Two mathematical formulations for the containers drayage problem with time windows

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1 It. Joural of Busess See ad Aled Maagemet, Volume 7, Issue 3, 2012 To mathematal formulatos for the otaers draage roblem th tme dos Mlorad Vdovć Uverst of Belgrade, Logsts Deartmet, Fault of Trasort ad Traff Egeerg Vojvode Stee 305, Belgrade, Serba Telehoe: Emal: Mloš Nolć Uverst of Belgrade, Logsts Deartmet, Fault of Trasort ad Traff Egeerg Vojvode Stee 305, Belgrade, Serba Telehoe: Emal: Draže Poovć Uverst of Belgrade, Logsts Deartmet, Fault of Trasort ad Traff Egeerg Vojvode Stee 305, Belgrade, Serba Telehoe: Emal: Abstrat The otaers draage roblem studed here arse ISO otaer dstrbuto ad olletg roesses, regos hh are oreted to otaer sea orts or lad termals. Cotaers of dfferet szes, but mostl 20ft, ad 40ft emt ad/or loaded should be delvered to, or olleted from the ustomers. Therefore, the roblem studed here s losel related to the vehle routg roblem th the tme dos that fds a otmal set of or routes vstg delveres ad us ustomers. The seft of the otaer draage roblem aalzed here les the fat that a tru ma smultaeousl arr oe 40ft, or to 20ft otaers, usg a arorate traler te. Ths meas that oe route to, three or four odes a be vsted, hh s euvalet to the roblem of mathg odes sgle routes hh rovde a total travel dstae shorter tha the ase he odes are vsted searatel. The aer resets to otmal MIP mathematal formulatos for the ase he u ad delver odes ould be vsted ol sef tme terals - tme dos. Proosed aroahes are tested o umeral examles. Keords: otaers draage, u delver, vehle routg roblems th TW Aoledgemets: Ths researh as artall suorted b the Mstr of Eduato ad See Reubl of Serba, through the rojets TR ad TR 36006, for the erod

2 Mlorad Vdovć, Mloš Nolć ad Draže Poovć 1 INTRODUCTION The routg roblem studed here s a tal for the termodal trasortato sstems here otaers are delvered b trus to ustomers oreted to a otaer sea ort or a lad otaer termal. I the termodal trasortato (Fg. 1a) the major art of the argo s joure s erformed b ral, lad atera or sea, hle the tal ad/or fal legs, dstrbuto ad olleto of otaers, are tall arred out b road (for more detals about termodal trasortato sstems see Cra ad Km, 2007). Tru should delver a loaded otaer that has arrved at a termal (mort.e. boud otaer) to a osgee, ad to u ad haul ba the loaded otaer from osgor to the termal (exort.e. outboud otaer). I the ase he a art of a otaer termal serves as a emt otaers deot, addto to u delver oeratos th loaded otaers, the emt otaers also eed to be delvered to a sher for loadg ad hauled ba emt to the termal after uloadg goods at the osgee ste. I addto, he the tme of subseuet shmet ad the sutablt of a emt otaer at the osgee ste, terms of te sze ad oersh, are satsfed, t s also ossble to move the emt otaers dretl to a sher stead of haulg them ba to the termal s deot ad havg them trasferred latter. From there, he osderg otaer trasortato th a loal rego oreted to a termodal termal that ludes emt otaers deot, fe ossble tes of otaer moves, also o as draage oeratos (Mahars & Boteog, 2004), a be reogzed (Fg. 1b). Draage oeratos are drve b the eed to fulfll ustomer demads hle satsfg varous ostrats mosed b the teholog ad ustomers reuremets. Draage ludes regoal movemets of loaded ad emt eumet (tralers ad otaers) b trators betee termals, shers, osgees, ad eumet ards. I geeral, draage oeratos volve ot ol the rovso of otaers but also emt tralers (Mahars & Boteog, 2004), hle ths researh ol the roblem of otaers u ad delver s osdered. Fgure 1: Itermodal trasortato sstem (1a) ad ossble tes of otaer moves otaer tru trasortato (1b) Most termodal otaers are szed aordg to Iteratoal Stadards Orgazato (ISO). Based o ISO, otaers are lassfed several grous (10ft, 20ft, 30ft, 40ft, 45ft ad se reetl 48ft ad 53ft), here 20ft ad 40ft otaers are the most freuetl used all over the orld. A mortat ssue otaers u ad delver s oordato of the dmesos of road trasort vehles th the dmesos of termodal otaers. I Euroe, exet Flad ad Sede, ad Asa, road vehles are restrted to trasort ol 20ft ad 40ft otaers, ol fe outres allo 45ft otaers, hle larger otaers are use ol the USA ad Caada (Nagl, 2007). I ojuto th the legth, the eght of otaer s also ver mortat. Most outres allo trasort of oe full loaded 20ft or 40ft otaer, but although trasort of to 20ft otaers ould be ossble regardg legth, the eght s a obstale. I most outres trasort of to loaded 20ft otaers b a stadard vehle s ot ermtted, exet the ase he the eght lmtato of 26 tos s ot exeeded. I the USA, Australa, Caada, Flad ad Sede the vehles use are the oes that offer the ossblt of trasortg to full loaded 20ft otaers, hle the EU has set u regulatos hh ermt erta tes of vehles alled modular oet vehles, offerg the ossblt of trasortg to full loaded 20ft otaers usg seal ombed hasss. I tur, the use of those tehal solutos rovdes dfferet oortutes for mrovg the effe of otaer trasortato b mergg dfferet u ad delver oeratos a sgle route. Draage oeratos ad eseall otaer tru trasortato aout for a sgfat orto of the total trasortato ost. Therefore, t s ver mortat to mrove the effe of otaer trasortato 24

3 It. Joural of Busess See ad Aled Maagemet / Busess-ad-Maagemet.org through the otmzato of suh trasortato roesses, hh leads to eesst of solvg the tru shedulg roblem otaer draage oerato (Zhag et al. 2010). Otmzato of otaer draage oerato has reeved reased atteto over the ast deade due to ts mortae termodal freght trasortato. Jula et al. (2005) formulated the roblem of otaer movemet th tme dos at orgs ad destatos as asmmetr multle travelg salesma roblem ad roosed three solvg aroahes. Coslovh et al. (2006) vestgated a otaer draage oerato th the reset ad future oeratg osts mmzed. Ima et al. (2007) formulated a otaer draage roblem as a u ad delver ad roosed Lagraga relaxato to solve the roblem. Chug et al. (2007) bult several mathematal models of otaer tru trasortato. The formulate the bas roblem here ever vehle a trasort exatl oe otaer at a tme, ad the mult-ommodt roblem th a ombed hasss used trasortg to 20ft otaers or oe 40ft otaer. To solve the roblem a soluto algorthm based o the Iserto Heurst as roosed. Namboothr ad Erera (2008) studed the maagemet of a fleet of trus rovdg otaer u ad delver serve (draage) to a ort th a aotmet-based aess otrol sstem. Zhag et al. (2010), osdered a tru shedulg roblem for otaer trasortato a loal area th multle deots ad multle termals. The roosed a aroah based o a teger rogrammg heurst determes u ad delver seuees for dal draage oeratos th mmum trasortato ost. Savelsbergh ad Sol (1995) sho that otaer trasortato roblems belog to u ad delver roblems, ad beause of the ature of the roblem, draage oeratos also orresods to mult-sto Vehle Routg Problems th Bahauls (VRPB). A more detaled sght VRPB, as ell as Vehle Routg Problems th Pu ad Delver (VRPPD), a be foud reet omrehesve overve gve b Parragh et al. (2008a, 2008b). The urose of ths aer s to roose mathematal formulatos for the otmal trus routg otaers draage oeratos the ase he u ad delver odes ma be vsted ol durg a erta redefed tme tervals. I ths a, our revous researh (Vdovć et al. 2011a, Vdovć et al. 2011b) has bee exteded b trodug addtoal mathematal formulato based o geeral mxed teger rogrammg model for the vehle routg roblem th smultaeous us ad delveres (VRP-SPDTW), roosed b Mgog ad Erbao (2010). Therefore, most of trodutoar art remaed the same, hle as revous researh, e osder both, emt ad loaded otaers moves ase he ombed hasss for trasortg to 20ft otaers or oe 40ft otaer are used. Dret moves of emt otaers from a osgee are to a sher s, as a relatvel rare tass, are ot osdered. I the otaer draage oeratos realzed b ombed hasss vehles, VRPBTW refers to the roblem here u to four odes a be vsted a sgle route startg ad edg otaer termal or deot hh s assumed here to be art of the termal. Our researh exteds the roblem aalzed b Zhag et al. (2010) to the mult-ommodt ase, but for the ase he ol oe termodal termal oerates the rego. Also, our researh exteds the roblem aalzed b Ima et al. (2007), to the mult-ommodt ase. Besdes resetg the mult-ommodt, ths aer also dffers the overall objetve hh s to fd otmal mathg ossbltes of odes that should be merged the same route formg bahaul loo. Aother haraterst of the roosed formulatos s resetg the smultaeous u ad delver oeratos. The remader of ths aer s orgazed as follos. Seto 2 resets to otmal roblem formulatos. Seto 3 resets omutatoal results, ad Seto 4 gves some oludg remars. 2 PROBLEM FORMULATIONS The roblem of dstrbutg olletg ISO otaers (20ft, ad 40ft) ma be desrbed as a varat of VRPB hh a tru vsts u to four odes utl retur to termal. Loaded otaers arrved termal (mort.e. boud otaers), or emt otaers from the termal deot should be delvered to ustomers, ad loaded (exort.e. outboud otaers), as ell as emt otaers should be ed u at a ustomers stes ad hauled ba to the termal. Therefore, he tru to ombed hasss, mathg ossbltes lude all feasble ombatos of 20ft, ad 40ft otaers that should be trasorted from/to termal ad ustomers (Fgure2). Obvousl, as t a be see from the Fgure 2, there are several ossble routes realzato oets ad t s orthhle to hoose those resultg th mmal legth. 25

4 Mlorad Vdovć, Mloš Nolć ad Draže Poovć Fgure 2: Some of routg ossbltes he draage s realzed b ombed hasss 2.1 Multle assgmet formulato of the otaer draage roblem G N,E be a grah, here N s the set of odes N th otaers move reuests, ad Let, j) j, E (, j N s the edge set. It s assumed that a ode ma smultaeousl have both, otaers demad ad sul move reuests. Number of reuests all odes ,,, hh orresod to 20ft otaers demad (20-) ad sul (20+), ad 40ft otaers demad (40-) ad sul (40+) are o advae. All otaers are avalable at the begg of the lag horzo (usuall oe da), ad all vehles start from the termal. The assumto that ode ma smultaeousl have demad ad sul move reuests, gves oortut of trasformg grah to the aother, hh eah ode N s relaed th tas odes. I ths a all tas odes of the trasformed grah, hose dexes are reumerated, have sgle move reuests, ether u or delver (20ft, or 40ft otaers). The set of all tas odes th reumerated dexes, a be o arttoed to four subsets, N, N, N ad N. Sets N 20-, ad N 40- ota ol delver, hle sets N 20+, ad N 40+ lude ol u odes th 20ft, ad 40ft otaers resetvel. I ths etor, he usg a ombed hasss, there are fftee ossble mathgs of tas odes to merged routes, ad four dret u or delver routes: Four odes mathgs: Termal Termal Termal Termal Three odes mathgs: Termal Termal Termal Termal Termal Termal Termal Termal Termal Termal Termal Termal To odes mathgs: Termal Termal Termal Termal Termal Termal Termal Termal Termal Termal Termal Termal Termal Termal Dret u or delver routes: Termal +20 Termal Termal -20 Termal Termal +40 Termal Termal -40 Termal 26

5 It. Joural of Busess See ad Aled Maagemet / Busess-ad-Maagemet.org The, the otaer draage roblem th tme dos, he ombed hasss s used, a be formulated as the follog roblem of assgg (mathg) odes the same route hh forms bahaul loo. m N tn N N N zn N N N N N subjet to N N en e e N N en e tz tz N e e N N N zn N en e N N en e N N N N en en e e 20 en e e N N 20 N e e en 20 N z 20 e N N en e e N N 1 N z e e e N N tn N N 1 zn N N N N N en en t N tn N en en te e z t N 20 e e N e e 1 tn N e z N e N N en e N N N N N zn 40 te tn N en e zn z N t z N N z e e e z en N tn zn t 40 e e 40 z tz z z (4) tn N Error! Objets aot be reated from edtg feld odes. (5) a e s s s s s t t t t t e M M M M M e (1 (1 (1 (1 (1 e e e ) ) ) ) ),,, e N,, N N N N 40 N s t M ( 1 ), N N N N (8) b N N N N (9) 1, f odes, j, ad l are merged the same route jl 0, otherse 1, f odes, j ad are merged the same route j 0, otherse 1, f odes ad j are merged the same route j 0, otherse 40 N t z e te (1) (2) (3) (6) (7) (10) (11) (12) 27

6 Mlorad Vdovć, Mloš Nolć ad Draže Poovć 1, f ode s served the dret route 0, otherse (13) Where,,,e dexes of ustomer odes th 20ft otaers sul or demad (,N 20-,,eN 20+ ) z,t dexes of ustomer odes th 40ft otaers sul or demad (tn40-, zn40+),j,,l dexes of a arbtrar ustomer odes N 20- set of 20ft otaers delver odes N 20+ set of 20ft otaers u odes N 40- set of 40ft otaers delver odes N 40+ set of 40ft otaers u odes,,, e otuous varable datg the tme at hh vehle starts servg ode,,,e resetvel a,b the left ad the rght boud of the tme do [a.b ] at the ode datg the tme terval he the ode s avalable for the servg s,s,s serve tmes at odes,, resetvel t,t,t e travel tmes betee odes -,-,-e, resetvel M,M,M e ostats used to learze tme dos ostrats, M =max[0, b +s +t -a ] e,,, osts of vstg odes a sgle route, ludg osts from/to termal (0) (umber of dexes deote umber of odes merged the same route), e = e + e0 Objetve futo (1) tres to mmze total trasortato osts of all routes that are used for servg all of sul/demad odes b solvg the set of odes mathg roblems. Terms 1-2 of the objetve futo (1) defe all alloable four, terms 3 7 three ad terms 8 12 all alloable to odes mathgs. Terms defe dret u ad delver routes, vstg ol oe ode. Sets of ostrat (2) (5) rohbt multle vsts of the same ode, ad rovde that eah ode must be vsted exatl oe, ether a route vstg four, three, to or oe ustomer ode. Costrats (6) to (9) are tme dos ostrats hh allo ode to be vsted ol durg the terval he the odes are avalable for servg. Costrats (10) to (13) defe bar ature of varables. Obvousl, the dea of defg tme dos ostrat, ths multle assgmet model formulato, s that the arbtrar seuee of odes vsted the same route should satsf seuee of ostrats datg that eah ode the seuee must be vsted durg ts avalable tme erod. 2.2 Geeral MIP formulato of the VRP th smultaeous us ad delveres for the otaer draage roblem The seod mathematal formulato e emloed here to solve the otaer draage roblem s based o the geeral mxed teger rogrammg model for the vehle routg roblem th smultaeous us ad delveres (VRP-SPDTW), roosed b Mgog ad Erbao (2010). Ths model otaed some lassal vehle routg roblems as seal ases, ad here s slghtl modfed ad adjusted th the objetve to rereset the otaer draage roblem. G N, E be a grah, here N s the set of odes N th As same as the revous model, let otaers move reuests, ad E, j) j,, j N ( s the edge set. Node th dex 0 reresets the termal ode, hle a ustomer ode ma smultaeousl have both, otaers demad ad sul move reuests. The assumto gves aga oortut of trasformg grah to the aother, hh eah ustomer ode s relaed th tas odes. I ths a all tas odes of the trasformed grah have sgle move reuests, ether u or delver. It s assumed that the trasort osts betee tas odes hh belog to the same ustomer ode ma be egleted. I the model e used the follog otato: umber of tas odes to be served total umber of vehles Q vehle aat j trasort osts betee ustomer odes ad j 28

7 It. Joural of Busess See ad Aled Maagemet / Busess-ad-Maagemet.org x j -bar deso varable euals to 1 f ad ol f vehle travel from the tas ode to the tas ode j j teger varable reresetg umber of otaers (trasort euvalet uts TEU, deotg 20ft otaers ) ed u from the tas odes u to tas ode, ad trasorted ar,je z j teger varable reresetg umber of otaers (TEU) to be delvered after tas ode, ad trasorted ar,je s tme of begg of serve at tas ode b the vehle t serve tme the tas ode t j travel tme betee tas odes ad j The model: m (14) st j1 0 0 j 0 j0 1 x j x j j 1, 0 j 1 xj x j 0 j 1, ; 1, x0 j 1 1, 0 0 j j j j 1, zj z j d j j 1, z j Q 1 x j 0, ; j 0, ; 1 x s 0, ; j 1, ; j; 1 s t tj M j j, (21) a s b 1, ; 1 (22) x, 0,1, 0, z 0, 0, ; j 0, ; j; 1 (23) jl j j, The objetve futo (14) sees to mmze total trasort osts. Costrats (15) esure that eah tas ode s vsted exatl oe, hle ostrats (16) guaratee that the same vehle arrves ad dearts from eah tas ode t serves. Costrats (17) revet multle deart of the vehle from the termal, the same route. Costrats (18) ad (19) are flo euatos for -u ad delver demads, resetvel. Costrats (20) revet vehle overloadg. Costrats (21) ad (22) are tme dos ostrats. Costrats (23) defe varables domas. 3 COMPUTATIONAL RESULTS Testg the ualt of roosed aroahes to solvg the otaer draage roblem he ombed hasss s used has bee arred out o the several roblem staes. Our dea as to test the to MIP models o the Solomo VRPTW behmar roblems (htt://eb.ba.eu.edu/~msolomo/roblems.htm). There are sx dfferet sets of roblem staes. Tas odes are radoml geerated roblem sets R1 ad R2, lustered roblem sets C1 ad C2, ad both, radomzed ad lustered roblem sets RC1 ad RC2. Also, roblem sets R1, C1 ad RC1 have a short shedulg horzo (fe ostumers er route) hle roblem sets R2, C2 ad RC2 have a log shedulg horzo (ma ostumers er route). Demad at eah tas ode orgal Solomo staes a have u to 50 uts. I the roblem of otaer draage e observe ol several otaers that eed to be ed from or delvered to eah tas ode. Therefore, e have trasformed the orgal Solomo staes to have both u ad delver tas, to have feer tass er eah ode ad to have four tes of tass. Total umber of tass j (15) (16) (17) (18) (19) (20) 29

8 Mlorad Vdovć, Mloš Nolć ad Draže Poovć eah ode s obtaed b dvdg the orgal Solomo demad b 20 ad roudg that umber to greater teger value. The, e radoml alloate derved tass to N, N, N ad N. Solomo staes have 100 tas odes ad otmal soluto for otaer draage roblem aot be obtaed from staes th so ma tas odes, ad therefore e observe follog three ases: small sale roblems th 10 ad 15 tas odes ad large sale roblems th 50 tas odes (ol frst 10, 15 or 50 tas odes are observed eah stae). I eah ase e observe ol the frst to staes from the roblem set (to rovde lart of the results resetato). The results for the small sale roblem staes are reseted Table 1 ad for the large sale roblem staes Table 2. Mathematal models ere mlemeted through Ptho 2.6 API of the CPLEX o the Itel(R) Core(TM) 3 CPU M Ghz th 6 GB RAM. Table 1: Results for smaller roblem staes Number of etor odes Istae Number of move reuests MULTIPLE ASSIGNMENT FORMULATION GENERAL MIP FORMULATION Soluto CPU tme CPU tme Soluto (se) (se) C C * C C R R R R RC RC RC RC * C C C C R R * R R RC RC * RC * RC * * Soluto obtaed after 1800 se of CPU tme (ga betee rmal ad dual as greater tha zero) 30

9 It. Joural of Busess See ad Aled Maagemet / Busess-ad-Maagemet.org Table 2: Results for larger roblem staes (geeral MIP formulato aot solve large sale roblem staes reasoable CPU tme) Number of etor odes 50 Istae Number of move reuests MULTIPLE ASSIGNMENT FORMULATION Soluto CPU tme (se) C C C C R R R R RC RC RC RC CONCLUDING REMARKS Ths aer rooses methods for the otmal trus routg otaers draage oeratos. We osder both, emt ad loaded otaers moves ase he ombed hasss for trasortg to 20ft otaers or oe 40ft otaer are used. To solve the roblem otaers draage s formulated as a multle assgmet roblem th tme dos, as ell as the VRP th smultaeous us ad delveres. That s, u ad delver odes are vsted ol sef tme tervals. Prelmar testg of the roosed aroahes shos that the frst model seems to be ver romsg, but more detaled aalss s left for the further hases. Ths oluso s dra from the observato of Wag ad Rega (2002), ho stated that tal sub-fleet of trus, ossts of less tha 20 trus ad s able to hadle at most 75 otaers a da. Hoever, havg md that Ima et al. (2007) have sho that eve the smler verso of the otaer draage roblem - routg roblem th full otaer load (VRPFC) s NP-hard, arorate heursts aroah develomet should be mortat dreto of future researh. Namel, the roblem studed b Ima et al. (2007), our otato orresods to the ombato of to odes mathg ad dret routes realzato, ad therefore t a be oluded that the otaer draage roblem osdered here s also NP-hard, se oe of ts arts s NP-hard. Develog adeuate softare suort, more detaled aalss of roosed aroah erformaes ad ossble adjustmets are thout a doubt dretos for the ver ear future, but mrovg algorthms th the other real lfe ostrats (ohomogeous fleet of vehles, multle use of vehles ) should be a mortat ssue for the future. Also, metaheursts alato s also osdered as a mortat dreto of future researh. REFERENCES Chug, K.H., Ko, C.S., Sh, J.Y., Hag, H. & Km, K.H. (2007). Develomet of mathematal models for the otaer road trasortato Korea trug dustres. Comuters & Idustral Egeerg, 53, Coslovh, L., Peset, R. & Uovh, W. (2006). Mmzg fleet oeratg osts for a otaer trasortato oma. Euroea Joural of Oeratoal Researh, 171, Cra, T.G. & Km, K.H. (2007). Itermodal trasortato. I: Barhart C, Laorte G (Eds) Hadboo OR & MS, 14 Elsever B.V., Iler, Y., Bazaraa, M., Gfford, T., Nemhauser, G., Sool, J. & Wum, E. (2006). A Otmzato Aroah for Plag Dal Draage Oeratos. Cetral Euroea Joural of Oeratos Researh, 14, Ima, A., Nshmura, E. & Curret, J. (2007). A Lagraga relaxato-based heurst for the vehle routg th full otaer load. Euroea Joural of Oeratoal Researh, 176, Jula, H., Dessou, M., Ioaou, P. & Chassaos, A. (2005). Cotaer movemet b trus metroolta etors: modelg ad otmzato. Trasortato Researh Part E: Logsts ad Trasortato Reve, 41,

10 Mlorad Vdovć, Mloš Nolć ad Draže Poovć Mahars, C. & Boteog, Y.M. (2004). Oortutes for OR termodal freght trasort researh: a reve. Euroea Joural of Oeratoal Researh, 153, Mgog L. & Erbao C. (2010). A mroved dfferetal evoluto algorthm for vehle routg roblem th smultaeous us ad delveres ad tme dos. Egeerg Alatos of Artfal Itellgee 23, Nagl, P. (2007). Loger ombato vehles (LCV) for Asa ad Paf rego: Some eoom mlatos. UNESCAP Worg Paers, No.2, Uted Natos. Namboothr, R. & Erera, A.L. (2008). Plag loal otaer draage oeratos gve a ort aess aotmet sstem. Trasortato Researh Part E: Logsts ad Trasortato Reve, 44, Parragh, S., Doerer, K. & Hartl, R. (2008a). A surve o u ad delver roblems Part I: Trasortato betee ustomers ad deot. Joural für Betrebsrtshaft, 58, 21-51, DOI: /s Parragh, S., Doerer, K. & Hartl, R. (2008b). A surve o u ad delver roblems Part II: Trasortato betee u ad delver loatos. Joural für Betrebsrtshaft, 58, , DOI: /s Savelsbergh, M.W.P. & Sol, M. (1995). The geeral u ad delver roblem. Trasortato See, 29, Vdovć, M., Radvojevć, G. & Ratovć, B. (2011). Vehle routg otaers u u ad delver roesses. Proeda Soal ad Behavoral Sees, 20, Vdovć, M., Radvojevć, G., Ratovć, B., Bjel, N. & Poov, D. (2012). Cotaers draage roblem th tme dos. CD boo of roeedgs of the 15 th Iteratoal oferee o trasort see ICTS 2012, Portoroz, Slovea. Wag, X. & Rega C.A. (2002). Loal truload u ad delver th hard tme do ostrats. Trasortato Researh Part B: Methodologal, 36, Zhag, R., Yu, W.Y. & Kofer, H. (2010). Heurst-based tru shedulg for lad otaer trasortato. OR Setrum, 32,

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