CURRENT TRENDS IN DETERMINISTIC SCHEDULING

1
CURRENT TRENDS IN DETERMINISTIC SCHEDULING

• IE573 Presentation
• by Merve Celen

2
Agenda

• Introduction
• Recent Developments in Scheduling Theory
• Recent Developments in Search Algorithms
• Recent Developments in Scheduling Practice
• Conclusion

3
Introduction

• Most practical problems are NP-Hard
• Local search methods have been studied during
  last decade
• Two recent extensions
• Scheduling with a 1-job on r-machine pattern
• Machine scheduling with availability constraints
• aß? notation (Pinedo(1995))

4
Recent Developments in Scheduling Theory

• Scheduling with a 1-job on r-machine pattern
• Jobs processed simultaneously on several machines
  (r is a positive integer)
• Several jobs processed by a single processor
  simultaneously ( 0 lt r 1)
• Machine scheduling with availability constraints
• Due to machine maintenance, time windows, etc.

5
n jobs to be processed on m machines notation
6
Scheduling with 1-job on r-machine pattern

• Where r is a positive integer (multiprocessor
  task system)
• Diagnosable microprocessor systems (Krawczyk and
  Kubale (1985))
• Semiconductor circuit design team workforce
  planning
• Berth allocation problem (Lee and Cai (1996))
• Two main classes
• Fixed number of machines, machines not specified
  (nonfix)
• Fixed set of machines for particular jobs (fix)

7
The machine set not fixed (nonfix)

• Pm nonfix Cmax
• Blazewicz, et al. (1984)
• Polynomial algorithms for Pm prmp, nonfix Cmax
  with one-machine and two-machine jobs
• Blazewicz, et al. (1986)
• For Pm nonfix Cmax polynomial algorithms only
  for unit processing times (by IP or DP)
• Polynomial algorithms for Pm prmp, nonfix Cmax

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(nonfix)

• Pm nonfix Cmax (contd)
• Du and Leung (1989)
• Parallel task system
• Processing time is a nonincreasing function of
  the number of machines used where the number of
  machines is decided before a job is processed
• P2 nonfix Cmax , P3 nonfix Cmax are NP-hard
  in the ordinary sense
• P5 nonfix Cmax is NP-hard in the strong sense
• P4 nonfix Cmax is an open question

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(nonfix)

• Pm nonfix Cmax (contd)
• Blazewicz et al. (1990)
• Processors with the same speed
• Only two types of tasks one-processor and
  two-processor tasks
• Polynomial algorithm to solve the problem
  optimally
• Later polynomial algorithm for two types of
  tasks one-processor and k-processor tasks

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(nonfix)

• Pm nonfix SwjCj
• Lee and Cai (1996)
• Strongly NP-hard even with two machines
• Two special cases
• wj w
• P2 nonfix SCj is NP-hard in the ordinary
  sense if the number of two-machine jobs fixed (
  O(nP3s1))
• pj p
• O(n log n) algorithm if pj p for all
  one-machine jobs
• Heuristic algorithm with an error bound of 1 for
  the Pm nonfix SwjCj problem
• Heuristic algorithm with an error bound of ½ for
  the Pm nonfix SCj problem

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(nonfix)

• Pm nonfix Lmax
• Plehn (1990)
• With release dates, due dates and preemption
  allowed
• Uses LP to check the existence of a feasible
  schedule
• Lee and Cai (1996)
• NP-hard in the strong sense even if m 2
• For the P2 nonfix Lmax
• One-m/c jobs b/w any two consecutive two-m/c jobs
  must follow the EDD rule on each machine
• Two-m/c jobs should follow the EDD rule
• Dynamic programming algorithm in O(nP3s1logP)
• O(n log n) algorithm for the special case P2
  nonfix, pj 1 Lmax

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The machine set fixed (fixed)

• Pm fix Cmax
• Bozoki and Richard (1970)
• A branch and bound algorithm for optimal solution
• Krawczyk and Kubale (1985)
• Diagnosable microprocessor systems
• P fix, pj 1 Cmax is NP-hard
• Kubale (1987)
• Pm fix Cmax is strongly NP-hard even if there
  are only two-m/c jobs
• Blazewicz et al. (1992)
• P3 fix Cmax is strongly NP-hard
• Some polynomially solvable special cases,
  heuristics for the general problem

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(fixed)

• Pm fix Cmax (contd)
• Hoogeven et al. (1994)
• P3 fix Cmax subject to block constraints is
  NP-hard in the ordinary sense
• A pseudopolynomial algorithm with O(nP)
• Pm fix, pj1 Cmax is polynomially solvable
• Pm fix, rj Cmax is NP-hard in the strong
  sense
• Pm fix, rj , pj1 Cmax is polynomially
  solvable
• Krämer (1995)
• Three branch and bound algorithms for the general
  problem with precedence constraints

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(fixed)

• Pm fix SwjCj
• Dobson and Karmarkar (1989)
• Two IPs for the general problem
• Heuristic algorithms and computational results
• m2, pj1, the same number of jobs to be assigned
  to m/c 1 and m/c 2, a polynomially optimal
  solution
• Hoogeven et al. (1994)
• P2 fix SCj is NP-hard
• P2 fix SwjCj , P3 fix SCj , P3 chain,
  fix, pj1 SCj are NP-hard in the strong sense
• P fix, pj 1 SCj is NP-hard in the strong
  sense

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(fixed)

• Pm fix SwjCj (contd)
• Cai, Lee and Li (1996)
• P2 fix SCj is NP-hard in the strong sense
• Heuristic for P2 fix SwjCj with a relative
  error at most 100
• For P2 prmp, fix SCj polynomial algorithm
• Brucker (1995)
• Pm fix, rj , pj1 SCj , Pm fix, pj1
  SwjCj , Pm fix, pj1 SCj are polynomially
  solvable

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(fixed)

• Pm fix Lmax , Pm fix STj , Pm fix
  SwjUj
• Brucker (1995)
• Pm fix, pj1 STj , Pm fix, pj1 SwjUj
  are polynomially solvable when m is fixed
• Pm fix Lmax is NP-hard in the strong sense
• Bianco et al. (1993)
• Polynomial algorithm based on LP for Pm prmp,
  rj, fix Lmax and Pm prmp, rj, set Lmax
• (set a job can choose a set of
  alternatives where each alternative contains
  several dedicated machines, fix is a special case
  of set when there is only one alternative for
  each job)

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Machine scheduling with availability constraints

• Machine breakdown (stochastic) or preventive
  maintenance (deterministic)
• m/c i is unavailable b/w sik and tik (0 sik
  tik), 0 k ni where ni is the number of
  unavailability periods for m/c i
• In most manufacturing cases ni 1

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Machine scheduling with availability constraints
(contd)

• Two main cases
• Resumable if a job cannot be finished before
  the next down period of a m/c and the job can
  continue after the m/c has become available again
  (r-a in the ß field)
• Nonresumable if the job has to restart rather
  than continue (nr-a in the ß field)

19
Machine scheduling with availability constraints
(contd)

• Studies on this subject
• Schmidt (1984)
• n job m parallel machines, each m/c has different
  availability intervals
• Polynomial time algorithm to find a feasible
  preemptive schedule whenever one exists
• Pm prmp, r-a Cmax can be solved polynomially
• Adiri et al. (1989)
• 1 nr-a SCj problem (both stochastic and
  deterministic cases)
• For deterministic case, NP-hard when there is
  only one unavailability period

20
Machine scheduling with availability constraints
(contd)

• Lee (1991)
• Parallel machine problem to minimize Cmax
• At most one unavailability period for each m/c,
  and it happens at the beginning of the time
  horizon
• Classical LPT has a tight error bound of 1/2
• Modified LPT has an error bound of 1/3
• Kaspi and Montreuil (1988) and Liman (1991)
• With the conditions in Lee (1991), SPT is optimal
  to minimize SCj

21
Machine scheduling with availability constraints
(contd)

• Lee and Liman (1993)
• P2 SCj where one m/c is available all the
  time and the other m/c is available from time 0
  up to a fixed point in time
• NP-hard and they use DP
• Mosheiov (1994)
• Same problem, m/c i is available in time window
  xi, yi where 0 xi yi
• SPT is asymptotically optimal for m m/cs in
  parallel problem
• Lee (1996b)
• F2 r-a Cmax and F2 nr-a Cmax , at least
  one m/c available
• F2 r-a(Mi) Cmax is NP-hard, pseudopolynomial
  DP algorithm

22
Machine scheduling with availability constraints
(contd)

• Lee (1996a)
• SPT solves 1 r-a SCj and EDD solves 1 r-a
  Lmax optimally
• 1 r-a SwjCj is NP-hard, solves by DP, a
  heuristic with an error bound analysis
• Pm r-a Cmax is NP-hard, analyzes the worst
  case performance of the LPT algorithm
• 1 nr-a Cmax , 1 nr-a Lmax and 1 nr-a
  SUj are NP-hard
• Solves P2 nr-a SwjCj optimally by a
  pseudopolynomial DP

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Recent Developments in Search Algorithms

• Scheduling problems are so complex to be
  formulated as mathematical programs
• Two types of search techniques
• Neighbourhood search techniques (ORIE)
• Constraint-guided heuristic search techniques
  (CSAI)

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Recent Developments in Search Algorithms (contd)

• Neighbourhood search techniques
• Local improvement
• Programming effort required is fairly modest
• First random swaps, then more sophisticated
• k-opt approach for TSP (Lin and Kernighan (1972))
• TSP is equivalent to 1 sjk Cmax
• Mainly three techniques
• Simulated annealing
• Tabu search (most often used in scheduling)
• Genetic algorithms (focused lately)

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Recent Developments in Search Algorithms (contd)

• Constraint-guided heuristic search techniques
• Not optimal schedules, seek to find a good
  feasible schedule
• A list of rules or constraints the schedule
  should satisfy
• Extend partial solutions to a feasible complete
  solution
• Based on measurements of flexibility and
  constraining factors
• Satisfy more stringent constraints first, then
  less stringent ones

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General concepts in neighbourhood search

• Techniques can be compared based on
• The mapping of the data (concise and unambiguous)
• The neighbourhood design
• The search process within the neighbourhood
• The acceptance-rejection criteria
• Set of all neighbours of a given solution
  (centered on the aspects that have the greatest
  impact)
• Pairwise swap, insertion or more complicated as
  in the critical path for Jm Cmax

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General concepts in neighbourhood search (contd)

• Ways to select schedules in the neighbourhood
• Random
• Most promising (e.g., swaps of jobs with the most
  effect)
• Acceptance-rejection criterion
• Probabilistic in simulated annealing
• Deterministic in tabu search
• Number of schedules in each iteration
• A set of different schedules in genetic
  algorithms
• Only a single schedule in simulated annealing and
  tabu search

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Simulated annealing and tabu search

• Simulated annealing (SA) appeared first
  (Kirckpatrick et al. (1983))
• Tabu search (TS) more widely used in production
  scheduling
• The difference b/w SA and TS is
  acceptance-rejection criterion
• for SA
• A tabu list with fixed number of entries
  (prevents cyclic depending on the length of the
  list)

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Simulated annealing and tabu search (contd)

• First applications of SA and TS focused on TSP,
  neighbourhood design based on 2-opt or k-opt
• SA used for job shop scheduling with makespan
  objectives (Matsuo et al. (1987))
• TS used for single m/c, parallel m/c, flow shop,
  flexible flow shop and job shop problems with
  objectives that include SwjCj , Cmax , SwjTj
• A number of heuristic approaches combining SA and
  TS have also been developed

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Genetic algorithms

• First suggested by Holland (1973, 1975)
• Generation population of feasible solutions at
  the end of each iteration
• Individual each single solution
• Individuals are referred to as chromosomes
• May consist of subchromosomes
• Each subchromosome the schedule of operations on
  a given machine
• Take a promising individual and perform a
  mutation (insertion, interchange, etc.)
• Create a new individual by combining two
  individuals

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Genetic algorithms (contd)

• TS and SA are special cases of genetic algorithm
  (GA) where only one individual is generated in
  each iteration
• GA is more powerful than TS and SA but slower
• The first applications focus on TSP, lately
  applied to the problem Jm Cmax
• Bean (1994) develops a sophisticated GA for a
  scheduling problem in automotive industry
• Mayrand et al. (1995) develop a GA for a
  scheduling problem that occurs in aluminum
  industry
• Herrmann et al. (1995) use a GA to develop a
  global job shop scheduler for semiconductor
  manufacturing test operations

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Constraint-guided heuristic search

• As a result of rule-based scheduling systems
• Focus on finding feasible solutions
• A list of rules or constraints the schedule has
  to satisfy
• Frequently, not possible to find a feasible
  solution without violating one or more
  constraints
• Make a distinction b/w soft and hard constraints
• Relax one or more soft constraints and try again
  (Cheng and Smith (1997))

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Constraint-guided heuristic search (contd)

• Constraint propagation techniques
• List all additional implied constraints
• Whenever a partial schedule has been extended,
  generate all additional constraints
• Consistency checking
• Verify if a feasible schedule exists
• Before search for a feasible schedule and
  whenever additional constraints are generated
• Deal with inconsistencies of constraints
  (conflict resolution)

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Recent Developments in Scheduling Practice

• Examples of recent areas
• Flexible-resource scheduling
• Scheduling variable-speed machines
• Scheduling with finite capacity input and output
  buffers
• Scheduling of machine and material handling
  operations
• Integrating scheduling with batching and
  lot-sizing

35
Recent Developments in Scheduling Practice
(contd)

• Machine scheduling with material handling
  operations
• Each resource could become a bottleneck
• Transportation is non-instantaneous and
  sequence-dependent
• To decrease material handling cost
• Sequencing order in which jobs are processed at
  machining centers
• Scheduling time-phased routing and dispatching
  of transporters for job pick-up and delivery
• Facility layout and flowpath design makes
  efficient operations possible

36
Recent Developments in Scheduling Practice
(contd)

• Expand the notation aß? to a(K)ß?
• K number of transporters
• J total number of job types
• n total number of jobs to be processed
• nmps number of jobs in a minimal part set (MPS)
• ?min objective of minimizing the production
  cycle time of an MPS in a repetitive process
• tw starting time of each material handling
  operation should be within a time window
• nwt jobs are not allowed to wait in process

37
Recent Developments in Scheduling Practice
(contd)

• A general model
• n jobs to be processed, m machining centers
• All jobs ready at time zero each with its own
  route and processing specifications
• The deliveries are performed by K, K 1,
  identical transporters
• A shared network, traffic collisions should be
  avoided
• All the operations by the transporters are
  non-instantaneous and non-preemptive
• The capacity of a m/c and a transporter is 1

38
Recent Developments in Scheduling Practice
(contd)

• Recent work related to this model
• Robotic cell scheduling (fewest constraints, most
  available analytical results, identify optimal
  job input sequence and robot operation sequence)
• Scheduling of AGVs (avoid collisions, minimize
  machine blocking)
• Cyclic scheduling of hoists subject to
  time-window constraints (most restrictive, time
  window constraint, nwt and collision-free
  constraints)

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The robotic cell scheduling problem

• In each cell, a single material handling robot,
  several flexible machines that produce MPSs
  repetitively
• Buffers b/w machines are limited
• Cell performance depends on the sequence of robot
  movements and job input
• Minimize steady-state cycle time
  Fm(1) Jgt1 ?min

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The no-buffer case

• Several polynomial algorithms for m 2
• Sethi et al. (1992)
• Polynomially solves F2(1) Jgt1 ?min assuming a
  fixed one-unit cycle used
• For any m-machine cell, m! one-unit cycles
• Hall, Posner and Potts (1993, 1994)
• The same one-unit cycle is not necessary to
  obtain optimal cycle time, even when m 2
• Hall et al. (1996a)
• MinCycle evaluates alternative cycle
  combinations and their optimal job sequences
  polynomially

41
The no-buffer case (contd)

• Hall et al. (1996b)
• QuickCycle considers only a subset of
  (promising) candidate combinations of cycles
• Generates solutions very close to optimal
• F2(1) Jgt1 Cmax
• Polynomial algorithm based on Gilmore and Gomory
  algorithm (Kise et al. (1991))
• If transportation times are job dependent, then
  similar to ATSP and NP-hard in the strong sense
  (Ganesharajah et al. (1995))
• Both Fm(1) Jgt1 Cmax and Fm(1) Jgt1 ?min
  even with no-buffer and a fixed one-unit cycle
  are NP-hard when m 3 (Hall et al. (1994))
• When jobs are identical, no job sequencing,
  problem becomes polynomially solvable for both
  Cmax and ?min

42
The finite buffer case

• Finite but non-zero input and output buffers,
  zero transportation times is NP-hard
• King, Hodgson and Chafee (1993)
• F2(1) Jgt1 Cmax with finite input buffer but no
  output buffer (blockage)
• Assume a fixed job input sequence
• Determine the sequence of robot moves by a BB
  procedure
• Jeng, Lin and Wen (1993)
• Assume input sequence of jobs is fixed
• BB procedure to find optimal robot moves with
  multiple parallel machines
• Minimize total flow time

43
Scheduling of automated guided vehicles (AGVs)

• Occurs in a FMS (CNCs, limited input and output
  buffers, material flow network)
• Can be viewed as Jm(K) Jgt1 ? with
  non-instantaneous material delivery
• Improper dispatching of AGVs will lead to
  congestion, collision, delays in manufacturing
• AGV flowpaths
• Unidirectional
• Bi-directional higher control and implementation
  cost but greater potential to improve
  productivity, fewer AGVs, reduce AGV travel time

44
Scheduling of automated guided vehicles (AGVs)
(contd)

• Two commonly used network configurations
• Single-loop
• Avoidance of collision is easier
• Most analytical studies for optimal solution
  assume single-loop
• Multi-loop
• Major concerns are AGV collision and the risk of
  m/c blocking

45
Analytical approaches to AGV scheduling

• Undirected flowpath case
• Blazewicz et al. (1991)
• Deadlines of deliveries fixed, single-loop, given
  fleet size, then existence of a feasible schedule
  can be determined in polynomial time
• Blazewicz et al. (1994)
• Two-loop network with a common stretch to switch
  b/w loops
• Analyze conditions for collision-free routing
• A DP approach to search for a feasible schedule
  with fixed deadlines for deliveries at each
  machine

46
Analytical approaches to AGV
scheduling (contd)

• Jaikumar and Solomon (1992)
• Each loop has a safety zone, all jobs returned to
  a warehouse b/w successive machining steps
• A polynomial algorithm to solve the minimum fleet
  size and AGV scheduling problem
• Ganesharajah et al. (1996)
• Objectives of minimizing cycle time, fleet size,
  AGV utilization
• Distinguish polynomially solvable problems from
  NP-hard ones
• Many results for the robotic cell scheduling
  problems can be applied to the cases with
  single-loop and zero buffer

47
Analytical approaches to AGV
scheduling (contd)

• Bi-directional flowpath case
• Kim and Tanchoco (1991)
• Polynomial procedure to find minimum-delay path
  to send an AGV from a source location to a m/c
  center
• Krishnamurthy et al. (1993)
• A heuristic approach
• Minimize makespan w.r.t vehicle interference
  constraints
• Langevin et al. (1994)
• A DP to solve two-AGV problem with minimizing
  makespan

48
Heuristic rules for AGV and machine scheduling

• Two classes of AGV dispatching rules
• Work center-initiated rules
• A work center selects an AGV whenever it finishes
  operation
• Vehicle-initiated rules
• An AGV selects a pick up when it becomes idle
• In a busy system, more effective
• Pull-based select the work center with the
  highest need for job replenishment then the job
• Push-based select a job first then a work center

49
Heuristic rules for AGV and machine scheduling
(contd)

• Taghabani and Tanchoco (1988)
• When a vehicle is selected for delivery, all
  pre-established routes for other vehicles are
  fixed
• A feasible route for the AGV is designed
• Intersections solved on a basis of FIFO
• Sabuncuoglu and Hommertzheim (1992a)
• Use job status information to schedule one AGV at
  a time
• A job should not be moved if it will have to wait
  for the next machine on its route
• Yim and Linn (1993)
• No significant difference in terms of output rate
  b/w pull and push based policies when an FMS is
  busy

50
The hoist scheduling problem

• A special case of Jm(K) Jgt1 ?min with tw and
  nwt constraints
• The job processing time at each machine is not
  fixed, must be selected from a given range
  (interval processing time)
• Often found in electroplating and chemical
  industries
• Large number of chemical tanks (machines)
• Each job is a barrel carrying identical parts
• Different job types may require different route
  and treatment process
• tw and nwt constraints are required
• Both the tank and hoist can hold only one job at
  a time
• Traffic collision must be eliminated
• Minimize cycle time of producing a given MPS

51
Fm(K) J1, nwt, tw ?min

• All jobs identical means each MPS consisting of a
  single job
• Even with K 1, the problem is NP-hard in the
  strong sense (Lei and Wang (1989))
• Mixed integer programs and branch bound
  procedures are proposed
• The special case of Fm(1) J1, nwt, tw ?min
  occurs when processing time in each tank is fixed
  and is polynomially solvable (Levner and
  Kats(1995))
• Another special case of Fm(1) J1, nwt, tw
  ?min occurs when the unit-cycle is fixed (Lei
  (1993a))
• O (m2log(m)log(B)) , B is the interval b/w a
  lower and upper bound on ?min

52
Fm(K) J1, nwt, tw ?min (contd)

• When K gt1
• Becomes more complicated due to additional
  single-track and collision-free constraints
• Known approaches are all heuristic based
• A special case when job processing times are
  fixed and single-track constraint is relaxed
• A pseudopolynomial algorithm that solves a
  sequence of associated assignment problems ( Lei
  (1993b))

53
Fm(K) Jgt1, nwt, tw ?min

• Computational effort increases tremendously even
  when K1
• Need for both job sequencing and hoist scheduling
• All available approaches are either heuristic
  dispatching rules or expert systems

54
Conclusion

• 1-job on r-machine pattern
• Changing r from 1 to a positive integer increases
  the complexity
• No clear relation b/w the complexity of fix and
  nonfix
• Both fix and nonfix are special cases of the set
  model
• A set problem is NP-hard if either the
  corresponding fix or nonfix is NP-hard
• Most of the nonpreemptive problems are NP-hard in
  the strong sense
• For preemptive problems, most polynomial
  algorithms are based on LP techniques
• Since most problems are NP-hard, BB techniques,
  DP or heuristic algorithms with an error bound
  analysis

55
Conclusion (contd)

• Machine scheduling with availability constraints
• A semi-resumable case
• Extension of existing models to more complicated
  job shop and open shop problems
• Combining machine availability constraints with
  human resource constraints
• Parameters to compare neighbourhood search
  techniques
• Quality of solution
• CPU time
• A ratio of the above two
• Development or the implementation time

56
Conclusion (contd)

• Issues that affect the outcome of each
  comparative study
• Initial solution
• Setting of parameters
• Language and manner in which the procedure is
  coded
• Platform on which the study is conducted
• Genetic algorithms are least effective among SA,
  TS and GA (Della Croce et al. (1992))
• Tabu search is the most efficient one (Aarts et
  al. (1994) and Morton and Ramnath (1995))

57
Conclusion (contd)

• Extensions for the scheduling applications
• Effective approaches that address m/c and
  transporter scheduling, as well as facility
  layout problems (easy to use heuristics)
• Transporter scheduling with dynamic job arrivals
• Optimal home position of a transporter after a
  delivery
• How to minimize the m/c, transporter, and job
  waiting time
• How to construct collision-free schedules when
  jobs arrive dynamically

58

• Thank you for your attention!