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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. REFERENCES [1] Blum A, Furst M. Fast planning through planning graph analysis. Proceedings of the 14th International Joint Conference on Artificial Intelligence; 1995; Montreal, Canada. AAAI Press; 1995. [2] Blum A, Furst M. Fast planning through planning graph analysis. Artificial Intelligence, 1997; 90:281-300. [3] Bonet B, Loerincs G, Geffner H. A robust and fast action selection mechanism for planning. Proceedings of the 14th National Conference on Artificial Intelligence (AAAI-97); 1997; Providence, Rhode Island. AAAI Press 1997. [4] Bonet B, Geffner H. HSP: Heuristic search planner. Entry at 4th International Conference on Artificial Intelligence Planning Systems (AIPS-98) Planning Competition; 1998 April; Pittsburgh, US. [5] Bonet B, Geffner H. Heuristic planning: New results. Proceedings of the 5th European Conference on Planning; 1999 September; Durham, UK. Springer; 2000. [6] Bonet B, Geffner H. Planning as heuristic search. Artificial Intelligence 2001; 129 (1-2): pp. 5-33. [7] Fikes RE, Nilsson NJ. STRIPS: A new approach to the application of theorem proving to problem solving. Artificial Intelligence 1971; 2:189-208. [8] Haslum P, Geffner H. Admissible heuristics for optimal planning. Proceedings of the 5th International Conference on Artificial Intelligence Planning and Scheduling Systems (AIPS2000); 2000 April; Breckenridge, Colorado. AAAI Press 2000. [9] Hoffmann J, Nebel B. The FF planning system: Fast plan generation through heuristic search. Journal of Artificial Intelligence Research 2001; 14:253-302. [10] Kautz H, Selman B. Planning as satisfiability. Proceedings of the 10th European Conference on Artificial Intelligence (ECAI-92); 1992 August; Vienna, Austria. Wiley and Sons, Chichester, 1992. [11] Kautz H, Selman B. Pushing the envelope: Planning, propositional logic and stochastic search. Proceedings of the National Conference on Artificial Intelligence; 1996 August; Portland, Oregon. AAAI Press 1996. [12] Kautz H, Selman B. BLACKBOX: A new approach to the application of theorem proving to problem solving. Proceedings of the AIPS-98 Workshop on Planning as Combinatorial Search, 1998 April; Pittsburgh, US. [13] McDermott D. A heuristic estimator for means-ends analysis in planning. Proceedings of the 3rd International Conference on Artificial Intelligence Planning Systems (AIPS-96); 1996; Edinburgh, UK. [14] McDermott D. Using regression-match graphs to control search in planning. Artificial Intelligence 1999; 109 (1-2):111-159. [15] Nigenda RS, Nguyen X, Kambhampati S. AltAlt: Combining the advantages of Graphplan and heuristic state search. Technical Report 2000; Arizona State University. [16] Pednault E. ADL: Exploring the middle ground between STRIPS and the situation calculus. Proceedings of the 1st International Conference on Knowledge Representation and Reasoning, 1989; Toronto, Canada. Morgan Kaufmann 1989. [17] Refanidis I. Heuristic Planning Systems. Dissertation thesis 2001; Aristotle University, Dept. of Informatics, Greece. [18] Refanidis I, Vlahavas I. GRT: A domain independent heuristic for STRIPS worlds based on greedy regression tables. Proceedings of the 5th European Conference on Planning; 1999 September; Durham UK. Springer; 2000. [19] Refanidis I, Vlahavas I. On determining and completing incomplete states in STRIPS Domains. Proceedings of the IEEE Intl. Conf. on Information, Intelligence and Systems. 1999 November; Washington DC. IEEE Computer Society 1999. [20] Refanidis I, Vlahavas I. A framework for heuristic planning using multiple criteria. Proceedings of the IJCAI-01 Workshop on Planning with Resources. 2001 August; Seattle, Washington. AAAI Press 2001. [21] Refanidis I, Vlahavas I. The GRT Planning System: Backward Heuristic Construction in Forward State-Space Planning. Journal of Artificial Intelligence Research 2001; to appear. [22] Refanidis I, Vlahavas I. The GRT Planner. AI Magazine 2001; to appear. [23] Refanidis I, Vlahavas I. Multiobjective heuristic state-space planning. Technical Report 2001; Aristotle University, Dept. of Informatics. [24] Veloso M. Learning by analogical reasoning in general problem solving. Ph.D. thesis 1992; Department of Computer Science, Carnegie Mellon University. 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).