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Transcript of 1 1 Slide Chapter 4 Linear Programming Applications nBlending Problem nPortfolio Planning Problem...
1 1 Slide Slide
Chapter 4 Chapter 4 Linear Programming ApplicationsLinear Programming Applications
Blending ProblemBlending Problem Portfolio Planning ProblemPortfolio Planning Problem Product Mix ProblemProduct Mix Problem
2 2 Slide Slide
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.
3 3 Slide Slide
Blending ProblemBlending Problem
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
4 4 Slide Slide
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
5 5 Slide Slide
Blending ProblemBlending 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 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
6 6 Slide Slide
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
7 7 Slide Slide
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.
8 8 Slide Slide
Portfolio Planning ProblemPortfolio Planning Problem
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.
9 9 Slide Slide
Portfolio Planning ProblemPortfolio Planning Problem
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.
10 10 Slide Slide
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
11 11 Slide Slide
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
12 12 Slide Slide
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
13 13 Slide Slide
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
14 14 Slide Slide
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 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
15 15 Slide Slide
Portfolio Planning ProblemPortfolio Planning Problem
Define the constraints (continued)Define the constraints (continued)
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
16 16 Slide Slide
Portfolio Planning ProblemPortfolio Planning Problem
Define the constraints (continued)Define the constraints (continued)
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
17 17 Slide Slide
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.
18 18 Slide Slide
Problem: Floataway ToursProblem: Floataway Tours
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
19 19 Slide Slide
Problem: Floataway ToursProblem: Floataway Tours
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 unit) Max: (Expected daily profit per unit)
x (Number of units)x (Number of units)
Max: 70Max: 70xx11 + 80 + 80xx22 + 50 + 50xx33 + 110 + 110xx44
20 20 Slide Slide
Problem: Floataway ToursProblem: Floataway Tours
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
21 21 Slide Slide
Problem: Floataway ToursProblem: 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: Nonnegativity of variables:
xxjj >> 0, for 0, for jj = 1,2,3,4 = 1,2,3,4
22 22 Slide Slide
Problem: Floataway ToursProblem: Floataway Tours
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
23 23 Slide Slide
Problem: Floataway ToursProblem: 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 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
24 24 Slide Slide
Problem: Floataway ToursProblem: 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).by 52 (surplus for constraint #4).
25 25 Slide Slide
Problem: Floataway ToursProblem: Floataway Tours
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
26 26 Slide Slide
Problem: Floataway ToursProblem: Floataway Tours
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