Current trends in deterministic scheduling
Chung-Yee Lee, Lei Lei and Michael Pinedo.
Presented by
Gökhan Metan
Outline
 Brief introduction to Deterministic Scheduling
 Recent developments in scheduling theory
 Recent developments in search algorithms
 Recent developments in scheduling practice
 Conclusion
Introduction
In deterministic scheduling, a set of jobs has to be
processed by a set of machines while optimizing a certain
performance measure.
Recent developments
focused on more practical
issues
Most practical problems
are NP-hard
models are extended!
Consequence: Local Search Methods (especially the
Meta-heuristics)
Notation
//
Machine
(resource)
configuration
Processing
restrictions
&
constraints
Performance
measure to
be optimized
Recent developments in
scheduling theory
Trend:
Results of classical algorithms
Extend
Models that are closely related to real problems.
Two areas reviewed:
Scheduling with 1-job-on-r-machine
Machine scheduling with availability constraints
Recent developments in
scheduling theory
Scheduling with 1-job-on-r-machine
Jobs may need to be processed simultaneously
on several machines (r is a positive integer), or
several jobs can be processed by a single
processor simultaneously (0<r1).
Machine scheduling with availability constraints
Machines may not be available at all times, due
to machine maintenance, or machines may only
be available in given time windows.
Recent developments in scheduling theory
Scheduling with 1-job-on-r-machine pattern
Notation:
//
: fix or nonfix
=fix : each job can be processed
simultaneously by several processors but these
machines are prespecified (a fixed set of
machines).
=nonfix: each job can be processed
simultaneously by several processors but the
number of machines is fixed (not the machine
set itself).
Recent developments in scheduling theory
Scheduling with 1-job-on-r-machine pattern
The machine set not fixed (=nonfix)
Versions of
Pm|nonfix|Cmax
Pm|nonfix|wjCj
Pm|nonfix|Lmax
studied in the literature.
Recent developments in scheduling theory
Scheduling with 1-job-on-r-machine pattern
The machine set fixed (=fix)
Versions of
Pm|fix|Cmax
Pm|fix|wjCj
Pm|fix|Lmax
Pm|fix|Tj
Pm|fix|wjUj
studied in the literature.
Recent developments in scheduling theory
Machine scheduling with availability constraints
Machines may not be available because of machine
breakdown (stochastic) or preventive maintenance
(deterministic) during the scheduling period.
Machine scheduling with availability constraints in
deterministic case is reviewed.
Two cases are discussed in the literature:
Resumable:
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. (=r-a)
Nonresumable: If the job has to restart rather than
continue. (=nr-a)
Recent developments in scheduling theory
Machine scheduling with availability constraints
Versions of
1|nr-a|  Cj
F2|r-a|Cmax
,
F2|nr-a|Cmax
1|r-a|  Cj
,
1|r-a| Lmax
1|r-a| wjCj
1|r-a|Uj
studied in the literature.
Search Algorithms
Most scheduling problems are so complex.
Cannot be formulated as Mathematical Programs.
Difficult to apply classical techniques (DP, B&B).
Increasing popularity of “Search Techniques” other
than classical operations research techniques.
Search Algorithms
Search Techniques
Neighbourhood S.T.
Frequently used by OR and IE
people.
Based on the concept of local
improvement.
Programming effort required for
the implementation is reasonable.
Structural knowledge needed
about the problem is significantly
less when compared with the MP
approach.
Constraint Guided
Heuristic S.T.
Often used by CS and AI people.
Do not try to find the optimal,
rather try to find good & feasible
solutions.
Require more programming
effort than Neighbourhood S.T.
Search Algorithms
Definition:
A heuristic is a technique which seeks good (i.e. near
optimal) solutions as a reasonable computational cost without being
able to guarantee either feasibility or optimality, or even in many cases
to state how close to optimality a particular feasible solution is.
(Reeves)
NOTE:
Many heuristics are problem specific, so that a
method which works well for one problem cannot be used to solve a
different one.
Search Algorithms
Neighbourhood Search Techniques
Four Important Design Concepts for Neighbourhood S.T.
1)
The mapping of the data in a format suitable for the
algorithm
Data is a problem specific issue and it must be well described
that how it should be represented in the search procedure.
2)
The neighbourhood design
The neighbourhood design specifies the set of all the
neighbouring solutions of a given solution.
(Note: For a more formal definition look to the “Definition-3” on page 2 of the
notes provided.)
It is a very critical decision which affects the solution quality
and the computational efficiency of the algorithm.
Search Algorithms
Neighbourhood Search Techniques
Job-1
Job-1
Job-2
Job-2
Job-3
Job-3
Job-(n-1)
Job-(n-1)
Job-n
Job-n
For any solution S, neighbourhood
of S, N(S), includes (n-1) different
alternative neighbouring solutions.
For any solution S, neighbourhood of S, N(S),
includes (n-1)! different alternative
neighbouring solutions.
(ex: for n= 20, 19 alternative
solutions.)
(ex: for n=20, 19!=121,645,100,408,832,000
alternative solutions)
Search Algorithms
Neighbourhood Search Techniques
3)
The search process within the neighbourhood
How to select a neighbouring solution as the new solution?
Different strategies (simple or complicated) can be employed.
(e.g. select any neighbouring solution at random, or select the
one that improves the objective much)
4)
The acceptance-rejection criterion
How to decide a temporarily seleted neighbouring solution to
be the new solution or not?
Depends on the user specified decision parameters. (e.g.
cooling parameter in SA, tabu length in TS)
Search Algorithms
Simulated Annealing(SA)
General Properties
SA algorithm is based on the analogy between the annealing
(cooling of material) process of solids and the problem of
solving combinatorial optimization problems.
Employes probabilistic approaches such as the probability of
accepting a worse solution than the current solution.
Has a memoryless property, that is, it does not keep track of
the information gained during the search process.
Simpler to implement (in terms of programming effort) than
TS & GA.
Note:
For a detailed information refer to the notes provided.
Search Algorithms
Tabu Search(TS)
General Properties
The philosophy of TS is to derive and exploit a collection of
principles of intelligent problem solving. TS is based on the
selected concepts that unite the fields of AI and optimization.
One of the main components of TS is its use of adaptive
memory(attributive memory, explicit memory etc.) (Memorybased strategies are employed!)
Not a random search technique!
A bad choice can yield more information than a good
random choice. In a system that uses memory, a bad choice
based on strategy can provide useful clues about how the
strategy may profitably be changed. (Glover)
Note:
For a detailed information refer to the notes provided.
Search Algorithms
Genetic Algorithms(GA)
General Properties
GA is the meta-heuristic technique that originates from the
analogy between the representation of a complex structure by
means of a vector of components, and the idea, familiar to
biologist, of genetic structure of a chromosome.
Employes probabilistic approaches such as the probability of
mutation, cross-over etc.
Has a memoryless property, that is, it does not keep track of
the information gained during the search process.
Works on a population of solutions, not a single current
solution.
Search Algorithms
Genetic Algorithms(GA)
NOTE (A mistake in the paper):
section-2.3, 1st line.
Meta-heuristic Classification:
Meta-heuristic
Classification 1
Genetic Algorithms
M/S/P
Simulated Annealing
M/S/1
Tabu Search
A/N/1
On page 15, 3rd paragraph of
Classification 2
M/N/P
M/N/1
A/N/P
A: Adaptive Memory
M: Memoryless
N: Systematic Neighbourhood Search
S: Random Sampling
1: if the method moves from one current solution to the next after each iteration
P: for a population based approach with a population of size P

Classification 1 is the traditional one and the Classification 2 is the most recent
implementations of the methods.
Search Algorithms
SA, TS & GA
A Comparision: What is going on in the literature?
Among the modern heuristic techniques, SA was the first one
that appears in the literature, but TS is the most popular one.
SA used for TSP and job shop scheduling problem with the
objective of makespan (Jm // Cmax).
TS used for TSP, single machine, parallel machine, flow shop,
flexible flow shop and job shop problems with the objective of
makespan, total weighted completion time and total weighted
tardiness.
A number of hybrid approaches, that is combining the SA &
TS philosophies, exist in the literature, as well.
Search Algorithms
SA, TS & GA
A Comparision: What is going on in the literature?
Cont’d
GA is used for TSP, job shop problem (Jm // Cmax).
Also, machine learning techniques from computer science
(AI) is combined with GA for a job shop problem. (Lee,
Piramuthu and Tsai, 1995.)
Recently, there exists a number of applitions and
implementations of GA in the real world problems.
Search Algorithms
Constraint-guided Heuristic Search
General Properties
Constraints of the problem are incorporated into the system as
rules.
Technique focuses on partial solutions of the problem and tries
to extend these partial solutions until a complete feasible
solution is obtained.
Technique first attempts to satisfy the most strict constraints at
the beginning of the search and the least strict ones at the end.
Recent Developments in
Scheduling Practice
Recent popular areas in Scheduling Practice:
Flexible-resourse 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
Our scope:
“machine scheduling with material handling
operations”
Recent Developments in
Scheduling Practice
Differences from classical machine scheduling problems:
Two types of resources are now involved:
Machines
Material handling transporters
Either resource could become a bottleneck (if not properly
scheduled).
Transportation operations are not instantaneous, depends on
the sequence in which material movement (between machines)
is executed.
Recent Developments in
Scheduling Practice
Issues that should be addressed simultaneously:
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.
Extended Notation:
(K) /  / 
: K denotes the number of transporters.
Recent Developments in
Scheduling Practice
Properties of the general model:
n jobs to be processed, m machining centers.
All jobs are ready at time zero, each with its own route and
processing specifications.
The deliveries of jobs between machining centers are
performed by K, K1, identical transporters.
Transporters travel on a shared network & collisions must
be avoided.
Operations of the transporters (loading, unloading, moving
jobs) are non-instantaneous & non-preemptive.
Recent Developments in
Scheduling Practice
Properties of the general model:
(Cont’d)
Neither a machine nor a transporter can hold more than 1
job at any time.
The problem is finding a simultaneous feasible schedule for
job sequencing and time-phased dispatching and routing
of transporters while optimizing a given objective.
Three sub-categories:
1- Robotic cell scheduling
2- Scheduling of AGVs
3- Cyclic scheduling of hoists subject to time-window constraints
Recent Developments in
Scheduling Practice
1- Robotic Cell Scheduling:
Typically arises in cellular manufacturing systems.
Cell
Material
handling
robot
Flexible
machines
MPS
Because of material handling cost, buffers between machines
are limited (zero or finite).
Cell performance depends on the sequence of robot moves and
the order in which jobs are loaded into the cell.
Recent Developments in
Scheduling Practice
The no-buffer case:
No buffer between machines.
Several polynomial
algorithms are available for 2machine cells (m=2).
For 3-machine cells (m=3),
efficient algorithms are
reported. (Hall, Kamoun,
Wan, 1994.)
The finite buffer case:
Finite (but non-zero) input
and output buffers at machine
centers.
2-machine scheduling with
finite input buffer but no
output buffer for each
machine is considered and
B&B procedure is employed
(King, Hodgson and Chafee,
1993.)
Recent Developments in
Scheduling Practice
2- Scheduling of AGVs:
Typically arises in flexible manufacturing systems.
Such an FMS contains NC machining centers, limited input &
output buffers, and a material flow network.
A major constraint that must be satisfied by any AGV schedule
is to avoid trafic collisions of AGVs during their operations.
There are two types of AGV flowpaths commonly used in
practice:
Unidirectional flowpath ()
Bi-directional flowpath ()
Recent Developments in
Scheduling Practice
Advantages & disadvantages of bi-directional flowpath designs over
the unidirectional networks:
Disadvantages:
Requires higher control.
Implementation cost is higher.
Advantages:
Have greater potential to improve productivity.
Require fewer AGVs.
Reduce AGV travel time.
Most analytical approaches that guarantee the optimal AGV schedule
wih respect to certain objective functions are limited to special cases.
Recent Developments in
Scheduling Practice
AGV dispatching rules:
Work center-initiated rules
A work center selects an AGV for a delivery operation
whenever it finishes an operation.
Vehicle-initiated rules
An AGV selects a pick up whenever it becomes idle.
Pull-based vehicle-initiated rules
Push-based vehicle-initiated rules
Recent Developments in
Scheduling Practice
Pull-based vehicle-initiated rules
2) A job is
selected to
be sent to
the work
center.
AGV
1)Selects
the one
with
highest
need.
Work Center
Work Center
Work Center
Work Center
Work Center
Set of candidate
Jobs
Highest need for
replenishment!
Recent Developments in
Scheduling Practice
Push-based vehicle-initiated rules
1) A job is
selected to
be moved.
AGV
2) selects
the work
center
which the
job should
be sent
Work Center
Work Center
Work Center
Work Center
Work Center
Recent Developments in
Scheduling Practice
3- Hoist Scheduling:
Typically deals with the scheduling of multiple hoists in a
flexible flow shop.
The most distinct feature is that the job processing time at each
machine is strictly limited by a lower and an upper bound. This
means that any hoist schedule that causes a hoist not to pick up
a job within this time window is infeasible.
Subject to nwt (jobs are not allowed to wait in process) and
collision free constraints.
Most of the real life problems are NP hard in the strong sense
and even they preclude a formal mathematical formulation.
Conclusion
Future research directions:
Nonpreemptive case with different job release times and
different machine available time windows.
In machine scheduling with availability constraints problem, it
may be interesting to study on the semi-resumable case where
some extra setup time may be required when a job restarts.
Studies comparing neighbourhood search techniques with
constraint-guided search techniques.
Studies that integrates the facility layout and flowpath design
problems with the machine and transporter scheduling.
Transporter scheduling with dynamic job arrivals.
References
C.-Y. Lee, L. Lei, M. Pinedo, Current trends in deterministic scheduling,
Annals of Operations Research 70(1997)1-41.
C. R. REEVES, Modern Heuristic Techniques for Combinatorial Problems,
New York, Toronto, GB: John Wiley & Sons INC., 1993.
F. Glover and M. Laguna, Tabu Search, Massachusetts, USA: Kluwer
Academic Publishers, 1997.