Multi-Cast Ant Colony System for Bus Routing

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This paper investigates a routing problem in a public transportation network, more precisely — the problem of finding an optimal journey plan in the Silesian Bus ...
MIC’2001 - 4th Metaheuristics International Conference

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Multi-Cast Ant Colony System for Bus Routing Problem Urszula Boryczka∗ ∗

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Mariusz Boryczka∗

Institute of Computer Science, University of Silesia ˙ Zeromskiego 3, 41-200 Sosnowiec, Poland Email: {uboryczk, boryczka}@us.edu.pl

Introduction

This paper investigates a routing problem in a public transportation network, more precisely — the problem of finding an optimal journey plan in the Silesian Bus Network, given a timetable. In our previous experiments we analysed the Ant System which was adapted to the Bus Routing Problem (BRP) [1]. Numerical experiments have shown that the evaluation function applied caused many difficulties and the system performance was unsatisfactory. At this time we try to improve the system achievement including the idea of Multi-Cast Ant Colony System (MCACS). Multi-Cast Ant Colony Optimisation is a new method, where two hierarchically connected casts of ants solve the Bus Routing Problem analysing two different optimisation criteria. Moreover, miscellaneous constraints may be imposed on the communication media between the casts of ants and on the rules used by the casts. Several scheduling problems in a transportation network have been extensively presented in the literature (e.g. [5, 6, 7, 13]). There are few commercial products aimed at assisting in train or bus scheduling and crew rostering, used or introduced nowadays, for instance GIST, HASTUS [8, 10, 11, 12]. In our paper we try to gain an insight into the routing problem, based on our investigation and experience gained during the experiments concerning the Silesian Bus Network.

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Bus Routing Problem

The problem of finding an optimal route in a transportation network in relation to a bus network can be formulated as follows: There is a set of points (bus stops) and bus lines, every one of a specific length, connecting pairs of these points. Several bus lines can connect a pair of points. There is a set of bus lines, each bus defined by a list of places where it halts, times of departures (except the terminal for every bus line) and times of arrivals (except the starting point for every bus line). Each bus is assigned a kind (e.g. slow, fast) and a special price of tickets. Given optimality criteria and additional constraints, find an optimal route between two points in a network subject to these constraints. The optimality criteria may concern, for instance: minimal time of journey, minimal number of bus changes, minimal distance travelled during the journey. The additional constraints may take into account following factors: the earliest time of departure from the starting point, the latest time of arrival at the ending point, assumptions about the kind and prices of the bus lines used in the connection, etc. Porto, Portugal, July 16-20, 2001

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MIC’2001 - 4th Metaheuristics International Conference

Multi-Cast Ant Colony System

Ant colony optimization (ACO) meta–heuristics defines a particular class of ant algorithms, called in this work ACO algorithms. Ant System (AS) [2, 3, 4, 5], which was the first ACO algorithm [2, 3] was designed as a set of three ant algorithms differing in the way the pheromone trail was updated by ants. Their names were: ant–density, ant–quantity, and ant–cycle. A number of algorithms, including meta–heuristic ones, were inspired by ant-cycle algorithm, the best performing of the three. The Ant Colony System (ACS) algorithm has been introduced by Dorigo and Gambardella [3] to improve the performance of Ant System [4], that allowed to find good solutions within a reasonable time for great problems. The ACS is based on 3 modifications of Ant System: (i) a different node transition rule, (ii) a different pheromone trail updating rule, (iii) the use of local and global pheromone updating rule (to favour exploration). For extension and adaptation this version of ACO to the BRP we take into account different optimality criteria and it causes new proposition of MACS firstly analysed for VRPTW [9]. We suggest the new idea of two casts of ants, each dedicated to the optimisation of a different objective function. IN MCACS– BRP the ants can co–operate/collaborate using the information about the best results obtained in the particular casts. The communication medium — pheromone trail laid on the appropriate edges belonging to the best tours plays the role of a reward measure for every leader of the cast. We must bear in mind that BRP is an extension of the bicriterion shortest–path problem, where we also minimise the length of tours for the weighted graph. Now, we consider an elementary version of this transportation network with two objective functions: (a) minimisation of the total travel time, comparative with the tour length, and (b) minimisation of the number of bus changes (the price is depended on this number). Basically, in MCACS–BRP both casts have optimised their functions using co–ordination of their activities. The goal of the first cast, CA–PRIME, is to diminish the number of bus changes, whereas the second cast, CA–SECOND minimises the total length of tours. Furthermore, each cast uses its own table of the pheromone, but casts can also influence each other by the pheromone. The collaboration between casts concerns sharing information about the best–known solutions obtained so far. More precisely, two casts search the same solution space, but they minimise different functions. We hope that double process of updating the pheromone improves the performance of the system searching for the best solutions for two objective functions. The node transition rules in both casts are similar and may be calculated as follows: when ant k is located in node i, it chooses the next node, j, randomly from the set of nodes Nik (not yet visited). The probabilistic rule used to construct a tour is the following: with probability q0 a node with the highest value of τijl (t) · [ηijl ]β , where j ∈ Nik , is chosen (exploitation), while with probability 1 − q0 the node j is chosen with measure pijl (t) proportional to τijl (t) · [ηijl ]β , j ∈ Nik (exploration):  τijl (t)·[ηijl ]β  if j ∈ Nik τimn (t)·[ηimn ]β k pijl = (1) m ∈N  n i 0 otherwise



where: β τijl ηijl

— — —

weight of the relative importance of the heuristic function, pheromone trail connected with bus line l and laid on the edge (i, j), attractiveness of the bus–line l between nodes i and j.

For the first cast, the attractiveness of the given edge we calculate according to the formula: ηijl = Porto, Portugal, July 16-20, 2001

1 δijl

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where δijl is a length of the edge (measured in time–units) connected with the bus line l. For CA– SECOND the attractiveness is represented by the formula: ηijl =

1 δijl

· rijl

where rijl is a coefficient determining the range of bus lines. We assume that Lk is a set of bus lines including the destination point. In this case the range of 3 is assigned to the bus line belonging to Lk . When the tour is created without any changes and not belongs to Lk we assign the value of 2 to appropriate bus line. Finally, the range of 1 is assigned to the bus line when there are changes and this line does not belong to Lk . In MCACS–BRP the pheromone trail is updated locally and globally. Local updating is performed during the process of solution construction whereas global updating is performed when the solution is completed. Consequently, the local updating rule effects in changing gradually the desirability of edges. In comparison, the global updating is used to intensify searching the neighbourhood of the best solution computed so far. The best result computed by each cast is used in the global pheromone–updating rule where, additionally, the number of changes, Cψgb and Cθgb , is considered: τijl (t + n) = (1 − ρ) · τijl (t) + ρ · τijl (t + n) = (1 − ρ) · τijl (t) + ρ ·

1 gb gb Jψ ·(Cψ +1) 1 Jθgb ·(Cθgb +1)

∀i, j, l ∈ ψ gb

— CA–PRIME

∀i, j, l ∈ θ

— CA–SECOND

gb

where: Jψgb Jθgb Cψgb Cθgb ψ gb θgb

— — — — — —

the the the the the the

length of the shortest tour for CA–PRIME, length of the best tour for CA–SECOND, number of changes in the shortest tour, number of changes in the best tour for CA–SECOND, shortest tour obtained so far, tour with the minimal number of changes obtained so far.

The local updating rule is used by each ant of the casts according to the formula: τijl (t + 1) = (1 − ρ) · τijl (t) + ρ · τ0 where τ0 =

1 h ·(C gb +1) s·Jψ ψ

is an initial value, Jψh is the length of the initial solution obtained by the greedy

algorithm, and s is the number of bus–stops.

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Experiments and results

This section reports our computational results showing the efficiency of MCACS for BRP. The real Silesian Bus Network consisting of 120 bus lines with about 1000 bus stops was analysed. Experiments were made by executing 5 runs for each starting point and destination point. The best solution was chosen for each cast and for MCACS and the results are reported in Table 1 and Table 2. Experiments have been done with the following parameters’ settings: mCA–PRIME = mCA–SECOND = 25, q0 = 0.9, β = 2, ρ = 0.25 (the same for each cast). Each tour consists in average of 100 nodes. Table 1 shows the best solutions obtained by the casts and MCACS and the number of bus line changes in different tours. It is characteristic that MCACS-BRP is able to designate the tours with a short time of travel and a small number of bus line changes. Table 1 reports the comparison of two parameters of the tour mentioned above (for casts and for the whole system). We analysed these results for CA– PRIME, CA–SECOND and MCACS. In addition, in Table 2 we compare the results obtained as an initial solution (we use a stochastic greedy algorithm) with the results produced by the MCACS–BRP. Porto, Portugal, July 16-20, 2001

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Tour no. 1 2 3 4 5

The best tour The best tour The best tour CA-PRIME CA-SECOND MCACS-BRP Travel Number of Travel Number of Travel Number of time changes time changes time changes 442 9 397 5 264 4 409 7 409 4 293 4 325 4 325 3 263 3 231 6 275 6 231 4 286 5 275 5 249 5

Table 1: Best results for casts and MCACS–BRP. Results obtained by MCACS–BRP significantly outperform the results obtained by the casts and by the stochastic greedy algorithm. The computational results for two different casts cannot be directly compared. According to these tables, we cannot show which cast is better. As can be seen from Table 1 (tour no. 4) the first cast finds the better result than the second one, but we can find another tours, showing that the second cast outperforms the first one. This property was exploited in our system. When one cast cannot manage to find a good result, another one helps it using a new heuristics in its searching process. Although the tables do not give us the complete results for every starting and destination points, nevertheless we may derive some conclusions. It may be possible to say that results obtained by MCACS–BRP increase by 25% the results obtained by CA–PRIME. There was a slight increase of results obtained by CA–SECOND. Bus-stop Start 1037 2 877 870 857 900 853 97 164 1151 2

Time of the tour [min.] Initial Destination MCACS-BRP solution 353 205 267 1188 403 486 1071 282 339 1039 341 376 939 73 110 819 234 262 808 277 277 445 118 141 752 263 271 525 140 205 540 264 414

Number of changes for tour Initial MCACS-BRP solution 3 8 7 12 5 12 5 9 2 5 3 12 6 8 2 7 3 8 3 6 3 14

Table 2: Comparison between MCACS–BRP and the initial solution.

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Conclusions

MCACS–BRP is a new meta–heuristic for optimising multiple objective functions. The basic idea of co–ordination and self–organisation among two casts belonging to the same colony really helped ants to find the good tours for different starting and destination points. MCACS–BRP as a system of finding the good tours for different starting and destination points shows Porto, Portugal, July 16-20, 2001

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that it is worth further experiments and extensions. The results obtained during our experiments tend to be more attractive than those, which were proposed in our previous version [1]. Therefore, we want to analyse different characteristics representing the special abilities of two casts, especially different number of ants and different parameters connected with the evaporation of the trail. The problem of co–operation between two casts is undoubtedly worth further analysis. Our suggestion is that we can exploit here game theory strategies of collaboration. To sum up, MCACS is similar to MACS and here we have adopted this idea to the Bus Routing Problem. This idea is really useful for multiple objective functions and our experiments have confirmed it. MCACS is shown to be interesting among many modern meta–heuristics for the more complicated problems when using different evaluating parameters.

References [1] U. Boryczka. Ant Colony System and Bus Routing Problem. In Proceedings of CIMCA’99, Vienna, 1999. [2] M. Dorigo and L. M. Gambardella. Ant colonies for the Traveling Salesman Problem. Biosystems, 43:73–81, 1997. [3] M. Dorigo and L. M. Gambardella. Ant Colony System: A cooperative learning approach to the Traveling Salesman Problem. IEEE Trans. Evol. Comp., 1:53–66, 1997. [4] M. Dorigo, V. Maniezzo and A. Colorni. Positive feedback as a search strategy. Tech. Rep. Politechnico di Milano, Italy, No. 91–016, 1991. [5] M. Dorigo, V. Maniezzo and A.Colorni. The Ant System: Optimization by a colony of cooperating agents. IEEE Trans. Syst. Man. Cybern., B26:29–41, 1996. [6] O. Engelhardt-Funke and M. Kolonko. Cost-Benefit-Analysis of investments into railway networks with randomly perturbed operations. In Proceedings of CASPT 2000, Berlin, Germany, 2000. [7] R. Freling. Scheduling Train Crews. A case study for the Dutch railways. In Proceedings of CASPT 2000, Berlin, Germany, 2000. [8] T. Galv˜ ao Dias and J. Falc˜ ao e Cunha. Evaluating DSS for operational planning in public transport systems: Ten years of experience with GIST system. In Proceedings of CASPT 2000, Berlin, Germany, 2000. [9] L. M. Gambardella, E. Taillard and G. Agazzi. MACS-VRPTW: A Multiple Ant Colony System for vehicle routing problems with time windows. Technical Report IDSIA, Lugano, Switzerland, 06-99, 1999. [10] A. S. K Kwan, R. S. K. Kwan, M. E. Parker and A. Wren. Proving the versatility of automatic driver scheduling on difficult train & bus problems. In Proceedings of CASPT 2000, Berlin, Germany, 2000. [11] M. Meilton. Selecting and implementing a computer aided scheduling system for a large bus company. In Proceedings of CASPT 2000, Berlin, Germany, 2000. [12] J.-M. Rousseau. Scheduling regional transportation with HASTUS. In Proceedings of CASPT 2000, Berlin, Germany, 2000. [13] C. R. Delgado Serna and J. Pacheco Bonrostro. MINMAX vehicle routing problems: Application to school transport in the province of Burgos (Spain). In Proceedings of CASPT 2000, Berlin, Germany, 2000.

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