From: AAAI Technical Report WS-00-02. Compilation copyright © 2000, AAAI (www.aaai.org). All rights reserved.
Building
dispatchable
networks involving consumable resources
within the HSTS framework.
R. J. Wallace,
E. C. Freuder
Constraint Computation Center
University of NewHampshire, Durham, NH 03824
decisions have caused it to reach a point where there
are no options (a ’dead end’), because these earlier
activities have already begun. For this reason, when
actual plan execution begins there must be guarantees that a schedule derived from the time-envelopes is
executable incrementally or "dispatchable’. In recent
work it has been shown that consistent temporal constraint networks, which are a basic component of the
Planner-Scheduler’s output, can be made dispatchable
(Muscettola, Morris, & Tsamardinos 1998).
Abstract
Earlier work on scheduling by autonomous systems
has demonstratedthat schedules in the form of simple
temporal networks, with intervals of values for possible event-times, can be made"dispatchable", i.e.,
executable incrementally in real time with guarantees
against failure due to unfortunate event-time selections. These schedules can be built prior to execution by an automated Planner/ Scheduler based on
the Heuristic Scheduling Testbed System (HSTS).
the present work we show howthe property of dispatchabflity can be extended to networks that include
constraints for consumableresources. Wefirst describe conditions under which a componentof the network composedof resource constraints will support
dispatchability. Then we showhow a network can be
produced during planning that supports dispatchabflity while allowing flexible resource use. This workindicates that flexible handling of consumableresources
can be safely extended to the execution layer to provide more effective deploymentof these resources.
In this paper we describe work we have done to extend the earlier work with temporal constraints to situations where there are also consumableresource constraints.
The goal of this work is to develop methods for deriving similar guarantees against running out
of options during plan execution, in this case because
some resource constraint cannot be satisfied. In this
report we restrict our analysis to modesof representation associated with the Heuristic Scheduling Testbed
System (HSTS) (Muscettola 1994) that provides
framework for the Planner-Scheduler for the NASARemote Agent (Muscettola et al. 1998). Wepresent in
some detail a system for building networks for the Executive that are dispatchable with respect to a consumable resource. In this case the dispatchability problem
can be defined in a way that makes it feasible to produce dispatchable networks, while allowing sufficient
flexibility to be of practical interest.
Introduction.
Like many other contemporary scheduling systems, the
scheduling system within of the Remote Agent architecture developed at NASA-Ames
(Muscettola et al.
1998) has two basic layers of operation. There is first,
deliberative layer, the "Planner-Scheduler", which creates a basic schedule of operation (Muscettola 1994).
In the Remote Agent, the output from this subsystem
consists of an envelope of scheduling times for each
operation, or activity, which allows someflexibility in
scheduling specific activities based on current conditions. The task of selecting specific times for operation is carried out in the second (reactive) layer
the plan executor, or "Smart Executive", which issues
the commandsignals for operating the physical system
and monitors the progress of the system to determine
whether activites have been performed successfully.
Because the Executive is operating in real time, the
constraints on its operation are severe. In particular,
during the instantiation of a schedule the Executive
cannot afford to backtrack. That is, it cannot go back
and reschedule earlier activities wheneverits previous
The next section gives an overview of how we are
dealing with the problem and describes how to represent resource usage and release as a "consumable resource network" (cRN) that can be associated with the
Executive STN. The section after that describes conditions that insure that the cRNwill support dispatchability in the combinednetwork. The next section discusses methods that can be used during the planning
stage to track resource use and detect oversubscription. The following section describes the construction
of a cRNbased on this information. The final section
gives conclusions.
54
What
The
Present
are
We Aiming
For?
types (along with an Idle state), instances of which are
inserted in the SSRtime-line.
Consumable resource constraints can be incorporated into the data structure sent to the Executive via
a separate subgraph that we will designate as the consumable resource network, or cRN. In Figure 1 the
cRN is shown as a subnetwork apart from the temporal subnetwork. This figure is to be taken strictly as
a means of showing the reader the relations between
parts of the overall structure. In particular, the ’null’
nodes and associated arcs in the cRNare not needed
in practice, but are included so that resource use can
be represented in a way that parallels the temporal
duration represented for the same activity in the STN.
In the cRNdepicted in Figure 1, intervals represent
bounds on resource use for a given activity. For example, in this figure each interval, [10,20], represents
a range of possible use of a resource between 10 and
20 units. In addition, k-ary constraints within sets of
variables prevent the resource capacity from being exceeded. In the present example, the capacity is 30 resource units, and the sum of the upper bounds exceeds
capacity by 10 units. Therefore, if activity x starts
before y, and if the duration stipulated for the former activity results in its using more than 10 resource
units, then the upper bound of y must be reduced by
the excess amount in order to satisfy the constraint
between x and y. Wewill refer to a set of activities
like this as a "bout" of resource use; in practice, bouts
are succeeded and possibly preceeded by an instance
of resource release (not shownin Figure 1).
In the full network, the cRNmust be associated with
STNnodes that correspond to the same activities. Resource use is assumedto be a nondecreasing function of
time, and for expository purposes we will assume a linear relation, specifically, multiplication of the temporal
bounds by a positive or negative quantity for resource
use and release, respectively, this quantity being constant for a given activity. Of importance is the fact
that the mapping from STNevent to associated resource use is both bijective, i.e., one-to-one and onto,
and monotonic. In Figure 1, the linkage between STN
and claN is shownexplicitly by means of dashed lines,
each labeled with its constant of proportionality. In
practice, a constraint of the form, 5(d - b) ÷ 2(g < 30 might be added to the STN, or it might only be
implicit in certain procedures carried out on the STN
(cf. below).
HSTS System
HSTS(for Heuristic Scheduling Testbed System) is
planning and scheduling system originally developed
at Carnegie-Mellon University and later put into use
at the NASA-Ames
Research Center as part of its program in automated scheduling. The system is designed
to create and schedule plans of action for systems such
as spacecraft and orbiting telescopes, that have multiple components whose activities must be coordinated.
This is done by building schedules along multiple timelines, or "state variables", one for each dynamic feature of the system. Activities are represented in terms
of types. For example, a power component might be
represented as a state variable with two types called
"power-low" and "power-high". Associated with each
type is a set of relevant constraint types. Instances
of each activity-type, called "tokens", are inserted on
the time lines along with their constraints; following
such insertion, constraint propagation is carried out.
To improve efficiency this is done using associated constraint networks. In particular, temporal events associated with tokens are represented in a separate network
that, for reasons of tractability, is in the form of a simple temporal network (STN) (Dechter, Meiri, ~z Pearl
1991).
In the present scheduling system, the PlannerScheduler sends the Executive an envelope of acceptable scheduling times, also in the form of an STN.
A critical characteristic of this network is that it is
not only consistent but dispatchable. This means that
events can be instantiated successively, whenever the
current time is within the present bounds of an event,
and the system will not encounter a dead-end, i.e., an
event that cannot be scheduled because the current
time is beyond its present time bounds (Muscettola,
Morris, & Tsamardinos 1998).
Incorporating
the Executive
Consumable
Network
Resources
Into
We begin with an example. Most if not all spacecraft use an onboard solid state recorder (SSR)
store data collected via cameras and other sensors temporarily, until the craft is in communicationwith the
Deep Space Network on earth, when the information
is downloaded. Data storage occurs during a Record
state; in this state resource capacity is used up. Data
transmission occurs during a Playback state, when information is relayed to earth; in this state resource
capacity is released. Change in the amount of storage used occurs at a fixed rate during each instance
of Record or Playback. The SSRis, therefore, typical
of consumable resources in that it has a fixed capacity that cannot be exceeded during the schedule. (It
is more general than manyin that it is also intermittently renewable.) In the HSTSrepresentation,
the
SSRis a state variable, with Record and Playback as
Making
(Executive)
cRN Support
Dispatchability.
At the outset, it is important to note that the tractability of the STN-cRNas a constraint satisfaction problem is not a fundamental problem. This is because
both temporal and consumable resource constraints
are closed under a binary function, min, that is associative, commutative, and idempotent. This CSP,
therefore, falls under tractability class 2 of (Jeavons,
55
the
Obviously, in this case the Executive does not need to
process the cRNat all, since whatever values it selects
from the STN,the resulting resource use will be within
capacity.
This condition puts limitations
on the range of
choice that can be given to the Executive. This can
be seen in Figure 1, where the sum of upper bounds
(40) is well above the capacity (30). At the same time,
the cRNin this figure is fully consistent, because for every value of resource usage chosen for activity x there
is a usage value for activity y within the designated
bounds. Given the constraint between these variables,
then any selection for one variable can be matched by
a selection for the other in such a waythat the resource
capacity is not exceeded.
This suggests that th.ere are weaker conditions for
dispatchability with the general form
j
k
[10,20],(~
y_< 30-x~ (\
(--)
[2,4]
, ~
tIo,2o]
*2\\
\\
,5 / ~ --5
\\
.hh \\
//
[7,101
[4,7] ~ [~,1ol
~
Figure 1:
Composite temporal-and-consumableresource network, in which intervals for duration
(STN) and resource use (cRN) associated with
same activity are linked together. Such links are indicated by dashed lines; the linked intervals are those
associated with arcs directed toward the nodes of origin and destination for the cross-links. For example,
activity x in the cRNis associated with arc (b,d)
the STN.See text for further explication.
i----1
+ r. tb, < C,.,,
i=j-}-i
(3)
for two mutually exclusive and exhaustive subsets of
resource activities within a bout of length k. In particular, the condition
k--1
E ubi + lbk <_ Cini,
(4)
i=1
holding for all subsets of k - 1 activities in a bout of
length k would allow any value of resource use to be
used for the first k- 1 activities, and still leave at least
one value available for the last one.
Unfortunately, it has been shown that conditions
such as these are not sufficient to support dispatchability (Wallace ~ Freuder 2000a)(Wallace & Freuder
2000b). In a nutshell, the problem is due to interactions between the STNand cRN: propagation over
resource constraints can exclude values in the STN,
while propagation in the STNcan exclude values for
resource use. As a result values may be lost that are
necessary to insure STN-dispatchability or, conversely,
that are necessary to fulfill conditions on resource usage of the kind just described. In previous work it has
been shown that dispatchability can be supported if,
for a critical subset of activities associated with lower
bounds in equation (2), the duration is independent
of the start-time. (Such activities will be referred to
as "buffers", since they serve to buffer the scheduling
procedure against going over resource capacity.) This
extra requirement insures that the lower bound values will be available regardless of the values (for times
or resource use) chosen for their predecessors and successors. This, together with the condition described
by inequation (2) will be referred to as a condition
type ii.
Conditions of this type mayappear to put severe restrictions on the values that must be chosen for some
activities, in order to meet the inequality condition.
Cohen, & Gyssens 1995).
To support dispatchability, a cRNmust allow any sequence of instantiations to be made in the associated
STNwithout resource capacity being exceeded. Given
the bijective mapping from STNevent to resource use,
this implies that any choice of value for an instance of
resource use must allow some values to be chosen for
all future (as yet uninstantiated) variables. Werefer
to this loosely as "cl:tN dispatchability". (The present
discussion is presented in terms of a single bout of resource use, between instances of release. Elsewhere the
analysis is extended to sequences of bouts (Wallace
Freuder 2000a)(Wallace & Freuder 20005).)
The only necessary condition for cRNdispatchability appears to be the following:
k
X:lb <
(1)
i---1
In words, the sum of the minimumusage for all activities in a bout must not exceed the current capacity.
Fortunately, there are weakersufficient conditions.
Weconsider two such conditions for cRN dispatchability, that we call condition (i) and (ii). The first
simply that the sum of the upper bounds on resource
use be less than or equal to the initial resource capacity,
k
<
(2)
56
Fortunately, these restrictions can be ameliorated in
various ways. First, it should be noted that one can use
the smallest possible buffer set consistent with inequality (2). Ideally, this wouldbe a singleton. Moreover,
there are more than one such minimal sets, then these
can be tracked during execution, so that the last one
is the one chosen as the actual buffer set.
In addition, it should be possible during execution
to track the difference between the upper bound on
resource for an activity and the actual usage. The
sum of these differences, called the "credit" (Wallace
& Freuder 2000a) (Wallace & Freuder 2000b), can then
be added to the lower bounds for the activities
in
the buffer set without exceeding capacity or otherwise
compromising dispatchability.
To summarize at this point, what we are aiming for
is an Executive cRN-STNnetwork that satisfies either
condition for dispatchability across each bout of resource use. The remainder of the paper describes how
such a network can be built within the HSTSframework.
¯ the dispatchability condition we wish to meet, i.e.
condition (i) or someversion of type (ii)
Wemust also take into account the fact that our task
is really two-fold, because we are always computing
in terms of bounds on events (the intervals in Figure
1) and, therefore, activity durations. This gives lower
and upper bounds on resource use for each relevant
activity, that need to be monitored separately. The
task of handling lower bounds is simpler, because any
instance of excess use must be avoided (inequality (1)
above).
Dynamic features include adding an activity that
goes at the end of the list, inserting a new activity
within the existing sequence, and adjusting the time
bounds for an activity. In addition, activities maybe
removedif the procedure is forced to backtrack. Placement of activities that release the resource (e.g., Playbacks) can be considered a further dynamic condition.
In some cases these activities may be inserted at our"
discretion, but in manycases there will probably be
fairly tight restrictions on the times of their occurrence.
Under such conditions, it appears unlikely that a series of sums could be used to track resource use. The
problem is that separate sums are based on a partition
of the set of activities into bouts. But this partitioning can only be done when one knows the locations
of instances of resource release. Unless these can be
specified a priori, neither the numberof sums nor the
association of an activity with a particular sum can be
determined. Moreover,if activities are shifted in time,
the Planner must recognize when this changes the partition, and this requires more apparatus than a series
Tracking Resource Use During
Planning
In the planning stage, a schedule is built by laying
down tokens on time lines, where some of these tokens represent activities that use the resources in question. With respect to consumable resources, our task
is to monitor resource use and use this information to
produce flexible, dispatchable schedules for the Executive that cannot result in oversubscription of these
resources.
In this context and perhaps more generally, we can
identify three approaches that might be used to monitor resource use to avoid oversubscription. These are:
1. maintaining sums of resource usage between instances of resource release
2. maintaining lists of activities that use the resource,
where these activities are ordered by time (especially
starttimes)
3. maintaining a graphical representation of resource
use, based on time lines that are isomorphic with
the time lines used in HSTS, to ’map’ resource use
by activities associated with different components
(state variables) of the system
These strategies are ordered by complexity of representation and processing. Obviously, we would like
to use the simplest strategy possible in a given situation. However, we must also consider computational
adequacywith respect to several criteria:
¯ whether the method will function adequately in the
dynamic setting of automated schedule production
¯ whether it can handle special constraints, in particular, those related to occurrences of resource release
¯ flexibility
of response to instances of potential
overuse
of sums.
Onthe other hand, it appears that a single list of activities ordered by start times can suffice, provided that
connections are maintained with appropriate activities
(tokens) on the time lines. Partial sums representing
minimal and maximal resource use by the end of each
activity must also be associated with each element in
the list. This strategy can be considered an amplified
version of the second approach listed above. It differs
from the third strategy not only in being restricted to
one sequence of activities that fall on different time
lines, but in excluding activities that do not affect resource capacity. (We have also developed a version of
the third approach, with a separate line for each state
variable associated with resource use, as well as isomorphism between activities on the temporal and resource
time lines, but in view of the above conclusion we will
not discuss it here.)
In the present conception, there is no special treatment of cases where activities that use a resource overlap in time. (Weassume, of course, that in such cases
the mechanicsof the system will allow this.) This is because the important information is how much resource
is used, which doesn’t depend on whether or not instances of use overlap in time. The same argument
holds for overlap of resource use and resource release,
57
since the relation being monitored is that between accumulated usage and amount of resource released.
Since adjustments to the changes mentioned above
appear to be straightforward in each case, they will not
be treated in detail, other than to note that rearrangement of the sequence may be necessary after bounds
have changed, and that after each alteration, the partial sums associated with successive activities must be
recalculated from the earliest change to the end of the
list.
In regard to flexibility of response, possible responses
to instances of oversubscription include reducing time
(and, therefore, resource) bounds, deleting activities
or shifting them ahead of an instance of resource release, adding an instance of resource release if this is
possible, or simply tracking excess usage in order to
add constraints to the Executive network. All of these
appear to be possible with the ordered list described
above.
Which response we choose will depend in part on
the condition we are trying to satisfy. Thus, the lastmentionedstrategy is only possible if our goal is to satisfy a condition of type (ii). (Hence, it can only be used
in the context of upper bounds.) In general, a rough
guiding principle might be to try to meet condition (i)
if this doesn’t have a marked effect on scheduling options or flexibility of execution. Another factor is the
place in a sequence of activities where capacity is first
exceeded. If this is late in the sequence, shifting activities might be practical, otherwise it maybe necessary
to reduce bounds or add a constraint. In manycases,
choice of strategy, especially betweenthe first two, will
depend on other criteria pertaining to planning goals.
If we are trying to satisfy conditions of type (ii),
important part of the tracking process involves determining minimal sets of buffers. This process is shown
schematically in Figure 2. Part of this process (subsumed by the test that the full set of buffers satisfies inequality (ii) - cf. figure) involves determining
that an activity is a buffer. This can be done during ~he planning phase by using the same dominance
rules, based on a triangle equality, that are used by
(Muscettola, Morris, & Tsamardinos 1998) to produce
a minimal dispatchable STNafter the plan has been
set up. Briefly, suppose that activity X uses the resource in question, and this activity is represented in a
temporal network by arc BC; in this case C represents
the end-time. If there is another arc, AC, ending at C,
then ifAB + BC --- AC, arc ACcan be removed at the
time when the Executive STNis made dispatchable. If
this can be done for all arcs other than BCthat end in
C, then activity X can be considered a buffer.
The basic procedure represented by the flowchart
can be augmented for special purposes. In particular, we maywant to insure that singleton buffers occur
as close to the end of each bout as possible in order
to maximizeflexibility during execution and/or to insure the effectiveness of execution procedures (cf. next
Figure 2: Flowchart for discovering a minimal set of
buffers for one bout of activities that satisfies condition
(ii).
section). In this case, the first buffer test and the procedure "adjust network" would include procedures for
manipulating the network to establish such conditions.
Creating
an Executive
network
Setting
up for execution
If the goal is to send feasible bounds to the Executive (in connection with condition (ii)), then the
work that the Executive receives must be elaborated
to include information about capacity bounds. Since
the critical decision during execution is the choice of
end-time, and therefore duration, for the last buffer
activity, all that is neededis to keep track of the credit
before this point. In general, this can be done with
a counter, in addition to a mechanismfor identifying
nodes associated with resource use. However, if the
buffer is the last activity in the bout, then resource
constraints like the one depicted in Figure 3 (part of
which is identical to the cRNin Figure 1). will be sufficient. (If the last buffer is earlier in the bout, then
a constraint of this sort will not be effective in setting
maximalusage consistent with inequation (2) or (3).)
In Figure 3 there is one bout composedof two activities, x and y, so there is a constraint associated with
this set. In addition, there is an instance of resource
release, z, and a constraint betweenit and the resourceusing activities to insure that release (e.g., Playback)
58
carried out during the planning and scheduling stage
(in Figure 2, within the "adjust network" procedure)
when time constraints are not so severe, it may need
to be done repeatedly as well as for every bout, so that
speed of execution will probably be important.
If the condition is not met, then determining an
adjustment that will meet the inequality is also easy.
(This argument does not consider restrictions due to
considerations outside those of trying to meet this condition.) Thus, if the sum for one of the subsets is < the
capacity, then the upper bound that does not appear in
the sum can be reduced to insure dispatchability. The
amount it must be reduced is equal to a difference of
differences. Supposea is the element in question, i.e.,
the activity whose lower bound is in the sum that is
< the limit. And suppose that b is the element whose
lower bound is in the sum that exceeds the limit by
the greatest amount. Viz,
© [lo,2o]
Figure 3: Example of an Executive cRN that could be
constructed by the Planner/Scheduler.
does not go on longer than necessary. (Note that this
constraint is of the same form as the first, to insure
that tractability conditions are met.)
With either approach, an additional complication
occurs if Playback does not release all of the capacity that has been used. In this case, the limit in the
resource constraint must be adjusted. This must be
done in part in the planning phase by considering lower
bounds on the capacity that each Playback will release
and upper bounds on preceding resource use (not exceeding capacity), although the resulting estimates will
be very conservative. Fortunately, it should be possible
to adjust these limits during execution, provided one
tracks actual resource use during each bout, together
with subsequent release.
An alternative method of employing resource constraints is to add constraints that include variables
representing amount of resource release. For example,
suppose activity y in Figure 2 occurs after activity z.
Westill need a constraint that x be less than capacity
C and one that specifies that z - x is _< 0. In addition,
we could add the constraint, x q- y + z _< C. (Note that
the latter is not sufficient by itself, because we cannot
let capacity be exceeded even temporarily.) However,
this approach becomesunwieldy with further instances
of usage and release, because further constraints must
cover the entire set of activites up to that point.
k-2
E ubi + ubb q- lba < Cinit
i=1
and
k--P.
ubi + uba + lbb > Cinit "
i=l
Then the upper bound of a can be reduced by the
following amountto insure dispatchability for this set:
(uo - lo) - - lb)
Since (ua - In) must be greater than (Ub -- lb), we know
that this difference must be positive, and obviously
it is _< the original difference between ua and la. If
there is no sum less than the limit, it is sufficient to
choose a sum, reduce one or more upper bounds until
the condition in equation (3) is met, and then use the
above procedure.
Given that the basic conditions for dispatchability
have been met, we can also add further constraints
that allow later activities to have more flexibility in
resource usage. For example, if we want to insure such
flexibility for the kth activity, we maybe able to add
a constraint based on the following formula:
Meeting dispatchability
conditions
for all
singleton activities
Provided that condition ii holds for some singleton
buffers, we may want to establish the condition de,
scribed by inequation (3) for all activities in the bout.
In this way, if the resource use for any activity within
the bout is within the bound set by the lower bound
plus the current credit, then the resource capacity cannot be exceed during the bout. This is another means
of increasing the flexibility of resource use during execution.
To determine whether this condition holds for a bout
of activities, we need to examine O(k) subsets of size
k- 1. These can be generated in sequence by swapping
single activities in and out; at the same time, respective
sumsafter the first can be generated by single additions
and subtractions. Therefore, the complexity of this
procedure is only O(k). Although this calculation
/¢-1
_< c-
(6)
i=1
where lb~ < t < ubk. In this case we must assure that dispatcha-bility conditions hold for this subset of activities. Of course, this idea maybe redundant
with credit-based strategies that do not require explicit
constraint-based processing; it is presented here as a
further possible strategy.
Summary and Conclusions.
In this work we have shown how to create a network
involving both temporal relations and consumable resource constraints that can support dispatchable execution. In the first place, this involves meeting limiting conditions that guarantee that resource capacity
59
will not be exceeded. Whenflexibility of usage is one
of the goals, then these conditions must also involve
resource constraints in some form, either in terms of
explicit constraints added to the Executive network
or credit-tracking procedures. Procedures of this sort
are described in this work. In addition, we have devised methods with low time complexity for establishing some conditions, that can be carried out prior to
setting up the Exec~’tive network.
The present work is pertinent to a large class of problems encountered in planning by autonomous systems
like the RemoteAgent, when consumable resources like
the Solid State Recorder are involved. In these systems, greater flexibility of resource use can makeoperations involving tasks such as data collection more effective, thus increasing the likelihood that overall mission goals are accomplished.
Acknowledgements.
This work was carried out in association with the
NASA-AmesReseach Center, Moffet Field, CA and
was partly supported by the National Science Foundation under Grant No. IRI-9504316.
References
Dechter, R.; Meiri, I.; and Pearl, J. 1991. Temporal
constraint networks. Artificial Intelligence 49:61-95.
Jeavons, P.; Cohen, D.; and Gyssens, M. 1995. A unifying framework for tractable constraints. In Montanari, U., and Rossi, F., eds., Principles and Practice
of Constraint Programming . CP ’95, volume 976 of
Lecture Notes in ComputerScience. Berlin: Springer.
276-291.
Muscettola, N. 1994. HSTS: Integrating
planning
and scheduling. In Zweben, M., and Fox, M. S., eds.,
Intelligent Scheduling. San Mateo, CA: Morgan Kaufmann. 169-212.
Muscettola, N.; Morris, P.; and Tsamardinos, I. 1998.
Reformulating temporal plans for efficient execution.
In Proceedings Kt~-98.
Muscettola, N.; Nayak, P. P.; Pell, B.; and Williams,
B. C. 1998. Remote agent: to boldly go where no AI
system has gone before. Artificial Intelligence 103:547.
Wallace, R. J., and Freuder, E. C. 2000a. Dispatchable execution of schedules involving consumable resources. In Proceedings Fifth International Conference on Artificial Intelligence Planning and Scheduling (AIPS-2000), 283-290. Menlo, CA: AAAIPress.
Wallace, R. J., and Freuder, E; C. 2000b. Dispatchability conditions for schedules with consumable resources. In Proceedings Fourteenth European Conference on Artificial Intelligence (ECAI-O0), (to appear).
6O