Implementation of a Column Generation Heuristic for Vehicle Scheduling in a Medium-Sized Bus Company

Transcription

1 7e Conférence Internationale de MOdélisation et SIMulation - MOSIM 08 - du 31 mars au 2 avril 2008 Paris- France «Modélisation, Otimisation MOSIM 08 et Simulation du 31 mars des Systèmes au 2 avril : 2008 Communication, Paris- France Cooération et Coordination.». 1 Imlementation of a Column Generation Heuristic for Vehicle Scheduling in a Medium-Sized Bus Comany Lucas Maia Alves de Lima Federal University of Minas Gerais (UFMG) 6627, Pres. Antônio Carlos Av , Belo Horizonte, MG, Brazil lucasmaia.bra@gmail.com Laure Thomä-Cosyns ESIEE Amiens 14, Quai de la Somme Amiens, France thoma@esiee-amiens.fr ABSTRACT: The aer reorts on the current status of a roject concerned with the imlementation of a Column Generation Heuristic Algorithm in co-oeration with a French medium-sized bus comany. We discuss the modeling aroach to the ractical roblem, as well as the mathematical bacground that suorts the strategy adoted. Finally some reliminary results for artificially-generated instances are resented and some ey actions to be taen in the near future are highlighted. INDEX TERMS: Bus Scheduling, Multile-Deots Vehicle Scheduling, Column Generation Heuristic. 1. INTRODUCTION In this woring aer we reort on the results obtained during the first stage of a Euroean Project concerned with the imlementation of a comutational tool for solving the multile-deots vehicle scheduling roblem in a French medium-sized bus comany. The Multile-Deot Vehicle Scheduling (hereafter referred to as MDVSP) is a classical roblem in the field of Transortation Science. It consists of determining a minimum-cost covering structure for a set of redefined tris using vehicles rovided by several deots. Let n be the cardinality of the set of tris T = {T 1, T 2,..., T n }, each one characterized by a starting and end time (s i and e i, resectively, for all i = 1,, n), and let m be the cardinality of the set of deots K= {D 1, D 2,..., D m } each one characterized by a number of available vehicles (v, for all = 1,, m). Now consider any two tris T i and T j, and let τ be the travel time between the end oint of T i and the starting oint of T j. We say that the air (i, is comatible if e i + τ + κ s j, where κ is some arameter defined a riori, ossibly zero. Moreover, throughout the aer A will denote the set of comatible airs of tris (ie. arcs, in a grah reresentation). Finally, for any two comatible tris, let c be the cost of covering T j just after T i using a vehicle rovided by deot = 1,, m. The MDVSP is roved to be NP-hard in Bertosi, Carraresi and Gallo (1987, cited in Ribeiro and Soumis, 1994). It has been the focus of academic researches for years (see Ribeiro and Soumis, 1994 for a brief literature revision) and the evolution of the aroaches develoed, as well as the imrovements in time and quality of solution, is evident. However, to our nowledge, the transosition of this now-how into the context of transortation comanies has not yet been well exlored. Our exerience has shown that the general feeling among managers, engineers and comuter scientists woring on these comanies is that the models are too comlicated and time-demanding to be used in real-life situations. According to our artners, many of the transortation comanies, both in rivate and ublic sectors, still do not formally consider asects of otimization theory in their vehicle scheduling lanning rocesses. In this sense, the goals of this aer are to rovide a general overview of the modeling and algorithmical aroach used to tacle the roblem in co-oeration with the comany, and to resent the comutational results obtained so far, as well as the ersectives for the continuation in the near future. We firstly resent in Section II two formulations for the MDVSP model as well as the aroach develoed to adat them to the real context. Section III highlights some imortant oints concerning the theoretical bacground of the field, mainly regarding the algorithmical dimension of the wor. Results for artificially-generated instances u to 600 tris and 10 deots are resented in section IV, as well as some imlementation issues to be addressed in the near future. Finally some conclusions are drawn in section V, mainly in what concerns the now-how generated so far and the contributions of the aer, both to theory and ractice. 2. THE MODEL This section is dedicated first, to the resentation of two mathematical formulations for the MDVSP and

2 MOSIM 08 du 31 mars au 2 avril 2008 Paris- France 2 second, to exlain the way this model was used to reresent the ractical roblem owned by the comany. Hereafter let () o ( be the nodes corresonding to the starting and end oints at deot K, resectively. Moreover, let G = ( T { o( ), o( )}, A ) be the grah defined o and ) by all the tris in T lus the deot the existing arcs. K 2.1. Multi-Commodity Flow Formulation, as well as We start by introducing the multi-commodity flow formulation of the MDVSP, which has been extensively used in the literature (Ribeiro and Soumis, 1994; Pein, Desaulniers, Hertz and Huisman, 2006). The decision variable is ( x, indicating whether the arc i, will be covered by a vehicle from the th deot, K. The maximum caacity of each deot is given by v and, as we had defined reviously, c denotes the cost of covering T j just after T i using a vehicle rovided by deot. For a more comlete descrition of this formulation and its underlying theory, we address to Ahuja, Magnanti and Orlin (1993). st min c x K ( i, x = 1 K ( i, = x ji x j:( j, i) j:( i, xo( ) j v j:( o( ), x {0,1} i T i T; K K ( i, A ; K (1) (2) (3) (4) (5) Model 1. MDVSP Multi-Flow Formulation In Model 1, the minimization of the total cost is established in (1). Constraints (2) imose that exactly one arc has to arrive (to deart) at (from) each node i T of the grah while constraints (3) imose that these two arcs (ie. the one arriving at and the one dearting from a given node) have to be covered by a vehicle rovided by the same deot K. In constraint (4) we have that the number of used vehicles er deot is not greater than the number of available vehicles and, finally, (5) imoses binary variables Set Partitioning Formulation Differently from the multi-commodity formulation, the set artitioning formulation considers a set Ω containing some set of feasible schedules rovided by deot. The otimization is therefore erformed over K Ω = K Ω. In the secific case of the MDVSP, these variables are reresented by ( n + m) 1 vectors where the first n ositions corresond to the set of tris to be covered and the last m ositions to the set of deots from where the vehicles may come. For a given variable, say, the i th comonent a i is equal to one if covers i, and zero otherwise. As for the last m ositions, a given n + 1 i n + m ), is equal to comonent, say i (with one if comes from deot i n, and zero otherwise. Finally, comuting the cost c of each variable in Ω, and defining θ equal to one if schedule is taen in the solution and zero otherwise, we can now formulate the set artitioning model as follows. st min K Ω c θ aiθ = 1 K Ω θ v Ω θ {0,1} i T K Ω ; K (6) (7) (8) (9) Model 2. MDVSP Set Partitioning Formulation We will briefly interret the constraints of Model 2 comaring its roles with those of Model 1. First, we see that (6) lays the same role of (1), ie. minimizing the total costs of the chosen schedules. Besides, (7) and (2) have also the same role, assuring that all services will be covering once. Finally, (8) lays the same role of (4) and (9) of (5). One may thin that (3) is not being considering, but in fact this constraint must be considered when building the set Ω. It means that all the schedules included is Ω already obey to (3). The way this is done will be clearer in Section III. Although the comarison above shows that both the multi-commodity and the set artitioning formulations are equivalent models for the MDVSP, one can mathematically roof this fact by alying the Decomosition Princile to Model 1. We byass this roof and refer to Ahuja, Magnanti and Orlin (1993) for details. We now address the way these mathematical formulations can be used to model the ractical roblem handled by the comany. Our artner ossesses three hysical deots, located in the surroundings of Amiens, France and erforms two tyes of tris: urban and inter-urban. The focus of the

3 MOSIM 08 du 31 mars au 2 avril 2008 Paris- France 3 current roject was on the latter, tyically longer tris. Besides, the articular characteristic of this alication that maes it slightly different from the usual is the fact that the driver may accomany the bus for several days without going bac to any of the three hysical deots. Instead the bus is taen with the driver to his residence. In this case, considering in the model only the three hysical deots would be obviously of no value from the ractical oint of view, however, on the other extreme, considering each driver s residence as a otential deot with unitary caacity would be imractical from the comutational oint of view. Finally, the solution roosed was to divide the region over where the tris are distributed La Somme, France in an aroriate number of smaller regions and consider the center of each one as a virtual deot. We resent the resulting division in Figure 1. The blue limits are circles in the sace of minutes, for which we determined a ray equal to 20 min from the geograhical center. The red and blac dots corresond to drivers residences. Figure 1. Definition of Virtual Deots 3. BACKGROUND In this section we resent some imortant theoretical concets which rovide the basis for understanding the Column Generation Aroach. We exlain briefly (i) the roles of the master and ricing roblems as well as the relationshi between them; (ii) the rounding rocedure used to raidly comute integer solutions for the master roblem; and (iii) the K-shortest ath algorithm chosen for solving the ricing roblem. traditional methods, lie the Simlex (Chvátal, 1980), may become useless. The reason lies on the fact that an exlicit search for riced-out variables to enter the basis is too difficult to be erformed efficiently. On the other hand, we observe that the otimal solution of these huge roblems usually contains a very tiny art of the feasible set. This observation gives rise to the idea of maniulating just a small subset of the most interesting variables among all the ossible ones 1. To tacle the question of how to determine this interesting set of variables the Column Generation Algorithm deals with two levels of roblems that exchange information throughout the rocedure. The first level corresonds to the Master Problem, reresented in our case by Model 2. This is the uer level of the algorithm and it manages the set Ω which contains variables from all the deots. The dual information rovided by the solution of the Master Problem is then used to modify the arc s costs of each G = 1,..., m grah, which will be considered indeendently in each of the Pricing Problems. As the title suggests, the Pricing Problems use the dual information rovided by the Master to determine a few riced-out variables (new schedules with negative reduced cost, in our case) that will hoefully imrove the current solution, if it is not otimal yet. One can roof that these Pricing Problems are actually shortest aths roblems over each G and with an adequate definition of arc s costs, the otimal solution of each shortest ath roblem also solves to otimality the multi-flow formulation and its equivalent setartitioning version. Let π i and β be the dual values corresonding to constraints (7) and (8), resectively. = c ), Then, choosing o( ), i o( i i T ; K and c = c π i i T c β, maes the otimality conditions of each shortest ath roblem equivalent to the ones of the multi-flow roblem, validating the resolution of the shortest-ath roblems indeendently as a searching engine for riced-out variables. This result is detailed in Lima (2007) based on the roof rovided by Ahuja, Magnanti, Orlin (1993). Now, having exlained the general strategy to deal with the difficulty of the multi-commodity-flow roblem and the role of the Master and Pricing Problems, we turn out efforts to understanding how to solve them Column Generation to solve huge linear rograms When dealing with linear rograms of very large scale the set of feasible solutions may be so immense that 1 This idea was firstly introduced by Ford and Fulerson (1958, cited by Lübbece, 2001) and further develoed by Dantzig and Wolfe (1960, cited by Lübbece, 2001) who ioneered the fundamental idea of decomosition.

4 MOSIM 08 du 31 mars au 2 avril 2008 Paris- France The Rounding Procedure to solve the Master Problem In the beginning of the algorithm, the set artitioning formulation given by Model 2 is initialized with some set of exensive artificial variables, which are then rogressively eliminated throughout the algorithm as new better variables are added to Ω. Imortant asects concerning this initialization hase were discussed in Lübbece (2001) and Lima (2007). We have chosen to solve the linear relaxation of Model 2 using the large scale linear rogramming algorithm described in Zhang (1997). However, if the solution contains some fractional variables, to raidly comute an integer solution avoiding going through a series of branches, we use the rounding rocedure roosed by Pein, Desaulniers, Hertz and Huisman (2006) and stated in the following. Algorithm 1. Rounding Procedure Define a rounding threshold γ; Let Π be the current master roblem and Γ be the current corresonding grah; while no integer feasible solution was obtained do for all variables θ do if ( ) do θ γ θ = 1; Delete the column corresonding to θ from Π and all lines corresonding to the tris covered by θ ; Also delete the nodes in Γ corresonding to the tris deleted; Udate vehicle availability; end if end for if no variable was rounded do Round u to 1 the greatest variable θ ; Delete its corresonding column from Π and all lines Corresonding to the tris covered in it; Also delete the nodes in Γ corresonding to the tris deleted; end if Solve the Master Problem over Π and the Pricing Problem over Γ; end while γ = 0, In our exeriments we used 7 as suggested by Pein, Desaulniers, Hertz and Huisman (2006) Using a K-shortest aths algorithm to solve the Pricing Problem The solution of the ricing roblem lies on the very heart of the column generation algorithm. For instance, Lübbece (2001) reorts on exhaustive tests showing that u to 80% of the CPU time may be sent on this hase. For being so exensive, instead of adding just the variable corresonding to the shortest ath between o () and o () we rather add a greater number of variables, all of them otentially caable of imroving the current solution. For doing so, we need to determine not only the shortest ath in G but some others shortest aths. This is done by means of a K- shortest ath algorithm. We have decided to use the MPS Algorithm (Martins, Pascoal and Santos, 1998) to solve the K-shortest ath roblem. According to the authors aforementioned, although the theoretical comlexity of this algorithm is still an oen question it has erformed well comared to its closest cometitors and this is the main reason for our choice. The MPS Algorithm requires the construction of a tree of shortest aths, for which we used the olynomial version of the Auction Algorithm with Grah Reduction roosed in Bertseas, Pallotino and Scutella (1995). This latter was imlemented using the second best tye of imlementation and the rerocessing algorithm, both suggested by Bertseas (1991). Moreover, to imrove the erformance of the K-Shortest Path algorithm we also used a sorted forward star form to organize the set of candidate aths. Finally, as the sub-roblems are solved sequentially, we have chosen to use the information contained in the rice vector of one sub-roblem to initialize the next, reducing the time sent in the rerocessing hase of the Auction Shortest Path algorithm. The interested reader is addressed to Lima (2007) for details about these erformance imrovement rocedures. Two interesting effects concerning the resolution of the Pricing Problem are reorted in the literature (Lübbece, 2001; Desaulniers and Desrosiers, 2005). The first one, nown as heading-in, is related with the ossibility of generating an imortant amount of bad columns in the beginning of the algorithm, when lots of artificial variables are still resent in the formulation. In this case, the dual information assed to the Pricing Problem is oor and therefore, the columns generated are not good. To avoid this effect, instead of maintaining a constant set oint for the K-shortest ath algorithm, we rather start with a small desired number of columns to be generated er sub-roblem and increase this number as new columns are added and artificial ones are deleted. The second tyical effect refers to the fact that imroving the total cost becomes more difficult as the algorithm evolves. As a consequence, at the end, the Pricing Problem continues generating columns with negative reduced cost but, the imrovement in the solution does not mae it worth to continue the rocess. This haens because roving otimality would be too exensive in terms of time. To avoid this tailing-off effect we introduce a arameter

5 MOSIM 08 du 31 mars au 2 avril 2008 Paris- France 5 that monitors the number of iterations without total cost imrovement and whenever this arameter becomes greater than a given threshold (3 to 5, in our exeriments), we allow the early termination of the algorithm. Table 2. Exerimental results 4. RESULTS Desite the early stage of the roject, we have already been able to come u with a first generation of reliminary results for ramdomly generated data sets. These results aim to test the suitability of the aroach develoed so far and have an imortant role as a way to identify gas where imrovement is needed. All codes used to generate the following results were imlemented on Matlab. For this reason the running times resented herein are greater on average than the ones achieved by imlementations carried out on other languages, mainly structured ones, lie C. However, as ointed out in Zhang (1997), Matlab allows a much faster develoment rocess and therefore is suitable for testing the concet of a solution framewor. Once validated, all the codes can be then converted into a faster structured language. The data used to test the code was generated randomly according to the statistical characteristics of real timetables obtained with artners. A brief descrition of the arameters used in this data-generation-rocess as well as the main comments about each one are rovided in Table 1. Table 1. Parameters for data generation Based on this reasoning, Table 2 resents the results obtained by the current version of the code for instances u to 600 tris and 10 deots. We remar that all instances were run in an Intel Xeon 2,66GHz Worstation, 1 Gb RAM, Windows XP, Matlab 6.5. In Lima (2007) we roosed a detailed analysis of the results obtained so far and list some ey actions to be taen in the near future to imrove the current erformance and strengthen its value to the comany. In the following we finish this section ointing out the most suggestions. i. Run of more intensive tests to imrove the code s arameters; ii. Tests with label setting shortest ath algorithms: Bertseas, Pallotino and Scutella

6 MOSIM 08 du 31 mars au 2 avril 2008 Paris- France 6 iii. iv. (1995) reort on results of comarative tests between the version of the Auction Algorithm used here and its label setting cometitor. In general the latter erformed better although the former is more suitable for arallel comutation. We believe that imlementing this version and comaring its results to the current ones could be of great value. Use arallel comutation to solve the Pricing Problem and the Auction Algorithm for the Shortest Path Problem: we could use the concet of arallel comutation to distribute the solution of each sub-roblem of the ricing hase among a given number of rocessors running in arallel. The result of each of these roblems would be then ut together and written in a common shared memory. Besides we can go even further imlementing the forward-reverse version of the auction algorithm for shortest aths in arallel. In doing so, two rocessors modifying the same rice vector are used; one running the forward and the other the bacward auction rocesses. The execution is stoed when the aths meet at a common node Polymenaos (1991) discusses some arallelization issues secifically designed for the Auction Algorithm. Imlementation in a non-interreted language: as discussed briefly before, after having a robust version of the current code, the imlementation of the codes in some noninterreted language (C or C++, for examle) will rovide a quantum lea in the current erformance, increasing the suitability of the code for even harder instances. v. Imlementation of a Large Neighborhood Search Heuristic (LNS): as suggested in the exerimental wor carried out by Pein, Desaulniers, Hertz and Huisman (2006), the Column Generation Heuristic can be combined with an algorithm for a slightly modified Single Deot Vehicle Scheduling Problem 2 giving rise to a more owerful tool for tacling the MDVSP. According to the results resented by the authors aforementioned the LNS heuristic yields better results in terms of the commitment between quality and solution time. 5. CONCLUSIONS 2 Desite not being reorted in this aer an algorithm for the Single Deot roblem was also imlemented in the first hase of the wor. The wor was mostly based on Freeling, Wagelmans and Pinto Paixao (2001). Throughout this aer we resented the main toics concerning the aroach used to tacle a ractical scheduling roblem in a medium-sized bus comany. Desite the early stage of the wor some exerimental results for real-size randomly generated instances were already resented. As for the continuation of the roject, some ey oints to be carried out in the near future to ensure the imrovement of the current erformance were ointed out and exlained. On the one hand, from the theoretical oint of view, the aer ut together a wide range of techniques from different areas of the Oerational Research literature, maing it easier for ractitioners to get started with similar alications. Last, given the relatively newness of the auction algorithm, its incororation within the scoe of roject contributes to evaluate its advantages and drawbacs when comared with its traditional label setting cometitors. On the other hand, from the ractical oint of view, the envisaged algorithm will mae it easier for the comany to rovide fast and recise answers to ublic demands, roviding the comany with a scientific tool of lanning, mainly when dealing with huge instances of thousand of tris. Last, but not least, the wor also shows how the co-oeration academy-industry can enrich the alication of tools develoed by the former, encouraging the latter to invest in their develoment and alication. 6. ACKNOWLEDGEMENT Lucas would lie to exress its gratitude to the Coordenação de Aerfeiçoamento de Pessoal de Nível Suerior (CAPES, Brazil), for its financial suort during his one-year exchange rogram in France. Together, the authors acnowledge the CAP staff for being continuously enriching the wor since its very beginning. Also we are both grateful to the Euroean Fond of Develoment for its suort through Interreg IIIA. 7. REFERENCES Ahuja, R.K., Magnanti, T.L., Orlin, J.B. Networ Flows : Theory, Algorithms and Alications. Prentice-Hall, Englewood Cliffs, New Jersey Bertseas, D.P. The Auction Algorithm for Shortest Paths. SIAM Journal on Otimization. Vol 1, Bertseas, D.P., Pallotino, S., Scutella, M.G. Polynomial Auction Algorithms for Shortest

7 MOSIM 08 du 31 mars au 2 avril 2008 Paris- France 7 Paths. Journal of Comutational Otimization and Alications. Vol Chvátal. Linear Programming. New Yor: W. H. Freeman and Comany Desaulniers, G., Desrosiers, J., Solomon, M.M. Column Generation. Gerad 25 th Anniversary Series. Sringer, 1 st edition Freeling, R., Wagelmans, A.P.M., Pinto Paixao, J.M. Models and Algorithms for Single-Deot Vehicle Scheduling. Transortation Science. Vol. 35, N 2, , May 2001 Lima, L.M.A. Otimisation Combinatoire: une alication à la gestion de flottes de bus. Technical Reort. ESIEE Amiens Lübbece, M. Engine Scheduling by Column Generation. PhD Thesis Lübbece, M., Desrosiers, J. Selected Toics in Column Generation. Oerations Research, Vol. 53, N 6, , Nov-Dec 2005 Martins, E.Q.V., Pascoal, M.M.B., Santos, J.L.E. Deviation Algorithms for Raning Shortest Paths. International Journal of Foundations of Comuter Science Pein, A.S., Desaulniers, G., Hertz, A., Huisman, D. Comarison of heuristic aroaches for the multile deot vehicle scheduling roblem Polymenaos, L. Analysis of Parallel Asychronous Schemes for the Auction Shortest Path Algorithm. MS Dissertation. MIT, Cambridge, MA Ribeiro. C.C., Soumis, F. A column generation Aroach to the multile-deot vehicle scheduling roblem. Oerations Research, Vol. 42, N 1, Jan-Fev 1994 Zhang, Y. Solving Large-Scale Linear Programs by Interior Point Methods Under the MATLAB Environment. 1997