1 1 slide chapter 4 linear programming applications nblending problem nportfolio planning problem...

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Chapter 4 Chapter 4 Linear Programming ApplicationsLinear Programming Applications

Blending ProblemBlending Problem Portfolio Planning ProblemPortfolio Planning Problem Product Mix ProblemProduct Mix 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 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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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, ,

wherewhere 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 least Floataway Tours would like to purchase at least 50 boats and would like to purchase the same 50 boats and would like to purchase the same number from Sleekboat as from Racer to number from Sleekboat as from Racer to maintain goodwill. At the same time, Floataway maintain goodwill. At the same time, Floataway Tours wishes to have a total seating capacity of Tours wishes to have a total seating capacity of at least 200. 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 Expected Maximum 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 $ Catman Racer $5000 2 $ 5050

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


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

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 Cells

Final Reduced Objective A llowable Allowable

Cell Name V alue Cost Coeffic ient 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

Final Reduced Objective A llowable Allowable

Cell Name V alue Cost Coeffic ient 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

Constraints

Final S hadow Constraint A llowable Allowable

Cell Name V alue P rice 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

Final S hadow Constraint A llowable Allowable

Cell Name V alue P rice 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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