chapter 3 linear programming applications 8000000.0000 0.0000 c2 0.0000 0.0453 c3 0.0000 0.0076 c4...

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ChapterChapter 33Linear ProgrammingLinear Programming ApplicationsApplications

The process of problemThe process of problem formulationformulation MarketingMarketing and mediaand media applicationsapplications Financial ApplicationsFinancial Applications Transportation ProblemTransportation Problem

1.1. Provide a detailed verbal description of the problemProvide a detailed verbal description of the problem2.2. Determine the overall objective that appears to beDetermine the overall objective that appears to be

relevant.relevant.3.3. Determine the factors (constraints) that appear toDetermine the factors (constraints) that appear to

restrict the attainment of the objective function.restrict the attainment of the objective function.4.4. Define the decision variables and state their units ofDefine the decision variables and state their units of

measurement.measurement.5.5. Using these decision variables, formulate anUsing these decision variables, formulate an

objective function.objective function.6.6. Formulate a mathematical equations for each of theFormulate a mathematical equations for each of the

identified constraints.identified constraints.7.7. Check theCheck the netirenetire formulation to ensure linearity.formulation to ensure linearity.

The process of problemThe process of problem formulationformulation

One application of linear programming in marketingOne application of linear programming in marketingisis media selectionmedia selection..

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

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

Restrictions on the allowable allocation usually ariseRestrictions on the allowable allocation usually ariseduring consideration of company policy, contractduring consideration of company policy, contractrequirements, and media availability.requirements, and media availability.

Marketing ApplicationsMarketing Applications

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Media SelectionMedia Selection

SMM Company recently developed a new instantSMM Company recently developed a new instantsalad machine, has $282,000 to spend on advertising.salad machine, has $282,000 to spend on advertising.The product is to be initially test marketed in the DallasThe product is to be initially test marketed 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 gameSunday game--time advertising. A mixture of onetime advertising. A mixture of one--minute TV spots is desired.minute TV spots is desired.


Estimated AudienceEstimated AudienceAd TypeAd Type Reached With Each AdReached With Each Ad Cost Per AdCost Per AdDaytimeDaytime 3,0003,000 $5,000$5,000Evening NewsEvening News 4,0004,000 $7,000$7,000Sunday GameSunday Game 75,00075,000 $100,000$100,000

SMM wants to take out at least one ad of each typeSMM wants to take out at least one ad of each type(daytime, evening(daytime, evening--news, and gamenews, and game--time). Further, theretime). Further, thereare only two gameare only two game--time ad spots available. There aretime ad spots available. There areten daytime spots and six evening news spots availableten daytime spots and six evening news spots availabledaily. SMM wants to have at least 5 ads per day, butdaily. SMM wants to have at least 5 ads per day, butspend no more than $50,000 on Friday and no more thanspend no more than $50,000 on Friday and no more than$75,000 on Saturday.$75,000 on Saturday.


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 Sundaynumber of daytime ads on SundayEFREFR == number of evening ads on Fridaynumber of evening ads on FridayESAESA == number of evening ads on Saturdaynumber of evening ads on SaturdayESUESU == number of evening ads on Sundaynumber of evening ads on SundayGSUGSU == number of gamenumber of game--time ads on Sundaytime ads on Sunday

Define the Decision VariablesDefine the Decision Variables

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Define the Objective FunctionDefine the Objective FunctionMaximize the total audience reached:Maximize the total audience reached:

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

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


Define the ConstraintsDefine the ConstraintsTake out at least one ad of each type:Take out at least one ad of each type:

(1)(1) DFRDFR ++ DSADSA ++ DSUDSU >> 11(2)(2) EFREFR ++ ESAESA ++ ESUESU >> 11(3)(3) GSUGSU >> 11

Ten daytime spots available:Ten daytime spots available:(4)(4) DFRDFR << 1010(5)(5) DSADSA << 1010(6)(6) DSUDSU << 1010

Six evening news spots available:Six evening news spots available:(7)(7) EFREFR << 66(8)(8) ESAESA << 66(9)(9) ESUESU << 66


Define the Constraints (continued)Define the Constraints (continued)Only two Sunday gameOnly two Sunday game--time ad spots available:time ad spots available:

(10)(10) GSUGSU << 22At least 5 ads per day:At least 5 ads per day:

(11)(11) DFRDFR ++ EFREFR >> 55(12)(12) DSADSA ++ ESAESA >> 55(13)(13) DSUDSU ++ ESUESU ++ GSUGSU >> 55

Spend no more than $50,000 on Friday:Spend no more than $50,000 on Friday:(14) 5000(14) 5000DFRDFR + 7000+ 7000EFREFR << 5000050000

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Define the Constraints (continued)Define the Constraints (continued)Spend no more than $75,000 on Saturday:Spend no more than $75,000 on Saturday:

(15) 5000(15) 5000DSADSA + 7000+ 7000ESAESA << 7500075000

Spend no more than $282,000 in total:Spend no more than $282,000 in total:(16) 5000(16) 5000DFRDFR + 5000+ 5000DSADSA + 5000+ 5000DSUDSU + 7000+ 7000EFREFR

+ 7000+ 7000ESAESA + 7000+ 7000ESUESU + 100000+ 100000GSUGSU77 << 282000282000

NonNon--negativity:negativity:DFRDFR,, DSADSA,, DSUDSU,, EFREFR,, ESAESA,, ESUESU,, GSUGSU >> 00


The Management ScientistThe Management Scientist SolutionSolution

Objective Function Value = 199000.000Objective Function Value = 199000.000

VariableVariable ValueValue Reduced CostsReduced CostsDFRDFR 8.0008.000 0.0000.000DSADSA 5.0005.000 0.0000.000DSUDSU 2.0002.000 0.0000.000EFREFR 0.0000.000 0.0000.000ESAESA 0.0000.000 0.0000.000ESUESU 1.0001.000 0.0000.000GSUGSU 2.0002.000 0.0000.000


Solution SummarySolution SummaryTotal new audience reached = 199,000Total new audience reached = 199,000

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

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Financial ApplicationsFinancial Applications

LP can be used in financial decisionLP can be used in financial decision--making thatmaking thatinvolves capital budgeting, makeinvolves capital budgeting, make--oror--buy, assetbuy, assetallocation, portfolio selection, financial planning, andallocation, portfolio selection, financial planning, andmore.more.

Portfolio selectionPortfolio selection problems involve choosing specificproblems involve choosing specificinvestmentsinvestments –– for example, stocks and bondsfor example, stocks and bonds –– from afrom avariety of investment alternatives.variety of investment alternatives.

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

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

Portfolio SelectionPortfolio Selection

Winslow Savings has $20 million availableWinslow Savings has $20 million availablefor investment. It wishes to investfor investment. It wishes to investover the next four months in suchover the next four months in sucha way that it will maximize thea way that it will maximize thetotal interest earned over the fourtotal interest earned over the fourmonth period as well as have at leastmonth period as well as have at least$10 million available at the start of the fifth month for$10 million available at the start of the fifth month fora high rise building venture in which it will bea high rise building venture in which it will beparticipating.participating.


For the time being, Winslow wishes to investFor the time being, Winslow wishes to investonly in 2only in 2--month government bonds (earning 2% overmonth government bonds (earning 2% overthe 2the 2--month period) and 3month period) and 3--month construction loansmonth construction loans(earning 6% over the 3(earning 6% over the 3--month period). Each of thesemonth period). Each of theseis available each month for investment. Funds notis available each month for investment. Funds notinvested in these two investments are liquid and earninvested in these two investments are liquid and earn3/4 of 1% per month when invested locally.3/4 of 1% per month when invested locally.

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Formulate a linear program that will helpFormulate a linear program that will helpWinslow Savings determine how to invest over theWinslow Savings determine how to invest over thenext four months if at no time does it wish to havenext four months if at no time does it wish to havemore than $8 million in either government bonds ormore than $8 million in either government bonds orconstruction loans.construction loans.


Define the Decision VariablesDefine the Decision Variables

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

CCii = amount of new investment in construction= amount of new investment in constructionloans in monthloans 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 ii,,(for(for ii = 1, 2, 3, 4)= 1, 2, 3, 4)


Define the Objective FunctionDefine the Objective Function

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

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

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Define the ConstraintsDefine the Constraints

Month 1's total investment limited to $20 million:Month 1's total investment limited to $20 million:(1)(1) GG11 ++ CC11 ++ LL11 = 20,000,000= 20,000,000

Month 2's total investment limited to principle andMonth 2's total investment limited to principle andinterest invested locally in Month 1:interest invested locally in Month 1:

(2)(2) GG22 ++ CC22 ++ LL22 = 1.0075= 1.0075LL11

oror GG22 ++ CC22 -- 1.00751.0075LL11 ++ LL22 = 0= 0


Define the Constraints (continued)Define the Constraints (continued)

Month 3's total investment amount limited toMonth 3's total investment amount limited toprinciple and interest invested in government bondsprinciple and interest invested in government bondsin Month 1 and locally invested in Month 2:in Month 1 and locally invested in Month 2:

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

oror -- 1.021.02GG11 ++ GG33 ++ CC33 -- 1.00751.0075LL22 ++ LL33 = 0= 0



Month 4's total investment limited to principle andMonth 4's total investment limited to principle andinterest invested in construction loans in Month 1,interest invested in construction loans in Month 1,goverment bonds in Month 2, and locally invested ingoverment bonds in Month 2, and locally invested inMonth 3:Month 3:(4)(4) GG44 ++ CC44 ++ LL44 = 1.06= 1.06CC11 + 1.02+ 1.02GG22 + 1.0075+ 1.0075LL33

oror -- 1.021.02GG22 ++ GG44 -- 1.061.06CC11 ++ CC44 -- 1.00751.0075LL33 ++ 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 >> 10,000,00010,000,000

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

(6)(6) GG11 << 8,000,0008,000,000(7)(7) GG11 ++ GG22 << 8,000,0008,000,000(8)(8) GG22 ++ GG33 << 8,000,0008,000,000(9)(9) GG33 ++ GG44 << 8,000,0008,000,000



No more than $8 million in construction loans atNo more than $8 million in construction loans atany time:any time:

(10)(10) CC11 << 8,000,0008,000,000(11)(11) CC11 ++ CC22 << 8,000,0008,000,000(12)(12) CC11 ++ CC22 ++ CC33 << 8,000,0008,000,000(13)(13) CC22 ++ CC33 ++ CC44 << 8,000,0008,000,000

NonNon--negativity:negativity:GGii,, CCii,, LLii >> 0 for0 for ii = 1, 2, 3, 4= 1, 2, 3, 4


The Management ScientistThe Management Scientist SolutionSolutionObjective Function Value = 1429213.7987Objective Function Value = 1429213.7987VariableVariable ValueValue Reduced CostsReduced Costs

GG11 8000000.0000 0.00008000000.0000 0.0000GG22 0.0000 0.00000.0000 0.0000GG33 5108613.9228 0.00005108613.9228 0.0000GG44 2891386.0772 0.00002891386.0772 0.0000CC11 8000000.0000 0.00008000000.0000 0.0000CC22 0.0000 0.04530.0000 0.0453CC33 0.0000 0.00760.0000 0.0076CC44 8000000.0000 0.00008000000.0000 0.0000LL11 4000000.0000 0.00004000000.0000 0.0000LL22 4030000.0000 0.00004030000.0000 0.0000LL33 7111611.0772 0.00007111611.0772 0.0000LL44 4753562.0831 0.00004753562.0831 0.0000

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Transportation ProblemTransportation Problem

TheThe transportation problemtransportation problem seeks to minimize theseeks to minimize thetotal shipping costs of transporting goods fromtotal shipping costs of transporting goods from mmorigins (each with a supplyorigins (each with a supply ssii) to) to nn destinationsdestinations(each with a demand(each with a demand ddjj), when the unit shipping), when the unit shippingcost from an origin,cost from an origin, ii, to a destination,, to a destination, jj, is, is ccijij..

TheThe network representationnetwork representation for a transportationfor a transportationproblem with two sources and three destinations isproblem with two sources and three destinations isgiven on the next slide.given on the next slide.


Network RepresentationNetwork Representation

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cc1111cc1212

cc1313

cc2121

cc2222cc2323

dd11

dd22

dd33

ss11

s2

SourcesSources DestinationsDestinations

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22

11

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LP FormulationLP FormulationThe LP formulation in terms of the amountsThe LP formulation in terms of the amounts

shipped from the origins to the destinations,shipped from the origins to the destinations, xxijij , can, canbe written as:be written as:

MinMin ccijijxxijijii jj

s.t.s.t. xxijij << ssii for each originfor each origin iijj

xxijij ≥≥ ddjj for each destinationfor each destination jjii

xxijij >> 00 for allfor all ii andand jj

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PowercoPowerco has three electric power plants that supplyhas three electric power plants that supplythe electric needs of fourthe electric needs of four cities.cities.

TheThe associated supply of each plant and demand ofassociated supply of each plant and demand ofeach city is given in the tableeach city is given in the table 1.1.

TheThe cost of sending 1 million kwh of electricity fromcost of sending 1 million kwh of electricity froma plant to a city depends on the distance thea plant to a city depends on the distance theelectricity must travelelectricity must travel..

A transportation problem is specified by the supply,A transportation problem is specified by the supply,the demand, and the shipping costs. So the relevantthe demand, and the shipping costs. So the relevantdata can be summarized in a transportation tableau.data can be summarized in a transportation tableau.The transportation tableau implicitly expresses theThe transportation tableau implicitly expresses thesupply and demand constraints and the shipping costsupply and demand constraints and the shipping costbetween each demand and supply pointbetween each demand and supply point..


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Transportation tableauTransportation tableau

From To

City 1 City 2 City 3 City 4 Supply(Million kwh)

Plant 1 $8 $6 $10 $9 35

Plant 2 $9 $12 $13 $7 50

Plant 3 $14 $9 $16 $5 40

Demand(Million kwh)

45 20 30 30

TransportationTransportationTableauTableau

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1.1. Decision Variable:Decision Variable:Since we have to determine how muchSince we have to determine how muchelectricity is sent from each plant to eachelectricity is sent from each plant to eachcity;city;XXijij = Amount of electricity produced at plant= Amount of electricity produced at plantii and sent to city jand sent to city jXX1414 = Amount of electricity produced at= Amount of electricity produced atplant 1 and sent to city 4plant 1 and sent to city 4


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3131

2. Objective2. Objective functionfunctionSinceSince we want to minimize the total cost ofwe want to minimize the total cost ofshipping from plants to cities;shipping from plants to cities;

Minimize Z = 8XMinimize Z = 8X1111+6X+6X1212+10X+10X1313+9X+9X1414

+9X+9X2121+12X+12X2222+13X+13X2323+7X+7X2424

+14X+14X3131+9X+9X3232+16X+16X3333+5X+5X3434


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3. Supply3. Supply ConstraintsConstraintsSinceSince each supply point has a limited productioneach supply point has a limited productioncapacity;capacity;

XX1111+X+X1212+X+X1313+X+X1414 <= 35<= 35XX2121+X+X2222+X+X2323+X+X2424 <= 50<= 50XX3131+X+X3232+X+X3333+X+X3434 <= 40<= 40


3333

4. Demand4. Demand ConstraintsConstraintsSinceSince each supply point has a limited productioneach supply point has a limited productioncapacity;capacity;

XX1111+X+X2121+X+X3131 >= 45>= 45XX1212+X+X2222+X+X3232 >= 20>= 20XX1313+X+X2323+X+X3333 >= 30>= 30XX1414+X+X2424+X+X3434 >= 30>= 30


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3434

5. Sign5. Sign ConstraintsConstraintsSinceSince a negative amount of electricity can not bea negative amount of electricity can not beshipped allshipped all Xij’sXij’s must be non negative;must be non negative;

XijXij >= 0 (>= 0 (ii= 1,2,3; j= 1,2,3,4)= 1,2,3; j= 1,2,3,4)


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LP Formulation of Powerco’s ProblemLP Formulation of Powerco’s Problem

Min Z = 8XMin Z = 8X1111+6X+6X1212+10X+10X1313+9X+9X1414+9X+9X2121+12X+12X2222+13X+13X2323+7X+7X2424

+14X+14X3131+9X+9X3232+16X+16X3333+5X+5X3434

S.T.:S.T.: XX1111+X+X1212+X+X1313+X+X1414 <= 35<= 35 (Supply Constraints)(Supply Constraints)XX2121+X+X2222+X+X2323+X+X2424 <= 50<= 50XX3131+X+X3232+X+X3333+X+X3434 <= 40<= 40XX1111+X+X2121+X+X3131>= 45>= 45 (Demand Constraints)(Demand Constraints)XX1212+X+X2222+X+X3232>= 20>= 20XX1313+X+X2323+X+X3333>= 30>= 30XX1414+X+X2424+X+X3434>= 30>= 30Xij >= 0 (i= 1,2,3; j= 1,2,3,4)Xij >= 0 (i= 1,2,3; j= 1,2,3,4)

chapter 3 linear programming applications 8000000.0000 0.0000 c2 0.0000 0.0453 c3 0.0000 0.0076 c4...

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