Download AI PLANNING FOR TRANSPORTATION LOGISTICS

AI PLANNING FOR TRANSPORTATION LOGISTICS
Ioannis Refanidis
Aristotle University of Thessaloniki, Dept. of Informatics
Thessaloniki, Greece
E-mail: [email protected]
Nick Bassiliades
Aristotle University of Thessaloniki, Dept. of Informatics
Thessaloniki, Greece
E-mail: [email protected]
Ioannis Vlahavas
Aristotle University of Thessaloniki, Dept. of Informatics
Thessaloniki, Greece
E-mail: [email protected]
Abstract: In the last decade the efficiency of the Artificial Intelligence Planning Systems has been
increased significantly. New systems appeared that are able to cope with planning problems being
orders of magnitude more complex than the ones solvable in early 90's. This vast improvement
increase was made possible mainly by three new approaches in plan generation: planning graphs,
satisfiability planning and heuristic state-space planning. The latter approach, which is the most
powerful one, derives a heuristic function from the specification of a planning problem,
independently of its domain, and uses it for guiding the search through the space of the states. During
the last years appeared many heuristic state-space planners, such as ASP, HSP, GRT and FF, which
were able to solve large transportation logistics problems, with numerous locations, trucks and objects
that have to be transferred, very efficiently, as it has been shown in the recent international planning
competitions.
This paper briefly presents the current status in domain-independent heuristic state-space
planning and concentrates on the GRT and MO-GRT planners, where the latter is a recent extension
of GRT being able to consider multiple criteria in the plan generation and evaluation process. Finally,
the paper outlines results of running MO-GRT in some transportation logistics problems and poses
directions for future research.
Keywords: Transportation Logistics, Artificial Intelligence, Planning, Heuristic Search, Multiple
Criteria
INTRODUCTION
In the last decade the efficiency of Artificial Intelligence Planning Systems has been
increased significantly. New systems appeared that are able to cope with planning problems,
which are orders of magnitude more complex than the ones solvable in early 90's. This vast
improvement increase was made possible mainly by three new approaches in plan generation:
planning graphs ([1], [2]), satisfiability planning ([10], [11], [12]) and heuristic state-space
planning ([3], [9], [15], [18], [21]). The latter, and the most powerful approach, derives a
heuristic function from the specification of a planning problem, independently of its domain,
and uses it for guiding the search through the space of the states.
The first domain-independent heuristic state-space planner was UNPOP ([13], [14]) and
was followed by ASP [3], HSP [4], HSPr [5], GRT ([18], [21], [22]), FF [9] and AltAlt [15].
These planners search for solutions either in the state-space or in the regression space. Most of
them use variations of a relatively simple idea as a guide: they estimate the distance between
two states, based on estimates of the distances between each fact of the problem and one of the
two states. These planners are able to solve large transportation logistics problems, with
numerous locations, trucks and objects that have to be transferred, very efficiently, as it has
been shown in the recent international planning competitions.
The heuristics derived by the above planners concern only plan length. It is implicitly
supposed that all actions have the same duration, so the length of the plan corresponds to the
time needed to execute it (no parallel execution is considered). Other factors, such as resource
consumption, deadlines, profit, safety etc., are completely ignored. Actually, this is a
consequence of using the pure STRIPS framework [7], where this kind of information cannot
easily be represented. On the other hand, the incorporation of this kind of information is not an
easy task, since this would demand a revision of the algorithms by which the heuristic functions
are constructed.
The first planner that used multiple criteria for plan generation and evaluation is MO-GRT
([17], [20], [23]). With the word 'criteria' are referred both resources and any other type of
measurable quantities that are of interest, such as profit, safety, etc. The criteria have to be
provided by the user, together with the problem definition. They can be organized in
hierarchies, where the leaves are the measurable criteria, and preferences, in the form of weight
factors, have to be defined among them. MO-GRT succeeds in trading off planning time and
plan quality, based on the criteria hierarchy and the assigned weights.
The paper is organized as follows: Firstly is defined the planning problem and are given
some preliminary but necessary definitions. Next is presented the current status on domainindependent heuristic planning and subsequently are presented the GRT and the MO-GRT
planners in some more detail. Finally, some experimental results of running MO-GRT on a
transportation logistics domain are outlined. The paper concludes by posing directions for future
work.
THE PLANNING PROBLEM
In STRIPS [7] each action a is represented by three sets of facts: the precondition list Pre(a), the
add-list Add(a) and the delete-list Del(a), where Del(a) ⊆ Pre(a). A state S is defined as a finite
set of facts. An action a is applicable to a state S if:
Pre(a) ⊆ S
(1)
The state resulting from the application of an action a to state S is defined as:
S' = res(S,a) = S \ Del(a) ∪ Add(a)
(2)
Inductively we can define the state resulting from the application of a sequence of actions
(a1, a2, ..., aN) to a state S as:
S' = res(S, (a1, a2, ..., aN)) = res( res(S, (a1, a2, ..., aN-1)), aN)
(3)
with the requirement that each action ai is applicable to the state res(S, (a1, a2, ..., ai-1)), for each
i=1, 2, ..., N. In the formalization used henceforth, the set of problem constants is assumed to be
finite and no function symbols are used, so the set of actions is finite.
A planning problem P is a triplet P=(O, Initial, Goals), where O is the set of ground actions,
Initial is the initial state and Goals is a set of facts. The task is to find a sequence of actions a1,
a2, ..., aN that can be applied to the initial state, so that the state resulting from their application
will be a superset of Goals. The sequences of actions are called Plans. A plan that can be
applied to the initial state is called a valid plan. A valid plan that achieves the Goals is called a
solution of the planning problem. A planning problem may have several or no solutions. In the
latter case the problem is described as unsolvable.
DOMAIN INDEPENDENT HEURISTIC PLANNING
The recent evolution of the domain independent heuristic planning started with the work of
Drew McDermott ([13], [14]) and the UNPOP planner. The planner is not restricted to pure
STRIPS representations, supporting the more expressive ADL language [16]. UNPOP proceeds
forwards in the state-space. Estimates of the distances between states are based on the so-called
regression graph, which is built from the goals using partially instantiated actions. UNPOP does
not consider subgoals interactions and reconstructs the regression graph from scratch for each
intermediate state.
The area of domain independent heuristic planning has been pushed forward by the ASP [3]
and HSP [4] planners. These planners traverse the state-space in a forward direction. In order to
estimate the distance between each state and the goals, the planners compute an estimate of the
distance between each fact of the problem and the current state. The estimates are computed in
an additive manner, so interactions are not taken into account. The sum of the estimates of the
goal facts is considered to be the cost for achieving them from the current state.
The GRT planner ([17], [18], [21], [22]) has many similarities to the ASP/HSP ones.
However, it differs in two ways. Firstly, it computes an estimate for the distance between each
fact and the goals. These estimates are computed once, in a pre-processing phase, and in a
backward direction, while the state-space is traversed forwards. A similar approach has been
adopted in HSPr [5], a descendent of HSP; however, HSPr computes the estimates forwards,
whereas it traverses the state-space backwards. Secondly, GRT introduces the notion of the
related facts, in order to track the interactions that arise when trying to achieve several facts
simultaneously, thus GRT produces more accurate estimates.
A recent variation of HSP, named HSP-2 [6], incorporates both HSP and HSPr, while it
supports a plethora of new heuristic functions, some of them being admissible [8]. However, the
need for admissibility imposes extra work in the construction of the heuristic.
The more recent planners are AltAlt [15] and FF [9]. They use a Graphplan-based approach
[2] to estimate distances between states. AltAlt is a regression planner based on HSPr. It creates
a planning graph in a pre-processing phase and uses several techniques to extract heuristic
estimates of the distances between the intermediate states and the initial state. For example, one
of them returns the level in the planning graph, where all the facts of the intermediate state
appear, without any mutual exclusion relation between them.
FF is a forward heuristic planner. In order to estimate the distance between an intermediate
state and the goals, it creates a planning graph from the state to the goals, using relaxed actions,
i.e. actions from which the delete lists have been removed. From this graph, FF extracts a
relaxed plan, the length of which is the distance estimate.
THE GRT PLANNER
GRT ([17], [18], [21], [22]) computes, in a pre-processing phase, estimates for the distances
between the domain's facts and the goals, i.e. the number of actions that should be applied to the
goals, in order to achieve backwards the various facts of the domain.
In order to apply actions to the goals, the domain actions have to be reversed. Suppose that
we have an action . and two states s and s', such that . is applicable in s and s' = res(s,a). The
reversed action a' of . is an action, such that s = res(s', a'). The reversed action is defined from
the original action as it follows:
Pre(a') = Add(a) ∪ Pre(a) \ Del(a)
Del(a') = Add(a)
(4)
Add(a') = Del(a).
Note that in many domains the sets of the normal and the reversed actions are identical.
If case where the goals do not constitute a complete state description, it is impossible to
apply the reversed actions to it. The solution adopted is to enhance the goals with all the facts
that are not in contradiction with the goals. The new goal set constructed in this way is called
the enhanced goal state. The enhanced goal state can be constructed either manually, or in an
algorithmic way, or finally by exploiting domain dependent knowledge ([19], [21]).
As mentioned above, in the preprocessing phase GRT assigns to each fact p of the domain
an estimate of the number of backwards actions needed to achieve this fact starting from the
goals. Moreover, each fact is also assigned with a list of other facts {r1, r2, ..., rN}, denoted as
the related facts. These are facts that are added by some action in the path that achieves p and
are not deleted by any subsequent action. The related facts are potentially co-achieved while
trying to achieve p and are taken into account when estimating distances.
We can outline the way in which GRT computes the above estimates with the following
steps:
1) All the facts of the enhanced goal state are assigned zero distances and empty lists of related
facts.
2) If a fact p can be achieved by a reversed action a, then:
a) The cost of achieving p is a function of the costs and the lists of related facts of a's
preconditions.
b) The list of related facts assigned to fact p is defined as:
rel(p)= Pre(a) ∪irel(qi:qi∈Pre(a))
∪Add’(a)-{Del(a),p}
(5)
where rel(f) denotes the related facts of any fact f and Add'(a) denotes the facts that are
first achieved by a (Add'(a) ⊆ Add(a)).
The related facts play an important role in step 2a, where the cost of achieving a set of facts
is computed. Specifically, the cost of achieving a set of unrelated facts is considered equal to the
sum of their individual costs, whereas the cost of achieving a set of related facts is considered
equal to their maximum cost. In the general case, the set of facts is partitioned in disjoint sets of
related facts and the above rules are applied. It has been proved that this partitioning is always
feasible [21].
All the information obtained by GRT in the pre-processing phase is stored in a table. Since
this table is obtained through greedy regression of the goals, it is called the "Greedy Regression
Table" of the problem, thus coming up the acronym GRT. During the search process, the
distances between the intermediate states and the goals are estimated as a function of the costs
and the lists of related facts of the state's facts (it is exactly the same function as the one
mentioned in rule 2a above).
An Example
We illustrate the GRT phases with the block world problem of Figure 1. Part of the Greedy
Regression Table for this problem is shown in Table 1.
A
A
C
B
B
C
Initial State
Goal State
Figure 1: A 3-blocks problem
Fact
Distance from goals
Related facts
(on C table)
0
-
(on B C)
0
-
(on A B)
0
-
(clear A)
0
-
(on A table)
1
(clear B)
(clear B)
1
(on A table)
(on B table)
2
(on A table) (clear A) (clear B) (clear C)
(clear C)
2
(on A table) (clear A) (clear B) (on B table)
(on C B)
3
(on A table) (clear A) (on B table) (clear C)
...
...
...
Table 1: Part of the Greedy Regression Table
for the 3-blocks problem
Let us compute the distance between the initial state and the goals. The initial state consists
of the following facts:
(on A table) (clear A) (on B table) (on C B) (clear C)
All the above facts are related to the fact (on C B), which has the maximum distance. So the
distance of all the facts is their maximum distance, i.e. 3, which in this case is also the actual
distance.
The above approach is followed to estimate the distances between all the intermediate
states, which arise during the forward search phase, and the goals. GRT always selects to
expand the state with the smallest estimated distance.
THE MO-GRT PLANNER
MO-GRT ([17], [20], [23]) extends the GRT one with the ability to take multiple criteria into
account. As criterion it is considered any type of measurable quantity, which is of interest in the
solution plan. The criteria are provided by the user, along with the definition of the problem.
Criteria can be organized in hierarchies, where the leaves are the measurable quantities, and
preferences have to be defined among them. An example of a criteria hierarchy is shown in
Figure 2.
evaluated entity (entire plan)
past plan
fuel
duration
remaining plan
free-volume
safety
length
fuel
duration
safety
Figure 2. An example criteria hierarchy
MO-GRT differs from GRT in that it assigns a set of non-dominated value-vectors to each
fact, which estimate the total cost of achieving the fact from the goals, with respect to the
various criteria, using alternative paths. The states are evaluated using both the known
accumulated value of the past plan, and the estimated value of the remaining plan, using the
value-vectors of the state's facts. The search-space is traversed using a weighted A* strategy,
which enables the planner to trade off planning time and plan quality.
A TRANSPORTATION LOGISTICS DOMAIN
In this section it is briefly presented the application of the MO-GRT planner in a transportation
logistics domain. A more detailed presentation can be found in [17] and [23].
The logisticsMO domain
In order to demonstrate the efficiency of the MO-GRT planner, it has been used as a basis the
logistics domain [24], which is commonly used in the bibliography and in the planning
competitions (AIPS-98 and AIPS-00). In this domain there are several cities, each one
containing several locations, some of which are airports. There are also trucks, which can move
within a single city, and airplanes, which can fly between airports. The goal is to get some
packages from their initial locations to their destinations.
In the original logistics domain there is a single means of transportation to transfer an object
between two cities: the airplane. In order to measure the effectiveness of MO-GRT, this
description has been extended with trains, which can only perform transportations between
different cities, and one location in each city has been characterized as a train station. This
extended logistics domain has been called logisticsMO.
Two criteria have been introduced: cost and duration, and application costs and durations
have been assigned to all domain actions schemas. Obviously, using planes for transportation
implies a greater cost but a lower duration than using trains instead. For both criteria the lower
values are preferable.
Experimental Results
Several experiments have been conducted, with different problems and different weights in the
criteria, and the following conclusions have been drawn:
•
As the weight of criterion length increases (decreases), the planner reaches faster (slower) a
solution and produces slightly worse (better) plans in terms of the other criteria.
•
As the weight of the remaining plan increases (decreases), with respect to the weight of the
past plan, the planner reaches faster (slower) a solution and produces slightly worse (better)
plans in terms of all criteria.
•
As the weight of a criterion (cost or duration) increases (decreases), the planner produces
better (worse) plans in terms of this criterion and worse (better) plans in terms of the rest of
the criteria. Moreover, solution time increases (decreases).
•
For each criterion and for each problem there is an ideal scale. As its scale diversifies from
this ideal scale, the resulting plan worsens in terms of this criterion and is improved in terms
of the remaining criteria. Solution time is affected accordingly.
•
If we zero all the weights, except for length and for the remaining plan, the planner behaves
as the single-objective one.
CONCLUSION AND FUTURE WORK
In this paper we have briefly presented the current status in the area of domain-independent
heuristic state space planning and we have concentrated in the GRT and MO-GRT planners. We
have overviewed some experimental results of running MO-GRT over some transportation
logistics problems and we have outlined some conclusions about how the various criteria affect
the efficiency of the planning process.
Possible directions for future work include embedding current planning systems in
integrated information systems, with friendly interfaces, databases, geographical information
systems and sensors for real-time monitoring of the environment. In this way we could exploit
their efficiency in every day life, for solving real-world logistics problems.
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Biographical sketches of the authors
Ioannis Refanidis received his BS degree in physics in 1992 from the Aristotle
University of Thessaloniki, Greece, his BS degree in computer science in 1997 from the
same institution, and his PhD degree from the Department of Informatics of Aristotle
University, in the research area of heuristic planning systems. He has published over 15
papers at international journals and conferences. He has been involved in several
projects in the areas of planning, expert systems, software evaluation and educational
software. His research interests include planning, scheduling, constraint solving,
parallelism, software engineering and educational software. He currently serves as a
member of the executive council of the Greek Association on Artificial Intelligence
(EETN), for the period 2000-02, he is a member of the American Association on
Artificial Intelligence (AAAI) and a member of the Greek Physics Society (HPS).
(http://www.csd.auth.gr/~lpis/people/giannis.html).
Nick Bassiliades received the BS degree in physics in 1991 from the Aristotle
University of Thessaloniki, Greece, the MS degree in applied artificial intelligence in
1992 from the University of Aberdeen, Scotland, and the PhD degree in parallel
knowledge base systems in 1998 from Aristotle University, Thessaloniki, Greece. He is
currently a part-time lecturer at the Department of Informatics, Aristotle University of
Thessaloniki. He has been involved in projects concerning expert systems for software
engineering, distributed expert systems and distributed constraint logic programming.
He has published over 15 papers at international journals and conferences and
coauthored a book on Parallel, Object-Oriented, and Active Knowledge Base Systems
by Kluwer Publishers. His research interests are deductive object-oriented databases,
active databases, knowledge base systems, parallel database systems and web databases.
He is a member of the Greek Physics, Computer and Artificial Intelligence Societies
and a member of the IEEE and the ACM. (http://www.csd.auth.gr/phd/bassil.html)
Ioannis Vlahavas received the PhD degree in computer science (Logic Programming
Machines) from the Aristotle University. He is an associate professor at the Department
of Informatics at the Aristotle University of Thessaloniki, Greece. His research interests
include logic programming, knowledge base systems, and AI applications. He has
published more than 50 papers, four book chapters, and coauthored two books in these
areas. He has led a project concerning an intelligent assistant for software evaluation
and a project for the development of a distributed expert system for the management of
a national network while he has been involved in many others. During 1997, he was a
visiting scholar at the Department of Computer Sciences at Purdue University. He is a
member of the Greek Physics and Computer Societies, a member of the IEEE, and a
member of the Assoc. for Logic Programming. Currently he serves as the conference
chair of the 2nd Hellenic Conference on Artificial Intelligence, which will be held on
April 2002, in Thessaloniki (http://www.csd.auth.gr/teachers/vlahavas.html).