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

39 randomised greedy sraegy, i.e. choosing a variable ha leads o he larges increase in he oal number of saisfied clauses. The parameer Maxflips is used o deermine he frequency of resars ha helps GSAT overcome local minima. The process coninues unil he parameer Maxries is reached. Mos local search algorihms for SAT (and also MAX- SAT) follow he procedure below and hus have a simple srucure of he algorihm. proc GSAT Inpu clauses C, Maxflips, and Maxries Oupu a saisfying oal assignmen of C, if found for i := o Maxries do A := random ruh assignmen for j := o Maxflips do if A saisfies C hen reurn A P := selec-variable ( C, A) A := A wih P flipped end end reurn No saisfying assignmen found end Figure 2.: Pseudo code for GSAT procedure (Selman e al, 992) Alhough he parameer Maxflips helps GSAT overcome local minima, i does no compleely eliminae his problem; because he algorihm can sill become suck on a plaeau (a se of neighbouring saes each wih an equal number of unsaified clauses). Therefore, i is useful o employ mechanisms ha escape from local minima or plaeaus by making uphill 28

40 moves (flips ha increase he number of unsaified clauses). The mechanism is for example GSAT wih walk (Selman e al, 994). The principle of GSAT wih walk is oulined as: Given a random number r ( 0 r ) and a fixed probabiliy p. Wih probabiliy p ( r p ), he algorihm randomly picks a variable appearing in some unsaisfied clauses and flips is ruh assignmen. Wih probabiliy - p ( r > p ), he algorihm randomly picks a variable from he lis of variables ha gives he larges decrease in he oal number of unsaisfied clauses WSAT WSAT (or Walk SAT) is based on ideas firs published by Selman e al (994) and i was laer formally defined as a local search for SAT by McAlleser e al (997). WSAT makes flips by firs randomly picking a clause ha is no saisfied by he curren assignmen, and hen picking (eiher a random or according o a greedy heurisic) a variable wihin ha clause o flip. Therefore, whils GSAT wih walk can be viewed as adding walk o a greedy algorihm, WSAT can be viewed as adding greediness as a heurisic o random walk. There is a suble difference beween GSAT wih walk and WSAT in he probabiliy ha a variable is chosen o be flipped. GSAT wih walk mainains a lis (wihou duplicaes) of he variables ha appear in unsaisfied clauses, and randomly picks a variable from ha lis; hus, every variable ha appears in an unsaisfied clause is chosen wih equal probabiliy. WSAT employs he wo-sep random process described above (firs randomly picking an unsaisfied clause, and hen picking a variable) ha favours variables ha appear in many unsaisfied clauses. 29

41 The main difference beween local search algorihms for SAT and MAX-SAT is he sraegy o selec he variable o be flipped. This sraegy is a key feaure of SAT local search algorihms and is of criical imporance for heir performance Complex neighbourhoods Mos local search algorihms for SAT and MAX-SAT rely on he -flip neighbourhood. One excepion is he 2 and 3-flip neighbourhood local search algorihm for MAX-SAT proposed by Yagiura and Ibaraki (999). Since he compuaional ime o examine he effecs of 2 and 3-flips is high, hey use a special daa srucure o speed-up as much as possible he neighbourhood evaluaion. In addiion, hey propose resricions o boh neighbourhoods ha allow pruning non-improving moves from he neighbourhood. A differen exension is he muli-flip approach by Srohmaier (998) for SAT, where several independen flips, ha is, only variables are flipped ha do no occur in he same clause, are execued in parallel. The advanage of he approach is ha he effec of independen flips is he sum of he single flips. The independen se of flips is deermined by a neural nework ype archiecure. In a similar idea, Roli (200), and Roli and Blum (200) proposed o perform flips in parallel wihou aking ino accoun possible ineracions amongs he variables. They divided he variables ino k subses of equal cardinaliy and, for each of he k ses, flip he variable having he highes score in he se; he evaluaion of a variable flip is done as if all he oher variables did no change. 30

42 2.4.5 Mea-heurisics for MAX-SAT In his secion, we review he mea-heurisics ha are inended o solve he MAX-SAT problem. The following mea-heurisics use a -flip neighbourhood. Baii and Proasi (997) proposed a hisory-based heurisic (reacive search) for MAX- SAT. The main idea is o have a wo-phase approach consising of a simple GSAT and a abu search phase. The abu search phase is run for a specified number of ieraions. Afer he abu search sops, GSAT is execued unil he search is rapped in a local opimum and hen he Hamming disance o he saring poin is measured. Based on he resuling disance, he abu enure is adjused and again he abu phase is iniiaed. Resende e al (997) applied a greedy randomised adapive search procedure (GRASP) o solve MAX-SAT problems. GRASP consiss of wo phases: consrucion and local search. The consrucion phase builds good feasible soluions (a se of saisfied clauses), whose neighbourhood is invesigaed unil a local opimum is found during he local phase. The bes oal weigh of saisfied clauses is kep as he resul. Pardalos e al (996) proposed a parallel GRASP for MAX-SAT problems. Each GRASP ieraion is regarded as a search in some region of he feasible space and a number of processors perform searching in parallel. When he specified number of ieraions has been reached, each processor gives he bes soluion found. The bes soluion amongs all processors is hen idenified and used as he soluion of he problem. Mills and Tsang (2000) proposed guided local search (GLS) for solving SAT and MAX-SAT problems. GLS uses a cos funcion including a se of penaly erms o guide he local search. 3

43 Each ime local search ges rapped in a local opimum, he penalies are updaed and local search is called again o maximise he modified cos funcion. Variable neighbourhood search (VNS) was recenly applied o he MAX-SAT problem (Hansen e al, 2000). VNS combines local search wih sysemaic changes of neighbourhood in he descen and escapes from local opimum phases. The search explores increasingly far neighbourhoods of he curren soluion, and allows he exchange of he curren bes soluion for a new one if and only if a beer one has been found. Therefore, he favourable characerisics of he curren soluion are kep and used o obain a promising neighbourhood, from which a furher local search is performed. 2.5 Move sraegies for local search The fundamenal principle of local search is o exploi he inerplay beween he diversificaion and inensificaion sraegies, where diversificaion drives he search o explore new regions, and he inensificaion focuses more inensively on regions previously found o be good or promising o find an opimal soluion Diversificaion One of he main problems of all mehods based on local search approaches is ha here end o be numerous local poins in he search space, i.e. he local search algorihms end o spend mos, if no all, of heir ime in a resriced porion of he search space. The negaive resul of his fac is ha, alhough good soluions may be obained, one may fail o explore he mos 32

44 promising regions of he search space and hus end up wih soluions ha are sill prey far from opimal. Diversificaion is one of he sraegies ha ry o reduce his problem. This can be done by forcing he search ino previously unexplored areas of he search space. Search diversificaion may be based on hisory of he search, e.g. frequency- based memory in abu search (Glover and Laguna, 997) in which he algorihm records he oal number of ieraions since he beginning of he search for which various soluion componens have been presen in he curren soluion or have been involved in he chosen moves. In cases where i is possible o idenify promising regions of he search space, he search hisory can be refined o rack he number of ieraions spen in differen regions. Diversificaion echniques may be classified ino hree groups. The firs, called resar diversificaion, involves assigning all variable values or forcing a few rarely used componens in he curren soluion or he bes known soluion and resaring he search from his poin. This echnique is used, for example, in muli-resars in GRASP (Feo and Resende, 995). The second echnique inegraes he diversificaion procedure direcly ino he regular searching process. This is achieved by perurbing or biasing he evaluaion of possible moves by adding o he objecive a small erm relaed o a componen of search hisory. Examples of his echnique are long-erm memory in abu search (Glover and Laguna, 997), perurbaion and bias sampling in ieraed local search (Lourenco e al, 2002). The las diversificaion echniques are heurisic mehods ha use mulineighbourhood srucures. For a given combinaorial opimisaion problem, several neighbourhood srucures may be used o diversify he search space and enable a 33

45 convergence of he search space. A decision on which neighbourhood is o be chosen in wha sequence during he search is an imporan sraegy. The diversificaion sraegy in SAT local search is ofen achieved by some noise sraegies, e.g. he random walk (Selman and Kauz, 993; Selman e al, 994), ha randomly picks a value of some variables. Ensuring proper search diversificaion is a criical issue in he design of local search mehods. I should be addressed wih care fairly early in he design phase and revised if he resuls obained do no reach expecaions Inensificaion The purpose of he inensificaion sraegy is o focus he search on promising regions of he search space in order o make sure ha he bes soluions in hese regions are found. However, inensive invesigaion on he search space is compuaionally expensive; very ofen he local search algorihm would sop he normal searching process o perform an inensificaion phase from ime o ime. Search inensificaion is used in many local search implemenaions, bu i is no always necessary (Gendreau, 2002). This is because here are many cases where he search performed by he normal searching process is horough (good) enough. Therefore, here is no need o spend ime exploring more inensively he porions of he search space so ha he compuaional effor is spen more effecively. For example, in abu search, inensificaion can be carried ou by giving a high prioriy o he soluions which have common feaures 34

46 wih he curren soluion. This can be done wih he inroducion of an addiional erm in he objecive funcion; his erm will penalise soluions disan from he presen one. This is done during a few ieraions and afer his i may be useful o explore anoher region so ha he diversificaion sraegy will be used nex. The inensificaion sraegy in local search for CSP has rarely been addressed. The inensificaion is carried ou by consisency echniques bu mosly embedded in sysemaic ree search algorihms, e.g. backracking algorihm (Kondrak and Beek, 997). A each ree node, consisency is enforced wih respec o he curren variable assignmens. As furher assignmens are made, he problem is divided ino sub-problems since more of he original variables have fixed assignmens wihin which consisency is enforced. Backracking is called when any variable domain becomes empy as a resul of consisency enforcing based on he curren assignmens, i.e. he sub-problem is inconsisen given hese assignmens. As consisency enforcing is also compuaionally cosly, here is always a debae on he radeoff beween how much consisency is mainained during he search and he qualiy of he soluion Move accepance crieria In many combinaorial opimisaion problems, consrains ofen resric he searching process oo much and can lead o low qualiy of soluions. This occurs, for example, in our rail scheduling problem and many oher problems where he resource capaciy is oo igh o allow assigning demands (resource consumed) effecively beween resources. In such cases, allowing non-improving moves (or relaxing consrains) is an aracive sraegy. This is 35

47 because i creaes a larger search space ha can be explored wih simple srucures of local move. Move accepance can be carried ou by several sraegies. The simples sraegy is o always allow improving and non-improving moves. The second simple one can be implemened by dropping seleced consrains from he search space definiion and adding o he objecive weighed penalies for consrain violaions. However, his raises he issue of finding correc weighs. An ineresing way of ackling his problem may use self-adjusing penalies (Frank, 997), i.e. penaly weighs are adjused dynamically on he basis of he recen hisory of he search: weighs are increased if only infeasible soluions were encounered in he las few ieraions, and decreased if all recen soluions were feasible. Penaly weighs can also be modified sysemaically o drive he search o cross he feasibiliy boundary of he search space and hus induce diversificaion. Anoher sraegy is probabilisic move accepance as used, for example, in simulaed annealing. A he beginning of he search, a high probabiliy of acceping any local moves is used wheher he algorihm improves he soluion or no. A some laer ieraions, he process is done wih respec o a probabilisic accepance funcion based on parameer called a emperaure. The emperaure is decremened unil i is small and herefore only few nonimproving moves are acceped. The way emperaure is conrolled is referred o as he cooling schedule Adapive conrol An adapive mechanism is always an aracive feaure for a beer conrol of a local search algorihm. For example algorihms using adapive mechanisms are: adapive abu search 36

48 (Glover and Laguna, 997), adapive simulaed annealing (Leser, 996), variable neighbourhood search (Hansen and Mladenovi, 200), adapive noise for SAT local search (Hoos, 2002), ec. Adapive echniques may range from a simple auomaed uning parameer o a complex learning mechanism. The formulaion and applicaion of he adapive echniques are also very differen, depending on where he echniques are used and he sae of he search. These adapive algorihms do no depend on specific designs of heurisics and do no need many uning parameers. Insead hey learn from he hisory of he search in order o conrol he search adapively. For example, Horviz e al (2002) proposed a Bayesian learning echnique for solving hard CSP and SAT problems. The algorihm explicily learns from he search hisory and predics he runime for resaring policies in randomised search. In our research a predicive choice learning model is proposed in order o inform he algorihm when he search space needs o be explored inensively and in which regions. Furher discussion on his echnique is given in Secion of Chaper Conclusions This chaper oulines some frameworks for combinaorial opimisaion, which can be mainly caegorised ino hree groups: 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.for large-scale combinaorial 37

49 opimisaion problems, local search may be used o find good soluions in reasonable ime. Local search can be described in erms of basic componens: a cos funcion of a soluion o he problem, a neighbourhood funcion ha defines he possible moves in he search space, and conrol sraegy according o which he local moves are performed. The fundamenal principle of local search is o exploi he inerplay beween diversificaion and inensificaion. Move accepance crieria are also of imporance for he performance of local search, in paricular when simple srucures of local move are used. The research presened in his hesis proposes an effecive solving algorihm for he conainer rail scheduling problem based on ideas discussed in his chaper, primarily hose of local search. The conainer rail scheduling problem is formulaed in Chaper 3 as a CSP. The soluion mehod and is exension will be discussed in Chaper 4 and Chaper 5. 38

50 Chaper Three Modelling he conainer rail scheduling problem 3. Inroducion There are many frameworks o represen and describe a conainer rail scheduling problem (Crainic and Lapore, 997; Cordeau e al, 998). However, wha we are ineresed in is no only he framework for he represenaion of he problem bu also an effecive way o solve he problem. As he conainer rail scheduling problem is ypically complex and large, a poenial compuaional difficuly arises in solving such a problem. Therefore, research in his area needs o consider boh how o model and o solve he problem. Many real life problems could naurally be represened by consrains and he saisfacion of hese consrains provides a soluion for he presened problem. In our conainer rail scheduling problem, rain capaciy, service resricions, and some cusomer requiremens are modelled by hard consrains, whils he objecives: minimum number of rains, maximum cusomer saisfacion, and minimum imeslo operaing coss are modelled by sof 39

51 consrains. We can herefore formulae he conainer rail scheduling problem as a consrain saisfacion problem. Then, we presen a consrain-based local search algorihm o solve his class of consrain saisfacion problem (described in Chaper 4). This Chaper is organised as follows: Secion 3.2 describes he problem s characerisics and assumpions. Secion 3.3 describes he echniques o quanify he cusomer saisfacion on a rail schedule. Secion 3.4 presens a formulaion of he conainer rail scheduling problem. A generalised cos funcion is presened in Secion 3.5, and finally conclusions are given in Secion Problem descripion and assumpions In he pas, he ransporaion of rail freigh was considered no o be an efficien mode of ranspor, paricularly in erms of physical accessibiliy and cargo handling. Since he adven of conainerisaion in he mid 940s, rail carriers have gained higher profiabiliy by ailoring conainerised freigh and have become more compeiive wih oher inland ranspor providers. Conainer rail service differs from convenional freigh rail in several imporan aspecs. Because of he high coss of conainer handling equipmen, conainer rail neworks have relaively few and widely spaced erminals. Neworks wih around en erminals are common and he nework flows are relaively simple, as illusraed in Figure 3.. A ypical conainer makes few or no sops and may be ransferred beween rains only up o a few imes on is journey. In addiion, small lo sizes of shipmen, frequen shipmen, and demand for flexible service are imporan characerisics in he ransporaion of rail conainers. 40

52 Depo Main erminal Main erminal Seapor Depo Terminal Main seapor Seapor Trucking services Rail links Ferry links Figure 3.: A ypical conainer erminal nework I is also noed ha conainer rail services are independen of one anoher in he sense ha demands for a conainer movemen in a specific roue do no inerac wih he demands in any oher roues (services). In addiion, complex neworks are no pracical for cusoms procedures as conainerised cargoes, in general, are moved wihin an inernaional conex. Even hough conainer raffic has increased, he increase in marke share of rail ranspor, paricularly in shor-haul and medium-haul, has no been successfully achieved. Therefore, here have been effors o invesigae he facors influencing modal choice. The resuls have shown ha he frequency and reliabiliy of service are he main facors influencing shippers decisions on he choice of ranspor mode (Indra-Payoong e al, 998). A rail carrier s profiabiliy is heavily influenced by he railway s abiliy o consruc schedules for which supply maches cusomer demands. For he ransporaion of conainerised freigh, shippers (cusomers) can ofen choose beween rail and ruck. A need 4

53 for responsive flexible schedules may become obvious no only because here is a risk ha some poenial cusomers may urn away if he cusomer s preferred iinerary is no aainable, bu also because he ake-up of some services in a fixed schedule may be low and herefore no profiable. In order o consruc a demand responsive schedule, a rail carrier needs o engage in a decision-making process wih muliple business crieria and a number of operaional consrains, which is very challenging. There is a large body of lieraure on freigh rail scheduling, using diverse modelling srucures. A recen survey by Cordeau e al (998) suggess mos of hem caer for fixed schedules. However, our proposed model incorporaes challenging pracical siuaions which involve:. Non-uniform arrivals wih disinc arge imes, i.e. no all conainers are available a he beginning of he planning ime horizon and mus be reaed as disinc cusomer bookings. 2. A demand responsive service providing he flexible schedules 3. A probabilisic decrease in cusomer saisfacion wih deviaion from arge ime A few aemps have been made o generae flexible rain schedules, which may be caegorised ino wo ypes according o how he overall demand is me. Hunley e al. (995), Gorman (998), and Arshad e al. (2000) aggregae cusomer demands wih minimum operaing coss hrough flexible scheduling. They do no propose o mee individual demands. Newman and Yano (2000), Yano and Newman (200), and Kraf (2002) share he same spiri of our sudy by being responsive o individual demand. Their models saisfy he operaional consrains fully for each cusomer. In conras, our framework models cusomer saisfacion, compued from preferred and alernaive booking ime ranges, which is considered as one of he rail business crieria. Therefore, some cusomers migh no be given 42

54 heir mos preferred booking ime range. This framework is a naural one for supporing decisions as a rail carrier can measure how well heir cusomers are saisfied and he implicaions of saisfying hese cusomers in erms of cos. We consider he conainer rail service from a conainer seapor o an inland conainer depo (ICD) in which he weekly schedule is provided and revised every week. Once conainers arrive a he seapor, hey can be ranspored o heir final desinaions by rail or ruck via an inland conainer depo, or direcly by ruck. This sudy assumes an advance booking scheme as illusraed in Figure 3.2. I also assumes ha all conainers are homogeneous in erms of heir physical dimensions, and hey will be loaded on rains ready for any scheduled deparure imes. Noe ha we consider a sandard conainer, which is measured in Tweny- Equivalen Uni (TEU). Tenaive schedule Slack ime Advance booking Confirm booking Week of operaion Time horizon Fixed schedule Figure 3.2: A shor-erm advance booking scheme The day is divided ino hourly slos for booking and scheduling. Cusomers are requesed o sae a preferred booking ime range (or an earlies booking ime range) in advance. A number of alernaive booking ime ranges for each shipmen may be specified, which migh be judged from experience or esimaed by he cusomer s delay ime funcions. These alernaives no only help a rail carrier consolidae cusomer demands o a paricular rain 43

55 service wih minimum oal coss, bu also provide flexible deparure imes for he cusomer s ranspor planning sraegy. A preferred deparure ime range and each alernaive booking ime range may cover a few hours, which is illusraed in Table 3.. This happens in pracice because he service ime needed o move conainers from he loading poin of a conainership o he rain conainer plaform may vary. In addiion, cusomers may have o allow more ime for unexpeced delays. Deparure Shipping companies (cusomers) Timeslo Evergreen Mearsk P&O Nedlloyd Misui OSK. P Sa: 0900 P A Sa: 000 P P A Sun: 500 Sun: 600 A A 2 A A A P P Wed: 00 Wed: 200 Wed: 300 A 3 A A 2 A 3 A A 2 A Table 3.: An example of poenial deparure ime ranges for cusomer In Table 3., P is a preferred deparure ime range, A is he firs alernaive deparure ime range, A 2 and A 3 are he second and he hird alernaive deparure ime range respecively. 44

56 Amongs hese alernaive deparure ime ranges, a cusomer has a more saisfacion wih a deparure ime in A han in A 2 and wih a deparure in A 2 han in A 3. Tha is, if one cusomer s alernaive deparure ime range is no possible, he nex alernaive deparure ime range is acceped bu wih less saisfacion. Blanks denoe infeasible deparure imes for he cusomer as conainer handling services may no be available eiher a he erminal of deparure or desinaion erminal, or a boh ends. I is noed ha here may be some cusomers ha book he conainer rail service close o he end of a week; herefore heir alernaive booking ime range may fall ino he following week. The proposed model only akes he booking ime ranges for hose cusomers which fall wihin he schedulable week and he oher alernaives are no considered direcly. 3.3 Cusomer saisfacion In a highly compeiive marke, assessing cusomer saisfacion wih he ranspor service is of grea imporance o a conainer rail carrier. A rail carrier could ake advanage of a knowledge of cusomer saisfacion o improve is service and o srenghen is compeiive posiion wih respec o he oher ranspor services. A rail carrier could increase he qualiy of service and marke share by ailoring a service ha saisfies individual cusomers. The rail schedule may be jus one of he mode-choice decision facors including cos, ravel ime, reliabiliy, safey, and so forh. As cusomers have differen demands, i is hard o find a single definiion of wha a good qualiy of service is. For example, some cusomers may be willing o olerae a delayed service in reurn for sufficienly low oal shipping coss. 45

57 3.3. Rail schedule facor We invesigae cusomer saisfacion wih respec o he rail schedule facor. To acquire he cusomer saisfacion daa, face-o-face inerviews were carried ou by he auhor. The ouline of he inerview is abulaed in Table 3.2. This survey includes 84 cusomers currenly using boh rail and rucking services or using only rail bu wih he poenial o swich heir shipmen o ruck in he fuure. The conainerised cargo is classified ino four caegories as follows:. Cargo ype I (perishable consumer goods): food and beverages, dairy producs, fruis and vegeables (24 cusomers) 2. Cargo ype II (durable consumer goods): household producs, and furniure (52 cusomers) 3. Cargo ype III (inermediae producs and raw maerials): exile fibres, obacco leaves, paper and paperboard, chemicals (67 cusomers) 4. Cargo ype IV (capial goods and ohers): iron and seel, meal manufacure, non-elecrical machinery and pars, consrucion maerials (4 cusomers) 46

58 Shipping informaion Type of company (shipping line, freigh forwarder, MTO,.ec) Type of conainer and commodiy value per on Conainer densiy measured Shelf life of he commodiy in days Annual conainer volume shipped Period of advance booking regularly used. Modal characerisics * (shipping ime) Arrival ime a conainer por Discharging ime a conainer por Waiing ime a he discharging poin Haulage ime from he discharging poin o he main erminal Waiing ime a por erminal Loading ime a rain/ruck erminal Travel ime Modal characerisics * (shipping cos) Freigh rae (TEU-on-km) and commodiy rae facor Terminal sorage cos a por erminal/shipside (TEU-on/day) (day = a consecuive 24-hour period) Free ime sorage period a por erminal (days) Reducion rae if conainers are moved from erminal/shipside wihin (daypercen) Terminal handling charge per TEU-on Overhead cos for waiing ime a por erminal/shipside (TEU-on/day) Bookings Preferred rain deparure ime range Alernaive deparure ime range I Alernaive deparure ime range II Ohers * Rail and ruck are he wo modes surveyed Table 3.2: The ouline of he survey inerview 47

59 3.3.2 Saisfacion Undersanding and quanifying cusomer saisfacion benefis a rail carrier. The cusomer saisfacion is linked o he probabiliy ha a cusomer will selec rail or ruck for a mode of ranspor. If cusomer saisfacion wih he rail service decreases, he probabiliy of cusomer choosing rail will also decrease and his will reduce he demand for he rail ranspor in he fuure. To quanify cusomer saisfacion, cusomer saisfacion funcions are developed. These use cusomer characerisics, shipping informaion and modal characerisics as primary inpu daa. Toal shipping coss associaed wih movemen by modes are expressed as a percenage of commodiy marke price or value of conainerised cargo, expressed in price per on. Average relaive shipping coss of he conainerised cargo from survey daa (see Appendix A) and he marke price are summarised in Table 3.3. The marke price for each cargo ype is esimaed by he Deparmen of Business Economics, Minisry of Commerce, Thailand (Minisry of Commerce, 2002). We assume ha all cusomers know a full se of shipping coss and can jusify he modal preferences on a basis of accuraely measured and undersood coss. The freigh rae may be adjused by he relaive coss ha a cusomer may be willing o pay o receive superior service. For example, some cusomers may have higher saisfacion using a rucking service even if he explici freigh rae is higher; speed and reliabiliy of he service may be paricularly imporan if he conainerised cargo has a shor shelf life. 48

60 Cargo ypes/cos Cos /uni price Marke Modal cos (%) ( 0 3 Bah /on) Truck Rail price Truck Rail C Freigh rae C T C R C -C T R Type I Type II Type III Type IV Terminal handling charge Type I Type II Type III Type IV Terminal sorage charges (Wihin free ime sorage) Overhead cos (Wihin free ime sorage) Toal shipping coss Type I Type II Type III Type IV Table 3.3: Average modal cos for each ranspor mode 49

61 To deermine cusomer saisfacion beween modes, we assume ha he difference beween modal cos percenages, i.e. C = CT -C R, follows a normal disribuion. The cusomer saisfacion is hen derived from cumulaive probabiliy densiy funcions (Appendix A). The cusomer saisfacion funcions for he conainerised cargoes are shown in Figure Saisfacion on rail Modal cos percenage C C Figure 3.3: Cusomer saisfacion funcion of cargo ype I Saisfacion on rail Modal cos percenage C Figure 3.4: Cusomer saisfacion funcion of cargo ype II 50

62 Saisfacion on rail Modal cos percenage C Figure 3.5: Cusomer saisfacion funcion for cargo ype III Saisfacion on rail Modal cos percenage C Figure 3.6: Cusomer saisfacion funcion of cargo ype IV From Figure , if here is no difference beween he modal cos percenages, i.e. = 0, cusomers end o sae heir saisfacion on he service beween rail and ruck equally. A C cargo ha has a low value of C has a high sensiiviy in he oal shipping coss. For insance, an arihmeic mean of C =.72 for cusomers shipping cargo I (see Appendix A); 5

63 when he ranspor of conainers are delayed by rail, i will resul in an increase in he oal shipping coss. For cusomers shipping his ype of cargo even a small cos increase can lower heir saisfacion using he conainer rail service quie subsanially. This is due o a high sensiiviy in he oal shipping coss. A probabiliy of 0.5 in he cusomer saisfacion funcion indicaes he lowes saisfacion level for he conainer rail service. If he saisfacion is below his level, cusomers may urn o use a rucking service insead; oherwise, hey would olerae he rail service. Nowadays, a rail carrier would ry o keep he cusomer saisfacion above his level. Once he saisfacion funcion has been developed, a cusomer s saisfacion measure can be obained from he modal saisfacion probabiliy. This probabiliy could also be used o predic he marke share beween ranspor modes and o es he modal sensiiviy when he rail schedule is changed. The cusomer saisfacion measure is a probabiliy of choosing he rail service and all cusomers have a saisfacion ranging from 0 o. Noe ha all cusomers currenly using conainer rail service may already hold a cerain level of saisfacion regardless of aking he qualiy of he rail schedule ino accoun. Once he rail carrier has been chosen as a choice of ranspor mode and laer he schedule is delayed, cusomers incur addiional oal shipping coss, i.e. erminal sorage and overhead coss involved a he seapor. This would resul in a decrease in cusomer saisfacion. An example of he calculaion of cusomer saisfacion is shown in Table

64 Shipping daa Uni Value Cargo ype Type IV Ship arrival ime Day: Time Mon: 0900 Discharging ime Hour 4 Free ime sorage allowance Day 3 Reducion rae on erminal handling charges - Scheme I Day - % - 25% - Scheme II Day - % 2-20% - Scheme III Day - % 3-0% Bookings Day: Time - Preferred ime range ( P ) Mon: Alernaive I ( A ) Tue: Alernaive II ( A 2 ) Wed: Alernaive III ( A 3 ) Thu: Modal cos (%) P A A 2 A 3 - Truck ( C T ) Rail ( C R ) C Saisfacion ( w ) Table 3.4: Cusomer saisfacion From Table 3.4, he cusomer has he modal cos percenages for ruck C and railc a he T R preferred booking ime range equal o 9.5 and 3.78 (he firs reducion rae on erminal handling charges is applied). Since he rucking service is always available, we use he same C T for A, A 2, and A 3 for a comparison wih C R. C R for A and A 2 akes he reducion 53

65 rae 20% and 0% respecively. For A 3 no reducion is applied and he erminal sorage charge is imposed. Afer geing he difference beween modal cos percenages C R, we can obain he cusomer saisfacion by applying he value of C = CT - C ino he cusomer saisfacion funcion (Figure 3.6), e.g. C = 5.73, we ge he saisfacion w = Alernaively, he saisfacion w can be obained by he following funcion: ( x) x µ 2 σ 2 f = e (3.) σ 2π C ( x) w = f dx (3.2) where: f ( x) is he normal probabiliy densiy funcion ha C akes he value x, µ and σ are he mean and sandard deviaion of he oal C (Appendix A). 3.4 Problem formulaion There are ofen differen ways of represening he same combinaorial opimisaion problem and his should provide some advanages in developing soluion procedures for such a problem. Since obaining an opimal soluion o a large-scale combinaorial opimisaion problem in a reasonable amoun of compuer ime may well depend on he way he problem is modelled, we need o consider boh how o model and o solve he problem. 54

66 3.4. Ineger programming formulaion Many combinaorial opimisaion problems can be formulaed as he problems in ineger programming in which all decision variables are required o ake inegral values. We firs model he demand-responsive conainer rail scheduling problem as an ineger programme. We consider he day divided ino hourly slos for weekly booking and scheduling. The following noaion will be used: Ses: T : se of schedulable imeslos M : se of cusomers j S j : se of poenial booking imeslos for cusomer j C : se of poenial cusomers for deparure imeslo R : se of service resricions for deparure imeslos Decision variables: x :, if a rain depars in imeslo, 0 oherwise y j :, if cusomer j is served by he rain deparing in imeslo, 0 oherwise Parameers: FC is a fixed cos of running a rain w j : cusomer j saisfacion in deparure imeslo N j : demand of cusomer j (number of conainers) 55

67 r : rain congesion cos in deparure imeslo g : saff cos in deparure imeslo P : minimum rain loading (number of conainers) P 2 : capaciy of a rain (number of conainers) The IP formulaion of he conainer rail scheduling problem is: Minimise FC x + w N y + ( r + g ) T T j C j j j T x (3.3) Subjec o S j y = ; j M (3.4) j j M j M N N j j y y j j P ; T (3.5) P2 ; T (3.6) x x y ; j C, T (3.7) j = 0 ; R (3.8) { 0, }; T j M x, y =, (3.9) j The objecive is o minimise he generalised cos represening he operaing coss and he virual loss of fuure revenue. The firs erm in he objecive funcion aims o minimise he number of rains on a weekly basis. The fewer rains, he greaer reducion on operaing coss a rail carrier can expec. The second erm is o maximise he oal cusomer saisfacion using values from a cusomer saisfacion funcions (Figure ). Each cusomer holds he 56

68 highes saisfacion a a preferred booking ime range, he saisfacion hen decreases probabilisically o he lowes saisfacion a he las alernaive booking ime range, i.e. deparure laer han he preferred booking ime range would cause a decrease in he fuure demand, and he rail carrier is expeced o ake a loss in fuure revenue. For he evaluaion of a schedule, he probabiliy of cusomer saisfacion is hen muliplied by demand N j. The las erm in he objecive funcion aims o minimise he imeslo-operaing cos. A rail carrier is likely o incur addiional coss in operaing a demand responsive schedule, in which deparure imes may vary from week o week. This may include rain congesion cos and saff cos. The rain congesion cos reflecs an incremenal delay resuling from inerference beween rains in a raffic sream. The rail carrier calculaes he marginal delay caused by an addiional rain enering a paricular se of deparure imeslos, aking ino accoun he speedflow relaionship of each rack segmen. The over-ime cos for crew and ground saff would also be paid when evening and nigh rains are requesed. Consrains (3.4) ensure ha no cusomer will be lef uncovered and each cusomer can only be served by one rain. These hard consrains are crucial for a rail carrier. This is because i does no make good business sense in a compeiive marke o offer a demand responsive service, requiring cusomers o sae boh heir preferred, and a se of alernaive, ime ranges regarding heir pracical conainer operaions (eiher a he erminal of deparure or desinaion erminal, or a boh ends) and business sraegies and hen o decline heir business. In addiion, cusomers shipmen canno be spli in muliple rains. This is because he desinaion erminal (inland conainer depo: ICD) locaed in he hear of he capial ciy is relaively small and all arrival conainers mus be sacked in he designaed area. Expensive equipmen is required o move he conainers from he rain o he ICD and laer ono ranspor o he final desinaion. Muliple shipmens are likely o resul in conainer 57

69 sacks having o be shuffled o assemble a paricular cusomer s load; hereby imposing subsanial conainer handling cos. Consrains (3.5) ensure ha he demand assigned o a deparure ime slo mus no be less han he minimum rain loading P. Seing a minimum rain loading ensures saisfacory revenue for a rail carrier and spreads ou he capaciy uilisaion on rain services. The carrier may wan o se he minimum rain loading as high as possible, ideally equal o he capaciy of a rain. Secion 4.4 explains how sensible values for P can be deermined. Consrains (3.6) ensure ha he demand assigned o a deparure ime slo mus no exceed he capaciy of a rain. These are hard operaional consrains; if load exceeds he capaciy, running he rain can damage he locomoive engine and railway infrasrucure, e.g. racks. Noe ha in case of a single cusomer s demand being more han he capaciy of a rain, we allow spliing his demand over muliple rains and rea he demand as differen subcusomers; however, his paricular case rarely occurs in pracice. Consrains (3.7) ensure ha if imeslo is seleced for cusomer j, a rain does depar a ha imeslo. On he oher hand, if deparure imeslo is no seleced for cusomer j, a rain may or may no run a ha ime. Consrains (3.8) are a se of banned deparure imes. The resricions may be pre-specified so ha a railway planner schedules rains o achieve a desirable headway or o avoid congesion a he conainer erminal. Consrains (3.9) require ha all decision variables in he model are binary. 58

70 The conainer rail scheduling problem was iniially solved by an ineger programming branch and bound mehod (CPLEX Solver, ILOG 2002). We failed o find an opimal soluion wihin he ime limi 2 hours, paricularly when he minimum rain loading P is close o he capaciy of a rain. Since all decision variables in our conainer rail scheduling model are binary (consrains 3.9) and fracional soluions o linear relaxaions are meaningless, igh bounds on he objecive funcion value canno be obained and used o reduce he size of he search space. Noe ha he marix of coefficiens in (3.4) consiss of only zeroes and ones and ha he consrains are equaions. Thus here is a relaxaion of he conainer rail scheduling problem o a se pariioning problem (SPP). Formally, he SPP is he problem of pariioning he rows i ( i =,..., M ) of a zero-one marix ( a ij ) by disinc subses of he columns j ( j =, K, N ) a minimal cos. Defining x j = if column j (wih cos c j > 0 ) is in he soluion and x j = 0 oherwise. The SPP is Minimise N j= c j x j (3.0) Subjec o N j= aij x j = ; i =, 2,3,..., M (3.) { 0,}; j =, 2,3, N x j..., (3.2) Consrains (3.) ensure ha each row is covered (served) by exacly one column and (3.2) are he inegraliy consrains. The SPP is known o be NP-hard (Balas and Padberg, 976); hence he conainer rail scheduling problem is NP-hard. In addiion, he conainer rail scheduling problem incorporaes oher consrains ha require he consisency beween 59

71 differen ses of decision variables which will make i more complex han SPP. Many curren approaches for NP-hard problems focus on finding good soluions wihin a reasonable ime using various local search heurisic mehods Consrain-based modelling Since he conainer rail scheduling problem is NP-hard, his implies ha proven opimal soluions may ake oo long o find, a leas for large insances. This leads us o he design of a local search ha can find good soluions wihin reasonable run-ime on a sandard personal compuer. As discussed earlier, wha we are ineresed in is no only he framework for he represenaion of he problem bu also an effecive way o solve he problem. Therefore, we need o consider boh how o model and o solve he problem. We model he conainer rail scheduling problem as a consrain saisfacion problem (CSP) and hen inroduce a consrain-based local search mehod for solving i (described in Chaper 4). The principal difference beween branch and bound based IP and CSP approaches o solving combinaorial opimisaion problems is ha: in IP, he inegraliy resricions on he variables are relaxed, he search space is he coninuous feasible region and he search is guided primarily by he objecive funcion; in CSP, some or all of he consrains are relaxed bu he inegraliy resricions are enforced, he search space is simply defined by he domains of he variables and he search is guided by he need o gain feasibiliy. 60

72 Boh approaches have me wih success in a variey of applicaions, neiher of he approaches being able o claim general superioriy over he oher. I is known ha branch and bound searches can be very lenghy if ineger feasibiliy is hard o achieve or if he opimal soluion o he coninuous relaxaion is no a good guide o good qualiy soluions of he IP (Darby-Dowman and Lile, 998; Brailsford e al,999). As noed in Secion 3.4., his appears o be he case for he conainer rail scheduling problem. A furher imporan difference is ha in IP, all consrains are global consrains, i.e. are always enforced, whereas consrain-based approaches can handle consrains locally, i.e. can decide which consrains o impose a various sages of he search. This allows differen consrains o be given differen prioriies, which allows problem-specific knowledge o be exploied o guide he search. Consrains can also be represened more compacly in consrain-based approaches as here is no need o make hem linear. Regarding he conainer rail scheduling problem, a consrain-based local mehod allows he search o move beween feasible and infeasible regions of he search space in a simple and flexible way. In addiion, each erm in he objecive funcion can be reaed separaely. This reduces he complexiy in he design and evaluaion of local moves. Any local search mehod can be applied o CSP and is hen someimes called a consrainbased local search mehod. In Chaper 2, we reviewed some local search mehods (heurisic and mea-heurisic mehods) for he railway scheduling problem and CSP. Many local search mehods are concepually simple, domain-independen frameworks; however hese mehods, applied oally independenly of problem specific knowledge, rarely work well for real indusrial problems. They are ofen enhanced by incorporaing inensive domain-knowledge 6

73 and use complex local moves. Thus, hey lose heir appeal as simple and general soluion mehods and quickly become algorihms highly specialised for he given problem. Our consrain-based local search (CLS) for he conainer rail scheduling problem is inspired by local search for he saisfiabiliy (SAT) problem. An aracive framework of SAT local search is ha he srucure of he local move is simple and his may be appropriae for our problem. The developmen of CLS is described in deail hrough Chapers 4 and 5. In a CSP, operaional requiremens are represened as hard consrains whils opimisaion crieria are handled by ransforming hem ino sof consrains. This is achieved by expressing each crierion as an inequaliy agains a bound on is ideal opimal value. As a resul, such sof consrains are rarely saisfied. A feasible soluion for a CSP represenaion of he problem is an assignmen o all decision variables in he model ha saisfies all hard consrains, whereas an opimal soluion is a feasible soluion wih he minimum oal sof consrain violaion (Winson, 994; Walser 999; Henz e al, 2000; Lau e al, 200). For simpliciy, we assume ha he violaion v i of consrain i is linear and is defined as follows: j N a ij x j bi ν i = max 0, aij x j bi (3.0) j N where aij are coefficiens, b i is a bound, x j are consrained variables. Noe ha violaions for oher ypes of linear and non-linear consrains can be defined in an analogous way. Furher, oher ways of defining v i are possible. 62

74 In his sudy, (3.0) is only used for sof consrains, i is laer modified for hard consrains as described in Secion Sof consrains In he conainer rail scheduling model, he sof consrains are minimum number of rains, maximum cusomer saisfacion, minimum imeslo-operaing cos. Number of rains. This consrain aims o minimise he number of rains. The number of rains consrain is defined as: T x θ (3.0) where: θ is a lower bound on he number of rains, i.e. ( N j ) / P2. j M The violaion of he number of rains consrain s is s = max 0, x θ (3.) T Cusomer saisfacion. This consrain aims o maximise he oal cusomer saisfacion. The cusomer saisfacion consrain is defined as: T j C w jn j yj Ω (3.2) 63

75 where: Ω is an upper bound on cusomer saisfacion, i.e. W N ; j j W j is he maximum saisfacion on a preferred booking ime range for cusomer j. j C The calculaion of upper bound on cusomer saisfacion Ω is illusraed in Table 3.5. Timeslo Cusomer saisfacion Sum ( ) C C 2 C 3 C 4 C 5 C 6 C ( Ω ) N j W j W j N j Table 3.5: The inpu daa and he calculaion of he upper bound Ω The violaion of he cusomer saisfacion consrain s 2 is s 2 = max 0, Ω wj N j yj (3.3) T j C 64

76 Timeslo-operaing cos. This consrain aims o minimise he imeslo-operaing cos. The imeslo-operaing cos consrain is defined as: T ( + g ) x ( λ + δ ) r (3.4) where: ( λ + δ ) is a lower bound on he imeslo-operaing cos, λ = T a r ; T a is he se of θ leas rain congesion coss, δ = T b g ; T b is he se of θ leas saff coss, θ is a lower bound on he number of rains. The calculaion of he upper bound on he imeslo-operaing cos ( λ + δ ) is illusraed in Table 3.6. In his example, he lower bound on he number of rains θ = 3, he uni cos 3 0 Bah. 65

77 Timeslo Congesion cos Saff cos Timeslo cos Sum ( ) ( r ) ( g ) ( r + g ) ( λ + δ ) Σ θ leas cos Table 3.6: The inpu daa and he calculaion of he upper bound ( λ + δ ) The violaion of he imeslo-operaing cos consrain s 3 is s 3 = max 0, ( r + g ) x ( λ + δ ) (3.5) T Given sof consrain violaions for he number of rains, cusomer saisfacion and imeslooperaing cos, he generalised cos for a rail carrier GC can be obained as: GC = FC θ + s ) + FRs + ( λ + δ + ) (3.6) ( 2 s3 66

78 where: FC is a fixed cos of running a rain, FR is a freigh rae per demand uni (onconainer), λ, δ are defined in Secion ; s, s 2, and s 3 are sof consrain violaions for (3.), (3.3), and (3.5) respecively Hard consrains In he conainer rail scheduling model, hard consrains are coverage consrain, rain capaciy consrain, consisency consrain, covering consrain, and service resricion consrain. These consrains are someimes called operaional or required consrains and mus be saisfied o ensure safey of service and pracical operaions. These consrains have been discussed in Secion 3.4.; however, some of hese will be modified for a more efficien solving algorihm (described in Secion 4.3.) Implied consrains The sof and hard consrains compleely reflec he requisie relaionships beween all he variables in he model, i.e. he operaional requiremens and business crieria. Implied consrains, derivable from he above consrains, may be added o he model. Whils implied consrains do no affec he se of feasible soluions o he model, hey may have compuaional advanage in he soluion algorihm as hey reduce he size of he search space (Proll and Smih, 998; Smih e al, 2000). The covering consrain is one such consrain, being implied by (3.4). Covering consrain. A covering consrain can be hough of as a se covering problem in which he consrain is saisfied if here is a leas one deparure imeslo x serving cusomer 67

79 j. This consrain favours a combinaion of seleced deparure imeslos ha covers all cusomers. The covering consrain is defined as: S j x ; j (3.4) 3.5 Conclusions In his Chaper, he conainer rail scheduling problem is modelled as a consrain saisfacion problem. We have presened a demand responsive scheduling model, in which service supply maches or responds o cusomer demands and opimises on booking preference whils saisfying hard consrains. The advance booking scheme is assumed in order o help a rail carrier consolidae he demands wih minimum oal coss and provide flexible bookings for he cusomer s ranspor planning sraegy. A consrain-based modelling framework is used in which rail business crieria and operaional requiremens are formulaed as sof and hard consrains respecively. The crieria are handled by ransforming hem ino sof consrains, which is achieved by expressing each crierion as an inequaliy agains a bound on is ideal opimal value. The cusomer saisfacion wih a rail schedule is quanified. I is compued from preferred and alernaive deparure ime ranges in which he saisfacion decreases wih deviaion from arge ime in a probabilisic scale. The cusomer saisfacion is hen maximised as one of he rail business crieria. Hence some cusomers migh no be given heir mos preferred ime ranges bookings. This framework is o suppor decision-makers in which a rail carrier can 68

80 measure how well heir cusomers are saisfied and he implicaions of saisfying hese cusomers in erms of cos. In Chaper 4, a consrain-based local search mehod is presened o solve he conainer rail scheduling problem. 69

81 Chaper Four Consrain-based local search for he conainer rail scheduling problem 4. Inroducion As discussed in Chaper 3, he conainer rail scheduling problem has been modelled as a consrain saisfacion problem. A feasible soluion for a CSP is an assignmen o all consrained variables in he model ha saisfies all hard consrains, and an opimal soluion is a feasible soluion wih he minimum oal sof consrain violaion. As he conainer rail scheduling problem is NP-hard, a poenial compuaional difficuly arises in solving such a problem. In his chaper, we presen a consrain-based local search algorihm o find a good soluion wihin reasonable compuaional ime. The algorihm 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 70

82 is used o evaluae local moves. A measure of consrain violaion is used o drive he search o he promising regions of he search space. This chaper is organised as follows. Secion 4.2 describes procedure of he consrain-based local search algorihm. Secion 4.3 describes how a consrain violaion scheme is used o improve he qualiy of he conainer rail schedule. Secion 4.4 describes he minimum rain loading as an adapive lower bound sraegy. Compuaional resuls are shown in Secion 4.5 and finally conclusions are given in secion A consrain-based local search algorihm The consrain-based local search algorihm (CLS) is inspired by local search for he saisfiabiliy (SAT) problem. An aracive framework of SAT local search is ha he srucure of he local move is simple. There are some similariies beween he ideas used in CLS and SAT local search. Boh mehods use a simple variable flip as a srucure of local move and employ a randomised sraegy for he selecion of consrains and variables o explore. However, many aspecs of CLS are differen from SAT local search. These are described in he following secions The main loop For he conainer rail scheduling problem, we firs apply a simple pre-processing procedure o ge rid of he model variables x and y j if here is no cusomer demand on imeslo, he oal demand on imeslo is less han he minimum rain loading P, or here is a service resricion banning x. Afer pre-processing consrains (3.8) can be removed as well. 7

83 Poins in he search space correspond o a complee assignmen of 0 or o all decision variables. The search space is explored by a sequence of simple randomised moves which are influenced by he violaed consrains a he curren poin. CLS sars wih an iniial random assignmen, in which some hard consrains in he model can be violaed. In he ieraion loop, he algorihm always selecs a violaed hard consrain a random, e.g. an assigned rain imeslo for which he demands exceed rain capaciy or an assigned imeslo ha is no consisen wih a cusomer s booking preferences. Having seleced a violaed hard consrain, he algorihm randomly selecs one variable in ha consrain and anoher variable, eiher from he violaed hard consrain or from he search space. Then wo flip rials are performed in which he curren value of he variable is changed o is complemenary binary value. Suppose ha V i akes he value v i a he sar of he ieraion so ha he curren soluion A ( v v v m h) =,,..., 2, where m is he oal number of variables and h is he oal violaion of all hard consrains. Suppose furher ha V, V 2 are chosen and ha heir flipped values are v, v 2 respecively. We hen look a he assignmens A = ( v, v,..., v h ), A ( v, v,..., v h ) 2 m 2 2 m = and selec he alernaive wih he smaller oal hard violaion. This alernaive becomes he new curren poin. I is noed ha CLS selecs he bes alernaive ( A or A 2 ) for a new curren soluion even if i is worse han A. The aim is o allow diversiy in he search so ha he search space may be fully explored. CLS erminaes when a feasible soluion A is found or when no improvemen o he bes oal hard consrain violaion found has been achieved for a specified number of ieraions Z. The procedure of he basic CLS is oulined in Figure

84 proc CLS A iniial random assignmen h iniial hard consrain violaion if h = 0 hen oupu A, exi CLS erminae false ry 0 while no erminae do C := selec-violaed-hard-consrain ( A ) P := selec-wo-variables ( C, A) A, 2 ry ry + A := flip ( A, P) if ( h < h 2 ) hen ( A A ) if h < h hen h h ry 0 else ( A A 2 ) end if end while end proc if h 2 < h hen h h 2 ry 0 if h = 0 hen oupu A, erminae rue if ry = Z hen erminae rue Figure 4.: The basic CLS procedure From Figure 4., in selec-violaed-hard-consrain, CLS randomly selecs a consrain ha is no saisfied by he curren assignmen, and hen in selec-wo-variables, CLS randomly 73

85 selecs one variable in ha consrain and anoher variable, eiher from he violaed hard consrain or from he search space o flip. The principal goal of CLS is o find a complee assignmen of values o variables which is consisen wih all he hard consrains. This ask is eased if parial consisency is mainained hroughou he search. The approach aken in applying CLS o he conainer rail scheduling problem is o decompose i ino a series of similar subproblems, each wih he number, bu no iming, of rains fixed in relaion o he minimum rain loading parameer P and mainaining he coverage consrains (3.4). Thus consisency is mainained wihin each se of variables bu enforced across differen ses. When CLS finds a feasible soluion A, he refined-improvemen procedure is called in an aemp o reduce he sof violaion for he cusomer saisfacion consrain s 2 ((3.3) in Secion ). The main concep of he refined-improvemen procedure is o search more inensively on A by fixing he number and iming of rains. This is because i is expensive o mainain he consisency beween he imeslo variables x and he cusomer s booking variables y j if he rain imeable is no fixed. In his procedure only feasible improving moves are allowed. Here, he refined-improvemen does no ry o reduce he sof violaion for he imeslo-operaing cos consrain s 3 ((3.5) in Secion ) for he same reason as above; in addiion, he variaion in imeslo-operaing cos is small. However, in Secion 4.3.2, he arificial sof violaion S * is inroduced in an aemp o reduce he oal sof violaion whils all hard consrains are sill o be fully saisfied by CLS. The refined-improvemen procedure is: 74

86 Sep Record he sof violaion s 2 for A. For each cusomer: Sep 2 Order he alernaive, acive imeslos for his cusomer in increasing order of saisfacion cos for his cusomer. For each possible imeslo: Sep 3 Swap his cusomer wih a cusomer currenly served in his imeslo. Sep 4 If he new assignmen is feasible and reduces he sof violaion s 2, replace A, s 2. Repea sep 2 for he nex cusomer, if any exis. If he new assignmen is no feasible or is feasible bu does no reduce s 2, repea sep 3 wih he nex cusomer if any exis served in his imeslo. Afer he refined improvemen is performed, he algorihm reduces he sof violaion cos s ((3.) in Secion ) by removing one rain ou of he curren feasible soluion. As a resul, he problem is more consrained and he curren soluion becomes infeasible. The value of sopping crierion parameer Z is refreshed and CLS is called again o find a new feasible soluion for he problem Violaed consrain selecion From he procedure of CLS in Figure 4., he remaining degrees of freedom in designing he search sraegy are how o selec a violaed consrain and which variables o flip. In CLS, i is hard o find good sraegies for he selecion of a violaed hard consrain and of variables 75

87 which have a srong impac on performance of he algorihm. Differen selecions of a violaed hard consrain have been invesigaed for SAT local search (McAlleser, 997 and Walser, 999). For insance, choosing he violaed consrain wih maximum or minimum consrain violaion; however none have been shown o improve over random selecion. Therefore, he quesion remaining is how o selec good variables o flip Variable selecion Once a violaed hard consrain has been chosen, CLS selecs wo variables in order o perform rial flips. This is differen from GSAT in which only one variable is chosen o flip in favour of less compuaion. Random selecion is used o achieve a diversified exploraion of he search space. CLS could also choose only one variable o flip in order o reduce he compuaional cos of each ieraion. On he oher hand, i could selec more han wo variables o improve he performance of he local move. However, ieraions become more compuaionally expensive as he number of variables seleced increases. Therefore a compromise of selecing wo variables is made. In addiion, he wo-variables selecion scheme is heoreically and compuaionally suiable for he predicive choice model (described in Chaper 5) which involves building a join probabiliy disribuion of he non-deerminisic ineracion beween wo variables in he combinaorial opimisaion model in order o predic a good choice of value for a variable. For he conainer rail scheduling model, here are wo ses of binary variables: a imeslo variable x represens wheher o operae a rain service or no in each poenial imeslo, and 76

88 cusomer s booking variable y j represens wheher a cusomer is served in each poenial imeslo. In his case, by randomly choosing one variable from he violaed consrain and anoher from he search space, diversified exploraion of he search space may no be achieved because he number of y j variables is much greaer han he x variables. Therefore, alernaive selecion schemes are inroduced as follows: - Scheme : Randomly selec any wo variables in he violaed consrain. - Scheme 2: Randomly selec one variable from he y j se and one x variable from he violaed consrain. - Scheme 3: Randomly selec one y j variable from he violaed consrain and anoher variable ( x or y j ) from he search space. - Scheme 4: Randomly selec one x variable from he violaed consrain and one x variable from he search space CLS selecs one of hese schemes a random so ha a wide exploraion of he search space can be achieved. I is noed also ha no all hese variable selecion schemes are applicable a all imes during he search process because of mainaining consisency wihin each se of variables, or across he wo ses of variables (described in Chaper 5). 4.3 Violaion sraegy In SAT local search and is varians, he number of violaed consrains (unsaisfied clauses) is used o evaluae local moves wihou accouning for how severely individual consrains are violaed. In CLS, a quanified measure of he consrain violaion is used o evaluae local 77

89 moves. In his case, he violaed consrains may be assigned differen degrees of consrain violaion. This leads o a framework o improve he performance of he solving algorihm. The consrains can be weighed according o heir relaive imporance in order o allow he search o give prioriy o saisfying some subses of he consrains. For conainer rail scheduling, sof and hard consrains in he model are reaed separaely. The hard consrains play he major role in guiding he search. When all hard consrains are saisfied, he sof consrain violaions are calculaed and used as a measure of he qualiy of he soluion. Neverheless, whils he hard consrains have no ye been fully saisfied, our scheme incorporaes an arificial consrain, and is weighed violaion measure is designed o exer some influence over he search process based on an esimaion of he sof consrain violaion (discussed in Secion 4.3.2) Hard violaion The principal goal of CLS is o find a feasible soluion o he problem, i.e. soluion poins a which he oal violaion of he hard consrains is zero. To improve he performance of he solving algorihm using a violaion sraegy, wo basic quesions arise:. If he same measure of he consrain violaion should be used wihin each se of hard consrains. 2. If he same measure of he consrain violaion should be used across all ses of hard consrains 78

90 Using differen measures of he violaion affecs he performance of he algorihm because he algorihm gives prioriy o saisfying some subses of consrains. For our conainer rail scheduling model, all ses of hard consrains use measures of violaion weighed according o some heurisic rules. For he rain capaciy consrains, any number of conainers in a poenial imeslo exceeding a rain capaciy is penalised wih he same violaion h m. Here, he violaion penaly for exceeding a rain capaciy does no depend on he amoun of overcapaciy. The violaion for overcapaciy imeslos represens he number of violaed capaciy consrains (or he number of over-loaded rains) and CLS ries o eliminae his violaion. However, his penaly measure will be esed when he compuaional experimens are given in Secion 4.5. The violaion penaly for a se of capaciy consrains is defined as: x j M N j y j P, violaion = 0 2 > P, violaion = h 2 m ; (4.) where: h m is a violaion penaly for a capaciy consrain, P 2 is he capaciy of a rain. I is noed ha he variable x is inroduced ino he capaciy consrain. This is because for CLS, a measure of hard consrain violaion should be quanified so ha i can be used o sensibly evaluae local moves. For insance, in (4.) if x is excluded, an over-loaded rain (overcapaciy imeslo) will be penalised wih he amoun of violaion even if a rain does no depar a imeslo, which is non-meaningful. Penalies for assigning cusomers for imeslos wihou an assigned rain are added via he consisency consrain (3.7). To do so 79

91 via (3.6) as well would effecively double coun his violaion. In addiion, wih he presence of x he algorihm needs no calculae rain capaciy usage when x = 0, reducing compuaional ime. For he consisency consrains, he algorihm assigns he violaion penaly h c if a rain does no depar a he imeslo bu here are some cusomers assigned o ha imeslo. The violaion penaly for a se of consisency consrains is defined as: j C y j x C > x C, violaion = 0, violaion = h c ; (4.2) where: h c is a violaion penaly for a consisency consrain, C is number of poenial cusomers for imeslo. For he covering consrains, he algorihm allocaes a penaly if he assigned rains do no serve all cusomer demands. In oher words, he covering consrains favour a combinaion of seleced imeslos ha covers all cusomers bookings. The violaion penaly wihin a se of covering consrains uses he same quanificaion, which is defined as: S j, violaion = 0 x, j = 0, violaion = hs (4.3) where: h s is a violaion penaly for a covering consrain. 80

92 Covering consrains do no affec he operaional requiremens for a rail carrier, and heir removal does no change a feasible soluion. However, heir presence may improve he performance of he algorihm as i guides he search o he feasible soluions. A quanified measure of he violaion h s should be se higher han hose of h m and h c so ha he search gives prioriy o saisfying a se of covering consrains. Therefore, he oal hard violaion is he sum of he violaion penalies for rain capaciy, imeslo consisency, and covering consrains, which can be wrien as: h = H + H + H (4.4) m c s where: h is he oal hard violaion; H m, H c, and H s are he oal hard violaions for rain capaciy, imeslo consisency, and covering consrains respecively Arificial sof violaion The arificial sof violaion S * has been inroduced so ha he search considers an esimaion of oal sof consrain violaion, while some hard consrains are sill o be fully saisfied. The arificial sof violaion is regarded as if i were a hard violaion unil he capaciy, consisency, and covering consrains have all been saisfied, hen he arificial violaion is se o zero. The algorihm assigns a violaion penaly s * if a imeslo is seleced as a rain deparure ime. This can be wrien as: 8

93 x =, violaion = s = 0, violaion * = 0, (4.5) Noe ha, * S is he sum of hese violaions. In conras o (4.) (4.3) which imply a fixed violaion penaly for each member of he associaed se of consrains, a violaion penaly for he arificial sof violaion s * varies from imeslo o imeslo. The arificial sof violaion penaly depends on he possibiliy of assigning a paricular imeslo on a rain schedule wih a minimum generalised cos. An aemp o derive he arificial sof violaion in moneary unis by rading off beween he business crieria is no possible. This is because a rain schedule is no a single imeslo, bu is a se of he imeslos. Therefore, considering only a single imeslo separaely from he ohers canno represen a oal cos for he rail carrier. However, as in pracice, some business crieria play more an imporan role han ohers, he relaive weighs for he crieria could be applied. A rail carrier may assign a relaive weigh o he violaion coss of number of rains N, cusomer saisfacion. Wih equal violaion coss for Q, and imeslo-operaing cos N, will give a smaller arificial sof violaion Q, and * s. E consrains in a seleced imeslo E he violaion cos wih he lower weigh 82

94 In pracice, given he relaive weighs 0.2, 0.5 and 0.3 by he Royal Sae Railway of Thailand, * s is herefore obained as: s * = 0.2N + 0.5Q + 0.3E ; (4.6) where: * s is he arificial sof violaion if imeslo is chosen (i.e. x = ) The violaion cos N We firs assume ha he higher he number of poenial cusomers in imeslo, he more likely i is ha assigning a rain o ha imeslo would lead o he minimum number of rains used. However, i is also necessary o consider he disribuion of cusomer shipmen size. Alhough here are a large number of poenial cusomers in a imeslo, each cusomer shipmen may be large. Therefore, such a imeslo could allow only a few cusomers o be served on a rain so giving a high violaion cos (or a prioriy) o his imeslo is no longer reasonable. N is defined by: a µ + σ = ; C (4.7) n a = ; T = a (4.8) N n = 00 ; n (max) (4.9) 83

95 where: n( max) = max{ n : =,2,3,..., T ), µ is he mean of cusomer shipmen size in imeslo, σ is he sandard deviaion of he cusomer shipmen size in imeslo, he number of cusomers in imeslo. C is The raionale of he formula for N is ha i aims o ackle he variaion of cusomer shipmen size in he poenial booking imeslo and hen o provide he esimaed chance for ha imeslo o be used. The variaion of cusomer shipmen size is simply handled using he sum of µ and σ (4.7) as a demand hreshold for he shipmen size. The remaining conceps are jus o reflec ha he higher he number of poenial cusomers in imeslo, he more likely i is ha assigning a rain o ha imeslo would lead o he minimum number of rains used. The calculaion of N is illusraed as in Table 4.. Timeslo Cusomer shipmen size (conainers) Mean STD ( ) C C 2 C 3 C 4 C 5 µ σ Sum a n N Table 4.: An example for he calculaion of violaion cos N 84

96 Table 4. shows ha when imeslo is seleced ( x = ), he algorihm assigns he highes violaion cos for ha imeslo = 00. The more cusomers in a imeslo, he lower he violaion cos in general. When imeslos serve an equal number of cusomers (e.g. imeslo 3 and 7). he lower violaion cos is assigned o he imeslo wih a more even spread of demand, here imeslo The violaion cos Q Alhough he virual loss of fuure revenue in he generalised cos funcion could represen cusomer saisfacion in erms of cos, i is an indirec cos. In pracice, he indirec cos is no obvious for rail expendiure as i affecs he long-erm financial plan. Therefore, in a compeiive ranspor marke, a direc cos ha affecs he shor-erm cash flow is regarded as more imporan. The saisfacion of cusomer j in imeslo is w ( 0 w ), and j j ( w 00 ) is he saisfacion score. W is a oal cusomer saisfacion score in imeslo, j i.e. = ( w j ) W 00. The higher he value of W, he more likely i is ha he imeslo j C would be used. The violaion cos Q is defined as: b T W = = W ; (4.0) q b = ; T = b (4.) 85

97 Q q = 00 ; q ( max) (4.2) where: q( ) = max{ q : =,2,3,..., T ). The calculaion of Q is illusraed as follows: max From Table 4.2, Q is no only affeced by he cusomer saisfacion score bu also is implicily dependen on he number of poenial cusomers using he imeslo, i.e. he more cusomers in he imeslo, he lower would be he violaion cos Q. When imeslos have he same number of cusomers (e.g. imeslo 3 and 7), he algorihm assigns a lower violaion cos o imeslo 3 because i has a higher value of W. Timeslo ( ) Saisfacion score C C 2 C 3 C 4 C Sum W b q Q Table 4.2: An example for he calculaion of violaion cos Q 86

98 The violaion cos E A rail carrier may have differen operaing coss for differen imeslos. The operaing coss comprise rain congesion cos and saff cos. Alhough a rain schedule is a se of imeslos, we could consider E for he operaing coss of each imeslo direcly. This is because he operaing cos is a cos uni and does no affec he number of imeslos in he opimal rain schedule. E is defined by: e U = ; T = U (4.3) E e = 00 ; e (max) (4.4) where: e( max) = max{ e : =,2,3,..., T ), E is a violaion cos for he operaing coss in imeslo, U is he operaing cos in imeslo ( U includes rain congesion cos and saff cos). In (4.3), e is derived from he proporion of he operaing coss in imeslo o he oal imeslo operaing cos. E in (4.4) reflecs ha he lower he operaing coss for he imeslo, he higher he esimaed chance ha he imeslo would lead o a schedule wih he minimum generalised cos, and he value of E is shown in a percenage scale. The calculaion of he violaion cos E is illusraed as follows: 87

99 Timeslo U e E ( ) 3 ( 0 Bah) Sum Table 4.3: An example for he calculaion of he violaion cos E Table 4.3 shows ha a rail carrier incurs operaing coss ha vary from imeslo o imeslo. Choosing a imeslo ha has high operaing coss would be penalised wih high violaion E. From (4.4) and (4.5), he oal consrain violaion h is he sum of oal hard violaion ( H, H c, H s ) and he weighed oal arificial sof violaion * S : m h = * ( H + H + H ) + ( S α ) m c s (4.5) The parameer α represens he violaion penaly of one uni of arificial sof violaion. This parameer can be adjused empirically in order o balance he rade-off beween he arificial 88

100 sof violaion * S and he hard violaions ( H m, c H, H s ), i.e. when α is increased, he search algorihm reas * S more imporanly relaive o H m, H c, and H s. Now, CLS uses his oal consrain violaion h o evaluae local moves. The modified procedure of CLS for he conainer rail scheduling problem is given as: proc CLS A iniial random assignmen h iniial oal consrain violaion if ( H m, H c, erminae false ry 0 H s ) = 0 hen while no erminae do * S 0, oupu A, exi CLS he same as he basic CLS in Figure 4. end while end proc if ( H m, H c, H s ) = 0 hen if ry = Z hen erminae rue * S 0, oupu A, erminae rue Figure 4.2: The modified CLS procedure 89

101 There are wo differences beween he basic CLS (Figure 4.) and he modified CLS for he conainer rail scheduling problem (Figure 4.2): he quanified measure of local moves and he sopping crierion. In Figure 4.2, h is he sum of ( H, m H c, H s ) and * S ; he algorihm sops when ( H m + H c + H s ) = 0 (i.e. a feasible soluion A is found) or when no improvemen o he bes oal consrain violaion found has been achieved for Z ieraions. A his ime, * S is discarded and is no longer used. The algorihm coninues wih he same procedure as described in Secion Minimum rain loading The consrain-based local search assigns a fixed number of rains according o he number of rains expeced, which is derived from he minimum rain loading. In oher words, a fixed number of imeslos used is mainained during he search process, which can be wrien as: T x = Texp (4.6) where: T exp is he number of rains expeced. Seing a minimum rain loading ensures saisfacory revenue for a rail carrier and spreads ou he capaciy uilisaion on rain services. The carrier may wan o se he minimum rain loading as high as possible, ideally equal o he capaciy of a rain. Noe ha he minimum rain loading is direcly relaed o T exp, which is defined as: 90

102 exp = N j / P (4.7) j M T where: N j is he demand of cusomer j, P is he minimum rain loading used (3.5). Apar from ensuring saisfacory revenue, minimum rain loading is a key facor in he performance of he search algorihm. The higher he minimum rain loading, he more consrained he problem is and hence he number of feasible soluions decreases. Using a high minimum rain loading allows he algorihm o focus on saisfying he hard consrains more han he sof consrains. In addiion, i increases he usefulness of he predicive choice model, i.e. he variables in he conainer rail scheduling model would ake more consisen values in all he feasible soluions during he search (described in Chaper 5). However, i would be very hard o prove wheher here exiss a feasible soluion o he problem consrained by a high minimum rain loading. If we could prove he exisence of a feasible soluion for he highes possible minimum rain loading, i implies ha he soluion is approximaely opimal. A good seing of he minimum rain loading helps limi he size of he search space. Alhough a few echniques for proving he exisence of feasibiliy have been proposed (Hansen, 992; Wolfe, 994; Kearfo, 998), implemenaions of hese echniques for pracical problems have no ye been achieved. In his research, he minimum rain loading is derived from some heurisic rules. Suppose ha he cusomer s booking daa is given as in Table

103 Day Cusomers Conainers Shipmen size Mean Sd. MON TUE WED THU FRI SAT SUN µ g = 7.48 σ g = 6.38 Table 4.4: An example cusomer s booking daa From Table 4.4 each schedulable day has an average cusomer demand (conainers) and is sandard deviaion (e.g. Mon: Mean = 5.20 and Sd. = 8.06). Noe ha in Table 4.4, he oal demand j M N is no he sum of he demand in all days (i.e ). This j is because a cusomer demand is assigned o a se of he preferred and alernaive booking imeslos in which a rain schedule has no ye been provided. However, we need he average size of cusomer demand in a scheduling week, here µ g = 7.48, in order o esimae roughly how many cusomers (demand unis) can be served by one rain (noe ha a cusomer demand canno be spli in muliple rains as described in Secion 3.4.), and herefore how many rains are required o serve all cusomer demands. 92

104 We also ake he cusomer demand variaion ino accoun. By assuming cusomer demand is normally disribued, we could say ha in mos cases cusomer demand is less han he demand hreshold ( µ g + σ g ). If we use his hreshold as he average cusomer demand, we could obain a number of rains needed o serve all cusomer demands and, from (4.7), a feasible minimum rain loading can be derived. An iniial value of P is defined as follows: T * P2 = µ + σ g g (4.8) P N j M = M /T j * (4.9) where: P 2 is he capaciy of a rain, number of cusomers. N j is he demand of cusomer j, M is he oal In (4.8), * T is used o express he proporion of he rain capaciy o he demand hreshold ( µ g + σ g ). In (4.9), he proporion of he oal demand o he esimaed number of rains * ( M / T ) gives he iniial minimum rain loading P. The calculaion of P is illusraed as follows: From Table 4.4, suppose ha M = 70, oal demands j M N = 370, a rain capaciy P 2 = 68, we will obain he minimum rain loading P = 56, and by subsiuing P ino (4.7), we can obain he iniial number of rains expeced T exp = 25. However, we noe ha his is a very rough esimae used simply o se he iniial value for he number of rains. j 93

105 Whenever all hard consrains are saisfied (a feasible rain schedule is obained), he minimum rain loading is increased by removing one rain from he curren sae of he feasible soluion, i.e. T exp = T exp, and CLS aemps o find a new feasible schedule. However if no feasible soluion is found, T exp is increased by. 4.5 Compuaional resuls The conainer rail scheduling model is esed on wo ses of four successive weeks daa from he easern-line conainer service, he Royal Sae Railway of Thailand (SRT), involving 84 shipping companies. Each rain has a capaciy of 68 sandard conainers. The problem insances are summarised in Table 4.5. In his able, θ denoes a lower bound on number of rains. Noe ha a cusomer may require several differen conainer rail services per week and we consider hese demands as arising from differen cusomers. 94

106 Tes Cusomers Conainers θ SRT s schedules Supply - Demand case Trains Capaciy Capaciy Trains W W W W W W W W Table 4.5: Problem insances From he consrain model s poin of view, Table 4.6 shows he size of he problem insances in erms of he number of he consrained variables (imeslo variable x and cusomer s booking variable y j ) and operaional consrains (rain capaciy, coverage, and imeslo consisency consrains). Noe ha hose x wihou any demands and imeslos ha cusomer j does no prefer are discarded and no couned. y j corresponding o 95

107 Tes Model variables Consrains case x W W W W W W W W y j Table 4.6: Problem size From Table 4.6, all es cases have a large number of binary decision variables. This shows ha in he wors case, here would be n 2 possible soluions; where n is he number of he model variables. Good shor cus may be obained o drive he search o he good soluions, or o reduce he size of he search space. However, for he conainer rail scheduling problem, he size of he search space canno be reduced easily because he consisency and capaciy consrains are enforced (Secion 3.4.2). In his secion, we presen he compuaional resuls of he implemened framework of he consrain-based local search algorihm (CLS), and compare resuls beween he demand responsive scheduling model we propose and curren pracice. Each es case is run en imes on a PC-Penium 2.4 GHz using differen random number seeds a he beginning of each run. 96

108 Now, we wan o find a good value of he sopping crierion parameer. The concep is ha he algorihm should no spend more compuaional ime han necessary; in conras, more ime should be given if soluion ends o be improved furher. We varied his parameer value from 500 o 5000 and observed ha he value = 2000 is a reasonable seing as soluions do no end o be improved afer his value. We se violaion penalies for capaciy, consisency, and covering consrains ( h m, h c, h s ) =,, 00 respecively, and he violaion parameer α is se o 0.5. The sensiiviy for hese parameers will be esed when he compuaional resuls are given. In SRT s schedules, he rail business crieria are represened by he operaing coss OC. However, in CLS s schedules, he crieria are expressed in erms of he generalised cos. Alhough he crieria for he CLS s schedules are convered ino a single generalised cos for more evaluaion, each componen of he generalised cos can be shown explicily. Tables compare he model resuls wih curren pracice. In hese ables, GC is he generalised cos, NC is he number of rains cos, TC is he imeslo-operaing cos, VC is he virual loss of fuure revenue, and he operaing coss OC = NC +TC. All coss are shown in a uni of ( 0 6 ) Bah. 97

109 Tes SRT CLS case Trains OC Trains GC Time Min Avg. Sd. Avg. Sd. (Sec) W W W W W W W W Table 4.7: The schedules obained by CLS Tes SRT CLS OC case OC GC NC TC VC Reducion (%) W W W W W W W W Table 4.8: Operaing cos comparison: SRT vs CLS 98

110 Tables 4.7 and 4.8 show, in all es cases, here are some reducions in he number of rains and operaing coss. In some es cases he reducions in operaing cos are no considerable; his is because in pracice he SRT s schedule is no fixed a he same service level everyday. The rail carrier may cu down he number of rain services wih shor noice if he rain supply is a lo higher han he cusomer demand. This is done by delaying some cusomer s deparure imes according o is demand consolidaion sraegy. However, he proposed model maximises cusomer saisfacion, in oher words, minimises he virual loss of fuure revenue wihin a generalised cos framework. Therefore, he schedule obained by CLS could reflec a high degree of cusomer saisfacion wih he minimum rail operaing coss hrough a demand responsive schedule. From a compuaional viewpoin, we perform furher experimens o es he performance of CLS using differen search sraegies and o es he sensiiviy of parameers in he algorihm Experimen I We firs es he performance of CLS wihou he refined improvemen procedure. When a feasible soluion is found by CLS, he algorihm coninues o reduce he sof violaion for he number of rains consrain by removing one rain from he curren sae of he feasible soluion. In addiion, wih he arificial sof consrain violaion * S in (4.5), he algorihm reduces he sof consrain violaion, whils some hard consrains are sill o be fully saisfied. Therefore, i seems ha CLS wihou he refined improvemen may be good enough and if so we could use he simpler algorihm and reduce compuaional effor. In his 99

111 experimen, each es case is run en imes, he parameers in CLS are he same as in he previous experimen. The compuaional resuls are shown in Table 4.9. Tes Trains Cos Time Case Min Avg. GC NC TC VC (Sec) W W W W W W W W Table 4.9: CLS wihou he refined improvemen procedure From Table 4.9, no surprisingly we can obain he number of rains as good as he one in Table 4.7 and he compuaional ime is beer in all es cases. However, in Table 4.9 he generalised cos GC and is componens ( NC,TC, and VC ) are worse han he resuls obained from CLS wih he refined improvemen. On average, GC is 7.6% more han GC in Table 4.8. Alhough he fac ha he difference is no subsanial, i could benefi he rail carrier in he long run. In addiion, given a fixed number of rains, he rail carrier can assess how well heir cusomers are saisfied in erms of VC ; he schedule wih more rains bu smaller VC may someimes be chosen depending upon he railway s sraegies. 00

112 4.5.2 Experimen II In his experimen, we es he sensiiviy of he violaion penaly parameers ( h m, for capaciy, consisency, and covering consrains (described in Secion 4.3). Each es case is run five imes, h m, h c, h s are varied whils he remaining parameers in he algorihm are fixed. The resuls are shown in Table 4.0, in which - denoes ha CLS failed o find a feasible soluion. h c, h s ) Tes CLS schedule s cos (GC ) case (,,0) (,0,0) (0,0,0) (00,,0) (,,0) (,,00) (,,200) (,,500) W W W W W W W W Table 4.0: Differen measures of he violaion penalies From Table 4.0, using he same value of he capaciy and consisency violaions ( h m, h c = ) produces beer qualiy schedules han does using differen values. This indicaes ha h m and h c are no sensiive and shows he robusness of he algorihm, which does no require 0

113 special uning of parameers for h m and h c. The violaion sraegy wih covering violaion ( h s > 0) shows superior qualiy of schedules. The values of h s have a compuaional advanage as i may influence he search for he feasible schedules. However, choosing oo large a value for h s can decrease is soluion qualiy. The experimens indicae ha h s = 00 is a reasonable seing Experimen III Anoher experimen, which is closely relaed o he experimen in Secion 4.5.2, is carried ou. We es he sensiiviy of he violaion penaly parameer h m bu, in his experimen, he violaion penaly for exceeding rain capaciy is given by he amoun of overcapaciy h m, i.e. from (4.), h m = N j y j P2. In his experimen, each es case is run en imes; he j M violaion penalies are h c = and h s = 00 and he remaining parameers in he algorihm are fixed. The resuls are shown in Table

114 Tes Trains GC case Min Avg. Sd. Avg. Sd. W W W W W W W W Table 4.: Parameer h m given by an amoun of overcapaciy From Table 4., he resuls obained by CLS using h m are worse han he resuls in Table 4.7 in which he violaion penaly for exceeding he rain capaciy is se equally, i.e. h m =. This may be because, given a fixed number of rains, i may be beer if CLS ries o reduce he number of over-capaciy rains, focusing on saisfying he consrain, raher han considering he number of over-capaciy conainers on an assigned rain. For insance, he larger violaion penaly may accoun for a smaller number of over-loaded rains because a cusomer shipmen is considered as one uni and canno be spli over muliple rains. However, in CLS, differen measures of violaion penaly across ses of hard consrains are used so ha he algorihm gives prioriy o saisfying some subses of consrains. 03

115 4.5.4 Experimen IV This experimen is o es he sensiiviy of he violaion parameer α. This parameer conrols he rade-off beween he hard violaions H m, H c, H s and he arificial sof violaion * S in (4.5). The sensiiviy es of α is shown in Table 4.2. In his experimen, each es case is run five imes by varying α. The violaion parameers h m, h c, hs are fixed a,, 00 respecively. Tes CLS schedule s cos (GC ) case α = W W W W W W W W Table 4.2: Sensiiviy analysis of he imeslo violaion parameer Table 4.2 shows he effec of inroducing he arificial sof violaion * S. Wih he exisence of * S, i.e. α > 0, we end o obain a slighly beer qualiy of soluion in erms of he 04

116 generalised cos and α = 0.5 seems o be he bes choice in general. The resuls also show ha a high value of α provides a low qualiy soluion as he search is mos likely dominaed by he arificial sof violaion eliminaed. * S whils he hard violaions H m, H c, H s have no been I is noed ha if he arificial sof violaion * S does no bring a significan improvemen in he schedule, i may no be worhwhile o use i. However, he ideas of how * S is obained and used could be adaped and applied o oher combinaorial opimisaion problems, especially problems ha have many feasible soluions and for which an effecive bound on he objecive funcion could no be achieved. In Chaper 6, we apply his idea o he generalised assignmen problem Experimen V We carry ou an experimen o es he performance of CLS using differen accepance sraegies of local moves. CLS always acceps boh improving and non-improving moves, and CLS incorporaing a simple simulaed annealing mehod (CLS+SA) always acceps an improving move and acceps a non-improving move only wih probabiliy P sa (Kirkparick, 984): P sa = exp (4.20) T where: is he change in an non-improving move, and T is he emperaure 05

117 A simple form of simulaed annealing (SA) was used in his experimen composing of four componens: iniial emperaure T 0, emperaure lengh, cooling schedule and sopping crierion. To ensure a fair comparison beween CLS and CLS + SA, we use approximaely he same number of ieraions for CLS + SA as CLS for each problem. A simple geomeric cooling schedule was used, wih he emperaure being muliplied by a consan facor (or cooling rae) α a every ieraion n, i.e. T + = αt. Some experimens were carried ou for n each problem in order o find good values of iniial emperaure T 0 and final emperaure and hen he cooling rae α can be calculaed direcly from T 0 and parameers T 0, T f, and he number of ieraions seleced are shown in Table 4.3. n T f T f. The values of Tes Case T 0 Conrol parameers α T f Ieraion W W W W W W W W Table 4.3: Conrol parameers for SA 06

118 To perform he experimen using CLS+SA, each es case is run en imes, he parameers in CLS are fixed. The compuaional resuls are shown in Table 4.4. Tes Trains GC Case Min Avg. Sd. Avg. Sd. W W W W W W W W Table 4.4: Compuaional resuls CLS+SA Table 4.9 shows ha he resuls obained from CLS+SA are relaively good; however hey are worse han he resuls obained from CLS alone (Table 4.7). This may be because in CLS+SA, he search may ge rapped in poor qualiy saes, in paricular when he emperaure is small. Assuming convergence o he opimal soluion in infinie ime; SA may cause problems because i uses approximaely he same number of ieraions as CLS; his could also imply ha SA needs o be run a lo longer. In CLS he search is more diversified as improving and non-improving moves are always acceped. 07

119 4.6 Conclusions The consrain-based local search algorihm, CLS, is presened in his chaper. The algorihm is inspired by SAT local search and is applied o he conainer rail scheduling problem. The abiliy o find a good operaional rain schedule along wih saisfying cusomer demand leads o some reducions in he operaing coss, and also enhances he level of cusomer service hrough he demand responsive planning model. CLS sars wih random iniial assignmens and uses a simple variable flip as a srucure of local move. The algorihm is easy o implemen and convenien o use. A violaion sraegy is inroduced in order o drive he search o he promising regions of he search space, differen measures of consrain violaion allow he search o give prioriy o saisfying some subses of he consrains. The arificial sof violaion is inroduced so ha he algorihm evaluaes he qualiy of he sof violaions implicily whils saisfying he hard consrains; hereby i helps o reduce significanly compuaional effor. In CLS, diversified exploraion of he search space is achieved by hree feaures:. Accepance crierion of local moves: CLS always acceps boh improving and non-improving moves o gain a diversified exploraion of he search space and o overcome local minima. 2. Randomised selecion: a violaed consrain and he variables o flip are seleced in diversified sequences, which carries lile compuaional cos. 3. Consisency enforcemen: consisency is only mainained wihin each se of variables in he model independenly (Secions and 4.4). A some laer 08

120 ieraions, he consisency will be enforced by he predicive choice model (described in Chaper 5). In he nex chaper, we will presen he consrucion and use of he predicive choice model as he search inensificaion sraegy. CLS will be incorporaed wih he predicive choice model so ha he algorihm can move around he soluion space more effecively, leading o beer qualiy soluions. 09

121 Chaper Five Search inensificaion using he predicive choice model 5. Inroducion As discussed in Chaper 4, he consrain-based local search uses a simple variable flip as he local move. When all variables in he model are assigned a value, he oal consrain violaion is calculaed. A quanified measure of he violaion is used o evaluae local moves. Using differen measures of he consrain violaion, he algorihm gives prioriy o saisfying some subses of consrains and hereby i guides he search o promising soluion regions. In his Chaper a predicive choice model is developed o improve furher he soluions o he problem of conainer rail scheduling. The predicive model is based on discree choice heory and he random uiliy concep. When sufficien rial hisory has been colleced for a variable, i is analysed o infer a good value for he variable. The variable is hen fixed a his value for a number of ieraions, deermined in a probabilisic manner. A his poin, consisency beween variables is enforced, leading o inensified exploraion of he search space. 0

122 This chaper is organised as follows. Secion 5.2 describes he moivaion for using he predicive choice model. Secion 5.3 describes he developmen of he predicive choice model. Secion 5.4 describes how he predicive choice model is used o improve he soluions of he conainer rail scheduling problem. The compuaional resuls obained by consrain-based local search incorporaing he predicive choice model are shown in Secion 5.5. Finally conclusions are given in Secion Moivaion for he predicive choice model For he purpose of search inensificaion, local search algorihms may ry o fix a value for some variables for a cerain number of ieraions, depending on he search hisory. To do his we need a quanified assessmen of each variable, called is preference measure, wih regard o is likely value in an opimal soluion. The preference measure may also help in deciding how long a variable should remain fixed. Fixing may be done eiher deerminisically or non-deerminisically, using some heurisic rules. The simples way would be, a some predeermined ieraion, o observe which value (0 or ) was chosen more ofen for a paricular variable in he search so far. The preference measure for his value would be he proporion of ime he variable was chosen o have ha value. Then we could fix ha variable a is preferred value for a number of ieraions proporional o he preference measure. If he proporion of ime a binary variable was chosen o have one of is values is much higher he proporion of ime he oher value was chosen, we could have some confidence in he value a which we fix he variable, Conversely, if he proporions are close in value, we hen lose some confidence in which value is o be chosen. Therefore, i may be helpful o

123 incorporae he amoun by which one value is preferable o he oher ino he preference measure. This preference measure could be analysed by saisical mehods so as o increase he confidence in choosing a value for a variable; in his research, we propose such an approach, he predicive choice model Predicive choice model The firs developmen of choice models was in he area of psychology (Luce and Suppes, 965). The developmen of hese models arose from he need o explain he inconsisencies of human choice behaviour, in paricular consumer choice in markeing research and mode choice in ransporaion research. If i were possible o specify he causes of hese inconsisencies, a deerminisic choice model could be easily developed. These causes, however, are usually unknown or known bu very hard o measure. In general, hese inconsisencies are aken ino accoun as non-deerminisic or random behaviour. Therefore, he choice behaviour could only be modelled in a probabilisic way because of an inabiliy o undersand fully and o measure all he relevan facors ha affec he choice decision Choice decision Deciding on a choice of value for a variable is no obviously similar o he consumer choice decision. However, we could se up he search algorihm o behave like he consumer behaviour in choice selecion. Tha is, we consider he behavioural inconsisencies of he algorihm in making a choice of good value for a variable. 2

124 For general combinaorial problems, a paricular variable may ake several differen values across he se of feasible soluions. Thus i is hard o predic a consisenly good value for he variable during he search. However, when he problem is severely consrained and has few feasible soluions, i may well be ha some variables would ake a more consisen value in all he feasible soluions during he search. For he conainer rail scheduling problem, he problem is more severely consrained by seing a high value of minimum rain loading (as described in Chaper 4). Once a variable has been seleced, he algorihm has o choose a value for i. The concep is o choose a good value for a variable, e.g. he one ha is likely o lead o a smaller oal consrain violaion in a complee assignmen. In our consrain-based search algorihm, wo variables are considered a each flip rial. The firs variable is randomly chosen from hose appearing in a violaed consrain and considered as a variable of ineres, he second variable is randomly seleced eiher from ha violaed consrain or from he search space and is o provide a basis for comparison wih he variable of ineres. Clearly, he inerdependency of he variables implies ha he effec of he variable value chosen for any paricular variable in isolaion is uncerain. Flipping he firs variable migh resul in a reducion in oal consrain violaion. However, i migh be ha flipping he second variable would resul in even more reducion in he violaion. In his case, he flipped value of he firs variable is no acceped. In Table 5., he variable of ineres is X and he compared variable is X j, he wo variables are rial flipped in heir values; he violaions associaed wih heir possible values are recorded and compared. In his Table, h is he oal violaion (in 4.5), * X is he value 3

125 of X chosen in he flip rial. Noe ha only h, h, and hisory of X. * X are recorded for he violaion Flip Variable of ineres X Compared variable X j rial Curren Flipped j Curren Flipped Val h Val h Val h 2 Val h 2 * X N Table 5.: Violaion hisory In flip rial he seleced variables are X (curren value ) and, separaely, X 5 (curren value ). The curren assignmen has violaion = 26. Flipping X, wih X 5 fixed a, gives violaion = 22; flipping X 5, wih X fixed a, gives violaion = 36. Hence in his rial he algorihm records X = 0 as he beer value. A some laer ieraion he algorihm chooses o flip X again, his ime (flip rial 2) wih compared variable X 9. Flipping X, wih X 9 fixed a 0, gives violaion = 2; flipping X 9, wih X fixed a, gives violaion = 6. Alhough flipping X o 0 gives a beer violaion han he curren assignmen, in his flip rial he algorihm records X = as he beer value as here is an assignmen wih X = which gives an even beer violaion. If we view he resuls of hese flip rials as a random 4

126 sample of he se of all assignmens, we can build up a predicive model o capure he inconsisency in he choice selecion and o predic wha would be a good value for X Proporional mehod The proporional mehod is based on a probabilisic mechanism in he sense ha he algorihm may selec he curren value of a variable even hough flipping ha variable o he oher value gives a lower violaion. The proporional mehod is sraighforward. The choice selecion is only affeced by he number of occurrences of choice values in The proporional mehod is defined as: * X, i.e. he consrain violaion is no considered. * ℵ0 P0 = (5.) N where: P 0 is a probabiliy for he algorihm choosing value 0, occurrences in * X choosing value 0, N is he number of flip rials. * ℵ 0 is he number of To invesigae he accuracy of forecas of he proporional mehod, he conainer rail scheduling problem is scaled down. The opimal soluion and heir assigned values are known by running an ineger programming branch and bound search o compleion (ILOG, 2002). Then, we run he CLS o collec he violaion hisory wih flip rials N = 20. The experimenal resuls are given in Table 5.2, in which h 0 and h are he average oal violaions when a variable is assigned a value 0 and respecively, * ℵ is he number of 5

127 occurrences of choice values in * X, * X is he value of variable X chosen in flip rial, Φ is he value of he corresponding variable in a known opimal soluion. Var Violaion * ℵ Probabiliy Φ no. h0 h 0 P 0 P Table 5.2: Value choice predicion by proporional mehod 6

128 Table 5.2 shows ha he proporional mehod predics a wrong opimal value for 8 of he 20 variables. We observe ha, in hese cases, alhough he number of occurrences of he choice values favours a paricular value, he average violaion, h, does no. I would seem ha a more complex predicor ha accouns for boh facors would be useful. Noe ha where here is a subsanially higher number of choices of a paricular value, he choice value is generally associaed wih a lower value of of h, e.g. variables 2, 4,, 6, 9. There are also cases, e.g. variable 9, for which boh he number of occurrences and h indicae he wrong choice of value. We inroduce an addiional mehod, he logi mehod, which migh be more saisfacory in considering boh facors. The selecion of wheher he proporional mehod or he logi mehod is o be applied is conrolled by a decision parameer D (%). For example, if D is se o 70, he logi mehod is called when he proporion of any one value in * X is less han 70%; oherwise, he proporional mehod is called insead Logi mehod The logi mehod is based on he random uiliy concep (Ben-Akiva and Lerman, 985). Choosing a good value for a variable in each flip rial is considered as a non-deerminisic ask of he search algorihm. In his research, he algorihm is designed o selec a choice of value for a variable ha has a maximum uiliy (or minimum disuiliy). However, he uiliy is no known by he algorihm wih cerainy and is defined as U 0 and U ; where U 0 and U are uiliies for he algorihm choosing value 0 and respecively. 7

129 For each flip rial, he algorihm selecs value 0 when flipping a variable o 0 is preferred o. This can be wrien as follows. 0 f U > U (5.2) 0 From his poin, he probabiliy for he algorihm choosing value 0 is equal o he probabiliy ha he uiliy of choosing value 0, U 0, is greaer han he uiliy of choosing value, U. This can be wrien as follows: P 0 = Prob[ U 0 > U] (5.3) where: P 0 is a probabiliy for he algorihm choosing value 0. When an occurrence of any choice value * X is no obviously dominaing, he logi mehod is used. The logi mehod is he predicive choice model wih an assumed probabiliy disribuion of he random uiliy. The search algorihm selecs a value 0 when he uiliy U 0 > U, and selecs a value oherwise. To derive a logi model, we require an assumpion abou he join probabiliy disribuion of he uiliies U and U 0. In his research, he difference beween he uiliies, i.e. U = U U 0, is used. We consider U as a random sample of he se of all assignmens for a variable. From he cenral limi heorem whenever a random sample of size n is aken from any disribuion wih mean µ and variance σ 2, he sample would be approximaely normally disribued (Troer, 959). We observe a real disribuion of U by running he 8

130 consrain-based local search algorihm o collec he violaion hisory wih 50 flip rials (samples). The oal consrain violaion h is used as a measure of he uiliies. To make a fair assumpion wheher U fis any specific probabiliy disribuion, we firs ake a look a he hisogram and probabiliy densiy race, which is illusraed in Figure 5. and 5.2 for a ypical case Densiy Violaion hisory Figure 5.: Probabiliy densiy race 9

131 2 80 Densiy Densiy Violaion hisory Uniform disribuion Violaion hisory Suden s disribuion Densiy 8 4 Densiy Violaion hisory Logisic disribuion Violaion hisory Normal disribuion Figure 5.2: Hisograms and possible probabiliy disribuions From Figure 5. and 5.2, i appears ha he logisic and normal disribuions are he mos likely candidaes o represen he probabiliy disribuion of U. Since he choice funcion assuming he normal disribuion is expressed in erms of an inegral (Ben-Akiva and Lerman, 985), i requires a significan compuaional effor o calculae he probabiliies in he choice funcion. Therefore, he choice model based on a logisic disribuion is considered because he logisic disribuion is an approximaion of he normal law (Kallenberg, 997). As shown in Figure 5.2, here is no a subsanial difference beween he normal and logisic disribuion. 20

132 The Kolmogorov-Smirnov (K-S) es (Chakravar e al, 967) is carried ou o es wheher U comes from a populaion wih a logisic disribuion. The es saisic D measures he larges absolue difference beween a heoreical logisic disribuion and he observed U disribuion. The es saisic D is defined as: D = max F( x ) i N i i N (5.4) where: F( x i ) is he heoreical cumulaive disribuion of he logisic disribuion, x i is ordered U from smalles o larges, and N is he number of flip rials. The null hypohesis H 0 is defined as follows: H 0 : There is no difference beween he disribuion of U and a heoreical logisic disribuion H 0 is rejeced if he es saisic D is greaer han he K-S criical value (for example, from he saisic able wih sample size n = 50 and significance level α = 0.05 based on a logisic disribuion, he K- S criical value = 0.250). Table 5.3 illusraes he Kolmogorov-Smirnov es for probabiliy disribuion of U for ypical cases. The saisical sofware, SAS, was used o obain he saisic value D (SAS, 2002). 2

133 Var no D Var no D Var no D Table 5.3: The K-S es for probabiliy disribuion of U The resuls from Table 5.3 show ha in general, he disribuion of U appears o be logisic because he es saisic D of a large majoriy of he variables is less han he K- S criical value Therefore, i may be reasonable o assume ha U is logisically disribued and o derive he predicive choice model from a logisic probabiliy densiy funcion. The choice model assuming a logisic disribuion is obained as follows (Ben-Akiva and Lerman, 985): P U 0 e = (5.5) 0 e + e 0 U U where: P 0 is a probabiliy for he algorihm choosing value 0 22

134 Uiliy funcion For any flip rial, he uiliy U may be characerised by many facors. In his research, he uiliy is only deermined by he oal consrain violaion h. This is because i can easily be measured by he algorihm and gives a reasonable hypohesis o he choice selecion. In oher words, we would like o use a funcion of uiliy for which i is compuaionally easy o esimae he unknown parameers. We define a funcion ha is linear in parameers. A choice specific parameer is inroduced so ha one alernaive is preferred o he oher when he oal violaion is no given, i.e. he choice decision may be explained by oher facors. The uiliy funcions for U 0 and U are defined as: U U = (5.6) 0 β + β 2h0 β 2h = (5.7) where: β is a choice specific parameer, β 2 is a consrain violaion parameer, h 0 and h are he oal violaions when a binary variable is assigned a value 0 and respecively Likelihood esimaion The parameers β and β 2 can be esimaed by several mulivariae mehods, such as maximum likelihood mehod, discriminan analysis, leas square esimaion, ec. (Johnson and Wichern, 996). Among hese mehods, he maximum likelihood mehod is mos popular. The aim of maximum likelihood mehod is o esimae he parameer values ha 23

135 make he observed daa a good fi o he likelihood funcion. The likelihood funcion over he N flip rials is a produc of individual likelihoods, which is wrien as: L N (, β 2 ) = n= y 0, n y, n 0, n, n β P P (5.8) where: L is a likelihood funcion, value 0 and in flip rial n, P, n P, 0 and n are probabiliies for he algorihm choosing y 0, n = if he algorihm selecs a value 0 in flip rial n; oherwise = 0, and y, n = if he algorihm selecs a value in flip rail n ; oherwise = 0. We simplify and ransform he likelihood funcion L o * L N [ y0, n log P0, n + y, n log P, n ] * L ( β, β ) = (5.9) 2 n= We hen solve for he maximum of he log likelihood funcion * L by differeniaing i wih respec o he parameers β and β 2, and seing he parial derivaives equal o zero: * L β k = / β N 0, n k y0, n + y, n n= P0, n, P P, n / β k = 0 P n (5.0) where: k =, 2 24

136 The maximum likelihood esimaion for uiliy s parameer values in he predicive choice model is illusraed as follows. In Table 5.4, h is he oal consrain violaion, * X is he value of variable X chosen in he flip rial. Flip Variable of ineres (X) rial Curren Flipped * X Value h Value h Table 5.4: The inpu daa for maximum likelihood esimaion From (5.9), we ge ( β ) = { log P } + { log( P )} + { log( P )} + { log( P )} * L k 0, 0,2 0,3 0,4 From he logi model assuming he logisic disribuion: L * ( β ) k log e + e + log e β+ ( β2 h0, ) β+ ( β2 h0, 2 ) e e = β+ ( β2 h0, ) β2 h, β+ ( β2 h0, 2 ) β2 h, 2 log e + e + log e + e + e β+ ( β2 h0,3 ) β+ ( β2 h0, 4 ) e e + β+ ( β2 h0,3 ) β2 h,3 β+ ( β2 h0, 4 ) β2 h, 4 25

137 where: h, n h, 0 and n are he oal violaions when a decision variable is assigned a value 0 and in flip rial n. Subsiuing h, n h, 0 and n from he Table 5.4 ( β ) * L k log e + e + log e β + 22β 2 β + 2β 2 = β + 22β 26β β + 2β 20β e log e e e + log e 2 + e 2 + e β + 5β 2 β + 4β 2 + β + 5β 2β β + 4β 5β e 2 2 e 2 2 Then we obain he values for * β which maximise ( ) k L * ( β ) β k k = 0 L β k via: To find he maximum likelihood esimae, we have o solve a sysem of wo non-linear equaions in wo unknowns. The parameer values can be obained by using non-linear unconsrained opimisaion mehods, such as Newon-Raphson mehod (Kallenberg, 997). An example of using he logi model for predicing a value of variable X wih flip rials N = 20 is shown in Table 5.5 in which h 0 and h are he oal violaions when a variable is assigned a value 0 and respecively, * X is he value of variable X chosen in he flip rial. To find he esimaes of he parameer ha bes fis he observed violaion hisory daa, he saisical package SAS (SAS, 2002) based on he Newon-Raphson mehod is used. For his daa se, he maximum likelihood mehod gives he choice specific parameer β = and he violaion parameer β 2 =

138 Flip Violaion * X Uiliy Probabiliy rial h 0 h U 0 U P 0 P Table 5.5: Probabiliies of a value choice selecion in flip rials 27

139 Aggregae predicion Up o his poin, we have focused on a model ha predics a choice of values for a variable in each flip rial n. However, he predicions for an individual flip rial may no reliably help he algorihm make a decision on wha a good value for a variable would be. Insead, we are ineresed in an aggregae quaniy, i.e. a predicion for he value choice based on a se of rials. We use he arihmeic mean of he oal violaion o represen he aggregae violaion of N flip rials, which can be wrien as: h N 0, n 0 =, and h = n= h N N h, n (5.) N n= where: h 0 and h are he average oal violaions when a variable is assigned a value 0 and respecively. To invesigae he accuracy of forecas of he logi mehod, he conainer rail scheduling problem is scaled down. The opimal soluion and heir assigned values are known by running an ineger programming branch and bound search o compleion (ILOG, 2002). Then, we run CLS o collec he violaion hisory wih flip rials N = 20. The experimenal resuls are given in Table 5.6, * ℵ is he number of occurrences of choice values in * X, is he value of variable X chosen in flip rial, Φ is he value of he corresponding variable in a known opimal soluion. P 0 is calculaed via (5.5) (5.7). * X 28

140 Var Violaion * ℵ Probabiliy Φ no. h0 h 0 P 0 P Table 5.6: Value choice predicion by logi mehod 29

141 The percen correcly prediced for a se of variables, PC, is calculaed as follows: PC = T c 00 (5.2) where: c is he number of variables correcly prediced, T is oal number of prediced variables. Table 5.6 shows ha our prediced value for each binary variable is relaively close o is known opimal value wih PC = 85%. For variables 4, 9, and 9, he choice model predics a wrong opimal value. However, PC is sufficienly high o allow us o have some confidence in he logi mehod Simplified esimaion Unil now, he parameers β and β 2 have been esimaed by he maximum likelihood mehod. Unforunaely, his mehod requires a significan compuaional effor and needs o be applied many imes during he search. In his secion, we inroduce a simplified esimaion of he uiliy s parameer values. Since he number of occurrences of choice values in * X will be reaed separaely by he proporional mehod, he logi mehod only accouns for he oal consrain violaion. However, in he logi mehod, we may have o se he choice specific parameer β o a small value, e.g. β = 0.05, so ha he uiliy of one alernaive is preferred o he oher. This 30

142 is because an equal uiliy lies ouside he assumpion of he choice heory (Ben-Akiva and Lerman, 985). Insead of considering h 0 and h separaely, he absolue difference beween h 0 and h, h is used in order o characerise he value choice selecion. h is defined as follows: 0 = (5.3) h h h 0 h + h where: h 0 and h are he average oal violaions when a variable is rial flipped or assigned a value 0 and respecively. From (5.3), when he value of h is large, he probabiliies of wo alernaives (value 0 and ) would be significanly differen, and when h = 0, he probabiliies of he wo alernaives would end o be equal. h is shown in a proporional scale so ha he formulaion could be generalised for he problem in which he oal violaion and he number of flip rials can be varied. Then we presen a simplified esimaion of he violaion parameer β 2 as follows: β = h (5.4) 2 Now finding he uiliy s parameer values only requires a very lile compuaional effor. I is noed ha in his sudy β could eiher be se o any small posiive or negaive value, 3

143 whils β 2 is always se o be negaive because he consrain violaion represens disuiliy, i.e. value 0 is preferred o when h 0 is less han h. Table 5.7 show he resuls obained by he predicive choice model using he simplified esimaion for logi mehod. The experimen uses he same es daa as shown in Table 5.6. For all es variables, he number of flip rials N = 20, he choice specific parameer β is se o 0.05, and he decision parameer D is se o 70 (more experimenal resuls are shown in Appendix B) Table 5.7 shows ha he prediced value for he es variable is close o is known opimal value wih PC = 75%. This PC value is sufficienly high o allow us o have some confidence in he predicive choice model, and indicaes ha he resul obained by he simplified esimaion for logi mehod serves almos as well as he resul from he more complicaed mahemaical one used in Table 5.6. In Tables (and Appendix B), we observe ha he predicions when P 0 is beween are no very accurae. Therefore, he predicive choice model discards hose predicions in order o increase he accuracy of forecas for all variables. 32

144 Var Violaion * ℵ Probabiliy Φ no. h0 h 0 P 0 P Table 5.7: Value choice predicion by he simplified esimaion for logi mehod. As an oucome of he predicive choice model is a probabiliy of choosing value for a variable, he variable will be fixed a is prediced value for a number of ieraions 33

145 deermined by he magniude of he probabiliy. During hese ieraions oher variables may become fixed. When he fixing ieraion for a variable is reached, i is freed and is violaion hisory is refreshed. Before we describe how he predicive choice model is used o improve he conainer rail schedule obained by CLS, relaed work on he consrucion and use of probabilisic models in he search algorihm will be discussed Relaed work The predicive choice model has some general similariies wih boh adapive echniques used in local search and populaion-based search echniques. For example, algorihms based on hese echniques are: adapive Tabu search (Glover and Laguna, 997), variable neighbourhood search (Hansen and Mladenovic, 200), an colony opimisaion (Blum e al, 200), esimaion of disribuion algorihms (EDAs) (Pelikan e al, 999), ec. These algorihms do no rely on problem specific designs of heurisics and do no need many uning parameers. Insead hey build probabilisic models o keep fixed some variables during he search or o predic he movemens of populaions in a probabilisic way. Such probabilisic models aemp o draw inferences specific o he complex combinaorial problem being solved and herefore may be regarded as adapive processes ha learn domain knowledge implicily. In mos local search algorihms based on probabilisic models, he problem specific ineracions amongs he decision variables in he combinaorial opimisaion model are kep in mind implicily, whereas in our predicive choice learning algorihm, as well as EDAs, he ineracions are reaed explicily hrough he probabiliy disribuion associaed wih he variables seleced a each sample. In EDAs, ofen incorporaed in geneic algorihm 34

146 (Goldberg, 989), a probabilisic model for selecing promising soluions is consruced, based on he observed probabiliy disribuion. The new soluions are generaed by he probabilisic model. The observed disribuions of he ineracions amongs variables are esimaed and are ofen displayed as complex chains or neworks. We consider he deerminisic and non-deerminisic (random) ineracions amongs he variables separaely. The logisic probabiliy disribuion of he non-deerminisic behaviour in choosing a good value for a variable is assumed in order o derive a specific probabilisic model; hereby i makes our model easy o develop and convenien o use. The model learns from he search hisory, he oucome in erms of a probabiliy is used o influence he search. In general, he probabiliy disribuion in EDAs is caegorised ino hree ypes according o he ineracions amongs he variables (Larraaga and Lozano, 2002). These are: no ineracions, pairwise ineracions, and mulivariae ineracions. No ineracions assumes ha all variables in a problem are independen, i.e. he search algorihm only looks a he values of each variable regardless of he remaining variables. The second ype assumes ha he variables in he combinaorial opimisaion model are largely independen excep for some pairwise ineracions. In his ype, he join probabiliy disribuion of he ineracions beween pairwise variables is consruced. The las is he mos complex and requires significan compuaional effor, bu in reurn for a poenially beer resul. I assumes mulivariae ineracions amongs he variables. The variables may be divided ino a number of independen clusers, and he condiional probabiliies are esimaed wihin each cluser. As he search hisory for our predicive choice model is based on a wo-variables selecion scheme, i is closely relaed o he pairwise ineracions of EDAs. In our model, he join 35

147 probabiliy disribuion of he ineracion beween variables is esimaed by observing he consrain violaions resuling from assigning values for wo variables. In addiion, he applicaion of our predicive choice model and EDAs is differen. In EDAs, he model is used o predic he movemens of he soluion (populaions) in he search space. For insance, when applied o geneic algorihms (Pelikan e al, 999), EDA generaes new soluions by sampling he consruced probabiliy disribuion of he promising soluions (i.e. EDAs do no use crossover and muaion). In our applicaion, he probabilisic model is used o predic a choice of good value for an individual binary variable. Wih sufficien rial hisory, he model predics likely opimal values for variables. The nex secion describes how he predicive choice model is used o improve he soluion of he conainer rail scheduling problem. 5.4 CLS incorporaing he predicive choice model In our algorihm, he predicive choice model is used as he search inensificaion sraegy in order o improve on soluions using CLS alone. As discussed in Secion 5.3., afer some ieraions, he search hisory is analysed, some variables may appear o have high preference measure of having cerain good values, and hey will be fixed for a number of ieraions. A ha poin, consisency beween variables is enforced. When he fixing ieraion limi is reached, he variable is freed, ogeher wih hose oher variables which were fixed by consisency enforcing. The procedure of CLS incorporaing he predicive choice model is oulined as follows: 36

148 Sep : The algorihm sars using CLS (as in Chaper 4) o perform he normal searching process and o collec he violaion hisory. Sep 2: For each variable separaely: when i has been flip-esed N imes, he algorihm calculaes he preferred value and proporional preference measure P ; where P = max( P 0, P ). If P is greaer han or equal o he decision parameer D, i uses his measure, oherwise i recalculaes P using he logi mehod. Now if P is less han he predicion error parameer E, here is no preferred value and no fixing is done. Oherwise he variable is fixed a is preferred value for a number of ieraions equal o P F ; where F is a predeermined number of fixing ieraions. However, if he number of fixed variables becomes greaer han he limi on he maximum number of fixing variables, hen a variable may become unfixed before his number of ieraions. In his case he variable chosen o be unfixed early is he one which has been fixed for he longes. This limi conrols how many variables can be fixed and is necessary o allow some flexibiliy in he search. Sep 3: Fixing a variable may imply ha oher variables should be fixed. A his poin he algorihm enforces he consisency beween variables. See he cases in Secions for how his should be done. Sep 4: When a variable becomes unfixed, i says unfixed unil i has been flipesed a furher N imes and he process above is repeaed using hese N observaions. Oher variables which were fixed a he same ime as his 37

149 variable in order o mainain consisency are also unfixed a he same ime as his variable. Sep 4: The algorihm sops when he ieraion limi Z is reached. In he conainer rail scheduling model, he imeslo and cusomer s booking variables ( x and y j ) are chosen for flip rials, bu propagaion of consisency may no be carried ou fully all he ime. A he beginning, he propagaion of consisency is only mainained wihin each of he se x and y j, bu no across he wo ses of variables, in order o promoe wider exploraion of he search space. However, afer a specified number of ieraions, he rial hisory is analysed. As a resul, some variables will be fixed a he preferred value given by he predicive choice model for a number of ieraions. A his poin, consisency beween imeslos and cusomer s booking variables is enforced, leading o inensified exploraion of he neighbouring search space Timeslo enforcing When he rial hisory is analysed and he imeslo variable x is prediced o have a paricular value, he variable will be fixed a ha value for he number of ieraions deermined by he magniude of he preference measure. In his case, all cusomers booking variables y j associaed wih imeslo x are assigned values consisen wih he preferred value of x. For example, if a preferred value of x = 0 (imeslo is no seleced), no cusomer s booking will be assigned o ha imeslo, i.e. y j = 0 for all poenial cusomers 38

150 in ha imeslo. On he oher hand, if he preferred value of x = (imeslo is seleced), any cusomer s booking may or may no be assigned o ha imeslo. As he imeslo variable x may ake a curren value 0 or and is preferred value can eiher be 0 or during he search processes, we caegorise he imeslo consisency enforcing ino four cases. To simplify our discussion, he following will be used. Saes of variables: N f : number of ieraions o be fixed (for each variable) ( N f = probabiliy P number of fixing ieraions F ) N x : number of fixing x variables N 0 : number of N : number of Parameers: y j variables fixed a 0 y j variables fixed a F nbfixx : number of fixing ieraions : limi on he max. lengh of he fixing lis, X, for x variables nbfixy 0 : limi on he max. lengh of he fixing lis, Y 0, for nbfixy : limi on he max. lengh of he fixing lis, Y, for y j variables fixed a 0 y j variables fixed a The seps of propagaion of consisency for x are given as follows: 39

151 Le K j be he imeslo chosen for cusomer j. U j be he se of alernaive imeslos for cusomer j no used, i.e. where S j is he se of poenial booking imeslos for cusomer j U = { } j S / ; j K j Case : curren value of x = 0, preferred value for x = 0 Sep If N x nbfixx, fix x a 0 for fixing lis X, N x N x +. Oherwise release he firs x variable in X, repea sep. N f ieraions, add x o he end of he N x N x -, reorder X and Sep 2 For all j : if = K j, flip U {} j ( ) { } U j \, y j, y j o 0, selec randomly U j, fix for N f ieraions, N 0 N 0 + y pj a 0; p U j U j. Sep 3 If oal hard violaion = 0, exi wih a feasible soluion. Sep 4 Release any x variable in X if is fixing ieraion limi has been reached and reorder X. 40

152 Case 2: curren value of x = 0, preferred value for x = Sep If N x nbfixx, flip x o, selec randomly an assigned imeslo,, x 0, fix x a for N f ieraions, add x o he end of X, Oherwise release he firs x variable in X, repea sep. N x N x N x +. N x -, reorder X and Sep 2 Sep 3 If oal hard violaion = 0, exi wih a feasible soluion. Release any x variable in X if is fixing ieraion limi has been reached and reorder X. Case 3: curren value of x =, preferred value for x = 0 Sep If N x nbfixx, flip x o 0, selec randomly an unassigned imeslo,, x, fix x a 0 for N f ieraions, add x o he end of X, Oherwise release he firs x variable in X, repea sep. N x N x N x +. N x -, reorder X and Sep 2 For all j : if = K j, flip U {} j ( ) { } U j \, y j, y j o 0, selec randomly U j, fix for N f ieraions, N 0 N 0 + y pj = 0 p U j U j. 4

153 Sep 3 Sep 4 If oal hard violaion = 0, exi wih a feasible soluion. Release any x variable in X if is fixing ieraion limi has been reached and reorder X. Case 4: curren value of x =, preferred value for x = Sep If N x nbfixx, fix x a for N x N x +. Oherwise release he firs x variable in X, repea sep. N f ieraions, add x o he end of X, N x N x -, reorder X and Sep 2 Release any x variable in X if is fixing ieraion limi has been reached and reorder X Cusomer s bookings enforcing For he cusomer s bookings enforcing, he booking variable y j may also ake a curren value 0 or and is preferred value can eiher be 0 or during he search. I is noed ha consisency wihin y j is mainained during he search by he coverage consrain, i.e. a cusomer s booking can only be assigned ino one imeslo. However, i may or may no be consisen wih a se of seleced imeslo variables. To beer conrol he cusomer s booking enforcing, we caegorise i ino wo groups: local and global fix. 42

154 Local fix The local fix is a process of prevening he algorihm selecing a cusomer s booking in imeslo ha may no lead o a feasible soluion, i.e. fixing y j a 0 for a number of ieraions. Alhough locally fixing y j a 0 will no enforce he consisency beween x and y j, he number of poenial booking imeslos for each cusomer is reduced quickly. The local fix is caegorised ino wo cases: Case : curren value of y j = 0, preferred value for y j = 0 Sep If N 0 nbfixy 0, fix y j a 0 for fixing lis Y 0, N 0 N 0 +. N f ieraions; add y j o he end of he Oherwise release he firs y variable in Y 0, N 0 N 0 -, reorder Y 0 and repea sep. Sep 2 Release any y variable in Y 0 if is fixing ieraion limi has been reached and reorder Y 0. Case 2: curren value of y j =, preferred value for y j = 0 Sep If N 0 nbfixy 0, flip U {} j ( ) { } U j y j o 0, selec randomly \, y j, U j,, 43

155 fix y j a 0 for N f ieraions, add y j o he end of Y 0, N 0 N 0 +. Oherwise release he firs y variable in Y 0, N 0 N 0 -, reorder Y 0 and repea sep. Sep 2 If oal hard violaion = 0, exi wih a feasible soluion. Sep 3 Release any y variable in Y 0 if is fixing ieraion limi has been reached and reorder Y Global fix The global fix provides a srong propagaion of consisency beween cusomer s booking and imeslo variables. When he global fix is called (i.e. a preferred value of y j = ), one poenial imeslo is assigned o cusomer j. A rain will run a ha ime in order o serve he demand for cusomer j, his prevens he algorihm selecing he remaining poenial imeslos for ha cusomer. The global fix is also caegorised ino wo cases: 44

156 Case : curren value of y j = 0, preferred value for y j = Sep If N nbfixy, flip y j 0; = K j, fix y j a for N N +. y j o, N f ieraions, add y j o he end of Y, Oherwise release he firs y variable in Y, N N -, reorder Y and repea sep. Sep 2 If x =, fix x a for N f ieraions, add x o he end of X, N x N x +. Oherwise flip x o, selec randomly an assigned imeslo,, x 0, fix x a for N x N x +. N f ieraions, add x o he end of X, Sep 3 If oal hard violaion = 0, exi wih a feasible soluion. Sep 4 Release any y variable in Y if is fixing ieraion limi has been reached and reorder Y. 45

157 Case 2: curren value of y j =, preferred value for y j = Sep If N nbfixy, fix N N +. y j a for N f ieraions, add y j o he end of Y, Oherwise release he firs y variable in Y, N N -, reorder Y and repea sep. Sep 2 If x =, fix x a for N f ieraions, add x o he end of X, N x N x +. Oherwise flip x o, selec randomly an assigned imeslo,, x 0, fix x a for N x N x +. N f ieraions, add x o he end of X, Sep 3 If oal hard violaion = 0, exi wih a feasible soluion. Sep 4 Release any y variable in Y if is fixing ieraion limi has been reached, and reorder Y. 5.5 Compuaional resuls In his secion, we demonsrae he performance of he consrain-based local search incorporaing he predicive choice model (PCM), and compare he resuls wih consrainbased local search (CLS) alone. The resuls are shown in erms of number of rains, and he 46

158 generalised cos. The generalised cos includes he operaing coss and he virual revenue loss represening cusomer saisfacion on he given schedules. Eigh weeks daa from a case sudy are run on he same PC-Penium 2.4 GHz. Each es case is run en imes using differen random numbers a he beginning of each run. Before running compuaional experimens, we need o find good values for parameers here. Afer some iniial experimenaion, he values of he parameers were chosen o be: he number of flip rials N = 20, decision parameer D = 75, he predicion error parameer E = 55%, number of fixing ieraions F = 00, and he limis on he number of fixed variables, nbfixx, nbfixy, nbfixy 0 = 50, 50, 200 respecively. The parameers nbfixx, nbfixy, nbfixy 0 govern he maximum number of variables which are allowed o be fixed in he search and are necessary o allow some flexibiliy in he search. These parameers are closely relaed o F ; if F is no oo large, he limi parameers may never be reached and hus hey become unnecessary. Oherwise, he limi parameers could help he algorihm spread ou he number of fixed variables o differen ses of variables so ha he diversified inensificaion of he search space may be achieved. In his experimen, we se nbfixx, nbfixy, nbfixy 0 = 50, 50, and 200. For nbfixx and nbfixy he values are esimaed by /3 o /4 of he schedulable imeslos x and cusomers M respecively, and nbfixy 0 is abou /5 o /0 of hese parameers have been chosen, we may hen only need a good value of F. y j. Once he values of We ried differen values of he sopping crierion parameer Z, ranging from 500 o 5000 ieraions for all es cases, and observed ha Z = 2000 ieraions is large enough as 47

159 soluions do no end o be improved afer his limi. Noe ha we do no claim here ha hese values of he parameers above are he bes values, bu some care was aken. In addiion, we will es he sensiiviy of he parameers D, E, N and F afer he compuaional resuls are given. The resuls for he es cases are shown in Table 5.8. The cos is a generalised cos ( 0 6 Bah). Tes CLS PCM case Trains GC Time Trains GC Time Min Avg. Avg. (Sec) Min Avg. Sd. Avg. Sd. (Sec) W W W W W W W W Table 5.8: Resuls obained by CLS and PCM Table 5.8 shows ha he resuls of PCM are beer han hose of CLS in erms of he number of rains and he generalised cos GC. Alhough he PCM learning from he search hisory implies a compuaional overhead over CLS, i is offse agains a lower run-ime required o find good schedules, in paricular for large es cases. 48

160 To es he sensiiviy of parameers in he algorihm, four addiional experimens are carried ou as follows Decision parameer D We firs es differen values for he decision parameer D. This parameer is especially imporan since he algorihm is swiching beween he proporional and logi mehods. We need o examine wheher swiching is of value. Alhough he resuls in Table 5.2 showed he proporional mehod alone predics a wrong opimal value for many variables and suggess he use of logi mehod, we may also wan o find a good seing of D. In his experimen, each es case is run five imes, D is varied and he remaining parameers in he algorihm are fixed. The resuls are shown in Table 5.9. Tes Trains (Avg.) Cos (Avg.) Case D = D = W W W W W W W W Table 5.9: Sensiiviy analysis of he parameer D 49

161 Table 5.9 shows ha in he exreme cases D = 55 (always use proporional mehod) and D = 00 (always use logi mehod) he algorihm finds lower qualiy resuls. When boh cases are compared, he logi mehod provides beer resuls han he proporional mehod and in some of he es cases i is as good as he resuls obained from he combinaion of he wo mehods (i.e. when D = 70 and D = 85). This may be because logi mehod considers he amoun by which one value is preferable o he oher in he preference measure which ends o be a main facor in value choice selecion, bu when an occurrence of any choice value * X is obviously dominaing, he logi mehod becomes less effecive as he measure of preferred value ends o be compensaed wih he measure of he oher value in a choice funcion. The resuls sugges ha choosing D beween is reasonable as he algorihm performs well in all es cases Predicion error parameer E This experimen is o es he predicion error parameer E. As in our experimens (in Tables and Appendix B) he predicions when P 0 is beween are no very accurae. In such cases, if P 0 < E % here is no preferred value and no fixing is done. The parameer E = 50% means ha here is always a preferred value for a variable o be fixed. In his experimen, each es case is run five imes, E is varied and he remaining parameers in he algorihm are fixed. The resuls are shown in Table

162 Tes Trains (Avg.) Cos (Avg.) Case E = E = W W W W W W W W Table 5.0: Sensiiviy analysis of he parameer E Table 5.0 shows ha seing E greaer han 60 gives a relaively low qualiy of resuls, bu i sill ges slighly beer resuls compared wih he resuls obained by using CLS alone, where no variable fixing is done. The smaller values of E ( E = 50 and E = 55) show some improvemen of soluions and choosing E = 55 is slighly beer han E = 50. However, no clear difference is observed and he iniially chosen value appears saisfacory Flip rial parameer N This experimen is o es he number of flip rials parameer N. In his experimen, each es case is run five imes, N is varied and he remaining parameers in he algorihm are fixed. The resuls are shown in Table 5.. 5

163 Tes Trains (Avg.) Cos (Avg.) Case N = N = W W W W W W W W Table 5.: Sensiiviy analysis of he parameer N Table 5. shows ha choosing N beween 0 and 30, he algorihm performs well in mos cases. In general, we ge slighly beer resuls when N is se o 20. I is noed ha a high value of N = 50 provides low qualiy resuls, boh in erms of he number of rains and he generalised cos. This may be because a large number of flip rials mus be colleced in he search hisory before he predicive choice model can be used; as a resul i is seldom used Fixing ieraions parameer F The oher experimen is o es sensiiviy of he number of fixing ieraions parameer F. This parameer is designed o guard agains inaccurae forecass by he predicive choice model. When F is no oo large, and he model predics a wrong value for a variable, ha variable will be refreshed afer a small number of ieraions; noe also ha he limis on he 52

164 number of fixing variables nbfixx, nbfixy, and nbfixy 0 may no be reached in his case. The sensiiviy es of F is shown in Table 5.2. In his experimen, each es case is run five imes, F is varied and he remaining parameers in he algorihm are fixed. Tes Trains (Avg.) Cos (Avg.) case F = F = W W W W W W W W Table 5.2: Sensiiviy analysis of he parameer F Table 5.2 shows ha he number of fixing ieraions F affecs he qualiy of schedule, boh posiively and negaively. Choosing F = 50, he resuls are relaively good, bu when F is increased o 00, he resuls end o be improved slighly. However, seing large a value for F beween 200 and 300 decreases he effeciveness of he algorihm in mos cases. This may be because an incorrec forecas by he predicive choice model for some variables prevens he search finding feasible soluions for he number of fixing ieraions. 53

165 5.6 Conclusions The consrucion and use of he predicive choice model is presened in his chaper. The predicive choice model is based on discree choice heory and he random uiliy concep. The model explicily considers he inconsisencies of he algorihm in choosing a good value for binary variable in a probabilisic way. We have shown ha he consrain-based local search algorihm incorporaing he predicive choice model is able o improve he conainer rail schedules obained by consrain-based local search alone. Wih sufficien rial hisory, he predicive choice model will predic a good choice of value for a variable. The variable will be fixed a is prediced value for a number of ieraions deermined by he magniude of an associaed probabiliy. A his poin, he propagaion of consisency beween he variables is enforced, leading o inensified exploraion of he search space. This inensificaion echnique is novel because i is he firs ime ha he predicive choice model has been ailored for a local search mehod (in his case, he consrain-based local search algorihm) and applied o a combinaorial opimisaion problem. Experimens based on real life daa demonsrae he srenghs and usefulness of he proposed echnique. Even hough he consrain-based local search incorporaing he predicive choice model has been developed o solve a specific conainer rail scheduling problem, i can be adaped o oher combinaorial opimisaion problems. In Chaper 6, he applicaion of his approach o he generalised assignmen problem is described. 54

166 Chaper Six Consrain-based local search for he generalised assignmen problem 6. Inroducion The compuaional resuls presened in Chaper 5 demonsrae ha he incorporaion of he predicive choice model in consrain-based local search leads o an effecive algorihm for he conainer rail scheduling problem. However, he soluion approach, especially he predicive choice model, is no srongly dependen on problem-specific knowledge. I seems appropriae o invesigae wheher his approach can be successfully applied o oher combinaorial opimisaion problems. In his chaper we apply he approach o one such problem, he generalised assignmen problem (GAP). GAP is a difficul combinaorial opimisaion problem, which is known o be NP-hard (Sahni and Gonzalez, 976). GAP considers he minimum cos assignmen of n jobs o m agens such ha each job is given o one and only one agen subjec o resource capaciy consrains on he agens. GAP has several applicaions in indusry such as compuer and 55

167 communicaion neworks, faciliy locaion, vehicle rouing, manufacuring sysems, resource scheduling, ec (Chu and Beasley, 997). The conainer rail scheduling problem (described in Chaper 3) has some similariies o GAP in ha he demand (resource consumed) assigned o a deparure imeslo mus no exceed he capaciy of a rain (resource capaciy) and each demand can only be served by one rain (job is processed by only one agen). However, he conainer rail scheduling problem has idenical rain capaciy consrain for all imeslos, whils in GAP, he capaciy of he agens are differen. In addiion, in he conainer rail scheduling problem, imeslo consisency has o be mainained, i.e. if a imeslo is seleced for a cusomer, a rain has o depar a ha imeslo. Each cusomer has a number of poenial booking ime ranges sparsely disribued hrough he available deparure imeslos and a rail carrier aemps o find he minimum number of imeslos o serve all he demand, in conras o GAP in which all agens are available o process jobs. Daa ses publicly available in OR-library (reproduced by Chu and Beasley, 997) are used o es our consrain-based local search algorihm (CLS) for GAP. This chaper is organised as follows: Secion 6.2 describes GAP and reviews some soluion mehods. A formulaion of he problem is given in his secion. Secion 6.3 describes he procedure of CLS for GAP. Secion 6.4 illusraes he search inensificaion echnique using he predicive choice model. Compuaional experimens wih consrain-based local search incorporaing he predicive choice model are given in Secion 6.5 and compared wih oher soluion mehods. Finally conclusions are given. 56

168 6.2 Generalised assignmen problem GAP is a problem of assigning jobs o agens wih minimum oal cos such ha - each job mus be assigned o exacly one agen - each agen requires a known amoun of a single resource and incurs a known cos o perform each job - he cos and resource requiremens of agen, job pairs may be differen - each agen has a limied amoun of he resource This problem is well-known and proved o be NP-hard. Finding a feasible soluion for GAP is also NP-hard (Sahni and Gonzalez, 976; Narciso and Lorena, 999). In his research, we consider GAP as a maximisaion problem, i.e. jobs are processed by agens wih maximum oal profi Relaed work In his secion, we presen a review of soluion mehods found in he lieraure for GAP. The soluion mehods can be caegorised ino wo groups: exac and heurisic mehods. An exensive survey on previous soluion mehods was done by Chu (997). However, exac mehods have a compuaional disadvanage in solving large-size problems. We focus on heurisic mehods ha can find good qualiy soluions wihin a reasonable compuaional ime. 57

169 Osman (995) proposed a hybrid algorihm, SA/TS, which combines simulaed annealing and abu search. The algorihm uses a λ - generaion mechanism which describes how a soluion can be alered o generae neighbour soluions. In SA/TS, he non-monoonic cooling scheme of simulaed annealing and he oscillaion sraegy of abu search are used, which are considered as a hybrid sraegy. Osman also proposed a abu search alone for GAP. Boh he SA/TS and TS use a frequency-based memory ha records informaion used for diversificaion purposes. Chu and Beasley (997) proposed a GA-based heurisic for solving GAP. The heurisic incorporaes a problem-specific encoding of a soluion srucure. Finess-unfiness pair evaluaions are used o handle boh feasible and infeasible soluions. Apar from using muaion and crossover operaors, a wo-phase heurisic improvemen operaion is used. In he firs phase, he operaor ries o recover feasibiliy by reducing he unfiness score. The second phase is o improve he cos of he soluion wihou furher violaing he capaciy consrains. Yagiura e al (999) proposed a variable deph search algorihm for GAP. The algorihm incorporaes an adapive use of modified shif and swap neighbourhoods where some moves are abu in order o overcome local minima. The mehod also allows he search o visi infeasible soluions, modifying he objecive funcion o penalise he overloaded capaciy of he agens. Yagiura e al (2004) also proposed a abu search wih an ejecion chain for GAP. The ejecion chain approach is embedded in a neighbourhood consrucion in order o creae more complex and powerful local moves. The ejecion chains are consruced by using he informaion from a Lagrangian relaxaion of he problem. The sored cos for each job is also mainained and used in a variable selecion process in order o make a move more 58

170 efficienly. In addiion, he algorihm incorporaes an auomaic mechanism for adjusing search parameers o mainain a balance beween visis o feasible and infeasible regions. Diaz and Fernandez (200) proposed a abu search heurisic for GAP. They presened a relaxed formulaion of GAP ha allows he search o cross he capaciy-infeasibiliy boundary using a penaly erm. A candidae move sraegy is also used in which he move wih lowes cos is seleced and performed. A sraegic oscillaion scheme is hen used o permi alernaing beween feasible and infeasible soluions. Search diversificaion and inensificaion sraegies are implemened by means of frequency-based memory. Lourenco and Serra (2002) proposed hybrid mea-heurisic search echniques for GAP. The algorihm is based on a greedy randomised adapive heurisic and a MAX-MIN an sysem ha akes ino consideraion he search informaion gahered in earlier ieraions of he algorihm in order o consruc a good iniial soluion. In his phase, a relaive resource consumed probabiliy is used by considering he raio of resource o agen capaciy. The agen ha has he highes probabiliy is chosen for an iniial soluion. Then, a descenden local search and abu search are used o improve he search. Several neighbourhoods are sudied and used, including one based on ejecion chains ha can produce good moves. Fell and Raidl (2004) proposed an improved hybrid geneic algorihm for GAP. The algorihm is based on he hybrid geneic algorihm proposed by Chu and Beasley (997). The algorihm includes wo iniialisaion heurisics; consrain-raio and linear programming heurisics o creae more promising soluions, which are mosly feasible. In addiion, a modified selecion and replacemen sraegy in GA is used, which assigns infeasible 59

171 soluions a finess value depending only on he relaive capaciy excess and always ranks hem worse han any feasible soluion Problem formulaion We model GAP as a consrain saisfacion problem (CSP) in order o inroduce a consrainbased local search (CLS) o solve his class of CSP. The following noaion will be used. Ses: I : se of agens, = {,2,3,, m } J : se of jobs, = {,2,3,, n } Parameers: p ij : assignmen profi of job j o agen i a ij : resource required for processing job j by agen i b i : capaciy of agen i Decision variable: x ij :, if job j is assigned by agen i, 0 oherwise In a CSP, opimisaion crieria and operaional consrains are represened as sof and hard consrains respecively. A feasible soluion for a CSP is an assignmen o all consrained variables in he model ha saisfies all hard consrains, whereas an opimal soluion is a feasible soluion wih he minimum oal sof consrain violaion. In GAP, he sof consrain is he maximum oal profi of assigning jobs o agens. The hard consrains are capaciy consrains and coverage consrains. 60

172 Sof consrain Maximum oal profi - his consrain aims o maximise oal profi of assigning jobs o agens, which is defined as: i I j J p θ (6.) ij x ij where: θ is an upper bound on oal profi, e.g. = max{ p ij : i I} θ. j J Hard consrains Coverage consrain - his consrain ensures ha each job is assigned o exacly one agen, which is defined as: i I x = ; j J (6.2) ij In CLS, he coverage consrains are mainained during he search; as a resul, hese consrains are never violaed. Resource capaciy - his consrain ensures ha he demand mus no exceed he capaciy of an agen, which is defined as: j J a x b ; i I (6.3) ij ij i 6

173 Violaion sraegy For GAP, he violaion of resource capaciy consrains is used o evaluae local moves. Any amoun of resource required in an agen exceeding is capaciy is penalised wih he same violaion h c. The violaion penaly for resource capaciy consrain h c is defined as: j J a ij x ij b, violaion = 0 i > b, violaion = h i c, i I (6.4) The objecive of he problem is ransformed ino he sof consrain (6.) which will never be saisfied. The sof violaion for he maximum oal profi consrain is defined as: θ, violaion = 0 pij xij i I j J < θ, violaion = h s (6.5) where: h s is a sof violaion penaly for maximum oal profi consrain, h = θ p x, and θ is an upper bound on oal profi. s i I j J ij ij When he search visis he infeasible region (i.e. h c >0), we evaluae he soluions by he oal consrain violaion, which can be wrien as: ( α ) h = H s + H c (6.6) 62

174 where: h is he oal consrain violaion, H s is he violaion for he oal profi consrain, H c is he violaion for he capaciy consrains, and α is a feasibiliy parameer. The parameer α represens he violaion penaly of using one uni of overloaded agen capaciy. The parameer can be adjused empirically in order o balance he rade-off beween hard and sof violaions, i.e. as α is increased he search algorihm reas hard consrains as more imporan relaive o he sof consrain. 6.3 Consrain-based local search The consrain-based local search algorihm (CLS) was originally designed o solve he conainer rail scheduling problem (Chaper 4). As concepually simple and convenien, CLS could be adaped and applied o GAP. CLS sars wih an iniial random assignmen, in which he capaciy consrain for each agen can be violaed. (Noe ha CLS mainains coverage consrains (6.2) during he search; herefore hese consrains are never violaed). In he ieraion loop, he algorihm randomly selecs a violaed consrain, i.e. an assigned agen for which he demand exceeds is resource capaciy. Having seleced a violaed consrain, he algorihm randomly selecs wo variables in ha consrain. For GAP, we caegorise he flip rial procedure ino wo cases: - Case : curren value of x ij = 0 (agen i does no process job j ) - Case 2: curren value of x ij = (agen i processes job j ) 63

175 For case, when eiher he firs or second seleced variable holds a curren value of 0, he algorihm changes he curren value of he variable o is complemenary binary value, i.e. flipping x ij o. A his poin, he agen previously processing ha job is released in order o saisfy he coverage consrain (6.2). For case 2, when he seleced variable has a curren value of, here are a number of poenial agens available o process job j (here are many alernaive agens k for flipping x k i, j o ). In his case, for each seleced variable, wo flip sub-rials are performed for wo randomly seleced agens k i o process job j. The subrial wih he smaller oal hard violaion is chosen for comparison wih he oher flip rial. Beween he wo flip rials corresponding o he wo seleced variables, he algorihm selecs he alernaive wih he smaller oal hard violaion. This alernaive becomes he new curren soluion. Whenever he hard consrain violaion H c is zero, a feasible soluion is found. The algorihm sops when no improvemen o he bes feasible soluion found has been achieved for a specified number of ieraions (he procedure of CLS is described in Chaper 4 in deail) Iniial experimen A sandard se of GAP problem insances were used o es he CLS. These problems are maximisaion problems S, publicly available in Beasley s OR-library (hp://mscmga.ms.ic.ac.uk /jeb/orlib/gapinfo.hml). These problems were used o es he se pariioning heurisic of Carysse e al. (994) as well as he hybrid simulaed annealing/ abu search heurisic, and abu search alone, proposed by Osman (995). The es problems have he following characerisics: 64

176 . The number of agens m is se o 5, 8 and The raio r = m / n is se o 3, 4, 5 and 6 o deermine he number of jobs n. 3. ij 4. ij a values are inegers generaed from a uniform disribuion ( 5, 25) U. p values are inegers generaed from a uniform disribuion ( 5, 25) U. 5. The i 0. j J b values are se o (.8 / m ) a ij The es problems S are divided ino 2 groups, gap o gap2, according o he size of he es problems. Each group conains five problems. The experimens were performed on a PC-Penium 2.4 GHz. The algorihm is coded in Visual Basic 6. For each of he es problems 0 runs are performed using differen random number seeds a he beginning of each run. The feasibiliy parameer α is se manually. For all es problems, we iniially performed a few runs in order o deermine good values of he parameer. Concepually, we ried o se he parameer relaively low so ha he search arges more on he qualiy of soluions han on feasibiliy. If no feasible soluion was found, we hen increased he value of he parameer. In addiion, afer he value of feasibiliy parameer has been chosen, we hen wan o find a good value of he sopping crierion parameer. The concep is ha he algorihm should no spend more compuaional ime han necessary; in conras, more compuaional ime should be given if he soluion could be improved furher. We varied his sopping crierion parameer from 500 o 3000 ieraions for each es problem, and observed ha he parameer value = 000 ieraions is a reasonable seing as in many cases, soluions do no end o be improved afer his limi. Noe ha his sopping crierion parameer will also be esed and discussed in more deail in Secion

177 Table 6. shows he resuls obained from CLS for he es problems. The resuls are shown in erms of he average percenage deviaion from opimal soluion values and average compuaional ime for each problem. The average percenage deviaion is defined as: N * S Si σ = 00 (6.7) * N S i= where: σ is he average percenage deviaion from * S, is he soluion value of he i -h run, and N is he number of runs. * S is he opimal soluion value, S i 66

178 Problem m n Opimal Bes σ Avg. ime gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap

179 Problem m n Opimal Bes σ Avg. ime gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap Table 6.: Resuls CLS alone Table 6. shows ha CLS can find feasible soluions for all problems. This indicaes ha CLS is a general solving mehod as i can readily apply o GAP wihou any modificaions. However, soluions obained by CLS can be far from he opimal soluion and special feaures o enhance he performance of CLS are necessary Variable selecion scheme In CLS, once a violaed consrain has been chosen, he algorihm randomly selecs wo variables in ha consrain in order o perform rial flips. For GAP here is only one se of 68

180 binary decision variables, i.e. a variable x ij represens wheher job j is processed by agen i or no. By randomly choosing variables from he violaed consrain, diversified exploraion of he search space may no be achieved because he number of assigned agens o jobs ( x ij = ) is significanly less han he number of unassigned agens o jobs ( x ij = 0). Therefore for GAP, wo alernaive variable selecion schemes are inroduced as follows: - Scheme : Randomly selec wo assigned jobs ( x ij = ) in he violaed consrain. - Scheme 2: Randomly selec any wo jobs in he violaed consrain ( x ij = 0) or ( x ij = ). The algorihm selecs one of he schemes a random so ha a wide exploraion of he search space may be achieved. To es he effec of he wo-alernaive variable selecion schemes, we carried ou compuaional experimens wih he same daa se and parameers as used in he iniial experimen (Secion 6.3.). The resuls are shown in Table

181 Problem m n Opimal Bes σ Avg. ime gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap

182 Problem m n Opimal Bes σ Avg. ime gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap Table 6.2: Resuls CLS wih wo-alernaive variable selecion scheme Table 6.2 shows ha CLS wih he wo-alernaive variable selecion schemes performs beer han wihou i. We obain beer resuls in erms of average percenage deviaion from opimal soluion for all es problems and, in almos all cases, in erms of average compuaional ime. In addiion, we observe ha he average compuaional ime ends o decrease quie significanly for he larger-sized problems and when he number of jobs increases. 7

183 6.3.3 Refined improvemen From previous experimens, alhough CLS can find feasible soluions for all problems, i lacks a feaure ha enables a move from a curren feasible soluion o a beer feasible soluion. This decreases he performance of CLS for GAP and mos likely for oher opimisaion problems in which here are many feasible soluions. Therefore, a refinedimprovemen procedure is inroduced o improve a feasible soluion found by CLS. The main concep is o allow only feasible improving moves and o search more inensively on a feasible soluion. When CLS finds a feasible soluion, he refined-improvemen procedure is called. The procedure of he refined improvemen is oulined as follows: Sep Record he oal profi for a feasible soluion A. For each job: Sep 2 Order he agens for his job in decreasing order of profi value p ij for his job. For each agen no processing his job: Sep 3 Swap his job wih a job currenly processed by his agen. This is illusraed in Figure 6. below: 72

184 Agen/Job n m Figure 6.: Inerchange assignmen Sep 3 If he new assignmen is feasible and increases he oal profi, replace A, oal profi. Repea sep 2 for nex job, if any exis. If he new assignmen is no feasible or feasible bu does no increase oal profi, repea sep 3 wih he nex job, if any exis processed by his agen. A firs glance, he refined improvemen procedure seems o be compuaionally cosly. However, he refined improvemen does no perform a complee assignmen o all decision variables, i.e. only he few variables associaed wih he inerchange assignmen are considered for checking feasibiliy and soluion qualiy. This reduces he compuaional effor spen by he refined improvemen process significanly. To es he performance of CLS wih he refined improvemen procedure, we performed he experimens wih he same daa se and parameers as used in he previous experimens. The resuls are shown in Table

185 Problem m n Opimal Bes σ Avg. ime gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap

186 Problem m n Opimal Bes σ Avg. ime gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap Table 6.3: Resuls CLS wih refined improvemen Table 6.3 shows ha he qualiy of soluions is improved by he refined improvemen procedure a he expense of some increase in compuaion ime. For small-sized problems, in many cases CLS finds opimal soluions or relaively close o opimal. The average percenage deviaion from he bes soluion is decreased subsanially compared o hose in Tables 6. and 6.2. This may be because he soluions are improved by he deerminisic process of he refined improvemen procedure and he search inensively invesigaes feasible regions recenly found o be good. In conras, in CLS wihou he refined improvemen procedure, finding good feasible soluions depends on he randomising sraegy. 75

187 6.3.4 Candidae agen lis The candidae agen lis is anoher feaure for CLS o solve GAP. From he problem daa ses, we observe ha resource consumed by a job for some agens may have a very high value. In his case, when ha demand is processed, i is very likely o exceed he capaciy of an agen. In conras, a job processed by some agens may have a very low profi compared wih he oher agens for ha job. In his case assigning ha agen o he job is very unlikely o lead o he maximum oal profi. We aim o exploi his informaion o ge rid of he unpromising agens a he beginning of he search. This helps increase he efficiency of CLS in wo ways: ) CLS sars wih a good iniial soluion and 2) i limis he size of he search space wihin promising regions. Since he rade-off beween geing high oal profi and using low demand is a major problem, o obain a good candidae agen lis we firs calculae a relaive demand index and a relaive profi index for all assignmens of job j o agen i. Then hese wo indices are combined ino a demand-profi index. To make use of his sraegy, a candidae agen lis for all jobs is sored in descending order of he demand-profi index. This index is used o resric he number of poenial agens for each job and guide he search in choosing good moves. A relaive demand index represens an esimaed chance of assigning he demand of job j o agen i in an opimal soluion. The index is weighed boh by he agen capaciy and by he oher agens demand in each job. A large demand consumed in each job ends o violae he agen capaciy; herefore i is unlikely o be seleced by he algorihm. To obain a relaive 76

188 demand index, he mean and sandard deviaion of he demand are used. The relaive demand index U ij is calculaed by he following seps: To simplify our discussion, le u ij be he proporion of agen capaciy i o demand j, i.e. ij = bi aij ; uj u / µ and σ uj be he mean and sandard deviaion of u ij over all agens; L uj be a variaion limi for u ij, i.e. Luj = µ σ. uj uj Sep If u ij is greaer han L uj, u = u L, oherwise u = 0. ij ij uj ij Sep 2 Calculae he relaive demand index U = u / u for all i, j ij ij i ij The variaion limi L uj handles he demand variaion of agens in each job. u ij greaer han his variaion limi means ha he demand placed on agen i by job j is small, and he assignmen ends o be chosen by he algorihm. The smaller u ij is he less likely i is ha agen i would process job j. The difference beween u ij and he variaion limi (i.e. u ij = u ij L uj ) is used o measure how much u ij varies from he variaion limi. Suppose ha a demand required o process job j by agen i, a ij is given in Table

189 a ij b i Table 6.4: An example of he demand marix Table 6.5 gives he corresponding values of he relaive demand index above procedure. U ij obained by he U ij Table 6.5: The relaive demand index A relaive profi index represens an esimaed possibiliy of he profi of job j processed by agen i conribuing o he maximum oal profi. The index is weighed boh by he maximum profi for agen i and by he oher agens associaed profi for ha job. The mean 78

190 and sandard deviaion of profi are used o handle profi variaion. The relaive profi index V ij can be obained similar o he relaive demand index: Le v ij be he proporion of profi i o he maximum profi for agen i, { p : j,2,3 n} vij = pij / max ij =,..., ; µ vj and σ vj be he mean and sandard deviaion of v ij over all agens; L vj be a variaion limi for v ij, i.e. Lvj = µ σ. vj vj Sep If v ij is greaer han L vj, v = v L, oherwise v = 0. ij ij vj ij Sep 2 Calculae he relaive profi index V = v / v for all i, j. ij ij i ij Suppose ha he profi of job j processed by agen i, p ij is given in Table 6.6. p ij p i(max) Table 6.6: An example of he profi marix Table 6.7 shows he relaive profi index is obained from he above procedure. 79

191 V ij Table 6.7: The relaive profi index Now, he demand-profi index W can be obained, i.e. W ( U + V )/ 2 ij ij =. A high value of W ij indicaes a high esimaed chance of assigning job j o agen i wih maximum feasible oal profi. The demand-profi index is shown in Table 6.8. ij ij W ij Table 6.8: The demand-profi index Then, he candidae agen lis for each job is sored in descending order of he demand-profi index W ij, which is shown in Table

192 Candidae lis Table 6.9: The candidae agen lis for each job The candidae lis parameer C is inroduced o le us limi he number of candidae agens empirically a he beginning of he search and he algorihm sars wih an iniial random assignmen wihin he number of candidae agens specified by he parameer C. Then he search algorihm only evaluaes a job assigned o he limied number of promising agens. This helps reduce he compuaional effor significanly and arges he search for an opimal soluion. For example, in Table 6.9, seing C = 3, agen 3, 4, and are only considered for job. For GAP, i is he firs ime ha he variaions of demand and profi are considered, and he rade-off beween hem is handled by he relaive demand-profi index. The design of his echnique is similar o ha of imeslo violaion discussed in Secion The use of he candidae agen lis is fruiful because praciioners wih lile compuer background can easily modify he relaive demand-cos index in order o improve he performance of he solving algorihm. To es he performance of CLS wih he candidae agen lis, we performed he experimens wih he same daa se and parameers as used in he previous experimens. The candidae lis parameer C is se manually. For all es problems, we 8

193 iniially performed a few runs in order o deermine good values of he parameer. Firsly, he parameer was roughly se o 50-60% of he number of agens for each es problem. We hen slighly increased he value of he parameer wheher i resuled in any improvemen on he qualiy of soluions. The resuls are shown in Table

194 Problem m n Opimal Bes σ Avg. ime gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap

195 Problem m n Opimal Bes σ Avg. ime gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap Table 6.0: Resuls CLS wih candidae agen lis From Table 6.0, we observe ha CLS wih candidae agen lis performs very well. For small-sized problems, in many cases, i finds opimal soluions or very near opimal soluions wihin lile compuaional ime. However, he candidae agen lis sraegy becomes less powerful when he number of agens and jobs increases. This may be because he rade-off beween oal profi and infeasibiliy becomes less clear and he search space canno furher be limied and inensified using he candidae agen lis sraegy. Noe ha from our experimens, all hree exensions (wo-variables selecion, refined improvemen, and candidae agen lis) improve on he resuls of CLS alone and do no 84

196 significanly increase run ime. This indicaes he usefulness of using CLS wih all hese exensions. 6.4 Search inensificaion echnique We adap he predicive choice model in discussed chaper 5 for GAP. The model learns from he search hisory and exracs problem specific knowledge auomaically. Afer a specified number of ieraions, he search hisory is analysed. The model predics good assignmens of jobs o agens. These assignmens will be fixed for a number of ieraions deermined in a probabilisic way, leading o inensified exploraion of he search space Violaion hisory In CLS, afer choosing a violaed consrain, a wo-variable selecion sraegy is used in each flip rial. The firs variable is randomly chosen from hose appearing in a violaed consrain (i.e. an overloaded agen) as he variable of ineres, he second variable is randomly seleced from ha violaed consrain, and provides a basis for comparison wih he variable of ineres. The procedure of he collecion of violaion hisory is described in Secion Variable fixing Afer a specified number of ieraions, he rial hisory is analysed. Some variables may have high probabiliy of a paricular value given by he predicive choice model. These variables will be fixed a heir prediced value for a number of ieraions deermined by he magniude of he probabiliy. The search space would be inensified and he algorihm arges an 85

197 opimal soluion. The decision variable x ij may hold a curren value 0 or and is prediced value can eiher be 0 or during he search, which we caegorise ino wo groups: local fix and global fix respecively. Local fix. The local fix is a process of prevening he algorihm assigning a job o an agen ha may no lead o an opimal soluion, i.e. fixing x ij = 0 for he number of ieraions. Alhough locally fixing x ij a 0 is no very effecive as in a complee assignmen here are many unassigned agens o jobs (i.e. x ij ha hold a curren value 0), he number of poenial agens o process a job is reduced quickly. Global fix. The global fix provides a srong propagaion of consisency wihin each job for he poenial number of agens. When he global fix is called (i.e. a prediced value of x ij = ), exacly one agen processes he job, hereby prevening he algorihm selecing he remaining poenial agens for ha job. 6.5 Compuaional experimens Apar from he firs se of es problems S, we es a second se of problems S 2, which conains 24 large-sized minimisaion problems. These problems were used o es he GA proposed by Chu and Beasley (997), he variable deph search proposed by Yagiura e al (999a), and abu search wih ejecion chain proposed by Yagiura e al (999b). The problems in S 2 are divided ino four classes according o he way hey were generaed. 86

198 . Type A. ij a are inegers generaed from a uniform disribuion ( 5, 25) U, c ij are inegers generaed from a uniform disribuion U ( 0,50), and ( n / m) 5 0. R b i = , where R = max i aij, and I j { i c c k I} = min, ij kj j J, I j = i 2. Type B. a ij and c ij generaed as in Type A and bi is se o 70% of he value given for Type A. 3. Type C. a ij and cij generaed as in Type A and 4. Type D. ij c b = 0.8 a / m. a are inegers generaed from a uniform disribuion (,00) i j J ij U, = a e, where e are inegers generaed from a uniform disribuion ij ij + ( 0,0) U and b i = 0.8 aij / m. j J Types B and C problems are more difficul han Type A because he resource capaciy consrains are igher. Type D problems are mos difficul o solve because a ij and c ij are inversely correlaed. Compuaional experimens, using CLS incorporaing he predicive choice model wih woalernaive selecion scheme, refined improvemen procedure and candidae agen lis are performed. For each of he es problems 0 runs are performed using differen random number seeds a he beginning of each run. For problems in S, we use he same daa se and parameers as used in he previous experimens. The sopping crierion is se o 3000 ieraions for problems S 2. For he predicive choice model, we se he number of flip rials N = 0, decision parameer D = 75, he number of fixing ieraions F ranges from 50 o 87

199 00, he number of fixing unassigned agens in each job ( x ij = 0) ranges from o 3, he number of fixing assigned agen ( x ij = ) ranges from 5 o 30. Noe ha he conceps of how o se good values for hese parameers and heir sensiiviy ess are given in Chaper 5. We do no claim here ha hese are he bes parameer values, bu some care was aken. The resuls for es problems S are shown in Table Problem m n Opimal Bes σ Avg. Time (s) gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap

200 Problem m n Opimal Bes σ Avg. Time (s) gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap Table 6.: Resuls for maximisaion problems S From Table 6., we observe ha CLS performs very well and almos finds he opimal soluion for all problems. However, for hose problems in which CLS fails o reach he opimal soluion, all soluions are very close o opimaliy. The resuls also demonsrae ha CLS is capable of producing good qualiy soluions in lile ime. 89

201 Prob. Se SA/TS TS GA TSH ASH CLS gap gap gap gap gap gap gap gap gap gap gap gap SA/TS: Osman (995), simulaed annealing + abu search TS: Osman (995), abu search GA: Chu and Beasley (996), GA wih heurisic operaor TSH: Diaz and Fernandez (200), abu search heurisic ASH: Lourenco and Serra (2002), adapive search heurisics, an + descenden local search + abu wih resriced ejecion chain Table 6.2: Average percenage deviaion from opimal soluion for S Table 6.2 shows he resuls compared wih some exising mehods in erms of he average percenage deviaion σ from opimal value for each problem. I can be seen ha he deviaion from opimal values obained by CLS are as good as or close o he values obained by he compared mehods. In cases when CLS is ouperformed by oher mehods, he gap is small, i.e. σ is less han %. In addiion, we noe ha for CLS he compuaional imes are very good, even hough i is implemened using a relaively less powerful programming language (Visual Basic: VB). The principal reason for using VB is ha, alhough i is less powerful han several oher 90

202 programming languages, e.g. Forran, C++, VB is sill a full language and offers much shorer developmen imes. This was expeced o be significan as many algorihmic varians were o be invesigaed. Nex, we perform experimens on large-sized minimisaion problems S 2. We compare our resuls o he bes known soluion (φ) for hese problems. For some problems in his se, he opimal soluion values were obained by a branch-and-bound algorihm proposed by Nauss (2003); hese are marked wih an aserisk. Table 6.3 shows he bes values (Bes), and he average percenage deviaion (σ) from he bes values obained from differen soluion mehods. The resuls show ha our mehod performs well paricularly in problem ypes A and B, in which he soluion values are close o he bes known soluion values. For problem ypes C and D, CLS is ouperformed by he exising mehods in general. However, he gap of he soluion values is no oo high and compuaional ime is very good. 9

203 Prob m n φ GA TSEC TSH HGA CLS Bes σ Bes σ Bes σ Bes SD Bes σ Time A * * * * * * B C * * * * * D * GA: Chu and Beasley (996), GA wih heurisic operaor EC: Yagiura e al (2004), abu search wih ejecion chain TSH: Diaz and Fernandez (200), abu search heurisic HGA: Fell and Raidl (2004), improved hybrid GA, SD is he sandard deviaion of he opimal value of he LP-relaxaion afer he CPLEX solver was erminaed due o he running ime or memory limis. Table 6.3: Resuls for minimisaion problems S 2 92

204 The resuls from he wo daa ses indicae ha CLS is a promising approach for GAP. We are able o obain soluions of good qualiy ha are as good as or close o he bes known soluions in lile compuaional ime. Alhough he exising soluion approaches ouperform CLS in erms of he opimal soluion values, we believe ha he simpliciy and compuaional advanage of our approach is a pay-off for he solving algorihm. Since CLS akes lile compuaional ime even wih a programming language, which is relaively slow, we are keen o run i for longer and expec o see beer resuls. We se he sopping crierion parameer o 0000 for problems S and for problems S 2. For each of he es problems 0 runs are performed using differen random number seeds a he beginning of each run and he remaining parameers in he algorihm are fixed. The resuls are shown in Table

205 Problem m n Opimal Bes σ Avg. Time (s) gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap

206 Problem m n Opimal Bes σ Avg. Time (s) gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap gap Table 6.4: Resuls for S - sopping crierion parameer = 0000 Table 6.4 shows ha CLS can find opimal soluions for all es problems excep wo cases, gap-3 and gap2-3. The resuls in erms of bes soluions found are only marginally beer han hose in Table 6. despie a en-fold increase in he sopping crierion parameer which naurally carries addiional compuaional cos. However, from Table 6.4, we obain much beer average percenage deviaion from opimal values han in Table 6.. These are as good as or very close o he values obained by he exising mehods. 95

207 Prob m n φ GA TSEC TSH HGA CLS Bes σ Bes σ Bes σ Bes SD Bes σ A * * * * * * B C * * * * * D * Table 6.5: Resuls for S 2 - sopping crierion parameer = Table 6.5 shows ha we obain beer resuls in erms of he bes soluions found and he average percenage deviaion from opimal values han in Table 6.3. For problem ypes A, B and C, he resuls are as good as or close o he resuls obained by he exising mehods. For problem ype D, he bes soluions found by CLS (and also GA and TSH) are sill relaively far from he bes known soluions φ, bu he average percenage deviaion has decreased. 96

208 We noe ha he sopping crierion parameer may no be of criical imporance o he performance of our algorihm. Since his parameer mainly serves he purpose of search diversificaion, when he search is sufficienly diversified wihin a high-enough number of ieraions, soluions may no be improved furher by diversificaion and good inensificaion is required o improve he soluions from his poin. 6.6 Conclusions The applicaion of consrain-based local search incorporaing wih he predicive choice model (CLS) o GAP is presened in his Chaper. The performance of he algorihm is evaluaed wih wo differen benchmark problem ses, and compared wih oher exising soluion mehods. The resuls obained wih CLS are very promising. In general we obain high qualiy soluions ha are as good as, or close o, he soluions obained from he exising mehods. CLS employs a random sraegy o achieve a diversified exploraion of he search space and incorporaes a self-learning feaure ha learns from he search hisory and implicily exracs problem knowledge. During he run, he search hisory is analysed and CLS predics good assignmens of jobs o agens. These assignmens will be fixed in a probabilisic manner, leading o inensified exploraion of he search space. I is he firs ime ha CLS has been applied o he soluion of GAP. Alhough CLS canno ouperform he exising mehods especially in erms of he soluion values, he soluions obained by CLS are as good as or very close o he soluions from he exising mehods. We believe ha he simpliciy and compuaional advanage of our mehod is a pay-off for he solving algorihm. The soluions for GAP ailored by CLS may be improved if CLS is coded in a more powerful programming language e.g. C or Forran, or using more sophisicaed 97

209 heurisics o explore complex local moves, i.e. mainaining a promising sored candidae lis during he search or incorporaing dynamic bound sraegies derived from LP soluions as used in mos exising mehods for GAP. Our proposed mehod may also be promising for oher assignmen ype problems which are compuaionally more demanding han GAP, such as he quadraic assignmen problem (Rardin, 998), he mulilevel generalised assignmen problem (Laguna e al, 995) and he blockmodel problem (Jessop, 2003). This is because he predicive choice model would be able o capure he complex ineracions amongs he variables in he model and o predic he movemens of he soluion in he search space. 98

210 Chaper Seven Conclusions 7. Summary In his research, he conainer rail scheduling problem has been presened and an opimisaion framework for is soluion has been proposed. The conainer rail scheduling problem is modelled as a consrain saisfacion problem in which a demand responsive scheduling service is considered in order o improve he service offered o cusomers and o reduce operaing coss for he rail carrier. A consrain-based local search algorihm is developed and applied o he conainer rail scheduling problem. The algorihm 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 hen used o evaluae local moves. Differen measures of he violaion are also used o drive he search o he promising regions of he search space. 99

211 In addiion, he consrain-based local search algorihm incorporaes a predicive choice model. The resuls using real-life daa ses show some reducions in oal operaing coss, and enhance he level of service hrough demand responsive schedules. The research also demonsraes he applicaion of he proposed algorihmic approach o he generalised assignmen problem. The consrain-based local search algorihm mainly employs a randomised sraegy o achieve a diversified exploraion of he search space and he predicive choice model predics good assignmen of jobs o agens. The performance of he algorihm has been assessed wih benchmark problem ses. 7.2 Achievemens of his research Generaing a profiable schedule is crucial o a conainer rail business because rail s profiabiliy is influenced by is service offerings. Opimisaion models o improve is operaions and advanced soluion echniques are significan. In his research, he major conribuions are divided ino wo areas as follows: From he applicaion poin of view, he demand responsive scheduling model incorporaes he following new feaures ha are more complex and no considered in previous work.. Non-uniform arrivals wih disinc arge imes, i.e. no all conainers are available a he beginning of he scheduling ime horizon and mus be reaed as disinc cusomer bookings. 2. A demand responsive conainer rail service providing flexible schedules 200

212 3. A probabilisic decrease in cusomer saisfacion wih deviaion from arge ime From he compuaional viewpoin, he proposed mehod conribues o he scheduling research in he following aspecs:. Local search for consrain saisfacion problems has been invesigaed. A consrain-based local search algorihm is developed o solve combinaorial opimisaion problems. The consrain-based local search plays a key role in diversifying he search. I incorporaes he predicive choice model for search inensificaion. Good inerplay beween he diversificaion and inensificaion sraegies is he main feaure of he proposed algorihm. 2. A novel learning mechanism, he predicive choice model, has been developed. We propose a heoreical discree choice learning model and hen make i applicable o combinaorial opimisaion problems, he conainer rail scheduling problem and he generalised assignmen problem. Alhough he predicive choice model needs o mainain and analyse he search hisory and hence imposes a compuaional cos, his is offse agains a lower run-ime required o find good soluions. Compuaional resuls demonsrae he robusness and usefulness of he proposed echnique. 3. The predicive choice model is no dependen on domain-specific knowledge; i does no depend on he naure of he objecive funcion or consrains, wheher linear or non-linear. This suggess ha he proposed algorihm would be applicable 20

213 o oher combinaorial opimisaion problems in which all variables in he model are binary. 7.3 Fuure work The following poins provide some of he issues ha could be invesigaed as he fuure work of his research.. Wih he demand responsive schedule, here migh be some cusomers ha book he rail service close o he end of a schedulable week. This may cause he rain schedule o be no profiable because he proposed model always insiss on saisfying all cusomer demand wihin he week. In his case, he auomaic scheduling sysem may be required o re-consolidae he cusomer shipmens and o compare he effec if hose cusomers are dropped and considered in he following week. The modes compuaional demands of he algorihm described in Chapers 4 and 5 make his approach viable. This needs more invesigaion on railway pracices and survey daa. 2. Wih respec o he algorihm proposed, here are always possible ways o improve he performance of he algorihm. For example, some parameers in he algorihm are deermined empirically, e.g. he feasibiliy parameer α ha balances he rade-off beween hard and sof violaions. They migh be uned effecively by adapive mechanisms based on saisical mehods. 3. The predicive choice model may be exended o muliple value choice decisions, i.e. he predicive model may be based on mulinomial discree choice heory ha can predic a good value for an ineger variable. However, in general discree 202

214 opimisaion problems, he variable domain can be large. Therefore, predicing every single value for a variable would be compuaionally expensive and is no reasonable. In his case he variable domain may be pariioned ino muliple groups, and he predicive choice model used o srenghen he variable domain. 4. The inensificaion sraegy used by he predicive choice model is soundly based on a saisical mehod; he consisency beween variables is enforced in a probabilisic way, leading o inensified exploraion of he search space. However, he diversificaion for he consrain-based local search is achieved by he randomised selecion of variables o explore and by no insising on complee consisency a some sages of he search. Alhough random selecion and variable selecion schemes are used, how well he diversificaion is achieved is kep in mind implicily. In addiion, i has o exploi domain knowledge o consruc effecive variable selecion schemes. Therefore, a variable selecion sraegy based on some saisical mehods may be incorporaed. For insance, using he cluser sampling echnique, he enire se of variables could be divided ino clusers and a random sample of hese clusers is seleced. This would make he consrain-based local search a more general solving mehod and a quanified measure of diversificaion can be done. The number of clusers o be used or he size of clusers is a key facor for he performance of diversificaion. 5. The consrain-based local search incorporaing he predicive choice model has been applied o GAP. The resuls demonsrae ha he mehod performs well and can obain high qualiy soluions. Therefore, he mehod may be promising for oher combinaorial opimisaion problems, paricularly hose for which i is possible o 203

215 mainain he consisency of a subse of he consrains hroughou he search. The mehod is simple and convenien o use; i does no depend on he naure of he objecive funcion or consrains, wheher linear or non-linear. As non-linear opimisaion problems are generally more compuaionally inensive han linear ones, i may be fruiful o apply our CLS approach in his area, e.g. he blockmodel problem (Jessop, 2003) which has assignmen-ype consrains bu for which he remainder of he consrains and he objecive funcion ake a quadraic form. The principal difficuly in solving his problem is ha is coninuous relaxaion has a nonconvex feasible region. Alhough his problem can be ransformed ino an ineger linear programme, is size expands rapidly and i has been shown o be difficul o solve (Proll, 2004). The predicive choice model may be applied as he uiliy funcion of having a cerain value for a variable is no affeced by he naure of he consrains and he objecive funcion. This would allow he original size of he problem o be reained. In addiion, i may be promising for problems when he consrains in he model keep changing over ime, e.g. dynamic vehicle rouing problems where vehicles on he preroue assignmen encouner unexpeced obsrucion and canno visi heir designaed cusomers on ime, and herefore vehicles have o be dynamically re-roued. Our CLS approach allows addiional consrains and can handle consrains locally. To make use of he predicive choice model, he search hisory may be defined ino wo pars: he firs par describes he global informaion guiding he search for qualiy soluions and he second is he local-updae informaion rying o recover he violaed consrains. 204

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227 Appendix A Toal shipping cos for conainerised cargoes as a percenage of commodiy marke prices. In all ables, C is he difference beween oal shipping cos of ruck and rail, i.e. C = CT - C R, Z is a sandardised score of of C. C, and CDF is a cumulaive probabiliy densiy funcion Cusomer Modal cos (%) Sor no. Truck, C T Rail, C C No. C Z CDF R µ =.72 σ =0.4 Table A.: Toal shipping cos for cargo ype I 26

228 Cusomer Modal cos (%) Sor no. Truck, C T Rail, C C No. C Z CDF R

229 Cusomer Modal cos (%) Sor no. Truck, C T Rail, C C No. C Z CDF R µ = 4.82 σ = 0.38 Table A.2: Toal shipping cos for cargo ype II Cusomer Modal cos (%) Sor no. Truck, C T Rail, C C No. C Z CDF R

230 Cusomer Modal cos (%) Sor no. Truck, C T Rail, C C No. C Z CDF R

231 Cusomer Modal cos (%) Sor no. Truck, C T Rail, C C No. C Z CDF R µ = 2.3 σ = 0.33 Table A.3: Toal shipping cos for cargo ype III Cusomer Modal cos (%) Sor no. Truck, C T Rail, C C No. C Z CDF R

232 Cusomer Modal cos (%) Sor no. Truck, C T Rail, C C No. C Z CDF R µ = 5.3 σ = 0.37 Table A.4: Toal shipping cos for cargo ype IV 22

233 Appendix B Tables B. B.5 show he accuracy of he predicive choice model. The same variables are esed in each able wih differen predicion no. Var Violaion * ℵ Probabiliy Φ no. h0 h 0 P 0 P PC = 85% Table B.: Predicion no. 222

234 Var Violaion * ℵ Probabiliy Φ no. h0 h 0 P 0 P PC = 80% Table B.2: Predicion no

235 Var Violaion * ℵ Probabiliy Φ no. h0 h 0 P 0 P PC = 80% Table B.3: Predicion no

236 Var Violaion * ℵ Probabiliy Φ no. h0 h 0 P 0 P PC = 70% Table B.4: Predicion no

237 Var Violaion * ℵ Probabiliy Φ no. h0 h 0 P 0 P PC = 80% Table B.5: Predicion no