In this section, we describe experiments that evaluate the use of summary information in coordinating a group of evacuation transports that must together retrieve evacuees from a number of locations with constraints on the routes. In comparing the EMTF and CFTF search techniques described in Section 5.2 against conventional HTN approaches, the experiments show that reasoning about summary information finds optimally coordinated plans much more quickly than prior HTN techniques.
We compare different techniques for ordering the expansion of subplans
of both and
plans to direct the decomposition of plan
hierarchies in the search for optimal solutions. These expansion techniques
are the
(for
subplans) and
(for
subplans) operators of the algorithm described in Section
5.1.
We compare EMTF's expansion of plans to the
ExCon heuristic and to a random selection heuristic. The ExCon
heuristic [Tsuneto, Hendler, Nau, 1998] first selects plans that can achieve an
external precondition, or if there are no such plans, it selects one
that threatens the external precondition. In the case that there are
neither achieving or threatening plans, it chooses randomly. Note
that EMTF will additionally choose to expand plans with only
threatened external preconditions but has no preference as to whether
the plan achieves, threatens, or is threatened. For the expansion of
plans, we compare CFTF to a depth-first (DFS) and a random
heuristic.
We also compare the combination of CFTF and EMTF to an FAF (``fewest
alternatives first'')
heuristic and to the combination of DFS and ExCon. The FAF heuristic
does not employ summary information but rather
chooses to expand and select
plans that have the fewest
subplans [Currie, Tate, 1991,Tsuneto, Hendler, Nau, 1997].
Since no summary information is used, threats are only
resolved at primitive levels. While it has been shown that the FAF
heuristic can be effectively used by an HTN planner [Tsuneto, Hendler, Nau, 1997],
the combination of DFS and ExCon has been shown to make great
improvements over FAF in a domain with more task interactions
[Tsuneto, Hendler, Nau, 1998]. We show in one such domain that the CFTF and EMTF
heuristics can together outperform combinations of FAF, DFS, and ExCon.
The problems were generated for an evacuation domain where transports are responsible for visiting certain locations along restricted routes to pick up evacuees and bring them back to safety points. Transports are allowed to be at the same location at the same time, but the coordinator must ensure that transports avoid collisions along the single lane routes. In addition, in order to avoid the risk of oncoming danger (from a typhoon or enemy attack), the transports must accomplish their goals as quickly as possible.
Suppose there are two transports, and
, located at safety
points
and
respectively, and they must visit the locations
0, 1, and 2 and 2, 3, and 4 respectively and bring evacuees back to
safe locations as shown in Figure 20. Because of
overlap in the locations they must visit, the coordinator must synchronize
their actions in order to avoid collision. The coordinator's goal network
includes two unordered tasks, one for each transport to
the
locations for which it is responsible. As shown in Figure
21, the high-level task for
(
)
decomposes into a primitive action of moving to location 0 on the ring
and an abstract plan to traverse the ring (
).
can
travel in one direction around the ring without switching directions,
or it can switch directions once.
can then either go clockwise
or counterclockwise and, if switching, can switch directions at any
location (
) and travel to the farthest location it needs
to visit from where it switched (
). Once it has
visited all the locations, it continues around until it reaches the
first safety point in its path (
and
).
The
plan is for the case where
is already at location
0. The task for
can be refined similarly.
Suppose the coordinator gathers summary information for the plan hierarchy
and attempts to resolve conflicts. Looking just at the summary
information one level from the top, the coordinator can determine that if
finishes evacuating before
even begins, then there will be
no conflicts since the external conditions of
's
plan
are that none of the routes are being traversed. This solution has a
makespan (total completion time) of 16 steps. The optimal solution is
a plan of duration seven where
moves clockwise until it reaches
location
, and
starts out clockwise, switches directions at
location 4, and then winds up at
. For this solution
waits
at location 2 for one time step to avoid a collision on the route from
location 2 to location 3.
We generated problems with four, six, eight, and twelve locations; with two, three and four transports; and with no, some, and complete overlap in the locations the transports visit. Performance was measured as the number of search states expanded to find the optimal solution or (if the compared heuristics did not both find the optimal solution) as the number of states each expanded to find solutions of highest common quality within memory and time bounds. We chose this instead of CPU time as the measure of performance in order to avoid fairness issues with respect to implementation details of the various approaches.
The scatter plot in Figure 22 shows the
relative performance of the combination of CFTF and EMTF (CFTF-EMTF) and
the combination of CFTF and random expansion (CFTF-Rand).
We chose scatterplots to compare results
because they capture the results more simply than trying to plot against three dimensions of problem size/complexity.
Note that
for all scatter plots, the axes are scaled logarithmically. Points above the diagonal line mean that EMTF (x-axis) is performing better than Rand (y-axis) because fewer search states were required to find the optimal solution. While
performance is similar for most problems, there are a few cases where
CFTF-EMTF outperformed CFTF-Rand by an order of magnitude or more.
Figure 23 exhibits a similar effect for CFTF-EMTF
and CFTF-ExCon. Note that runs were terminated after the expansion of
3,500 search states. Data points at 3,500 (the ones forming a horizontal line at the top) indicate that no solution
was found within memory and time
constraints. While performance is
similar for most problems, there are four points along the
top where CFTF-ExCon finds no solution. Thus,
although EMTF does not greatly improve performance for many problems,
it rarely performs much worse, and almost always avoids getting stuck
in fruitless areas of the search space compared to the ExCon and the
random heuristic. This is to be expected since EMTF focuses on
resolving conflicts among the most problematic plans first and avoids
spending a lot of time reasoning about the details of less problematic
plans.
The combination of CFTF with EMTF, pruning inconsistent abstract plan
spaces, and branch-and-bound pruning of more costly abstract plan
spaces (all described in Section 5.2) much more
dramatically outperforms
techniques that do not reason at abstract levels.
Figure 24
shows DFS-Rand expanding between one and three orders of magnitude
more states than CFTF-Rand. Runs were terminated after the
expansion of 25,000 search states. Data points at 25,000 (forming the horizontal line at the top) indicate that no
solution was found within memory and time constraints. By avoiding search spaces with greater
numbers conflicts, CFTF finds optimal or near-optimal solutions much
more quickly. In Figures 25
and
26,
CFTF-EMTF outperforms FAF-FAF (FAF for both
selecting and
plans) and DFS-ExCon by one to two orders of
magnitude for most problems. These last two comparisons especially
emphasize the importance of abstract reasoning for finding optimal
solutions. Within a maximum of 3,500 expanded search states (the
lowest cutoff point in the experiments), CFTF-EMTF and CFTF-Rand found
optimal solutions for 13 of the 24 problems. CFTF-ExCon and FAF-FAF
found 12; and DFS-ExCon and DFS-Rand only found three.
A surprising result is that FAF-FAF performs much better than
DFS-ExCon for the evacuation problems contrary to the results given by
tsuneto:98 that show DFS-ExCon dominating for problems with
more goal interactions. We believe that this result was not
reproduced here because those experiments involved hierarchies with no
plans. The experiments show that the selection of
subplans
more greatly affects performance than the order of
subplans to
expand. So, we believe DFS-ExCon performed worse than FAF-FAF not
because FAF is better at choosing
subplans than ExCon but
because FAF is stronger at selecting
subplans than DFS.
However, the main point of this section is that each of the heuristic combinations that use summary information to find solutions and prune the search space at abstract levels (CFTF-EMTF, CFTF-ExCon, and CFTF-Rand) greatly outperform all of those that do not (FAF-FAF, DFS-ExCon, and DFS-Rand) when searching for optimal solutions.