Linear Programming Linear Programming ApplicationsApplications

in Marketing, Finance and in Marketing, Finance and OperationsOperations

Marketing ApplicationsMarketing Applications Financial ApplicationsFinancial Applications Operations Management ApplicationsOperations Management Applications Blending ProblemsBlending Problems Data Envelopment AnalysisData Envelopment Analysis

One application of linear programming in One application of linear programming in marketing is marketing is media selectionmedia selection..

LP can be used to help marketing LP can be used to help marketing managers allocate a fixed budget to various managers allocate a fixed budget to various advertising media.advertising media.

The objective is to maximize reach, The objective is to maximize reach, frequency, and quality of exposure.frequency, and quality of exposure.

Restrictions on the allowable allocation Restrictions on the allowable allocation usually arise during consideration of usually arise during consideration of company policy, contract requirements, company policy, contract requirements, and media availability.and media availability.

Marketing ApplicationsMarketing Applications

Media SelectionMedia Selection SMM Company recently developed a new SMM Company recently developed a new instantinstantsalad machine, has $282,000 to spend on salad machine, has $282,000 to spend on advertising. The product is to be initially test advertising. The product is to be initially test marketed in the Dallasmarketed in the Dallasarea. The money is to be spent onarea. The money is to be spent ona TV advertising blitz during onea TV advertising blitz during oneweekend (Friday, Saturday, andweekend (Friday, Saturday, andSunday) in November. Sunday) in November. The three options availableThe three options availableare: daytime advertising,are: daytime advertising,evening news advertising, andevening news advertising, andSunday game-time advertising. A mixture of one-Sunday game-time advertising. A mixture of one-minute TV spots is desired. minute TV spots is desired.

Media SelectionMedia Selection Estimated AudienceEstimated AudienceAd TypeAd Type Reached With Each AdReached With Each Ad Cost Cost Per AdPer Ad Daytime Daytime 3,000 3,000 $5,000 $5,000 Evening News Evening News 4,000 4,000 $7,000$7,000Sunday Game Sunday Game 75,000 75,000 $100,000$100,000

SMM wants to take out at least one ad of each SMM wants to take out at least one ad of each type (daytime, evening-news, and game-time). type (daytime, evening-news, and game-time). Further, there are only two game-time ad spots Further, there are only two game-time ad spots available. There are ten daytime spots and six available. There are ten daytime spots and six evening news spots available daily. SMM wants evening news spots available daily. SMM wants to have at least 5 ads per day, but spend no to have at least 5 ads per day, but spend no more than $50,000 on Friday and no more than more than $50,000 on Friday and no more than $75,000 on Saturday.$75,000 on Saturday.

Media SelectionMedia Selection

DFRDFR = number of daytime ads on Friday = number of daytime ads on FridayDSADSA = number of daytime ads on Saturday = number of daytime ads on SaturdayDSUDSU = = number of daytime ads on Sunday number of daytime ads on Sunday EFREFR = = number of evening ads on Friday number of evening ads on Friday ESAESA = = number of evening ads on Saturday number of evening ads on SaturdayESUESU = = number of evening ads on Sunday number of evening ads on SundayGSUGSU = = number of game-time ads on Sunday number of game-time ads on Sunday

Define the Decision Define the Decision VariablesVariables

Media SelectionMedia Selection Define the Objective Define the Objective

FunctionFunctionMaximize the total audience reached:Maximize the total audience reached:

Max (audience reached per ad of each Max (audience reached per ad of each type)type) x (number of ads used of each x (number of ads used of each type)type)

Max 3000Max 3000DFRDFR +3000 +3000DSADSA +3000 +3000DSUDSU +4000+4000EFREFR +4000+4000ESAESA +4000 +4000ESUESU +75000 +75000GSUGSU

Media SelectionMedia Selection Define the ConstraintsDefine the Constraints

Take out at least one ad of each type:Take out at least one ad of each type: (1) (1) DFRDFR + + DSADSA + + DSUDSU >> 1 1 (2) (2) EFREFR + + ESAESA + + ESUESU >> 1 1 (3) (3) GSUGSU >> 1 1

Ten daytime spots available:Ten daytime spots available: (4) (4) DFRDFR << 10 10 (5) (5) DSADSA << 10 10 (6) (6) DSUDSU << 10 10

Six evening news spots available:Six evening news spots available: (7) (7) EFREFR << 6 6 (8) (8) ESAESA << 6 6 (9) (9) ESUESU << 6 6

Media SelectionMedia Selection Define the Constraints Define the Constraints

(continued)(continued)Only two Sunday game-time ad spots available:Only two Sunday game-time ad spots available: (10) (10) GSUGSU << 2 2

At least 5 ads per day:At least 5 ads per day: (11) (11) DFRDFR + + EFREFR >> 5 5 (12) (12) DSADSA + + ESAESA >> 5 5 (13) (13) DSUDSU + + ESUESU + + GSUGSU >> 5 5

Spend no more than $50,000 on Friday:Spend no more than $50,000 on Friday: (14) 5000(14) 5000DFRDFR + 7000 + 7000EFREFR << 50000 50000

Media SelectionMedia Selection Define the Constraints Define the Constraints

(continued)(continued)Spend no more than $75,000 on Saturday:Spend no more than $75,000 on Saturday: (15) 5000(15) 5000DSADSA + 7000 + 7000ESAESA << 75000 75000

Spend no more than $282,000 in total:Spend no more than $282,000 in total: (16) 5000(16) 5000DFRDFR + 5000 + 5000DSADSA + 5000 + 5000DSUDSU + + 70007000EFREFR + 7000+ 7000ESAESA + 7000 + 7000ESUESU + 100000 + 100000GSUGSU7 7 < < 282000 282000

Non-negativity: Non-negativity: DFRDFR, , DSADSA, , DSUDSU, , EFREFR, , ESAESA, , ESUESU, , GSU GSU >> 0 0

Media SelectionMedia Selection The Management ScientistThe Management Scientist

SolutionSolutionObjective Function Value = Objective Function Value = 199000.000199000.000

VariableVariable ValueValue Reduced Reduced CostsCosts DFRDFR 8.000 8.000 0.000 0.000 DSADSA 5.000 5.000 0.000 0.000 DSUDSU 2.000 2.000 0.000 0.000 EFREFR 0.000 0.000 0.000 0.000 ESAESA 0.000 0.000 0.000 0.000 ESUESU 1.000 1.000 0.000 0.000 GSUGSU 2.000 2.000 0.0000.000

Media SelectionMedia Selection Solution SummarySolution Summary

Total new audience reached = Total new audience reached = 199,000 199,000

Number of daytime ads on Friday Number of daytime ads on Friday = 8= 8 Number of daytime ads on Saturday Number of daytime ads on Saturday = 5= 5 Number of daytime ads on Sunday Number of daytime ads on Sunday = 2= 2 Number of evening ads on Friday Number of evening ads on Friday = 0= 0 Number of evening ads on Saturday Number of evening ads on Saturday = 0= 0 Number of evening ads on Sunday Number of evening ads on Sunday = 1= 1 Number of game-time ads on SundayNumber of game-time ads on Sunday = 2= 2

Financial ApplicationsFinancial Applications LP can be used in financial decision-making LP can be used in financial decision-making

that involves capital budgeting, make-or-buy, that involves capital budgeting, make-or-buy, asset allocation, portfolio selection, financial asset allocation, portfolio selection, financial planning, and more.planning, and more.

Portfolio selectionPortfolio selection problems involve choosing problems involve choosing specific investments – for example, stocks and specific investments – for example, stocks and bonds – from a variety of investment bonds – from a variety of investment alternatives.alternatives.

This type of problem is faced by managers of This type of problem is faced by managers of banks, mutual funds, and insurance banks, mutual funds, and insurance companies.companies.

The objective function usually is maximization The objective function usually is maximization of expected return or minimization of risk.of expected return or minimization of risk.

Portfolio SelectionPortfolio Selection

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 month $10 million available at the start of the fifth month forfor

a high rise building venture in which it will bea high rise building venture in which it will be

participating.participating.

Portfolio SelectionPortfolio Selection

For the time being, Winslow wishes to investFor the time being, Winslow wishes to invest

only in 2-month government bonds (earning 2% overonly in 2-month government bonds (earning 2% over

the 2-month period) and 3-month construction loansthe 2-month period) and 3-month construction loans

(earning 6% over the 3-month period). Each of these(earning 6% over the 3-month period). Each of these

is available each month for investment. Funds notis available each month for investment. Funds not

invested in these two investments are liquid and earninvested in these two investments are liquid and earn

3/4 of 1% per month when invested locally.3/4 of 1% per month when invested locally.

Portfolio SelectionPortfolio Selection

Formulate a linear program that will helpFormulate a linear program that will help

Winslow Savings determine how to invest over theWinslow Savings determine how to invest over the

next four months if at no time does it wish to havenext four months if at no time does it wish to have

more than $8 million in either government bonds ormore than $8 million in either government bonds or

construction loans.construction loans.

Portfolio SelectionPortfolio Selection Define the Decision VariablesDefine the Decision Variables

GGii = amount of new investment in government = amount of new investment in government bonds in monthbonds in month i i (for (for ii = 1, 2, 3, 4) = 1, 2, 3, 4)

CCii = amount of new investment in construction = amount of new investment in construction

loans in month loans in month ii (for (for ii = 1, 2, 3, 4) = 1, 2, 3, 4)

LLii = amount invested locally in month = amount invested locally in month i i, ,

(for(for i i = 1, 2, 3, 4) = 1, 2, 3, 4)

Portfolio SelectionPortfolio Selection Define the Objective FunctionDefine the Objective Function

Maximize total interest earned in the 4-month period:Maximize total interest earned in the 4-month period:

Max (interest rate on investment) X (amount Max (interest rate on investment) X (amount invested)invested)

Max .02GMax .02G11 + .02 + .02GG22 + .02 + .02GG33 + .02 + .02GG44

+ .06+ .06CC11 + .06 + .06CC22 + .06 + .06CC33 + .06 + .06CC44

+ .0075+ .0075LL11 + .0075 + .0075LL22 + .0075 + .0075LL33 + .0075 + .0075LL44

Portfolio SelectionPortfolio Selection 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

Portfolio SelectionPortfolio Selection 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 2:bonds in Month 1 and locally invested in Month 2:

(3) (3) GG33 + + CC33 + + LL33 = 1.02 = 1.02GG11 + 1.0075 + 1.0075LL22

or - 1.02or - 1.02GG11 + + GG33 + + CC33 - 1.0075 - 1.0075LL22 + + LL33 = 0= 0

Portfolio SelectionPortfolio Selection 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 = 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

Portfolio SelectionPortfolio Selection Define the Constraints (continued)Define the Constraints (continued)

No more than $8 million in government No more than $8 million in government bonds at any time: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

(9) (9) GG33 + + GG44 << 8,000,000 8,000,000

Portfolio SelectionPortfolio Selection Define the Constraints (continued)Define the Constraints (continued)

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

Non-negativity:Non-negativity:

GGii, , CCii, , LLii >> 0 for 0 for ii = 1, 2, 3, 4 = 1, 2, 3, 4

Portfolio SelectionPortfolio Selection The Management ScientistThe Management Scientist

SolutionSolution Objective Function Value = Objective Function Value =

1429213.79871429213.7987 VariableVariable ValueValue

Reduced CostsReduced Costs GG1 1 8000000.0000 8000000.0000

0.00000.0000 GG2 2 0.0000 0.0000

0.00000.0000 GG3 3 5108613.9228 5108613.9228

0.00000.0000 GG4 4 2891386.0772 2891386.0772

0.00000.0000 CC1 1 8000000.0000 8000000.0000

0.00000.0000 CC2 2 0.0000 0.0000

0.04530.0453 CC3 3 0.0000 0.0000

0.00760.0076 CC4 4 8000000.0000 8000000.0000

0.00000.0000 LL1 1 4000000.0000 4000000.0000

0.00000.0000 LL2 2 4030000.0000 4030000.0000

0.00000.0000 LL3 3 7111611.0772 7111611.0772

0.00000.0000 LL4 4 4753562.0831 4753562.0831

0.00000.0000

Blending ProblemBlending ProblemFerdinand 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.

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

Ferdinand is interested in producing the 8-Ferdinand is interested in producing the 8-ounceounce

mixture at minimum cost while meeting the mixture at minimum cost while meeting the minimumminimum

daily requirements of 6 units of vitamin C, 5 daily requirements of 6 units of vitamin C, 5 units ofunits of

protein, and 5 units of iron.protein, and 5 units of iron.

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

Blending ProblemBlending Problem Define the constraintsDefine the constraints

Total weight of the mix is 8-ounces (.5 Total weight of the mix is 8-ounces (.5 pounds):pounds):

(1) (1) xx11 + + xx22 + + xx33 + + xx44 = .5 = .5Total amount of Vitamin C in the mix is at least Total amount of Vitamin C in the mix is at least 6 units: 6 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 Total amount of iron in the mix is at least 5 units:units:

(4) 14(4) 14xx22 + 15 + 15xx33 + 7 + 7xx44 > 5 > 5

Non-negativity of variables: Non-negativity of variables: xxjj >> 0 for all 0 for all jj

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 of grain 2, .09 lb. of grain 3, and .10 lb. of grain 4. The4. The

mixture costs Frederick’s 40.6 cents.mixture costs Frederick’s 40.6 cents.

Blending ProblemBlending Problem