In an ongoing project at Honeywell SRC, we are constructing a prototype scheduling system for a NASA domain using the 'Time Map Manager' (TMM). The TMM representations are flexible enough to permit the representation of precedence constraints, metric constraints between activities, and constraints relative to a variety of references (e.g., Mission Elapsed Time vs. Mission Day). The TMM also supports a simple form of causal reasoning (projection), dynamic database updates, and monitoring specified database properties as changes occur over time. The greatest apparent advantage to using the TMM is the flexibility added to the scheduling process: schedules are constructed by a process of 'iterative refinement,' in which scheduling decisions correspond to constraining an activity either with respect to another activity or with respect to one time line. The schedule becomes more detailed as activities and constraints are added. Undoing a scheduling decision means removing a constraint, not removing an activity from a specified place on the time line. For example, we can move an activity around on the time line by deleting constraints and adding new ones

Boddy, Mark

Carciofini, Jim

Hadden, George D.

English

NASA Technical Reports Server

N93-32130
Scheduling with Partial Orders and a Causal Model
Mark Boddy Jim Carciofmi George D. Hadden
{boddy Jc_rciofiJhadden}Osrc.honeywell.com
Honeywell Systems _ Research Center,MN65-2100
3660 Technology Drive
Minneapolis,MN 55418
Abstract
In an ongoing project at Honeywell SRC, we are constructing a proto.
type acheduling system for a NASA domain using the "Time Map Manqer"
(TMM). TMM representations are flexlble enough to permit the representation
of precedence constraints, metric constraintS between activities, and constraints
relative to a variety of references (e.g., Mimion Elaimed Time vs. Mhmion
Day). The TMM also supports a simple form of eausal reasoning (projection),
dynamic database updates, and monitoring specified database properties u
changes occur over time.
The greatest apparent advantage to using the TMM is the flexibility added
to the scheduling process: schedules are constructed by a process of "iterative
refinement, I in which scheduling decisions correspond to constraining an ac-
tivity either with respect to another activity or with respect to some timeline.
The schedule becomes more detailed u activities and constraints are added.
Undoing a scheduling decision means removing a constraint, not removing an
activity from a specified place on the timeline. For example, we can move an
activity around on the time]ine by deleting constraints and adding new ones,
and other activities constrained with respect to the one we move will move as
well.
1 Introduction
We are interested in the solution of large, complex scheduling problems. Examples
of the kinds of domains we are interested in include several NASA scheduling prob-
lems (e.g. Spacelab, Space Station operations, Shuttle ground processing), and the
Transportation Planning problem being addressed by the joint DARPA/Air Force
Planning Initiative.
A "solution" as we use the term is not simply an implementation of an algorithm
for solving a particular constraint satisfaction or constrained optimization problem.
For many domains, constructing schedules is an extended, iterated process that may
involve negotiation among competing agents or organizations, scheduling choices
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made for reasons not easily implementable in an automatic scheduler, and last-
minute changes when events do not go as expected. In such an enviroment, the
process by which a schedule is constructed must be considered in any attempt to
provide a useful scheduler for a given domain.
Even the more limited problem of generating a single schedule is becoming in-
creasingly complex. Simple models solvable by straightforward application of stan-
dard operations research techniques such as linear programming are less and less
relevant to current scheduling problems. For example, many NASA scheduling
domains involve large problem instances (hundreds to thousands of activities and
constraints), context-dependent activity effects (including context-dependent tran-
sitions or setup times as a special case), complex resource structures (e.g., a power
bus that is divided into sub-busses), and user preferences on where activities appear
in the final schedule (e.g., "as late as possible=). To provide an effective solution, a
scheduling system must be expressive enough to represent or reflect these domain
complexities as well as supporting the process by which a schedule is constructed.
In an ongoing project at Honeywell SI_C, we are implementing a prototype
scheduling system for a NASA domain using the "Time Map Manager _ (TMM).
TMM representations permit the expression of precedence constraints, metric con-
straints between activities, and constraints relative to a variety of references (e.g.
Mission Elapsed Time vs. Mission Day). The TMM also supports causal reasoning
(projection and persistence), dynamic database updates, and monitoring certain
database properties as changes occur over time.
The greatest apparent advantage to using the TMM is the flexibility added to the
scheduling process: schedules are constructed by a process of "iterative refinement,"
in which scheduling decisions correspond to constraining 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 scheduling decision
means removing a constraint, not removing an activity from a specified place on the
timeline. For example, we can move an activity around on the timeline by deleting
constraints and adding new ones, and other activities constrained with respect to
the one we move will move as well.
In the rest of this paper, we provide a brief introduction to the TMM, describe
the application of the TMM to scheduling, and describe some related work.
2 TMM Overview
As part of the DAI_PA/I_ Planning Initiative, Honeywell has developed a new
implementation of Dean's Time Map Manager (TMM) [5], involving improvements
in robustness, user interface, and documentation, in addition to a number of ex-
tensions in functionality. The TMM provides users and application programs (e.g.,
planners and schedulers) with the following functionality:
• Metric and ordering constraints between any two points.
• Causal reasoning.
252
===1
P I I..............-->
P I
o I [
Figure 1: A simple temporal database
• Database monitors for temporal conditions and protections.
• OptimJzations for large temporal databases.
The structure and capabilities of the TMM are described in more deta£1 below.
2.1 Temporal Relations
The TMM lets users assert constraints between pairs of time points, resulting in a
partial ordering among the points. TMM supports queries regarding necessary and
possible temporal relations among the time points. The truth of facts over intervals
of time is represented by tokens, which may include properties of pers/stence be-
yond their observed endpoints. In the current implementation, tokens may persist
both forward and backward in time. The truth of a proposition over an interval is
determined based on the ordering of token endpoints and the token's persistence
properties: For example, Figure 1 is a simple temporal database, involving three
tokens of three different types. In this example, P is true over the interval bounded
by the vertical lines, mad persists into the future. (not P) becomes true at a later
time, and clips the forward persistence of P. The statement up mad Q" is true for
an interval defined by the overlap of the tokens labelled P and Q.1
2.2 Causal Reasoning
The TMM supports reasoning about the changing state of the world as activities
occur using three forms of inference:
• The persistence assumption. As described above, users of the TMM specify
that certain facts are believed to be true over specific intervals of time. In
addition, they can specify that those facts can be assumed to remain true
until something occurs to make them false.
• Projection. This is inference of the form: given an event E and a set of precon-
ditions Px, P2,... Ph, and a result R, whenever the preconditions are believed
to be true for the entire event E, R is believed to become true immediately
following E.
IWe are in the process of developing • formal semantics for the TMM. A drdt version is available
by request.
253
• Overlap chaining. Given a set of preconditions Pl, P2,... P_, and a result R,
R is believed to be true for any interval for which all of the preconditions are
true.
All of these forms of inference are handled completely automatically: the user spec-
ifies which f_cts are persistent and asserts a set of projection and overlap rules, and
the requisite inference is performed by the system.
2.3 Nonmonotonic Reasoning and Database Monitors
TMM supports two basic kinds of nonmonotonic reasoning:
• Possibly true temporal relations between time points (which may be invali-
dated by additional constraints), and
• Assumed truth of a temporal proposition over an interval based on a time to-
ken's persistence (which may be invalidated by the addition of a contradictory
token, which clips the proposition during that interval).
In addition, the database itself is "nonmonotonic', in the sense that information
can be deleted, and the inference performed by the system thus far will be checked
to ensure that it continues to be supported by the current state of the database. 2
The existence of specified database properties as changes are made over time
can be tracked through the use of monitors. The existing types of TMM database
monitors are temporal conditions and protections. Temporal conditions monitor
whether specified relations among points can be derived from the current state of
the database, maintaining this information as the database changes. Protections
do the same thing for the truth of some fa_t over an interval. Between them, these
two mechanisms provide support for monitoring the continued validity of previ-
ous inference, or triggering demons based on complex properties of the temporal
database
2.4 Efficiency
Current and planned TMM optimizations for handling large databases include the
use of & global reference point where appropriate (rather than forcing its use as
some systems do), limiting search to that necessary to prove or disprove a query,
ca_hing search results for later use, graph decomposition, temporal indexing, lazy
monitor evaluation, and algorithms that are designed to search only those parts of
the database that may result in useful answers.
3 Scheduling Using the TMM
The assumptions underlying our scheduling work are as follows:
2Tkia capability (temporal reason maintenance) is described in detail elsewhere [5].
254
SV1
SV2
, ,"
I I I I [
Figure 2: Time-line scheduling
sv11 I I
SV21 I
I I
Figure 3: Constraint-posting scheduling and the resulting partial order
I. Explicitly mode]ling the constraints resulting from specific scheduling deci-
sions makes the schedule easier to construct and modify.
2. P_epresenting only those relationships required 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 utilization. Instead, the evolving
schedule consists of a partially ordered set of activities, becoming increasing ordered
as additkmai constraints are added (or less so, as those decisions are rescinded).
Timeline schedules can be represented using linear sequences of tokens, one
sequence for each resource. Figure 2 depicts a simple timeline schedule Arrows be-
tween the sequences represent constraints on parts of the two sequences that must
obey the indicated ordering relationship. In contrast, schedules constructed by ac-
cumulating constraints have a structure like that in Figure 3. Here, the current
set of constraints is insufficient to force a totally-ordered sequence of activities. KI-
though providing increased flexibility (through delaying commitment), the explicit
representation of partiaUy-ordered activities in the time map makes reuoning about
resource usage and other state changes more complicated. It is no longer possible to
construct a single thne-line representing (e.g.) changing resource availability over
time. Instead, the system computes bounds on the system's behavior.
255
P7
P_PS _PIO
PS P7
I I I I ; __ - _t._lpPl P2 PS P4 PS 10
P7
Figure 4: Gradual hardening of a partial order
Causal reasoning and resource profiles both depend on precise orderings of facts
and activities in time, that is, on what propositions are true and what activities
occur when. For a partial order, we can determine what facts might possibly or
neceasar//y hold at a point, in some or all of the total orders consistent with the given
partial order. With even a very simple causal model, this is an NP-complete problem
[4]. The solution we have implemented (first presented in [3]) is to approximate the
necessary quantification, implementing strong and weak reasoning as approximations
for what is possibly or necessarily true, given the current partial order, s
Figure 4 depicts the process by which a partially ordered schedule is gradually
refined into an executable, totally ordered schedule. In our approach to constructing
the final schedule, this is one of the ways in which a partial or incomplete schedule
is modified, the other being the addition of new activities.
The TMM's strong and weak reasoning provides a partial solution to the problem
of reasoning about what will happen. For certain classes of inference, in particular
problems involving resource capacity or the aggregate duration of mutually exclu-
sive activities, strong and weak (even exact "necessary" and "possible") reasoning
occasionally provides insuliicient information. For these cases, there are two pos-
sible approaches: simulation (sampling) of totally-ordered sequences, or some kind
of static graph analysis to determine better bounds on the system's behavior. The
end result in either case is a measure of how likely it is that further constraints on
the partial order will cause problems, requiring the scheduler to backtrack to earlier
choices.
Despite the approximate nature of this reasoning, we are still ahead of the
game: where the least-commitment approach to scheduling can at least provide
8See [5], or [13] for details.
256
approximate answers in support of scheduling decisions (e.g. what order activities
should occur in), timeline schedulers 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.
4 Related Work
The idea that schedules should be constructed Ufrom the side," looking at part or all
of the schedule history rather that just sweeping forward or backward in time, has
been implemented in several scheduling systems, e.g. [7, 18, I, 14]. Typically, these
systems also support an iterative process of schedule refinement or repair. Recent
work on COMPASS provides a protocol for allowing different agents to modify the
same schedule, wherein commitments made by one agent cannot be affected by
the actions of any other. Research in constraint-based scheduling [8, 15, 12] has
demonstrated the advantages of considering the structure of problem constraints
over time and using this structure to dynamically focus decision-making on the
most critical decisions. However, these systems have historically had a weak model
of the interaction of activities and the evolving state of the domain.
Research in generative planning has focused on the construction of activity net-
works that bring about desired goal states, given basic representations of the e.qects
of actions in the world. Classical domain modeling assumptions [6] [17] make it
difficult to reason about the duration of activities, continuously varying quantities,
and resource consumption. The consequence of these limitations is that automatic
planners have not had any great success in applications to significant planning and
scheduling problems [2, 16].
In addition to the TMM, several other temporal database systems have been
implemented with an ability to reason about time _from the side" [Ii, 9, i0]. To
date, the TMM's combination of expressive flexibility, precise semantics, support for
database operations, and extensive optimization set it apart from other temporal
reasoning systems.
5 Summary
Effective support for current scheduling domains requires a focus on the scheduling
process as well as the scheduling problem. Using the TMM, we are in the process of
constructing a novel form of scheduler that constructs schedules through the accu-
mulation of constraints on the relations between activities, and between activities
and various reference points (e.g. calendar time, mission elapsed time, etc.).
The TMM's support for flexible temporal relations, dynamic updates in large
databases, and causal reasoning provide an effective base for building schedulers for
complex problems. The TMM is freely available and written in Common Lisp. To
obtain a copy of the software or a more detailed description of the system's design
and capabilities, contact the authors at the address given on the first page.
257
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