Current trends in Current trends in deterministic schedulingdeterministic scheduling by by

Chung-Yee Lee, Chung-Yee Lee, Lei LeiLei Lei, , Michael PinedoMichael Pinedo

Emrah ZarifoğluEmrah Zarifoğlu

9702173097021730

Deterministic SchedulingDeterministic Scheduling

• Set of jobs Set of jobs

• Set of machinesSet of machines

• Certain performance measuresCertain performance measures

NotationNotation

• αα││ββ││γγ– ααmachine configurationmachine configuration– β β processing restrictions and processing restrictions and

constraintsconstraints– γ γ performance mesure to be optimizedperformance mesure to be optimized

NotationNotation

• JJjj : job : job jj, , j j = 1,…, = 1,…, nn

• MMjj : machine : machine ii, , i i = 1,…, = 1,…, mm

• CCjj : the completion time for : the completion time for J J j j

• wwjj : the weight for : the weight for JJjj

• ddjj : the due date of : the due date of JJjj

• LLjj : the lateness of : the lateness of JJjj = = CCjj – – ddjj

• LL max max : maximum lateness = max{: maximum lateness = max{LLjj , , j j = 1,…, = 1,…, nn}}

• C C maxmax : makespan = max{: makespan = max{CCjj : : j j = 1,…, = 1,…, nn}}

ComplexityComplexity

• Polynomial time algorithmPolynomial time algorithmA known A known algorithm algorithm that is guaranteed to terminate within a number that is guaranteed to terminate within a number of steps which is a of steps which is a polynomialpolynomial function of the size function of the size of the problem of the problem

• NPNPnon-deterministic polynomial time, non-deterministic polynomial time, A set or A set or property of computational decision problems property of computational decision problems solvable by a nonsolvable by a non--deterministic Turing Machine in deterministic Turing Machine in a number of steps that is a polynomial function of a number of steps that is a polynomial function of the size of the input the size of the input

• NP-hardNP-hard if solving if solving a problem a problem in in polynomial timepolynomial time would make it possible to solve all problems in would make it possible to solve all problems in class class NPNP in polynomial time in polynomial time

Recent Developments in Recent Developments in Scheduling TheoryScheduling Theory

• New trend: extending classical algorithms to New trend: extending classical algorithms to models related to real problemsmodels related to real problems

• Two popular areasTwo popular areas– Scheduling with a 1-job-on-r-machine patternScheduling with a 1-job-on-r-machine pattern

• Jobs processed simultaneously on several machines (r Jobs processed simultaneously on several machines (r positive integer)positive integer)

• Several jobs processed by a single processor (0<rSeveral jobs processed by a single processor (0<r≤1)≤1)

– Machine scheduling with availability constraintsMachine scheduling with availability constraints• Possibility of machine unavailability due to maintenancePossibility of machine unavailability due to maintenance

• Machine availability in giventime windowsMachine availability in giventime windows

Scheduling with 1-job-on-r-Scheduling with 1-job-on-r-machine patternmachine pattern

• r positive integerr positive integer– Diagnosable microprocessor systemsDiagnosable microprocessor systems– semiconductor circuit design team semiconductor circuit design team

workforce planningworkforce planning– Berth allocation (one vessel for several Berth allocation (one vessel for several

berths)berths)

• 0<r0<r≤1≤1– Berth allocation (several vessels share one Berth allocation (several vessels share one

berth)berth)

1-job-on-r-machine (r positive 1-job-on-r-machine (r positive integer)integer)• multiprocessor task systemmultiprocessor task system• Nonfix-fixed number of machines working Nonfix-fixed number of machines working

simultaneously but machines required are not simultaneously but machines required are not specifiedspecified– Example: PmExample: Pm││nonfixnonfix││CC max max an an mm-parallel-machine -parallel-machine

scheduling problem where each job can be processed scheduling problem where each job can be processed simultaneouslyby a fixed number of machines with the simultaneouslyby a fixed number of machines with the objective minimizing the makespan.objective minimizing the makespan.

• Fix-fixation of the set of machines for particular jobsFix-fixation of the set of machines for particular jobs– PP22││fixfix││ΣΣwwj j CCjj denotes a 2-parallel-machine scheduling denotes a 2-parallel-machine scheduling

problem where each job can be processed simultaneously problem where each job can be processed simultaneously by a specific set of machines, and the objective is to by a specific set of machines, and the objective is to minimize the total weighted completion time.minimize the total weighted completion time.

The machine set not fixedThe machine set not fixed

• PmPm│nonfix││nonfix│C C max max

– PmPm│prmp,nonfix││prmp,nonfix│C C max max (r=1,2)(r=1,2) polynomial algorithms polynomial algorithms (Blazewicz et al. 1984)(Blazewicz et al. 1984)

– PmPm│nonfix,p│nonfix,pjj=1│=1│C C max max (m(m≥≥rr≥2)≥2) linear integer linear integer programming or dynamic programming polynomial in n programming or dynamic programming polynomial in n (Blazewicz et al. 1986)(Blazewicz et al. 1986)

– PmPm│nonfix││nonfix│C C max max (m=2,3,4; (m=2,3,4; ppjj is a function of is a function of nonincreasing function of number of mehines used and nonincreasing function of number of mehines used and determined before)determined before) NP-hard (Du and Leung 1989) NP-hard (Du and Leung 1989)

– PmPm│nonfix││nonfix│C C maxmax (r=1,2; same speed processors) (r=1,2; same speed processors) O(nm O(nm + n logn) (Blazewicz et al. 1990)+ n logn) (Blazewicz et al. 1990)

– PmPm│nonfix││nonfix│C C maxmax (r=1,k; same speed processors) (r=1,k; same speed processors) O(nm O(nm + n logn) (Blazewicz et al. 1990)+ n logn) (Blazewicz et al. 1990)

The machine set not fixed The machine set not fixed (cont’d)(cont’d)• PmPm│nonfix│ │nonfix│ ΣΣwwj j CCjj

– (m=2)(m=2) NP-hard (Lee and Cai 1996) NP-hard (Lee and Cai 1996)– P2P2│nonfix│ │nonfix│ ΣΣCCj j dynamic programming NP-hard dynamic programming NP-hard

O(nPO(nP3s+13s+1) () (Lee and Cai 1996)Lee and Cai 1996)– P2P2│nonfix│ │nonfix│ ΣΣCCj j (p(pjj=p)=p) O(nlogn) ( O(nlogn) (Lee and Cai Lee and Cai

1996)1996)• PmPm│nonfix││nonfix│L L maxmax

– PmPm│nonfix,r│nonfix,rjj,d,djj,prmp│,prmp│L L max max -> linear programmşng -> linear programmşng to check feasibility (Plehn 1990)to check feasibility (Plehn 1990)

– P2P2│nonfix││nonfix│L L maxmax (EDD rule) (EDD rule) dynamic dynamic programming NP-hard O(nPprogramming NP-hard O(nP3s+13s+1logP) (Plehn 1990)logP) (Plehn 1990)

– P2P2│nonfix, │nonfix, ppjj=1=1 │ │L L maxmax O(nlogn) (Plehn 1990) O(nlogn) (Plehn 1990)

The machine set fixedThe machine set fixed

• PmPm│fix││fix│C C max max

– Branch and bound algorithm (Bozoki and Richard 1970)Branch and bound algorithm (Bozoki and Richard 1970)– PP│fix, │fix, ppjj=1=1 │ │C C maxmax Np-hard (Krawczyk and Kubale 1985) Np-hard (Krawczyk and Kubale 1985)– PmPm│fix││fix│C C max max (only 2-machine jobs)(only 2-machine jobs) NP-hard (Kubale NP-hard (Kubale

1987)1987)– P3P3│fix││fix│C C maxmax NP-hard (Blazewicz et al. 1992) NP-hard (Blazewicz et al. 1992)– P3P3│fix││fix│C C max max (block-constraints)(block-constraints) pseudopolynomial O(nP) pseudopolynomial O(nP)

(Hoogeveen et al. 1994)(Hoogeveen et al. 1994)– PmPm│fix, │fix, ppjj=1=1 │ │C C max max polynomial (Hoogeveen et al. 1994) polynomial (Hoogeveen et al. 1994)– P2P2│fix, r│fix, rjj││C C max max NP-hard (Hoogeveen et al. 1994) NP-hard (Hoogeveen et al. 1994)– PmPm│fix││fix│C C max max (precedence constraints)(precedence constraints) branch and bound branch and bound

algorithms (Krämer 1995)algorithms (Krämer 1995)

The machine set fixed The machine set fixed (cont’d)(cont’d)• PmPm│fix│ │fix│ ΣΣwwj j CCjj

– PmPm│fix│ │fix│ ΣΣwwj j CCjj integer programming (Dobson integer programming (Dobson and KArmarkar 1989)and KArmarkar 1989)

– P2P2│fix│ │fix│ ΣΣCCjj NP-hard (Hoogeveen et al. 1994) NP-hard (Hoogeveen et al. 1994)

– PP│fix, │fix, ppjj=1=1 │ │ ΣΣCCjj NP-hard (Hoogeveen et al. NP-hard (Hoogeveen et al. 1994)1994)

– P2P2│prmp, fix│ │prmp, fix│ ΣΣCCjj O(nlogn) (Cai et al. 1996) O(nlogn) (Cai et al. 1996)

– PmPm│fix, │fix, ppjj=1=1 │ │ ΣΣCCjj polynomial (Brucker 1995 polynomial (Brucker 1995

Machine Scheduling with Machine Scheduling with Availability ConstraintsAvailability Constraints• Mostly assumed available machines but may not Mostly assumed available machines but may not

be true (e.g., machine breakdown-stochastic, be true (e.g., machine breakdown-stochastic, preventive maintenance-deterministic)preventive maintenance-deterministic)

• Assume machine i unavailable from sAssume machine i unavailable from sikik until t until tikik (0 (0≤ ≤ ssikik ≤ ≤ ttikik, 0≤k≤n, 0≤k≤nii))

• ““unavailability constraints”↔”machines are unavailability constraints”↔”machines are available in time windows”available in time windows”

• Resumable (r-a)Resumable (r-a)If a job cannot be finished If a job cannot be finished before the next down period of a machine and the before the next down period of a machine and the job can continue after the machine has become job can continue after the machine has become available againavailable again

• Nonresumable (nr-a)Nonresumable (nr-a) if the job has to restart if the job has to restart rather than continuerather than continue

Machine Scheduling with Machine Scheduling with Availability Constraints Availability Constraints (Cont’d)(Cont’d)• PmPm│prmp│ │prmp│ feasibility (different availability intervals) feasibility (different availability intervals)

O(nlogm) (Schmidt 1984)O(nlogm) (Schmidt 1984)• 1│nr-a│1│nr-a│ΣΣCCj j (one unavailability period)(one unavailability period) NP-hard (Adiri et al. NP-hard (Adiri et al.

1989)1989)• P││P││CCmax max (at most one unavailability period that is at the (at most one unavailability period that is at the

beginning) beginning) classical LPT by tight error bound ½, modified classical LPT by tight error bound ½, modified LPT by bound 1/3 (Lee at al. 1991)LPT by bound 1/3 (Lee at al. 1991)

• P││P││ΣΣCCj j (at most one unavailability period that is at the (at most one unavailability period that is at the beginning) beginning) SPT algorithm (Kaspi and Montreuil 1988, SPT algorithm (Kaspi and Montreuil 1988, Liman 1991)Liman 1991)

• P2││P2││ΣΣCCj j (one machine always available, other available (one machine always available, other available from time zero to a fixed oint)from time zero to a fixed oint) NP-hard dynamic NP-hard dynamic programming (Lee and Liman 1993)programming (Lee and Liman 1993)

Machine Scheduling with Machine Scheduling with Availability Constraints Availability Constraints (Cont’d)(Cont’d)• Pm││Pm││ΣΣCCj j (machine i available in a time window)(machine i available in a time window)

SPT (Mosheiov 1994)SPT (Mosheiov 1994)• F2│r-a│F2│r-a│CCmax max (at least one machine always (at least one machine always

available)available) NP-hard pseudopolynomial dynamic NP-hard pseudopolynomial dynamic programming algorithm (Lee 1996b) programming algorithm (Lee 1996b)

• F2│nr-a│F2│nr-a│CCmax max (at least one machine always (at least one machine always available)available) NP-hard pseudopolynomial dynamic NP-hard pseudopolynomial dynamic programming algorithm (Lee 1996b)programming algorithm (Lee 1996b)

• 1│r-a│1│r-a│ΣΣCCj j SPT (Lee 1996a) SPT (Lee 1996a)• 1│r-a│L1│r-a│Lmaxmax EDD (Lee 1996a) EDD (Lee 1996a)• 1│nr-a│C1│nr-a│Cmaxmax NP-hard (Lee 1996a) NP-hard (Lee 1996a)• Pm│prmp, r-a│CPm│prmp, r-a│Cmaxmax O(n + m logm) (Schmidt 1984) O(n + m logm) (Schmidt 1984)

Recent developments in search Recent developments in search algorithmsalgorithms• Complexity Complexity not easy to formulate as mathematical not easy to formulate as mathematical

programmingprogramming• Classical techniquesClassical techniquesoften not solvable in real timeoften not solvable in real time• By improvement of computing technologyBy improvement of computing technology

– Neighbourhood search techniqueNeighbourhood search technique• Local improvement with minor changesLocal improvement with minor changes• Simulated annealing, tabu search, genetic algorithmsSimulated annealing, tabu search, genetic algorithms

– Constraint-guided heuristic search techniqueConstraint-guided heuristic search technique• Not to find optimal but to find good feasible schedulesNot to find optimal but to find good feasible schedules• Formulationbased on list of rules and consraintsFormulationbased on list of rules and consraints• Focus on partial solutions and extending them to a complete Focus on partial solutions and extending them to a complete

feasible solutionfeasible solution• Based on measurements of flexibility and constraining factorsBased on measurements of flexibility and constraining factors• Expert systems-scheduling systems based on this search techniqueExpert systems-scheduling systems based on this search technique

General concepts in General concepts in neighboorhood neighboorhood searchtechniquessearchtechniques• The mapping of the data in a format suitable for the The mapping of the data in a format suitable for the

algorithmalgorithm– the description of a schedule has to be both concisethe description of a schedule has to be both concise and and

unambiguousunambiguous• The neighbourhood designThe neighbourhood design

– The knowledge required centers mainly on those aspects of The knowledge required centers mainly on those aspects of the schedule thatthe schedule that have the greatest impact on the objectivehave the greatest impact on the objective

• The search process within the neighbourhoodThe search process within the neighbourhood– Given all the schedules in the neighbourhood, a search has Given all the schedules in the neighbourhood, a search has

to be conducted thatto be conducted that leads to the next schedule in the leads to the next schedule in the search processsearch process

• The acceptance-rejection criterionThe acceptance-rejection criterion– Whenever a schedule within the neighbourhood is selected, Whenever a schedule within the neighbourhood is selected,

a decisiona decision has to be made whether or not to accept the has to be made whether or not to accept the scheduleschedule

Simulated Annealing and Tabu Simulated Annealing and Tabu SearchSearch• Very similarVery similar• Difference in acceptance-rejection criteriaDifference in acceptance-rejection criteria

– Simulated annealing-probabilistic processSimulated annealing-probabilistic process– Tabu search-deterministic processTabu search-deterministic process

• Simulated annealingSimulated annealing job shop scheduling job shop scheduling problems with the makespan objectiveproblems with the makespan objective

• Tabu searchTabu search single machine, parallel machine, single machine, parallel machine, flow shop, flexible flow shop and job shop flow shop, flexible flow shop and job shop problems with objectives that include the problems with objectives that include the makespan, the total weighted completion time, as makespan, the total weighted completion time, as well as the total weighted tardinesswell as the total weighted tardiness

Simulated Annealing and Tabu Simulated Annealing and Tabu Search (Cont’d)Search (Cont’d)• Jm││Jm││CCmax max simulated annealing (Matsuo et al. 1987) simulated annealing (Matsuo et al. 1987)• 11││ ││ ΣΣwwj j CCj j tabu search (Potts and Van Wassenhove 1997) tabu search (Potts and Van Wassenhove 1997)• 11│ │ ssjkjk│ │ ΣΣwwj j CCj j tabu search (Laguna et al. 1991, 1993) tabu search (Laguna et al. 1991, 1993)• 11│ │ ssjkjk│ │ ΣΣTTj j tabu search (Laguna et al. 1991, 1993) tabu search (Laguna et al. 1991, 1993)• Pm││ Pm││ ΣΣwwj j CCj j tabu searcH (Barnes and Laguna 1992, tabu searcH (Barnes and Laguna 1992,

Barnes et al. 1995)Barnes et al. 1995)• Flow shop problem Flow shop problem tabu search (Adenso-Dias 1992, tabu search (Adenso-Dias 1992,

Nowicki and Smutnicki 1994)Nowicki and Smutnicki 1994)• Jm││Jm││CCmax max tabu search (Dell’Amico and Trubian 1993, tabu search (Dell’Amico and Trubian 1993,

Nowicki and Smutnicki 1993, Taillard 1994, Nowicki and Smutnicki 1993, Taillard 1994, Dauzère-PérèsDauzère-Pérès and Paulli 1997)and Paulli 1997)

• Job shop problem with multiple-purpose machines Job shop problem with multiple-purpose machines tabu tabu search (search (Hurink, Jurisch and TholeHurink, Jurisch and Thole 1994) 1994)

Genetic AlgorithmsGenetic Algorithms

• Mimics natural evolutionary processMimics natural evolutionary process• Population, generation, mutation, crossover, Population, generation, mutation, crossover,

fitness, reproduction, chromosomefitness, reproduction, chromosome• Jm││Jm││CCmax max genetic algorithms genetic algorithms (Lawton 1992, (Lawton 1992,

Della CroceDella Croce et al. et al. 1992,1992, BeanBean 1994,1994, BierwirthBierwirth 1995), Herrmann1995), Herrmann et al. et al. 1995)1995)

• Job shop with machine learningJob shop with machine learning genetic genetic algorithms (Lee et al. 1995)algorithms (Lee et al. 1995)

• Real life applications Real life applications genetic algorithms (Bean genetic algorithms (Bean 1994, Kettani and Jobin 1995, Herrmann et al. 1994, Kettani and Jobin 1995, Herrmann et al. 1995)1995)

Constraint-guided Heuristic Constraint-guided Heuristic SearchSearch• popularization of artificial intelligence techniquespopularization of artificial intelligence techniques and and

languages (e.g., PROLOG)languages (e.g., PROLOG)• Focus on finding feasible schedules rather than optimal Focus on finding feasible schedules rather than optimal

onesones• The most severe constraints in the beginning, the least The most severe constraints in the beginning, the least

severe constraints for the final partsevere constraints for the final part• Sometimes necessary to break some constraintsSometimes necessary to break some constraints

– Soft constraints (constraint relaxation)Soft constraints (constraint relaxation)– Hard constraintsHard constraints

• List implied constraints as soon as possible (constraint List implied constraints as soon as possible (constraint propagation)propagation)

• Consistency checkingConsistency checking• Dealing with inconsistenciesDealing with inconsistencies conflict resolution conflict resolution

Recent Developments in Recent Developments in Scheduling PracticeScheduling Practice• Flexible-resource scheduling (Daniels and Mazzola 1993, Flexible-resource scheduling (Daniels and Mazzola 1993,

1994, Ozdamar and Ulusoy 1995, Daniels et al. 1996, 1997, 1994, Ozdamar and Ulusoy 1995, Daniels et al. 1996, 1997, Alidaee and Ahmadian 1997, Alidaee and Kochenberger Alidaee and Ahmadian 1997, Alidaee and Kochenberger 1997, Armstrong et al. 1997a, 1997b)1997, Armstrong et al. 1997a, 1997b)

• Scheduling variable-speed machines (Trick 1994)Scheduling variable-speed machines (Trick 1994)• Scheduling with finite capacity input and output buffers Scheduling with finite capacity input and output buffers

(Hall et al. 1993, 1994,1997, Nawijn and Bass 1994)(Hall et al. 1993, 1994,1997, Nawijn and Bass 1994)• Scheduling of machine and material handling operations Scheduling of machine and material handling operations

(Egbelu 1987, Matsuo, Shang and Sullivan (1991), Hall et al. (Egbelu 1987, Matsuo, Shang and Sullivan (1991), Hall et al. 1993, Lei et al. 1993, 1995, Blazewicz et al. 1994, Hall et al. 1993, Lei et al. 1993, 1995, Blazewicz et al. 1994, Hall et al. 1994, and Crama 1995)1994, and Crama 1995)

• İntegrating scheduling with batching and lot sizing İntegrating scheduling with batching and lot sizing (Potts (Potts and Van Wassenhove 1992)and Van Wassenhove 1992)

Machine scheduling with Machine scheduling with material handling operationsmaterial handling operations• ResourcesResourcesmachines and machines and

materialhandling transportersmaterialhandling transporters• Cost of material handling 80%Cost of material handling 80%• To reduce cost deal with the issues:To reduce cost deal with the issues:

– SequencingSequencing that specifies the order in which that specifies the order in which jobs are processed at machiningjobs are processed at machining centerscenters

– SchedulingScheduling that makes time-phased routing that makes time-phased routing and dispatching of transporters forand dispatching of transporters for job pick-up job pick-up and deliveryand delivery

– Facility layout and flowpath designFacility layout and flowpath design that that makes efficient operations possible.makes efficient operations possible.

Machine scheduling with Machine scheduling with material handling operations material handling operations (Cont’d)(Cont’d)• KKnumber of transporters in a systemnumber of transporters in a system• JJtotal number of job typestotal number of job types• nntotal number of jobs to be processedtotal number of jobs to be processed• nnmpsmpsthe number of jobs in a minimal part setthe number of jobs in a minimal part set• ΠΠminminthethe objective of minimizing the production objective of minimizing the production

cycle time of an MPS in a repetitive processcycle time of an MPS in a repetitive process• twtwa manufacturing environment where the a manufacturing environment where the

starting time of each material handlingstarting time of each material handling operation operation must be confined within a must be confined within a time windowtime window

• nnwtwt the constraint that jobs arethe constraint that jobs are not allowed to not allowed to wait in processwait in process

Machine scheduling with Machine scheduling with material handling operations material handling operations (Cont’d)(Cont’d)• The problem is to find a simultaneous feasible scheduleThe problem is to find a simultaneous feasible schedule for for

job sequencing and time-phased dispatching and routing of job sequencing and time-phased dispatching and routing of transporters so that atransporters so that a given objective is optimizedgiven objective is optimized

• Work divided into:Work divided into:– Robotic cell schedulingRobotic cell scheduling

• has the fewest constraints, and is also thehas the fewest constraints, and is also the one for which most one for which most analytical results are availableanalytical results are available

• identify the optimal job inputidentify the optimal job input sequence and the robot operation sequence and the robot operation sequence with respect to certain objective functionssequence with respect to certain objective functions

– Scheduling of Automated Guided Vehicles (AGVs)Scheduling of Automated Guided Vehicles (AGVs)• deals with an automated job shop withdeals with an automated job shop with non-zero buffers at non-zero buffers at

machining centers and multiple AGVs traveling on a sharedmachining centers and multiple AGVs traveling on a shared network.network.

– Cyclic scheduling of hoists subject to time-window constraintsCyclic scheduling of hoists subject to time-window constraints• deals withdeals with the scheduling of multiple hoists in a flexible flowshopthe scheduling of multiple hoists in a flexible flowshop• tw, nwt and collision-free constraintstw, nwt and collision-free constraints

The robotic cell scheduling The robotic cell scheduling problemproblem• The no-buffer caseThe no-buffer case

– F2(1)│J>1│F2(1)│J>1│ΠΠminmin polynomial (Sethi et al. 1992) polynomial (Sethi et al. 1992)– F2(1)│J>1│F2(1)│J>1│ΠΠminmin O(n O(n44

mpsmps) to optimize robot moves and ) to optimize robot moves and job sequence (Hall et al. 1996a)job sequence (Hall et al. 1996a)

– F2(1)│J>1│F2(1)│J>1│CCmaxmax Gilmore and Gomory algorithm O(n Gilmore and Gomory algorithm O(n33) ) (Kise et al. 1991)(Kise et al. 1991)

– F2(1)│J>1│F2(1)│J>1│CCmaxmax (transportation between machines job (transportation between machines job dependent)dependent) NP-hard (Ganesharajah et al. 1995) NP-hard (Ganesharajah et al. 1995)

• The finite buffer caseThe finite buffer case– F2(1)│J>1│F2(1)│J>1│CCmaxmax (fixed job inputsequence)(fixed job inputsequence) Np-hard Np-hard

branch-and-bound algorithm to determine the sequence branch-and-bound algorithm to determine the sequence of robot moves (King et al. 1993)of robot moves (King et al. 1993)

Scheduling of automated Scheduling of automated guided vehiclesguided vehicles• process of flexible manufacturingprocess of flexible manufacturing• circulate on a network of guidepathscirculate on a network of guidepaths

connecting machine centers, and transport connecting machine centers, and transport tools and jobs among the centerstools and jobs among the centers

• AGV flowpathsAGV flowpaths– UnidirectionalUnidirectional ((→→)) – BBi-directional i-directional ((↔↔))

• Netwok configurationsNetwok configurations– Single-loopSingle-loop– Multi-loopMulti-loop

Analytical approaches to AGV Analytical approaches to AGV schedulingscheduling• Unidirected flowpath caseUnidirected flowpath case

– Deadlines are fixedDeadlines are fixed single-loop network determined in single-loop network determined in O(nlogn) (Blazewicz et al. 1991)O(nlogn) (Blazewicz et al. 1991)

– Deadlines are fixed, collision-free routingDeadlines are fixed, collision-free routing two-loop two-loop network determined with dynamic programming network determined with dynamic programming (Blazewicz et al. 1994)(Blazewicz et al. 1994)

• Bi-directional flowpath caseBi-directional flowpath case– Send an AGV from a source location to a machine Send an AGV from a source location to a machine

centercenter Dijkstra’s algorithm polynomial O(K Dijkstra’s algorithm polynomial O(K44mm22) (Kim ) (Kim and Tanchoco 1991)and Tanchoco 1991)

– Column generation based heuristic approach Column generation based heuristic approach (Krishnamurthy et al. 1993)(Krishnamurthy et al. 1993)

– Two-AGV scheduling problem to minimize makespanTwo-AGV scheduling problem to minimize makespan dynamic programming (LAngevin et al. 1994)dynamic programming (LAngevin et al. 1994)

Heuristic rules for AGV and Heuristic rules for AGV and machine schedulingmachine scheduling• AGV dispatching rulesAGV dispatching rules

– Work center-initiatedWork center-initiated– Vehicle-initiatedVehicle-initiated

• Pull-based-vehicle selects a work center with the highest need for job Pull-based-vehicle selects a work center with the highest need for job replenishmentreplenishment

• Push based-Push based-vehicle first selects a job to move and then a work center tovehicle first selects a job to move and then a work center to which the job should be sent.which the job should be sent.

• Plan conflict-free vehicle routesPlan conflict-free vehicle routes(Taghaboni and Tanchoco 1988)(Taghaboni and Tanchoco 1988)• TestTesting ing various machine and AGVvarious machine and AGV scheduling rules against scheduling rules against

different scheduling criteria via simulation different scheduling criteria via simulation experimentsexperiments((Sabuncuoglu and Hommertzheim Sabuncuoglu and Hommertzheim 1989, 1989, 19921992bb))

• HHierarchical approach forierarchical approach for real-time on-line AGV scheduling real-time on-line AGV scheduling problemsproblems ( (Sabuncuoglu and Hommertzheim 1992Sabuncuoglu and Hommertzheim 1992aa))

• Artificial intelligence and expert systems techniques Artificial intelligence and expert systems techniques review review (Kusiak 1989)(Kusiak 1989)

• Approach based on a Hopfield neural network with simulated Approach based on a Hopfield neural network with simulated annealingannealing (Chung and Fischer 1995) (Chung and Fischer 1995)

The hoist scheduling The hoist scheduling problemproblem• Considered as a special class of Considered as a special class of

Jm(K)│J>1│Jm(K)│J>1│ΠΠminmin problems with problems with tw tw and and nwt nwt constraints.constraints.

• Interval processing timeInterval processing time a decision a decision variable selected from a given range as job variable selected from a given range as job processing timeprocessing time

• A common objective of hoist scheduling inA common objective of hoist scheduling in practicepractice to minimize the cycle time of a to minimize the cycle time of a repetitive process for producing a givenrepetitive process for producing a given MPSMPS

The hoist scheduling problem The hoist scheduling problem (Cont’d)(Cont’d)• Fm(K)│J=1, nwt, tw│Fm(K)│J=1, nwt, tw│ΠΠminmin

– Fm(1)│J=1, nwt, tw│Fm(1)│J=1, nwt, tw│ΠΠminmin NP-hard (Lei and Wang NP-hard (Lei and Wang 1989)1989)

– Mixed integer program (Philips and Unger 1976)Mixed integer program (Philips and Unger 1976)– Branch-and-bound procedure that solves a large Branch-and-bound procedure that solves a large

number of LP subproblems (Shapiro and Nuttle number of LP subproblems (Shapiro and Nuttle 1988)1988)

– Branch-and-bound procedure that solves Branch-and-bound procedure that solves relaxations of LPs (Armstrong et al. 1991, 1994)relaxations of LPs (Armstrong et al. 1991, 1994)

• Fm(K)│J>1, nwt, tw│Fm(K)│J>1, nwt, tw│ΠΠminmin– Heuristic dispatching rules and expert systemsHeuristic dispatching rules and expert systems– Expert systemsExpert systems (Yih 1990, Yih and Thesen 1991) (Yih 1990, Yih and Thesen 1991)

Some ConclusionsSome Conclusions

• Changing r Changing r from 1 to a positive integerfrom 1 to a positive integer increases the complexity of problemincreases the complexity of problem

• no relationship between the complexity of no relationship between the complexity of the the nonfix nonfix and the and the fixfix models.models.

• Most nonpreemptive problems are NP-hardMost nonpreemptive problems are NP-hard

• For preemptive problems, most polynomialFor preemptive problems, most polynomial algorithms based on linear integer algorithms based on linear integer programming techniquesprogramming techniques

Future Research DirectionsFuture Research Directions

• BBranch-and-bound techniques, dynamic programming, heuristicranch-and-bound techniques, dynamic programming, heuristic algorithms with an error bound analysisalgorithms with an error bound analysis

• The nonpre-emptiveThe nonpre-emptive case with different job release times and case with different job release times and different machine available timedifferent machine available time windowswindows

• semi-resumable case where somesemi-resumable case where some extra setup time may be required extra setup time may be required when a job restarts.when a job restarts.

• Extension of the existing modelsExtension of the existing models to more complicated job shop and to more complicated job shop and open shop problemsopen shop problems

• combining machine availability constraints with human resourcecombining machine availability constraints with human resource constraintsconstraints

• comparing neighbourhood search techniquescomparing neighbourhood search techniques with constraint-guided with constraint-guided heuristic search techniquesheuristic search techniques

• where the optimal home positionwhere the optimal home position of a transporter after a deliveryof a transporter after a delivery is is• How to coordinate the machine and material handlingHow to coordinate the machine and material handling operations to operations to

minimize the machine, transporter, and job waiting timeminimize the machine, transporter, and job waiting time• How to con-structHow to con-struct collision-free schedules when jobs arrive collision-free schedules when jobs arrive

dynamicallydynamically