An Integer Optimization Model for the Time Windows Periodic Vehicle Routing Problem with Delivery, Fleet, and Driver Scheduling

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1 Inernaona Journa on Reen and Innovaon rends n Compung and Communaon ISSN: Voume: 4 Issue: An Ineger Opmzaon Mode for he me Wndows Perod Vehe Roung Probem wh Devery Fee and Drver Shedung Syafar Graduae Shoo of Mahemas Sae Unversy of Medan Medan Indonesa Herman Mawengkang Graduae Shoo of Mahemas Unversy of Sumaera Uara Medan Indonesa hmawengkang@yahoo.om Absra he vehe roung probem (VRP) s a we-known ombnaora opmzaon probem whh desrbes a homogeneous se of vehes and roues n whh eah vehe sars from a depo and raverses aong a roue n order o serve a se of usomers wh known geographa oaons. If he devery roues are onsrued for perod of me he VRP s generazed as perod VRP (PVRP). hs paper deveops a mode for he opma managemen of perod deveres of meas of a aerng ompany. he PVRP norporaes me wndows deveres fee and drver shedung n he perod pannng.. he objeve s o mnmze he sum of he oss of raveng and eapsed me over he pannng horzon. We mode he probem as a near mxed neger program and we propose a feasbe neghbourhood dre searh approah o sove he probem. Keywords-Caerng probem modeng devery shedung eapsed me dre searh ***** I. INRODUCION Companes whh serve her usomers hrough devery goods or serves needs a sysem o pan her serves. he sysem aed ogs sysem s expeed oud gve benef no ony o he ompanes bu aso o her usomers. Vehe Roung Probem (VRP) s one of he mporan ssues reaed o he ogs sysem. hs we known ombnaora opmzaon probem ams a mnmzng he oa rave os (proporona o he rave mes or dsanes) and operaona os (proporona o he number of vehes used) suh ha usomers demand s fufed n me. [13] who were he frs o nrodue a verson of VRP n he ruk Dspahng Probem. hey modeed a probem for a fee of homogeneous ruks ha serve he demand for o of a number of gas saons from a enra hub and wh a mnmum raveed me. [32] gve a omprehensve overvew of he VRP whh dsusses probem formuaons souon ehnques mporan varans and appaons. Due o s haengng naure many researhers have been workng n hs area o dsover new approahes n seeng he bes roues n order o fnd he beer souons. Some neresng survey of VRP an be found n [33] [18] [24] [19] [23] [22] and [21]. By assumng ha he usomers for deveres of goods and serves are known n advane he assa vehe roung probem (VRP) an be defned as foows: vehes wh a fxed apay Q mus dever order quanes q ( 1 n) of goods o n usomers from a snge depo ( = 0). Gven he dsane d j beween usomers and j ( j 1 n ) he objeve of he probem s o mnmze he oa dsane raveed by he vehes. here are requremens are neessary o be sasfed: a) ony one vehe handes he deveres for a gven usomer b) he oa quany of goods ha a snge vehe devers s no arger han Q ) eah usomer s assgned o one and ony one vehe d) every vehe sars and end a he depo ( = 0) akng no aoun shedued me of pannng for roung probems s anoher varan of he VRP known as he prod VRP (PVRP). In PVRP whn a gven me horzon here s a se of usomers needs o be devered one or severa mes. here woud be a deverng shedues assoaed wh eah usomer. A fee of vehes s avaabe and eah vehe eaves he depo serves a se of usomers when s work shf or apay s over reurns o he depo. he probem s o mnmze he oa engh of he roues raveed by he vehes on he me horzon. hs probem has severa mporan rea word appaons suh as dsrbuon for bakery ompanes [27] bood produ dsrbuon [20] or pk-up of raw maeras for a manufaure of auomobe pars [1]. [5] desrbe a ase for perod manenane of eevaors a dfferen usomer oaons. Furher ase sudes onernng wase oeon and road sweepng an be found n [4] and [15]. [10] desrbe a mk oeon probem where s mporan ha he goods are oeed when fresh. [7] mpemen perod apaaed PVRP for rea dsrbuon of fue os. A survey on PVRP and s exensons an be found n [17]. Due o he ombnaora naure of he probem mos of he works presen heurs approahes neverheess [3] proposed an exa mehod based on a se paronng neger near programmng formuaon of he probem. he frs work on he PVRP s proposed by [4]. hen here are oher works by [9] [31] and [28. [7] are he frs o provde a wo-phase heurs ha aows esapng from oa opma. hey use an neger near program o assgn vs day ombnaons o usomers n order o naze he sysem. Moreover he apay m of he vehes s emporary. [6] addressed a ombned of heurs and exa mehod for sovng PVRP. Eary formuaons of he PVRP were deveoped by [4] and by [28]. hey proposed heurss ehnque apped o wase oeon probem. [21] use he dea of he generazed IJRICC January 2016 hp:// 126

2 Inernaona Journa on Reen and Innovaon rends n Compung and Communaon ISSN: Voume: 4 Issue: assgnmen mehod proposed by [14] and assgn a devery Seon 3 desrbes he mahemas formuaon of he shedue o eah verex. Evenuay a heurs for he VRP s (PVRPDFDS). he bas approah s gven n Seon 4. he apped o eah day. [29] deveoped a heurs organzed n agorhm s desrbed n Seon 5. Fnay Seon 6 desrbes four phases.[11] presen anoher agorhm: he souon he onusons. agorhm s a abu searh heurs whh aow nfeasbe souons durng he searh proess. hey onsdered wo ypes II. HE BASIC FRAMEWORK MODEL OF HE PVRPW neghborhood operaors. Smary good resus were obaned Frsy we onsder he bas frame work mode for he n he work of [14] [2] and [5] who provde spef praa aerng probem as a PVRPW By usng graph he appaons of he PVRP. [25] used pare swarm formuaon of he mode an be wren as foows (based on opmzaon o ake he probem. [26]). Le G ( V A) be a graph where V = { n} s [17] nrodue he Perod Vehe Roung Probem wh he verex se and A = {( j) : j V j} s he se of Serve Choe (PVRP-SC) whh aows serve eves o be deermned endogenousy. hey deveop an neger roue. For eah roue ( j) A a dsane (or rave) os j s programmng formuaon of he PVRP-SC wh exa and defned. he ener of aerng aed depo verex 0. heurs souon mehods. Due o he ompuaona Defne V s he se of usomers verex. Eah usomer ompexy of he probem souons o he dsree PVRP-SC are med by nsane sze. Connuous approxmaon modes V has a known day demand q 0 whn he pannng are beer sued for arge probem nsanes ye he use of horzon of me n a day. As he probem has me wndows onnuous approxmaon modes for perod roung hen he serve me s 0 requres a as he eares me of probems has been med. serve may sar and b s he aes me. Anoher requremen [13] presens modeng ehnques for dsrbuon s he frequeny number of vss f need o be performed probems wh varyng serve requremens. [30] deveop aordng o one of he aowabe vs-me paerns aken onnuous approxmaon modes for dsrbuon nework from he s of shedue L. A he ener of aerng ( 0 ) desgn wh mupe serve eves. hese referenes show ha he nerva of me for vehes o eave and o reurn o he onnuous approxmaons an be powerfu oos for sraeg depo s gven by [a 0 b 0 ]. here are m vehes eah wh and aa desons when serve hoe exss. In onnuous approxmaon modes aggregaed daa are used apay Q k s based a he enra. Eapsed me u wh nsead of more deaed npus. [26] presen PVRP wh me V k K may our. Vehes are grouped no wndows (PVRPW). he probem requres he generaon of a med number of roues for eah day of a gven pannng se. Vehe roues are resred o a maxmum duraon of horzon. he objeve s o mnmze he oa rave os D k k = 1... m. whe sasfyng severa onsrans. Assume ha he vehe fee s homogen wh Q k = Q and hs paper s abou o dever day meas from a aerng a ommon duraon resron D k = D k = 1... m. he ompany o usomers spread aross he y of Medan. he PVRPW an hen be regarded as he probem of generang requess from usomers are vared n me. herefore he (a mos) m vehe roues for eah me n a day suh ha o ompany needs o shedue he me of devery vehe o be mnmze he oa os over he enre pannng horzon. here used and he drvers. Coverage area of he operaon of hs are some onsrans ha mus be sasfed suh as 1) eah aerng ompany s arge n suh a way he ompany dvdes verex s vsed he requred number of mes f he whoe area no severa sub-area wh a onsequene ha orrespondng o a snge me shedue of vs hosen from L s neessary o nude he shedung of he sub-area n he and s served whn s me wndow.e. a vehe may PVRP. [17] address hs ype of PVRP as PVRP wh serve arrve before a and wa o begn serve; 2) eah roue sars hoe (PVRPSC). However he shedue of serve s whn from he enra vss he usomers verex seeed for ha a se of days. In our ase he shedue of serve s a se of me wh a oa demand no exeedng Q and reurns o he me n a day. We a our PVRP as PVRP wh devery enra afer a duraon (rave me) no exeedng D. shedung (PVRPDS). Le be 1 f me beongs o s of shedue he aerng ompany has a med number of fee of and 0 oherwse. Bnary varabes referred o Roue-seeon vehes and drvers. herefore needs o pan a shedue ha me s-seeon and onnuous mng deson varabes an organze hese vehes and drvers n order o sasfy her are defned as foows usomers. hs paper onerns wh a omprehensve f vehe k K raverses roue ( j) A modeng for he PVRP norporaed wh me wndows fee 1 x = on me ; and drver and devery shedung (PVRPDFDS). Due o he 0 oherwse; fa ha heerogeneous vehe wh dfferen apaes are f shedue L s assgned avaabe he bas framework of he vehe roung an be 1 y r = o usomer V ; vewed as a Heerogeneous Vehe Roung Probem wh 0 oherwse; me Wndows (HVRPW). We address a mxed neger w ndae he serve me begns for vehe k programmng formuaon o mode he probem. A feasbe K a usomer V a me neghborhood heurs searh s proposed o ge he neger feasbe souon afer sovng he onnuous mode of he z = 1 f vehe k K vs usomer V on me ; probem. 0 oherwse; Seon 2 revews he mxed neger programmng formuaon of he PVRP wh me wndows from [26] IJRICC January 2016 hp:// 127

3 Inernaona Journa on Reen and Innovaon rends n Compung and Communaon ISSN: Voume: 4 Issue: he PVRPW for he aerng probem an be wren shedued me from he s L for drver D s denoed by mahemaay as foows. he objeve funon s o mnmze rave me for eah vehe and eapsed me.. he sar workng me and aes endng me for mnmze j x u z (1) drver D on me are gven by g and h ( j) A kk V kk respevey. For eah drver D e H denoe he maxmum day workng duraon. here are onsrans ha mus be fufed formuaed as Le K denoe he se of vehes. For eah vehe k K foows e Q k and P k denoe he apay n wegh and n voume y 1 V (2) respevey. We assume he number of vehes equas o he L number of drvers. Denoe he se of n usomers (nodes) by N 12 n 0. Eah vehe x x j V k K (3) x y V (4) kk L kk x m 0 x0 1 k K (6) q x Q k K (7) V w s M (1 x ) w ( j) A k K (8) j (9) e x w x V k K w s M(1 x ) D V k K (10) 0 0k x {01} ( j) A k K (11) y {01} V L (12) z {01} V k K (13) w 0 V k K (14) Expresson (2) ensures ha a feasbe paern s assgned o eah usomer. Eq. (3) shows he enformen of fow onservaon o ensure ha a vehe arrvng a a usomer on a gven me eaves ha usomer on he same me. Eq. (4) guaranee ha eah usomer s vsed on he mes orrespondng o he assgned shedue whe expresson (5) makes sure ha he number of vehes used on eah me shedued does no exeed m. o ensure ha eah vehe s used a mos one n me shedue s presened n Eq. (6) whe (7) guaranee ha he oad harged on a vehe does no exeed s apay. Consrans (8) enfore me feasby.e. vehe k anno sar servng j before ompeng serve a he prevous usomer and raveng from o j.e. no before j (5) w s. Consrans (9) ensure ha usomer me wndow resrons are respeed whe (10) onsrans he roue engh. Expressons (11) - (14) defne he ses of deson varabes. III. MODELING HE PVRDFDS Now we formuae he aerng probem as a Perod Vehe Roung wh me Wndows onsderng Devery Fee and Drver Shedung (PVRPDFDS) Defne he pannng horzon by and he se of drvers by D. he se of and he enra depo by sars from enra of aerng 0 and ermnaes a enra 0. Eah usomer N spefes a se of mes o be vsed denoed by. A eah shedued me usomer N requess serve wh demand of q n wegh and wndow p n voume serve duraon d and me IJRICC January 2016 hp:// a b. Noe ha for he enra depo me we se q p d 0 0 a. he rave me beween usomer and j s gven by j. Denoe he os oeffens of he rave me of he drvers by μ and he eapsed me by ρ. We defne bnary varabe x o be 1 f vehe k raves he roue (j) V on me. Oher varabes an be found a he foowng noaons. hs noaons used are gven as foows : Se: he se of shedued me n he pannng horzon D he se of drvers D he se of workng me for drver D K he se of vehes N he se of usomers he se of me on whh usomer N Parameer: orders Q k he wegh apay of vehe k K P k he voume apay of vehe k K j he rave me of roue ( j) V [a b ] he eares and he aes vs me a node V d he serve me of node V on me on me q he wegh demand of node V p he voume demand of node V on me 128

4 Inernaona Journa on Reen and Innovaon rends n Compung and Communaon ISSN: Voume: 4 Issue: [ g h ] he sar me and he aes endng me of p x Pk k K (18) drver D on me N Devery quany for usomer on me u u j M (1 x ) Mz F he maxmum eapsed drvng me he os faor on he oa rave me of j V k K (19) drvers u j M (1 x ) j N k K he os faor on he oa workng hour of (20) drvers λ he os faor for he oa eapsed me u j x F Mz V k K (21) Varabes: jn0 x k Bnary varabe ndang wheher vehe k K raves from node V o j V on me v he me a whh vehe k K a node V on me z sars serve Bnary varabe ndang wheher vehe k K akes break afer servng node N on me u he eapsed drvng me of vehe k K node V on me y k a Bnary varabe ndang wheher vehe k K s assgned o drver D r he oa workng duraon of drver D me s he oa rave dsane of drver D me on me on on Number of devery demands of usomer j served by vehe k K on me v v d M (1 x ) j j V k K (22) b v a N k K (23) v0 k ( g yk ) k K (24) D vn 1 k ( h y ) k K (25) D s j x M (1 y ) D I k K (26) V r vn 1 k g M (1 yk ) D k K (27) j kk j N (28) x w z y {01} j N 0 D k K (29) k v u r s 0 j N 0 D k K (30) {012...} j N k K (31) he mahemaa formuaon for hs probem s presened as foows. he objeve of he aerng ompany s o mnmze he os of workng hours rave me and eapsed me of drvers. (15) mn s r u Consrans D D V kk x 1 N kk (16) q x Qk k K (17) N Consrans (16) sae ha eah usomer mus be devered by one vehe on eah of s devery me.) he vehe apaes whh are resred o her wegh and voume expressed n Consrans (17-18). In Consrans (19-24) we defne he eapsed drvng me and he me spen a eah devery. Agebraay for he vehe (k) raveng from usomer o j on day he eapsed drvng me a j equas he eapsed drvng me a pus he drvng me for roue o j (.e. u u + j ) f he drver does no ake a break when devers o usomer (.e. z = 0); Oherwse f he drver akes a break (.e. z = 1) he eapsed drvng me a j w be onsraned by (20) whh make sure s greaer han or equa o he rave me beween and j (.e. u j ). Consrans (21) sae ha he eapsed drvng me never IJRICC January 2016 hp:// 129

5 Inernaona Journa on Reen and Innovaon rends n Compung and Communaon ISSN: Voume: 4 Issue: exeeds an upper m F by mposng a break a usomer I s now ear ha a nonbas varabe pays an mporan (.e. z = 1. roe o negerzed he orrespondng bas varabe. herefore he foowng resu s neessary n order o onfrm ha mus be a non-neger varabe o work wh n negerzng proess. Consrans (23) deermne he me o sar he devery a eah usomer. If j s devered mmedaey afer he me v o sar he devery a j shoud be greaer han or equa o he devery sarng me v a pus s devery duraon d he rave me beween he wo usomers j and he break me G f he drver akes a break afer servng I (.e. z = 1). Consrans (23) make sure he serves sar whn he usomers me wndow. Consrans (24-25) ensure ha he sarng me and endng me of eah roue mus e beween he sar me and aes devery job endng me of he assgned drver. Consrans (26) auae he oa rave me for eah drver. Consrans (27) defne he workng duraon for eah drver on every workday whh equas he me he drver reurns o he enra mnus he me he drver sars he devery. Consrans (28 29) defne he devery for eah usomer. Consrans (30-31) defne he varabes used n hs formuaon. IV. HE BASIC APPROACH Consder a Mxed Ineger Lnear Programmng ( MILP) probem wh he foowng form Mnmze P = x (32) Subje o Ax b (33) x 0 (34) x j neger for some j J (35) A omponen of he opma bas feasbe veor x B k o MILP soved as onnuous an be wren as x B k = β k α k1 (x N ) 1 α kj (x N ) j α kn m(x N ) N n (36) Noe ha hs expresson an be found n he fna abeau of Smpex proedure. If (x B ) k s an neger varabe and we assume ha β k s no an neger he paronng of β k no he neger and fraona omponens s ha gven β k + f k 0 f k 1 (37) suppose we wsh o nrease (x B ) k o s neares neger ( + 1). Based on he dea of subopma souons we may eevae a paruar nonbas varabe say x N j above s bound of zero provded kj as one of he eemen of he veor j s negave. Le j be amoun of movemen of he non varabe x N j suh ha he numera vaue of saar (x B ) k s neger. Referrng o Eqn. (17) j an hen be expressed as f = 1 f k (38) α kj whe he remanng nonbas say a zero. I an be seen ha afer subsung (37) no (38) for x N j and akng no aoun he paronng of β k gven n (18) we oban hus (x B ) k s now an neger. (x B ) k = + 1 heorem 1. Suppose he MILP probem 32 -(35) has an opma souon hen some of he nonbas varabes. (x N ) j j = 1 n mus be non-neger varabes. Proof: Sovng probem as a onnuous of sak varabes (whh are non-neger exep n he ase of equay onsran). If we assume ha he veor of bas varabes onsss of a he sak varabes hen a neger varabes woud be n he nonbas veor x N and herefore neger vaued. V. HE ALGORIHM Sage 1. For he negerzng proess. Sep 1. Ge row * he smaes neger nfeasby suh ha mn{ f 1 f } * (hs hoe s movaed by he desre for mnma deeroraon n he objeve funon and eary orresponds o he neger bas wh smaes neger nfeasby). Sep 2. Do a prng operaon * * IJRICC January 2016 hp:// v e B 1 Sep 3. Cauae v j * j Wh j orresponds o d j mn j j Cauae he maxmum movemen of nonbas j a ower bound and upper bound. Oherwse go o nex non-neger nonbas or superbas j (f avaabe). Evenuay he oumn j* s o be nreased form LB or dereased from UB. If none go o nex *. Sep 4. Sove B j* = j* for j* Sep 5. Do rao es for he bas varabes n order o say feasbe due o he reeasng of nonbas j* from s bounds. Sep 6. Exhange bass Sep 7. If row * = {} go o Sage 2 oherwse Repea from sep 1. Sage 2. Pass1 : adjus neger nfeasbe superbass by fraona seps o reah ompee neger feasby. Pass2 : adjus neger feasbe superbass. he objeve of hs phase s o ondu a hghy oazed neghbourhood searh o verfy oa opmay. VI. CONCLUSIONS hs paper was nended o deveop a mode of Perod vehe Roung wh me Wndows Devery Fee and Drver Shedung Probem hs mode s used for sovng a 130

6 Inernaona Journa on Reen and Innovaon rends n Compung and Communaon ISSN: Voume: 4 Issue: aerng probem oaed n Medan y Indonesa. he resu mode s n he form of mxed neger near programmng and hen soved by empoyng neares neghbor heurs agorhm. hs agorhm offers approprae souons n a very sma amoun of me. REFERENCES [1] Aegre J. M. Laguna J. Paheo. Opmzng he perod pk-up of raw maeras for a manufaure of auo pars. European Journa of Operaona Researh [2] Angee. E. & Speranza. M.G. he perod vehe roung probem wh nermedae faes. European Journa of Operaona Researh [3] Bada R. E. BAron A. Mngozz A. Vaea An Exa agorhm for he perod roung probem Operaons researh 59 (1) [4] Beram E. Bodn L. Neworks and vehe roung for munpa wase oeon. Neworks 4(1) [5] Bakey F. Bozkaya B. Cao B. Ha W. Knomajer J. Opmzng perod manenane operaons for Shnder Eevaor Corporaon. Inerfaes 33(1) [6] Cahan V. V. C. Hemmemayr F. rore. A se overng based heurs agorhm for he perod vehe roung probem. Dsree Apped Mahemas [7] Caroenuo P. S. Godarn S. Massar F. Vagaggn. Perod apaaed vehe roung for rea dsrbuon of fue os. ransporaon Researh Proeda [8] Chao I.M. Goden B. Was E. An mproved heurs for he perod vehe roung probem. Neworks 26(1) [9] Chrsofdes N. Beasey J. he perod roung probem. Neworks 14(2) [10] Caassen G. D. H. h. H. B. Hendrs An appaon of spea ordered ses o a perod mk oeon probem. European Journa of Operaona Researh [11] Cordeau J.F. Gendreau M. Lapore G. A abu Searh heurs for perod and mu-depo vehe roung probems. Neworks 30(2) [12] Daganzo C. F. Modeng dsrbuon probems wh me wndows: Par II. ransporaon Sene 21(3) [13] G.B. Danzg and J.H. Ramser he ruk dspahng probem Managemen Sene Vo [14] Dj onsannuo. Ha & Bada. E. A mu-depo perod vehe roung probem arsng n he ues seor. Journa of Operaons Researh Soey [15] Egese R. W. H. Murdok Roung road sweepers n a rura area. Journa of he Operaona Researh Soey [16] Fsher. M.L. & Jaumar. R. A generazed assgnmen heurs for vehe roung. Neworks ±124. [17] Frans P. Smowz K. zur M. he perod vehe roung probem wh serve hoe. ransporaon Sene [18] B. L. Goden A. A. Assad and E. A. Was Roung Vehes n he rea word: Appaons n he sod wase beverage food dary and newspaper ndusres. In P. oh and D. Vgo edors he Vehe Roung Probem pages SIAM Phadepha PA [19] B. L. Goden S. Raghavan and E. A. Was he vehe roung probem: aes advanes and new haenges. Sprnger [20] Hemmemayr V. K. F. Doerner R. F. Har M. W. P. Savesbergh Devery sraeges for bood produ suppes. OR Sperum [21] De Jaegere N. M. Defraeye I. van Neuwenhuyse Fauy of Eonoms and Busness Researh Repor Fauy of Eonoms and Busness Leuven Begum [22] Kumar S. N. R. Pannersevam A survey on he vehe roung probem and s varans Inegen Informaon Managemen [23] Lahyan R. M. Khemakhem F. Seme. Rh vehe roung probems: From a axonomy o a defnon European Journa of Operaona Researh 241(2) [24] G. Lapore and F. Semen. Cassa Heurss for he Vehe Roung Probem. In oh P. and Vgo D. edors he vehe Roung Probem SIAM Monographs on Dsree Maheams and Appaons pages SIAM Phadepha PA [25] Moghaddam B. F. Ruz R. & Sadjad S. J. (2012). Vehe roung probem wh uneran demands: An advaned pare swarm agorhm. Compuers & Indusra Engneerng 62(1) do: /j.e [26] P. K. Nguyen. G. Cran M. ououse A Hybrd Gene Agorhm for he Perod Vehe Roung Probem wh me Wndow. CIRREL [27] Paheo J. A. Avarez I. Gara F. Ange-Beo Opmzng vehe roues n a bakery ompany aowng fexby n devery daes. Journa of he Operaons Researh Soey [28] Russe R. Igo W. An assgnmen roung probem. Neworks 9(1) [29] Russe. R.A. & Grbbn. D. A muphase approah o he perod roung probem. Neworks ±765. [30] Smowz K. Daganzo C. F. Cos modeng and souon ehnques for ompex ransporaon sysems. IEMS Workng paper Norhwesern Unversy [31] an. C.C.R. & Beasey. J.E. A heurs agorhm for he perod vehe roung probem. OMEGA Inernaona Journa of Managemen Sene ±504. [32] P. oh and D. Vgo. he Vehe Roung Probem Soey for Indusra and Apped Maheams Phadepha PA [33] Vda.. G. Cran M. Gendreau C. Prns. Heurss for mu-arbue vehe roung probems: A survey and synhess. CIRREL IJRICC January 2016 hp:// 131