change occurs, for instance, when an agent observes that a city has ... In the TSP the objective is to find the shortest tour, that is, a permutation n of the cities {c\ ...
A Multiagent Approach To Solving Dynamic Traveling Salesman Problem Andrea Varga, Camelia Chira, and Dan Dumitrescu Citation: AIP Conference Proceedings 1117, 189 (2009); doi: 10.1063/1.3130623 View online: http://dx.doi.org/10.1063/1.3130623 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1117?ver=pdfcov Published by the AIP Publishing
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A Multi-agent Approach To Solving Dynamic Traveling Salesman Problem Andrea Varga"", Camelia Chira and Dan Dumitrescu" * 'Department of Computer Science, Babe^-Bolyai University, No. 1, Str. M. Kogalniceanu, 400084 Cluj-Napoca, Romania ''varga_andy84@yahoo. com, cchira@cs. ubbcluj.ro, 'ddumitr@cs. ubbcluj. ro Abstract. A multi-agent approach to solving dynamic Traveling Salesman Problem (TSP) is presented. In the dynamic version of TSP cities can be dynamically added or removed. Proposed multi-agent approach is based on the Sensitive Stigmergic Agent System model refined with new type of messages between agents. The agents send messages every time a change occurs, for instance, when an agent observes that a city has disappeared or appeared. The system is tested for various pheromone sensitivity level values and learning ability for agents. Numerical results and comparisons with the Ant Colony System and Ant System models indicate a good performance of the proposed model. Keywords: Dynamic TSP, Sensitive Stigmergic Agent System, Multi-agent System, Stigmergy. PACS: 07.05.Mh, 02.60.Pn
1. I N T R O D U C T I O N Dynamic optimization problems play an important role in practical applications and are a challenging field for optimization methods. Problem instances that may change over time require that optimization methods adapt the solution to the changing optimum. One key aspect refers to whether preliminary solutions already identified can be re-used to quickly find a new good solution after the problem data has changed. In this paper we investigate several strategies for Sensitive Stigmergic Agent System model [1] for combinatorial optimization. These strategies are studied for a dynamic problem where changes occur in time and the algorithm reacts explicitly to these changes. The various model strategies of the Sensitive Stigmergic Agent System algorithm are tested for the Dynamic Traveling Salesman Problem where changes in time refer to the deletion or insertion of some cities. The simplest way to handle the change of a problem instance would be to restart the algorithm after the change has occurred. However, if one assumes that a change of the problem is relatively small, it is likely that the new optimum will be in some sense related to the old one. Therefore it would probably be beneficial to transfer knowledge from the old optimization ran to the new ran. On the other hand, if too much information is transferred, the ran basically starts near a local optimum, and will stuck there. Thus, a reasonable compromise between these two opposing approaches has to be found. CPl 117, BICS 2008, Proceedings of the f^ International Conference edited by C. Enachescu, B. L. lantovics, and F. Gh. Filip ©2009 American Institute of Physics 978-0-7354-0654-4/09/$25.00
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Based on the general idea of stigmergy and sensitivity on agent-based models [1], we propose and test strategies to make Sensitive Stigmergic Agent System algorithms more suitable for the optimization in dynamic environments. The paper is structured as follows: Section 2 gives a definition of the Dynamic Travehng Salesman Problem. The Section 3 describes the Sensitive Stigmergic Agent System model used. The proposed algorithm is presented in Section 4. Further on, numerical experiments are discussed in Section 5, and conclusions are drawn in Section 6.
2. DYNAMIC TRAVELING SALESMAN PROBLEM 2.1. Traveling Salesman Problem In the Traveling Salesman Problem (TSP) which is closely related to the Hamiltonian cycle problem, a salesman must visit n cities. Modeling the problem as a graph with n vertices we can say that the salesman wishes to make a tour, or a Hamiltonian cycle, visiting each city exactly once and finishing at the city he starts from. There is an integer cost c(i, j) to travel from city / to city j and the salesman wishes to make the tour whose total cost is minimum, where the total cost is the sum of the individual costs along the edges of the tour. The formal language for the travehng salesman problem is TSP = : G = {V, E) is a complete graph, c is a function from F x V ^- Z,k e Z, and G has a traveling-salesman tour with cost at most k. In the TSP the objective is to find the shortest tour, that is, a permutation n of the cities {c\, C2, . . . , c„}, such that/( ;?•) is minimal, where/(;7-) is given as:
/ ( ; r ) = X (c(^(0, ^{i + m + c{7T{n), 7T{1)) . 2.2. Dynamic Traveling Salesman Variants The Dynamic Traveling Salesman Problem (DTSP) is a generalization of TSP where G is time-dependent. This problem has got several practical applications such as traffic jams [5] or fluctuating set of active machines [4]. Two main versions of Dynamic Traveling Salesman Problem exist in the literature. The first one consists of inserting or deleting cities into a given problem instance [3-4]. A different approach [5] consists of keeping constant the number of cities, allowing distance changing among them. It can be applied to address traffic jams and motorways. In this paper, we focus on the first approach.
2.3. Related Work In the following we consider the current state of research for Dynamic Traveling Salesman Problem using ant algorithms. A Population-based Ant Colony Optimization (P-ACO) algorithm was proposed in [3] where (as in Genetic Algorithm) a population
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of solutions is directly transferred to the next iteration. The solutions are then used to compute pheromone information for the ants of new iteration. For every solution in the population some amount of pheromone is added to the corresponding edges of the construction graph (every edge has been initialized with the same nonzero initial value before). In [4] several strategies were proposed for modiiying the pheromone information in reaction to a change for a dynamic Traveling Salesman Problem.
3. SENSITIVE STIGMERGIC AGENT SYSTEM MODEL The Sensitive Stigniergic Agent System (SSAS) is a metaheuristic [I] that combines multi-agent systems, stigmergic behavior (inspired by Ant Colony System) and a new concept of gradual pheromone sensitivity of agents. The system emphasizes a more robust and flexible system obtained by considering that not all agents react in the same way to pheromone trails. These autonomous, reactive agents are able to interoperate on the following two levels in order to solve problems: - direct communication: agents are able to exchange different types of messages; - indirect stigmergic communication: agents have the ability to produce pheromone trails that influence future decisions of other agents in the system. In this model each agent is characterized by a pheromone sensitivity level denoted by PSL, a real number in the unit interval [0, I]. The Sensitive Stigmergic Agent System model facilities a good balance between search exploitation and search exploration: - the PSL = 0 denotes that the sensitive stigmergic agent completely ignores the stigmergic information. Agents of this class - called small PSL agents (sPSL) will normally choose very high pheromone level moves. These agents are more independent and can be considered environment explorers; - the PSL = I denotes that the sensitive stigmergic agent has maximum pheromone sensitivity. In this case agents - called high PSL agents (hPSL) will choose any pheromone marked move and are able to intensively exploit the search regions already identified.
4. SSAS MODEL FOR SOLVING DYNAMIC TSP In this section we describe the generic decision process which forms the basis for SSAS algorithm, including SSAS-DTSP. When constructing solutions to a problem-instance, agents with high pheromone sensitivity level proceed in an iterative fashion, making a number of local decisions which result in a global solution. 4.1. State transition rule Agents pro-actively make decisions based on stigmergic strategy or direct communication.
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High PSL agents use stigmergic strategy to determine the next move: hPSL agents build a tour by repeatedly applying stochastic greedy rule (the state transition rule) [6]: ^^|argmax„^^^(^){[r(r,M)]" •[?^(r,u)f}Jfq 0). The state transition rule resulting from Eq. 1 and Eq. 2 is called pseudo-randomproportional rule [6]. [T{r,s)r.[ij{r,u)f
(2)
Pkir,s)0, otherwise This state transition rule favors transition toward nodes connected by short edges and with a large amount of pheromone. The parameter qo determines the relative importance of exploitation versus exploration: every time an agent in city r has to choose a city s to move to, it samples a random number 0 < q< l.lfq^qo then the best edge, according to Eq. 1, is chosen (exploitation), otherwise an edge is chosen according to Eq. 2 (biased exploration). Small PSL agents use direct communication or stigmergic strategy to determine the next move: sPSL agents consider the information received from other agents about what was the last visited city and decide to choose a different city to better explore the search space. When no message is received, the sPSL agents decide as hPSL agents.
4.2. Local Updating Rule While constructing its tour, an agent also modifies the amount of pheromone on the visited edge by applying the local updating rule (Eq. 3). High PSL and small PSL agents perform the same local updating rule: T{r,s)4^{l-p)T{r,s)+pro
(3)
where Q
