Service Network Design with Asset Management: Formulations and Comparative Analyzes

Transcription

1 Service Network Design with Asset Management: Formulations and Comarative Analyzes Jardar Andersen Teodor Gabriel Crainic Marielle Christiansen October 2007 CIRRELT

2 Service Network Design with Asset Management: Formulations and Comarative Analyzes Jardar Andersen 1,2,*,Teodor Gabriel Crainic 3, Marielle Christiansen 1 1 Deartment of Industrial Economics and Technology Management, Norwegian University of Science and Technology, 7491 Trondheim, Norway 2 Institute of Transort Economics, Gaustadalléen 21, 0349 Oslo, Norway 3 Interuniversity Research Centre on Enterrise Networks, Logistics and Transortation (CIRRELT), Université de Montréal, C.P. 6128, succursale Centre-ville, Montréal, Canada H3C 3J7 and NSERC Industrial Research Chair in Logistics Management, Université du Québec à Montréal, C.P. 8888, succursale Centre-ville, Montréal, Canada H3C 3P8 Abstract. In this aer, we address the Service Network Design with Asset Management roblem, SNDAM. This roblem arises when vehicles need to be considered while designing service networks, and integrates a new layer of vehicle management decisions to the service network design roblem that traditionally has considered selection of services (design arcs) and routing of flow. We introduce extended asset management constraints to the roblem and roose alternative formulations of the SNDAM roblem based on new cycle design variables. In the comutational study, we solve test cases reresenting real lanning roblems in order to analyze the strengths and weaknesses of the various model formulations and the imact of asset management considerations on the transortation lan. Exerimental results indicate that formulations based on cycle variables outerform traditional arc-based formulations, and that considering asset management issues may significantly imact the outcome of the service lanning models. Keywords. Service network design, freight transortation, caacitated multicommodity network design, asset management. Acknowledgements. This work has received financial suort from The Norwegian Research Council through the Polcorridor Logchain roject. Partial funding has also been sulied by the Natural Sciences and Engineering Research Council of Canada (NSERC) through its Discovery and Industrial Research Chair grant rograms. Results and views exressed in this ublication are the sole resonsibility of the authors and do not necessarily reflect those of CIRRELT. Les résultats et oinions contenus dans cette ublication ne reflètent as nécessairement la osition du CIRRELT et n'engagent as sa resonsabilité. * Corresonding author: jardar.andersen@iot.ntnu.no Déôt légal Bibliothèque nationale du Québec, Bibliothèque nationale du Canada, 2007 Coyright Andersen, Crainic, Christiansen and CIRRELT, 2007

3 Service Network Design with Asset Management: Formulations and Comarative Analyzes 1 Introduction Transortation is among the most vital industries in a modern economy. Increased cometition and reduced rofit margins in the transort industry have increased the need for efficient oerations for carriers. Simultaneously, there has been a change in attitude among customers with increased focus on service quality. Service roviders in the transort industry are thus facing a more challenging situation than just a few decades ago, and survivability is becoming more deendent of intelligent lanning, design, and execution of oerations. For this urose, decision suort systems and otimization models have been increasingly used to assist in decision-making and resource allocation. There has traditionally been a distinction between three lanning levels in transortation lanning (Crainic and Laorte, 1997, Macharis and Bontekoning, 2004). Strategic lanning deals with long-term issues (several years) like major investments and general develoment olicies. Examles of strategic lanning issues are location of terminals, network configurations, and design and layout of terminals. Tactical lanning covers medium-term issues (tyically weeks, months) such as efficient allocation and utilization of resources. One examle is service network design covering issues like determination of service schedules and routing of flow. Oerational lanning covers short-term issues (real-time, days, weeks) in a dynamic environment where the time factor is articularly imortant. Examles are imlementation and adjustment of schedules for services and dynamic allocation of scarce resources. Service network design models address issues like route selection and terminal olicies, whether to use direct or consolidating services, and how often to oerate (Crainic, 2000). Service network design issues are usually considered to be at a tactical lanning level, but asects of strategic or oerational lanning may also be considered. The resulting formulations are often fixed charge caacitated multicommodity network design (CMND) models. Services are reresented as arcs or aths in the network that may be oerated or not. The fixed charges reresent costs that must be aid for each network arc that is oened, while the network is caacitated because several flows need to share caacities of the arcs that are oened. In order to meet requirements for more intelligent lanning, service network design models are increasingly being made more sohisticated. Simultaneously, advances in comutational caacity and methodological develoment have increased the range of roblems that can be solved. Traditional service network design studies have included two layers: The design layer includes decisions on what services to oerate, while the flow layer focuses on routing the demand in the system. From the alication side, a need for inclusion of a third layer to service network design model has been exressed, namely a layer reresenting assets that are needed to oerate services. The standard aroach has however been to manage such assets in an a osteriori analysis. Vehicles that are needed for oeration of transort services are an examle of such assets. When assets are considered in the service network design rocess, we refer to the resulting models as Service Network Design with Asset Management (SNDAM). In this aer, we study the imact of asset management on service network design. One asset management issue that has been studied in the literature is ensuring that there is an equal number of assets entering and leaving each terminal (node) in the network. Constraints ensuring this are referred to as design balance constraints (Pedersen, 2006; Pedersen et al., 2007). Design balance constraints have been modeled for various modes of transortation, see e.g. Smilowitz et al. (2003), Lai and Lo (2004), Barnhart and Schneur (1996), Kim et al. (1999) and CIRRELT

4 Service Network Design with Asset Management: Formulations and Comarative Analyzes Andersen et al. (2007). However, these constraints and their imact on the service network design models and lans have not been extensively studied in the literature, and most other asset management issues have not been addressed at all. This aer aims to contribute to fill this ga. The goal of this aer is to discuss how service network design models can incororate various asset management issues, and to analyze how the exlicit consideration of asset management issues imacts the solutions that are roduced by service network design models. We resent four alternate formulations of an extended model by introducing cycles as design variables in addition to the more common ath variables for the flow. We comare exerimentally the four model formulations on roblem instances that can be handled through a riori enumeration of columns (cycles and aths). The same set of roblem instances is used as a test bed for evaluating the imact of constraints reresenting various asset management issues. The contributions of the aer are thus twofold. First, it offers a comrehensive modeling framework for SNDAM, together with cycle and ath generation methods exloring the secial structure of the formulations. Second, the aer contributes with an extended analysis of how asset management issues affect service network design. The outline of the aer is as follows. In Section 2, we recall service network design and associated earlier work on asset management. Section 3 resents an extended model for SNDAM. We develo three alternate formulations of the extended SNDAM model in Section 4, and resent algorithms for model solving in Section 5. The comutational study, comaring the different formulations of the extended model and analyzing the imact of asset management issues on service design, is resented in Section 6. Finally, concluding remarks are given in Section 7. 2 Service network design formulations In this section, we recall formulations of service network design from the literature. We introduce the fixed charge caacitated multicommodity network design (CMND) roblem in Section 2.1, CMND being the traditional formulation for service network design roblems. In Section 2.2, we recall the basic formulation of the Service Network Design with Asset Management (SNDAM) roblem, incororating the first asset management issue considered in the service network design literature. 2.1 Fixed charge caacitated multicommodity network design The directed grah G= ( NxA, ) reresents the network, where N is the set of nodes and A the set of arcs. Without loss of generality, we assume that all arcs (, i j) A are design arcs. In this aer, we refer to commodities as roducts P = { } that need to be transorted through the network. Each roduct has a demanded volume w that has to be transorted from the unique origin node of the roduct o to its unique destination node d. Design variables y are binary variables indicating whether an arc is used or not, while the flow variables x are nonnegative real numbers. Each design arc y has an associated caacity u, and a fixed cost f associated with the use of the arc. For each unit of roduct, there is a flow cost + c for traversing arc ( i, j) A. Sets N () i = { j N :(, i j) A } and N () i = { j N :( j,) i A } b = min w, u. The roblem of outward and inward neighbors are defined for each node. Let { } CIRRELT

5 Service Network Design with Asset Management: Formulations and Comarative Analyzes consists of minimizing the sum of fixed costs and flow costs while satisfying all demand, and the arc-based formulation of the CMND roblem is formulated in (1) (5). Min z = f y + c x (, i j) A (, i j) A P (1) w, i = o x x = w, i = d 0, otherwise, ji + j N () i j N () i P x x u y 0 i Nv, Pv, (2), (, i j) Av, (3) b y 0, (, i j) Av, P, (4) x 0, (, i j) Av, P, (5) { } y 0,1, (, i j) Av. (6) The objective function (1) minimizes the sum of fixed costs for oening design arcs and flow costs for all roducts that are transorted through the network. Constraints (2) are node balance constraints for the flow. Constraints (3) and (4) are weak and strong forcing constraints, resectively, defining arc caacities and forcing flows to zero when arcs are not oened. Constraints (4) are redundant in the MIP-formulation, but have aeared to imrove the lower bounds obtained through relaxations (Crainic et al., 2001). The formulation including (4) is the strong formulation, while removal of (4) results in the weak formulation. Constraints (5) and (6) are variable-tye constraints, restricting x to nonnegative real numbers and y variables to binary values. An equivalent network design formulation is based on ath variables for the flow (e.g., Ahuja et al., 1993; Crainic et al., 2000). We define the set of aths L that roduct may use from its origin node to its destination node. For these aths, we define arameters (, i j) A belongs to ath l h a l =1 if arc, while the flow cost for transorting roduct on ath l is l L for roduct, 0 otherwise. The flow of roduct on ath l is l l l k, k = c a. The athbased formulation can then be resented as in (7)-(12). (, i j) A CIRRELT

6 Service Network Design with Asset Management: Formulations and Comarative Analyzes Finally, we removed all asset management constraints and, as reorted in the last column of Table 3, we observed dramatic imrovements in comutational erformance together with a decrease in objective function value in the 16% to 25% range. To conclude, asset management issues imact the design of the services and the efficiency of the solution rocedure of the standard MIP-solver of CPLEX. The most significant imact on the solution times was found by removing the route length constraints. Moreover, it aears as if the design balance constraints in articular have an imact on the objective function values that are obtained. The latter is not a surrising result, as these constraints considerably affect the flexibility in the design decisions. 7 Concluding remarks In this aer, we have studied the imact of considering asset management in service network design. Asset management reflects imortant issues connected to vehicle management and utilization in real-world design studies. We have roosed an extended model for Service Network Design with Asset Management (SNDAM), introduced several formulations for the model, and erformed an extensive set of analyzes both on the relative erformance of the formulations and the imact of asset management-based constraints on the service network design roblem. Based on a cyclic time-sace reresentation caturing the scheduling asect of real-world transortations services, we defined cycle design variables. A cycle reresents a feasible asset (vehicle) route through the lanning horizon suorting selected services. It is thus a joint selection of design arcs that together cover the lanning horizon exactly once. In addition, we introduced ath variables for flows as an alternative to arc variables. Combining the two reresentations of the design variables and the two reresentations of the flow variables, we introduced four alternative formulations of the extended SNDAM roblem. The comutational study comared model solving based on the four formulations on roblem instances for which aths and cycles were generated a riori. The comutational study showed that the formulations of SNDAM based on design cycle variables may be solved significantly faster than the formulations based on design arc variables. One interesting observation in this resect is that the cycle-based formulations dislay a significantly stronger strong linear relaxation than arc-based formulations. This is encouraging for the develoment of efficient solution methods, because oor lower bounding is one of the challenges in traditional network design (Gendron and Crainic, 1996). We also analyzed the effect of the considered asset management issues on the efficiency of model solving and on the values of the integer solutions that were obtained by solving the models. The overall conclusion is that these issues indeed affect the otimal solutions obtained and that, in general, they increase the comutational effort required to solve the extended formulations. This emhasizes the need to focus future research on develoing efficient solution aroaches for realworld-dimensioned service network design roblems that include asects of asset management. The test roblems that we use in this aer are smaller than many instances from real-world lanning roblems. The number of cycles that have to be considered in a full enumeration scheme increases considerably when the number of time eriods increases for a given hysical network. As a rule of thumb, we observed that increasing the number of time eriods by five increased the number of cycles by a factor of ten. We have tested an alternative cycle reresentation where holding arcs are removed and waiting instead is introduced as art of movement arcs. With this CIRRELT

7 Service Network Design with Asset Management: Formulations and Comarative Analyzes aroach we were able to solve slightly larger instances than with the aroach used in this aer, but no significant imrovements in tractability were found. Different solution methods are thus required in order to handle larger instances. The definition of cycle variables naturally suggests dynamic column generation as art of these methods. The combinatorial nature of the roblem and revious work on network design hints that meta-heuristics and arallel comuting will robably also be art of the solution. The algorithms that we roosed for the a riori ath and cycle generation contain ideas that may be exlored in this context. We are currently working on these toics and exect to reort results in the near future. Acknowledgements This work has received financial suort from The Norwegian Research Council through the Polcorridor Logchain roject. Partial funding has also been sulied by the Natural Sciences and Engineering Research Council of Canada (NSERC) through its Discovery and Industrial Research Chair grant rograms. CIRRELT

8 Service Network Design with Asset Management: Formulations and Comarative Analyzes References Ahuja, R.K., Magnanti, T.L., Orlin, J.B., Network Flows. Prentice Hall, Uer Saddle River, NJ. Andersen, J., Crainic, T.G., Christiansen, M., Service network design with management and coordination of multile fleets. Euroean Journal of Oerational Research. Barnhart, C., Schneur, R.R, Air network design for exress shiment service. Oerations Research 44 (6), Crainic, T.G., Service network design in freight transortation. Euroean Journal of Oerational Research 122(2), Crainic, T.G., Laorte, G., Planning models for freight transortation. Euroean Journal of Oerational Research 97, Crainic, T.G., Gendreau, M., Farvolden, J.M., A Simlex-based Tabu Search for Caacitated Network Design. INFORMS Journal on Comuting 12(3), Crainic, T.G., Frangioni, A., Gendron, B., Bundle-based relaxation methods for multicommodity caacitated fixed charge network design. Discrete Alied Mathematics 112, Gendron, B., Crainic, T.G., Bounding rocedures for multicommodity caacitated fixed charge network design roblems. Publication CRT-96-06, Centre de recherche sur les transorts, Université de Montréal, Quebec, Canada. Gendron, B., Crainic, T.G., Frangioni, A., Multicommodity caacitated network design. In B. Sansò, P. Soriano (Eds.), Telecommunications Network Planning, Kluwer Academic Publishers, Doordrecht, 1999, (Chater 1). Kim, D., Barnhart, C., Ware, K., Reinhardt, G., Multimodal exress ackage delivery: A service network design alication. Transortation Science 33(4), Lai, M.F., Lo, H.K., Ferry service network design: otimal fleet size, routing and scheduling. Transortation Research Part A: Policy and Practice 38, Macharis, C., Bontekoning, Y.M., Oortunities for OR in intermodal freight transort research: A review. Euroean Journal of Oerational Research 153, Magnanti, T.L., Wong, R.T., Network design and transortation lanning: models and algorithms. Transortation Science 18(1), Pedersen, M.B., Otimization models and solution methods for intermodal transortation. PhD thesis, Centre for Traffic and Transort, Technical University of Denmark. Pedersen, M.B., Crainic, T.G., Otimization of intermodal freight train service schedules on train canals. Publication, Centre de recherche sur les transorts, Université de Montréal. Pedersen, M.B., Crainic, T.G., Madsen, O.B.G., 2007 Models and tabu search meta-heuristics for service network design with asset-balance requirements. Publication, Centre de recherche sur les transorts, Université de Montréal. CIRRELT

9 Service Network Design with Asset Management: Formulations and Comarative Analyzes Smilowitz, K.R., Atamtürk, A., Daganzo, C.F., Deferred item and vehicle routing within integrated networks. Transortation Research Part E: Logistics and Transortation 39, CIRRELT