Appunti riassuntivi secondo parziale Optimization and data science for management

Travelling salesman TSP problemi

The directed associated estsquaple /N with and problem Given G avea = , ,, permutation I that determine /modecycle to asks a :at starting aubuituamily depotepointmode designed Begin.

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• Exactly mode Visit each
• One twowelledest Isum est Have the the minimum of of

Mathematical formulation

min dijxijj)ri en, Vien1Xij =jefst VjEN1Tij =ebsiji Sne susN0stAsse NXijk1 = ,.Kijiealse) integerF/i j) A and&Xij 0 ,

Special TPS is VRP

Of casea ITSP The Travelling Problem With loaderistic Salesman the VRP is following adenotsingle· a to the/whielvelviele salesmantcorresponssingle k 1· a =, Nconstraintcapacity the luncapacitated) divelvicle ,no on· comstwaintsoflueu abditionalno·

Vehicle routing definition problem VRP

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The deportwhichbydescribedroad guaple socesmondmodesis ina ,toclient colucespandlocations Anc be:andand readsand cam., timefiowelling Directed andsosteachyou.

• = And Umdirected knowm
• Auecliemtsclients requirementswith specificmodesa re :metworkclient

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Assumption Quiveus vehicles: Capacitated COVRP Vehicle Problem Routing

Assumptions

DeliveryClients anddemand commodityandask knownsingleservice is1. you a ,unsplittable athomogenedus centralbased denotVehicles. velubles.

Additional capacityvelicle.

Notation

A) complete/V guaplisG· a= , my50 WhereV 1· = ...,,, denotthe"d')row is ando· m) flue11 clientsidentifiesset n· , ...,twavelling toj)A est : jFli grom0cij· , the pudmeutymote most pactical applications trangularein satusfysosts= Sjisik, exj: .

Preprocessing

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Lower Kmin

DiA fuinial boundlower (B) Kninon :This be inascuratecan .Kmin the optimal solution PackingBin problempuduitedalso ofbe -1acan as* Bin Packing Problem0-1

Two formulation

index 11 Flightmarteinj) isif (iXij some= .ofleuwiseOcomstwaintscapacityOut 22- twogoedConstwaint Is) urleplayesc a:, , the elimination constraintsguarantee commestion with bubtodenotThey.

• Thetheguarantee constwaintThey capacity vehicles
• Onthe theparameter computedis pleasedo minimumre representsandpremesessingbe ina to the byvisitrequired alientsvehicles set /S) computedbeallnumber Sinof can

Staffordable instancedsolving Bin Packing real-sizeeachO-1 problem VRDyou on22 GSEdVS the Constraintsequivalentreplacedbe by EliminationGeneralized Subto2 GSEScam sevisorTijzi-vis se,sties jesNigea , at mostThey thesetheupperbound number Sinsideofimpose ison auss soan a,the thatthe within Equivalentlylimit thisroads usednumberon of group means. ,leastat /S) veluidles seti leave .5a re s e ,, .,.Milleu-Tucken-Zemlin (MT) comstwaintsfoalternativeThey Theyused amb/C polynonialEdandbe in numberas a reamcam . ,theformulated thebeliverywithers nickbe for gourancasecan caseun .disuise ijfoViel /EA-di Fliexijuj-ui + ,the the visitsitwhere load ivehicle begane modeisht ofcomstwaints

Coursestness MTE of iconstwaints

The Klij)-di Aexij jouj-ui + ,twogredPlay mole :a diIf.1 Xij 1 uij += , theis due comstwaintsredundant thetoUj-uixG-di box variablesIf 02 Xij on= ,. Viene(di uj) andbetween

Equivalence GSEe22220-GSEG

(1) andeach toconsider theconstraint CRP j incoming equalmode isin gou flow,Tij 1=jeB/j) toconstraints the whichthese belongeall modes 5 TijSum 1you : =jesijessg)Separate eterand flows followsinner as : 15tijies tijjes tijt =isjeslijiessijjes / inner flox auter flowconstraintsthe Wald vistijksince ese : jessthus 161- Xij1(5) -(s)Xij -=, jesissins jesthis complete theand mudgGSES 200>- thatSymmetrically know 16isjestities jesti:we =, that 1151-vis(GSEC)In guaranteeaddition Tij, isjesThus , 161-isjesTij11-115) Tij =isjesgetmulfiplying by se1 we,

Two-index goumulation

glowmin dijxijlijet Vjzv1809Xij 1=li jeBsj, Vjer1jagji 1(ilefs(j) = KXio =11 012A, kXoi =il10 &A, VievljogdiYijYa Tislessi =bilebs . diYouYjo =- Tenjoyla jiefsioljosebsiol /i j)O A&Oxij&Yij ,1)Xijeg0 V(i j) A, ,theWhile advancedexpomential/GSE andmodels requireprevious aretheBianchlike and problegolumulationsandalgotlums Out FlowMTS modea, bothmathematical that limits andstructure luamblespolynomialLizest subtocapacity, simultaneouslyelimination .

Set partitioning model VRP

HIGH Hay the byclesset routesalldegine geasible isHjWe Ginof ias a ae.. ,... . .,at thebenet VIcapacitystarting satuogies comstwaintsroute the whichendingand 1 q...9min j1 jxj9 juVieV1119ijxj =j =9 kxj =j 1= 1)(0 Vi 1, ...,xj = 9, =foumulation

Three texin mathematical model

Kmin dij Xijk/i j) Ak 1= ,K View1Fijk =dietsir= 1 Yx kTojk 1 1= = , ...,10 Esposj), KVk VienTijk 1,oLiebsi ik = = ...,injiefsir Xx k1 1Tjn = = ....,1k+Ijimt1/ebsim + VK FliiglFij/Wik AWjx) K &0 1Xijk + +Si - = ...,, VKbi VieNTijk KWik Xijkai 1= ...,,vijiefsir vigetsi 19Vk ViegoEWik L k1= +m= ,i, . . . YkDi K=dXijk 1= , ...,ien Vi jefslip, 19&20 Yk j)VliK AXijk 1,=, ,...,

Observations

Comstwaintstij-wjx/comstwaintsObservation mon-linearXijk(Wik if1 1oSi +: Xijkave+· =limeaufluetheybutthem by followingreplacedWiksittij beWik camcomstwaints tij-Wjk/1-Xijk/Mij VK Vi &AkWiksi 1,+ = ,...,

With time URP Windows-VRPTW

The timingcapacitatedIt extende constraintsWith URP .Two trime /TW)Windowskinds of :they flue butoutsidewislated allowedSoft be it1 service isTW ini s u e scan ae .,,. .pemalty begarec lientattheyHand cammot theabdition vehiclebe avveswidlated In.2 if a: ,.theirthe wait.beginning mustTW itof ,

VRPTW-Haud TWonstwaints

& bidSai atthe timestopsand unitsat veluidei stauts iThe service client in.1 gou si, unfulmustat waitthe vehicle itIf beforeianives ai a2 ,.

Hynothesis assumptions

Andtime theWindow benotassociated with 0 andmodes withHP1 also modesis i .1+e: a n.,, .Cam 1)But ESao L)buspecifically =. = ,, +, latestearliest bepot thethewhere start andtime theis isLE gram· to thetimeavving deport .nawameters the deporttime withassociateddemandand alsomodesService iHP2 , . ,a re: e.with and Andomodes SpecificallyO non so 0In = += + = =1,

Feasibility conditions

Ibi-toilconditions WoldtheIf following EXmaxienf ac. = timesbu laitminierL sit· =mightThem existsolutionsgeasible, .

VRPTW - Soft Windows

Kmin PiyiTiendijijkvijieaKen Viengizo K VienWik-biyik 1= ViewVK1? kWik Xijkairije ...,,stie VKbi Vie NkTijk 1,Wik yi = ...,rigers