1 1 Slide © 2005 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS ST. EDWARD’S UNIVERSITY

1 1 Slide

Slide

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

Slides Prepared bySlides Prepared by

JOHN S. LOUCKSJOHN S. LOUCKSST. EDWARD’S UNIVERSITYST. EDWARD’S UNIVERSITY

2 2 Slide

Slide


Chapter 4 Chapter 4 Linear Programming ApplicationsLinear Programming Applications

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

3 3 Slide

Slide


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.

4 4 Slide

Slide



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.

5 5 Slide

Slide



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

6 6 Slide

Slide



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

7 7 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.


8 8 Slide

Slide


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.

9 9 Slide

Slide



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.

10 10 Slide

Slide



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.

11 11 Slide

Slide




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

12 12 Slide

Slide



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

13 13 Slide

Slide



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

14 14 Slide

Slide



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

15 15 Slide

Slide




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

16 16 Slide

Slide




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

17 17 Slide

Slide




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

18 18 Slide

Slide


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.

19 19 Slide

Slide


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


20 20 Slide

Slide



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


21 21 Slide

Slide


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


22 22 Slide

Slide



(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


23 23 Slide

Slide


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


24 24 Slide

Slide


Partial Spreadsheet Showing Problem DataPartial Spreadsheet Showing Problem Data

A B C D E F12 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

LHS Coefficients


25 25 Slide

Slide


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


26 26 Slide

Slide


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).


27 27 Slide

Slide


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


28 28 Slide

Slide


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


Documents

1 1 Slide © 2005 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS ST. EDWARD’S UNIVERSITY