Robust Scheduling: A General View

1
Robust Scheduling A General View

• Heng-Soon GAN
• and
• Andrew WIRTH

When scheduling information is moderately
incomplete and will deviate during the execution
phase, a proactive-reactive scheduling method,
such as robust scheduling is preferred. In this
seminar, I will define and discuss (analytically
and empirically) five different robust scheduling
performance measures, namely schedule
effectiveness, schedule predictability, heuristic
efficiency, heuristic robustness and schedule
nervousness. If time permits, I will make a
comparison between stochastic and robust
scheduling techniques and empirically justify the
use of deterministic, robust and online
scheduling techniques via the entropy measure.
2
Outline

• The scheduling environment
• Terminologies
• Schedule execution costs
• Heuristic robustness (or stability)
• Schedule robustness (effectiveness and
  predictability)
• Schedule nervousness (frequent rescheduling)
• Integer program formulation
• A more practical robust scheduling approach
• Some empirical results
• Stochastic scheduling
• Scheduling and uncertainty (the entropy concept)
• Conclusions and future directions

3
Scheduling Environment
Schedule Planning Phase
Schedule Execution Phase
Schedule Deployment Time
Local disruption or information update from other
dependent sources.
Information sent to other dependent sources.
4
Terminologies

• Initial schedule
• the schedule generated in the planning phase
  (off-line)
• referred to as initial off-line schedule
• OR the schedule prior to a perturbation event
• Perturbed schedule
• the schedule produced after a decision is made
  and executed in reaction to a perturbation event
• Perturbation event
• may occur during the planning and execution
  phases
• described by the event time and disruption
  magnitude
• e.g. machine breakdowns, change in operation
  processing times, arrival and removal of new
  operations

5
Terminologies (contd)

• Perturbation scenario
• a set of perturbation events
• In-process operations, completed operations and
  operations that have not started

current time
completed operations
in-process operations
not-started operations
6
Terminologies (contd)

• Shift-rescheduling (Sh)
• regarded as the simplest possible repair procedure

a
b
c
d
Processing time of operation a is updated,
replaced with a.
a
b
c
d
Shift operation b to the left.
a
b
c
d
0
current time
7
Terminologies (contd)

• Heuristic-rescheduling (H)
• repair/regeneration of schedule using algorithms
  or heuristics or local search methods.

a
b
c
d
Processing time of operation a updated, replaced
with a.
b
a
c
d
Use LPT to reschedule.
a
d
c
b
0
current time
8
Schedule Execution Costs

• Schedule effectiveness
• the degree of optimality of a perturbed schedule,
  e.g. makespan, flowtime, earliness, tardiness
  etc.
• this is the main cost to be optimised if no
  disruption occurs
• Schedule predictability
• the closeness of the perturbed schedule
  performance relative to the initial off-line
  schedule performance
• reduces costs of under-utilisation or overtime
• Heuristic efficiency
• the computational complexity of the schedule
  generation/repair method
• timeliness of response to perturbation events
• Degree of re-arrangement (heuristic robustness or
  stability)
• the degree of alteration to the operations
  arrangement
• reduces costs of replanning and rerouting
• Schedule nervousness
• the frequency of H-rescheduling
• reduces the number of plan revisions in other
  parts of the supply chain

9
Schedule Execution Costs (contd)

• If no rescheduling is allowed (only perform
  shift), we want to minimise
• If rescheduling is allowed, we want to minimise

10
Heuristic Robustness

• A heuristic is said to be robust if the sequences
  of the operations do not change drastically when
  this heuristic is used for rescheduling after a
  disruption.
• If local search method is used to generate/repair
  schedules, this measure can be embedded in the
  objective function.
• Possible measures
• Sum of the absolute changes in start-time and
  completion times of operations
• Minimal Perturbation (El Sakkout et al.-2000)
• Neighbourhood-based Robustness (Jensen-1999,2000,2
  001,2003 Jensen and Hansen-1999)
• Predictable Scheduling (ODonovan et al.-1999)
• Rescheduling with effectiveness and stability as
  criteria (Wu et al.-1993)
• Sum of the absolute changes in the precedence of
  operation
• Neighbourhood-based Robustness (Jensen-1999,2000,2
  001,2003 Jensen and Hansen-1999)
• Rescheduling under random disruptions (Abumaizar
  and Svestka-1997)

11
Heuristic Robustness (contd)

• Sum of operations reassigned
• Matchup Scheduling (Bean et al.-1991)
• Sum of the absolute changes in sequence/positions
  of operations
• Spearmans footrule as measure of disarray
  (Diaconis and Graham-1977)
• Artificial Immune System (Hart et al.-1997)
• Most of the work provide definitions, but lack of
  analyses of the measure provided.

12
Heuristic Robustness (contd)
Increasing start time
1
2
3
4
Before perturbation
5
6
2
4
1
1
4
2
After perturbation and application of some
heuristic
3
6
2
5
1
4
Increasing start time
13
Heuristic Robustness (contd)
.
.
(Sh) (H) (H) (H) (Sh)
(H) (H) (Sh)
time
.
.
?j ?j1 ?j2 ?j3 ?j4
?j5 ?j6 ?j7
14
Heuristic Robustness (contd)

• General bounds

Diaconis and Graham (1977)
15
Heuristic Robustness (contd)

• Longest Processing Time heuristic (P Cmax)
• change in processing time of one operation
  (unbounded) or removal of one operation or
  addition of one operation
• change in processing time of k operations
  (unbounded)

16
Heuristic Robustness (contd)

• change in processing time of one operation
  (bounded)

17
Heuristic Robustness (contd)

• mean analysis (all scenarios are equally likely)
• a change in processing time
• addition of one operation
• removal of one operation
• addition or removal or change
• in processing time of one
• operation
• Diaconis and Graham (1977)

18
Heuristic Robustness (contd)

• Abdekhodaee and Wirth equal length algorithm
  (P2si pi aCmax)
• the algorithm

S2k-1
S2k-2
S2k-3
S3
S2
S1
S2k
Even number of jobs
S2k1
S2k
S2k-1
S2k-2
S3
S2
S1
Odd number of jobs
19
Heuristic Robustness (contd)

• change in processing time of one operation
  (unbounded) or removal of one operation or
  addition of one operation
• mean analysis

20
Heuristic Robustness (contd)

• Johnsons algorithm (F2 Cmax)
• Nt refers to jobs (equivalent to nt 2Nt
  operations)

21
Heuristic Robustness (contd)

• Multifit heuristic
• (P Cmax)

RLPT,A 18
n 10, m 4
RMF7,A 42
22
Heuristic Robustness (contd)
23
Heuristic Robustness (contd)
24
Schedule Robustness

• In a weaker sense, an initial off-line schedule
  is said to be robust if
• the perturbed schedule is effective
• low cost
• the absolute deviation of the perturbed schedule
  performance relative to that of the initial
  off-line schedule is small
• predictability

Z
time
0
DZ
0
time
25
Schedule Robustness (contd)
time
.
.
?j ?j1 ?j2 ?j3
?j4 ?j5 ?j6 ?j7
26
Schedule Robustness (contd)

• Suppose that and are quantities to be
  minimised, the schedule produced by heuristic A
  is more robust than that of heuristic B (in a
  weaker sense) if,

27
Schedule Nervousness
.
.
(Sh) (H) (H) (H)
(Sh) (H) (H) (Sh)
time
.
.
?j ?j1 ?j2 ?j3
?j4 ?j5 ?j6 ?j7
28
Integer Program Formulation

• A robust initial off-line schedule (in a stronger
  sense) is a schedule which
• minimises the total schedule execution cost and
  do not require any H-rescheduling when disruption
  occurs
• costs consist of effectiveness, predictability
  and stability (shift robustness)
• More formally, a robust initial off-line schedule
  is a schedule S which minimises
• and only Sh is performed when disruption
  occurs.

29
Integer Program Formulation (contd)

• Previous integer program formulation attempts
• Daniels and Kouvelis (1995)
• single machine problem (SPT is optimal for the
  deterministic case)
• minimising maximum absolute deviation of
  perturbed schedule total flowtime from the
  optimal schedule
• suggested solution procedures (for processing
  time intervals) BB algorithm and 2 heuristics
  (endpoint sum and endpoint product pairwise
  interchange)
• Book Kouvelis and Yu (1997)
• described robust formulation for various problems
  such as scheduling (single machine and flowshop),
  facility layout etc.
• presented 3 variations of objective function
  formulations
• minimise the maximum perturbed schedule
  performance over all perturbation scenarios

30
Integer Program Formulation (contd)

• minimise the maximum absolute deviation of
  perturbed schedule performance from the optimal
  schedule over all perturbation scenarios
• minimise the maximum relative deviation of
  perturbed schedule performance w.r.t the optimal
  schedule over all perturbation scenarios
• Kouvelis, Daniels and Vairaktarakis (2000)
• two-machine flowshop problem (Johnsons algorithm
  provide optimal schedule for the deterministic
  case)
• absolute deviation robust schedule (makespan)
• suggested solution procedures BB algorithm and
  a heuristic approach
• Kuo and Lin (2002)
• single machine problem, an extension of Daniels
  and Kouvelis (1995)
• relative deviation robust schedule
• solution procedure BB algorithm
• Yang and Yu (2002)
• single machine problem, also an extension of
  Daniels and Kouvelis (1995)

31
Integer Program Formulation (contd)

• revealed that three types of robust formulation
  (absolute, absolute deviation and relative
  deviation) can be solved using a common solution
  procedure - generalisation of Daniels and
  Kouvelis(1995) and Kuo and Lin(2002)
• suggested solution procedures (for discrete
  processing times) dynamic programming, surrogate
  relaxation procedure and greedy heuristic
• Conclusions from the literatures and some open
  questions
• the problem (single machine and two machine
  flowshop) is NP-hard.
• most solution procedures use extreme processing
  time information (lower and upper bounds) and
  this has been proven to be sufficient.
• is this sufficient for more complex scheduling
  problems?

32
Integer Program Formulation (contd)

• objective function assumes the existence of
  optimal solution to the problem
• challenge the optimal solution to most (more
  complex) scheduling problems is unknown, even for
  identical parallel machines.
• can lower bounds be used?
• only effectiveness is considered
• need to include other measures such as
  predictability and stability
• perturbation scenarios are not time dependent
• if only effectiveness and predictability is
  considered, perturbation scenarios need not be
  time dependent
• but if stability is to be included, perturbation
  scenarios has to be time dependent.

33
Practical Robust Scheduling

• Since finding a robust schedule is NP-hard (even
  for the simplest scheduling problem), we propose
  the following scheduling procedure
• create a initial off-line schedule using
  heuristic or local search (in consideration of an
  objective)
• create a rescheduling policy, i.e. decide to use
  either H or Sh when disruptions occur
• decide the robust scheduling scheme (which
  initial off-line schedule and rescheduling
  policy) to be used, i.e. the scheme which
  minimises the average or maximum cost

34
Practical Robust Scheduling (contd)

• cost to be minimised
• in real time, to decide whether to shift or to
  regenerate the schedule
• map the current state of disruptions (magnitude,
  time etc.) to the database of the robust
  scheduling scheme chosen OR
• use the best heuristic 0-look-ahead procedure
  and apply it myopically at each disruption OR
• game-theoretic control approach (Leon, Wu and
  Storer-1994)

35
Practical Robust Scheduling (contd)

• Other practical scheduling approaches
• contingency schedules
• Artificial Immune System (Hart et al.-1997)
• Proactive rescheduling analysis (Guo and
  Nonaka-1999)
• least commitment scheduling
• Preprocess-First-Schedule-Later (Byeon et
  al.-1998 Kutanoglu and Wu-1998 Wu et al.-1999)
• Generating initial off-line schedule
• choice of deterministic-(near-)optimal OR
  robust-(near-) optimal initial off-line schedule
• attempts (mostly for machine breakdowns)
• ARS, ADRS and RRS (Daniels and Kouvelis etc.)
  as discussed earlier
• capacity hedging method (Yellig and
  Mackulak-1997)
• schedule sensitivity analysis (Morikawa et
  al.-1993)

36
Practical Robust Scheduling (contd)

• neighbourhood-based robustness (Jensen-1999,2000,2
  001,2003 Jensen and Hansen-1999)
• slack-based techniques (Chiang and Fox-1990 Gao
  et al.-1995 Davenport et al.-2001)
• fuzzy evaluation of expected delay (Dorn et
  al.-1995 Chen and Muraki-1997) for uncertain
  processing times
• Assuming the perturbation scenario is known,
  rescheduling policies can be constructed via
  methods such as
• IP formulation (no attempts yet)
• the problem is likely to be intractable
• BB algorithm computationally exhaustive
• Genetic Algorithm
• easy coding of chromosomes 1010 ? H,Sh,H,Sh
• The a-look-ahead heuristic
• 2(a 1) possibilities
• for a 0, i.e. 0-look-ahead heuristic can be
  used in real-time scheduling

37
Practical Robust Scheduling (contd)
. . . .
. .
. . . .
. .
38
Practical Robust Scheduling (contd)

• Off-line procedures to create a robust scheduling
  scheme
• When heuristic (e.g. LPT, MFk etc.) is used,
• use heuristic to generate initial off-line
  schedule and repair schedule when disruptions
  occur
• the initial off-line schedule created is myopic.
• the rescheduling policy can be constructed via
  methods described earlier.
• this procedure is myopic if 0-look-ahead is used
  (but suitable in real-time).
• When local search method (e.g. GA, SA etc.) is
  used,
• if stability (heuristic robustness) is important,
  embed this measure into the objective function.
• initial off-line schedule created can be
  long-sighted
• application of LSM similar to that of a heuristic
• It is possible to combine both heuristic and
  local search methods into the robust scheduling
  scheme.

39
Some Empirical Results

• Heuristic A is better than heuristic B if C(A,B)
  ? 1, where

40
Some Empirical Results (contd)

• Use a-look-ahead heuristic, where a 0.
• Perform Sh if

41
Some Empirical Results (contd)

• Compare the use of LPT, MF7 and SPT on identical
  parallel machines
• minimising makespan
• subjected to changes in processing times and
  machine breakdowns.
• 10 sets of n 30 operations, where processing
  times are randomly generated from U(1,100).
• m 6 identical parallel machines.
• 10 sets of perturbations with 20 events each,
• change in processing time
• probability of 0.5 that an operation will change
  its processing time
• range U(0.1pi, 2pi)
• occurrence time U(0, 200)
• machine breakdown
• Time between failure neg-exp(0.005)
• Downtime neg-exp(0.08)

42
Some Empirical Results (contd)

• Cost coefficients used

43
Some Empirical Results (contd)

• Results

44
Stochastic Scheduling

• Stochastic dominance
• almost surely larger
• P(X1 ? X2) 1
• larger in likelihood ratio sense
• P(X1 t)/P(X2 t) is nondecreasing in t, t ? 0
  and f1(t) and f2(t) are p.d.f.s.
• stochastically larger
• P(X1 gt t) ? P(X2 gt t) for all t
• larger in expectation (often used in stochastic
  scheduling)
• E(X1) ? E(X2)
• Types of policies
• static list policy
• puts all operations in a list at time 0 and this
  list does not change during schedule execution
  (perform Sh whenever disruptions occur)
• dynamic list policy
• no fixed list the decision maker allowed to make
  decisions during schedule execution (perform H
  whenever disruptions occur)
• could be preemptive or non-preemptive

45
Stochastic Scheduling (contd)

• Most results for stochastic scheduling depends on
  the following
• optimality in expectation (the crudest form of
  stochastic optimality)
• simple distribution
• some nice results (extracted from Pinedos
  Scheduling Theory, Algorithms and
  Systems-1995)
• 1pi generalSwiCi (nonpreemptive
  static/dynamic list policies)
• WSEPT is optimal in expectation (also optimal for
  general machine breakdowns on single machines)
• 1pi generalLmax (dynamic nonpreemptive
  static list policies)
• EDD is optimal almost surely
• P2 pi exp(lj)Cmax (nonpreemptive static list
  policies)
• LEPT is optimal in expectation
• PpreemptCmax (preemptive dynamic list policies)
• LEPT is optimal in expectation
• Ppi generalSCi (preemptive dynamic list
  policies)
• SEPT is optimal stochastically

46
Stochastic Scheduling (contd)

• Stochastic vs Robust scheduling
• robust hedge against uncertainty in expectation
  and/or worst case optimisation of other
  criteria such as efficiency, stability,
  predictability and nervousness
• stochastic hedge against uncertainty in
  expectation (usually)
• some comments
• both stochastic and robust scheduling are
  addressing the same problem, i.e. uncertainty in
  scheduling
• which is preferable? depends on what is to be
  optimised and the availability of optimal
  solutions
• the stochastic analysis only provide optimal
  solutions for simple shop-floor configurations
  and restricted uncertainty distributions
• but at least this formulation gives more
  optimistic results than the robust formulation
• both stochastic and robust formulations are
  difficult to solve
• need a more practical approach, e.g. the more
  practical robust scheduling, contingency
  schedules etc.
• need more flexibility in deciding whether to
  reschedule or not (from the stochastic scheduling
  point of view, this is in fact switching between
  static and dynamic list policies)

47
Scheduling and Uncertainty

• Certain event
• Event happens with no variability (I AM
  ABSOLUTELY SURE)
• Uncertain event
• Some information on the event available, but with
  variability (MAYBE..)
• Unexpected event
• Information on the event revealed at the time it
  occurs (I DONT KNOW..)

48
Scheduling and Uncertainty (contd)
Low Uncertainty
Medium Uncertainty
High Uncertainty
Unexpected
Deterministic Scheduling
Robust Scheduling
On-line Scheduling
Reactive
Proactive
49
Scheduling and Uncertainty (contd)

• Heuristic applied in a deterministic sense
  (static list policy)
• all operations committed to the initial off-line
  schedule
• perform shift when disruption occurs
• Heuristic applied in a robust sense
• all (or partial) operations are committed to the
  initial off-line schedule
• perform shift or H-rescheduling when disruption
  occurs
• perform shift when (0-look-ahead heuristic)
• Heuristic applied in a online sense (dynamic list
  policy)
• operations not committed to the initial off-line
  schedule
• operation assigned over time according to a
  specified rule

50
Scheduling and Uncertainty (contd)

• We measure the uncertainty associated with
  scheduling information using the entropy concept.
• schedule stability radius Sotskov et al.
  (1997,1998), Lai et al. (1997)
• empirical testing on static and dynamic
  applications of optimal and heuristic solution to
  job shop problem Lawrence and Sewell (1997)
• Recalling the entropy concept
• finite scheme with mutually exclusive events, A1,
  A2, , An with probabilities p1, p2, ,pn
  respectively, where Si pi 1
• the amount of uncertainty associated with the
  finite scheme is given by
• and if pk 0, pk log pk 0

51
Scheduling and Uncertainty (contd)

• Recalling the entropy concept (contd)
• for m mutually independent schemes, M S1S2
  Sm, the entropy is given by
• Applying to scheduling problem where operation
  processing times are uncertain,
• let fi(wi) be the p.d.f. of the processing time
  of operation i, such that

52
Scheduling and Uncertainty (contd)
Event Aik
Assuming independence of Gi,
53
Scheduling and Uncertainty (contd)

• Simulation setup
• operation processing time uniformly distributed
  between ai and bi i.e. fi(wi) 1/(bi ai), and
  hence
• assume Di D 0.001 for all i
• two cases investigated bi ai c bi ai
  ci
• initial data randomly chosen within ai, bi

54
Scheduling and Uncertainty (contd)
equal ci
unequal ci
55
Scheduling and Uncertainty (contd)

• compare LPT, SPT and MF7 applied in both
  deterministic and robust sense and LPT in an
  online sense using normalised cost
• only consider effectiveness, heuristic robustness
  and nervousness costs
• efficiency and predictability costs omitted
• display results on Nervousness Cost versus
  Heuristic Robustness Cost
• order of preference
• online, deterministic, robust

56
Scheduling and Uncertainty (contd)
unequal c n30
equal c n30
57
Scheduling and Uncertainty (contd)
equal c n50
equal c n30
58
Conclusions and Future Directions

• Creating robust schedule is known to be NP-hard
  (even for single machine problems)
• more investigations needed for the parallel
  machine problem
• A more lazy alternative is to use
  proactive-reactive scheduling approach (a more
  practical robust scheduling)
• account for effectiveness, predictability,
  efficiency, stability and nervousness
• a robust scheduling scheme consists of robust
  initial off-line schedule and rescheduling
  policies
• using a more robust initial off-line schedule
  will improve all five measures mentioned above
• Real-time scheduling
• based on the robust scheduling scheme
• further investigations needed
• reaction to disruptions based on what we have
  simulated
• Artificial Intelligence fuzzy systems, neural
  network etc.
• use the 0-look-ahead heuristic OR game-theoretic
  control approach

59
Conclusions and Future Directions

• Entropy concept used to justify the use of
  deterministic, robust and online scheduling
  techniques
• conjecture that bi ci c can be used and the
  measure is scalable
• detailed analysis and more simulation needed
• added an extra dimension to sensitivity analysis
• proactive approach to deal with changes in
  information uncertainty and costs
• extension to other disruptions such as
• machine breakdowns described by the mean time
  between failure and the duration of breakdown
• arrival of new operations described by the
  arrival rate, number of operations at each
  arrival and the parameters of operations upon
  arrival
• removal of operations described by the
  probability that an operation will be removed