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 = + l l Min z f y k h (, i j) A P l L (7) l L h l P l L l L a h = w a h, u y l l b y l l 0 P, (8), (, i j) Av, (9) 0, (, i j) Av, P, (10) l h 0, Pv, l L, (11) { 0,1} y, (, i j) Av. (12) The objective function (7) minimizes the sum of fixed costs for oening design arcs and flow costs on aths. Constraints (8) now state that the sum of flow on all aths for a roduct has to equal the demand for this roduct. The other constraints are similar to those of (1)-(6), with the new variables and arameters substituted into the formulation: Strong and weak forcing constraints are found in (9) and (10), resectively, while variable-tye constraints are resented in (11) and (12). We refer to model (1)-(6) as aa-cmnd, because both design variables and flow variables are arcs. Model (7)-(12) is referred to as a-cmnd, because the flow variables are defined in terms of aths. 2.2 Basic SNDAM model In this subsection we define the basic SNDAM model that corresonds to what was labeled Design Balanced Multicommodity Network Design in Pedersen (2006). The basic idea is that there should be an equal number of vehicles leaving and entering a node. We introduce design balance constraints (13) to ensure that all nodes have equal in- and out-degree of oened arcs. y y = 0, + j N () i j N () i ji i Nv. (13) The balancing requirement exressed in (13) has been resent in most design models within maritime and air transortation. For these modes of transortation, the focus is on routing exensive assets in an otimal way, which leads to the requirements for vehicle balance at terminals. Examles of such design studies with balancing requirements are Lai and Lo (2004) who model ferry service network design with balancing constraints for the ferries, and Barnhart and Schneur (1996), who model a balanced exress shiment design roblem for air transort. The literature with design balance constraints is however sarse in traditional service network design. Smilowitz et al. (2003) resent a model for multimodal ackage delivery with design balance constraints for ground vehicles, while Andersen et al. (2007) and Pedersen (2006; see also Pedersen and Crainic, 2007; Pedersen et al., 2007) resent service network design models with design balance constraints for locomotives. CIRRELT

7 Service Network Design with Asset Management: Formulations and Comarative Analyzes All the model formulations resented in Section 2 may be used on a static or a time-sace network. In a time-sace reresentation, the lanning horizon is divided into a set of time eriods, and the nodes in the static network are relicated in each time eriod. Service arcs reresent movements in both time and sace. In addition, holding arcs between consecutive time realizations of hysical nodes reresent the ossibility that assets and flow wait at nodes. The time-sace network increases the size of the network considerably, but allows for analysis of dynamic asects within the lanning horizon. 3 Extended SNDAM model The basic SNDAM model resented in Section 2.2 reresents an imortant ste for imroved reresentation of real-world decisions. There are however additional challenges from real-world roblems that have to be addressed in service network design models. In this subsection we develo an extended SNDAM model with constraints reresenting several asset management issues integrated within the service network design. We introduce relevant definitions and notation in Section 3.1, while the different asset management issues are discussed in Section 3.2. The extended SNDAM model is resented in Section Definitions and notation In this subsection, we resent definitions and notation for the extended SNDAM model. We first discuss the time-sace reresentation of network oerations including holding and reositioning decisions, before resenting the notation in the second subsection Time-sace reresentation of network oerations The asset management asects that we introduce require that we work in a time-sace setting. In the rest of this aer we assume an initial static network, but use a time-sace reresentation where the nodes in the static network are relicated in each time eriod. Figure 1 illustrates a time-sace network with three hysical nodes and four time eriods. The usual reresentation in service network design is to let services with intermediate stos between the origin and destination nodes be reresented as aths in the network, while non-sto services are reresented with single arcs. In this aer, we assume that all services can be reresented as network arcs, as illustrated in Figure 1. This does not affect the generality of the results, but allows us to kee the resentation simler than if we introduced service aths. The solid lines in Figure 1a) reresent design arcs that may be chosen for oeration. In Figure 1b), six design arcs are oened, and these satisfy design balance constraints (13), as all nodes have the same in- and out-degree. CIRRELT

8 Service Network Design with Asset Management: Formulations and Comarative Analyzes t=1 t=2 t=3 t=4 t=1 t=2 t=3 t=4 Node 1 Node 1 Node 2 Node 2 Node 3 Node 3 1a) 1b) Figure 1. Time-sace diagram for a network with three nodes and four time eriods (1a) and examle of feasible service lan (1b). In time-sace networks, holding arcs reresent assets or flow units ket at a hysical node from one time eriod to the next and, thus, link consecutive reresentations in time of the same hysical node. In the following, we assume that the models always have holding arcs for the flow, so that flow units may be ket at a hysical node from one time eriod to the next. Holding arcs for assets should be included unless it is required that the assets should oerate without any idle time. The latter is not usually an advantage, because it considerably restricts the management of the assets. Holding arcs are assumed to have infinite caacity both for assets and for flow. We do not comlicate the model formulation with articular notation and variables for holding arcs. In the rest of this aer we label arcs in the time-sace network that are not holding arcs as movement arcs. We do not include reositioning arcs for the assets. Notice that the design balance constraints (13) ensure that an equal number of assets enter and leave all nodes. Thus, reositioning may be reresented by a service without flow. The restriction imosed by this aroach is that direct reositioning is only ermitted between nodes that are connected with services. This does not imact the generality of the develoment, however, because including exlicit reositioning arcs does not change the structure of the model (but makes the resentation significantly more comlex) Notation We define a time-sace reresentation by dividing the lanning horizon into a set of time eriods T = { t} = { 1,.., TMAX} =. We refer to the grah G' = ( N ', A') for the static service network that is not time-exanded, and to the grah G= ( NxA, ) for the time-sace network. A node i ' N ' in the static service network is referred to as a hysical node, and has T MAX time realizations i N in the time-sace network. Each node i N has an associated time eriod, Ti T, and reresents a hysical node Ni ' N '. We allow services to leave at any time eriod. The service arcs (', i j') A ' are thus available in TMAX realizations, one for each time eriod. The duration of arcs (, i j) A in the time-sace network is the same for all realizations in time of the arcs in the static service network and is equal to Tj Ti. The flow costs on arcs, c, reresent time costs. They are CIRRELT

9 Service Network Design with Asset Management: Formulations and Comarative Analyzes calculated as the roduct of the value of time for a roduct and the traversal time of the arc. The maximum number of vehicles available is MAX V. Products have secific origin nodes in the time-sace network, but no unique destination nodes, because no exlicit delivery due-dates are considered. We introduce sink-destination nodes and zero-cost arcs from all time realizations of a hysical node to the sink-destination node for the roduct considered. Notice that, because roduct arc costs generally are reresented as time costs, this aroach will result in bringing the roducts to their destinations as early as ossible, even though there are no hard constraints reresenting due times. Delivery due dates for roducts may be incororated, however, by increasing the costs on the artificial arcs for the time eriods when a roduct is not allowed to arrive at its destination 3.2 Asset management issues We refer to the model formulation in Section 2.2 as the basic SNDAM model. We resent three extensions based on ideas from real-world alications Fleet size and asset costs In many service network design studies, secific fleet sizes for articular assets have to be resected. This is tyically the case for roblems that are closer to oerational-level lanning, where the degrees of freedom are fewer than in higher-level lanning. By introducing a set of new constraints to the model, we are able to connect the design variables to vehicle utilization. We assume that there is an uer bound on the number of vehicles, V MAX, and state in (14) that V MAX is an uer bound on the number of oerations that may be erformed in each time eriod. y VMAX (, i j) A : T i t< T j, t T. (14) The fleet of vehicles is assumed to be given and the objective function in the model is not directly affected by how these vehicles are managed. It may still be useful to include an otion to not utilize all vehicles in the fleet. By introducing a vehicle utilization cost, f, we let the model determine the minimum number of vehicles required to erform the transortation services for the given demand. The idea is not to include asects of investment analysis with trade-offs between fixed costs for vehicles and flow costs for roducts, but rather to investigate whether the fleet size could be reduced. Vehicles (assets) oerate not only when roviding service, but also to move from one service to the next. Thus, introducing vehicle utilization costs require the reresentation of the comlete vehicle workload. We resent this reresentation in Section We take this oortunity, however, to notice that, in this case, the fixed costs for oening arcs associated in the objective functions (1) and (7) to the cost of selecting/oerating services, is not really required any more. Therefore, in the rest of this aer, we will relace these fixed costs with the vehicle (asset) utilization costs Frequency constraints Secific lower and uer bounds may be imosed on service frequencies, i.e., on the number of occurrences of each service arc in the time-sace setting. These bounds may come from oerational requirements like this service should be oerated at least three times, but not more than six times a CIRRELT

10 Service Network Design with Asset Management: Formulations and Comarative Analyzes week (Andersen et al., 2007). We label lower and uer frequencies on service arcs ( i', j') A ', L i' j' and U i' j', resectively. The constraints are formulated in (15). L y U i' j' i' j' (, i j) A: N = i' UN = j' i j, (', i j') A ' v. (15) Several model runs would rovide the means to evaluate the cost of meeting these requirements, which might be imortant inut to decision-makers. If the costs for meeting such requirements are high, it might be worthwhile to reconsider whether these constraints are necessary for the case that is considered Route length constraints In many lanning roblems there is the need for reetitiveness in asset rotations. Considering crew management, for examle, this reresents the requirements that eole should leave their hometown every Monday morning. For vehicles, it may be that individual vehicles should oerate the same attern of services in all relications of the lanning horizon, even if these vehicles belong to fleets of homogenous vehicles. We use Figure 2 to illustrate this idea. Time TMAX Node 1 Nodes in hysical network Node 2 Node 3 Node 4 Node 5 Figure 2. Time-sace network with asset routes with length equal to two lanning horizons. In Figure 2, we have two assets that are used to oen design arcs in a cyclic time-sace network, and node balance constraints (13) are satisfied at all nodes. However, we observe that the assets interchange the arcs they cover at every second time eriod. It thus takes two reetitions of the lanning horizon for the assets to return to their initial attern. In other words, the route length for the assets corresonds to two times the length of the lanning horizon. The requirement that we want to address is that these assets routes are limited to the length of the lanning horizon. The oerations in Figure 2 will not be feasible if we imose such route length constraints. In order to cature the route length asect, the model needs to have a more exlicit V = v = 1,.., VMAX =. In addition, we reresentation of the vehicles. We introduce a set of vehicles { } { } need binary decision variables for each vehicle, δ v =, indicating whether it is used or not. The decision variables for design arcs y, have to be indexed for vehicles v, thus the new decision variables are y v. When this change is imlemented, we will have node balance constraints for each CIRRELT

11 Service Network Design with Asset Management: Formulations and Comarative Analyzes individual vehicle, not for the fleet of vehicles. The route length constraints (16) ensure that in each time eriod, each vehicle that is selected for oeration is engaged in one activity only. (, i j) A : T t< T : i j y v δ = 0, v t T, v V. (16) Observe that, with route length requirements, the asset movements constitute cycles in the time-sace network with length equal to the length of the lanning horizon, as is the case in Figure 1. It might seem that the strict cycle length of a single lanning horizon imoses a rigid structure, but, as illustrated in Figure 3, the reresentation allows the oeration of cycles shorter than this length. Time TMAX Node 1 Nodes in hysical network Node 2 Node 3 Node 4 Node 5 Figure 3. Examles of asset cycles in a time-sace network. The black arcs in Figure 3 corresond to oerations that satisfy the route length constraints, the length of the resulting cycle being equal to one lanning horizon. In the static service network, this time-sace cycle consists of two cycles, one from node 3 at time 1 to node 3 at time 4, and one from node 3 at time 4 to node 3 at time 1. The grey arcs in Figure 3 reresent another examle of a feasible vehicle rotation. With three holding arcs, including the one from time T MAX to time 1, the oerating cycle lasts for four time eriods only. There is thus flexibility in the system even when route length restrictions are imosed. 3.3 Extended SNDAM model Adding the extensions of Section 3.2 to the basic SNDAM formulation defined in Section 2 yields the model defined by exressions (17)-(27). The rest of this aer will be based on this formulation, which we label extended SNDAM. CIRRELT

12 Service Network Design with Asset Management: Formulations and Comarative Analyzes Min z = c x + f δ v (, i j) A P v V (, i j) A : T t< T : i y j y v v + j N () i j N () i v V y v 1, δ = 0, v y = 0, jiv L y U i' j' v i' j' (, i j) A : N = i' UN = j' i j, (17) t T, v V, (18) i Nv, v V, (19) (, i j) A, v (20) (', i j') A ' v, (21) w, i = o x x = w, i = d 0, otherwise, ji + j N () i j N () i P x x u y v v V b y v v V 0 P, i Nv, (22), (, i j) A, v (23) 0, (, i j) Av, P, (24) x 0, (, i j) Av, P, (25) v { 0,1} y, (, i j) A, v V, (26) v { 0,1} δ, v V. (27) The objective function (17) minimizes the total system cost comuted as the sum of the flow costs and of the costs associated with using the assets. Constraints (18) state for each time eriod that if an asset is utilized, it should be engaged in one activity only. These are the route length constraints that were introduced in Section Constraints (19) are the design balance constraints, while relations (20) ensure that only one vehicle can oerate an arc( i, j) A, which is analogous to the binary y variables in the basic SNDAM formulation (1)-(6) and (13). Lower and uer bounds on service frequencies are enforced by constraints (21), while flow balance is ensured through equations (22). Constraints (23) and (24) are the weak and strong forcing constraints, resectively, where we now aggregate over all vehicles. Finally, variable-tye constraints are given in (25) to (27). CIRRELT

13 Service Network Design with Asset Management: Formulations and Comarative Analyzes 4 Alternate formulations of the extended SNDAM model Solution aroaches for CMND models are based on either arc- or ath-based formulations referring to the decision variables associated to roduct flows, as discussed in Section 2.1. Obviously, a athbased formulation also exists for the basic SNDAM model. Notice that, when design balance constraints are introduced, the selection of the design arcs follows closed ath-like structures. Indeed, while roduct aths must start and end at different nodes, namely the origin and destination nodes of the roduct demands, the aths of design variables must return to their starting nodes, at a different time eriod, to satisfy the design balance constraints. Selected design variables thus form cycles and, when route length constraints are imosed, these cycles cover all time eriods exactly once. Examles of such cycles were found in Figure 3. Four different reresentations of the extended SNDAM model can be written using the two different reresentations for the flow variables and the two reresentations for the design variables. To distinguish them, we introduce the following naming scheme for the SNDAM formulations: yx-sndam, where y = a or c, deending of whether design variables are defined in terms of arcs or cycles x = a or, deending of whether flow variables are defined in terms of arcs or aths The extended model formulation (17) - (27) is therefore aa-sndam. In the following subsections we resent the three other formulations to the extended SNDAM model. 4.1 Arc-Path formulation of SNDAM We define aths L for roduct as in Section 2.1, and arameters to ath l L for roduct, 0 otherwise. The flow of roduct on ath l is l l l cost for transorting roduct on ath l is k, k = c a SNDAM (arc-based design, ath-based flow variables): (, i j) A l a =1 if arc ( i, j) A belongs l h, while the flow. We can then formulate a- CIRRELT

14 Service Network Design with Asset Management: Formulations and Comarative Analyzes l l Min z = k h + f δv P l L v V (28) (, i j) A : T t< T : i y j y v v + j N () i j N () i v V y v 1, δ = 0, v y = 0, jiv L y U i' j' v i' j' (, i j) A : N = i' UN = j' l L h l P l L l L a h i = w a h j, u y l l v v V b y l l v v V, 0 t T, v V, (29) i Nv, v V, (30) (, i j) A, v (31) (', i j') A ' v, (32) P, (33), (, i j) A, v (34) 0, (, i j) A, P, v (35) l h 0, Pv, l L, (36) v { 0,1} y, (, i j) A, v V, (37) v { 0,1} δ, v V. (38) The objective function (28) minimizes the sum of flow costs on aths and fixed costs for assets. Constraints (29)-(32) are identical to (19)-(22). The major change from model formulation (17)-(27) is reresented by constraints (33) stating that for each roduct, the sum of the flows on all aths need to equal the demand. These constraints relace the flow balance constraints (22) of aa- SNDAM, which are ensured by the ath definition. As in revious formulations, inequalities (34) and (35) are the weak and strong forcing constraints, resectively, while the variable-tye constraints are resented in (36)-(38). 4.2 Cycle-Arc formulation of SNDAM We introduce cycle design variables k g,{ k} Kv, equal to 1 if cycle kvis in the solution, and 0 k otherwise. We also define arameter r vequal to 1 if arc ( i, j) A is in cycle kv, 0 otherwise. We may then formulate the ca-sndam model (cycle-based design, arc-based flow variables), where route length constraints enforce the requirement that selected cycles cover all time eriods in the time-sace reresentation. The cycles in Figure 3 are examles of feasible asset cycles. Notice that, CIRRELT

15 Service Network Design with Asset Management: Formulations and Comarative Analyzes we no longer need the asset decision variables, because each selected cycle reresents a unit of asset (a vehicle). The cycle-arc version of the extended SNDAM model is resented in (39)-(47): Min z = c x + f g k (, i j) A P k K (39) k K k K k g VMAX rg k k, 1, (, i j) A, (41) L rg U k k i' j' i' j' k K (, i j) A: NOD = i' UNOD = j' i j w, i = o x x = w, i = d 0, otherwise, ji + j N () i j N () i P x x u r g k k k K b r g k k k K 0, (40) (', i j') A ' v, (42) P, i Nv, (43), (, i j) A, v (44) 0, (, i j) Av, P, (45) x 0, (, i j) Av, P, (46) { 0,1} k g, k Kv. (47) The objective function (39) minimizes the sum of the flow distribution costs and fixed costs for the assets that are selected and used. The structurally new constraint is (40). This constraint restricts the cycle selection, stating that the number of selected cycles is limited by the size of the V = 1,.., VMAX =in the aa- available fleet of assets. This roerty was ensured by the dimension of set { } SNDAM and a-sndam formulations. Constraints (41) state that each arc( i, j) A can be chosen by at most one cycle, which is analogous to constraints (20). In relations (42) we set the lower and uer bounds on service frequencies, while flow balance is ensured through constraints (43). Weak and strong forcing constraints are formulated in (44) and (45), resectively, while variable-tye constraints are found in (46) and (47). 4.3 Cycle-Path formulation of SNDAM We can combine ath variables for the flow and cycle variables for the design arcs. The cycle-ath formulation is resented in (48)-(56), where the objective function (48) minimizes the sum of the flow costs on aths and fixed costs for assets utilized on cycle routes. The fleet size restriction aears in (49), while selection of arc disjoint cycles is enforced by constraint (50). Lower and uer bounds on service frequencies are formulated in constraints (51), while demand satisfaction is CIRRELT

16 Service Network Design with Asset Management: Formulations and Comarative Analyzes rovided by relations (52). Weak and strong forcing constraints are found in (53) and (54), resectively, while variable-tye constraints aear in constraints (55) to (56). Min z = k h + f g k K P l L k g VMAX l l k, k K (48) (49) k K rg k k 1, (, i j) A, (50) L rg U k k i' j' i' j' k K (, i j) A: NOD = i' UNOD = j' i j, (', i j') A ' v, (51) l L h l = w, P, (52) P l L a h u r g l l k k k K 0, (, i j) A, v (53) l L a h b r g l l k k k K 0, (, i j) Av, P, (54) l h 0, Pv, l L, (55) { 0,1} k g, k Kv. (56) 5 Algorithms for ath and cycle generation CMND models may be solved in a variety of ways, see for instance Magnanti and Wong (1984) and Gendron et al. (1999). The develoment of articular methods for the models roosed is well beyond the scoe of this aer. Our goal is rather to comare the different formulations of the extended SNDAM model in order to identify their strengths and weaknesses with resect to comutational efficiency, and to study the imact of asset management considerations. We erform this comarison on roblem instances where all aths and cycles have been generated a riori. This avoids the unnecessary burden of develoing rogressive variable generation-based rocedures that, together with meta-heuristics, are required to address larger instances and should be the scoe of future work. We describe algorithms for generation of flow aths and design cycles in Sections 5.1 and 5.2, resectively. CIRRELT

17 Service Network Design with Asset Management: Formulations and Comarative Analyzes 5.1 Path generation We now resent an algorithm for comlete a riori generation of flow aths. We assume that there are no connections between nodes of the same time eriod and that the ath length cannot exceed the length of the lanning horizon. The algorithm considers all outgoing arcs from the origin node of a roduct. For each of these arcs, a ath, reresented as a linked list is created. These aths constitute an initial ool of candidate aths. The algorithm then traverses all nodes, starting from the closest successor among the nodes that are reached by the outgoing arcs from the origin node. For all candidate aths whose last node so far is the current node, the candidate aths are udated with the outgoing arcs from these nodes. When a node belonging to the roduct hysical destination node is reached, a feasible roduct ath from origin to destination is found. A dominance rule is introduced to avoid an exonentially growing number of aths as a consequence of the existence of holding arcs: If a candidate ath has already visited a hysical node, it should not return to the same hysical node later, since using the uncaacitated holding arcs in this case is at least as chea as using movement arcs. The algorithm is resented in Figure 4. forall( in Product) do forall(arcs starting in the roduct s origin) do create candidate list based on the arc; end-do define candidate nodes to be nodes from the time eriod after the roduct becomes available to the end of the lanning horizon, and again starting from the first eriod until two time eriods before the roduct becomes available; forall(candidate nodes) do forall(candidate lists where last node = this node) do forall(emanating arcs of the candidate node) do if(destination node of arc belongs to the roduct s destination node) then a ath has been found, store it; else if(visited=false for the aurtenant hysical node) then if(destination of arc is reached without crossing the time eriod where the roduct becomes available) then add arc to candidate ath; visited of aurtenant hysical node of candidate ath=true; end-if else if(the origin and destination of the arc have the same aurtenant hysical node) then add arc to candidate ath; end-if end-if end-if end-do end-do end-do end-do Figure 4. Pseudo-code for the ath generation algorithm. 5.2 Cycle generation A cycle is a set of consecutive arcs in the time-sace network, a ath, with length equal to the length of the lanning horizon, and closed such that one will return to the initial node after traversing all the arcs. To build cycles, we start with all arcs emanating from nodes in the first time eriod, and define each of these arcs to be the first arc in a candidate ath. Then, for nodes in time eriod 2 and CIRRELT

18 Service Network Design with Asset Management: Formulations and Comarative Analyzes later, we check if the candidate aths have this node as their current last node. In the affirmative, the outgoing arcs of this node are added to these candidate aths. Thus, for examle, if there are three emanating arcs at a node, each candidate ath that has this node as its current last node yields three candidate aths, one for each emanating arc. If the destination node of such an outgoing arc is the origin node of the candidate ath, a cycle has been found. However, if the destination node of the arc is such that the length of the candidate ath is longer than or equal to the length of the lanning horizon and no cycle has been established, then this candidate ath is not udated with the arc, because it cannot result in a feasible cycle covering all time eriods in the lanning horizon exactly once. When all arcs have duration of one time eriod, this aroach is sufficient to create all ossible cycles. However, when arcs with longer durations exist, aths that do not include nodes at eriod 1 may be overlooked. To avoid such a case, we include a second hase where candidate aths are established from all arcs that are cross-horizon arcs (origin node has a smaller node number than the destination node) and end after time eriod 1. The algorithm for cycle generation thus consists of three hases, where the two first hases generate initial candidate aths and the third hase traverses the network and searches for cycles. In Figure 5 we illustrate one cycle, solid lined, which would be created from the first hase of the algorithm, and a dotted cycle that would be created from the second hase of the algorithm. Time TMAX Nodes in hysical network Node 1 Node 2 Node 3 Node 4 Figure 5. Cycles that are created from the first (solid) and second (dotted) hase of the algorithm. There are no restrictions on asset movement during the cycle generation. Thus, for instance, a vehicle may enter and leave a terminal (node) several times throughout a weekly lanning horizon. The algorithm for cycle generation is summarized in Figure 6. 6 Comutational study The urose of the comutational study is to evaluate the erformance of the different formulations of the extended SNDAM model with resect to solution times - and to study the effect of asset management on solution times and on the lans obtained from the models. For cycles and aths, we use the a riori generation rocedures resented in Section 5. We are interested, in articular, whether the introduction of cycles seems useful and, if this is the case, under what circumstances do the associated formulations erform well. CIRRELT

19 Service Network Design with Asset Management: Formulations and Comarative Analyzes To exlore these issues, we create a set of roblem instances with varying dimensions. On the one hand, we create instances with 15, 20, and 25 time eriods, and 15, 20, and 25 roducts, roducing 9 scenarios. On the other hand, we exlore two different network densities, reresenting sarse and dense networks, building networks with 5 hysical nodes and 10 and 15 arcs in each time eriod, resectively. Finally, we generate three 15 arcs and 15 time eriods scenarios where we increased the number of roducts to 50, 100 and 200. The 21 data sets and corresonding scenarios are labeled nwaxtyz, where w is the number of hysical nodes, x is the number of hysical arcs excluding holding arcs, y is the number of time eriods, and z is the number of roducts. We work with the strong formulations of the roblem throughout the comutational study, imlying that the strong forcing constraints are included in the formulations. This increases the size of the roblems considerably, which again reduces the tractable roblem dimensions. Yet, because for general CMND models the strength of the linear relaxations is imroved significantly when strong forcing constraints are considered, we find it relevant to consider the strong formulations for the SNDAM as well. Initialization hase 1 forall(nodes in first time eriod) do forall(emanating arcs) do create candidate ath based on the arc; end-do end-do Initialization hase 2 forall(nodes after first time eriod) do forall(emanating arcs) do if(the arc is a cross-horizon arc and ends after the first eriod) then create candidate ath based on the arc; end-if end-do end-do Joint cycle generation hase forall(nodes after first time eriod) do forall(candidate aths where last node = this node) do forall(emanating arcs of this node) do if(destination of arc = the aths origin) then a cycle has been found, store it; else if(destination of arc > the aths origin) then ski arc because we cannot make a cycle; else add arc to candidate ath; end-if end-if end-do end-do end-do Figure 6. Pseudo-code for the cycle generation algorithm. We comare the four SNDAM formulations in Section 6.1 using the 21 scenarios defined above. To make a stronger claim on the reresentativeness of the results, we also comare the formulations on five additional data sets generated for a subset of the scenarios. Section 6.2 is dedicated to the study of the imact of the asset management issues. We choose a subset of the scenarios from Section 6.1 and examine how solution times and otimal integer solution values are affected by the various asset management issues of the extended SNDAM model. CIRRELT

20 Service Network Design with Asset Management: Formulations and Comarative Analyzes The solution times of the arc-based formulations are sensitive to the fleet size, because a large number of vehicles introduces symmetry and increases the number of binary design variables. To mitigate this imact, we always solve the cycle-based formulations first, and use the resulting otimal number of vehicles as the asset fleet size when solving the arc-based formulations. All formulations are modeled in XPRESS-IVE version and matrix files are created on a comuter with an Intel Dual Core 2,3Ghz rocessor and 2GB memory. The matrix files are imorted to CPLEX and solved using a 2.4 GHz AMD Oteron 250 rocessor with 16GB memory. The algorithms for a riori generation of flow aths and design cycles are imlemented in Visual C Comaring the SNDAM formulations Table 1 dislays solution times for finding the otimal integer solution for the four formulations of the extended SNDAM roblem in columns 1 to 4, resectively. All roblems were run for 10 hours ( CPU seconds). No integer is marked when no integer solution was found within 10 hours of CPU time. The ga between the lower and uer bounds at termination is reorted when integer solutions were found, but otimality could not be roven within 10 hours of CPU time. The last two columns of Table 1 dislay the gas between the strong linear relaxations at the root node of the branch-and-bound with the two arc- and the two cycle-based formulations, resectively, and the best integer solution obtained within 10 hours of CPU time with any of the four formulations. For most of the instances, this integer solution was the otimal integer solution of the roblem. These gas gave an indication of the tightness of the strong linear relaxations of the formulations. We observe from Table 1 that the cycle-based formulations erform very well comared to the formulations with design arcs. In all scenarios, the fastest solution time was obtained with one of the cycle-based formulations. It is not so that both cycle-based formulations are the best for all the roblems but, in general, they outerform the arc-based formulations. One interesting observation is that for given numbers of hysical arcs and roducts, increases in the number of time eriods increases the solution times for the cycle-based formulations. The reason for this is that the number of cycles grows exonentially with the number of time eriods. For arc-based formulations, however, we observe the oosite effect in several cases; the solution times are reduced with an increased number of time eriods. To exlain this, we observe that for a given demand and duration of the arcs, one vehicle can oerate more services during the lanning horizon with more time eriods, and fewer vehicles are thus needed to cover the demand. The latter reduces the roblem with symmetry in the arc-based formulations, which are based on vehicle-indexed design arcs. However, if we increase the number of roducts for a given number of time eriods, the solution time increases also for the arc-arc and arc-ath formulations. To summarize the discussion, cyclebased formulations aear to erform significantly better than the formulations with design arcs, and the difference in solution times increases if the ratio between volumes shied and number of time eriods increases. CIRRELT

21 Service Network Design with Asset Management: Formulations and Comarative Analyzes Table 1. Solution times (CPU seconds) and gas between strong linear relaxation at root node and best integer solution. Otimality ga is reorted if solution time exceeds sec. Problem Arc-arc Arc-ath Cycle-arc Cycle-ath Strong LP-ga Strong LP-ga solution time solution time solution time solution time aa and a ca and c n5a10t % 9 % n5a10t % 6 % n5a10t % 5 % n5a10t % 3 % n5a10t % 4 % n5a10t % 4 % n5a10t1525 ga 3.6% ga 4.5% % 4 % n5a10t2025 ga 6.0% ga 3.9% % 2 % n5a10t % 3 % n5a15t <0.5 <0.5 9 % 0.5 % n5a15t % 6 % n5a15t % 0.0 % n5a15t % 1 % n5a15t % 1 % n5a15t2520 ga 1.5% ga 3.0% % 2 % n5a15t1525 ga 4.6% ga 4.8% % 5 % n5a15t2025 ga 4.1% ga 4.1% % 4 % n5a15t % 0.2 % n5a15t1550 ga 10.9% ga 8.4% % 2 % n5a15t15100 No integer ga 10.4% ga 0.8% ga 0.7% 9 % 2 % n5a15t15200 No integer No integer ga 0.6% ga 0.5% 20 % 1 % Caacitated multicommodity fixed charge network design roblems usually have oor linear relaxations (Gendron and Crainic, 1996), which is one imortant reason for the difficulty associated with addressing CMND models. In the last two columns of Table 1, we reort the ga between the strong linear relaxation at the root node of the branch-and-bound tree with arc- and cycle-based formulations, and the best integer solution that was obtained for the secific instance with any of the four formulations. Table 1 indicates that there is a significant difference between the formulations with arc-based and cycle-based design variables when it comes to the strength of the linear relaxation. While the arc-arc and the arc-ath formulations tyically yield gas from 5% to 20%, the cycle-arc and cycle-ath formulations dislay gas from 1% to 5%. This weakness of the lower bounds contributes certainly to the long solution times of the arc-arc and the arc-ath formulation reorted in Table 1. The results of the comutational study thus indicate that the introduction of cycle-based design variables constitutes a romising research direction for efficient solution methods for service network design roblems with asset management. For the last three scenarios reorted in Table 1, the number of roducts is increased towards more realistic values. We observe that the cycle-based formulations become more difficult to solve when the dimensions of the roblems increase; otimal solution cannot be roved for the 100 and 200 roduct roblem instances within 10 hours of CPU time. However, the cycle-based formulations still outerform arc-based formulations. For 200 roducts, none of the arc-based formulations roduce integer solutions, while for 100 roducts, only one integer solution is found for the arc-ath formulation, with an otimality ga of 10.4% after 10 hours of CPU time. We thus conclude that cycle-based formulations offer the most attractive ersective for designing solution methods for this class of roblems. CIRRELT