C Prins

E Fatnassi

M Mrad

O Chebbi

R Lahyani

S Sivanandam

English

Olfa Chebbi

Ezzeddine Fatnassi

Hadhami Kaabi

Crossref

An Adaptive Heuristic Approachfor the Multiple Depot AutomatedTransit Network ProblemOlfa Chebbi(B), Ezzeddine Fatnassi, and Hadhami KaabiInstitut Supérieur de Gestion de Tunis, Université de Tunis,41, Rue de la Liberté-Bouchoucha, 2000 Bardo, Tunisiaolfaa.chebbi@gmail.comAbstract. Automated Transit Networks (ATN) are innovative trans-portation systems where fully driverless vehicles offer an exclusive onde-mand transportation service. Within this context of ATN, this studytries to deal with a specific routing problem arising in the context ofa ATN’s network with a multiple depot topology. More specifically, wepresent an optimization routing model for automated transit networkswhich can be used to strategically evaluate depots locations. Our modelextends the basic Multi-depot Vehicle Routing Problem (MDVRP). Inthis paper, the proposed model is tackled using an heuristic approach asthe proposed problem is NP-Hard. Experiments are run on a carefullygenerated instances based on the works from the literature. The numer-ical results show that the proposed algorithm is competitive as it foundsa small gap relative to a lower bound values from the literature.Keywords: Automated transit network · Multi-depot vehicle routingproblem · Heuristics · Genetic algorithm1 IntroductionNowadays, public rapid transit systems provide an interesting way for reducingthe distinctive negative impact of transportation tools in urban areas. In fact,public rapid transit systems help to improve the access of lower income groupsin societies to transportation tools as well as reducing the environmental impactof urban mobilities. Public rapid transit systems consists of light rapid transit(LRT), bus rapid transit (BRT), Automated Transit Networks (ATN), metro,commuters rail and so on. Recently, several models has been put forward tojustify the operational, tactical and strategic implementation of rapid transitsystems. In this paper, we focus on the implementation of ATN. We extendthe operational model of Mrad and Hidri [10] which is used as a base of ouroperational ATN model.In the operational model of Mrad and Hidri [10], the optimized variablesare the energy consumption, the objective function is the minimization of totalc© The Author(s) 2016T. Czachórski et al. (Eds.): ISCIS 2016, CCIS 659, pp. 3–11, 2016.DOI: 10.1007/978-3-319-47217-1 14 O. Chebbi et al.energy consumption of ATN. The ATN’network is assumed to have a single un-capacitated depot [7]. We extend this model to account for a multiple depottopology network. We introduce also a maximum allowable distance constraintrelated to the electric battery capacity of the ATN vehicles. We study the effectof these constraints on the operational level for the multiple depot topologyATN’network. In spite of its relative complexity, the proposed operational modelcould be solved heuristically based on approximate methods which could yieldssome analytical insight on the structure of its optimal solutions. In particular,we found that introducing multiple depots topology helps to reduce the totalservice time for rapid transit users. Also, the proposed heuristic approach wasproven to found good quality solutions in a fast computational time.The remainder of this paper is as follows: Sect. 2 presents the ATN systemand its related literature review which motivates our work. Section 3 presents theoptimization model. Our proposed heuristic approach is introduced in Sect. 4.Section 5 provides numerical results analysis of our approach. Conclusions arereported in Sect. 6.2 The Automated Transit NetworksATN (also called Personal Rapid Transit (PRT)) consists mainly on a set of smallautomated driverless electrical vehicles running on a set of exclusive guideways.ATN is implemented to provide an interesting mode of urban transportationservice which could address the need of urban mobility based on specific set-tings. Table 1 provides an overview of the several needs related to urban mobilityand how could ATN satisfy them. In the literature, there is a general consen-sus that the key characteristics of ATN includes [2]: (i) Fully automated vehi-cles; (ii) Small and dedicated guideways; (iii) On-demand, origin-to-destinationservice; (iv) Off-line stations; and (v) A network or system of fully connectedguideways.Table 1. ATN main featuresNeed ATN featureProvide faster service Non-stop, on-demand serviceReduce congestion Faster and personalized service toattract private automobile usersReduce pollution Electric vehiclesReduce energy use Small vehiclesOn-demand and Non-stoptransportation service to eliminateempty vehicle movementsAn Heuristic Approach for the Multiple Depot ATN Problem 52.1 Literature ReviewATN as a conceptual mode of public rapid transit systems has a history of over60 years. Since, its first introduction in 1953 [2], it was studied by governments,universities, research organization and so on. Literature of ATN includes severalbooks, scholar papers and technical reports. These studies proposed to treat sev-eral features related to ATN such as technical and operational analysis, systemdesign, environmental impact, cost performance and so on.A literature review published in 2005 [5] states that there is more than 200research papers related to ATN. More recently, several operational and strategicoptimization studies related to ATN were published such as simulation [7], energyminimization [10], total traveled distance [6,8], optimized operational planning[4] and so on. However and from our literature review, many optimization routingmodels related to ATN considered a single depot network topology [4,6,10].Consequently, it becomes of a high interest to study optimization routingproblem related to ATN based on a multiple network topology. Therefore in thenext section, we extends the single depot based optimization model of Mrad andHidri [10] to propose a multiple depot optimization model which would aim atreducing the total travel time of ATN vehicles while serving a set of known staticdeterministic list of passengers travels.3 The Optimization ModelIn this section, we present the multiple depots ATN optimization model whichextends the works of Mrad and Hidri [10]. We first start by presenting the set ofassumptions related to our model. Then, we give a graph based model. Finally,we present the complexity of our problem.3.1 The Set of AssumptionsLet suppose that we have a ATN N with a finite number of stations M .N satisfies connectivity constraints. Therefore, a ATN vehicle could travelbetween any pairs of stations in N . We suppose that N has a set of depotsκ = {d1, d2, d3, ....dk} where k represents the number of depot in N . In eachdepot, there exists an unlimited number of ATN vehicles. The exact numberof vehicles needed from each depot is considered as a decision variable. Eachvehicle has a limited battery capacity denoted B. We supposed to have a staticpre-deterministic list of trips to serve denoted T . |T | = n. Each trip i is identifiedby a quadruplet:(i) a depart time Dti,(ii) a depart station Dsi(iii) an arrival time Ati and(iv) an arrival station AsiFinally, let SP be a matrix cost which defines the shortest time travel pathbetween each pair of stations.6 O. Chebbi et al.3.2 Graph Based FormulationOur problem has an objective to find a set of least cost roads starting andending at one of the depots in N which minimizes the total travel time of theATN vehicle while serving each trip exactly once. To model our problem, let usdefine G = {V,E} where V is a set of nodes and E is a set of edges. Each trip iis represented by a node in V . Also, each depot di is represented by two nodessi and ti. Also, we have n trips and k depots. The cardinality of V is equal ton + 2k. V ∗ = V \{s1, s2, ...sk, t1, t2, ..., tk}. As for the set of edge E, it will bedefined as following:– For each pair of nodes i and j ∈ V ∗, we add an edge (i, j) to E if Ati +SP(Asi, Dsj) ≤ Dtj . The edge has a cost cij , representing the total time neededto move from arrival station Asi of trip i to depart station Dsj of trip j.– For each node i and each depot k, we add an edge (k, i). This edge has a costcki which is equal to total traveled time to reach the depart station Dsi oftrip i, from the depot k.– For each node i and each depot k, we add an edge (i, k). This edge has as acost the total travel time needed to move from the arrival station Asi of tripi to the depot k.Let us also denote E∗ = E{(i, j) where i ∈ κ or j ∈ κ}.3.3 The Complexity of Our ProblemStarting from our graph modeling of the problem, we could note that it extendsthe asymmetric distance constrained vehicle routing problem (ADCVRP) [1].Our problem is asymmetric as the cost of edge (i, j) = (j, i). The DCVRP isa vehicle routing problem where each road is subject to total distance, time orcost constraints. The ADCVRP is not well studied in the literature. In fact andas Almoustafa et al. state [1], only two papers studied this problem [1]. Thework related to ADCVRP are based on a single depot topology. Therefore, ourproposed ATN problem could be considered as an extension to the ADCVRPby adding multiple depots to its basic version. Thus, it represents an interestingworth to study extension to the works in the literature. The ADCVRP is provento be an NP-Hard problem [8]. Consequently, our proposed extension to theADCVRP is an NP-Hard problem. In the next section, we present details of oursolution approach proposed to solve our problem.4 Genetic Algorithm ApproachAs mentioned earlier, the proposed multiple depot ATN routing problem is anNP-Hard optimization problem which has its own difficulties to solve. Conse-quently, this paper presents an heuristic approach based on the implementa-tion of genetic algorithm (GA) to solve the proposed optimization problem. GApresents a good solution approach for the proposed ATN problem as it could dis-cover many different zones in the search space [4]. Consequently, it could reachAn Heuristic Approach for the Multiple Depot ATN Problem 7Algorithm 1. Pseudo Code of Genetic Algorithm1: Initialize-parameter()2: while Not reach termination criterion do3: for all Individual in the population do4: parent1 ←− Select-at-random(pop)5: parent2 ←− Select-at-random(pop)6: offspring ←− One point Crossover(parent1, parent2)7: offspring ←− Insertion mutation(offspring)8: Evaluate(offspring)9: if the offspring is better than the worst individual then10: The offspring replace the worst individual in the population)11: end if12: end for13: end while14: individual ←− Best-individual(pop)a good quality solution in a fast computational time. A high level overview ofour GA is presented in Algorithm1.The choice of developing GA1 for this problem is motivated by the fact thatlarge number of studies adopted this solution approach to solve routing problems.One could note for instance [9].Similarly and starting from a population of individuals, a GA applies geneticoperators like crossover and mutation in each iteration in order to generate newoffsprings. Consequently, the key issue to successfully develop GA is to selectthe appropriate genetic operators and solution representation.In the next subsections, we focus more closely on the proposed GA. We firstdescribe the individual’representation and evaluation function. Then, we discussthe implemented genetic operators and the parameters used therein.4.1 Solution Representation and Evaluation FunctionIn our GA, a solution is represented using a vector of trips to perform. In this vec-tor, each trip is represented by a single gene only once. Therefore, each solutionis in a form of a permutation of trips. As for the evaluation function, we adaptthe split function of Prins [11] to our context. More specifically and starting froma permutation, the split function constructs an auxiliary graph where each noderepresents a trip in addition to a node representing the different depots in theATN’network. Each edge in the auxiliary graph represents a feasible road basedin the permutation at hand. Next, the algorithm uses the shortest path in theauxiliary graph to find the related set of roads. Thus, we obtain the set of roadsstarting and ending at one of the depots in the network covering each trip onlyonce. More details could be found in [11].1 Non expert readers can for instance refer to [12] for more details about GA.8 O. Chebbi et al.4.2 Crossover and Mutation OperatorsAfter deciding the representation form of the individuals in the GA, two parentsare selected randomly according to Algorithm 1 in order to create new offspringsusing crossover operator. Our crossover operator applied in our algorithm is theone point crossover. For the first parent, we choose randomly a cut point. Thetrips that are present before the cut point in the first parent are copied to theoffspring. The missing trips in the resulted offspring are copied from the secondparent while following their order of appearance. More details could be found inFig. 1.Fig. 1. Example of one point crossoverAlso, mutation helps GAs to preserve diversification in the population. Inour algorithm, the mutation procedure is applied on the new generated offspringafter the crossover operator. In our approach, we use the insertion mutationoperator. This operator chooses at random one trip from the permutation andinsert it at a random position.5 Computational ResultsIn this section, we present the computational results related to the proposedGA. The algorithms proposed in this paper were coded in C++ language. Theexperiments are performed on a PC with a 3.2 GHZ CPU and 8 GB of RAM.5.1 Test InstancesTo test our proposed approach, we generated 100 ATN multiple depot instances.The size of the problem (i.e. the number of trips) in our testing bed variesbetween 10 and 100 trips by a step of 10. For each number of trips, 10 instancesAn Heuristic Approach for the Multiple Depot ATN Problem 9were generated. To generate the different instances, the ATN’s instances gener-ator from the literature of Mrad and Hidri [10] is adapted to our context. Toassert the quality of the obtained solutions we used the GAP metric. The GAPis obtained as follows:GAP =((SOL − LB)LB)× 100 (1)We should note that SOL is the solution of LB represents the linear relax-ation of the valid mathematical formulation presented in the literature [1]. Themathematical models related to the linear relaxation were implemented usingthe IBM ILOG CPLEX Optimizer 12.2.5.2 Result of the Genetic AlgorithmAs for the parameter tuning, we used a specific method from the literatureto effectively tune our proposed GA [3]. Based on this method, we found thefollowing parameters: (i) Number of generations:800, (ii) population size:20;(iii) crossover rate:0.9 and (iv) mutation rate: 0.3. Table 2 presents the resultsof our approach. It should be noted the good quality of our proposed GA as wefound an average GAP of 2.859 % in 0.231 s.We should note also that the average GAP grows steadily. The maximumGAP was equal to 6.435 % which is still represents good quality results. As forthe average time, our algorithm proved to be very effective as the average compu-tational time was still below 1 s. These results comfort our choice in the selectionof a GA for solving our hard combinatorial optimization problem related to ATN.These results are encouraging in term of problem solvability.Table 2. The Obtained ResultsNumber of travels Average GAP % Average time in seconds10 0 0.83320 0.575 0.03930 0.438 0.48540 0.832 0.06350 1.771 0.08260 3.327 0.10370 4.263 0.12280 4.628 0.14890 6.327 0.191100 6.435 0.241Average 2.859 0.23110 O. Chebbi et al.6 ConclusionsIn this paper the Multi-Depot automated transit network problem is evoked andmodeled. A genetic algorithm is proposed and implemented to solve it. The pro-posed algorithm integrates an effective genetic operators and evaluation functionfor solving the combinatorial optimization problem. The algorithm constructs aset of ATN’vehicles routes starting and ending at any of the proposed depotswith minimum routing costs. Computational experiments on a set of carefullygenerated instances show that the proposed heuristic is very effective. As anextension to this work, a more adapted meta-heuristic approach such as beecolony algorithms, ant colony algorithm could be adapted to our context. Alsothe inclusion of additional constraints such as mixed fleet with varying maximumallowable distance and multi-compartment vehicles is under investigation.Open Access. This chapter is distributed under the terms of the Creative Com-mons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, duplication, adaptation, distribution and reproduction in anymedium or format, as long as you give appropriate credit to the original author(s) andthe source, a link is provided to the Creative Commons license and any changes madeare indicated.The images or other third party material in this chapter are included in the work’sCreative Commons license, unless indicated otherwise in the credit line; if such mate-rial is not included in the work’s Creative Commons license and the respective actionis not permitted by statutory regulation, users will need to obtain permission from thelicense holder to duplicate, adapt or reproduce the material.References1. Almoustafa, S., Hanafi, S., Mladenovi, N.: New exact method for large asymmetricdistance-constrained vehicle routing problem. Eur. J. Oper. Res. (2012)2. Carnegie, J.A., Hoffman, P.S.: Viability of personal rapid transit in New Jersey.Technical report (2007)3. Chebbi, O., Chaouachi, J.: Effective parameter tuning for genetic algorithm tosolve a real world transportation problem. In: 2015 20th International Conferenceon Methods and Models in Automation and Robotics (MMAR), pp. 370–375. IEEE(2015)4. Chebbi, O., Chaouachi, J.: Reducing the wasted transportation capacity of per-sonal rapid transit systems: an integrated model and multi-objective optimizationapproach. Transp. Res. Part E Logistics Transp. Rev. 89, 236–258 (2015)5. Cottrell, W.D.: Critical review of the personal rapid transit literature. In: Proceed-ings of the 10th International Conference on Automated People Movers, pp. 1–4,May 20056. Fatnassi, E., Chebbi, O., Chaouachi, J.: Discrete honeybee mating optimizationalgorithm for the routing of battery-operated automated guidance electric vehiclesin personal rapid transit systems. Swarm Evol. Comput. 26, 35–49 (2015)7. Fatnassi, E., Chebbi, O., Siala, J.C.: Two strategies for real time empty vehicleredistribution for the personal rapid transit system. In: 2013 16th InternationalIEEE Conference on Intelligent Transportation Systems (ITSC), pp. 1888–1893.IEEE (2013)An Heuristic Approach for the Multiple Depot ATN Problem 118. Fatnassi, E., Chebbi, O., Siala, J.C.: Comparison of two mathematical formulationsfor the offline routing of personal rapid transit system vehicles. In: The Interna-tional Conference on Methods and Models in Automation and Robotics (2014)9. Lahyani, R., Khemakhem, M., Semet, F.: Rich vehicle routing problems: from ataxonomy to a definition. Eur. J. Oper. Res. 241(1), 1–14 (2015)10. Mrad, M., Hidri, L.: Optimal consumed electric energy while sequencing vehicletrips in a personal rapid transit transportation system. Comput. Ind. Eng. 79, 1–9(2015)11. Prins, C., Lacomme, P., Prodhon, C.: Order-first split-second methods for vehiclerouting problems: a review. Transp. Res. 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An Adaptive Heuristic Approach for the Multiple Depot Automated Transit Network Problem

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An Adaptive Heuristic Approach for the Multiple Depot Automated Transit Network Problem

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