shipping among the three equal distribution of shipping among the three

carriers. carriers.

The shipping costs per pound for truck, railroad, The shipping costs per pound for 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 to determine Formulate and solve a linear program to determine the shipping arrangements (mode, destination, and the shipping arrangements (mode, destination, and

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Problem: U.S. Navy

 DataData

DestinationDestination

Mode Mode San Diego Norfolk San Diego Norfolk Pensacola

Pensacola

Truck Truck $12 $ 6 $ $12 $ 6 $ 5

5

Railroad Railroad 20 11 20 11 9

9

Airplane Airplane 30 26 30 26 28

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Problem: U.S. Navy

 Define the Decision VariablesDefine the Decision Variables

We want to determine the pounds of material,

We want to determine the pounds of material, xx_ij_ij, , to be shipped by mode

to be shipped by mode ii to destination to destination jj. The . The

following table summarizes the decision variables:

San Diego Norfolk PensacolaSan Diego Norfolk Pensacola

TruckTruck xx₁₁₁₁ xx₁₂₁₂

xx₁₃₁₃

Railroad Railroad xx₂₁₂₁ xx₂₂₂₂

xx₂₃₂₃

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Problem: U.S. Navy

 Define the Objective FunctionDefine the Objective Function

Minimize the total shipping cost.Minimize the total shipping cost.

Min: (shipping cost per pound for each mode Min: (shipping cost per pound for each mode per destination pairing) x (number of pounds

per destination pairing) x (number of pounds

shipped by mode per destination pairing).

Min: 12Min: 12xx₁₁₁₁ + 6 + 6xx₁₂₁₂ + 5 + 5xx₁₃₁₃ + 20 + 20xx₂₁₂₁ + 11 + 11xx₂₂₂₂ + + 9

9xx₂₃₂₃

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Problem: U.S. Navy

 Define the ConstraintsDefine the Constraints

Equal 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 = 3000 = 3000

Destination material requirements:Destination material requirements:

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

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

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Problem: U.S. Navy

 Partial Spreadsheet Showing Problem DataPartial Spreadsheet Showing Problem Data

A 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 #6 1 1 1 2500

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

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Problem: U.S. Navy

 Partial Spreadsheet Showing SolutionPartial Spreadsheet Showing Solution

A 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

Constraints

Truc Rail

Minimized Total Shipping Cost 142000

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Problem: U.S. Navy

 The Management Scientist OutputThe Management Scientist Output

OBJECTIVE FUNCTION VALUE = 142000.000OBJECTIVE FUNCTION VALUE = 142000.000

VariableVariable ValueValue Reduced CostReduced Cost

xx₁₁₁₁ 1000.000 0.000 1000.000 0.000

xx₁₂₁₂ 2000.000 0.000 2000.000 0.000

xx₁₃₁₃ 0.000 1.000 0.000 1.000

xx₂₁₂₁ 0.000 3.000 0.000 3.000

xx₂₂₂₂ 500.000 0.000 500.000 0.000

xx₂₃₂₃ 2500.000 0.000 2500.000 0.000

xx₃₁₃₁ 3000.000 0.000 3000.000 0.000

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Problem: U.S. Navy

 Solution SummarySolution Summary

•

San Diego will receive 1000 lbs. by truckSan Diego will receive 1000 lbs. by truck and 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.

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Data 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

E

E, 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

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Data 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 = 1

Sum of weights = 1 E

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DEA Example: Roosevelt High

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 School.

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 Input

Roosevelt Lincoln

Washington

Senior Faculty 37 25 23

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

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DEA Example: Roosevelt High

 Output

Roosevelt Lincoln Washington

Average SAT Score 800 830

900

High School Graduates 450 500

400

College Admissions 140 250

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 Decision VariablesDecision Variables

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

w₁ = Weight applied to Roosevelt's input/output resources by the composite high school

w₂ = Weight applied to Lincoln’s input/output resources by the composite high school

w₃ = Weight applied to Washington's

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DEA Example: Roosevelt High

 Objective FunctionObjective Function

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

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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:

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DEA Example: Roosevelt High

 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)

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DEA Example: Roosevelt High

 Management ScientistManagement 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

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DEA Example: Roosevelt High

 Management ScientistManagement 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

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DEA Example: Roosevelt High

 ConclusionConclusion

The 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

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