At the Honeywell Technology Center (HTC), we have been working on a scheduling problem related to commercial avionics. This application is large, complex, and hard to solve. To be a little more concrete: 'large' means almost 20,000 activities, 'complex' means several activity types, periodic behavior, and assorted types of temporal constraints, and 'hard to solve' means that we have been unable to eliminate backtracking through the use of search heuristics. At this point, we can generate solutions, where solutions exist, or report failure and sometimes why the system failed. To the best of our knowledge, this is among the largest and most complex scheduling problems to have been solved as a constraint satisfaction problem, at least that has appeared in the published literature. This abstract is a preliminary report on what we have done and how. In the next section, we present our approach to treating scheduling as a constraint satisfaction problem. The following sections present the application in more detail and describe how we solve scheduling problems in the application domain. The implemented system makes use of Ginsberg's Dynamic Backtracking algorithm, with some minor extensions to improve its utility for scheduling. We describe those extensions and the performance of the resulting system. The paper concludes with some general remarks, open questions and plans for future work

Boddy, Mark S.

Goldman, Robert P.

English

NASA Technical Reports Server

N95- 23761
Empirical Results on Scheduling and Dynamic Backtracking
Mark S. Boddy and Robert P. Goldman
Honeywell Technology Center, MN65-2200
3660 Technology Drive
Minneapolis, MN 55418
612-951-7403 612-951-7436 (FAX)
{boddy, goldman}@src.honeywell.com
KEY WORDS AND PHRASES
Constraint Satisfaction, Scheduling, Dynamic
Backtracking
INTRODUCTION
At the Honeywell Technology Center (HTC),
we have been working on a scheduling problem
related to commercial avionics. This applica-
tion is large, complex, and hard to solve. To
be a little more concrete: "large" means al-
most 20,000 activities, "complex" means sev-
eral activity types, periodic behavior, and
assorted types of temporal constraints, and
"hard to solve" means that we have been un-
able to eliminate backtracking through the use
of search heuristics. At this point, we can gen-
erate solutions, where solutions exist, or report
failure and sometimes why the system failed.
To the best of our knowledge, this is among
the largest and most complex scheduling prob-
lems to have been solved as a constraint satis-
faction problem, at least that has appeared in
the published literature.
This abstract is a preliminary report on
what we have done and how. In the next
section, we present our approach to treating
scheduling as a constraint satisfaction prob-
lem. The following sections present the ap-
plication in more detail and describe how
we solve scheduling problems in the applica-
tion domain. The implemented system makes
use of Ginsberg's Dynamic Backtracking al-
gorithm [2], with some minor extensions to
improve its utility for scheduling. We de-
scribe those extensions and the performance
of the resulting system. The paper concludes
with some general remarks, open questions
and plans for future work.
CONSTRAINT ENVELOPE
SCHEDULING
We are interested in the solution of large, com-
plex scheduling problems. A "solution" as
we use the term is not simply an implemen-
tation of an algorithm for solving a particu-
lar constraint satisfaction or constrained opti-
mization problem. For many domains, con-
structing schedules is an extended, iterated
process that may involve negotiation among
competing agents or organizations, schedul-
ing choices made for reasons not easily imple-
mentable in an automatic scheduler, and last-
minute changes when events do not go as ex-
pected. In such an environment, the process
by which a schedule is constructed must be
considered in any attempt to provide a useful
scheduler for a given domain.
In our approach, which we call constraint
envelope scheduling, schedules are constructed
by a process of "iterative refinement," in which
scheduling decisions correspond to constrain-
ing an activity either with respect to another
activity or with respect to some timeline. The
schedule becomes more detailed as activities
and constraints are added. Undoing a schedul-
ing decision means removing a constraint, not
removing an activity from a specified place on
the timeline.
The assumptions underlying our schedul-
ing work are as follows:
431
https://ntrs.nasa.gov/search.jsp?R=19950017341 2020-06-16T08:32:26+00:00Z
. Explicitly modelling the constraints re-
suiting from specific scheduling decisions
makes the schedule easier to construct
and modify.
. Representing only those relationships re-
quired by the current set of constraints
(the decisions made so far) provides a
more useful picture of the current state
of the scheduling effort.
The main consequence of this approach is that
the scheduler does not manipulate totally-
ordered timelines of activities and resource uti-
lization. Instead, the evolving schedule con-
sists of a partially ordered set of activities, be-
coming increasing ordered as additional con-
straints are added (or less so, as those decisions
are rescinded). This approach is common to a
number of scheduling systems, e.g., [1, 5, 4, 3]
Figure 1 depicts the process by which
a partially ordered schedule is gradually re-
fined into an executable, totally ordered sched-
ule. Although providing increased flexibility
(through delaying commitment), the explicit
representation of partially-ordered activities in
the time map makes reasoning about resource
usage and other state changes more compli-
cated. It is no longer possible to construct a
single time-line representing (e.g.) changing
resource availability over time. Instead, the
system computes bounds on the system's be-
havior.
Despite the approximate nature of this rea-
soning, we are still ahead of the game: where
the least-commitment approach to scheduling
can at least provide approximate answers in
support of scheduling decisions (e.g. what or-
der activities should occur in), timeline sched-
ulers make the same decisions arbitrarily--
putting an activity on the timeline is a
stronger commitment than constraining it to
occur (say) between two other activities, or
within a given time window.
STATIC SCHEDULING FOR
AVIONICS
P7
P$ P7
I I I I [ p_ "- _"-_pPI P2 PS P4 P8 10
I)7
Figure 1: Gradual hardening of a partial order
I' ..... '11" ..... "11" ..... 'll" .... 'l
i ! ! I I j
t t
Figure 2: System architecture
One of the applications to which we have ap-
plied constraint envelope scheduling is static
scheduling of processing time and bus com-
munications in a distributed environment.
This application involves safety-critical appli-
cations running on flight hardware on a com-
mercial airplane. Figure 2 is a simple diagram
of the architecture involved. The arrows at the
bottom of the picture indicate that commu-
nication also occurs into and out of the cab-
inet in which the bus and processors reside.
The schedule is static for reasons having to do
with verifiability and repeatability of behav-
ior, and ultimately with FAA certification for
flight safety.
As we have already suggested, this problem
is both large and complex. In a typical prob-
lem instance, there are approximately 6000 ac-
tivities representing slices of processor time,
and 14000 activites representing the transmis-
sion of data messages on the bus. There are six
processors, which are between 80% and 90%
loaded. The processes running on these pro-
432
cessors are periodic at rates between 5 Hz and
80 Hz. This makes the problem more com-
plicated, in that data communication is spec-
ified between processes, not between process
instances. One of the decisions to be made
in constructing a schedule is to determine the
mapping from instances of data producers to
instances of data consumers. To make matters
worse, we are constructing a schedule for a 200
mS "frame" which itself runs at 5 Hz. Com-
munication from one instance of this frame to
the next is entirely legal, and so we have in
some sense a circular model of time, in which
constraints on activites late in the frame may
affect activities early in the frame.
Processes are to a limited extent pre-
emptible, with minimum slice times and
context-dependent context-switch times (i.e.,
it matters who you were preempted by). Inter-
process constraints include jitter (bounds on
how far from perfectly periodic instances of
a process may be) and latency (limits on
the time between producer and consumer in-
stances for a given data message). There are
data cycles, where process A gives a message
to process B gives a message to process C,
which sends a message back to process A. The
interaction of these cycles with latency and jit-
ter has complex effects on schedule feasibility.
In fact, much of the work that we have done
on this application has been the definition and
derivation of conditions under which a given
set of constraints was or was not consistent.
SCHEDULING AND DYNAMIC
BACKTRACKING
The scheduler we have applied to this problem
uses Ginsberg's Dynamic Backtracking algo-
rithm [2], with some minor extensions. One of
these extensions was to enable the search en-
gine to report the set of inconsistent variables
involved, should it fail to find a solution. For
this application, knowing what constraints are
in conflict is crucial: it enables us to go back to
the system designers and tell them that their
requirements cannot be met.
The second extension that we made was
necessitated by the nature of the scheduling
problem, or at least of how we have repre-
sented it. Ginsberg's algorithm involves gen-
erating eliminations: explanations of why a
given value for some variable is ruled out given
the current partial assignment. The assump-
tion that eliminations are available by inspec-
tion does not work for complex temporal con-
straints: frequently we discover that a given
ordering is infeasible by trying it. Accordingly,
we have extended the algorithm to handle un-
successful attempts to assign a given value to
a variable. In this case, the search engine un-
does the assignment (including removing any
added constraints), records an elimination ex-
planation for that value, and reports failure
back to the scheduler.
Empirically, this extended implementa-
tion of Ginsberg's algorithm has been invalu-
able. A typical scheduling problem involves
some tens of thousands of variables represent-
ing choices on ordering, preemption or pro-
ducer/consumer pairing. Given the difficulty
of localizing variable interaction, sorting re-
lated variables to be close to each other is
impractical or impossible. Despite consider-
able effort, we have not managed to find vari-
able or value ordering heuristics that result in
backtrack-free solutions (we are currently us-
ing a variant of Smith's "slack" heuristic for
value ordering [6]).
For these reasons, having a search method
that leaves intact that part of a partial assign-
ment not involved in a given inconsistency is
crucial. One of the ways in which we might
have run into trouble using dynamic back-
tracking has not materialized, either: inconsis-
tencies typically involve less than 30 variables.
This means that the elimination bookkeeping
is kept within bounds, as well.
There is one feature of the current al-
gorithm which has been inconvenient, how-
ever. The requirement that it be the most
recently assigned variable that is re-assigned
first clashes with the fact that in scheduling
433
ORIGINAL F:::GE ;S
OF POOR QUALITY
applications there are frequently qualitative
differences between variable types. For exam-
ple, changing the ordering of an activity with
respect to other activities using the same vari-
able is in some sense a more local change to
the schedule than changing the resource as-
signed to that activity. In the latter case, the
activity must be ordered with respect to a dif-
ferent set of activities (those using the new re-
source). Any orderings remaining from the old
resource assignment may now be for no pur-
pose. For these reasons, we might like more
flexible choices about variable ordering when
backtracking.
cation.
References
[1] Fox, M.S. and
Smith, S.F., ISIS: A Knowledge-Based Sys-
tem for Factory Scheduling, Expert Systems,
1(1) (1984) 25-49.
[2] Ginsberg, Matthew L., Interpreting Proba-
bilistic Reasoning, Proceedings of the 1985
AAAI/IEEE Sponsored Workshop on Uncer-
tainty and Probability in Artificial Intelli-
gence, 1985.
[3]
CONCLUSIONS
The bottom line for this project is that we have [4]had a successful impact on the solution of a
hard problem that is a critical part of a multi-
billion dollar investment. In the process of
solving that problem, we have provided some
empirical evidence that dynamic backtrack-
ing, suitably modified, is useful for nontriv-
ial scheduling problems. We have also gained [5]
some useful experience in how to exploit the
structure of the problem: heuristics are still
critical to generating solutions or finding fail-
ures in a reasonable amount of time. "Rea-
sonable" for this application currently means [6]
a small number of hours. Minutes would be
better, days would be unworkable.
There is a lot of work yet to be done on
this problem. For example, the problem is [7]
currently being solved in phases, with proces-
sor schedules being generated before the bus
schedule. There are indications that heuristic
repair techniques as in [7] might be useful for
data scheduling.
One of the things we are hoping to arrange
in the next few months is to release an instance
or instances of this scheduling problem to the
research community. Generation or accumula-
tion of standard scheduling problems has been
difficult. This problem has the advantages of
being fairly challenging in both scale and com-
plexity, and of having its roots in a real appli-
Muscettola, N., HSTS: Integrating Planning
and Scheduling, Technical Report CMU-RI-
TR-93-05, The Robotics Institute, Carnegie
Mellon University, 1993.
Sadeh, N. and Fox, M.S., Variable and Value
Ordering Heuristics for Activity-based Job-
shop Scheduling, Proceedings of the Fourth
International Conference on Expert Systems
in Production and Operations Management,
Hilton Head Island, S.C., 1990.
Smith, S.F., Ow, P.S., Potvin, J.Y., Muscet-
tola, N., , and Matthys, D., An Integrated
Framework for Generating and Revising Fac-
tory Schedules, Journal of the Operational Re-
search Society, 41(6) (1990) 539-552.
Smith, Stephen F. and Cheng, Cheng-Chung,
Slack-Based Heuristics for Constraint Sat-
isfaction Scheduling, Proceedings AAAI-g&
Washington, DC, AAAI, 1993, 139-144.
Zweben, M., Deale, M., and Gargan, R., Any-
time Rescheduling, Proceedings of the DARPA
Workshop on Innovative Approaches to Plan-
ning, Scheduling, and Control, San Diego,
DARPA, 1990.
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