BUS 660 Topic 3 Introduction to Linear Programming - Homework

Problem 8-25(Algorithmic) GeorgiaCabinetsmanufactureskitchencabinetsthataresoldtolocaldealersthroughouttheSoutheast.Becauseofalargebacklogofordersfor oakand cherrycabinets,the company decidedto contract with threesmaller cabinetmakersto do the final finishing operation. For thethree cabinetmakers, the number of hours required to complete all the oak cabinets, the number of hours required to complete all the cherrycabinets,thenumberofhoursavailableforthefinalfinishingoperation,andthecostperhourtoperform theworkare shownhere: For example, Cabinetmaker 1 estimates that it will take 47 hours to complete all the oak cabinets and 60 hours to complete all the cherrycabinets.However,Cabinetmaker1onlyhas40hoursavailableforthefinalfinishingoperation.Thus,Cabinetmaker1canonlycomplete40/47=0.85,or85%,oftheoakcabinetsifitworkedonlyonoakcabinets.Similarly,Cabinetmaker1canonlycomplete40/60=0.67,or67%,ofthecherrycabinetsifitworkedonlyoncherrycabinets. a. Formulate a linear programming model that can be used to determine the proportion of the oak cabinets and the proportion of thecherry cabinetsthatshould be given to each ofthe three cabinetmakersin orderto minimize the total cost of completing bothprojects. LetO1=proportionofOakcabinetsassignedtocabinetmaker 1O2=proportionofOakcabinetsassignedtocabinetmaker2O3=proportionofOakcabinetsassignedtocabinetmaker3C1 = proportion of Cherry cabinets assigned to cabinetmaker 1C2 = proportion of Cherry cabinets assigned to cabinetmaker 2C3=proportionofCherrycabinetsassignedtocabinetmaker3 b. Solve the model formulated in part (a). What proportion of the oak cabinets and what proportion of the cherry cabinets should beassignedtoeachcabinetmaker?Whatisthetotalcostofcompletingbothprojects?Ifrequired,roundyouranswersfortheproportions tothreedecimalplaces,andforthetotalcosttotwodecimalplaces. c. IfCabinetmaker1hasadditionalhoursavailable,wouldtheoptimalsolutionchange? d. IfCabinetmaker2hasadditionalhoursavailable,wouldtheoptimalsolutionchange? e. Suppose Cabinetmaker 2 reducedits cost to$39 perhour.Whateffect would thischangehaveon the optimal solution? If required,roundyouranswersfortheproportionstothreedecimalplaces,andforthetotalcosttotwodecimalplaces. 1. Problem8-19 Better Products, Inc., manufactures three products on two machines. In a typical week, 40hoursareavailableoneachmachine.Theprofitcontributionandproductiontimeinhoursperunitareasfollows: Two operators are required for machine 1; thus, 2 hours of labor must be scheduled for eachhour of machine 1 time. Only one operator is required for machine 2. A maximum of 100 labor-hours is available for assignment to the machines during the coming week. Other productionrequirements are that product 1 cannot account for more than 50% of the units produced andthatproduct3mustaccountforatleast20%oftheunits produced. a. How manyunitsofeachproductshouldbe producedtomaximize thetotalprofitcontribution? What isthe projected weekly profitassociated withyour solution? b. Howmanyhoursofproductiontime willbescheduledoneachmachine?Ifrequired,roundyouranswerstotwodecimalplaces. c. What isthe value ofanadditionalhouroflabor?Ifrequired,roundyouranswerstotwodecimalplaces. d. Assumethatlaborcapacitycanbeincreasedto120hours.Developtheoptimalproductmix,assumingthattheextrahoursaremadeavailable.Wouldyoubeinterestedinusingtheadditional20hoursavailableforthisresource? Problem 7-49(Algorithmic) PharmaPlus operates a chain of 30 pharmacies. The pharmacies are staffed by licensed pharmacists and pharmacy technicians. The companycurrently employs 85 full-time-equivalent pharmacists (combination of full time and part time) and 175 full-time-equivalent technicians. Eachspring management reviews current staffing levels and makes hiring plans for the year. A recent forecast of the prescription load for the nextyear shows that at least 250 full-time-equivalent employees (pharmacists and technicians) will be required to staff the pharmacies. Thepersonnel department expects 10 pharmacists and 30 technicians to leave over the next year. To accommodate the expected attrition andprepare for future growth, management states that at least 15 new pharmacists must be hired. In addition, PharmaPlus’s new service qualityguidelines specify no more than two technicians per licensed pharmacist. The average salary for licensed pharmacists is $40 per hour and theaveragesalaryfortechniciansis$10perhour. a. Determineaminimum-coststaffingplanforPharmaPlus.Howmanypharmacistsandtechniciansareneeded? LetP=numberoffull-timeequivalentpharmacists T = numberoffull-timeequivalent technicians b. Givencurrentstaffinglevelsandexpectedattrition,howmanynewhires (ifany)mustbemadetoreachthelevelrecommendedinpart(a)?Whatwillbe theimpacton thepayroll? Problem7-15 Par, Inc., is a small manufacturer of golf equipment and supplies. Par’s distributor believes a market exists for both a medium-pricedgolf bag, referred to as a standard model, and a high-priced golf bag, referred to as a deluxe model. The distributor is so confident ofthe market that, if Par can make the bags at a competitive price, the distributor will purchase all the bags that Par can manufactureover the next three months. A careful analysis of the manufacturing requirements resulted in the following table, which shows theproduction time requirements for the four required manufacturing operations and the accounting department’s estimate of the profitcontributionperbag: The directorof manufacturing estimates that 630hours ofcuttingand dyeing time, 600hours of sewing time, 708hours of finishingtime, and135 hoursof inspectionandpackagingtimewillbeavailablefortheproductionofgolfbagsduringthenextthreemonths. SelectthecorrectgraphthatshowstheoptimalsolutionifPar'smanagementencountersthefollowingindependentsituations: a. The accountingdepartmentrevisesits estimateof theprofitcontributionfor thedeluxebagto$18perbag. b. Anewlow-costmaterialisavailableforthestandardbag,andtheprofitcontributionperstandardbagcanbeincreasedto$20perbag.(Assumethattheprofitcontributionofthedeluxebagistheoriginal$9value.) c. Newsewingequipmentisavailablethatwouldincreasethesewingoperationcapacityto750hours.(Assumethat10A+9Bistheappropriateobjectivefunction.) Problem7-41 SouthernOilCompany producestwo gradesof gasoline: regularand premium. The profit contributions are $0.30 per gallon for regulargasolineand$0.50pergallonforpremiumgasoline.Eachgallonofregulargasolinecontains0.3gallonsofgradeAcrudeoilandeachgallonof premium gasoline contains 0.6 gallons of grade A crude oil. For the next production period, Southern has 18,000 gallons of grade A crude oilavailable. The refinery used to produce the gasolines has a production capacity of 50,000 gallons for the next production period. Southern Oil'sdistributorshaveindicated that demand for thepremiumgasolinefor thenext production period willbeat most 20,000gallons. a. Formulatealinearprogrammingmodelthatcanbeusedtodeterminethenumberofgallonsofregulargasolineandthenumberofgallons of premium gasoline that should be produced in order to maximize total profit contribution. If required, round your answers totwodecimalplaces. LetR=numberofgallonsofregulargasolineproduced P=numberofgallonsofpremiumgasolineproduced b. Whatistheoptimalsolution? c. Whatarethevaluesandinterpretationsoftheslackvariables? d. Whatarethebindingconstraints? Problem 7-25(Algorithmic) GeorgeJohnsonrecentlyinheritedalargesumofmoney;hewantstouseaportionofthismoneytosetupatrustfundforhistwochildren.Thetrustfundhastwoinvestmentoptions:(1)abondfundand(2)astockfund.Theprojectedreturnsoverthelifeoftheinvestmentsare6%for the bond fund and 10% for the stock fund. Whatever portion of the inheritance George finally decides to commit to the trust fund, he wantstoinvestatleast30%ofthatamountinthebondfund.Inaddition,hewantstoselectamixthatwillenablehimtoobtainatotalreturnofatleast7.5%. a.Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possibleinvestmentalternatives.Ifrequired,roundyouranswerstothreedecimalplaces. LetB= percentageoffundsinvestedinthebondfund S= percentageoffundsinvestedinthestockfund b. c. Solve the problemusingthegraphicalsolutionprocedure.If required,roundtheanswerstoonedecimalplace. Problem 8-21(Algorithmic) RoundTreeManorisahotelthatprovidestwotypesofroomswiththreerentalclasses:SuperSaver,Deluxe,andBusiness.Theprofitpernightforeachtypeofroomandrentalclassisasfollows: TypeIroomsdonothavewirelessInternetaccessandarenotavailablefortheBusinessrentalclass. RoundTree'smanagementmakesaforecastofthedemandbyrentalclassforeachnightinthefuture.Alinearprogrammingmodeldevelopedtomaximizeprofitisusedtodeterminehowmanyreservationstoacceptforeachrentalclass.Thedemandforecastforaparticularnightis125rentalsintheSuperSaverclass,50rentalsintheDeluxeclass,and40rentalsintheBusinessclass.RoundTreehas110TypeIroomsand120TypeIIrooms. a. Uselinearprogrammingtodeterminehowmanyreservationstoacceptineachrentalclassandhowthereservationsshouldbeallocatedtoroomtypes. Isthedemandbyanyrentalclassnotsatisfied? b. Howmanyreservationscanbeaccommodatedineachrentalclass? c. ManagementisconsideringofferingafreebreakfasttoanyoneupgradingfromaSuperSaverreservationtoDeluxeclass.IfthecostofthebreakfasttoRoundTreeis$5,shouldthisincentivebeoffered? d. With a little work, an unused office area could be converted to a rental room. If the conversion cost is the same for both types ofrooms,wouldyourecommendconvertingtheofficetoaTypeIoraTypeIIroom? e. Couldthe linearprogrammingmodelbe modifiedtoplanforthe allocationofrentaldemandforthe nextnight? Problem 8-29(Algorithmic) La Jolla Beverage Products is considering producing a wine cooler that would be a blend of a white wine, a rose wine, and fruit juice. To meettastespecifications,thewinecoolermustconsistofatleast45%whitewine,atleast20%andnomorethan35%rose,andexactly25%fruitjuice. La Jolla purchases the wine from local wineries and the fruit juice from a processing plant in San Francisco. For the current productionperiod,10000gallonsofwhitewineand 6500gallonsofrosewinecanbepurchased;anunlimitedamountoffruitjuicecanbeordered.Thecostsforthewineare$1pergallonforthewhiteand$1.5pergallonfortherose;thefruitjuicecanbepurchasedfor$0.5pergallon.LaJollaBeverageProductscansellallofthewinecooleritcanproducefor$3pergallon. a. Isthecostofthewine and fruit juiceasunkcost ora relevant costin thissituation? b. Formulatealinearprogramtodeterminetheblendofthethreeingredientsthatwillmaximizethetotalprofitcontribution.SolvethelinearprogramtodeterminethenumberofgallonsofeachingredientLaJollashouldpurchaseandthetotalprofitcontributionitwillrealizefromthisblend.Ifrequired,roundyouranswerstoonedecimalplace.Forsubtractiveornegativenumbersuseaminussignevenifthereisa+signbeforetheblank. c. IfLaJollacouldobtainadditionalamountsofthewhitewine,shoulditdoso? d. IfLaJollaBeverage Productscouldobtainadditionalamountsofthe rosé wine,shoulditdoso? e. Interprettheshadowpricefor theconstraintcorrespondingtotherequirementthatthewinecoolermust containatleast50%whitewine.Whatisyouradvicetomanagementgiventhisshadowprice? f. Interprettheshadowpricefor theconstraint correspondingtotherequirementthatthe winecoolermust containexactly20% fruitjuice.Whatisyouradvicetomanagementgiventhisshadowprice?