1 1 slide © 2001 south-western college publishing/thomson learning anderson sweeney williams...

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© 2001 South-Western College Publishing/Thomson Learning© 2001 South-Western College Publishing/Thomson Learning

Anderson Sweeney Anderson Sweeney WilliamsWilliams

Anderson Sweeney Anderson Sweeney WilliamsWilliams

Slides Prepared by JOHN LOUCKSSlides Prepared by JOHN LOUCKS

QUANTITATIVE QUANTITATIVE METHODS FORMETHODS FORBUSINESS 8eBUSINESS 8e

QUANTITATIVE QUANTITATIVE METHODS FORMETHODS FORBUSINESS 8eBUSINESS 8e

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Chapter 9 Chapter 9 Linear Programming ApplicationsLinear Programming Applications

Blending ProblemBlending Problem Portfolio Planning ProblemPortfolio Planning Problem Product Mix ProblemProduct Mix Problem Transportation ProblemTransportation Problem

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Blending ProblemBlending Problem

Frederick's Feed Company receives four Frederick's Feed Company receives four raw grains from which it blends its dry pet food. raw grains from which it blends its dry pet food. The pet food advertises that each 8-ounce can The pet food advertises that each 8-ounce can meets the minimum daily requirements for meets the minimum daily requirements for vitamin C, protein and iron. The cost of each raw vitamin C, protein and iron. The cost of each raw grain as well as the vitamin C, protein, and iron grain as well as the vitamin C, protein, and iron units per pound of each grain are summarized on units per pound of each grain are summarized on the next slide.the next slide.

Frederick's is interested in producing the Frederick's is interested in producing the 8-ounce mixture at minimum cost while meeting 8-ounce mixture at minimum cost while meeting the minimum daily requirements of 6 units of the minimum daily requirements of 6 units of vitamin C, 5 units of protein, and 5 units of iron.vitamin C, 5 units of protein, and 5 units of iron.

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Vitamin C Protein Iron Vitamin C Protein Iron

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

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

2 16 2 16 10 10 14 14 .90 .90

3 83 8 10 10 15 15 .80 .80

4 10 4 10 8 8 7 7 .70 .70

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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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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 Total amount of protein in the mix is at least 5 units: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 > 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 1, .21 Thus, the optimal blend is about .10 lb. of grain 1, .21 lb.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 Frederick’s 40.6 cents.mixture costs Frederick’s 40.6 cents.

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Portfolio Planning ProblemPortfolio Planning Problem

Winslow Savings has $20 million available Winslow Savings has $20 million available for investment. It wishes to invest over the next for investment. It wishes to invest over the next four months in such a way that it will maximize four months in such a way that it will maximize the total interest earned over the four month the total interest earned over the four month period as well as have at least $10 million period as well as have at least $10 million available at the start of the fifth month for a high available at the start of the fifth month for a high rise building venture in which it will be rise building venture in which it will be participating.participating.

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For the time being, Winslow wishes to For the time being, Winslow wishes to invest only in 2-month government bonds invest only in 2-month government bonds (earning 2% over the 2-month period) and 3-(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 3-month period). Each of these is available each month for investment. Funds not invested in month for investment. Funds not invested in these two investments are liquid and earn 3/4 of these two investments are liquid and earn 3/4 of 1% per month when invested locally.1% per month when invested locally.

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Formulate a linear program that will help Formulate a linear program that will help Winslow Savings determine how to invest over Winslow Savings determine how to invest over the next four months if at no time does it wish to the next four months if at no time does it wish to have more than $8 million in either government have more than $8 million in either government bonds or construction loans.bonds or construction loans.

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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, , where where j j = 1,2,3,4 = 1,2,3,4

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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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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 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 - 1.02or - 1.02gg22 + + gg44 - 1.06 - 1.06cc11 + + cc44 - 1.0075 - 1.0075ll33 + + ll44 = = 00

$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 at No more than $8 million in government bonds at any time: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 at No more than $8 million in construction loans at any time: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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Problem: Floataway ToursProblem: Floataway Tours

Floataway Tours has $420,000 that may Floataway Tours has $420,000 that may be used to purchase new rental boats for hire be used to purchase new rental boats for hire during the summer. The boats can be during the summer. The boats can be purchased from two different manufacturers. purchased from two different manufacturers. Floataway Tours would like to purchase at Floataway Tours would like to purchase at least 50 boats and would like to purchase the least 50 boats and would like to purchase the same number from Sleekboat as from Racer to same number from Sleekboat as from Racer to maintain goodwill. At the same time, maintain goodwill. At the same time, Floataway Tours wishes to have a total seating Floataway Tours wishes to have a total seating capacity of at least 200. capacity of at least 200.

Pertinent data concerning the boats are Pertinent data concerning the boats are summarized on the next slide. Formulate this summarized on the next slide. Formulate this problem as a linear program.problem as a linear program.

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DataData

Maximum Maximum Expected Expected

Boat Builder Cost Seating Daily Boat Builder Cost Seating Daily ProfitProfit

Speedhawk Sleekboat $6000 3 $ 70Speedhawk Sleekboat $6000 3 $ 70

Silverbird 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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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)

x (Number of units)x (Number of units)

Max: 70Max: 70xx11 + 80 + 80xx22 + 50 + 50xx33 + 110 + 110xx44

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(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

xxjj >> 0, for 0, for jj = 1,2,3,4 = 1,2,3,4

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Partial Spreadsheet Showing Problem DataPartial Spreadsheet Showing Problem Data

A B C D E F12 Constr. X1 X2 X3 X4

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

A B C D E F12 Constr. X1 X2 X3 X4

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 Solution

A B C D E F9

10 X1 X2 X3 X4

11 28 0 0 28

1213 5040

1415 LHS RHS

16 420.0 <= 420

17 56.0 >= 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

A B C D E F9

10 X1 X2 X3 X4

11 28 0 0 28

1213 5040

1415 LHS RHS

16 420.0 <= 420

17 56.0 >= 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 Science OutputThe Management Science Output


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 Cells

Final Reduced Objective Allowable AllowableCell 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

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

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

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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Problem: U.S. NavyProblem: U.S. Navy

The Navy has 9,000 pounds of material in The Navy has 9,000 pounds of material in Albany, Georgia which it wishes to ship to three Albany, 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, The shipping costs per pound for truck, railroad, and airplane transit are shown on the railroad, and airplane transit are shown on the next slide. Formulate and solve a linear program next slide. Formulate and solve a linear program to determine the shipping arrangements (mode, to determine the shipping arrangements (mode, destination, and quantity) that will minimize the destination, and quantity) that will minimize the total shipping cost.total shipping cost.

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DataData

DestinationDestination

Mode Mode San Diego Norfolk San Diego Norfolk PensacolaPensacola

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

Railroad Railroad 20 11 20 11 9 9

Airplane Airplane 30 26 30 26 28 28

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Define the Decision VariablesDefine the Decision Variables

We want to determine the pounds of material, We want to determine the pounds of material, xxij ij , to , to be shipped by mode be shipped by mode ii to destination to destination jj. The following . The following table summarizes the decision variables:table summarizes the decision variables:

San Diego Norfolk PensacolaSan Diego Norfolk Pensacola

TruckTruck xx1111 xx1212 xx1313

Railroad Railroad xx2121 xx2222 xx2323

Airplane Airplane xx3131 xx3232 xx3333

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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).shipped by mode per destination pairing).

Min: 12Min: 12xx1111 + 6 + 6xx1212 + 5 + 5xx1313 + 20 + 20xx2121 + 11 + 11xx2222 + + 99xx2323

+ 30+ 30xx3131 + 26 + 26xx3232 + 28 + 28xx3333

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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

(6) (6) xx1313 + + xx2323 + + xx3333 = 2500 = 2500

Nonnegativity 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 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

LHS Coefficients

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

LHS Coefficients

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Partial Spreadsheet Showing SolutionPartial Spreadsheet Showing Solution

A B C D E F G H I J

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

Nor

Pen

San

Air

Constraints

Truc

Rail

Minimized Total Shipping Cost 142000

A B C D E F G H I J

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

Nor

Pen

San

Air

Constraints

Truc

Rail

Minimized Total Shipping Cost 142000

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The Management Scientist OutputThe Management Scientist Output


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

xx3232 0.000 2.000 0.000 2.000

xx3333 0.000 6.000 0.000 6.000

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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. • The total shipping cost will be $142,000.The total shipping cost will be $142,000.

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The End of Chapter 9The End of Chapter 9

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