Constraint-Based Local Search for Container Rail Scheduling

Transcription

1 Consrain-Based Local Search for Conainer Rail Scheduling By Nakorn Indra-Payoong Submied in accordance wih he requiremens for he degree of Docor of Philosophy THE UNIVERSITY OF LEEDS SCHOOL OF COMPUTING February 2005 The candidae confirms ha he work submied is his own and ha appropriae credi has been given where references have been made o he work of ohers This copy has been supplied on he undersanding ha i is copyrigh maerial and ha no quoaion from he hesis may be published wihou proper acknowledgemen

2 Acknowledgemens I owe a grea deb of graiude o my supervisors, Dr. Raymond R.S. Kwan and Dr. Les Proll for heir consan guidance and supervision as well as grealy assising in various oher ways. I would like o hank all my friends in Leeds for heir various kinds of help. Finally, I will never forge o hank my wife, Kanchana, who has given me infinie love and always cheered me up during he research, and hanks very much o all members of my family in my homeland which have ransmied heir encouragemen across coninens. i

3 Absrac A radiional conainer rail service is based on regular imeables. This causes a risk ha some cusomers may urn away if heir preferred iinerary is no aainable and he ake-up of some services in a fixed schedule may be low and herefore no profiable. To increase railway s profiabiliy and compeiiveness, a demand responsive schedule would be advanageous. The decision suppor model and algorihms for producing a schedule in advance of he weekly operaion is he main subjec of his hesis. The conainer rail scheduling problem is modelled as a consrain saisfacion problem in which he rail business crieria and operaional consrains are represened as sof and hard consrains respecively. A consrain-based local search algorihm is developed o solve problems of realisic size. The algorihm includes sraegies for acceping non-improving moves and randomised selecion of violaed consrains and variables o explore. These sraegies aim o achieve diversified exploraion of he search space. Differen measures of he consrain violaion are also used o drive he search o promising soluion regions. A predicive choice model is inroduced for search inensificaion o improve furher he qualiy of soluions for he problem. Wih sufficien rial hisory, he model will predic a good choice of value for a variable. The variable will be fixed a is prediced value for a dynamically deermined number of rials. A his poin, he propagaion of consisency beween he variables is enforced, leading o inensified exploraion of he search space. Experimenal resuls, based on real daa from he Royal Sae Railway of Thailand, have shown good compuaional performance of he approach and suggesed benefis can be achieved for boh he rail carrier and is cusomers. Finally, he proposed algorihm for rail scheduling has been adaped o solve he generalised assignmen problem, a well-known hard combinaorial opimisaion problem. The experimenal resuls have shown ha he proposed mehod can obain high qualiy soluions ha are as good as or close o he soluions obained from he exising mehods, bu wih using significanly less compuaional ime. This suggess ha generalising he mehod may be a promising approach for oher combinaorial problems in which all decision variables in he model are binary and where quick and high qualiy soluions are desirable. ii

4 Declaraions Some pars of he work presened in his hesis have been published or will appear in he following aricles: Indra-Payoong, N., Kwan, R.S.K. and Proll, L.G. Rail conainer service planning: a consrain-based approach, (o appear in) Journal of Scheduling. Indra-Payoong, N., Kwan, R.S.K. and Proll, L.G. (2004) An adapively relaxed consrain saisfacion approach for a demand responsive freigh rail imeabling problem, (presened a) The 5h Inernaional Conference on Pracice and Theory of Auomaed Timeabling, Augus 8-20, Pisburgh, USA. Indra-Payoong, N., Kwan, R.S.K. and Proll, L.G. (2004) Demand responsive scheduling of rail conainer raffic, Proceedings of he 0h World Conference on Transpor Research. Indra-Payoong, N., Kwan, R.S.K. and Proll, L.G. (2003) A randomised algorihm wih predicion of rail conainer service planning (invied alk a) Scheduling Workshop on Applicaion of Consrain Programming, 9-0 Sepember, Universiy of Huddersfield. Indra-Payoong, N., Kwan, R.S.K. and Proll, L.G. (2003) Consrain-based local search for rail conainer service planning, in: Kendall G, Burke E & Perovic S (Eds), Proceedings of he s Mulidisciplinary Inernaional Conference on Scheduling: Theory and Applicaions, vol. 2, pp Indra-Payoong, N., Kwan, R.S.K. and Proll, L.G. (2002) Sraegic scheduling of conainerised rail freigh using consrain programming, (presened a) ORS Local Search Workshop, 6-7 April, London. iii

5 Conens Acknowledgemens Absrac Declaraions Conens Lis of Tables Lis of Figures i ii iii iv viii x Inroducion. Problem background.. Overview of freigh rail planning process 2..2 Conainer rail scheduling problem 5.2 Research proposal 7.3 Thesis ouline 8 2 Lieraure review 0 2. Inroducion Frameworks for combinaorial opimisaion 2.2. Ineger programming Consrain programming Local search Soluion mehods for railway scheduling Heurisic mehods Mahemaical programming mehods Mea-heurisic mehods Simulaed annealing Geneic algorihm Tabu search 23 iv

6 2.4 Local search for consrain saisfacion problems Min-conflic heurisic GSAT WSAT Complex neighbourhoods Mea-heurisics for MAX-SAT Move sraegies for local search Diversificaion Inensificaion Move accepance crieria Adapive conrol Conclusions 37 3 Modelling he conainer rail scheduling problem Inroducion Problem descripion and assumpions Cusomer saisfacion Rail schedule facor Saisfacion Problem formulaion Ineger programming formulaion Consrain-based modelling Sof consrains Hard consrains Implied consrain Conclusions 68 4 Consrain-based local search for he conainer rail scheduling problem Inroducion A consrain-based local search algorihm The main loop Violaed consrain selecion Variable selecion Violaion sraegy 77 v

7 4.3. Hard violaion Arificial sof violaion Violaion cos N Violaion cos Q Violaion cos E Minimum rain loading Compuaional resuls Experimen I (Refined improvemen) Experimen II (Violaion penaly) Experimen III (Penaly for overcapaciy rain) Experimen IV (Violaion parameer) Experimen V (SA) Conclusions 08 5 Search inensificaion using he predicive choice model 0 5. Inroducion Moivaion for he predicive choice model 5.3 Predicive choice model Choice decision Proporional mehod Logi mehod Uiliy funcion Likelihood esimaion Aggregae predicion Simplified esimaion Relaed work CLS incorporaing he predicive choice model Timeslo enforcing Cusomer s bookings enforcing Local fix Global fix 44 vi

8 5.5 Compuaional resuls Decision parameer D Predicion error parameer E Flip rial parameer N Fixing ieraions parameer F Conclusions 54 6 Consrain-based local search for he generalised assignmen problem Inroducion Generalised assignmen problem Relaed work Problem formulaion Sof consrain Hard consrains Violaion sraegy Consrain-based local search Iniial experimen Variable selecion scheme Refined improvemen Candidae agen lis Search inensificaion echnique Violaion hisory Variable fixing Compuaional resuls Conclusions 97 7 Conclusions Summary Achievemens in his research Fuure work 202 vii

9 Bibliography 205 Appendix A 26 Appendix B 222 Lis of Tables 3. An example of poenial deparure ime range for cusomer The ouline of he survey inerview Average modal cos for each ranspor mode Cusomer saisfacion The inpu daa and he calculaion of he upper bound Ω The inpu daa and he calculaion of he upper bound ( λ + δ ) An example for he calculaion of violaion cos N An example for he calculaion of violaion cos Q An example for he calculaion of violaion cos E An example cusomer s booking daa Problem insances Problem size The schedules obained by CLS Operaing cos comparison: SRT vs CLS CLS wihou he refined improvemen procedure Differen measures of he violaion penalies 0 4. Parameer hm given by given by an amoun of overcapaciy Sensiiviy analysis of he imeslo violaion parameer Conrol parameers for SA Compuaional resuls CLS +SA Violaion hisory Value choice predicion by proporional mehod The K-S es for probabiliy disribuion 22 viii

10 5.4 The inpu daa for maximum likelihood esimaion Probabiliies of a value choice selecion in flip rials Value choice predicion by logi mehod Value choice predicion by simplified esimaion for logi mehod Resuls obained by CLS and PCM Sensiiviy analysis of he parameer D Sensiiviy analysis of he parameer E 5 5. Sensiiviy analysis of he parameer N Sensiiviy analysis of he parameer F Resuls CLS alone Resuls CLS wih wo-alernaive variable selecion scheme Resuls CLS wih he refined improvemen An example of demand marix The relaive demand index An example of profi marix The relaive profi index The demand-profi index The candidae agen lis for each job Resuls - CLS wih candidae agen lis Resuls for maximisaion problems S Average percenage deviaion from opimal soluion for S Resuls for minimisaion problems S Resuls for S - sopping crierion = Resuls for S 2 - sopping crierion parameer = A. Toal shipping cos for cargo ype I 26 A.2 Toal shipping cos for cargo ype II 28 A.3 Toal shipping cos for cargo ype III 220 A.4 Toal shipping cos for cargo ype IV 22 B. Predicion no. 222 ix

11 B.2 Predicion no B.3 Predicion no B.4 Predicion no B.5 Predicion no Lis of Figures. Freigh rail planning process 2.2 An example of pah formulaion 3.3 A ypical conainer ranspor 6 2. Pseudo code for GSAT procedure A ypical conainer erminal nework A shor-erm advance booking scheme Cusomer saisfacion funcion of cargo ype I Cusomer saisfacion funcion of cargo ype II Cusomer saisfacion funcion for cargo ype III Cusomer saisfacion funcion of cargo ype IV 5 4. The basic CLS procedure The modified CLS procedure Probabiliy densiy race Hisograms and possible probabiliy disribuions The inerchange assignmen 73 x

12 Chaper One Inroducion. Problem background In Thailand, pors are linked wih inland by sae-owned rail, which is mainly single-racked and used by boh passengers and freigh. We consider he problem of he easern-line conainer rail service in Thailand which serves conainer raffic beween Bangkok and he easern region. In 2000, i was roughly esimaed ha more han a million conainers a year were moved beween hese areas, of which 30 percen was carried by rail ranspor (Naional Economic and Social Developmen Board, 2000). In a laer year, because of raffic congesion and environmenal concerns in Bangkok, he capial ciy of he counry, he Thai governmen decided o limi he number of conainers via Bangkok por o one million conainers per year. As a resul, Laem Chabang por, locaed in he easern region, will have o serve increasing conainer flow beween ha region and Bangkok. How o mainain and increase profiabiliy is a major concern for he easern rail line in order o say compeiive wih he growing number of rucking companies as providing conainer ranspor service becomes a lucraive business. In general, he rail carrier can increase

13 profiabiliy in wo ways: ) he railway could generae he same or similar amoun of revenue wih lower operaing coss, 2) he railway may saisfy more cusomer demand in order o mainain he curren level of revenue or receive addiional profi. The quesion may hen be how o run he rail business as effecively as possible... Overview of freigh rail planning process The ransporaion of rail freigh is a complex domain, wih several processes and levels of decision, where invesmens are capial-inensive and usually require long-erm sraegic plans. In addiion, he ransporaion of rail freigh has o adap o rapidly changing poliical, social, and economic environmens. In general, freigh rail planning involves five main processes: pah formulaion, flee assignmen, schedule producion, crew scheduling and flee reposiioning. Figure. presens he freigh rail planning process. Pah formulaion Flee assignmen Schedule producion Crew scheduling Flee reposiioning Figure.: Freigh rail planning process 2

14 Pah formulaion. The firs sep in he planning process is pah formulaion, as shown in Figure.2. The formulaion process compues conainer flows in he nework using he shores pah or minimum generalised cos, based on he hisorical demand daa. The pah formulaion is generally no changed frequenly, and his sep hardly occurs in pracice. Noe ha in case he railway is privaised, roues are usually fixed by conrac and herefore his sep is performed only a he sraegic level and no a he operaional level. T T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 0 Pah formulaed T T 2 Passing poins Terminals (T) Figure.2: An example of pah formulaion Flee assignmen. The second sep in he planning process is flee assignmen. The goal of his sep is o allocae he available flee (locomoives and wagons) o service pairs so ha he capaciy maches he average ranspor demand a boh ends. For insance, a a erminal, he flee cycle sars when cusomers reques services and hen compaible locomoive and wagons are grouped and moved o a loading poin. Once he conainers have been loaded, he rain formaion will hen be adjused o he requiremens a he desinaion. A his poin, he flee is available wih more or less he same capaciy for a new shipmen in he reverse 3

15 direcion and he cycle may repea. Flee assignmen involves huge capial invesmen, which is done infrequenly. If he rail carrier assigns a flee o a paricular origin-desinaion (OD) pair i wans o serve, he commimen is relaively long erm. Therefore, mos of he ime, he remaining seps in he planning process use a fixed given flee. Noe ha in conras o road and air ransporaion, a rail carrier uses fixed racks. Changes o he service nework canno be done easily because hey require huge capial invesmen and involve a number of operaional consrains, such as rack availabiliy, handling equipmen, cusoms procedures and so forh. Schedule producion. Schedule producion ypically sars several monhs before a schedule goes ino operaion. In his process, significan amouns of ime and resources are used o produce a profiable schedule. In general, he scheduling process begins wih an exising schedule, which will hen be developed or changed, based on hisorical ranspor demand daa. The aim of his sep is o deermine, under operaional consrains, he appropriae service frequency hroughou he monh or several monhs. This sep is he mos difficul. If he services are oo frequen, he rail carrier bears high operaional coss. If he services are oo infrequen, some poenial cusomers may urn away, resuling in los revenue. Crew scheduling. Crew scheduling is concerned wih he developmen of duy schedules for crews, in order o cover a given rain schedule. This sep involves he shor-erm (ypically one day or one week) acical scheduling of crew, wih he aim of developing a se of duies ha will be performed by each crew o cover adequaely he rain schedule. The objecive is o find he minimum cos assignmen of crews and aendans o service lines, subjec o some resricions. For insance, rain drivers are qualified for cerain locomoive classes and service lines; he schedules mus saisfy resricions on maximum working hours and so on. 4

16 Flee reposiioning. The las sep of rail planning process is flee reposiioning. The imbalance beween demand and supply is refleced in he fac ha a any poin in ime here are erminals wih a surplus of locomoives and wagons of a cerain ype, whils some oher erminals show a shorage. Moving empy locomoives and wagons does no direcly conribue o he profi of rail carrier bu i is essenial o is coninuing operaions. Nowadays, he rail indusry has a pooling agreemen o consolidae is resources. Under his agreemen, he rail carriers agree o pool he locomoives and wagons of each ype, so ha he empy locomoives and wagons would be redireced o oher desinaions, raher han being sen back o he poin of origin. The operaion faced by each rail carrier in he pool is o deermine he flee size i needs o acquire for any given period, and o deermine he apporionmen of responsibiliy for he capial invesmen amongs he paricipaing rail carriers. In his research we address an issue in schedule producion, ha of consrucing profiable schedules for he rail service...2 Conainer rail scheduling problem Typically he ransporaion of rail conainers involves an inernaional conex. Freigh forwarders or mulimodal ranspor operaors, who ac on behalf of heir cusomers, regularly book hose services in advance o make sure ha he shipmen is delivered o heir cusomers wihin expeced ime. In general, shipping lines canno provide frequen sailing services. They ofen provide a weekly service or more. This is because conainer ships are big (greaer han 8,000 weny- 5

17 equivalen uni (TEU) conainer) in order o achieve economy of scale; hereby significan cos reducion can be obained. Once conainers arrive a he seapor, here is a need o move hem o heir final cusomers, which can basically be done eiher by rail, via inermediae erminals, or by ruck direc o he final desinaions (see Figure.3). Shippers would like o minimise he lead-ime of he ranspor chain. Figure.3: A ypical conainer ranspor A presen, he easern line conainer rail carrier provides a weekly fixed schedule, in which a cerain number of rain services are provided in fixed deparure imeslos. This creaes a risk ha ake-up of some services in a fixed schedule may be low and no profiable, and some cusomers may urn away if heir requiremens are no saisfied. In order o creae a profiable schedule, a conainer rail carrier needs o engage in a decision-making process wih muliple business crieria and numerous consrains, which is challenging. Furher descripions of he problem are given in Chaper 3. 6

18 .2 Research proposal Generaing a good schedule is of umos imporance o a conainer rail business because rail s profiabiliy is heavily influenced by is service offerings. The conainer rail scheduling problem has araced much research ineres over several decades; from he view poins of boh problem modelling and soluion echniques. Alhough a grea deal of effor has been made in his area, here are sill many aspecs ha need o be furher invesigaed and improved. A model ha incorporaes challenging pracical siuaions and advanced soluion echniques is significan. The research proposal may be divided ino wo principal areas: Applicaion domain. A rail carrier s profiabiliy is influenced by he railway s abiliy o consruc schedules for which supply maches cusomer demands. The need for responsive flexible schedules is obvious no only because here is a risk ha some poenial cusomers may urn away if a desirable schedule is no available, bu also because he ake-up of some services in a fixed schedule may be low and no profiable. We propose an opimisaion model ha quanifies cusomer saisfacion, which is maximised as one of he rail business crieria. This framework is a necessary ool for supporing decision-makers, hrough which a rail carrier can measure how well heir cusomers are saisfied and he implicaions of saisfying hese cusomers in erms of cos. Soluion approach. The conainer rail scheduling problem is complex and large and we need a mehod ha can solve he problem effecively. Two goals in designing a soluion mehod are: 7

19 . I provides good qualiy of soluion wihin reasonable ime. 2. I is simple o implemen and convenien o use. We aemp o achieve hese goals by invesigaing and exending exising mehods and developing several new echniques o improve heir performance. We propose a consrainbased local search algorihm incorporaing a predicive choice model for solving he conainer rail scheduling problem. The soluion algorihm is simple and convenien o use, whils providing good qualiy of soluion..3 Thesis ouline Following his inroducory chaper, Chaper 2 oulines frameworks for combinaorial opimisaion. Chaper 3 describes he conainer rail scheduling model. Chaper 4 and 5 presen he research on he invesigaion and consrucion of an efficien soluion mehod for he conainer rail scheduling problem. Chaper 6 adaps his mehod for anoher combinaorial opimisaion problem, he generalised assignmen problem. Conclusions drawn from he research are given in Chaper 7. In Chaper 3, he conainer rail scheduling problem is modelled as a consrain saisfacion problem. The rail business crieria and operaional requiremens are considered as sof and hard consrains respecively. A demand responsive scheduling model is proposed in which service supply maches or responds o cusomer demands and opimises on booking preference whils saisfying railway operaions. Chaper 4 describes he consrain-based local search algorihm for solving he conainer rail scheduling problem. The consrain-based local search sars wih random iniial assignmens 8

20 and uses a simple variable flip as a srucure of local move. When all variables in he model are assigned a value, he oal hard violaion is calculaed; a quanified measure of he violaion is used o evaluae local moves. Differen measures of he violaion are also used in order o drive he search o he promising regions of he search space. Chaper 5 develops a novel predicive choice model o improve he soluion obained by consrain-based local search alone. The predicive choice model is based on discree choice heory and he random uiliy concep. Learning from search hisory, he model will predic a good choice of value for a variable. The variable will be fixed a is preferred value for a number of ieraions deermined by he magniude of he preference measure. A his poin, he propagaion of consisency beween variables is enforced, leading o inensified exploraion of he search space. Chaper 6 demonsraes he applicaion of he proposed algorihm o he generalised assignmen problem (GAP). A se of diversified feasible soluions o GAP is obained by he consrain-based local search. The predicive choice model learns from he search hisory and predics good assignmens of jobs o agens. The search focuses more inensively on regions which promise o find beer soluions. The performance of he algorihm is evaluaed wih wo differen benchmark problem ses. Finally, Chaper 7 gives he conclusions, discusses he achievemen of his research and suggess some fuure work. 9

21 Chaper Two Lieraure review 2. Inroducion This chaper oulines he basic principles of he opimisaion frameworks which could be applied o our conainer rail scheduling problem. Furher discussion on relaed work will also be given in laer chapers as appropriae. The opimisaion frameworks discussed in his chaper are: ineger programming, finie domain consrain programming, and local search. Ineger and consrain programming can be considered as general-purpose opimisaion mehods, whereas local search may be viewed as an approach ha can be ailored o many differen combinaorial opimisaion problems by adaping is simple concepual componens o he respecive problem conex. The local search mehod is aracive, and ofen used, when proven opimal soluions may ake oo long o find. The effecive performance of local search mainly depends on wo sraegies: diversificaion and inensificaion. Diversificaion drives he search o explore new regions so ha he search space is fully covered, inensificaion focuses he search more 0

22 inensively on regions previously found o be good or promising o find an opimal soluion. Good inerplay beween he diversificaion and inensificaion sraegies is he criical issue in he design of local search mehods. This chaper is organised as follows: Secion 2.2 oulines he frameworks for combinaorial opimisaion. Secion 2.3 reviews soluion mehods for rail scheduling. Secion 2.4 reviews local search mehods for consrain saisfacion problems, especially for hose ha lead o he developmen of our proposed mehod, which will be used in laer chapers. Secion 2.5 describes move sraegies for local search and inroduces conceps for he adapive conrol used in local search. Finally conclusions are given in Secion Frameworks for combinaorial opimisaion Many problems arising from diverse areas can be considered as combinaorial opimisaion problems. Combinaorial opimisaion problems are concerned wih he efficien use or allocaion of limied resources o mee desired crieria. In his secion, we ouline he frameworks for combinaorial opimisaion problems which are relaed and applied o our rail scheduling problem (Chaper 3) as well as he generalised assignmen problem which will be described in Chaper 6. Three opimisaion frameworks will be discussed: ineger programming (IP), consrain programming (CP), and local search. These frameworks are well esablished, comprising a variey of echniques, and many successful applicaions have been repored.

23 2.2. Ineger programming Combinaorial opimisaion problems are considered as ineger programming problems when he decision variables in he model are required o be inegers. IP is used in pracice for solving many indusrial problems, for example in ransporaion and manufacuring: airline crew scheduling, vehicle rouing, producion planning, ec (Nemhauser and Wolsey, 988). IP branch-and-bound is concerned wih finding opimal soluions o he IP problem. A general ineger linear programming formulaion is defined as: z IP { cx x S} = min : (2.) S, and x Ineger where: = { Ax b : x 0} In branch-and-bound mehod, he original problem is divided ino sub-problems, and subproblems are creaed by resricing he range of he ineger variables. For binary variables, here are only wo possible resricions, i.e. seing he variable o 0 or. Lower bounds are provided by he linear-programming relaxaion o he problem, i.e. keep he objecive funcion and all consrains, bu relax he inegraliy resricions o derive a linear programme. If he opimal soluion o a relaxed sub-problem is inegral, i is a feasible soluion o he problem; and he opimal value can be used o erminae searches of subproblems whose lower bound is higher. Many issues need o be considered o develop efficien branch-and-bound mehods, such as he selecion of branching variables and he node o develop nex. In oher words, sraegies o explore he search ree need o be defined. Efficien branch-and-bound implemenaions may furher add valid inequaliies (cus), which 2

24 are inferred from specific classes of consrains implicily presen in he original consrains (Michell, 2000). Using hese componens, in addiion o efficien algorihms for solving and re-solving LP relaxaions, IP branch-and-bound provides a general and efficien echnique for many combinaorial opimisaion problems. A variey of efficien commercial branchand-bound solvers are available on he marke, e.g. CPLEX, LINDO, XPRESSMP, MINTO (Nemhauser and Wolsey, 988) Consrain programming Consrain programming or finie domain consrain programming (CP) has araced much aenion amongs researchers from many areas because of is poenial for solving hard combinaorial opimisaion problems. Real-life problems end o have a large number of consrains, which may be hard or sof. Hard consrains require ha any soluion will never violae he consrains, whereas sof consrains are more flexible, consrain violaion is oleraed bu aracs a penaly. Naurally, combinaorial opimisaion problems can be hough of as consrain saisfacion problems (CSP). A CSP is ypically defined in erms of () a se of variables, each ranging over a finie discree domain of values, (2) a se of consrains, which are relaions over subses of he variable domains. The problem is o assign values o all variables from heir domains, subjec o he consrains (Tsang, 993). When combinaorial problems are solved by CP, he consrain sore sores informaion on he consrained variables in he form of he se of possible values ha a variable can ake. This se is called he curren domain of he variable. Compuaion sars wih an iniial domain for each variable as given in he CSP-model. Some consrains can be direcly enered in he consrain sore by srenghening he consrain on a variable, e.g. he 3

25 consrain x y can be expressed in he consrain sore by removing he curren value of y from he domain of x. Anoher componen in CP is called propagaors. Each propagaor observes he variables given by he corresponding consrain in he problem. Whenever possible, i srenghens he consrain sore wih respec o he variables by excluding values from heir domains according o he corresponding consrain, e.g. a propagaor of consrain x y observes he upper bound and lower bounds of he domains of x and y. A possible srenghening consiss of removing all values from he domain of x ha are greaer han he upper bound of he domain of y. The process of propagaion coninues unil no propagaor can furher srenghen he consrain sore, i.e. he consrain sore is said o be sable. However, variables in many CSP problems ypically canno be reduced o a singleon domain. Therefore, he consrain sore does no represen a soluion and search becomes necessary. Search for CSP soluions is implemened by choice poins. A choice poin generaes a branching consrain c. From he curren sable consrain sore cs, wo new consrain sores are creaed by adding c and sores are no sable, and c and c o cs respecively. Typically, he new consrain c rigger some propagaors in heir respecive new sores. Afer sabiliy is reached again, he branching process is coninued recursively on boh sides unil he resuling sore is eiher consisen or represens a soluion o he problem. CP is bes considered as a sofware framework for combining sofware componens o achieve problem-specific ree search solvers. These componens can be organised ino hree pars (Marrio and Suckey, 998): 4

26 . Propagaion: implemens individual consrains by describing how he consrains can be employed o srenghen he consrain sore. 2. Branching: selecs branching consrains a each node of he search ree afer all propagaion has been done. Branching sraegies define he size and shape of he search ree. 3. Exploraion: describes which par of a given search ree is explored and in which order. CP has seen much success in a variey of applicaion domains, e.g. planning and scheduling. Various echniques have been inegraed ino consrain programming, propagaors and branching sraegies o make he solving algorihm powerful (Prosser, 993; Jussien and Lhomme, 2002). Example of general CP solvers are Oz, CHIP, ECLiPSe, and ILOG (Marrio and Suckey, 998) Local search Many combinaorial opimisaion problems are NP -hard, i.e. may no be solved wihin polynomial compuaion ime (Nemhauser and Wolsey, 988). This implies ha proven opimal soluions may ake oo long o find, a leas for large insances. However, subopimal soluions are someimes easy o find. Therefore, here is much ineres in local search ha can find good soluions wih reasonable imes. Local search mehods have successfully been applied o many combinaorial opimisaion problems. Local search can be described in erms of several basic componens: a cos funcion of a soluion o he problem, a neighbourhood funcion ha defines he possible moves in he search space, and a conrol sraegy according o which he moves are performed. 5

27 - Cos funcion: a combinaorial problem is defined by he se of feasible soluions and a cos (finess) funcion ha maps each soluion o a quanified cos. The search algorihm is o find an opimal feasible soluion, i.e. a feasible soluion ha opimises he cos funcion. - Neighbourhood funcion: local search proceeds by making moves from one soluion poin o anoher. The se of poins includes feasible soluions, bu may also include infeasible soluions. Given a combinaorial problem, he neighbourhood funcion is defined by mapping from he se of poins o is neighbours, i.e. he subses of he se of poins. A soluion is locally opimal wih respec o a neighbourhood funcion if is cos is no worse han he cos of each of is neighbours. - Conrol sraegy: defines how he search space is explored. For insance, a basic conrol sraegy of local search is ieraive improvemen, i.e. one sars wih an iniial soluion and searches is neighbourhood for a soluion of lower cos. If such a soluion is found, he curren soluion is replaced and he search coninues. Oherwise, he algorihm reurns he curren soluion, which is locally opimal. A main problem of local search is local opima, i.e. poins in he search space where no neighbour improves over he curren poin, bu which may be far from he global opima. Many sraegies have been proposed o overcome his problem. In many cases, nonimproving local moves are acceped based on a probabilisic rule or based on he hisory of he search (Aars e al, 997). 2.3 Soluion mehods for railway scheduling The soluion mehods for railway scheduling may be classified ino hree groups: heurisic 6

28 mehods, mahemaical programming mehods, and mea-heurisic mehods Heurisic mehods The early age of solving railway scheduling problems only relied on heurisic mehods. Mos heurisics a ha ime were similar o he mehods used by manual schedulers. The refinemen of he service plan (schedule) was made complemenarily beween a planner and a compuer. Crainic e al. (984) proposed a acical model for rail service planning. They decomposed he planning model ino rouing and scheduling models. A local search heurisic was used o solve each wo sub-models of he problem in succession and o obain a good feasible soluion offering a rough framework for producing a rail schedule. Haghani (989) used a heurisic decomposiion echnique for railway scheduling. The heurisic based on a special srucure was used o solve he problem wihin a small nework. However, his approach failed o solve larger problem insances. Gualda and Murgel (2000) considered he rain formaion problem. The objecives are o maximise revenue from he ranspor of cargoes, o safeguard he relaive prioriies of cargoes, and supply he services wih he minimum oal operaing cos under operaional consrains. The heurisic begins wih he formulaion of direc rains ha ravel loaded from origin o desinaion and come back empy. This soluion is hen submied o a refinemen procedure o combine rains and minimise he movemen of empy wagons, and he algorihm seeks a beer use of he rolling sock. The heurisic incorporaes a shores pah algorihm and a sraegy based on he knapsack problem. 7

29 The heurisics used for railway scheduling were heavily problem-specific. A heurisic which works for one problem canno be used o solve a differen one, or canno easily be adaped o new problem condiions. Purely heurisic mehods for railway scheduling rarely appear nowadays. Ofen hey are now used o gear up mahemaical programming mehods ino a more flexible and general problem solvers Mahemaical programming mehods Mahemaical programming (MP) has been well known and developed in he operaions research sociey for several decades. Mos railway scheduling problems have been modelled based on mahemaical formulaions. Keaon (989) and Keaon (992) used he Lagrangian relaxaion mehod o simplify he rail rouing and scheduling problem. He incorporaed he rain capaciy, ravel ime and demand flow consrains ino he objecive funcion wih Lagrangian mulipliers. Relaxing he consrains allows him o decompose he problem ino separable rain demand flow problems. By relaxing he rain capaciy consrain, he demand flow problem can be viewed as a collecion of shores-pah problems, one for each origin - desinaion pair. Using a dual adjusmen approach, he arrived a an infeasible lower bound o he problems. Aferwards, he used a simple heurisic o obain a feasible soluion. Schrijver (993) considered he problem of minimising he number of rain unis of differen ypes for an hourly rain line in he Neherlands, given ha he passenger s sea demand and rain capaciy consrain mus be saisfied. The resricion on he ransiion beween wo composiions on wo consecuive rips is ha he required rain unis mus be available a he 8

30 righ ime and he righ saion. Coupling and uncoupling resricions relaed o he feasibiliy of shuning movemens are ignored. He proposed an algorihm based on graph heory and ineger programming. The algorihm concerns he circulaion of differen ype of rain unis, which can be linked more ogeher. I can be described as a mulicommodiy flow problem, and is solved using ideas from polyhedral combinaorics. Newon e al (998) considered he freigh rail blocking plan problem. The objecive is o choose he blocks o be buil a each cargo yard and o assign sequences of blocks o deliver each shipmen o minimise oal mileage, handling cos, and delay coss. They developed a column generaion approach in which aracive pahs for each shipmen are generaed by solving a shores pah problem. They also disaggregaed some of he consrains in he model o provide a igher lower bound. Newman and Yano (2000) considered he rains and conainers scheduling problem. The objecive is o minimise oal operaing coss, whils meeing on-ime delivery requiremens. They formulaed he problem as an ineger programme. A decomposiion procedure o find near-opimal soluions and a mehod o provide relaively igh bounds on he objecive funcion values were proposed. Yano and Newman (200) considered he conainer rail scheduling problem wih due daes and dynamic arrivals. The objecive is o minimise he sum of ransporaion and holding coss. They inroduced a definiion of a regeneraion sae, which derived from a srong characerisaion of he shipmen schedule wihin he regeneraion inerval properies of an opimal soluion. The opimal assignmen of cusomer orders o rains can hen be found by solving a linear programme. 9

31 Kraf (2002) considered he shipmen rouing problem. He formulaed he problem as a muli-commodiy nework flow problem, where each shipmen is reaed as a separae commodiy. A Lagrangian heurisic was used o obain a primal feasible soluion by ranking all flows based on prioriy. Then he algorihm sequenially assigns flows on a shores pah based on adjused link coss. A primal feasible soluion was used o validae he qualiy of he dual prices by esablishing heir prices leading o a igh upper bound on he objecive funcion. MP incorporaing heurisics is ofen used for many pracical railway scheduling. Heurisics are used o enhance MP o obain he opimal soluion or near-opimal soluion in a viable ime. Mos of MP relies on bound sraegies, e.g. linear relaxaion, linear dualiy, and Lagrangian relaxaion. A good bound helps limi he size of he search. However, he heurisics and bound sraegies depend on he presence of special srucures in he model; he adapaion of which for new pracical aspecs migh be non-rivial Mea-heurisic mehods Mea-heurisics are widely used o solve imporan pracical combinaorial opimisaion problems. Basically, a mea-heurisic is a op-level sraegy ha guides an underlying heurisic solving a given problem. Tha is, a mea-heurisic is an ieraive maser process ha guides and modifies he operaions of subordinae heurisics o efficienly produce highqualiy soluions. I may manipulae ieraively a complee (or incomplee) single soluion or a collecion of soluions. The subordinae heurisics are e.g. high- (or low-) level procedures, simple local search, or jus a consrucion mehod. Mea-heurisics may use learning 20

32 sraegies o srucure informaion in order o find opimal or near-opimal soluion effecively (Osman and Kelly, 996; Glover and Laguna, 997) Simulaed annealing Simulaed annealing is a mea-heurisic echnique for combinaorial opimisaion problems which is designed as a simple and robus algorihm (Kirkparick, 984). The erm simulaed annealing derives from he physical process of heaing and cooling a subsance o obain a srong crysalline srucure. A simulaed annealing algorihm repeas an ieraive procedure ha looks for beer soluions, whils offering he possibiliy of acceping, in a conrolled manner, worse soluions. This second feaure allows he algorihm o escape from he local opima. Hunley e al (995) used simulaed annealing o solve a railway scheduling problem a he CSX ransporaion company. They used a perurbaion move operaor ha insers or delees a sop from he roue and adjuss he deparure imes of he rains. The compuaional resuls showed ha he algorihm was useful for analysing a variey of scenarios, and producing rain schedules having similar properies o hose of soluions in use by he CTX company, bu wih a smaller cos. Brucker e al. (999) used simulaed annealing for freigh rail rouing. They defined neighbourhoods using he ideas from he nework simplex mehod for min-cos flow problems. Aferwards, hey proposed a wo-phase local search mehod based on simulaed annealing which execues a series of local search applicaions o single commodiy problems. In he firs phase, he algorihm ries o cover a large par of he search space and o idenify 2

33 a good soluion. In he second phase, he algorihm sars wih he bes soluion found in he firs phase and ries o improve his soluion. They applied he wo phases several imes (muliple resars) Geneic algorihm A geneic algorihm is a heurisic search algorihm premised on he evoluionary ideas of naural selecion and geneics (Holland, 975). The algorihm sars wih a se, called a populaion, of soluions (represened by chromosomes). Soluions from one generaion are aken and used o form a new populaion. Soluions which are hen seleced o form new soluions (offspring) are seleced according o heir finess; he more suiable hey are he more chances hey have o reproduce. Salim and Cai (997) used a geneic algorihm o schedule rail freigh ransporaion. The algorihm begins wih randomly generaing he iniial populaion, and hen finds he arrival ime and deparure ime of each rain a every loop by using sopping and saring marix schedules and evaluaes he cos of he populaion. Aferwards, he algorihm performs a crossover operaion on he randomly chosen individuals o yield wo new srings and replace he duplicaes in he populaion wih he newly formed individuals. The algorihm erminaes if he bes individual in he populaion has no changed for a predefined number of ieraions. Arshad e al (998) used a geneic algorihm combined wih consrain programming for conainer ranspor chain scheduling. The objecive funcion is o minimise he empy conainers beween erminals, depos, and cliens under operaional consrains. Consrain 22

34 programming was used o compue feasible soluions on a subse of search space. A geneic algorihm was used o explore he space formed by soluions provided by CP, and o perform opimisaion. The feasibiliy of he soluions was defined inrinsic o he chosen represenaion and inegraed wihin he creaion of he chromosomes in he differen seps (iniialisaion, crossover, and muaion), and wihin he finess Tabu search Tabu search is a heurisic mehod proposed by Glover (986) for solving combinaorial opimisaion problems. Tabu search allows accepance of non-improved soluions in order o avoid being rapped in local opima. To preven going back o recenly visied soluions, a memory scheme is used o record he moves made in he recen pas of he search. This recorded search hisory is usually represened by a abu lis of moves, which are forbidden for a cerain number of ieraions. Marin and Salmeron (996) used abu search o plan freigh rail services. The algorihm was based on he decomposiion of he planning model in wo problems; rouing for he freigh cars and grouping of cars in he rains. Heurisic rouing and sequenial loading algorihms were proposed. In abu search, recency-based memory wih frequency was used o preven he search going back o recenly visied soluions. They also compared abu search wih simulaed annealing and descen mehods. The comparison amongs hese mehods was made wih he help of saisical analysis. They assumed he hypohesis ha he disribuion of local minima can be represened by he Weilbull disribuion in order o obain an approach o he global minimum and a confidence inerval. The global minimum esimaion was used o compare he heurisic mehods. 23

35 A combinaion of geneic algorihm and abu search was used by Gorman (998) o solve he rail scheduling problem. In GA, he populaion was formed by all possible rain schedules. Every ime an individual schedule is generaed, is finess (oal operaing cos) is evaluaed. Muaions are obained by eiher adding or deleing a rain, or by shifing a rain o an earlier or a laer ime in he schedule. To improve he performance of he geneic algorihm, each soluion is cloned and modified wih a abu search algorihm, hus simulaing he use of knowledge based muaion operaors. However, implemening random saring soluions and simple abu moves sill suffers from misdireced search. Much aenion has been focused on mea-heurisic mehods as concepually simple, domainindependen frameworks for solving railway scheduling problems. However, classical meaheurisics, applied oally independenly of problem domain knowledge, rarely work well for real indusrial problems. Ofen mea-heurisics are enhanced by incorporaing inensive domain-knowledge, and good soluions may be obained by fasidious uning of various parameers. The mea-heurisics hen lose heir appeal as general soluion approaches and quickly become algorihms highly specialised for he given problem. Mea-heurisics may be hybridised o be more effecive. However, he resuling algorihms would be complex, and hey ofen sill have o exploi domain knowledge o be effecive. 2.4 Local search for consrain saisfacion problems The saisfiable problem in proposiional logic (SAT) is o decide wheher a given Boolean formula is saisfiable and was he firs problem proved o be NP -complee (Cook, 97). To explain he saisfiable problem, he following erms are given: 24

36 - A lieral is a proposiional variable or is negaion, e.g. x or x - A clause is a disjuncion of lierals, e.g. ( x y z) - A formula in conjuncive normal form (CNF) is a conjuncion of disjuncions, e.g. ( x y z) ( a b c)... The goal of he SAT problem is o find an assignmen of values o variables, if one exiss, where all clauses are saisfied or o prove i is unsaisfiable if no valid assignmen exiss. MAX-SAT and weighed MAX-SAT are he opimisaion varians of SAT. Given a se of clauses, MAX-SAT is he problem o find a variable assignmen ha maximises he number of saisfied clauses. In weighed MAX-SAT, a weigh is assigned o each clause and he goal is o maximise he weigh of he saisfied clauses. Alernaively he goal could be defined as o minimise he weigh of he unsaisfied clauses. In MAX-SAT and weighed MAX-SAT, all clauses need no be saisfied and i may be considered as he unsaisfiable problem. Noe also ha in case he weighs are no specified, MAX-SAT can be called unweighed MAX- SAT and herefore, when he erm MAX-SAT is used in general, i refers o he general form of he problem including clause weighs. The SAT problem can be viewed as a 0- ineger consrain problem, i.e. a Boolean clause (disjuncion of lierals) is ranslaed ino an arihmeic consrain. For insance, he clause ( y) x would be ranslaed o x + y =. For a consrain saisfacion problem (CSP), if he variable domain is Boolean and he consrains are expressed in conjuncive normal form, hen he CSP is equivalen o he SAT problem; in oher words, CSP is a generalisaion of SAT in wo aspecs which are he domains of variables and he arihmeic consrains. There are many local search mehods for solving a consrain saisfacion problem. An overview of hese mehods is given in Hoos and Suzle (2004). 25

37 Hill-climbing is he local search mehod inroduced in he pas decades for a hard combinaorial problem (Nilsson, 980). I sars from a randomly generaed assignmen of variables. A each sep, i changes he value of some variables in such a way ha he resuling assignmen saisfies more consrains. If a sric local minimum is reached hen he algorihm resars a anoher randomly generaed poin. The algorihm sops when all consrains are saisfied, or he compuaional resource is exhaused. However, he hillclimbing algorihm has o explore all neighbours of he curren sae in choosing he move. To avoid his problem, heurisics are inroduced as described nex Min-conflic heurisic The min-conflic heurisic has been inroduced as a mehod for solving consrain saisfacion problems (Minon e al, 992). This heurisic chooses randomly a variable in a violaed consrain, and hen picks a variable value which minimises he number of violaed consrains. If no such a value exiss, i picks randomly one value ha does no increase he number of violaed consrains (he curren value of he variable is picked only if all he oher values increase he number of violaed consrains). The min-conflic heurisic allows sideway moves, i.e. he curren soluion is allowed o move o anoher soluion wih he same soluion cos. This les he procedure raverse plaeaus in he soluion landscape. By doing his, he search algorihm can find is way off he plaeau and coninue he gradien descen. The min-conflic heurisic is briefly oulined as follows: Given: a se of variables, a se of consrains, and an assignmen of a value for each variable; wo variables conflic if hey boh occur in a consrain which is violaed a he curren poin. 26

38 Procedure: selec a variable ha is in conflic, and assign i a value ha minimises he number of conflics. Empirical ess obained from Minon e al (992) using he min-conflic heurisic for hill climbing showed ha he heurisic obained similar resuls o an exising neural nework mehod. The resuls also showed ha he local search min-conflic heurisic works well on some problems, e.g. he n-queens problem, graph colouring problem and he real world problem of scheduling he Hubble space elescope. However, he min-conflic heurisic can easily be rapped in local minima. Since he min-conflic heurisic alone canno overcome he problem of local minima, several echniques have been inroduced o solve his problem. These mehods ofen diversify he search and can be caegorised ino wo ypes: ) mehods ha add randomness, such as noise, using random walks (Selman and Kauz, 993), simulaed annealing and 2) mehods ha resrucure he neighbourhood, ha is he search is no allowed o move o some poins for a number of ieraions, resuling in a smaller neighbourhood size, e.g. Tabu search (Glover and Laguna, 997) GSAT Local search for he SAT problem became popular when Selman a al (992) inroduced GSAT. The procedure of classical GSAT is shown in Figure 2.. From Figure 2., GSAT searches for a saisfying variable assignmen A for a se of clauses C. Local moves are flips of variables which are chosen by selec-variable according o a 27