Linear Programming for Cost Minimization in Survey Sampling at AmeriBank

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Linear Programming for Cost Minimization in Survey Sampling atAmeriBankCASE3.2 NEW FRONTIERSRob Richman, president of AmeriBank is bothered about the state of his bank.AmeriBank, amedium-sizedbankwithbranchesthroughouttheUnitedStates,isheadedfordireeconomicstraits.Thebankissimplylosingcustomerstononbankandforeignbankcompetitors.ManyAmericanbankshavebeensufferingsignificantlossesbecauseofincreasingcompetitionfromnonbankandforeignbankcompetitorsofferingservicestypically in the domainof American banks.RobhasonestrategyfordistinguishingAmeriBankfromitscompetitorsistoimproveservicesthatnonbankandforeignbankcompetitorsdonotreadilyprovidetransactionservices. Internet allows this convenient transactionmethod. Over the Internet, customersare able to performtransactions on their desktop computers either at home or atwork.Rob needs to understand the market forWeb banking and the services AmeriBank shouldprovideovertheInternet.Likebalanceinquiries,transactions,limitations,recaptureaportionoftheinvestmentmarketbycontinuouslyrunningstockpricesandallowingcustomers to make stock transactions over the Internet for a minimal fee.Rob has decided to outsource the survey projectto a professional survey company. He hasopened theproject up for bidding by several survey companies and willaward the project tothe company which is willing to performthe survey for the least cost.Sophisticated Surveys is one of three survey companiescompeting for the project. Robprovided each survey companywith a list of survey requirements to ensure thatAmeriBank receives the needed information for planning theWeb banking project.Hetriestoacquire informationabouttheage groupsof accountholders,theirinternetusage ability, income levels,and investmentbehavior. AmeriBank wants to interview2,000customers with at least 20 percent from the firstage group, at least 27.5 percent from thesecond age group,at least 15 percent from the third age group, and at least 15percentfrom the fourth age group. Rob wants to ensure that the survey includes a mix of customerswho know the Internet well and those that have less exposureto the Internet. To ensurethatAmeriBankobtainsthecorrectmix,hewantstointerviewatleast15percentofcustomersfromtheSiliconValleywhereInternetuseishigh,atleast35percentofcustomersfrombigcitieswhereInternetuseismedium,andatleast20percentofcustomers from smalltowns where Internet use is low.Sophisticated Surveys has performed an initial analysisof these survey requirements todetermine the cost of surveyingdifferent populations. The costs per person surveyedarelisted in the following table:

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1Sophisticated Surveys explores the forthcoming options cumulatively.(a)Formulate a linear programming model to minimize costs while meeting all surveyconstraints imposed by AmeriBank.Ans.Let,X1 is the no of people fromSilicon Valleyunder age group 18-25X2 is the no of people fromSilicon Valleyunder age group 26-40X3 is the no of people fromSilicon Valleyunder age group 41-50X4 is the no of people fromSilicon Valleyabove age group 51X5 is the no of people from big-cities under age group 18-25X6 is the no of people from big-cities under age group 26-40X7 is the no of people from big-cities under age group 41-50X8 is the no of people from big-cities above age group 51X9 is the no of people from small towns under age group 18-25X10 is the no of people from small towns under age group 26-40X11 is the no of people from small towns under age group 41-50X12 is the no of people from small towns above age group 51Min z=4.75 x1+ 6.50 x2+ 6.50 x3+5 x4+5.25 x5+5.75 x6+ 6.25 x7+ 6.25 x8+ 6.50 x9+7.50 x10+ 7.50 x11+ 7.25 x12Constraints:x1+ x2+ x3+ x4+ x5+ x6+ x7+ x8+ x9+ x10+ x11+ x12=2000 ……………………….……(1)x1+ x2+ x3+ x4>=300………………………………………………………………………………..…. (2)x5+ x6+ x7+ x8>=700 …………………………………………………………………………………. (3)x9+ x10+ x11+ x12=>400……………………………………………………………………………... (4)Region18 to 2526 to 4041 to 5051 and overSilicon Valley$4.75$6.50$6.50$5.00Big cities$5.25$5.75$6.25$6.25Small towns$6.50$7.50$7.50$7.25

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2x1+x5+x9>=400 …………………………………………………………………………………….……. (5)x2+x6+x10>=5500………………………………………………………………………………………. (6)x3+x7+x11>=300………………………………………………………………………………………... (7)x4+ x8+ x12>= 300……………………………………………………………………………….….…. (8)x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12>=0………………………………….……..…..(9)X1=600X2=0X3=0X4=300X5=0X6=550X7=150X8=0X9=250X10=0X11=150X12=0

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