maths university

Exam

Problem 1:

(a)The company Optimiza SUN makes two types of products:Sunscreen Lotion and Sun- screen Spray. Each bottle of Sunscreen Lotion is made from 30grams(g) of Gross Chemical and 40g of Grease,whileeachbottleofSunscreenSprayismadefrom50gofGrossChemicaland10g of Grease .Sunscreen Lotionsells for$15 a bottle,while Sunscreen Spray is priced at $20 per bottle. These data are summarized in the following table:

Optimiza SUN has a daily supply of 1000g of Gross Chemical(already paid for).It canpurchase as much Grease as it want sat$0.01 per gram from a nearby restaurant.

Given the data above,give a linear program formulation that finds the daily production plan that maximizes Optimiza SUN’s profit(i.e. total value of bottles produced minus production costs). De- fine your decision variables clearly, and briefly describe your objective function and each of your constraints .Solutions without adequate explanations will not receive full credit.

(b)Suppose,in addition to the datain(a),Optimiza SUN now wants to make sure that the number of bottles of Sunscreen Lotion and Sunscreen Spray produced per day are within 10 of each other.Also,it can now sell unused Gross Chemical for $0.05 per gram.

Modifyyourlinear programin(a)tofindthemostprofitabledailyproductionplanforOptimizaSUN thattakesthesenewconditionsintoaccount.Briefly explain eachofyourmodifications.Solutions withoutadequateexplanationswillnotreceive fullcredit.

Problem 2:

Ford hasfourautomobileplants. Each iscapableofproducingtheTaurus, Lincoln,orEscort, butitcan onlyproduce oneofthesecars.Thevariable costofproducingacarofeachtypeateachplant isgivenin the followingtable:

VariableCost($) Plant Taurus Lincoln Escort

Fordfacesthefollowingrestrictions:-

(i)Eachplant canproduceonlyonetypeo fcar.

(ii)Thetotalproductionofeachtypeofcarmustbeatasingleplant; thatis,forexample, ifany Taurusesaremadeatplant1,thenallTauruses mustbemadethere. Eachyear,Fordmustproduce 500000ofeachtypeofcar.

(b)Fordfacesthefollowingrequirementsinadditionto(i)and(ii)above.

(iii)Ifplant2isused,thenTauruses needtobemanufactured atthisplant. (iv)Ifbothplants3and4areused,thenplant1mustalsobeused.

Also,inadditiontothe variable costincurredbytheproductionofacar,Fordalsoincursaone-time, fixedopeningcostforproducingcarsateachplant.Thesefixedcostsaregiveninthefollowingtable.

Hereisanexample forthesakeofillustration. Suppose thatPlant1isusedtoproduce Escorts, Plant2isusedtoproduceTauruses, andPlant3isusedtoproduceLincolns.Thisproduction plan wouldincurafixedcostof(7×109+6×109+4×109) =17 ×109,plusavariable costof (9×103×5×105)+(15×103×5×105)+(19×103×5×105)=21.5×109,resultinginatotal costof$38.5billion.

Howwould your IPfrom(a)changeinordertoincorporatethenewrequirements(iii)and(iv),and inordertominimizethesumoffixedandvariablecosts? Brieflyexplaineachofyourmodifications. Solutionswithout adequateexplanationswillnotreceive fullcredit.