Current trends in deterministic scheduling by ChungYee Lee, Lei Lei, Michael Pinedo

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Current trends in deterministic scheduling by
Chung-Yee Lee, Lei Lei, Michael Pinedo

• Emrah Zarifoglu
• 97021730

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Deterministic Scheduling

• Set of jobs
• Set of machines
• Certain performance measures

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Notation

• aß?
• a?machine configuration
• ß ?processing restrictions and constraints
• ? ?performance mesure to be optimized

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Notation

• Jj job j, j 1,, n
• Mj machine i, i 1,, m
• Cj the completion time for J j
• wj the weight for Jj
• dj the due date of Jj
• Lj the lateness of Jj Cj dj
• L max maximum lateness maxLj , j 1,, n
• C max makespan maxCj j 1,, n

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Complexity

• Polynomial time algorithm?A known algorithm that
  is guaranteed to terminate within a number of
  steps which is a polynomial function of the size
  of the problem
• NP?non-deterministic polynomial time, A set or
  property of computational decision problems
  solvable by a non-deterministic Turing Machine in
  a number of steps that is a polynomial function
  of the size of the input
• NP-hard? if solving a problem in polynomial time
  would make it possible to solve all problems in
  class NP in polynomial time

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Recent Developments in Scheduling Theory

• New trend extending classical algorithms to
  models related to real problems
• Two popular areas
• Scheduling with a 1-job-on-r-machine pattern
• Jobs processed simultaneously on several machines
  (r positive integer)
• Several jobs processed by a single processor
  (0ltr1)
• Machine scheduling with availability constraints
• Possibility of machine unavailability due to
  maintenance
• Machine availability in giventime windows

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

• r positive integer
• Diagnosable microprocessor systems
• semiconductor circuit design team workforce
  planning
• Berth allocation (one vessel for several berths)
• 0ltr1
• Berth allocation (several vessels share one berth)

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1-job-on-r-machine (r positive integer)

• multiprocessor task system
• Nonfix-fixed number of machines working
  simultaneously but machines required are not
  specified
• Example PmnonfixC max ?an m-parallel-machine
  scheduling problem where each job can be
  processed simultaneouslyby a fixed number of
  machines with the objective minimizing the
  makespan.
• Fix-fixation of the set of machines for
  particular jobs
• P2fixSwj Cj denotes a 2-parallel-machine
  scheduling problem where each job can be
  processed simultaneously by a specific set of
  machines, and the objective is to minimize the
  total weighted completion time.

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The machine set not fixed

• PmnonfixC max
• Pmprmp,nonfixC max (r1,2)? polynomial
  algorithms (Blazewicz et al. 1984)
• Pmnonfix,pj1C max (mr2)? linear integer
  programming or dynamic programming polynomial in
  n (Blazewicz et al. 1986)
• PmnonfixC max (m2,3,4 pj is a function of
  nonincreasing function of number of mehines used
  and determined before)? NP-hard (Du and Leung
  1989)
• PmnonfixC max (r1,2 same speed processors)?
  O(nm n logn) (Blazewicz et al. 1990)
• PmnonfixC max (r1,k same speed processors)?
  O(nm n logn) (Blazewicz et al. 1990)

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The machine set not fixed (contd)

• Pmnonfix Swj Cj
• (m2)? NP-hard (Lee and Cai 1996)
• P2nonfix SCj ? dynamic programming NP-hard
  O(nP3s1) (Lee and Cai 1996)
• P2nonfix SCj (pjp)? O(nlogn) (Lee and Cai
  1996)
• PmnonfixL max
• Pmnonfix,rj,dj,prmpL max -gt linear programmsng
  to check feasibility (Plehn 1990)
• P2nonfixL max (EDD rule)? dynamic programming
  NP-hard O(nP3s1logP) (Plehn 1990)
• P2nonfix, pj1 L max ? O(nlogn) (Plehn 1990)

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The machine set fixed

• PmfixC max
• Branch and bound algorithm (Bozoki and Richard
  1970)
• Pfix, pj1 C max ? Np-hard (Krawczyk and Kubale
  1985)
• PmfixC max (only 2-machine jobs)? NP-hard
  (Kubale 1987)
• P3fixC max ? NP-hard (Blazewicz et al. 1992)
• P3fixC max (block-constraints)?
  pseudopolynomial O(nP) (Hoogeveen et al. 1994)
• Pmfix, pj1 C max ? polynomial (Hoogeveen et
  al. 1994)
• P2fix, rjC max ? NP-hard (Hoogeveen et al.
  1994)
• PmfixC max (precedence constraints)? branch and
  bound algorithms (Krämer 1995)

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The machine set fixed (contd)

• Pmfix Swj Cj
• Pmfix Swj Cj ? integer programming (Dobson and
  KArmarkar 1989)
• P2fix SCj ? NP-hard (Hoogeveen et al. 1994)
• Pfix, pj1 SCj ? NP-hard (Hoogeveen et al.
  1994)
• P2prmp, fix SCj ? O(nlogn) (Cai et al. 1996)
• Pmfix, pj1 SCj ? polynomial (Brucker 1995

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Machine Scheduling with Availability Constraints

• Mostly assumed available machines but may not be
  true (e.g., machine breakdown-stochastic,
  preventive maintenance-deterministic)
• Assume machine i unavailable from sik until tik
  (0 sik tik, 0kni)
• unavailability constraints?machines are
  available in time windows
• Resumable (r-a)?If a job cannot be finished
  before the next down period of a machine and the
  job can continue after the machine has become
  available again
• Nonresumable (nr-a)? if the job has to restart
  rather than continue

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Machine Scheduling with Availability Constraints
(Contd)

• Pmprmp feasibility (different availability
  intervals) ? O(nlogm) (Schmidt 1984)
• 1nr-aSCj (one unavailability period)? NP-hard
  (Adiri et al. 1989)
• PCmax (at most one unavailability period that
  is at the beginning) ? classical LPT by tight
  error bound ½, modified LPT by bound 1/3 (Lee at
  al. 1991)
• PSCj (at most one unavailability period that is
  at the beginning) ?SPT algorithm (Kaspi and
  Montreuil 1988, Liman 1991)
• P2SCj (one machine always available, other
  available from time zero to a fixed oint)?
  NP-hard dynamic programming (Lee and Liman 1993)

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Machine Scheduling with Availability Constraints
(Contd)

• PmSCj (machine i available in a time window)?
  SPT (Mosheiov 1994)
• F2r-aCmax (at least one machine always
  available)? NP-hard pseudopolynomial dynamic
  programming algorithm (Lee 1996b)
• F2nr-aCmax (at least one machine always
  available)? NP-hard pseudopolynomial dynamic
  programming algorithm (Lee 1996b)
• 1r-aSCj ? SPT (Lee 1996a)
• 1r-aLmax ? EDD (Lee 1996a)
• 1nr-aCmax ? NP-hard (Lee 1996a)
• Pmprmp, r-aCmax ? O(n m logm) (Schmidt 1984)

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Recent developments in search algorithms

• Complexity ?not easy to formulate as mathematical
  programming
• Classical techniques?often not solvable in real
  time
• By improvement of computing technology
• Neighbourhood search technique
• Local improvement with minor changes
• Simulated annealing, tabu search, genetic
  algorithms
• Constraint-guided heuristic search technique
• Not to find optimal but to find good feasible
  schedules
• Formulationbased on list of rules and consraints
• Focus on partial solutions and extending them to
  a complete feasible solution
• Based on measurements of flexibility and
  constraining factors
• Expert systems-scheduling systems based on this
  search technique

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General concepts in neighboorhood searchtechniques

• The mapping of the data in a format suitable for
  the algorithm
• the description of a schedule has to be both
  concise and unambiguous
• The neighbourhood design
• The knowledge required centers mainly on those
  aspects of the schedule that have the greatest
  impact on the objective
• The search process within the neighbourhood
• Given all the schedules in the neighbourhood, a
  search has to be conducted that leads to the next
  schedule in the search process
• The acceptance-rejection criterion
• Whenever a schedule within the neighbourhood is
  selected, a decision has to be made whether or
  not to accept the schedule

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Simulated Annealing and Tabu Search

• Very similar
• Difference in acceptance-rejection criteria
• Simulated annealing-probabilistic process
• Tabu search-deterministic process
• Simulated annealing? job shop scheduling problems
  with the makespan objective
• Tabu search? single machine, parallel machine,
  flow shop, flexible flow shop and job shop
  problems with objectives that include the
  makespan, the total weighted completion time, as
  well as the total weighted tardiness

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Simulated Annealing and Tabu Search (Contd)

• JmCmax ? simulated annealing (Matsuo et al.
  1987)
• 1 Swj Cj ? tabu search (Potts and Van
  Wassenhove 1997)
• 1 sjk Swj Cj ? tabu search (Laguna et al. 1991,
  1993)
• 1 sjk STj ? tabu search (Laguna et al. 1991,
  1993)
• Pm Swj Cj ? tabu searcH (Barnes and Laguna
  1992, Barnes et al. 1995)
• Flow shop problem ? tabu search (Adenso-Dias
  1992, Nowicki and Smutnicki 1994)
• JmCmax ? tabu search (DellAmico and Trubian
  1993, Nowicki and Smutnicki 1993, Taillard 1994,
  Dauzère-Pérès and Paulli 1997)
• Job shop problem with multiple-purpose machines ?
  tabu search (Hurink, Jurisch and Thole 1994)

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Genetic Algorithms

• Mimics natural evolutionary process
• Population, generation, mutation, crossover,
  fitness, reproduction, chromosome
• JmCmax ?genetic algorithms (Lawton 1992, Della
  Croce et al. 1992, Bean 1994, Bierwirth 1995),
  Herrmann et al. 1995)
• Job shop with machine learning? genetic
  algorithms (Lee et al. 1995)
• Real life applications ? genetic algorithms (Bean
  1994, Kettani and Jobin 1995, Herrmann et al.
  1995)

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Constraint-guided Heuristic Search

• popularization of artificial intelligence
  techniques and languages (e.g., PROLOG)
• Focus on finding feasible schedules rather than
  optimal ones
• The most severe constraints in the beginning, the
  least severe constraints for the final part
• Sometimes necessary to break some constraints
• Soft constraints (constraint relaxation)
• Hard constraints
• List implied constraints as soon as possible
  (constraint propagation)
• Consistency checking
• Dealing with inconsistencies? conflict resolution

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Recent Developments in Scheduling Practice

• Flexible-resource scheduling (Daniels and Mazzola
  1993, 1994, Ozdamar and Ulusoy 1995, Daniels et
  al. 1996, 1997, Alidaee and Ahmadian 1997,
  Alidaee and Kochenberger 1997, Armstrong et al.
  1997a, 1997b)
• Scheduling variable-speed machines (Trick 1994)
• Scheduling with finite capacity input and output
  buffers (Hall et al. 1993, 1994,1997, Nawijn and
  Bass 1994)
• Scheduling of machine and material handling
  operations (Egbelu 1987, Matsuo, Shang and
  Sullivan (1991), Hall et al. 1993, Lei et al.
  1993, 1995, Blazewicz et al. 1994, Hall et al.
  1994, and Crama 1995)
• Integrating scheduling with batching and lot
  sizing (Potts and Van Wassenhove 1992)

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Machine scheduling with material handling
operations

• Resources?machines and materialhandling
  transporters
• Cost of material handling 80
• To reduce cost deal with the issues
• Sequencing that specifies the order in which jobs
  are processed at machining centers
• Scheduling that makes time-phased routing and
  dispatching of transporters for job pick-up and
  delivery
• Facility layout and flowpath design that makes
  efficient operations possible.

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Machine scheduling with material handling
operations (Contd)

• K?number of transporters in a system
• J?total number of job types
• n?total number of jobs to be processed
• nmps?the number of jobs in a minimal part set
• ?min?the objective of minimizing the production
  cycle time of an MPS in a repetitive process
• tw?a manufacturing environment where the starting
  time of each material handling operation must be
  confined within a time window
• nwt? the constraint that jobs are not allowed to
  wait in process

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Machine scheduling with material handling
operations (Contd)

• The problem is to find a simultaneous feasible
  schedule for job sequencing and time-phased
  dispatching and routing of transporters so that a
  given objective is optimized
• Work divided into
• Robotic cell scheduling
• has the fewest constraints, and is also the one
  for which most analytical results are available
• identify the optimal job input sequence and the
  robot operation sequence with respect to certain
  objective functions
• Scheduling of Automated Guided Vehicles (AGVs)
• deals with an automated job shop with non-zero
  buffers at machining centers and multiple AGVs
  traveling on a shared network.
• Cyclic scheduling of hoists subject to
  time-window constraints
• deals with the scheduling of multiple hoists in a
  flexible flowshop
• tw, nwt and collision-free constraints

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The robotic cell scheduling problem

• The no-buffer case
• F2(1)Jgt1?min ? polynomial (Sethi et al. 1992)
• F2(1)Jgt1?min ? O(n4mps) to optimize robot moves
  and job sequence (Hall et al. 1996a)
• F2(1)Jgt1Cmax ? Gilmore and Gomory algorithm
  O(n3) (Kise et al. 1991)
• F2(1)Jgt1Cmax (transportation between machines
  job dependent)? NP-hard (Ganesharajah et al.
  1995)
• The finite buffer case
• F2(1)Jgt1Cmax (fixed job inputsequence)? Np-hard
  branch-and-bound algorithm to determine the
  sequence of robot moves (King et al. 1993)

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Scheduling of automated guided vehicles

• process of flexible manufacturing
• circulate on a network of guidepaths connecting
  machine centers, and transport tools and jobs
  among the centers
• AGV flowpaths
• Unidirectional (?)
• Bi-directional (?)
• Netwok configurations
• Single-loop
• Multi-loop

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Analytical approaches to AGV scheduling

• Unidirected flowpath case
• Deadlines are fixed? single-loop network
  determined in O(nlogn) (Blazewicz et al. 1991)
• Deadlines are fixed, collision-free routing?
  two-loop network determined with dynamic
  programming (Blazewicz et al. 1994)
• Bi-directional flowpath case
• Send an AGV from a source location to a machine
  center? Dijkstras algorithm polynomial O(K4m2)
  (Kim and Tanchoco 1991)
• Column generation based heuristic approach
  (Krishnamurthy et al. 1993)
• Two-AGV scheduling problem to minimize makespan?
  dynamic programming (LAngevin et al. 1994)

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Heuristic rules for AGV and machine scheduling

• AGV dispatching rules
• Work center-initiated
• Vehicle-initiated
• Pull-based-vehicle selects a work center with the
  highest need for job replenishment
• Push based-vehicle first selects a job to move
  and then a work center to which the job should be
  sent.
• Plan conflict-free vehicle routes?(Taghaboni and
  Tanchoco 1988)
• Testing various machine and AGV scheduling rules
  against different scheduling criteria via
  simulation experiments?(Sabuncuoglu and
  Hommertzheim 1989, 1992b)
• Hierarchical approach for real-time on-line AGV
  scheduling problems? (Sabuncuoglu and
  Hommertzheim 1992a)
• Artificial intelligence and expert systems
  techniques ? review (Kusiak 1989)
• Approach based on a Hopfield neural network with
  simulated annealing? (Chung and Fischer 1995)

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The hoist scheduling problem

• Considered as a special class of Jm(K)Jgt1?min
  problems with tw and nwt constraints.
• Interval processing time? a decision variable
  selected from a given range as job processing
  time
• A common objective of hoist scheduling in
  practice? to minimize the cycle time of a
  repetitive process for producing a given MPS

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The hoist scheduling problem (Contd)

• Fm(K)J1, nwt, tw?min
• Fm(1)J1, nwt, tw?min? NP-hard (Lei and Wang
  1989)
• Mixed integer program (Philips and Unger 1976)
• Branch-and-bound procedure that solves a large
  number of LP subproblems (Shapiro and Nuttle
  1988)
• Branch-and-bound procedure that solves
  relaxations of LPs (Armstrong et al. 1991, 1994)
• Fm(K)Jgt1, nwt, tw?min
• Heuristic dispatching rules and expert systems
• Expert systems? (Yih 1990, Yih and Thesen 1991)

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Some Conclusions

• Changing r from 1 to a positive integer increases
  the complexity of problem
• no relationship between the complexity of the
  nonfix and the fix models.
• Most nonpreemptive problems are NP-hard
• For preemptive problems, most polynomial
  algorithms based on linear integer programming
  techniques

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Future Research Directions

• Branch-and-bound techniques, dynamic programming,
  heuristic algorithms with an error bound analysis
• The nonpre-emptive case with different job
  release times and different machine available
  time windows
• semi-resumable case where some extra setup time
  may be required when a job restarts.
• Extension of the existing models to more
  complicated job shop and open shop problems
• combining machine availability constraints with
  human resource constraints
• comparing neighbourhood search techniques with
  constraint-guided heuristic search techniques
• where the optimal home position of a transporter
  after a delivery is
• How to coordinate the machine and material
  handling operations to minimize the machine,
  transporter, and job waiting time
• How to con-struct collision-free schedules when
  jobs arrive dynamically