# Logistic Modelling

## Problem:RoutingofFruitandVegetableDistribution

Afruitandvegetable wholesalerdistributesfruits andvegetables toretailers primarily locatedin the SouthEastofEngland.Thefruitsandvegetablesarepackagedinto handerecttrays.Ahomogeneous fleetof9vansdeliverthetraysoffruitsand vegetablesto64retailers.Eachretailerhasaspecificquantityofdemandforthefruits andvegetables,andeachvanhas alimitedcapacityfor carrying100trays. Thevans depart fromthedistributioncentretoservicetheretailerswithknownlocationsand demands, andthenreturntothedistributioncentre. Thelocations oftheretailersare expressedusing(x,y)coordinatesontheEuclideanplane. Thecoordinatesofthe locationsandthedemandsaredescribedinthefile“fruit-vegetable-distribution.xls”on Moodle.

Themainobjectiveistoassignthevanstoroutesto deliverthe fruits andvegetables totheretailerssoastominimisethetotaldistancetravelledbythevans,subjecttothe followinghardconstraints:

Eachvandepartsfrom andreturnstothedistributioncentre.

Eachretailer is visitedexactlyonce byasingle van.

Eachvanhasalimitedcapacityof 100trays.

Thenumber ofvansavailableatthe distributioncentreis 9.

1. 1. Formulate mathematicallythe optimisationmodel of the fruit and vegetable distributionroutingproblemtominimisethetotaldistancetravelledby thevan Describeclearlyindetailyouroptimisationmodel:fitness function,decision variables, andconstraints.Provide anyreferences intheliterature youhave used.

## [Maximum 300words excluding references] [10 marks]

2.Consider thedata described inthefile“fruit-vegetable-distributiong.xls”onMoodle.

The datadescribes thecoordinates ofthelocations andthe demands ofretailers, coordinatesofthelocationofthedistribution centre,capacityofthevans,andthe numberofvansavailable atthedistributioncentre.

SetuptheExcel spreadsheetandoptimisationmodelsonEvolverandexplain clearlytheimplementationofthedecisionvariables,theconstraints,andthefitness function.IndicatewhichEvolversolvingmethodyouhaveusedtosolvetheproblem andtheinitial solution you have used.

Solvetheproblemtofindthe best minimum distancetravelledby experimenting with thefollowingparameters:

(a) Populationsize:30and 50

(b) Foreachpopulationsize,considerthefollowingcrossoverandmutation rates:

(i)  Crossover rate=0.8,mutationrate= 0.01. (ii) Crossover rate= 0.9,mutationrate= 0.02. (iii)Crossover rate= 0.6,mutationrate= 0.35. (iv)Crossoverrate= 0.4,mutationrate= 0.8.

(c)Thestoppingcriterionfor alltheexperimentsshouldbesetto20000 generations.

Foreachcombinationprovideadetaileddescriptionofthebestsolution:totaldistance travelled,set ofroutesto betravelledby thevans,computationtimerequiredto generatethebestsolution.For eachvan,specify theretailerstovisitandtheorderin whichtheyhave tobevisited.

Provideagraphicalillustrationoftheroutesofthevans fortheinitialsolutionandthe overall bestsolutionfor eachpopulationsize.

### [Maximum 1200words] [35marks]

1. 3. Thebestsolution,b-best,foundintheliteratureforthedataprovidedis1174.

Analyseyourresultsanddiscussthe effectofthepopulation size,crossover and mutationrates onthe qualityofthebestsolutionsforthedifferentcombinations and how thesesolutionscomparetob-best. Provideagraphicalillustrationto comparethequalityofthebestsolutions foundforthe differentpopulationsize, crossover andmutationratecombinationsand howtheycomparetob-best.

[Maximum 500words]

[15marks]

1. 4. SolvetheproblemusingtheTabusearch(TS)methodwithOptQuestand thesamestoppingconditionasGA, 20000iterations.Comparethesolution qualityandthecomputationtimeofthebestsolutionfoundwithGAandTS. Discussandargumentthedifferenceinthesolutionqualityandcomputationtime ofbothmethods,andhowthesolutionofTScomparestob-best.

#### [Maximum 500words] [15marks]

5.GA,TS,andSimulatedAnnealing(SA)havebeenwidelyusedintheliterature to        solve  the  capacitated  vehicle  routing  problem  with  wide-ranging applications inthefields oflogistics and transportation.Discussandcompare theperformance ofGA,TS, andSAinterms solutionqualityandcomputation time insolving thecapacitatedvehiclerouting problem.  Illustratethe performanceanalysisofthesemethodsandthecomparativestudywithsome experimental resultsoncasestudies fromtheliterature.