Network transformation algorithm for Supply chain plant location problem

Transcription

1 Ntwork transformation algorithm for Supply chain plant location problm Shihua Ma, Xiaoqun Liu School of Managmnt, Huazhong Univrsity of Scinc and tchnology 1037# Luoyu Strt, Wuhan , P.R.China Abstract: - Through analyzing th product structur tr, th complt supply chain ntwork (SCN) including matrials supplying, products manufacturing and distribution is stablishd. Th dirctions of all th flows in th original SCN ar narrowd down into on singl way by dividing up all th possibl circulation of both manufacturing and transportation. Each nod of th SCN is simplifid according to its diffrnt functions, outputs and raw matrial sourcs. According to th layr of th product structur tr, all th dividd nods ar combind so that ach rout includs all th raw matrial sourcs of th finishd product. Th most suitabl plants location undr th tim constraints ar workd out through th application of gnralizd prmannt labling algorithm on th combind SCN. Finally, by tracing backward to th original SCN according to th most suitabl plants location and routs slction, th ral optimal plants location and routs slction can b found out. Ky-Words: - Supply chain managmnt, Plant location, Rout slction, Ntwork transformation 1 Introduction Nowadays, living in a complx markt nvironmnt whos situations including firc comptition, quickly changd consumr's tast, shortr and shortr product's lif cycl, all th ntrpriss ar forcd to mt th consumr's dmand as soon as possibl with lowr cost and shortr production cycl. To achiv ths goals, ntrpriss ar no longr only absorbd in improving thir intrior production procdurs, but xtnd th flr of managmnt to othr ntrpriss in th SCN. That is to say, ntrpriss want to control, oprat and plan th total supply chain corrlatd with its products ntirly. With th popularization of th supply chain and th dvlopmnt of th information tchnology, supply chain managmnt (SCM) hav bn gray sprad and applid. How to plan and dsign th SCN from th viw of th whol product, such as th supplirs slction, plants location, distribution ntwork dsigning, bcoms th focal issu in th study of th SCM [1. According to th classification by Cohn and Mallik [, th problm studid in this papr rfrs to th plant location of th SCN modl. To th plant location problm, thr ar two kinds of solutions mainly. On is using various kinds of optimization algorithms to find out th most suitabl plant combination. And th othr is to confirm th plant combination, assss th tactics by using computr to simulat th situation of th actual supply chain. All thos kinds of optimization algorithms can b dividd into two catgoris roughly as following. Th first on includs thos various kinds of mathmatics analytic mthods basd on th linar programming or mixd-intgr programming, which contribut to find out th bst solution of th maximal or minimal objct functions undr som assumptions and limitations. Cortinhal [3, Tjndra [4 hav don grat contribution rsarchs to this. And Lagrangan huristic algorithm, which brought forward by Corria [5, is commonly usd in larg-scal multi-facility location problms, such as Yuri [6. Bsids th aformntiond mthods, simulation, which usd by Jung [7, t ac, is anothr on to confirm th plant location and policy valuation. But th following qustion gnrally xists in ths mthods. Firsy, th plant location blongs to NP-hard problm usually, and th algorithm complxity prsnts th indx incras with th rang. Scondly, most solutions do not considr th product structur tr. Thirdly, many rsarchrs divid th supply chain into diffrnt sub-ntwork, and calculat ach sub-ntwork as th optimum solving of th total supply chain. But th concatnation of ach sub-ntwork optimum solving might not th bst solving of th ntir supply chain. Last, th tim factor is sldom considrd. But actually, th tim factor should b studid as an important goal in th nw markt nvironmnt [8. From all abov-mntiond rsarchs, it lacks a simpl, gnral and fasibl mthod to solv th plant location in th SCN. In that, Chn tris to transform th original supply chain and applis th shortst path algorithm, th maximal flow algorithm to solv th plant location [9. Th most dfct of Chn's rsarch dos not considr th tim rstriction among supply chain nods. So th plant location and routs

2 slction of th SCN mrgd with th product structur tr undr th tim window is studid mphatically. Modl and mthodology.1 Dscription, assumptions and notations For any supply chain with a final product, thr is a dndrit product containing all th matrials/moduls usd in th supply chain to produc th final product, and showing th hirarchical composition rlationships btwn ths matrials/moduls, as th xampl shown in Fig.1.Th raw matrial P 01 and P 0 ar usd to produc th smi-final product P 03. And th raw matrial P 04 and th P 03 ar usd to mak th final product P 0. In addition, th lttr in th parnthss aftr ach componnt in th product tr indicats th quantity of that componnt ndd to mak on abov lvl componnt. Fig.1. Product structur tr of P 0 SCN, shown in Fig., is th ntwork of facilitis that prforms th functions of procurmnt of matrial, transformation of matrial to intrmdiat and finishd products, and distribution of finishd products to customrs [10. In this ntwork, thr ar four modality nods: supplir (S), manufacturr (M), distributor (D) and rtailr (R). 3 1 [ t ( S M ) t ( S M ) 1 1 l [ t ( S M ) t ( S M ) l 1 1 [ t ( S M ) t ( S M ) 3 1 l 3 1 C SM 4 1 [ t ( S M ) t ( S M ) l [ t ( S M ) t ( S M ) 5 1 l 5 1 C M 1 D 1 [ t ( M D ) t ( M D ) C M M l 1 1 l 1 1 [ t ( M M ) t ( M M ) C M D 1 [ t ( M D ) t ( M D ) l Fig.. SCN of P 0 C DR 1 C DR 1 1 [ t ( DR ) t ( DR ) 1 l l 1 1 C DR [ t ( D R ) t ( D R ) l C DR 1 [ t ( DR ) t ( D R ) [ t ( D R ) t ( D R ) 1 l 1 In Fig., th M 1 rcivs th P 01, P 0 and P 04 supplid by th S 1, S and S 3 to produc th P o and th P o3. And th M rcivs th P 04 supplid by th S 4, S 5 and th P o3 supplid by th M 1 to produc th P o. Th P o manufacturd by th M 1 and th M ar transportd to th D 1 and th D rspctivly. Last, th D 1 and th D distribut th P o to th R 1 and th R). Th lttrs bsid ach arc ar th componnts producd or shippd from on ntity to anothr, and according to th actual nds, th production or transportation cost, tim rstriction ar addd bsid ach arc. Howvr, in a SCN, not all ntitis ar ncssary to transport, produc, and distribut from th raw matrial to th final product. For instanc, on possibl way to produc and dlivr th P o is S 1, S, S 3, M 1, D 1, R 1, as arrows shown in bold lins in Fig.. Anothr possibl way, as shown in Figur with bold dottd arrow lins, would go through S 1, S, S 4, M 1, M, D, and R 1. Nvrthlss, as th xampl shown in Fig.1, all componnts, from th raw matrial P 01, P 0 and P 04 to th smi-finishd product P 03, must b usd in th procss of manufacturing th final product P 0, thrfor it is ncssary for ths componnts to appar in th production sub-ntworks for th corrsponding SCN, such as th xampl sub-ntworks shown in Fig.. A nod in a SCN may b a raw matrial supplir, a factory, or a distribution cntr. Hnc, in this study, such a nod is said to hav ithr production function or distribution function or both [9. If a nod prforms manufacturing work, which mans transforming som matrials, it will hav rcivd into anothr smi-finishd or final product in a supply chain, it is said to hav th production function, such as M 1 and M in Fig.. Morovr, a production nod, which is a nod with production function, is furthr classifid into an xtrnal production nod, an intrnal production nod, or a production nod with both functions. An xtrnal production nod uss only what it rcivs as matrials to produc itms without using what it producs as matrials. At last, a on-itm production nod is th on that ships out xacy on kind of itm, such as M in Fig.. An intrnal production nod is a production nod, and morovr, it dos not only us th componnts rciving as matrials for production, but also taks th itms producing again as matrials to furthr produc othr itms. For xampl, nod M 1 in Fig. is a good xampl of both xtrnal nod (it imports th componnt P 01 and P 0 to produc th P 03 and transports th P 03 to th M ) and intrnal production nod (it imports th P 01 and P 0 to produc th P 03, and still mor, th nod M 1 again taks th P 03 as on raw matrial, along with th componnt P 04 that it imports, to mak th final product P 0 ). If a nod ships out what it rcivs without manufacturing work, it is said to hav th distribution function, such as D 1 and D in Fig.. For instanc, th D 1 in Fig. is a nod with distribution function only.. SCN transformation: dcomposition and composition Bfor applying any ntwork flow algorithms, a SCN hav to b transformd into anothr topology graph

3 to satisfy th rgarding rquirmnts of thos algorithms. And th basic idas of th SCN transformation ar as following. - Analyzing th product structur tr and stting up th original SCN; - Simplifying ach nod of th SCN according th diffrnt nodal functions, outputs, and raw matrial sourcs; - Combining th nw supply chain nod according to th layr of th product structur tr; - Applying th optimization algorithm to th SCN aftr dcomposition and composition and sking out th most suitabl plants and routs; - Tracing backward to th original SCN to find th ral optimal plant location and routs...1 Product structur tr and SCN analyzing As th dscription and assumption, diffrnt nods ar connctd with dirctd arcs. Bcaus th nodal typ is not appointd at ahad, it is ncssary to chck out th circulation in th original SCN bfor analyzing th function of supply chain nods. Th transformation of product structur and SCN whn dcomposing supply chain circulations is dividd into two parts: on is to chck and transform th manufacturing procss circulation and th othr is to chck and transform th distribution procss circulation. Manufacturing procss circulation-start and nd in th sam nod. All nods in this circulation ar manufacturs, so this circulation can b viwd as th manufacturing procss to th sam ntity. Just as shown in Fig.3, th M 1 M and M 3 form a circl to transport P 03. Firsy, it nds to dcompos this circl to xprss as th manufacturing procss. And bcaus both th M 1 and M contains th P 01 and P 0 to produc P 03, so both th M 1 and M can b actd as th starting of th manufacturing procss. Transforming this sub-ntwork in Fig.3 and rvising th corrsponding cost and tim window on ach rout, th sub-ntwork and th cost and tim window aftr dcomposition ar prsntd in Fig.4 rspctivly. S 1 M C SM [ t ( 1 S ) 1M ( S1M ) S M C S ) M ( SM ) SM M M 3 C [ t ( 3 M ) M3 M 3 M 4 C M 3M 4 M ) 3M4 ( M3M 4) M 4 M C M 4M M ) 4M ( M4M ) M M 5 C 5 M ) M5 ( MM 5) Rout S 3 M 1-1 S 4 M 1-1 M 1-1 M -1 M -1 M 3-1 M 3-1 M 1- S 1 M - S M - M - M 3- M 3- M 1-3 M 1-3 M -3 Cost Tim Window C S ) 3M 1 S3M1 C S ) 4M 1 S4M1 C M ) 1M M1M C M ) M 3 3 C M ) 3M 1 ( M3M 1) M 3M1 C S ) 1M ( S1M ) SM 1 C S ) M ( SM ) SM C M ) M 3 3 C M ) 3M 1 ( M3M 1) M 3M1 C M ) 1M M1M Fig.4. Manufacturing procss ntwork and its data aftr transformation Manufacturing procss circulation--start from th nod A, transit and rturn to th nod B. All nods in this kind circulation ar also manufacturs, so this kind circl can also b viwd as th manufacturing procss to th sam ntity. Shown in Fig.5, th procss rout of P 03 is M 1 M M 3 M 4 M M 5. Th start nod of this procss rout is M 1, and th nd nod is M 5. Transforming this sub-ntwork and rvising th cost and tim window on ach rout, th sub-ntwork and th cost and tim window aftr dcomposition is prsntd in Fig.6 rspctivly. Transportation procss circulation. In th actual SCN, th product transshipmnt among distributors to support rtailrs dmands is xistd gnrally. Shown in Fig.7, th products ar transshippd from D to D 1 to support th rtailr R 1 whn th products is out of stock in D 1, and so as to th rtailr R. Dcomposing ths transportation nods, which transships th products ach othr, and th transformd SCN is Fig.8. S 1 M C SM 1 1 l S M C SM l S 3 M 1 C S3M l S 4 M 1 C S4M l M 1 M C M1M 1 M M 3 C 3 3 M 3 M 1 C M 3M t S M t ( S1M ) t S M t ( SM ) t S M t t S M t t M M t M M t M M ( M3M 1) Fig.5. Original manufacturing circulation ntwork and its data (Typ ) Fig.3. Original manufacturing circulation ntwork and its data

4 S 1 S P 01 P 0 M -1 P 03 M 3 P 03 M 5 M 4 P 03 M - S 1 M -1 C SM [ t ( 1 S ) 1M ( S1M ) S M -1 C SM S ) M ( SM ) M -1 M 3 C [ t ( 3 M ) M3 M 3 M 4 C M 3M [ t ( 4 M ) 3M4 ( M3M 4) M 4 M -1 C M 4M M ) 4M ( M4M ) M - M 5 C [ t ( 5 M ) M5 ( MM 5) Fig.6. Manufacturing procss ntwork and its data aftr transformation (Typ ) M 1 D 1 C D 1 R 1 C D1R DR ) 1 1 ( DR 1 1) D 1 D C D1D DD ) 1 ( DD 1 ) M D C M D M ) D ( MD ) D R C DR D ) R ( DR ) D D 1 C DD D ) D1 ( DD 1) Fig.7. Original transportation circulation ntwork and its data M 1 D 1-1 C D 1-1 R 1 C D1R DR ) 1 1 ( DR 1 1) M 1 D 1- C D 1- D - C D1D DD ) 1 ( DD 1 ) D - R C DR D ) R ( DR ) M D -1 C M D M ) D ( MD ) D -1 R C DR D ) R ( DR ) M D -3 C M D M ) D ( MD ) D -3 D 1-3 C DD D ) D1 ( DD 1) D 1-3 R 1 C D1R DR ) 1 1 ( DR 1 1) Fig.8. Transportation procss ntwork and its data aftr transformation Othr supply chain circulations. To th othr circulation including manufacturrs, distributors or rtailrs, it just happns in th situations as follows; (1) th infrior products rturnd to r-work, or () th advrs logistics considrd th castoff rcycl. In this papr, just on dirctd SCN is studid, so ths situations abov-mntiond do not considrd. And in this papr, ach nod in th SCN should link to at lst on nod. It is to nsur that no unattachd nod in th SCN. To a gnral, common SCN, this assumption is rasonabl and accord with practic... Nod function dcomposition Aftr analyzing and stting up th product structur tr and th SCN, thn th primitiv ntwork structur nds to b changd into a nw SCN according to th composition of th products and nodal function to mak any nod on this nw ntwork only bar th singl function of singl output. But to th production function nod, th cost and tim of obtaining raw matrials from diffrnt sourcs ar diffrnt. So whn th production function nod which has a lot of sourcs is of obtaining th sam raw matrials, this nod should b spittd too. In othr words, in th nw SCN, ach nod is rsponsibl for th singl function. Th distribution function nod can snds th products to diffrnt nods, and th production function nod can only hav on output, and to ach componnt, this nod can only b rcivd from on supplir. And just as th distribution function nod, th production function nod snds th products to diffrnt nods. Among all SCN nods, only th manufacturr nod might possss two kinds of functions togthr. So whil simplifying th nodal functions, all nods which ar involvd th manufactur procss only nd to b chckd. Th mthod to gnrat th singl function nod is dividd into thr stps: - Chcking and isolating all nodal distribution functions; - Isolating diffrnt outputs to th rmaindr production function nods; - Chcking th production nod aftr isolating diffrnt outputs and splitting th sam raw matrials with diffrnt sourcs. Aftr ths thr stps, bcaus nw nods hav rplacd all old nodal functions and outputs, all ths old nods and thir arcs ar dltd and th nods with th sam inputs and outputs ar amalgamatd. Exampling as th nod M 1 in Fig.9, P 0 is producd by P 03 and P 04, P 03 is composd by P 01 and P 0. Sing Fig.9, M 1 has thr functions: (1) transports P 0 to M, () producs and transports P 03 to M 3, (3) transports P 0 to D 1. Th Function 1 of M 1 blongs to th distribution function (S M 1 M ), so it nds to gnrat a nw nod to rplac this function, as shown in Fig.10(a). Th function of M 1 is to produc P 03 (S 1 +S M 1 M 3 ), so it nds to gnrat anothr nw nod to rplac this function, as shown in Fig.10(b). Th function 3 of M 1 is to produc P 0, so it nds to gnrat th third nod M 1-3 to rplac this function, as shown in Fig.10(c). And bcaus thr ar two diffrnt supplis (S 3 and S 4 ) to offr th sam matrial P 04 to produc P 0, so th nod M 1-3 nds to b split into two nods (M 1-4 and M 1-5 ) on bhalf of diffrnt supplis combination, as shown in Fig.10(d). Chcking th nw ntwork, th inputs and outputs of

5 th nod M 1- and M 1-3 ar compltly th sam, so on nod of both can b amalgamatd and som rlatd arcs ar dltd. S 1 M 1 C SM [ t ( 1 1 S ) 1M1 ( S1M 1) S M 1 C SM S 3 M 1 C SM [ t ( 3 1 S ) 3M1 S 4 M 1 C S4M S ) 4M1 M 1 D 1 C M 1 M C M ) 1M M 1 M 3 C [ t ( 3 M ) 1M3 ( M1M 3) Fig.9. Original SCN and its data of M 1 nod st, and also combin all th rlating arcs into on arc that imports th ntir componnt. In addition, bfor applying th optimization algorithms on th SCN, two xtra virtual nods (th sourc nod S th sink nod T) and som arcs hav to b addd in th SCN. Aftr th SCN composition, vry nodal input contains all componnts for its outputs and ach componnt is just offrd by on supplir. Th amalgamating procss is to amalgamat th nods of sam layr according to th product structur downstram from th root to th bottom. Only a rout amalgamatd out lik this includs producing all compositions of th finishd product. All manufacturr nods changd in th SCN composition nd to b dltd. Taking Fig.11 as an xampl, P 0 is th final product and th transformd ntwork aftr composition is shown in Fig.1. (a) Ntwork of of M 1 (b) Ntwork of of M 1 (c) Ntwork of of M 1 aftr splitting function 1 aftr splitting function 1, splitting function 1, and 3 (d):ntwork of M1 aftr combination th sam function nods S M 1-1 C SM M 1-1 M C M ) 1M S M 1- C SM S 1 M 1- C SM [ t ( 1 1 S ) 1M1 ( S1M 1) M 1- M 3 C [ t ( 3 M ) 1M3 ( M1M 3) M 1- M [max ( ( SM1), ( S1M max ( ( S3M1), ( S4M M 1- M [max ( ( SM1), ( S1M max ( ( S3M1), ( S4M S 3 M 1-3 C SM [ t ( 3 1 S ) 3M1 M 1-3 D 1 C 1 S 4 M 1-4 C S4M S ) 4M1 M 1-4 D 1 C Fig.10. Function dcomposition of th nod M 1..3 SCN composition basd on product structur tr Th SCN composition basd on th product structur tr has two main stps: (1) th composition of production nod with th sam matrial, () th composition of production nod without th sam matrial. For a on-itm production nod-producing componnt, it nds to compos all th nodal upstram nods that provid th componnt into on S 1 S P 01 P 0 M 1- S 3 P 03 P 03 S 4 P 04 P 04 M 1-3 M 1-4 P 0 P 0 D 1 S 1 S P 01 P 01 P 0 P 0 M 1- S 3 M 1- S 4 P 03 P 04 P 03 P 04 M 1-3 M 1-4 P 0 D 1 P 0 S M 1- C SM1 S 1 M 1- C SM 1 1 S ) 1M1 ( S1M 1) S 3 M 1-3 C SM 3 1 S ) 3M1 M 1- M [max ( ( SM1), ( S1M max ( ( S3M1), ( S4M M 1- M [max ( ( SM1), ( S1M max ( ( S3M1), ( S4M S 4 M 1-4 C S4M S ) 4M1 M 1-3 D 1 C M 1-4 D 1 C Fig.11. Original ntwork and its data of M 1 S 1 S CSM [max ( t( SM1), t( S1M M 1- S 3 CSM 1 max ( ( SM1), ( S1M S 1 S CSM [max ( t( SM1), t( S1M M 1- S 4 CSM 1 ( t ( S M ), t ( S M )) M 1- S 3 M 1-3 M 1- S 4 M 1-4 max l 1 l 1 1 ( t ( S M ), t ( S M ), t ( S M )) ( t ( S M ), t ( S M ), t ( S M )) SM C [max l 1 l max l 1 l 1 1 l 3 1 ( t ( S M ), t ( S M ), t ( S M )) ( t ( S M ), t ( S M ), t ( S M )) S M C [max l 1 l max l 1 l 1 1 l 4 1 M 1-3 D 1 C M 1-4 D 1 C 1 Fig.1. Ntwork composition and its data of M 1..4 Appling th optimization algorithm to th transformd SCN Aftr all convrsion oprations abov-mntiond in th SCN, th rlvant algorithm can b applid to th

6 transformd ntwork to find out th bst nod association. Th optimization algorithm adoptd in this rsarch is th gnralizd prmannt labling algorithm (GPLA) of th singl sourc shortst path with tim window (SSP-TW) [11. That is to say, th spcific rout of th minimum cost undr th tim constrains from th starting point to th trminal point should b sarchd out. And all nods in this rout ar th optimal plant location in production and distribution...5 Sking th optimal plant location of original SCN Th shortst rout found out aftr using th GPLA to th transformd ntwork includs all nods of th optimal plant location. Now, th ral plant location should b confirmd from this rout according to th amalgamatd nods, outputs and componnts. Th corrsponding mthod to find ths is to tak apart and rturn to th original SCN. 3 Discussion and conclusion Unlik many studis don in th SCM fild, this rsarch taks a diffrnt approach by transforming th SCN with th product structur tr undr th tim rstriction of th SCN, thn maks us of th GPLA algorithm to dvlop supply chain algorithms that ar good for SCM dcision-makings. Th most important rsults larnt from th rsarch ar summarizd as follows. First, ths supply chain algorithms provid a simpl yt practical fashion to build supply chain modls that ar usful for SCM without a larg amount of mony or profssional knowldg. Scond, in th acadmic rsarch for a mor gnral modl to rprsnt and discuss th SCM, th algorithms built hr and th way to transform th SCN is a grat candidat that srvs as a good starting point. It is possibl to add mor nw considrations into th rprsntation and algorithms to improv its rprsntativ ability. Third, sinc ths supply chain algorithms, along with th basic dcomposition and composition of th SCN ar dsignd closly connctd to th product structur tr, it can b sn that thr sms to b diffrnt sctions or cuts xisting in th supply chain aftr composition, as xampls shown in prvious chaptrs. Last, a managr in a cntralizd SCN can us this supply chain algorithm to gnrat initial rsults rgarding output, cost or both factors for his SCM dcisions. In a SCN that xists a dominat playr, such as Acr in its prsonal computr supply chains, it is also a similar situation sinc th powrful playr is abl to choos his supplirs or customrs to optimiz th bnfits for th whol supply chain. Acknowldgmnts This work was supportd by National Natural Scinc Foundation of China (NO ). Rfrncs: [1 Tan K.C. A framwork of supply chain managmnt litratur, Europan Journal of Purchasing & Supply Managmnt, Vol.7, No.1, 001, pp [ Cohn M., Mallik S. Global supply chains: rsarch and applications, Production and Oprations Managmnt, Vol.6, No.3, 1997, pp [3 Cortinhal M.J.,Captivo M.E. Uppr and lowr bounds for th singl sourc capacitatd location problm, Europan Journal of Oprational Rsarch, Vol.151, No., 003, pp [4 Tjndra S., Shabbir A., Marc G., tc. A stochastic programming approach for SCN dsign undr uncrtainty, Europan Journal of Oprational Rsarch, Vol.167, No.1, 005, pp [5 Corria I., Captivo M.E. A Lagrangan Huristics for a Modular Capacitatd Location Problm, Annals of Oprations Rsarch, Vol.1, 003, pp [6 Yuri L., Adi B.-I. A huristic mthod for larg-scal multi-facility location problms, Computrs and Oprations Rsarch, 004, Vol.31, No., pp [7 Jung J.Y., Blau G., Pkny J.F., tc. A simulation basd optimization approach to supply chain managmnt undr dmand uncrtainty, Computrs and Chmical Enginring, Vol.8, No.10, 004, pp [8 Yusuf Y.Y., Gunaskaran A., Adly E.O., tc. Agil supply chain capabilitis: Dtrminants of comptitiv objctivs, Europan Journal of Oprational Rsarch, Vol.159, No., 004, pp [9 Chn S. Y., Ntwork Flow Problms for Supply Chain Managmnt with Product Tr Structur, Doctor dissrtation of National Taiwan Univrsity, Taiwan, 000. [10 L H.L., Cory B. Matrial Managing in Dcntralizd Supply Chains, Oprations Rsarch, Vol.4, No.5, 1993, pp [11 da Cunha C.B., Swait J. Nw dominanc critria for th gnralizd prmannt lablling algorithm for th shortst path problm with tim windows on dns graphs, Intrnational Transactions in Oprational Rsarch, Vol.7, No., 000, pp