Discrete approximationheuristicsforthecapacitatedcontinuouslocation–allocation problemwithprobabilisticcustomerlocations

The capacitatedcontinuouslocation–allocationproblem,alsocalledcapacitatedmultisourceWeberprob- lem (CMWP),isconcernedwithlocating m facilities intheEuclideanplane,andallocatingtheircapacity to n customers atminimumtotalcost.Thedeterministicversionoftheproblem,whichassumesthat customer locationsanddemandsareknownwithcertainty,isanonconvexoptimizationproblem.Inthis work, wefocusonaprobabilisticextensionreferredtoastheprobabilisticCMWP(PCMWP),andconsider the situationinwhichcustomerlocationsarerandomlydistributedaccordingtoabivariateprobability distribution. Wefirstformulatethediscreteapproximationoftheproblemasamixed-integerlinear programming modelinwhichfacilitiescanbelocatedonasetofcandidatepoints.Thenwepresentthree heuristics tosolvetheproblem.Sinceoptimalsolutionscannotbefound,weassesstheperformanceof the heuristicsusingtheresultsobtainedbyanalternatelocation–allocationheuristicthatisoriginally developed forthedeterministicversionoftheproblemandtailoredbyusforthePCMWP.Thenew heuristics dependontheevaluationoftheexpecteddistancesbetweenfacilitiesandcustomers,which is possibleonlyforafewnumberofdistancefunctionandprobabilitydistributioncombinations.We therefore proposeapproximationmethodswhichmaketheheuristicsapplicableforanydistancefunction and probabilitydistributionofcustomercoordinates.