Train Timetabling on double track and multiple station capacity railway with useful upper and lower bounds Afshin Oroojlooy Jadid 1, Kourosh Eshghi 2

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1 Train Timeabling on double rack and muliple saion capaciy railway wih useful upper and lower bounds Afshin Oroojlooy Jadid 1, Kourosh Eshghi 2 1 Deparmen of Indusrial and Sysems Engineering, Lehigh Universiy 2 Deparmen of Indusrial Engineering, Sharif Universiy of Technology, Tehran, Iran 1 Corresponding auhor Afshin Oroojlooy jadid Deparmen of Indusrial and Sysems Engineering, Lehigh Universiy Harold S. Mohler Laboraory 200 Wes Packer Avenue Behlehem, PA , USA Tel: ; Cell: afo214@lehigh.edu 2 Deparmen of Indusrial Engineering, Sharif Universiy of Technology, Zip code 14588/89694, Azadi Ave, Tehran, Iran, Tel.: ; Cell: ; fax: eshghi@sharif.edu Train scheduling is one of he significan issues in he railway indusry in recen years since i has an imporan role in efficacy of railway infrasrucure. In his paper, he imeabling problem of a muliple racked railway nework is discussed. More specifically, a general model is presened here in which a se of operaional and safey requiremens is considered. The model handles he rains overaking in saions and considers he saions capaciy. The objecive funcion is o minimize he oal ravel ime. Unforunaely, he problem is NP-hard and real size problems canno be solved in an accepable amoun of ime. In order o reduce he processing ime, we presened some heurisic rules, which reduce he number of binary variables. These rules are based on problem's parameers such as ravel ime, dwell ime and safey ime of saions and ry o remove he impracicable areas of he soluion space. Furhermore, a Lagrangian Relaxaion algorihm model is presened in order o find a lowerbound. Finally, comprehensive numerical experimens on he Tehran Mero case are repored. Resuls show he efficiency of he heurisic rules and also he Lagrangian Relaxaion mehod in a way ha for all analyzed problems he opimum value are obained. Key words: Transporaion, Train imeabling, Mahemaical Programming, Heurisic, Lagrangian relaxaion 1- Inroducion Railway is a fas and economic mode of ransporaion; according o he Associaion of American Railroad s sudy, rail companies move more han 40 percen of he US s oal freigh [1] and is prediced o expand as double of he curren amoun on So, he railways managers have o expand he infrasrucures or manage he curren faciliies more efficienly. The consrucion of he new infrasrucures is very expensive and ime consuming so ha he uilizaion efficiency of he curren nework faciliies by opimized line planning, nework imeabling, crew scheduling and mainenance scheduling is very imporan. One of he mos influenial majors in his lis is imeabling, which has engendered a big field of sudy by iself. 1

2 In a general poin of view, i can be divided ino hree main fields: mahemaical programing, simulaion based opimizaion mehods and exper sysems. In he field of he mahemaical programing, he aim is o creae he global opimal imeable or reschedule he exising imeable. This aricle is focused on creaing new schedule and imeables, based on mahemaical programming. [2] presened a mahemaical model for single-rack railway wih dynamic ravel imes. They considered delay and operaional coss as he objecive funcion and proposed a Branch and Bound (B&B) o solve he problem. [3] presened a model based on Periodic Even Scheduling Problem (PESP) in which all of heir parameers such as dwell ime, headway ime and rip ime are dynamically considered. [4] developed a muli-objecive nonlinear model o minimize fuel consumpion and oal passenger-ime. They found he Pareo fronier and hen used a disance-based mehod o find he soluion. [5] used he Pareo soluion o solve a double objecive model and, a combinaion of he expeced waiing ime and he oal ravel ime as an objecive funcion. A beam search in a B&B algorihm is proposed o solve he MIP problem. In anoher sudy on muli-objecive problems [6] proposed a Paricle Swarm Opimizaion (PSO) for dealing wih he oal ravel ime and he variaion of iner deparure. [7] proposed an objecive funcion consiss of differen ypes of waiing ime and he lae arrival. Furhermore, hey presened a wo phase algorihm, obaining an ideal buffer ime, and hen a imeable is creaed by using an LP model. Finally, a simulaion compares differen imeables. [8] proposed ideal running ime insead of summaion of acual ravel and dwell ime. Their objecive is based on he delay disribuion of rains, he passengers coun and differen ypes of waiing ime and lae arrivals. They buil differen imeable wih a LP model and a simulaion evaluaes hem. [9] proposed a mixed ineger model wih a fuzzy muli objecive funcion, o minimize energy minimizaion, carbon emission cos and oal passenger ime. The model considered an improved version of objecive funcion compared o [4] and in he siuaion ha all rains powered by elecriciy, he proposed model degeneraes o [4] model. [10] presened a complicaed B&B algorihm o solve he proposed problem in [5]. They developed hree mehods for node selecion and proposed some oher complicaed rules for he branching process in B&B mehod. Also, hey designed a Lagrangian Relaxaion mehod and anoher heurisic mehod o find a lower bound. [11] proposed a heurisic mehod ha provides rain pah and imeable simulaneously on mixed single and double rack neworks. The mehod has four phases ha ieraively creaes and adjuss a imeable. [12] for a single line nework proposed a hree sage mehod which decomposes he model o find a soluion wih he maximum relaive ime, i.e. he raio of ravel ime o minimum possible ravel ime, he minimum sum of he deparure ime, and he minimum fuel consumpion. In order o solve he problem, hey proposed a bicrieria algorihm o minimize he relaive ravel ime. [13] proposed a model o minimize he oal passenger rip ime, considering he number of passengers as a sochasic funcion. They used he expeced value, pessimisic and opimisic values for he number of passengers and designed a B&B algorihm o solve he model. [14] proposed a bisecion mehod wih objecive funcion of relaive ravel ime and used some heurisic mehods o reduce he number of inacive binary variables in single rack and double racks neworks. Furhermore, hey proved ha he soluion of he algorihm is opimal. [15] presened a nonlinear model which considers alernae double single rack (ADST) lines. Their objecive is o minimize he consrucion cos, maximum relaive ravel ime of all rains, and he sum of oal relaive ravel imes such ha he obained deparure imes be close o he desired ones. They linearized he iniial model and proposed some binary reducion. By he graph approach, [16] modeled he rain imeabling problem as a blocking parallel-machine job shop scheduling problem and inroduced an improved Shifing Boleneck Procedure. Then, hey used an alernaive graph o solve he main problem. [17] inroduced a 2

3 graph model for parallel rails wih linked rails in sidings, capaciaed buffer, acceleraion and deceleraion ime which also does no allow unforced idle ime. In order o solve he problem, hey used a consrucive algorihm (CA), simulaed annealing (SA) and local search (LS) meaheurisics. [18] addressed he adjusing of imeables o handle perurbaions and unnecessary muliple overaking conflics. They used he disjuncive graph o represen he problem and used LS and SA o obain a good correcion. [19] considered he imeabling problem as a job-shop problem and addressed a cusomized disjuncive graph o consruc he imeable. In addiion, hey proposed some CA o creae feasible soluions. [20] inroduced a disjuncive graph model, considering rains and secions lengh, headways, and blocking condiions, non-delay scheduling policy and passing loops. They proposed a CA, based on NEH algorihm for Job shop problem, an SA, and LS mehods o creae and improve soluions. [21] proposed a new model for solving microscopic scale rain imeabling problems. In heir defined problem, saions have muli line and gae capaciies and he arrival and deparure ime of rains are known. They proposed a new mehod for deermining large conflic cliques in conflic graph and pu hem in an ILP model, relaxing srongly he relaed LP problem. [22] proposed a disjuncive graph, considering capaciaed saions and sidings on single line neworks, no-wai, and blocking properies. A hybrid algorihm is proposed o consruc he feasible rain imeable combined by a LS algorihm, minimizing he makespan. [23] proposed a heurisic algorihm using relaxed dynamic programming, based on he acyclic space-ime graph. The algorihm sars wih an ideal imeable and ries o resolve he conflics beween rains. Their model considers single corridor neworks, parallel racks in he saions, and also he presence of juncions on he nework. In he caegory of mea-heurisic mehods, [24] proposed a Geneic Algorihm (GA) for he PESP ha included a guided process o build iniial populaion. [25] proposed a model o deal wih rain imeabling in single rail neworks which is based on PESP. They also proposed a hybrid algorihm consising of SA and PSO. [26] proposed a new robus periodic model which is based on PESP on a single rail nework. In heir model, saion capaciy and headway consrains are considered. A fuzzy approach is used o consider he robusness and is radeoff wih he rain delay and he ime inerval beween deparures of rains from same origin. Also, a SA is used o solve he problems. In he robus problem caegory, [27] presened a complee survey on he robus models and heir feaures. Some new robus research also is presened in [28]. [29] proposed a mahemaical programming model, considering oal energy consumpion and oal raversing ime opimizaion in railway nework wih muliple rains and muliple links in saions. An inegraed GA and simulaion is used o obain an approximae opimal sraegy. [30] proposed a cusomized An Colony Opimizaion o minimize oal weighed ardiness in single rack railways. [31] proposed a ravel advance sraegy (TAS) mehod combined wih GA for dealing wih single rack railways. The algorihm searches for a imeable wih minimal delay raio, i.e. he oal delay ime over he oal free-run ime. [32] proposed a model o obain imeabling on one-way high speed double rack neworks. In order o solve he model, hey proposed an improved GA and used simulaion o analyze he accuracy of he algorihm. [33] also proposed a GA o provide he imeable of an urban rail ransi sysem. The model adjuss he headway o obain he bes rade-off beween he passenger ravel ime and energy consumpion wih a guaraneed ransi capaciy. In a lile bi differen conex, [34] proposed a linear formulaion of cyclic imeabling problem for single rack railways in which minimizaion of cycle lengh is he objecive. [35] proposed a rain imeabling model which deals wih dynamic demand. Considering dynamic demand for differen roues in differen imes, hey proposed hree binary models and a branch and cu algorihm o minimize he oal waiing of passengers in saions. 3

4 We could no find any sudy represening a linear mahemaical model o obain imeabling in a nework wih parallel uni-direcional racks and limied number of saion's plaforms or siding's capaciies. In his way we propose an approach ha models overaking decisions a saions/ sidings, as opposed o oher approaches which model precedence's (i.e. sequencing) on single racks as limied sources. Also, we show ha he problem is NP-hard and he real size problems canno be solved in a reasonable amoun of ime. In order o obain beneficial soluions in a reasonable ime, we provide some new upper bound and lower bound rules. By concenraing on hese wo opics, he srucure of his aricle is as follows. In secion 2 we provide he mahemaical model. Then in secion 3 we propose he upper bound rules and in secion 4 propose he lower bound rule and i is followed by numerical experimen s resul in a real world problem. 2- Problem Definiion In his secion, we define he problem and is assumpions, give he noaion and inroduce our model. This sudy considers a general siuaion of a double-lines neworks. In all lines and corridors in he nework, here are wo parallel racks and each rack faciliaes uni-direcional flow and no bi-direcional, i.e. here is no opposie direcion rain, which is a common siuaion in mero, subway and high speed neworks; and is of ineres as [36, 37] covered some earlier research. Each saion or siding can have one or more capaciy, which some of hem may have no plaform for passengers. Also, in sidings here is no linked rail beween opposie direcions and each direcion has is separae sidings. In addiion, overaking is allowed and rains can only overake each oher in he siding (passing loops) or in he saions wih more han one plaform, i.e. rain roues in saions are no fixed. Moreover, he main line is he line ha connecs he saions or siding wih each oher. A general view of siding and main line are showed in Figure 13. [Figure 1 should be here] There may be differen ypes of rains wih differen speeds and dwell ime. The rains may be freigh or passenger specific or any mix of hem. I is assumed ha for each rain in each saion of he nework, here is a minimum and maximum dwell ime and also ravel ime beween wo consecuive saions or sidings is known. Roues and he dispaching sequence of rains beween saions and sidings are given. In addiion, he ravel ime beween wo saions or sidings is he difference beween he deparure imes of he las saion unil he arrival ime of he curren saion. Considering he ype of rains and heir inrinsic naure in mero and high speed neworks, here is no significan ime for acceleraion and deceleraion ime, bu we have considered he minimum ravel ime beween wo saions o include hese imes. Noe ha, he considered parameers, e.g. ravel ime, dispaching sequence, and dwelling ime are known in any railway nework and hey are he mos common feaures of he MIP-models for he imeabling problem. A mahemaical model is presened as follows. In his model oupu is he imeable of a railway nework. The mos imporan assumpions of our model are as below. All parameers of he model are deerminisic. Each corridor of he nework has one line in each direcion. The saion capaciy which is he number of plaforms in he saions can be any posiive ineger number. There is a same siuaion abou sidings. Acceleraion and deceleraion ime are considered in he ravel ime. 4

5 Number of rains and heir roues are given. 2-1 Noaion Throughou he paper, we reserve l o denoe line index in a nework, L is he lis of all lines, defines rain index, Tl is he lis of rains in line l, s is he saion index in each line and Sl defines he oal number of saion(s) in line l. Also, q is he nex saion in line l,.i.e. q = s + 1. Finally, ml denoes he las saion in line l. The major parameers of our problem are as follows: SFl,s: Minimum headway (Safey ime) beween wo adjacen rains a saion s a line l. SCs: Capaciy of saion s, which is he number of plaform racks and capaciy of sops. D l,s : Minimum dwell ime of rain in saion s of line l. D l,s: Maximum dwell ime of rain in saion s of line l T l,s,q : Minimum ravel ime of rain beween saion s and q (s+1) of line l T l,s,q : Maximum ravel ime of rain beween saion s and q (s+1) of line l rl,,s: Earlies sar ime. Minimum sar ime of rain in saion s of line l M : shows a large posiive number. Also, we use hese wo noaions hroughou he paper: ψ,',q: The ime inerval beween he deparure ime of rain from saion q and he arrival ime of rain ' o his saion ϑ,',s: The ime inerval beween he deparure ime of wo rains from saion s For safey reasons rains are no permied o close o each oher in case of collision. Beween deparure and arrival of rains in each saion or siding minimum headway mus be saisfied. Oher siuaions are, deparure of wo rains from a saion or siding and arrival of wo rains in a saion or siding. Saions' headway is a predefined parameer ha is inherenly relaed o geographical and echnical characerisics of each saion. In order o idenify he decision variables, firs, we define he siuaions ha every wo rains may encouner in a saion or siding. These siuaions are he evens ha may bring some changes o he sequence of any wo arbirary rains. Wih regard o he siuaions of he problem, for each wo rains, four differen siuaions can occur. Even (1) defines he siuaion ha rain overakes rain ' in saion s. In even (2) rain ' overakes rain in saion s. In even (3) rain ' arrives and depars saion s afer rain. Finally, even (4) demonsraes he siuaion ha rain ' arrives and depars saion s before rain. In oher words, in his even rain has overaken rain ' before saion s or is scheduled before rain ' from he firs saion. Figure 14 shows and clarifies he menioned siuaions for he rains and '. [Figure 2 should be here] By considering hese evens and siuaions, he decision variables are as below ha cover any siuaion in a nework: cl,,s = Deparure ime of h rain from saion s of line l sl,,s = Arrival ime of h rain in saion s of line l x,,s = 1 if rain ' overakes rain in saion s as even 1 and 2, oherwise 0 5

6 x,,s= 1 if rain ' scheduled afer rain in saion s, oherwise 0. This variable represens he siuaions of even 3 and 4. y,,s = 1 if rain ' and rain use simulaneously saion s, oherwise 0 As he definiion of decision variables shows, binary variables will model overaking decisions in a double-line uni-direcional railway nework. By his approach, our decision variables define he sequence of rains in he nework and deermine when and in which saion or siding he rains will overake each oher. Somehow, his is a differen approach compared o he curren models ha define precedence and sequencing on single racks. Wih regard o hese variables, consrains of problem are defined as follows. 2-2 Objecive Funcion Our model's aim is o minimize he summaion of he deparure ime a he las saion for all rains, which is he minimum oal ravel ime as is shown in Equaion (1). L T l=1 =1 ; min z = c lml (1) This objecive funcion ries o increase he efficacy of he nework's infrasrucure and also minimize he ravel ime. Therefore, i saisfies he crieria of wo main groups in he railway neworks, i.e. passengers and railway owners. I is also a commonly used objecive funcion in research of [3, 5, 10, 12, 36, 38, 39]. 2-3 Travel Time Consrains The consrains below ensure ha he running ime of a rain does no violae a given limi of ime, which also defines he speed limiaion in he line. Furhermore, hey do no allow he running ime o drop below a specific amoun of ime defined by he desire of passengers. For all T l ; l L; s S l he consrains are as follow: s l,,s+1 c ls + T l,s,q s l,,s+1 c ls + T l,s,q (2) (3) 2-4 Dwell Time Consrains The following consrains ensure ha he dwell ime of a rain does no violae a given limi of ime. The upper and lower bounds mus be gahered as real values for each saion and rain. Moreover, for rains ha have no been scheduled o sop a a given saion, boh bounds are se o zero for ha saion. For all T l ; l L; s S l he consrains are as below: c ls s ls + D l,s c ls s ls + D l,s (4) (5) 2-5 Overaking Consrains The consrains below aim o define he sequence of rains in each corridor and saion and also deermine he arrival and deparure ime of rains. Using he defined binary variables, relaed consrains o each even have been defined. In even (3) and (4), consrains (6) and (7) ensure 6

7 ha he arrival and deparure ime of rain ' be greaer han rain if rain ' scheduled afer rain and vice versa. Also, in his siuaion consrain (8) does no allow rains overake each oher excep in he saions. Consrain (9) and (10) ensure ha he arrival ime of rain ' be greaer han rain and is deparure be lower han rain, if rain ' scheduled for overaking of rain. They ac he same as descripion of evens (1) and (2). Consrain (11) acs similarly o he consrain (8) and does no allow in evens (1) and (2) rains overake each oher excep in he saions. Consrain (12) ensures ha in each saion and for each pair of rains and ', exacly one even can occur. In order o simplify he noaion, we used ϕ l,s (, ) o represen SF l,s (1 x,,s)m. For all l L;, T l ; s S l he consrains are as follow. s l s s ls + ϕ l,s (, ) ; c l s c ls + ϕ l,s (, ) ; s l s s ls + ϕ l,s (, ) ; s > 1 ; s l s s ls + ϕ l,s (, ) ; c ls c l s + ϕ l,s (, ) ; s ls s l s + ϕ l,s (, ) ; s > 1 ; x,,s + x,,s + x,,s + x,,s = 1 ; (6) (7) (8) (9) (10) (11) (12) 2-6 Saion Capaciy Consrains This se of consrains is proposed o guaranee he maximum capaciy consrain of each saion or siding. Consrain (13) assures ha in a saion wih one capaciy, only one rain sops in saion a any ime (even ype (3) and (4)). Thus he arrival ime of one rain mus be greaer han anoher one which resuls in he condiion ha more han one rain canno be sopped a one momen in he saion. s ls c l s + ϕ l,s (, ) s l s c ls + ϕ l,s (, ) ; SC s = 1 ; > (x 1, 2,s + x 2, 1,s) + (x 1, 3,s + x 3, 1,s) + + (x 1, SC s+1,s + x SC s+1, 1,s) + + l L (x SC s, SCs+1,s + x SC s+1, SC s,s ) ( (SC s+1) ) 1 ; { 2 1, 2 SCs +1 T l s S l and SC s > 1 Consrain (14) considers he capaciy consrain of saion s wih capaciy of SCs, for he overaking evens (even ype (1) and (2)). In saions ha heir capaciy is one, consrain (13) does no allow any overaking, i.e. he consrain is (x,,s + x,,s) 0 for all of T l and s S l in line l. In saions wih one more capaciy, he consrain considers all pair combinaions of hree rains (because ( SC s+1 2 ) = 3) as (x,,s + x,,s) + (x,,s + x,,s) + (x,,s + x,,s) 2 for all T l and s S l in line l. This consrain ensures a mos one rain can overake from anoher rain a each ime, i.e. a each ime only wo rains can sop in he saion s. To furher illusrae he issue, consider he siuaion ha in saion s, which has wo plaforms, rain is scheduled o overake rain ', i.e. (13) (14) 7

8 x,,s = 1, and (x,,s + x,,s) = 1, and rain " is enering he saion s. Three siuaions may occur: 1- Assume rain " is scheduled o depar he saion s afer rain '. In his siuaion here is no addiional overaking and rain " may incur some delay. 2- Assume ha wihou regarding he saion capaciy, rain " is scheduled o overake rain in he saion s. If rain " overakes rain, i.e. x,,s = 1 and (x,,s + x,,s) = 1, rain " arrive laer han rains and ' and mus depar before hem. In anoher word, rain " also has o overake rain ' ha means x,,s = 1 and (x,,s + x,,s) = 1. So, (x,,s + x,,s) + (x,,s + x,,s) + (x,,s + x,,s) = 3 and considering he RHS of he consrain, he siuaion is no allowed and he assumpion is no rue. So, consrain (14) does no allow exceeding he capaciy of he saion. 3- Assume rain " is scheduled o overake rain ' in he saion s and rain depar he saion before arrival of rain ". If rain " and rain overake rain ' in saion s, i.e. x,,s = 1 and (x,,s + x,,s) = 1 and x,,s = 1 and (x,,s + x,,s) = 1 he consrain saisfies. Wih a similar approach for saion wih capaciy of SCs, here are (SC s+1)sc s pairs of 2 variables and on he righ hand side, he number of pairs minus one is replaced. This se does no allow violaion in saions' capaciy wih a larger capaciy. Furhermore, consrains (15), (16) and (17) consider he saion capaciy when he wo rains are scheduled wih evens ype (3) or (4). s l s c ls + SF l,s (1 x,,s)m My,,s ; SC s 2 ; > T, (y,,s + y,,s + x,,s + x,,s) SC s 1 ; SC s 2 y,,s x,,s ; SC s 2 ; > As menioned y,,s is he auxiliary variable o consider he saion capaciy in he menioned evens. Consrains (15) and (17) insure ha he auxiliary variables can ake value when even ypes (3) and (4) occur. For insance in even ype 3 ha rain scheduled before rain ', i.e. x,,s = 1, y,,s can ake value one. Regarding Consrain (15), if y,,s akes 1, s l s will be greaer han M, his means ha he rains can sop simulaneously in saion s. The same siuaions can be assumed for even ype (4). Also, Consrain (16) insures ha he number of rains ha can sop in each saion will no be more han is capaciy. These consrains wih coninues ime variables handle he saion capaciy in each of he four evens, and i is a new formulaion in he lieraure. The curren papers mosly consider ime sloing approach, e.g. [41], which consider a ime horizon for he nework s operaion ime and discreize i ino some imesamps and propose some consrains o consider he saion capaciy. A similar slo based scheduling is used by [39] in an inegraed bi-level saion layou design and scheduling model. 2-7 Deparure Time Consrain I is necessary ha some rains may have a predefined deparure ime for some saions. I may occur in he firs saion of he corridor. For all l L; T l ; s S l he consrain is as equaion (18). 8 (15) (16) (17)

9 c ls r ls Compared o he curren models of rain imeabling problem in he lieraure, our model has some advanageous poins. The firs feaure of our model is o define even base decision variables and consrains. These variables make i possible o use hem in decomposiions or relaxaion algorihms so ha each sub-problem consiss of a decision making problem relaed o each even. Consrain (12) which inegraes he decision variables allows he occurrence of only one even and creaes a suiable siuaion for decomposiion. In his siuaion a difficul problem can be decomposed o some simpler sub-problems and increases he possibiliy of solving real size problems. On he oher hand, as anoher benefi of he model, planning of he saion capaciy is simulaneously possible in our imeabling model. Consrains (14) and (16) make i possible o consider he capaciy of saions and sidings. Mos of models in he lieraure decompose he problem o wo separae problems. One o define and solve a general imeable and anoher o deal wih he complex saion capaciy planning problem [2, 5, 10, 12, 14, 37, 42, 43, 44]. The saion capaciy planning problem needs special aenions, so some research and algorihms in recen years focused on his field of sudy [21, 45, 46, 47, 48, 49, 50, 51]. By his poin of view, he proposed model inegraed he wo models and provides an inegraed answer. Moreover, he new definiion of decision variables classifies he soluion space and gives possibiliy o define and remove he areas of soluion space ha are no raionally as par of opimal soluion area. A comprehensive invesigaion of his opic will follow. 2-8 Complexiy Analysis Alhough he problem is generally considered as a NP-hard problem, almos every paper considers a differen version of he problem. For his reason, we give a NP-hardness proof for he specific problem we consider. Our proof will show ha our problem is a reducion of FFS rj C j. In order o show he complexiy of he problem, we used a reducion from a flexible flowshop model. Our proposed model is a complicaed version of a flexible flow-shop model wih capaciaed buffer and some limiaions for he job assignmen o machines. According o he definiion of flowshop problem by [52], here are m machines in series and each job passes each machine in a same roue. If he number of idenical machines in a leas one sage be greaer han 1, he problem is classified as a Flexible Flow Shop problem (FFs). Moreover, when machine M in a specific sage is no capable for processing all jobs, he Mj defines his machine eligibiliy consrain. Finally, if here is a limied buffer beween wo machines, in a way ha when he buffer is full he machine is no allowed o release a new job, he block consrain obligaes he compleed job o remain in he previous machine. To furher illusrae he issue, consider he rains as jobs and he saions as machines which each job (rain) mus pass hrough machines (saions); as [10, 20] have proposed. As he definiion of flowshop problem by [52] his problem can be classified as flow-shop problem wih block consrain. Moreover, he number of plaforms in each saion or siding defines he capaciy of he saion or siding ha may be greaer han one, so he problem classifies as a flexible flowshop problem or muli-sage parallel machine problem [20]. On he oher hand, some express rains are scheduled o no sop in some saions and he saions also have some specific plaforms for non-sopping rains. This siuaion defines he machine eligibiliy consrain. Furhermore, here are inrinsically some limiaions abou he sar ime of each rain. Thus, he problem can be shown as FFS MJ; rj; block γ according o he noaion of [52]. Since he problem of FFS C j is NP-hard [53], our problem is NP-hard. Therefore, i is no (18) 9

10 possible o solve large scale problems in a poly-nominal ime and some policies should be conduced o reduce complexiy of he problem. 3- Upper Bound Rules In his secion, we define some rules o efficienly limi he number of binary variables and also creae efficien upper bounds. Binary variables, which increase he complexiy of he problem, are used o idenify he sequence and overaking deails of rains. Therefore, decreasing he number of binary variables decreases he number of decisions and make he problem easier o solve. There exiss some research in order o limi he number of unnecessary over-akings. [18] inroduced some algorihms o idenify and correc some conflics such as muliple overaking and compound moves. In heir aricle hey resric rains from muliple overaking and here is no poin abou resricing general siuaions of overaking. In he following we propose some general rules o resric undesirable overaking among he rains in a nework. We use rains' parameers in order o define usefulness or undesirabiliy of overaking among every wo rains. In each railway nework, some rains have same characerisics such as ravel and dwell ime. I is clear ha he sudy of his group of rains can lead o rules ha may relax he binary decision variables among he similar rains. For example, in German railway nework here are eigh ypes of rain such as Inerciy Express (ICE), RegionalBahn (RB) and S-Bahn (S) and mos rains in each group have same parameers in he nework. Consider wo rains A and B in a group ha have exacly same speed and dwell ime parameers in all nework saions. The idea is ha overaking of each of hese wo rains from each oher is no beneficial. Consider he imeable ha rain A is scheduled o overake rain B in saion s. Because he dwell ime and ravel ime of wo rains are idenical, rain B has o delay a he saion and as a resul he objecive funcion increases. By his idea, we ry o expand he simple rule o he whole of he rains and considering he general siuaions we will obain he rules ha fix unnecessary overaking binary variables. In a word, we wan o obain he siuaions among every wo rains ha occurrence of evens (1) and (2) may resul in a delay in he imeable. We consider wo random rains and he objecive funcion of wo imeables in order o find he siuaions and relaed rules ha overaking is no beneficial. We compare he objecive funcions of he imeable ha rains go ahead wihou overaking wih he imeable ha one rain overakes anoher one. In hese invesigaions, we assume ha here is no deliberae delay. The resuls of he invesigaions can be used in reducion of binary variables and he CPU ime. Now, assume wo rains ' and, which are now in saions s and q (= s+1) consecuively as shown in Figure 15, which saion q has wo plaforms. In he Figure 15, he wo horizonal lines show he scheduling of he relaed saions. Poins k and w define he arrival and deparure ime of rain in saion q respecively. The ime inerval beween R and K represens ravel ime of rain beween wo consecuive saions s and q. Also, he ime inerval beween K and W defines he dwell ime of rain in saion q. [Figure 3 should be here] A firs, we define wo ime inervals in order o characerize differen siuaions. We define ψ,',q as he ime inerval beween he deparure ime of rain from saion q and he arrival ime of rain ' o his saion, i.e. ψ,',q = Sl'q - Clq. Also, ϑ,',s is he ime inerval beween he deparure ime of wo rains from saion s, i.e. ϑ,',s = Cl's Cls. If ψ,',q be greaer han zero, he overaking of rain ' from rain in saion q is no recommended, because overaking will resul in a delay for rain. So, he imeable wihou overaking is seleced for he saion 10

11 and rains. Moreover, wihou loss of generaliy, we can assume ha ϑ,',s is greaer han zero. The menioned condiion can be wrien as follows: If 0 < ψ,',q and 0 < ϑ,',s hen x,',s = 0. In some siuaions his rule does no give he opimal soluion, alhough i fixes los of binary variables. To invesigae furher, we analyze he siuaion ha he menioned condiion is no saisfied, i.e. wo rains simulaneously are in he saion and one rain can overake anoher one, ha means ψ,',q be lower han zero. This condiion mees when he ime inerval beween he arrival and deparure of wo rains in saion q is lower han he required saion safey ime. The firs derived condiion is: ψ q 0 S l q C lq + SF l,q C l s + T l,s,q C l s C ls T l,s,q S l q (C ls + T l.s,q T l,s,q + D l,q + D l,q ) + SF l,q + SF l,q In he equaion (20), i is obvious ha, he deparure ime of rain from saion q is equal o is deparure ime from saion s plus is ravel ime beween wo saions and is dwell ime in he saion q, i.e. C lq = C ls + T l,s,q + D l,q. Also, as anoher fac, he arrival ime of rain ' o saion q is greaer han is deparure ime from saion s and he ravel ime beween wo saions, i.e. S l q C l s + T l,s,q. On he oher hand, i is obvious ha ϑ,',s is greaer han dwell ime of rain ' in saion s plus safey ime of his saion, i.e. D l,s + SF l,s C l s C ls. Thus, we can wrie: D l,s + SF l,s C s C s T l,s,q T l,s,q + D l,q + SF l,q The derived condiion in equaion (22) describes he siuaion ha wo rains are simulaneously in he saion q so ha overaking may be beneficial and we will invesigae i laer on. On he oher hand, if he condiion in equaion (22) does no mee (like he siuaion ha is shown in Figure 15), overaking of rain ' from rain in saion q resuls a leas D l,s + SF l,s (T l,s,q T l,s,q + D l,q + SF l,q ) obligaory delay for rain. This siuaion causes he increasing of objecive funcion and a grasp vision consrain (23) can be added o he problem under he below condiions: l L ; s, q S l ; T l D l,s x,,s 0; + SF l,s > T l,s,q T l,s,q + D l,q + SF l,q and he consrain is as follows ha forbids overaking of wo rains in saion s. In a specific condiion he added consrain brings he opimal soluion as follows. Proposiion 1: Suppose ha here is a railway nework wih hree saions and no opposie direcion rains. Furhermore, here are wo consecuive rains, and ', ha a he firs saion rain precedes rain '. Train will finish is pah firs, if ϑ,',1 be greaer han ψ,',2, i.e. he ime inerval beween he deparure ime of wo rains in firs saion (ϑ,',1) be greaer han he ime inerval beween he arrival ime of rain ' o he second saion and he deparure ime of rain from second saion (ψ,',2). Proof: As described in he proposiion, here are hree saions and he only saion where overaking is possible is he second saion. Moreover, as described in he proposiion, (19) (20) (21) (22) (23) 11

12 he condiion in equaion (22) is no saisfied. If rain ' overakes rain in he second saion, as shown in equaion (22), he objecive funcion a leas increases by D l,s + SF l,s (T l,s,q T l,s,q + D l,q + SF l,q ). So he opimal roue is he curren roue ha rain ends is roue firs. On he oher hand, we consider he siuaion ha he condiion in equaion (22) mees, i.e. ψ,',q < 0 and 0 < ϑ,',s and in a word, D l,s + SF l,s T l,s,q T l,s,q + D l,q + SF l,q. In his siuaion rain ' arrives in saion q before rain depars he saion. In order o simplify he invesigaion and also make he equaions more clear, we ry o classify differen circumsances and hen check heir condiions. As he firs classificaion facor, we consider he relaion beween ψ,',q and SFl,q. In he siuaion ha he equaion (22) is valid, we consider he condiion ha ψ,',q be lower han he saion's safey ime, i.e. ψ,',q < SFl,q, so rain ' will be delayed unil he safey ime mees. Wih regard o his condiion wo differen siuaions in which he safey ime can/ canno saisfy will be discussed. In order o obain he relaion beween ψ,',q and SFl,q more clearly, we use equaion ψ,,q = C l s + T l,s,q C ls T l,s,q which describes he ime inerval beween arrival ime of wo rains o saion q and according o inequaliy D l,s + SF l,s C s C s is wrien as ψ,,q = SF l,s + D l,s + T l,s,q T l,s,q. So, he firs classificaion can be wrien as SF l,s + D l,s + T l,s,q D l,s + T l,s,q T l,s,q > SF l,q. T l,s,q SF l,q or SF l,s + Moreover, as he second classificaion facor, we consider he siuaion ha rain ' overakes rain in saion q which causes scheduled delays for rain. In anoher siuaion ha here is no any scheduled delay for rain, i.e. he ime inerval ha rain sops in he saion q is greaer han he required safey ime for rain ' o ener he saion q (SF l,q ) plus is dwell ime in ha saion (D l,q ) plus anoher required safey ime (SF l,q ) afer rain ' depars saion q. So, SF l,q + D l,q + SF l,q D l,q and SF l,q + D l,q + SF l,q < D l,q are he wo classificaions. According o hese wo classificaions, four differen condiions are obained, heir characerisics and assumpions are summarized in (24). Also, one example from each of hem is shown in Figure 16. For all l L; T l ; s, q S l he condiions are as: D l,s + SF l,s T l,s,q T l,s,q + D l,q + SF l,q { SF l,s + D l,s + T l,s,q T l,s,q SF l,s + D l,s + T l,s,q T l,s,q SF l,q { D l,q D l,q > SF l,q { D l,q D l,q + 2SF l,q D l,q (1) + 2SF l,q < D l,q (2) + 2SF l,q D l,q (3) + 2SF l,q < D l,q (4) (24) [Figure 4 should be here] Similar o proposiion 1, for each condiion we ry o find he siuaions ha overaking is no beneficial. So, we should compare he objecive funcions of he wo imeables ha in he firs, rain ' overakes rain and in he second, he siuaion ha no overaking occurs. In hese comparisons, he aim is o find he relaion of he parameers, in a way ha overaking by a grasp view resuls in increasing of he objecive funcion. Therefore, we obain he objecive funcion of he imeable ha rain arrives and depars saions firs. We will use his amoun in each of he four condiions in order o make comparisons. Then, we calculae he objecive funcion in each of he menioned classified 12

13 condiions in a way ha rain ' overakes rain in saion q. These objecive funcions will be used in he menioned comparison in order o obain some heurisic rules. In he following hree equaions, we calculae he objecive funcion of he no overaking siuaion. C lq = C ls + T l,s,q + D l,q C l q = C ls + (C l s C ls ) + T l,s,q C ls )) + D l,q C lq + C l q = 2C ls + 2T l,s,q + 2D l,q + (T l,s,q T l,s,q + SF l,q + D l,q 13 + D l,q + SF l,q (C l s Equaion (25) and (26) respecively show he deparure ime of rains and ' from saion q and equaion (27) is he summaion of he wo deparure imes. In equaion (25), C lq = C ls + T l,s,q + D l,q defines he deparure ime of rain from saion q wihou any delay. In equaion (26), he deparure ime of rain ' consiss of he deparure ime from saion s, he ravel ime beween wo saions, he amoun of ime ha rain ' mus be delayed unil rain depars saion q, i.e. T l,s,q T l,s,q + D l,q + SF l,q (C l s C ls ) and is dwell ime a saion q. The scheduled delay amoun was obained in equaion (22).As shown in equaion (27), he resul is C lq + C l q = 2C ls + 2T l,s,q + 2D l,q + SF l,q + D l,q and will be used laer for he comparisons. On he oher hand, we have o calculae he objecive funcion of he imeable ha overaking occurs for each of he menioned condiions. Regarding he firs se of assumpions in equaion (24), he deparure ime of he wo rains calculaed in he siuaion ha rain ' overakes rain in saion q. The deparure imes of rains, ' are shown in equaions (28) and (29) respecively and he objecive funcion is as equaion (30). In order o clarify he origin of he equaions, he elemens of each deparure ime is described as below: Deparure ime of rain = is arrival ime o saion q + safey ime of he saion q + dwell ime of rain ' in he saion q + safey ime of saion q Deparure ime of rain ' = is deparure ime from saion s + ravel ime among wo saion s and q + obligaory delay o insure he safey ime of saion q + dwell ime of rain ' in saion q C lq = (C ls + T l,s,q ) + SF l,q + D l,q + SF l,q C l q = (C ls + (C l s C ls )) + (T l,s,q D l,q C lq + C l q = 2C ls + 2T l,s,q + 3SF l,q + 2D l,q + SF l,q (C l s + T l,s,q C ls T l,s,q )) + By comparing equaion (27) and (30), equaion (32) shows he relaion of parameers in which overaking is no recommended and he objecive funcion of he imeable wih overaking is greaer han he one wihou overaking. Equaion (31) shows he deail of he comparison. 2C ls + 2T l,s,q 2D l,q D l,q + 2D l,q + 2SF l,q + SF l,q + D l,q 2C ls + 2T l,s,q + 3SF l,q + 2D l,q Thus, in he circumsance ha he se of assumpions of he firs condiion is rue and he inequaliy (32) is valid, we can propose ha overaking is no a good decision and we recommend adding consrain (33) o he problem under he below condiions: (28) (29) (30) (31) (32) (25) (26) (27)

14 T l ; l L; s, q S l ; D l,s x,,s 0 ; + SF l,s T l,s,q T l,s,q + D l,q + SF l,q SF l,s + D l,s + T l,s,q T l,s,q SF l,q ; 2SF l,q + D l,q D l,q ; 2D l,q 2SF l,q + D l,q and he consrain is as follows ha forbids overaking of wo rains in saion s. By a similar approach, he derived equaion in condiion (2) is D l,q 0, in condiion (3) is 0 D l,q + SF l,q and in condiion (4) is D l,q D l,q. Relaed consrains are described in equaions (34), (35) and (36) respecively. For all l L; T l ; s, q S l and in he condiion ha he parameer of problem saisfies D l,s + SF l,s T l,s,q T l,s,q + D l,q + SF l,q, he consrains are as follow. x,,s 0; SF l,s + D l,s + T l,s,q T l,s,q SF l,q ; 2SF l,q + D j l,q < D l,q ; D l,q 0 x,,s 0; SF l,s + D l,s + T l,s,q SF l,q x,,s 0; SF l,s + D l,s + T l,s,q D l,q T l,s,q > SF l,q ; 2SF l,q + D l,q D l,q ; 0 D l,q + T l,s,q > SF l,q ; 2SF l,q + D l,q < D l,q ; D l,q As saed before, hese rules are able o fix some binary variables. Thus, by he reducion of binary variables a soluion in a shorer ime can be obained; however, here is no guaranee abou is opimaliy and i can ac as an upper-bound for he problem. In addiion, if we consider he rains in a group, ha hey have same ravel and dwell imes, he phrase T l,i,s,s+1 T l,j,s,s+1 will relax and he menioned condiions for each consrain will simplify. Furhermore, he dwell imes for mos of he saions in a nework are equal and can be relaxed from some of he consrains. Only in he saions ha here is a crossing poin of wo or more lines, dwell ime is slighly differen wih oher saions. In addiion, he safey ime, which is based on he geographical specificaions of a saion and he line, usually is equal for saions. Thus, he menioned consrains are simplified for mos rains and saions so ha hey are able o decrease he complexiy of he problem. (33) (34) (35) (36) 4- Lower Bound In his secion, we presen a Lagrangian Relaxaion (LR) lower bound algorihm o esimae a powerful lower bound of he objecive funcion. Lagrangian Relaxaion algorihm is one of he mos efficien algorihms, obaining lower bound. In his algorihm complex consrains are relaxed from he se of consrains and wih a penaly muliplier are added o he objecive funcion. Selecing he relaxed consrains wih non-zero inegraliy gap and updaing he muliplier of relaxed consrain are of high imporance [54]. Here, he overaking consrains and capaciy relaed consrains increase he complexiy of he problem. The binary decision variables are he elemens which increase he complexiy of he consrains. Consrain (12) binds all he binary decision variables and i seems ha i is he mos difficul consrain in he curren se. Thus, consrain (12), which inegraes he evens ogeher, is he bes candidae, saisfies he menioned crieria, and should be seleced so ha he problem efficienly relaxes. Considering he seleced consrain, he objecive funcion of LR model is as follows. 14

15 L T min z = l=1 =1 c lm + l L T T s S u l,,,s(x ", l ; E l l,s + x,,s + x,,s + x,,s 1 ) k which, u l,,,s are he Lagrangian mulipliers and he sub-gradien mehod updaes hem because of is efficacy [55, 56, 57]. Oher consrains, excep he consrain (12), are embedded in he LR model. The mulipliers iniially are inerpreed as he price (marginal cos) of he relaxed consrain in a feasible soluion. Because he consrain is in he equaliy form, he mulipliers in a feasible soluion always are zero. The mulipliers are ieraively adjused using he resul of he model in a way ha helps o improve he amoun of he lower bound. Therefore, in each sep amoun of x ",,s + x,,s + x,,s + x,,s 1 is calculaed and called as γ l,,,s, and i is used o updae he mulipliers. Equaions (38), (39) and (40) demonsrae his process as sub-gradien mehod. γ l,,,s = x ",,s + x,,s + x,,s + x,,s 1 Sep_Size k θ k+1 u l,,,s k = u l,,,s (UB Objecive k ) l L s S (γ 2 l,,,s ) E T l T l ; l + γ l,,,s Sep_Size k where k is he ieraion index used in he LR model, UB is he objecive funcion of a feasible soluion, "Objecive k" is he amoun of he objecive funcion of he LR model in ieraion k and θ is a parameer by an iniial value as 2. Equaion (38) defines he amoun ha relaxed consrain is no saisfied. Equaion (39) calculaes he amoun of he sep-size parameer in ieraion k and equaion (40) updaes he amoun of Lagrangian mulipliers for he nex ieraion. Using hese parameers, he LR ieraively updaes he parameers o reach he opimal soluion. The relaed procedure is shown in Figure 17. [Figure 5 should be here] k (37) (38) (39) (40) A he firs sep, in order o define UB, which is used in equaion (39), we find a feasible soluion in which rains go hrough he saions wih a lexicographical order and wihou any overaking. The objecive funcion of his soluion ha muliplies o 1.05 defines UB. Moreover, we ge he objecive funcion of he problem wih relaxed binary variables. The objecive funcion of his Relaxed Mixed Ineger Problem (RMIP) will be used as a benchmark for calculaing he improvemen of he LR model. Afer he firs ieraion, he parameers and mulipliers will updae and he nex ieraion will sar unil one of hese condiions mee: max l,,, s (u k+1 k l,,,s u l,,,s) < Number of ieraions exceeds 100. Moreover, in each sep if he objecive funcion of LR model does no increase, he amoun of θ will be halved, which helps i o increase he lower bound. 5- Experimenal Resuls In order o demonsrae he compuaional efficiency of he mahemaical programming model and he proposed upper and lower bound rules, a series of numerical experimens are illusraed. The following case sudy is based on he real parameers of Tehran Mero line 5 beween Tehran and Golshahr wih 12 saions and 40 kilomeers lengh, which is general view is magnified in Figure 18. The acual Tehran Golshahr line is a double-racked line, in which 15

16 every 11-minues rains run from Tehran o Golshahr and vice versa. There are some express rains ha sop only in hree saions and hey have no dwell ime in oher saions. The ravel ime for he express rains is abou 32 minues and i is 52 minues for oher rains. Each saion has a leas one plaform in which loading and unloading of passengers occurs. Excep for he firs and las saions, he main line of each saion has no plaform for passengers o board or aligh from rains. Trains can overake each oher in he sidings or in he saions. There are differen headways in he nework in a day. The nework headway in he peak-hours alers and he minimum amoun of he deparure ime beween wo rains reaches 480 seconds. Regarding he complexiy of imeabling in he peak-hours, we consider relaed parameers of he peak-hours o obain a imeable. The compuaional ime of rain imeabling algorihms can be affeced by number of siding's and saion's capaciies, number of rains, number of saions and sidings and he variabiliy of rains' parameers. In he following numerical experimens, we focus on he impac of he variabiliy of saion capaciy, rains and number of rains, since hese facors mainly influence he srucure of a rain imeable and he resuling number of possible soluions. However, he real-world problem used in his sudy only offers wo ypes of rains, and a fixed number of rains ha creaes a simple problem o solve. To allow a comprehensive and sysemaic assessmen, we consruc random insances o evaluae he performance of he proposed algorihm. In his way, we increased he capaciy of some middle saions. The saions were chosen randomly and he number of faciliies increased randomly up o hree. As anoher facor for increasing he complexiy of he problem, we increased he variabiliy of he ravel ime for some rains in a way ha for 50% of randomly seleced rains we increased he ravel ime up o 100% for all saions. In he following experimens, consciousness and auheniciy of he model, he effeciveness of he heurisic upper bound rules and he qualiy of he proposed Lagrangian Relaxaion mehod are invesigaed. In hese invesigaions five problems wih differen number of rains have been solved wih he proposed models. We have used GAMS worksaion and CPLEX solver o solve he problems on a PC equipped wih 3 GHz sixeen cores processor and 18 GB of RAM. [Figure 6 should be here] Figure 19 illusraes he resuling opimal imeable for nine rains for he case of Tehran- Golshahr line. As shown in he Amosfer saion, hree rains 3, 4, and 5 simulaneously are in he saion, in saion Vardavard and Garmdareh rain 3 overakes rains 5 and 4, and he oher general consrains of he problem are saisfied. 5-1 Performance of Upper-Bound Rules [Figure 7 should be here] In order o analyze he five inroduced upper-bound rules, we solved each problem by GAMS using he CPLEX solver, as an efficien and exac solver, and he opimal soluions gahered as a benchmark. The opimal soluions are shown in Table 4. The qualiy of he rules is measured by percenage gap beween he obained upper-bound and he corresponding opimal value. In addiion, improvemen in CPU ime is he second mos imporan crierion in analyzing he obained rules. 16

17 In order o analyze he effeciveness of upper-bound rules, he obained consrains (consrains (23), (33), (34), (35) and (36)) are embedded in he problem. This problem is named as Upper-Bound Rule Problem (UBRP) in he resuls and hereafer. The objecive funcion and CPU ime of UBRP is also shown in Table 4. [Table 1 should be here] As shown in Table 4, in all discussed problems when five upper-bound rules are added o he se of consrains he objecive funcion of he problem is equal o he opimal soluion, which is obained by he CPLEX solver. Thus, here is no opimaliy gap beween soluions. This shows ha no area of opimal soluion space is removed by hese rules. On he oher hand, he resuls show ha he CPU ime is considerably reduced. This reducion in CPU ime is he resul of reducion in he number of branchings and also increasing he speed a which UBRP closes he opened nodes in he Branch and Bound ree. For more explanaion on Branch and Bound see [52]. These are he benefis of he proposed rules which are described as follows. I. Reducion in he number of branchings on he nodes. II. In order o clarify, consider a node ha he CPLEX solver branches on o creae he wo new sub-problems. The branching is on he value of variable x,,s which is value is fixed in he UBRP. Therefore, in UBRP he branching on his variable does no occur. If he value of variable x,,s in CPLEX solver is ineger, he node in he UBRP is also feasible and compared o he CPLEX solver, he ree size from ha node halves. Also, he node is infeasible in UBRP if he value of x,,s is no ineger and so he ree does no expand more. Thus, by fixing binary variables he number of branching decreases and as a resul he CPU ime decreases. Reducion of ime required o fahom he opened nodes. Fixing he binary variables decreases he required ime o fahom he opened nodes. Afer a node opens, regarding he rules, he node is infeasible or some of heir binary variables are known. Thus, in UBRP he nodes fahom in a shorer ime compared o CPLEX. This process increases he speed of Branch and Bound algorihm and in less ime more nodes can be invesigaed. For more invesigaion, hree diagrams of "number of opened nodes versus dualiy gap", "CPU ime versus dualiy gap" and "CPU ime versus number of opened nodes" are shown in Figure 20, Figure 21 and Figure 22 consecuively. The dualiy gap in hese figures refers o he gap beween he upper and lower bounds, which can be obained a each sep of Branch and Bound algorihm. [Figure 8 should be here] [Figure 9 should be here] As shown in Figure 20, in all problems wih opimaliy gap equal o zero, he number of opened nodes in UBRP is lower han he CPLEX solver (as benefi I). Also, as you can see in Figure 21 he dualiy in UBRP converges o zero quicker han in CPLEX. This is because in a shorer ime more nodes opened and fahomed as benefi II which we have menioned earlier. As menioned in benefi II, fixing binary variables increases he speed a which he nodes fahom, i.e. a higher number of nodes fahom in an equal ime. For more invesigaions we proposed a measuremen crierion as ξ= CPU ime/ Opened nodes ha measures he speed a which he opened nodes close. This crierion measures he impac of he proposed rules on he node fahoming speed. The amoun of ξ for each problem is shown in Table 5. 17

18 [Table 2 should be here] As is shown in Table 5 he crierion ξ is reduced for all problems. In all problems in order o reach a given dualiy gap, UBRP opens fewer nodes in a shorer period of ime (excep he problem wih 12 rains ha is sopped in seconds and resuls of CPLEX is no known). The reason for his, is as described in benefi I and shows ha he proposed rules reduce he number of branching. In order o clarify he improvemen, consider he problem wih nine rains. In his problem in order o reach he dualiy gap abou 0.5%, CPLEX opened nodes in 206 seconds, however he UBRP opened nodes (48% of opened nodes in CPLEX) in 107 seconds (is equal 51% of CPLEX CPU ime). From anoher view poin, in he problem wih eleven rains UBRP afer abou 3775 seconds of solving process opened nodes and he gap is abou 0.99% versus opened nodes (is equal 51% of UBRP) in seconds (is equal 7.1%,i.e. 1 o 14 of CPLEX) and 0.98% gap in CPLEX. On he oher hand, in an equal period of ime UBRP opens more nodes because of improvemen in ξ. Figure 23 shows "CPU ime versus number of opened nodes". As shown, in all problems excep he problem wih eigh rains, in an equal period of ime UBRP opens more nodes and in ha period of ime creaes an smaller dualiy gap. E.g. in he problem wih en rains afer abou 1290 seconds of solving process UBRP opened nodes and he gap is abou 0.74% versus opened nodes (is equal 21% of UBRP) and 1.13% gap in CPLEX (1.52 imes greaer han UBRP). In sum, we can conclude ha UBRP opens more nodes in an equal period of ime while achieving a beer dualiy gap and also required number of he opened nodes o ge he opimal soluion is lower han CPLEX. [Figure 10 should be here] Las bu no leas, he number of discree variables in CPLEX and UBRP is shown Figure 23. [Figure 11 should be here] Figure 23 also shows he rend of discree variables in UBRP and CPLEX. I is obvious ha he wo rends go ahead while he difference beween he numbers of discree variables increases. In oher words, by increasing he number of rains he number of relaxed discree variables increases and ha engenders he reducion of CPU ime. Regarding he figure, he improvemens in CPU ime can be explained more clearly. 5-2 Lower Bound Resuls In order o invesigae he resuls of Lagrangian Relaxaion algorihm for a lowerbound, we used he same problem, which is described in las secion. The measuremen crieria are he dualiy gap beween opimal value and he lower bound and he CPU ime. In order o obain UB, a feasible soluion is obained as he menioned procedure. Our algorihm sars wih resul of he relaxed mixed ineger programing (RMIP) model as a base for LR comparison and ries o improve i. Maximum number of he ieraions in our invesigaion se o 25. In our invesigaions, here is only a lile amoun of improvemen afer ieraion 15 and he algorihm ries o prove he opimaliy of he soluion. 18

19 Resuls of LR model in Table 6 show ha i has achieved he opimal value of problem and heir value are he same as he values ha have been repored by CPLEX in Table 4. The achievemen sipulaes he idea ha he relaxed consrain has been correcly seleced. [Table 3 should be here] To invesigae furher he qualiy of he algorihm, he rend of he lower-bound in ieraions of he LR model for problems on 6 and 7 rains is shown in Figure 24. [Figure 12 should be here] Considering he fac ha he seleced relaxed consrain is in he form of equaion and bind all he defined evens ogeher, i seems o be a very powerful consrain. As Figure 24 shows, he LR algorihm ieraively improves he lower bound unil i ges he opimal soluion. In addiion, because he relaxed consrain is in an equaliy form here is no flucuaion in he sign of γ l,,,s; which, resuls in a sraighforward rend of improvemen wihou flucuaion. Noneheless, he CPU ime as anoher qualiy measuremen crierion has no improved. In order o improve he CPU ime, we defined anoher model using consrain (16), which is a special ordered se (SOS) of consrains and considers he plaforms capaciy. A same experimen analysis was performed on ha model. The resul showed a flucuaion wihou any improving rend. Also, as anoher model we relaxed consrains (2) and (3). The idea was from he problem of Seoul meropolian Railway nework ha [37] relaxed he consrain ha connecs he saions wih each oher. The resul of heir analysis was hopeful and encouraging; neverheless, he model ha he connecing consrains - consrains (2) and (3)- relaxed does no resul in an admirable soluion in our case. The resul again showed a flucuaion wihou any improved rend. Conclusion In his sudy a new MIP model for he railway scheduling problem is proposed ha uilizes evens of he railway neworks. The general specificaions and resricions of railway neworks are considered and also he saion capaciy consrains are embedded in he imeabling problem simulaneously. The model is NP-hard and is reduced o a flow-shop problem wih block, machine assignmen resricion and sar ime resricions. Therefore, solving large scale problems in an accepable amoun of ime is no possible. In order o reduce CPU ime some upper-bound rules based on analysis of model's parameers have been presened. These rules consider he relaion among parameers of he model and wih a grasp view ry o eliminae some evens ha raionally are no wise o occur. The resuls has been added as five consrains o he maser problem and experimenal resuls esify ha CPU ime reduces up o 94% and he opimal soluion also has been achieved. Furhermore, he analysis of he opened nodes shows ha he rules can fahom more nodes in a shorer ime, which describes is efficiency. These rules can be used on any oher models by mapping he relaed variables. By relaxing he consrain ha ensures only one of he evens occurs each ime, a Lagrangian Relaxaion algorihm is proposed. In order o updae he Lagrangian mulipliers, we used he sep size mehod. Numerical analysis admied accuracy and efficiency of he proposed algorihm, in a way ha in all of he samples he opimal soluion were achieve afer abou 20 ieraions. However, he solving ime of LR model is higher han he main problem, 19

20 i creaes some hopes o improve he processing ime by focusing more on he proposed Lagrangian Relaxaion algorihm. Our on-going research has focused in heurisic mehods in order o find upper bounds. Wih regard o he new definiion of decision variables, furher research can be done on implemenaion of a decomposiion mehod as Danzig-Wolfe algorihm, Benders decomposiion or Branch and Price. Also, inegraing of he proposed rule in his aricle by a pricing algorihm will be worhwhile or using hem in Bisecion algorihm. Acknowledgemens The auhors have benefied from Tehran Mero Agency's moivaion and helpful commens in he model. This aricle has benefied from he advice and encouragemen of Prof. N. Salmasi. Also, we wish o hank he auhors of all he cied papers who moivaed our work in his area. References [1]. Mu, S., & Dessouky, M. Scheduling freigh rains raveling on complex neworks, Transporaion Research Par B: Mehodological, 45(7), pp (2011). [2]. Higgins, A., Kozan, E., & Ferreira, L. Opimal Scheduling of Trains on a Single Line Track, Transporaion Research Par B: Mehodological, 30(2), pp (1996). [3]. Kroon, L. G., & Peeers, L. W. P. A Variable Trip Time Model for Cyclic Railway Timeabling, Transporaion Science 2003 INFORMS, 37(2), pp (2003). [4]. Ghoseiri, K., Szidarovszky, F., & Asgharpour, M. J. A muli-objecive rain scheduling model and soluion, Transporaion Research Par B: Mehodological, 38, pp (2004). [5]. Zhou, X., & Zhong, M. Bi-crieria rain scheduling for high-speed passenger railroad planning applicaions, European Journal of Operaional Research, 167 (3), pp (2005). [6]. Ping, R., Nan, L., Liqun, G., Zhiling, L., & Yang, L. Applicaion of Paricle Swarm Opimizaion o he Train Scheduling for High-Speed Passenger Railroad Planning, In IEEE Inernaional Symposium on Communicaions and Informaion Technology, 1, pp (2005) [7]. Vanseenwegen, P., & Oudheusden, D. V. Developing railway imeables which guaranee a beer service, European Journal of Operaional Research, 173(1), pp (2006). [8]. Vanseenwegen, P., & Van Oudheusden, D. Decreasing he passenger waiing ime for an inerciy rail nework, Transporaion Research Par B: Mehodological, 41(4), pp (2007). [9]. Li, X., Wang, D., Li, K., & Gao, Z. A green rain scheduling model and fuzzy muliobjecive opimizaion algorihm, Applied Mahemaical Modelling, 37(4), pp (2013). [10]. Zhou, X., & Zhong, M. Single-rack rain imeabling wih guaraneed opimaliy: branch-and-bound algorihms wih enhanced lower bounds, Transporaion Research Par B: Mehodological, 41(3), pp (2007). [11]. Lee, Y., & Chen, C.-Y. A heurisic for he rain pahing and imeabling problem, Transporaion Research Par B: Mehodological, 43(8 9), pp (2009). [12]. Casillo, E., Gallego, I., Ureña, J. M., & Coronado, J. M. Timeabling opimizaion of a single railway rack line wih sensiiviy analysis, TOP, 17 (2), pp (2009). 20

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22 [32]. Sun, Y., Cao, C., & Wu, C. Muli-objecive opimizaion of rain rouing problem combined wih rain scheduling on a high-speed railway nework, Transporaion Research Par C: Emerging Technologies, 44(0), pp (2014). [33]. Huang, Y., Yang, L., Tang, T., Cao, F., & Gao, Z. (2016). Saving energy and improving service qualiy: bicrieria rain scheduling in urban rail ransi sysems, IEEE Transacions on Inelligen Transporaion Sysems. [34]. Heydar, M., Peering, M. E. H., & Bergmann, D. R. Mixed ineger programming for minimizing he period of a cyclic railway imeable for a single rack wih wo rain ypes, Compuers & Indusrial Engineering, 66(1), pp (2013). [35]. Barrena, E., Canca, D., Coelho, L. C., & Lapore, G. Exac formulaions and algorihm for he rain imeabling problem wih dynamic demand, Compuers & Operaions Research, 44(0), pp (2014). [36]. Khan, M. B., & Zhou, X. Sochasic Opimizaion Model and Soluion Algorihm for Robus Double-Track Train-Timeabling Problem, IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, 11(1), pp (2010). [37]. Min, Y.-H., Park, M.-J., Hong, S.-P., & Hong, S.-H. An appraisal of a columngeneraion-based algorihm for cenralized rain-conflic resoluion on a meropolian railway nework, Transporaion Research Par B, 45, pp (2011). [38]. Li, F., Gao, Z., Li, K., & Yang, L. Efficien scheduling of railway raffic based on global informaion of rain, Transporaion Research Par B: Mehodological, 42(10), pp (2008). [39]. Dessouky, M. M., Lu, Q., Zhao, J., & Leachman, R. C. An exac soluion procedure o deermine he opimal dispaching imes for complex rail neworks, IIE Transacions, 38(2), pp (2006). [40]. Qi, J., Yang, L., Gao, Y., Li, S., & Gao, Z. Inegraed muli-rack saion layou design and rain scheduling models on railway corridors, Transporaion Research Par C: Emerging Technologies, 69, pp (2016). [41]. Yang, L., Zhou, X., & Gao, Z. Credibiliy-based rescheduling model in a double-rack railway nework: a fuzzy reliable opimizaion approach, Omega 48 (2014): [42]. Acuna-Agos, R., Michelon, P., Feille, D., & Gueye, S. A MIP-based Local Search Mehod for he Railway Rescheduling Problem. Neworks, 57(1), pp (2011). [43]. Törnquis, J. Railway raffic disurbance managemen, Transporaion Research Par A: Policy Prac, 41(3), pp (2007). [44]. Törnquis, J., & Persson, A. N-racked railway raffic re-scheduling during disurbances, Transporaion Research Par B: Mehodological, 41 (3), pp (2007). [45]. Billionne, A. Using Ineger Programming o Solve he Train-Plaforming Problem, Transporaion Science, 37(2), pp (2003). [46]. Caimi, G., Burkoler, D., & Herrmann, T. Finding Delay-Toleran Train Rouings hrough Saions, In H. Fleuren, D. den Herog & P. Kor (Eds.), Operaions Research Proceedings 2004, Vol. 2004, pp : Springer Berlin Heidelberg, (2005). [47]. Carey, M., & Crawford, I. Scheduling rains on a nework of busy complex saions, Transporaion Research Par B: Mehodological, 41 (2), (2007). [48]. Chakrobory, P., & Vikram, D. Opimum assignmen of rains o plaforms under parial schedule compliance, Transporaion Research Par B: Mehodological, 42(2), pp (2008). [49]. Kroon, L. G., Edwin Romeijn, H., & Zwaneveld, P. J. Rouing rains hrough railway saions: complexiy issues, European Journal of Operaional Research, 98(3), pp (1997). 22

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24 Lis of Figures Figure 1- Main line and Siding Figure 2- The siuaion of wo rains in each even ype, (a) refer o he even (1), (b) is he second even, (c) is aligned wih he hird even and (d) is equivalen o he even (4) Figure 3 - The siuaion of rains and ' in saion s and q Figure 4- Four condiions in obaining upper-bound, (a), (b), (c) and (d) respecively define he firs, second, hird and fourh defined condiions. Figure 5-Lagrangian Relaxaion algorihm procedure Figure 6- General view of he Tehran Mero and Line 5 Figure 7- ime-disance graph for 12 rains and six las saions Figure 8- number of opened nodes versus dualiy gap for opimal soluion and UBRP, (a) refers o he problem wih 7 rains, (b), (c), (d), (e), and (f) refer he problems wih 8, 9, 10, 11, and 12 rains respecively Figure 9-CPU ime versus dualiy gap in Opimal soluion and UBRP, (a) refers o he problem wih 7 rains, (b), (c), (d), (e), and (f) refer he problems wih 8, 9, 10, 11, and 12 rains respecively Figure 10-CPU ime versus number of opened nodes in Opimal soluion and UBRP, (a) refers o he problem wih 7 rains, (b), (c), (d), (e), and (f) refer he problems wih 8, 9, 10, 11, and 12 rains respecively. X axis represen number of node and anoher axis is CPU ime in second. Figure 11- Number of discree variables versus number of rains for CPLEX and UBRP Figure 12- Trend of he Lagrangian relaxaion lower bound- (a) for 5 rains, (b) for 6 rains Lis of Tables Table 1- The comparison resul of CPLEX and UBRP Table 2- The amoun of ξ and number of binary variables for each problem Table 3- Resuls of LR model Figure 13 24

25 ` ` ` ` (a) (b) ` ` ` ` (c) (d) Figure 14 Saion q K w ψ,',q s ϑ,',s R Figure 15 ime Saion Saion q q s ' (a) ime ' ' SF 2; D 2; D 3 ' (b) SF 2; D 2; 7 l, q l, q l, q l, q, D l q l, q s ime Saion 0 q 5 10 Saion 0 q 5 10 s ' (c) ime ' ' SF 1; D 2; D 3 (d) F 1; 2; 7 l, q l, q l, q ' S D D l, q l, q l, q s ime Figure 16 25

26 Sar Find a feasible soluion 1-Calculae he mulipliers 2-Increase he ieraion couner Solve he LR model using MIP solver Yes Solve he model using RMIP solver Halve θ No Is here any improvemen? No Saisfy he erminaion crieria Yes Sop Figure 17 Figure 18 26

27 Figure 19 Figure 20 27

28 Figure 21 28

29 Figure 22 29

30 CPLEX UBRP Figure 23 Figure 24 30

31 Table 4 Number of rains Opimal soluion Objecive (CPLEX) CPU Time Objecive UBRP CPU Time Opimaliy gap Improvemen in CPU Time 44% 61% 45% 50% 73% 94% 79% Table 5 Number of rains CPU Time (second) Number of Opened Nodes CPU Time/ Opened Nodes (ξ) Number of Discree Variables CPU Time (second) CPLEX UBRP Number of Opened Nodes CPU Time/ Opened Nodes (ξ) Number of Discree Variables Improvemen percenage in ξ 82% 67% 64% 96% 99% 88% 91% Table 6 Number of rains Bes Bound LR Model Number of Ieraion CPU Time >3.35 >4.02 >

32 Auhor s Biographical Skech Dr. Kourosh Eshghi is a professor of Indusrial Engineering Deparmen a Sharif Universiy of Technology, Tehran, Iran. He received his Ph.D. in he field of Operaions Research from Universiy of Torono in He was he Head of IE Dep. for 7 years. His field of ineress include graph heory, ineger programming and faciliy locaion. He is he auhor of 4 books and over 80 inernaional journal papers. Afshin Oroojlooy jadid: is a PhD Suden of Indusrial Engineering in he deparmen of Indusrial Engineering a Lehigh Universiy, Behlehem, PA, USA. He received his MSc from he Sharif Universiy of Technology in he Indusrial Engineering and his BSc in indusrial Engineering from Isfahan Universiy of Technology. His main areas of research ineress include Reinforcemen Learning, Supply Chain Managemen, and Mixed Ineger Programming. 32