Combining Vehicle Routing with Forwarding

Transcription

1 Combnng Vehcle Routng wth Fowang Extenon of the Vehcle Routng Poblem by Dffeent Type of Sub-contacton Hebet Kopfe Xn Wang Cha of Logtc, Unvety of Bemen, WHS 5, D Bemen, Gemany Coeponng autho: Pof. D. Hebet Kopfe, Cha of Logtc, Depatment of Bune Stue & Economc, Unvety of Bemen, Wlhelm- Hebt- Staße 5, D Bemen, Gemany, Tel , mal: Abtact: The effcency of tanpotaton equet fulfllment can be nceae though extenng the poblem of vehcle outng an cheulng by the poblty of ubcontactng a pat of the equet to extenal cae. Th poblem extenon tanfom the uual vehcle outng an cheulng poblem to the moe geneal ntegate opeatonal tanpotaton poblem. In th contbuton, we analyze the motvaton, the chance, the ealzaton, an the challenge of the ntegate opeatonal plannng an epot on expement fo extenng the plan Vehcle Routng Poblem to a coeponng poblem combnng vehcle outng an equet fowang by mean of ffeent ub-contacton type. The extene poblem fomalze a a mxe ntege lnea pogammng moel an olve by a commecal mathematcal pogammng olve. The computatonal eult how temenou cot avng even fo mall poblem ntance by allowng ubcontactng. Atonally, the pefome expement fo the opeatonal tanpotaton plannng ae ue fo an analy of the econ on the optmal fleet ze fo own vehcle an egulaly he vehcle. Keywo: VRP wth ubcontactng, own fleet, extenal cae, balancng ffeent moe of fulfllment, tuc fleet ze, chey-pcng Acnowlegement Th eeach wa uppote by the Geman Reeach Founaton (DFG) a pat of the Collaboatve Reeach Cente 637 Autonomou Coopeatng Logtc Pocee A Paagm Shft an t Lmtaton (Subpoject B9). 1

2 1. Intoucton Tanpotaton plannng eque both, econ on the avalable tanpot eouce an econ on the eployment of the ue eouce. Theefoe, tanpotaton oe (epeente by cutome tanpotaton equet) mut be agne to eouce fo fulfllment whle the opeaton o conton fo the uage of each eouce have to be etemne. Mot feght fowang compane have to cope wth a tongly fluctuatng eman on the tanpotaton maet whch vae coneably ove tme. Ae fom thee long-tem fluctuaton they have to manage the aly vaaton of the volume of oe. Each ay a vayng numbe of equet ae eceve fom cutome on hot call. Theefoe, feght fowang compane have to enue that enough eouce wll be pove. On the othe han, the fxe cot of the own vehcle fleet (contng e.g. n the wage fo ve, taxe fo vehcle, an amotzaton cot) foce to eep the own fleet mall n oe to each a maxmal utlzaton of the fleet. Thu, the numbe of own vehcle often euce nce t mae no ene fo a fowae to pove enough tanpotaton capacty able to cove the pea of a volatle volume of oe. Uually, only a pat of the upcomng equet fulflle by own tanpotaton eouce. All the emanng oe ae outouce. Ung own vehcle fo the executon of ta calle elf-fulfllment, whle the outoucng of tanpotaton equet to extenal cae calle ubcontactng. A goou eucton of the own fleet ze motly poftable becaue t allow the o calle chey-pcng whch mean to pefom only the mot utable equet n a vey effcent manne by elf-fulfllment. But chey-pcng oe not eally mae ene unle the emanng equet ae fulflle n a cot-effcent way a well. The clacal Vehcle Routng Poblem (VRP) ha ft been ntouce an nvetgate by Dantzg an Rame (1959). Ball et al. (1983) have popoe the opton fo tanpotaton equet fulfllment by ung extenal cae. Chu (2005) peente a moel coneng multaneouly the etemnaton of outng a heteogeneou own fleet an the electon of 2

3 fowang ome equet ngly to extenal cae. Boluc et al. (2007) have eve ome eo n Chu pape an popoe an avance heutc fo the poblem. Extenng the uual plannng poblem of vehcle outng an cheulng by the atonal poblty of ubcontactng a pat of the equet ae two man queton fo the eeach on opeatonal tanpotaton plannng. The ft queton concen the long-tem plannng hozon an efe to the optmal ze of the own fleet. The econ queton affect the hot-tem plannng hozon an apple to the electon of equet to be pefome by elf-fulfllment an thoe to be fulflle by ubcontactng. Th electon poce cannot be euce to a mple ethe-o altenatve n the ene of an olate mae-o buy econ fo each ngle equet uppote by an aequate an ealy applcable compaon metho. Intea, the complex econ fo elf- fulfllment o ubcontacton ha to tae nto account epenence among all avalable equet nce they ae to be clutee to common bunle an the eultng tanpotaton cot epen on the bunlng pefome by the patche of the feght fowang company. The poce of contuctng an ente fulfllment plan fo elf-fulfllment an ub-contacton wth the hghet eachable qualty coepon to olvng the combne vehcle outng an fowang poblem whch alo calle the ntegate opeatonal tanpotaton poblem (IOTP). Although th poblem vey mpotant fo fowae n pactce thee ext only few appoache that nvetgate an olve that poblem n lteatue. A uvey of extng appoache can be foun n Kopfe an Kajewa (2007). Caue by the tenency to outouce a temenou pat of the aly tanpotaton equet to extenal cae, the nee fo olvng the IOTP n pactce even oang. The IOTP concen almot all fowae wth an own fleet of vehcle. Fo each equet to be execute ung the next plannng peo they have to chooe an appopate moe of fulfllment (moe-electon),.e. they mut ece whethe a equet houl be execute ung own eouce o whethe t houl be fowae to an extenal cae. In oe to mnmze the 3

4 cot of the own fleet, the fowae have to olve a uual vehcle outng an cheulng poblem fo all thoe equet that ae ecate fo elf-fulfllment. The fulfllment cot ncue by the engagement of cae can alo be nfluence fo the et of all fowae equet by mean of a llful opeatonal plannng of the employment of ubcontacto. Th uually can be eache by bulng favoable bunle of equet whch ae te togethe an ae agne to be fowae to an electe cae. The coeponng plannng poce calle feght conolaton. The goal of the fowae ung the feght conolaton poce to mnmze the ncung extenal feght cot. The oluton pace of the feght conolaton poblem bult by all feable choce on ffeent poblte of concentatng equet to bunle an all choce on agnng the contucte bunle to electe cae of vee type. The pupoe of th pape to entfy the ffeent moe of ub-contacton an to nvetgate the nteepenency of thee moe an elf- fulfllment. The objectve to tae ava n- tage of ncopoatng vee type of ub-contacton n the clacal vehcle outng an cheulng poblem by mnmzng the total fulfllment cot whch ae compoe of both, the vaable an fxe cot of the own fleet an the total extenal cae cot. In oe to get meanngful an llumnatng eult the nvetgaton concentate on a type of the IOTP whch nclue all cue fulfllment moe, but whch eay enough to be olve to optmalty. Th pape oganze a follow. In Secton 2 the expanon of the uual poblem of vehcle outng an cheulng to the IOTP by ncopoatng ffeent type of ubcontactng cue. Appoache evelope to olve uch poblem ae evewe n Secton 3. A mathematcal moel fo the expanon of the plan VRP to a bac type of the IOTP ntouce n Secton 4. Computatonal eult ae peente n Secton 5. Fnally, concluon an an outloo fo futue eeach ae awn n Secton 6. 4

5 2. Expanon of Vehcle Routng Poblem by Subcontactng The IOTP a complex econ poblem whch cont of eveal plannng level wth ffeent ub-poblem hown n Fgue 1. The ub-poblem of moe-electon, vehcle outng an cheulng, a well a the ub-poblem of feght conolaton ae tongly epenent on each othe. That why the optmal oluton of the ente poblem can efntely be eache only by appoache whch pefom the oluton poce fo all nvolve ub-poblem multaneouly. Goo uboptmal oluton can only be geneate by heutc whch tae the epenence between the ub-poblem nto account, fo ntance by a tabu each algothm olvng the outng poblem n the ffeent ub-poblem an allowng move fo wap an neton whch co the boe of the nvolve ub-poblem. The multaneou optmzaton of both ub-poblem of the IOTP (.e. elf-fulfllment an ub-contacton) ape. Snce the IOTP an extenon of the uual vehcle outng the oluton pace of the IOTP geate than that of a coeponng VRP an that the eaon fo the upeoty of the oluton of the IOTP compae to thoe fo plan elf-fulfllment. Opeatonal Tanpotaton Plannng Moe- Selecton Self- Fulfllment Sub-contacton Touoente Flowoente Integate Poblem Vehcle Routng & Scheulng Feght Conolaton Fgue 1. Sub-poblem of the IOTP (cf. Kopfe an Schönbege (2009)) 5

6 Due to the applcaton of complex an vee metho fo feght calculaton, the complexty of the IOTP vey hgh. Fgue 1 how the elaton between the nvolve ubpoblem. Applyng tou-oente contact fo ub-contacton complete tou ae tanfee to extenal cae an the tanpotaton fee epen on the attbute of the tou planne fo the executon of fowae equet. In contat, flow oente contact ae bae on the flow of goo caue by tanfeng bunle of fowae equet. Mot feght fowang compane ue eveal, altenatve fom fo compenatng extenal cae fo the employment. Of coue, fo each ngle cae the fom of compenaton etemne n avance by mean of a contact between the fowae an the cae. But nce we cone the employment of eveal cae fo the fulfllment of the whole et of upcomng equet, we have to chooe between eveal fom of compenaton multaneouly,.e. we have to agn each equet to an appopate type of ub-contacton wth t pecfc type of feght calculaton fo the payment of cae. The type of ub-contacton (type of payment) to be apple epen on the choce of the entute cae, nce the contact wth the cae ae fxe. We cone thee type of ub-contacton n th pape. The ft two type ae touoente an the th one flow-oente. In cae of a tou-oente type of ub-contacton le-than-tucloa equet ae combne to full-tucloa oe an the eultng tou ae fowae to extenal cae. Applyng the ft type of ub-contacton, the cae entute wth a complete tou an the payment fo the executon of the tou epen on the length of the oute to be pefome. The calculaton of the tanpotaton fee bae on a fxe taff ate pe tance unt,.e. the amount of payment calculate by multplyng the length of the entute oute wth the agee taff ate. Th type of fowang equet calle ub-contacton on oute ba. If ub-contacton on oute ba apple, thee ae no fxe cot fo the fowang company ncue by the uage of extenal vehcle. But compae to the uage of own vehcle the vaable cot fo oute bae ub-contacton ae 6

7 hghe than thoe fo elf-fulfllment a the payment of the fowae ha to cove a pat of the fxe cot of the cae. The econ type of ub-contacton eult fom payng the ubcontacto on a aly ba nepenent of the ze of the fowae tou. In th cae an extenal cae get a aly flatate an ha to fulfll all the eceve equet of a ngle ay up to agee tance an tme lmt. Th type calle ub-contacton on aly ba. Cot elate to both tou-oente ub-contacton type (.e. oute bae an aly bae ub-contacton) a well a the compaon of thee type of ub-contacton to the typcal cot fo elf-fulfllment ae hown n Fgue 2. Wth epect to the cot the egee of actvty n Fgue 2 can be eplace by the length of the execute tou. Fgue 2 mplfe the tuaton by aumng that the tunove alo woul lnealy epen on the egee of tanpotaton actvty, pobly meaue by the total length of all execute tou. Subcontacto on the tou ba Subcontacto on the aly ba Self-fulfllment maxmal capacty tunove maxmal capacty tunove maxmal capacty tunove cot cot cot egee of actvty egee of actvty egee of actvty Fgue 2. Compaon of cot fo ffeent type of ub-contacton an elf-fulfllment (cf. Kopfe an Kajewa (2007)) The th type of ub-contacton chaacteze by a payment fo the pue tanpotaton evce pefome by the cae an not on the ba of tavelle tance. The tanpotaton evce meaue by the extenvene of the flow epeentng the tanpot of the goo of the tanfee equet. Th type calle ub-contacton on flow ba. The fee ue fo payment epen on the flow of goo elate to the fowae equet. The tanpotaton 7

8 flow each fom the ouce of the goo of the fowae equet to the etnaton of thee goo an mght be combne to conolate flow accong to the oluton of an unelyng flow poblem (cf. Kajewa an Kopfe (2009)). The amount of payment fo flow bae ub-contacton ae fom the length of the tanpotaton flow an fom the amount of goo to be tanpote on thoe flow. An analy of extng opeatonal tanpot optmzaton ytem on the oftwae maet fo feght fowae ha hown that the poblem uneetmate. Thee no utable ytem fo feght conolaton on the maet, an a ytem fo ntegatng elf-fulfllment an ubcontactng not avalable, anyway. Due to the lac of oftwae, the poblem of plttng the equet potfolo nto a elf-fulfllment an a ub-contacton clute olve manually by the patche of the fowae. An appopate oftwae uppot only avalable fo the ub-poblem of elf-fulfllment (.e. fo vehcle outng an cheulng). But fnng goo oluton fo the global plannng ta of plttng the combne poblem nto ub-poblem even moe mpotant than geneatng hgh qualty oluton fo a ngle ub-poblem, nce an unfavoable agnment of equet to fulfllment moe may have a moe evee mpact on the total oluton qualty than the geneaton of moeate plan fo vehcle outng o feght conolaton. In pactce, plannng of the combne outng an fowang poblem mae heachcally. In the ft place the mot attactve equet wth hgh contbuton magn ae planne nto the elf-fulfllment clute untl all own vehcle ae chage to capacty. Hee, the cheule can be uppote by oftwae that optmze the ub-poblem of bulng oun oute fo a gven et of vehcle n the own fleet. Then the othe type of ub-contacton ae alo planne heachcally. They ae conee n a equental fahon, ft plannng the fo - wang accong to the oute bae ub-contacton type completely, followe by the plannng of the aly bae ub-contacton type an fnally by the flow-bae type. Followng the above poceue commonly ue n pactce, the mutual epenence between the plannng 8

9 fo the ffeent fulfllment-moe ae gnoe an conequently the avantage of multaneou plannng ae lot. 3. Appoache fo the Integaton of the Clute The IOTP cont of thee ub-poblem: plttng the equet nto jont clute fo ffeent fulfllment moe, cot optmzaton fo the et of equet pefome by elffulfllment (.e. agnment of equet to vehcle a well a equencng fo vehcle outng an cheulng), an cot optmzaton (o calculaton) fo the et of equet ecate fo ub-contacton (wth eveal ffeent type of ubcontactng). Dffeent metho of combnng thee thee ub-poblem eult n ffeent type of the IOTP wth ffeent elaton between elf-fulfllment an ub-contacton. Thee ae thee man appoache fo combnng the nvolve ub-poblem: heachcal, em-heachcal an global ntegaton. Thee thee appoache of ntegaton ae hown n Fgue 3. In cae of a heachcal ntegaton (mult-tage plannng), the total equet potfolo ft plt nto two ubet that ae agne to the clute by applyng a mple econ ule. Th ule oe not antcpate the attbute of optmal o nea-optmal oluton of the nvolve ubet. Afte the plttng nto two ubet fo elf-fulfllment an ub-contacton ha been complete, the cot optmzaton poce (pobly cot calculaton fo the ub-contacton) pefome ne each clute nepenently. Such an appoach peente by Chu (2005). When the volume of equet excee the avalable capacty of the own fleet, whle tme wnow contant pevent the extenon of the oute, ubcontacto have to be nvolve. Thu, the man ea of the heachcal plannng of Chu (2005) to chooe a many equet a poble fo elf-fulfllment an to elect them n avance on the ba of a cot aement of tou, an then to optmze the oute fo the own fleet. Aftewa, the cot fo ubcontact- 9

10 ng the emanng equet ae jut calculate, nce the feght calculaton pefome nepenently fo each equet applyng a taff ate fo ngle equet. Due to the level of the taff ate, ub-contacton alway moe expenve than elf-fulfllment. The heutc popoe by Boluc et al. (2007) ue the ame geneal appoach a Chu algothm whle the mpovement concentate on a bette oluton of the ub-poblem to be olve fo elffulfllment. a. Heachcal (mult-tage) equet potfolo agnment to the clute cot optmzaton et of equet n the elf-fulfllment clute et of equet n the ub-contacton clute cot calculaton / optmzaton b. Sem-heachcal (epeately) equet potfolo (e-) agnment to the clute cot optmzaton et of equet n the elf-fulfllment clute et of equet n the ub-contacton clute cot calculaton / optmzaton c. Global (flat,.e., ngle-tage) equet potfolo ntal agnment to the clute et of equet n the elf-fulfllment clute global cot optmzaton et of equet n the ub-contacton clute Fgue 3. Dffeent type of clute ntegaton (cf. Kopfe an Kajewa (2007)) The em-heachcal appoach, e.g. n Panatz (2002), un epeately by eagnng the equet to clute n an teatve poce. In the ft tep the oluton poce bul et of equet (bunle) whch ae to be hanle n common. Then thee bunle ae agne ethe to the elf-fulfllment clute o to the ub-contacton clute. Next, ffeent optmzaton poceue un n the clute fo each bunle epaately. They pefom the equencng 10

11 an cheulng fo each bunle n the elf-fulfllment clute an the cot optmzaton fo each bunle n the ub-contacton clute. Aftewa, ung a Genetc Algothm, new popoal fo plttng an bunlng ae geneate by eagnng the equet to the clute. The bunle of thee new oluton ae alo optmze an evaluate, an o on. A the optmzaton ta fo the bunle n both clute caue hgh tme-conumpton, the em-heachcal plannng appoach allow change concenng the von of the equet potfolo nto the clute only on the ba of cot etmaton an not on the ba of exact optmzaton of the ub-poblem of the two clute. The global (flat) ntegaton, e.g. n the appoache of Schönbege (2005) an Kajewa (2008), oente towa the global vew at the total cot of elf- fulfllment an ubcontacton an te to mnmze thee cot holtcally. The meta-heutc ue n the peente flat appoache aume that the equet ae ntally agne to one of both clute. Then the cot optmzaton poceue tae place by alteng th ntal oluton n eveal teaton. In ngle teaton of the optmzaton poceue, the ntegate poblem not ve nto ffeent ub-poblem whch ae olve by aement, but thee ext a unfom poblem epeentaton wth a complete mplementaton plan fo all equet. The mofcaton of uch a plan fo the next teaton un on the global level. In oe to get the next mofe plan the equet ae hfte not only at othe poton wthn one clute, but ae alo pobly hfte fom one clute to a poton n anothe clute. Conequently, a equet can be planne out of the ub-contacton clute an agne to a oute of an own vehcle, an vce vea. In patcula, Stumpf (1998) a well a Savelbegh an Sol (1998) can be clafe a global plannng appoache, a thee ext no ffeence between the tatege fo plannng the own vehcle an the vehcle of ubcontacto n thoe algothm;.e. the ame plannng poceue ae apple fo all fulfllment-moe. The optmal oute ae ape fo each moe an the equet ae hfte between all the oute a well a wthn one patcula oute. 11

12 Almot all appoache fo the IOTP peente n lteatue concentate on the extenon of a pecfc type of vehcle outng an cheulng by only one ngle type of ub-contacton. An avance appoach combnng eveal concuent type of ub-contacton wth vehcle outng an cheulng peente n Kajewa (2008) an Kajewa an Kopfe (2009). In that appoach the PDP-TW extene by eveal type of ub-contacton bae on the payment fo tou an on the payment fo flow of goo. The eultng complex econ poblem fo tanpotaton plannng olve ung a tabu each algothm. A compaon between the elf- fulfllment an ub-contacton moe a well a an nvetgaton of a competng uage of thee moe can be foun n Schönbege an Kopfe (2009) an n Kajewa (2008). Schönbege an Kopfe (2009) analyze the beneft of the combnaton of ub-contacton wth elf-fulfllment n volatle oe tuaton. They ue the atonal moe of ub-contacton fo an enhancement of the flexblty an evce qualty n cae of an ovetane own fleet. Kajewa (2008) ecbe an olve an IOTP contng n a global ntegaton of the PDP-TW wth the followng thee type of ub-contacton: oute ba, aly ba, an flow ba. She popoe a tabu each algothm fo the oluton of that opeatonal plannng poblem allowng a mxe uage of elf-fulfllment an ub-contacton. The popoe tabu each algothm alo ue fo expement on the m-tem plannng level by compang poblem ntance wth ffeent fleet ze. Snce the IOTP ha not yet been olve exactly by any optmzaton algothm t ha not been poble to pefom a benchma fo the popoe algothm. So, t cannot be juge to whch exten the eult ae tube by the abeaton fom the exact oluton. The plannng tuaton nvetgate by Kajewa (2008) nclue tme wnow an typcal fo feght fowang compane n pactce. Tme wnow have a petubng an complcate effect on the abolute compaon of fulfllment moe nce they ae teate ffeently n the vaou moe wth epect to cot an feablty. Becaue of the tong an unpectable nfluence on the mx of ffeent fulfllment moe tme wnow ae omtte n th pape. Th wll concentate the analy- 12

13 on the nvetgaton of the bac eaon fo choong a fulfllment moe an wll eep computatonal expement a mple a poble. None of the extng appoache fo ntegatng elf-fulfllment an ubcontactng peente n lteatue eally te to olve the ente IOTP multaneouly fo all nvolve ubpoblem. Th ue to the hgh complexty of th poblem. But an exact optmzaton of mall poblem ntance may ene ome mpotant theoetcal nght on the elaton between ffeent fulfllment moe. In the followng ecton of th pape a totally ntegatng appoach puue by ung a mxe ntege lnea pogammng (MILP) moel. 4. A Mathematcal Moel fo Combne Vehcle Routng an Fowang In oe to allow a taght competton between the conee fulfllment moe, to enable expemental computaton wth exact oluton, an to nvetgate the eultng mx of moe n an unbae tuaton, the plan VRP choen fo the combnaton wth the above mentone type of ub-contacton. The combne poblem moele a an MILP an olve to optmalty fo mall tet ntance. Gven a et of vetce V { 0,..., n} =, the VRP concene wth the optmum outng of a fleet of tuc between the epot ( = 0) an a gven et of n cutome ( V \ { 0} ae to be elvee wth goo avalable at the epot. The tance j all cutome locaton (, j) a well a the tance j 0 ( j V \ { 0} (, j V \ { 0} ) whch ) between ) fom the epot to each cutome ae nown. Cycle fom a cutome locaton to telf ae pohbte,.e. = + ( V ), an all tance ae ymmetc,.e. j = j (, j V ). The quantty q of the eman fo goo gven fo each cutome. Each vehcle ha a lmte capacty Q. Theefoe, n geneal eveal vehcle ae neee fo cutome atfacton. The plannng ta 13

14 of the VRP to fn an agnment of cutome to vehcle an to fn fo each vehcle a equence of evng t cutome n uch a way that all cutome eman ae atfe an the total mleage tavelle by the fleet a mnmum, whle the etcton of the capacty lmtaton of the vehcle ae met. Now, the extenon of vehcle outng to a combne IOTP wll be emontate an nvetgate fo the VRP,.e. the VRP extene to the Vehcle Routng an Fowang Poblem (VRFP). Self-fulfllment an thee ub-contacton type ae ue fo the executon of equet. Fo elf- fulfllment a homogeneou fleet wth a lmte numbe of vehcle avalable. The own fleet epeente by a et K holng m equal vehcle. A cot ate c pe tavel unt ue to calculate the vaable cot fo the elf-fulfllment of a tou. The maxmal tou length of any own vehcle lmte by max. Atonally, each own vehcle aocate wth fxe cot efne a cf. Of coue, thee fxe cot cf o not affect the optmal oluton eultng fom the opeatonal plannng but they ae of mpotance fo the long-tem analy of the oveall cot tuctue fo equet executon. Long-tem ageement etablh the conton an the amount of payment fo the employment of ubcontacto. The employe vehcle of extenal cae ae equal to the fo - wae vehcle wth epect to type an capacty. Moeove, t aume that wthn one type of ub-contacton all cae ae equal wth epect to the apple taff. On the ba of tou-oente contact, vehcle can be he fom ubcontacto fo an excluve ue by the fowae. Not all of the avalable vehcle owne by a ubcontacto have to be n evce. Thu, a payment mae only fo thoe extenal vehcle that ae actually ue. On the ba of flow-oente contact, the cot fo fowang equet epen on the length an amount of tanpotaton. Uually, the equet fowae to a cae by flow bae ub-contacton ae le-than-tucloa tanpotaton oe. The cae wll ty to combne the eceve e- 14

15 quet togethe wth futhe equet fom othe hppe to full-tucloa bunle. Th plannng poce of the cae not vble to the fowae. Fo the ft type of fowang (oute ba) the et K contng of m vehcle of ubcontacto pa on oute ba poable. The taff ate c pe tavel unt of a vehcle K coepon to the cot ate c fo the own fleet, but t hghe than c > c. The tou length of any vehcle fom K not allowe to excee the lmt c,.e. max. Fo the econ type of fowang (aly ba) a et K of m vehcle avalable. Thee vehcle can be he fom ubcontacto pa on aly ba. Only a flat-ate cf pe ay ha to be pa fo any actual ue vehcle of K. The maxmal tou length of any vehcle fom K lmte by max. The th type of fowang ealze by ub-contacton on flow ba. The peente moel oe not tae nto account a feght conolaton by mean of flow optmzaton,.e. all equet n the flow clute ae fowae epaately. The payment C f epen on the length an the volume of tanpotaton. The length of tanpotaton gven by the tance 0 fom the epot to the cutome locaton of equet. The tanpotaton cot ae aume to epen on the tanpotaton length wth a lnea cot ate c f pe tance unt fom the epot to the cutome. The volume of tanpotaton gven by the eman q. Wth epect to the amount of tanpotaton the ub-contacton cot ae calculate on a po-ata ba. The po-ate functon p ( q ) epen on q an eflect the egee of utlzaton of a vehcle. In oe to get a mple MILP the po-ata functon p q ) aume to be lnea o ( pecewe lnea, e.g. q p1 ( q ) =, p 2 ( q ) = 1 o p3( q ) = max{ p1, α p2} wth a utable Q paamete α. Altogethe, wth epect to the length an amount of tanpotaton the fowang cot ae compute to C f = p q) 0 ( c. f 15

16 The oluton of the VRFP cont n a feable total fulfllment plan wth the mnmal executon cot. The objectve functon C compehen the ente cot nclung the cot C fo elf- fulfllment, the cot C fo ub-contacton on oute ba, the cot C fo ubcontacton on aly ba, an the cot C f fo ub-contacton on flow ba: mn C = C + C + C + C (1) f Altogethe thee a et K = K K K wth m vehcle ( m m + m + m = ) that can be ue fo bulng tou. Let x be a bnay vaable uch that x = 1 f an only f any vehcle j j K tavel between cutome locaton an j. Fo the fomulaton of the contant of the VRFP we nee two atonal bnay vaable y an z. The bnay vaable y enote the agnment of cutome to vehcle,.e. y = 1 f cutome eve by vehcle. Let z be the bnay vaable uch that z = 1 f an only f the equet of cutome agne to be fulflle by flow bae ub-contacton. The component of the fulfllment cot C, C, C an C f can be calculate accong to the equaton (1a), (1b), (1c) an (1). C = V j V K x j j c + m cf (1a) C = V j V K x j j c (1b) C = x cf (1c) V \ { 0} K 0 C f = z p( q) 0 c f (1) V \ { 0} The feablty of the ente fulfllment plan aue f each equet agne to exactly one fulfllment moe an f the contant fo each fulfllment moe ae mantane. Fo apect of contant analy, all vehcle K can be conee togethe, a only the objectve functon ffe, whle all etcton ae ale. Snce the et K, K, K ae jont, oun oute have to be contucte fo all own an extenal vehcle n a mla way le 16

17 n a uual VRP wth a homogeneou fleet. The ffeence between the vehcle out of ffeent pool ealze by the component of the objectve functon ummng up the cot fo each type of ub-contacton nclung all vehcle whch belong to th type. Equaton (2) aue that each cutome ethe agne to exactly one vehcle K o othewe that the cutome eve by mean of flow bae ub-contacton. Snce K = m, K = m an K = m, the numbe of actually ue vehcle fo elf-fulfllment, fo oute bae ub-contacton, an fo aly bae ub-contacton cannot excee the numbe of avalable vehcle of each type epectvely. y + z = 1 V \ { 0} K (2) In equaton (3) t aue that each cutome agne to a vehcle appoache by that vehcle exactly once an that t left by the ame vehcle once. Atonally, equaton (3) guaantee that a cutome only eve by that vehcle that he agne to an that cutome eve by flow bae ub-contacton ae not vte by any vehcle at all. j V x = x = y j j V j V, K (3) Equaton (4) an (5) enfoce that the um of the eman of all cutome eve by a vehcle oe not excee the capacty lmt Q of the vehcle. Atonally (4) an (5) pohbt fo each tou of a vehcle the executon of hot cycle,. e. they pevent cycle wthout vtng the epot. So, (4) an (5) guaantee that each vehcle pefom a Hamltonean cycle wthout exceeng the capacty lmt Q. u u + Q x Q q, j V \ { 0}, K (4) j j j q u Q V \ { 0}, K (5) Contant (6), (7), an (8) enfoce that the lmt fo the maxmal length of tou ae obeve. Fnally, the contant chaactezng x j, y an z a bnay vaable ae epeente by (9), (10), an (11). 17

18 V j V V j V V j V xj j max K (6) xj j max K (7) xj j max K (8) { 0, 1} x, j V, K (9) j { 0, 1} y V, K (10) { 0, 1} z V (11) The objectve functon (1) togethe wth the contant (2) to (11) conttute a complete MILP moel fo the hot-tem plannng of the VRFP. Fo the tategcal fleet ze poblem, the fxe cot of the own fleet woul tongly nfluence t cale. The bet value of the numbe of own vehcle m ha then to be egae a a econ vaable an to be etemne fo the long-tem. Th can be acheve by ubttutng the tem m cf by V \{ 0} 0 K x cf. Th ubttuton yel the new objectve functon (1 ) applcable fo the etemnaton of the bet fleet ze, both fo the own fleet an fo the fleet ubcontacte fom extenal cae. mn x c j j V j V K V \ 0 V \ + x + { 0} K V \{ 0} { } K 0 cf + z p q ) x 0 0 cf f + V j V K x j j c ( c (1 ) Fo opeatonal plannng the optmal uage of the avalable fleet eache whle the ze ( m, m, m ) of the own fleet an of the fleet avalable fom ubcontacto etcte accong to the econ aleay mae by the fowae at the tategcal level. Fo long-tem plannng the value of m, m, an m ae et to be geat enough o that fo all fulfllment moe thee wll be an oveupply on avalable vehcle. The objectve functon (1 ) then wll not only offe the bet executon plan, but alo the bet fleet compoton. A the et of contant allow ome n K not to be egnate to any equet an the objectve functon 18

19 (1 ) enue that thee vehcle wll not caue any cot, the actual quantty of ue vehcle n the optmal executon plan can be ectly ue fo the etemnaton of m, m, an m. 5. Computatonal Expement In th Secton ome computatonal expement fo balancng ffeent fulfllment moe at the hot-tem an at the long-tem level ae pefome. The commecal mathematcal pogammng olve ILOG CPLEX 11 wa ue to fn optmal oluton of all the tet poblem, whch ae geneate fom the eal tuaton of a fowae n Cental Gemany. The compaon between VRP an VRFP ncate how much cot coul be ave by ncopoatng extenal cae an to whch exten chey-pcng coul mpove the utlzaton of the own fleet. Fo the ealzaton of computatonal expement, fve tet ntance mulatng the cutome eman n one wee ae geneate (D1, D2, D3, D4 an D5). Thee ae altogethe 20 geogaphcally cattee cutome locaton. Fom thee 20 cutome, a anom ntege numbe n ( n = 9,..., 12 ) of anomly choen cutome ae electe fo each ntance. Deman of choen cutome ae then geneate accong to the Poon tbuton q ~ Poon(8). Detal of thee tet poblem can be foun n the appenx. Dtance between two vetce ae oune to the neaet malle ntege n ou computaton. Value of the paamete ae choen a follow. It aume that the fowae ha fou vehcle avalable fo the executon of all thee tanpotaton equet. Two of them ae own vehcle ( m = 2), one vehcle pa on oute ba ( m = 1) an one vehcle pa on aly ba ( m = 1). The cot ate pe tance unt fo an own vehcle et to c = 0. 8 monetay unt. The value of the othe cot/taff paamete ae a follow: c = 1. 7 monetay unt pe tance unt; c = 3 monetay unt pe tance unt, p ( ) = 1; cf = 500 monetay unt; f 19 q

20 cf = 630 monetay unt. The value et fo the emanng paamete ae Q = 25 ton, max = max = 850 tance unt an 400 tance unt. Fgue 4 llutate the optmal plan of tet ntance D1. max = Fgue 4. Fulfllment plan fo the exemplay Vehcle Routng an Fowang Poblem D1. In oe to exploe the potental of cot avng by ub-contacton, the total fulfllment cot fo the VRP an the VRFP ae compae. In cae of the VRP, the fowae ha to hol an own fleet wth at leat 5 vehcle f he want to execute all the equet on h own. In contat, by ntegate plannng whch moele a the VRFP, he can euce h fleet ze ncopoatng extenal cae. Ung the ame fleet ze a hown n Fgue 4,.e. two own 20

21 vehcle, one vehcle pa on oute ba an one vehcle pa on aly ba, the total executon cot fo each ay (D1, D2, D3, D4 an D5) ae peente n Table 1. The total cot fo the whole wee (um) ae alo hown fo the vaable cot, the fxe cot an the total cot. The weely total cot fo the VRP ae egae a efeence an et a 100%. The lat ow n Table 1 how the pecentage of the ang expene n elaton to that efeence value. The utlzaton facto epeent the facton of the eally ue own vehcle to the numbe of avalable own vehcle fo each ay. In th mulaton, ub-contacton an chey-pcng can euce the total executon cot fo one wee by moe than 20 pecent a well a mpove the aveage utlzaton of the own fleet fom 88% to 100%. Tet Poblem Table 1. Compaon between VRP an VRFP at the opeatonal level Vaable cot Fulfllment only wth own fleet (VRP) m = 5 Fxe cot Total cot No. of utlze own vehcle Utlzaton of the own fleet η Integate tanpotaton plannng (VRFP) m = 2, m = 1, m = 1 Total cot No. of utlze own vehcle Utlzaton of the own fleetη (%) (%) D D D D D um % η : η : 100 To hol an own fleet nceae the elf-uffcency of the fowae. H own fleet pmaly ue fo chey-pcng. The cae pa on oute ba ae pemum ubcontacto whch ae compenate fo the evce on a elatve hgh level. Uually, they ae employe by the fowae evey ay. They enue a elable executon of the equet agne to them an enable the fowae to pactce the chey-pcng tategy fo the own fleet. The fo - wae te to upply the cae engage on oute ba wth well-bunle tou becaue he pay them on the ba of the tou length. Compae to h own fleet he wll ty to agn ela- 21

22 tvely hot tou to oute bae ub-contacton a the taff ate fo each tavel unt hghe than n the cae of elf-fulfllment. The cae pa on aly ba ae manly ue to manage the aly fluctuaton n the volume of oe an they ae npenable fo coveng the loa pea. The elate cot ae elatvely hgh compae to the vaable cot of elf-fulfllment. Theefoe, the bea-even pont cannot be eache unle the aly lmt actually utlze. In pactce, th ubcontacton type ue fo equet whch o not match vey well but ae combne to a tou whch explot the maxmal tou length fo aly pa vehcle. The flow bae ub-contacton motly apple fo le-than tucloa oe fowae to nepenent cae. Fom the fowae pont of vew t favoable to entut thee cae wth equet whch cannot effcently combne by the fowae to oute, fo ntance becaue they un nto ecton whee no favoable cluteng poble. Fo the long-tem plannng poblem at the tategcal level, t an eental ue fo the fowae, how many vehcle n the own fleet houl be hel (cf. e.g. Ball et al.) an how many vehcle fom extenal cae houl be obtane by gnng ffeent type of contact wth ubcontacto. We nvetgate that tategcal plannng poblem fo the cenao peente n th pape by compang the oluton of the VRFP at the hot-tem level wth that at the long-tem level. In ou expement, we aume that the numbe of own vehcle coul have been change eveyay of the wee an we etemne the optmal numbe of vehcle fo each tet poblem (D1 to D5). The eult of th n of expement yel a lowe boun fo the optmal oluton fo the fleet ze at the tategcal level, becaue the oluton pace of the tagetcal plannng enlage by allowng to vay the numbe of own vehcle n ou expement ntea of an optmal but fxe numbe of vehcle n the ognal longtem plannng. Fo ou expement etemnng the optmal fleet ze fo each ay anew, we can ue the objectve functon (1 ) peente n Secton 4 by gvng 22 m, m, an m value geat enough

23 o that they wll not have an mpact on the etcton of the oluton pace. Howeve, becaue of the enomou computatonal expene, we have obtane the optmal oluton fo only one ntance (D2) by olvng the complex poblem wth CPLEX. Thu, we mae expement fo ffeent cenao of fleet ze mpong ome moe etcton: K K y 0 = m (12a) y 0 = m (12b) Thee two contant mply that fo a gven cenao, all the avalable vehcle n the own fleet an thoe ubcontacte on oute ba have to execute ome of the equet. We then compae the optmal plan of ffeent cenao wth ffeent poble et of paamete value an foun the bet cenao fo the long-tem. The eult ae hown n Table 2. The total cot fo the whole wee ae alo compae wth both, the VRP an VRFP tet cae. The aumpton fo th compaon that the fowae coul have fo each ay the optmal fleet ze. Even une th unealtc aumpton, t obvou fom Table 2 that lttle cot coul be futhe euce fom the tuaton wth the fleet ze m = 2, coul be een a a vey eaonable oluton fo the long-tem opeaton. m = 1 an m =1, whch Tet Poblem Table 2. Compaon between opeatonal plannng an tategcal plannng Fulfllment only wth own fleet (VRP) m = 5 Opeatonal plannng Integate tanpotaton plannng (VRFP) m = 2, m = 1, m = 1 Total cot Total cot No. of utlze vehcle: m / m / m ) ( Lowe boun fo tategcal plannng Total cot No. of utlze vehcle: m / m / m ) ( D /1/ /2/1 D /1/ /1/0 D /1/ /1/1 D /1/ /2/0 D /1/ /1/0 um %

24 On the othe han, th mulaton aume etemntc ata et fo each ngle ay, uually not avalable fo the tategcal econ-mang. Fo the long-tem plannng, ome aggegate nfomaton houl be gathee. The objectve functon (1 ) can then facltate the econ-mang fo the etemnaton of the bet fleet ze. 6. Concluon an Futue Reeach We have analyze an extenon of the VRP to the VRFP allowng the combnaton of elffulfllment an ub-contacton of ffeent type. Th extenon nlne wth an often pactce tategy of feght fowae. By applyng optonal ub-contacton the total executon cot fo tanpotaton can be euce. Th not upng, nce the oluton pace of the VRP enlage. But the amount of avng that can be acheve n ou expement mpeve. Futue eeach ha to tae nto account mpotant ealtc aumpton of the combne vehcle outng an fowang. The cot functon fo ub-contacton on flow ba often ha a nonlnea, egeve hape n pactcal applcaton. Atonally, flow ae mege to jonte flow, contanng ffeent combne equet. Thu, the megng of flow yel a mnmum cot flow poblem to be combne wth the outng poblem. Fo futhe nvetgaton on the effect of combnng vehcle outng an fowang t neceay to be able to olve meum-ze an lage poblem of that type. Of coue, th can only be acheve by heutc appoache. Thu, the evelopment of poweful heutc an mpotant challenge fo the next tep n the eeach on the IOTP. 24

25 Appenx Detal of poblem D1 No x y q Total eman: 112 Mn. numbe of vehcle neee fo the VRP: 5 Detal of poblem D2 No x y q Total eman: 72 Mn. numbe of vehcle neee fo the VRP: 3 Detal of poblem D3 No x y q Total eman:95 Mn. numbe of vehcle neee fo the VRP: 4 Detal of poblem D4 No x y q Total eman: 94 Mn. numbe of vehcle neee fo the VRP: 4 25

26 Detal of poblem D5 No x y q Total eman: 84 Mn. numbe of vehcle neee fo the VRP: 4 Fo cutome, ( x, y ) ae the coonate of h locaton an q epeent h eman. Refeence Ball, M.O., Golen, B.L., Aa, A.A., Bon, L.D. (1983), Plannng fo tuc fleet ze n the peence of a common-cae opton, Decon Scence, Volume 14(1), Chu, C. (2005), A heutc algothm fot he tucloa an le than tucloa poblem, Euopean Jounal of Opeatonal Reeach, 165, Dantzg, G. an Rame, J. (1959), The tuc patchng poblem, Management Scence, 6, Boluc, M.-C., Renau, J., Bocto, F. (2007), A heutc fo the outng an cae electon poblem, Euopean Jounal of Opeatonal Reeach, 183, Kopfe, H. an Kajewa, M.A. (2007), Appoache fo moellng an olvng the ntegate tanpotaton an fowang poblem. In: Coten, H. an Mbaue, H. (E.): Pouton- un Logtmanagement, Spnge, Beln Heelbeg New Yo, Kopfe, H. an Schönbege, J. (2009), Logtc: The complexty of opeatonal tanpot optmzaton. In: Luca, P. an Rooen, P. (E.): Emege Analy an optmzaton of tuctue concept an tatege aco cplne, Spnge, Beln Heelbeg New Yo, to appea. 26

27 Kajewa, M.A. (2008), Potental fo effcency nceae n moen feght fowang, Gable Eton Wenchaft, Webaen, Gemany. Kajewa, M.A. an Kopfe, H. (2009), Tanpotaton plannng n feght fowang compane Tabu each algothm fot the ntegate opeatonal tanpotaton plannng poblem, Euopean Jounal of Opeatonal Reeach, to appea. Panatz, G. (2002), Spetonelle Tanpotpoton, Deutche Unvetätvelag (DUV), Webaen, Gemany. Savelbeg, M. an Sol, M. (1998), Dve: Dynamc outng of nepenent vehcle, Opeaton Reeach 46, p Schönbege, J. (2005), Opeatonal feght cae plannng, Spnge, Beln Heelbeg New Yo. Schönbege, J. an Kopfe, H. (2009), Onlne econ mang an automatc econ moel aaptaton, Compute & Opeaton Reeach, to appea. Stumpf, P. (1998), Touenplanung m petonellen Gütefenveeh, Schftehe e Geellchaft fü Veehbetebwtchaft, Nünbeg, Gemany. 27