BranchandPrice for Service Network Design with Asset Management Constraints


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1 BranchandPrice for Servicee Network Design with Asset Management Constraints Jardar Andersen Roar Grønhaug Mariellee Christiansen Teodor Gabriel Crainic December 2007 CIRRELT
2 BranchandPrice for Service Network Design with Asset Management Constraints Jardar Andersen 1,*, Roar Grønhaug 1, Marielle Christiansen 1, Teodor Gabriel Crainic 2 1 Deartment of Industrial Economics and Technology Management, Norwegian University of Science and Technology, 7491 Trondheim, Norway 2 Interuniversity Research Centre on Enterrise Networks, Logistics and Transortation (CIRRELT), and NSERC Industrial Research Chair on Logistics Management, Université du Québec à Montréal, C.P. 8888, succursale Centreville, Montréal, Canada H3C 3P8 Abstract. We resent a branchandrice algorithm for solving the service network design roblem with asset management constraints, SNDAM. The asset management constraints may result from vehicle management in alications from transortation. The roblem is a multicommodity network design roblem with additional constraints restricting and connecting the integer design variables. As a result of these constraints, design variables are cycles in a cyclic timesace network, while the flow variables are aths in the same network. Searate subroblems generate design cycles and aths for commodity flows to the restricted master roblem. The comutational study shows that we are able to find nearotimal solutions for large network instances. Keywords. Service network design, branchandrice, column generation, vehicle management, 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 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: Déôt légal Bibliothèque nationale du Québec, Bibliothèque nationale du Canada, 2007 Coyright Andersen, Grønhaug, Christiansen, Crainic and CIRRELT, 2007
3 Introduction Imrovements in solution techniques and comutational caacity continuously increase the range of tractable otimization roblems. Simultaneously, realworld lanning roblems challenge the OR community to study increasingly larger and more comlex otimization roblems. New requirements from ractical alications result in new constraints that have to be modeled, and this also encourages develoment of solution methods that can solve the new models. In articular, there has recently been an increase in new service network design models for lanning roblems in transortation. What these models have in common is an increased focus on utilization and management of vehicles in addition to the traditional decisions on service selection and flow routing. One issue that has been studied in the literature is ensuring that there is an equal number of vehicles entering and leaving each node in the network. Constraints ensuring this are referred to as design balance constraints (Pedersen and Crainic, 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) and (Kim et al., 1999). Andersen et al. (2007a) introduce more asects of vehicle management in network design, including fleet sizing and management. Andersen et al. (2007b) resent a framework for Service Network Design with Asset Management constraints (SNDAM), which is a richer roblem than the roblem studied in (Pedersen and Crainic, 2007). Asset is a generic term for something a erson or an organization owns. In carrier terms, it refers to what we usually call resources. It may be a ower unit (a tractor or a locomotive), a carrying unit (a railcar, a trailer, a truck, a shi, etc.), a loading/unloading unit (cranes in terminals), a crew, etc. The term asset management reflects that assets have to be managed intelligently in order to obtain efficient oerations. The SNDAM model of (Andersen et al., 2007b) addresses the case when only one asset is controlled, and to increase readability we refer to the assets as vehicles. The SNDAM model allows for cycles as design variables corresonding to vehicle rotations in timesace networks, and the comutational study of (Andersen et al., 2007b) indicates that cycle variables contribute to efficient model solving. Cycle variables have also been imlemented for other roblems. For instance, Sigurd et al. (2005) work on a cycle reresentation of shi schedules in a icku and delivery roblem, and use column generation to generate schedules. Agarwal and Ergun (2006) introduce cycle variables for a cargo routing roblem in liner shiing, where cycles reresent shi schedules. They reort romising comutational results based on column generation and benders decomosition. The focus in (Andersen et al., 2007b) was on model develoment, as the main objective was to comare alternative SNDAM formulations. The comutational study was based on a riori enumeration of cycles and aths, which imoses huge memory requirements for larger instances, and the enumeration of cycles is imractical for large roblems. There is thus a need for more advanced solution strategies that can solve realisticallysized roblems of this character. The urose of this aer is to address the need for advanced solution methods for the SNDAM roblem that was resented in (Andersen et al., 2007b), and we therefore develo a tailormade branchandrice algorithm for the SNDAM roblem. A variablefixing technique is introduced to enhance the erformance of the algorithm. We are able to find good integer solutions for roblems that are considerably larger than what was solved in (Andersen et al., 2007b). CIRRELT
4 The contribution of the aer is a tailormade solution aroach for the SNDAM roblem that can solve realisticallysized roblems. The comutational study indicates that the algorithms are able to roduce good integer solutions within reasonable comutational time. The algorithms thus lay the ground for imroved lanning of roblems where these additional constraints occur, for instance in service network design studies where fleet management is considered. The outline of this aer is as follows. In Section 1, we recall the SNDAM formulation. In Section 2, we roose a branchandrice algorithm for solving SNDAM. The comutational study is resented in Section 3, while concluding remarks follow in Section 4. 1 Service Network Design with Asset Management In this section we recall the service network design roblem with asset management constraints, SNDAM, that was resented in (Andersen et al., 2007b). A roblem descrition follows in Section 1.1, while we in Section 1.2 recall two SNDAM formulations from (Andersen et al., 2007b). 1.1 Problem descrition In the classical service network design formulation (see e.g. (Crainic, 2000)) we consider a set of services that may be oened to accommodate a demand for flow through the system. Major decisions are selection of services and routing of the flow on services. Crainic (2000) differentiated between frequency and dynamic service network design models. Frequency models address strategic/tactical issues like what services to oerate and how often to oerate them. Dynamic models target lanning of schedules and are tyically concerned with when services deart. The SNDAM model is develoed within a dynamic service network design setting. A given lanning horizon is assumed, and the services have to be oerated in a reetitive manner reresenting fixed schedules of realworld transortation services. Demand in the system is defined in terms of commodities (roducts) requiring transort through the network. Each commodity has an associated volume that has to be transorted from the commodity s origin node to its destination node. The commodities have secific times where they become available, but they may arrive at their destinations at any time, as long as they are transorted to their destinations within the length of the lanning horizon. Oeration of services requires vehicles, which are available in a limited quantity. There is thus a need to consider a set of vehicle management issues when the services are designed, which extend the formulations in (Crainic, 2000). Firstly, we have to make sure that for all terminals, there is an equal number of vehicles entering and leaving, referred to as vehicle balance or design balance. Moreover, there may not be more simultaneous activities taking lace than the given fleet of vehicles allow for. A third issue from (Andersen et al., 2007b) is the existence of lower and uer bounds on the number of occurrences of each service, for instance that a service should be oerated at least 3 times and at most 7 times each week. A fourth asect from (Andersen et al., 2007b) is limited durations of vehicle routes, which may be catured by a route length requirement. The idea is that vehicles should not have longer routes than the lanning horizon considered, as illustrated in Figure 1. In Figure 1, we have three vehicles oerating in a cyclic timesace network, which is reresenting five terminals and a lanning horizon divided into seven time eriods. In Figure 1, design balance is satisfied for all nodes. However, we observe that the vehicles reresented with dotted arcs interchange the arcs they cover in every second realization of the lanning horizon. It thus takes two reetitions of the lanning CIRRELT
5 horizon for a vehicle to return to its initial attern. The oerations in Figure 1 reresented with dotted arcs will not be feasible if we imose route length requirements. However, the combination of solid arcs in Figure 1 reresents an oeration of a vehicle that satisfies the route length requirements. Time T MAX Nodes in hysical network Figure 1. Timesace network with two twohorizon cycles and one 1horizon cycle. In traditional service network design studies, one considers service selection cost associated with oeration of services, and flow costs associated for commodities utilizing services (Crainic, 2000). In the SNDAM roblem of (Andersen et al., 2007b), it is assumed that there are high fixed costs associated with vehicles, and that these costs dominate the service selection costs. This is based on a lanning roblem in the rail freight industry, where the acquisition costs for locomotives aeared to be the dominant cost factor. Inclusion of service selection costs would however not introduce any structural changes to the SNDAM models that we recall in the next subsection. 1.2 SNDAM models We assume that we have a static network G' = ( N ', A') with nodes reresenting terminals and intersections, and arcs reresenting connections. The lanning horizon is divided into a set of time eriods T= = { 1,.., TMAX}, and we introduce the grah G= ( NxA, ) for the timesace network. A node i ' N ' in the static network hast MAX realizations in the timesace network. Each of these nodes i N in the timesace network has an associated time eriod, Ti T, and reresents a hysical node NOD i N '. The arcs (', i j') A ' in the static network are also available in TMAX realizations, one for each time eriod. We consider one reetition of the lanning horizon with time eriod TMAX receding time eriod 1, giving a cyclic timesace network. In timesace networks, holding arcs are required when modeling realworld lanning roblems. Holding arcs reresent vehicles or flow units ket at a static node from one time eriod to the next. 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. Thus, the holding arcs link consecutive reresentations in time of the same hysical node. Holding arcs are assumed to have infinite caacity both for vehicles and for flow. We do not comlicate the model formulation with notation and variables for the holding arcs. CIRRELT
6 Without loss of generality we assume that all service arcs ( i, j) A are design arcs. To simlify the resentation, we assume that the arcs ( i', j') A ' reresent services, hence we do not consider services consisting of multile arcs. The different time realizations( i, j) A reresent alternative dearture times of the service connections (', i j') A '. Lower and uer bounds on the number of occurrences of each service are labeled Li' j' and U i' j', resectively. For the demand, we define for each commodity P the volume destination node d. w to be transorted from the commodity s origin node o to its In order to model the asset management constraints, we introduce vehicles v V, available in a quantity of V MAX = V. Any service arc that is oened has to utilize one of these vehicles. For each vehicle v V we introduce a binary decision variable δ v, which is 1 if vehicle v is utilized, and 0 otherwise. The cost for utilizing a vehicle is f. In classical service network design formulations, there are binary variables indicating whether a service arc is oened or not. In order to satisfy the route length requirements, these variables have to be indexed by vehicle in the SNDAM formulation, and are labeled y v. Flow variables x are nonnegative real numbers. Each design arc y v has an associated caacity u. For each unit of commodity there is a flow cost c for traversing arc (, i j) A. For each node we define sets + N () i = { j N :(, i j) A } and () i = { j :( j,) i } also define b min { w, u} N N A of outward and inward neighbors. We =. The SNDAM model is recalled in (1) (11): Min z = c x + f δ v (, i j) A P v V (1) (, 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, L y U i' j' v i' j' (, i j) A: NOD = i' UNOD = j' v V i jiv j t T, v V, (2) i Nv, v V, (3) (, i j) A, v (4), (', i j') A ' v, (5) CIRRELT
7 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, (6), (, i j) A, v (7) 0, (, i j) Av, P, (8) x 0, (, i j) Av, P, (9) v { 0,1} y, (, i j) A, v V, (10) v { 0,1} δ, v V. (11) The objective function (1) minimizes the sum of flow costs and costs associated with use of vehicles. Constraints (2) state that for each time eriod, if a vehicle is utilized, it should be engaged in one and only one activity, corresonding to the route length requirements. Constraints (3) are the vehicle balance constraints, stating that for each node, there is an equal number of vehicles entering and leaving. In constraints (4) we ensure that only one vehicle can oerate an arc ( i, j) A, which reflects the roerty that design arcs in traditional network design formulations are binary variables. In (5) we resent the lower and uer bounds on the number of occurrences of services while flow balance, as found in traditional network design formulations, is ensured in (6). Constraints (7) and (8) are weak and strong forcing constraints, resectively, where we need to aggregate over all vehicles used. Finally, variabletye constraints are given in (9) (11). In (Andersen et al., 2007b), four different formulations of the SNDAM model were resented In this section we recall the formulation with design cycles and flow aths as decision variables, which in (Andersen et al., 2007b) roduced the most romising comutational results, and which was significantly faster solved than (1) (11). A design cycle consists of a set of design arcs satisfying design balance constraints (3) and covering each time eriod exactly once. The latter is a necessary condition for avoiding multilanninghorizon cycles, which would violate the route length requirements. The design variable g k for cycle k Kv is 1 if cycle k is in the solution and 0 k otherwise. Moreover, arameter m vis 1 if arc (, i j) A is in cycle k. The cycles have to cover all time eriods in the timesace reresentation, as exemlified with the cycle indicated with solid arcs in Figure 1. As is frequently done in network design (see e.g. (Ahuja et al., 1993)), we further define the set of aths L that commodity may use from its origin node to its destination node. For these aths, we define arameters l a =1 if arc (, i j) A belongs to ath 0 otherwise. The flow of commodity on ath l is l h l L for commodity,, while the flow cost for transorting CIRRELT
8 commodity on ath l is l k, variables is recalled in (12) (20): Min z = k h + f g P l L l l k k K k = c a l l (, i j) A. The SNDAM model with cycle and ath (12) k K k K k g VMAX mg k k, 1, (, i j) A, (14) L mg U k k i' j' i' j' k K (, i j) A: NOD = i' UNOD = j' l L h l P l L l L a h = w a h, l l k k k K i u m g b m g l l k k k K j 0 (13), (', i j') A ' v, (15) P, (16), (, i j) A, v (17) 0, (, i j) Av, P, (18) l h 0, Pv, l L, (19) { 0,1} k g, k Kv. (20) The objective function (12) minimizes the sum of flow costs on aths and fixed costs for vehicles that are utilized. The structurally new constraint comared to model (1) (11) is (13). This constraint restricts the cycle selection, stating that the number of selected cycles is limited by the V = 1,.., VMAX =in model (1) (11). fleet of vehicles. This roerty was ensured by the size of set { } Constraints (14) state that each arc( i, j) A can be chosen by at most one cycle, analogous to (4). Lower and uer bounds on the number of realizations of static arcs are formulated in (15), and demand satisfaction is ensured in (16). Weak and strong forcing constraints are found in (17) and (18), while variabletye constraints aear in (19) (20). Note that the introduction of cycles and aths has removed the need for design balance constraints (3) and flow balance constraints (6). Moreover, because each cycle corresonds to a vehicle and fleet size is accounted for in (13), we no longer need the binaryδ v variables. Magnanti and Wong (1984) show that the uncaacitated fixed charge network design roblem is NPhard. As the caacitated version is even harder (Balakrishnan et al., 1997), this roblem also belongs to the class of NPhard roblems. There is no reason to believe that the roblem becomes easier to solve when additional asset management constraints are introduced. CIRRELT
9 2 Solving SNDAM with branchandrice Among four formulations of the SNDAM model in (Andersen et al., 2007b), formulation (12) (20) aeared to be more easily solved than the other formulations. However, the comutational study was limited to instances that could be handled with a riori enumeration of cycles and aths. For realworld lanning roblems, comlete enumeration is imracticable. In this section we develo a branchandrice algorithm for the SNDAM roblem, consisting of a column generation algorithm embedded in a branchandbound framework. The algorithm is based on formulation (12) (20), and the columns that are generated are design cycles and flow aths. This branchandrice framework addresses the need for an exact solution algorithm that may solve realworld instances of the SNDAM roblem. Several toics in column generation are covered in (Desaulniers et al., 2005), and general introductions to branchandrice and column generation can be found in (Barnhart et al., 1998) and (Lübbecke and Desrosiers, 2005). We recall basic rinciles of column generation and branchandrice in Section 2.1. Sections are devoted to the solution of the linear relaxation of the roblem; we define the restricted master roblem in Section 2.2 and subroblems in Section 2.3 and 2.4, resectively. Asects of the branchandrice algorithm for the integer roblem are resented in Sections Branching strategies are discussed in Section 2.5, while strategies for obtaining lower and uer bounds are resented in Sections 2.6 and 2.7, resectively. 2.1 Branchandrice Column generation usually refers to the solution of a linear roblem, and for (mixed) integer roblems this may corresond to solving the linear relaxation of the underlying roblem. To obtain integer solutions, the column generation is embedded in a branchandbound framework. Because of the ricing of columns within the column generation at each branchandbound node, we refer to the overall aroach as branchandrice. In this subsection we recall the fundamentals of column generation and branchandrice Column generation Column generation is a owerful tool for solving otimization roblems with a huge number of variables. The basic rincile is to work on a restricted version of the roblem including all the rows, but only a subset of the variables. This restricted roblem is referred to as the Restricted Master Problem (RMP). The variables that are not exlicitly reresented in the RMP are considered imlicitly by evaluation of one or more subroblems (ricing roblems) that evaluate what imact it would have to include new variables in the RMP given the current solution of the RMP, and add variables to the RMP if that contributes to an imroved solution of the RMP. If no new variables will contribute to an imroved solution of the RMP, we can conclude that the existing solution of the RMP is the otimal (linear) solution of the original formulation. Because the variables occur as columns in the master roblem, this rocess is referred to as column generation. We sketch the column generation rocess in Figure 2. We first have to initialize the RMP with a feasible basis, and then dual rices are transferred to the subroblems. If the subroblems identify columns that would imrove the solution of the RMP, these columns are added to the RMP and the rocess is reeated. Otherwise, the rocess is ended and the current solution to the RMP is the otimal one. CIRRELT
10 Initialize with a feasible basis Solve restricted master roblem Transfer dual rices to subroblems Solve subroblems Imroving columns found? no Add columns to restricted master roblem yes Sto Figure 2. Flow chart for linear column generation Branchandrice algorithm We resent a branchandrice flowchart in Figure 3. We start by adding the linear relaxation of the initial roblem to a ool of unexlored nodes, this is the root node roblem. In a major iteration of the algorithm, one node is extracted from this ool and solved. If the solution is worse than the current uer bound, we fathom the node and extract a new node from the ool. Otherwise, we check whether the solution is integral. If so, we have a new uer bound, otherwise we have to branch to create new roblems that are added to the ool of unexlored nodes. The box in Figure 3 with thick bold frame contains the column generation algorithm illustrated in Figure 2. In other words, the column generation is run for each branchandbound node. If the ool of unexlored nodes is emty or if the lower bound is equal to the uer bound, the search is terminated. If the uer bound has been udated during the search, this is the otimal integer solution. Otherwise, no feasible integer solution exists. The lower bound is based on solution of the linear roblem, and the lower bound is initially set to the solution value of the first roblem solved (corresonding to the root node). Udates deend on the search strategy; with a bestfirst search the lower bound can be udated at each node. In contrast, with a dethfirst strategy, the lower bound can only be udated each time the current to of the tree is reached after backtracking. We infer from the branchandrice literature (see e.g. (Lübbecke and Desrosiers, 2005)), that branching directly on the integer variables is not likely to succeed in branchandrice. If we for instance branch on the binary cycle variables in (12) (20), we fix the cycle to 1 in one branch and 0 in the other branch. In the branch with the cycle fixed to 0, that same column will immediately be regenerated, and we will not have reached further towards an integer solution. In addition, the search tree will be unbalanced. The standard branching technique in branchandrice is to branch on the underlying network structure, which for the SNDAM roblem means branching on network arcs. We elaborate on branching strategies in Section 2.5. CIRRELT
11 Add initial roblem to node ool Pool of nodes emty? yes no lower bound=uer bound? yes uer bound <? yes no No integer solution no Select a node from the ool Otimal integer solution = uer bound Solve linear relaxation by column generation and udate lower bound Sto Fathom node no Solution < uer bound? yes Udate uer bound yes Integer solution? no Branch to create child nodes and add them to node ool Figure 3. Flowchart for branchandrice algorithm. 2.2 Restricted master roblem The restricted master roblem (RMP) is a linear relaxation of the SNDAM roblem with cycle and ath variables, formulated in (12) (20). However, in order to reduce the size of the matrix, we initially exclude the strong forcing constraints (18) from the formulation. For cycles and aths, we work on subsets from the total amount of cycles and aths; % K K and %L L, resectively. To obtain the linear relaxation we relace integrality constraints (20) with (20b). We also reformulate (15) to (15b) by introducing slack variables si' j' (', i j') A ', 0 si' j' ( Ui' j' Li' j' ). The RMP can be solved with commercial LPsolvers, but the solution times may be substantial for large instances. k K (, i j) A: NOD = i' UNOD = j' i j mg + s = U k k i ' j ' i ' j ', (', i j') A ' v, (15b) k 0 g 1, k k K. % (20b) CIRRELT
12 We associate dual variables α, β, θ i' j', σ, η and ρ with constraints (13) (14), (15b) and (16) (18). From the solution of the RMP, we transfer the dual information to subroblems for generation of design cycles and flow aths. There is one subroblem for each commodity generating flow aths, and one subroblem for design cycles. If the subroblems identify variables with negative reduced costs, these are added to the RMP which is resolved. Otherwise, the current solution of the RMP is an otimal solution for the linear roblem. Because we have two different subroblem categories, there are basically two different aroaches of the column generation. The first aroach is to solve both cycle and ath generation subroblems for a given set of dual multiliers before resolving the RMP. The second aroach is to generate either aths or cycles first, and then resolve the RMP before solving the other subroblem(s). Presumably, the second aroach would erform relatively better if the RMP is solved efficiently. In the next two subsections we define the two categories of subroblems. 2.3 Subroblem for design cycle generation A design cycle consists of a set of arcs satisfying the design balance constraints, and not covering a time eriod more than once. In the subroblem for generation of design cycles, we need to make sure that these roerties are maintained. The reduced cost of a cycle is calculated in (21), and a negative reduced cost imlies that inclusion of the cycle in the RMP would imrove the objective function value of the current RMP. The subroblem identifies the design cycle with smallest reduced cost, and this cycle is added to the RMP if its associated reduced cost is negative. The subroblem for cycle generation is formulated in (21) (24): Min z = f α β u η b ρ + θ y U C i' j' i' = NOD ' OD (, ) i j = N i j A P j (, i j) A : T t< T : i y j y 1, y = 0, + j N () i j N () i { 0,1} ji (21) t T, (22) i N, (23) y, (, i j) A. (24) We use binary variables y reresenting whether an arc( i, j) A in the underlying network is selected to be in the cycle or not. The objective function (21) minimizes the sum of fixed vehicle costs f and all dual cost of the arcs constituting a cycle. Constraints (22) ensure that cycles do not cover multile lanning horizons by stating that at most one arc can be oened in each time eriod. Constraints (23) are the design balance constraints, while (24) restrict the y s to take binary values. We solve the cycle generation roblem with shortest ath calculations using a labelcorrecting algorithm. We extend the timesace network beyond the lanning horizon by the duration of the longest arc in the network. With the new network, we are able to roduce any feasible cycle by one CIRRELT
13 traversal of the network. The construction of cycles can be handled by two different hases that together cature all otential cycles. Firstly, if we start from all nodes in time eriod 1 and create aths through the network, we obtain cycles when we find arcs that have their destination node in the origin node of that ath, i.e. crossing the lanning horizon. If all arcs have duration of one time eriod, this aroach would be sufficient to generate all ossible cycles. However, for arcs with longer duration, we may have the case that arcs ass by time eriod 1, for instance if an arc has its origin in the last time eriod, asses the end of the lanning horizon and has its destination node in time eriod 2. This arc will not be evaluated with the aroach described above. In order to cature such arcs, we identify with the second aroach all arcs having the roerty that they ass by time eriod 1. We illustrate these ideas in Figure 4. Time Nodes in hysical network Planning horizon Figure 4. Illustration of cycle generation based on arcs originating in the first time eriod (solid arcs) and on crosshorizon arcs (dotted arcs) In Figure 4, solid arcs form cycles that are established from nodes in the first time eriod, while dotted arcs form cycles that are established based on the crosshorizon arcs that ass by time eriod 1. Because of constraints (22) we do not continue on the extension of a ath if the time eriod reresenting the origin node of the ath has been reached or assed. In the cycle generation, we have to be careful with alying dominance rules. The calculation of reduced costs in (21) includes all arcs constituting a cycle, including the last arc that closes a cycle, and we are therefore not able to aly dominance rules between aths with different origins. We therefore kee one label for each node belonging to time eriod 1, and one additional label for each node that has at least one arc emanating from it that asses by time eriod 1. In the end, this aroach is very well suited for roducing several cycles simultaneously. 2.4 Subroblems for flow ath generation The flow ath subroblems generate aths that contribute to imroved solutions of the current RMP. Requirements to aths are that they have their origins in the commodity s unique origin node in the timesace network, node balance has to be ensured, and the aths have to end in one of the time realizations of the commodity s given hysical destination node. The subroblem for flow ath generation finds the origindestination ath with smallest reduced cost, as defined in (25). This ath CIRRELT
14 is added to the RMP if the associated reduced cost is negative. The subroblem for generation of commodity aths for commodity is defined in (25) (27): ( ) Min z = σ + c η ρ x P (, i j) A (25) 1, i = o x x = 1, i = d 0, otherwise, ji + j N () i j N () i i Nv, (26) { 0,1} x, (, i j) Av. (27) The x variables are now binary variables reresenting flow on service arcs in the underlying network, and x = 1 indicates that arc (, i j) A is in the ath. The objective function (25) minimizes the sum of dual cost of the demand satisfaction constraints (16) and reduced costs for all arcs constituting the ath. Constraints (26) are flow balance constraints, while we in (27) restrict the variables to take binary values. Paths are established by shortestath comutations using labelcorrecting algorithms. Shortest aths are created from the commodity s unique origin node to the nodes in the timesace network corresonding to the commodity s hysical destination node. In rincile, all these time realizations need to be examined in order to find the ath with the most negative reduced cost. However, as the costs on the arcs are ositive, we can terminate if the cost at any stage is higher than the comuted cost of reaching the current best destination node. Moreover, if ath costs exceed σ, we can sto evaluating that ath because it cannot result in a ath with negative reduced costs. 2.5 Branching strategies for the integer roblem In order to obtain integer solutions for the SNDAM roblem, we embed the column generation in a branchandbound framework, where the solution obtained from the linear relaxation of the roblem reresents the root node of a search tree. In this subsection we resent branching strategies for the roblem. As described in Section 2.1, it is not desirable to branch directly on the cycle variables. Instead we consider four alternative branching strategies. The first aroach (BB1) is to branch on service arcs in the underlying network structure, i.e. arcs in the timesace network. In evaluating the solutions of the RMP, we choose the arc with fractional value closest to 0.5, and fix this arc to 1 in one branch and to 0 in the other branch. The interretation of this branching is that fixing an arc to 1 means that the sum of the cycle variables using this arc is 1. In the 0branch, all cycles using this arc have to be 0. The second aroach (BB2) is an adatation of constraint branching (Ryan and Foster, 1981), where the basic rincile is to branch on two arcs, letting at most one of them be 1 in one branch, and both or none of them be 1 in the other branch. x CIRRELT
15 The third branching strategy (BB3) is to use the slack variables solutions to (12)(20), integrality of Ui' j' and of all r k also ensures integrality of the s i' j' from (15b). For integer si' j' variables. When we obtain fractional si' j' variables in the linear solution, we branch by identifying the s i ' j ' variable with fractional value closest to 0.5 and round it down to the nearest integer in one branch and round it u to the nearest integer in the other branch. However, even if all s i' j' are integer, we do not necessarily have integer values on the cycles. We therefore need to switch branching entity if all k s are integer, but the solution is not integer in the g variables. In such cases, we branch on the i' j' arcs in the underlying network structure, which was the first aroach. The last branching strategy (BB4) is a cyclebased greedy heuristic. We select a fractional cycle and fix all the arcs in the cycle to 1, this allows for more flexibility than fixing the cycle itself to 1. This aroach is not a roer branching, because we would never consider cycles with a subset of the arcs that have been fixed further u in the tree, and would thus not cover the comlete solution sace. We refer to this aroach as heuristic branching. 2.6 Lower bounds Caacitated multicommodity network design roblems have in general oor linear relaxations. For SNDAM, it was shown in (Andersen et al., 2007b) that the introduction of cycle variables imroved the tightness of the linear relaxations. We therefore use the linear relaxation as the lower bound in the branchandrice aroach. However, in contrast to many other roblems that have been solved with column generation, even solving the linear roblem constitutes a significant challenge when the dimensions increase. The lower bounds for the SNDAM roblem are significantly tighter when they are based on the strong linear relaxation including strong forcing constraints (18), comared to the weak linear relaxation that is obtained when only weak forcing constraints (17) are included. For large networks with many arcs and commodities, the number of strong forcing constraints would grow extremely large if all of them were included exlicitly in the roblem. In order to overcome this difficulty, we generate dynamically those strong forcing constraints that are violated in the otimal solution of the RMP. To do this, we first solve the weak linear relaxation to otimality with column generation. Then, we verify whether strong forcing constraints are violated in the otimal linear solution, and if so, these are added to the formulation. Then the new roblem is solved to otimality with column generation, and the rocess is reeated if additional strong forcing constraints are needed. When no new strong forcing constraints are needed, we have obtained the strong linear relaxation. The drawback of this aroach is that we send time on testing if these constraints are violated and on reotimization after new constraints have been added. For large instances, these reotimizations may become comutationally exensive, and an imortant issue is thus to which degree strong forcing constraints should be generated. These constraints may be generated at all branchandbound nodes, at the root node only, each time the lower bound is udated, or none of these. To strengthen the bound, we introduce the following idea from a vehicle routing setting in (Dumas et al., 1991): The number of vehicles used has to be integer in an integer solution. If the linear solution has a fractional sum of design cycles, we round this fractional value u to the nearest integer, catured in (28) with m being the sum of cycle variables in the current LP solution. In addition, we introduce the comlement of (28) in (29), and round the fractional sum of cycle CIRRELT
16 variables down to the nearest integer. The effect of the strengthening with (28) and (29) is most significant if introduction of (29) imlies an infeasible roblem. k K k K g g k k m, m. (28) (29) 2.7 Uer bounds The uer bound is udated each time a new best integer solution is found. In order to encourage fathoming, it is of high imortance to obtain good integer solutions early in the traversal of the tree. We introduce an accelerating technique with the aim of finding integer solutions quickly and thus imrove the erformance of the branchandrice algorithm. k From the solution of the linear RMP, we fix to 1 cycle variables g with fractional values above a threshold. Then, new columns are generated until the solution of the RMP cannot be further k imroved. Again, the fractional g variables with values above the threshold are fixed to 1, and this rocedure is reeated until an integer solution has been found, the solution is worse than the current best integer solution, or the roblem becomes infeasible. 3 Comutational study In this section we resent a comutational study based on the branchandrice algorithm that was introduced in Section 2 for the SNDAM roblem. We have rogrammed the algorithms in C++, and the models have been solved on comuters with 3 GHz rocessor and 8 GB RAM running on Rock Cluster v oerating system. The restricted master roblems are solved with the LPsolver of XPRESS Otimizer v We first discuss imlementation and calibration issues in Section 3.1, before giving an introduction to the data sets used in Section 3.2. Comutational results aear in Section Imlementation and calibration In the comutational study, we exlore three different aroaches A, B and C, which are targeted at different instance sizes. One imortant difference between the aroaches is the role of strong forcing constraints. These constraints imrove the bound and may also contribute to better branching decisions, so it is desirable to include them. However, for large roblems, it is timeconsuming to solve even the linear version of the roblem if many constraints are added to the roblem. The rationale underlying imlementations AC is that strong forcing constraints are generated more often for small instances. There are a significant number of arameter settings and issues that could be exlored in a comutational study, but we have ket a focus on the role of the strong forcing constraints in the imlementations. The three solution aroaches AC are develoed through extensive testing in a calibration hase, and the arameter selections are based on what has been observed to give reasonable erformance over a set of instances. It was however not ossible to erform fullscale CIRRELT
17 testing of all combinations of the different arameters and issues. We summarize significant imlementation issues and arameter selections in Table 1. Table 1.Imlementation issues and arameter selections. Issues and arameters Solution aroach A Solution aroach B Solution aroach C Search strategy Deth first Maximum search time seconds How often is the master roblem solved All subroblems are solved for given dual information before the master roblem is resolved Strategy for solving RMP at each BBnode Save basis before adding columns, and reotimize with dual simlex from this basis when columns have been added to the roblem Max number of aths generated in each iteration One er commodity Branching strategies BB3 (slack variables) / BB3 (slack variables) BB4 (heuristic) BB4 (heuristic) Max and min values for fixing cycles to 1 in acc. technique 0.7 and and 0.55 Max number of cycles generated in each iteration 5* 10*/20*/30* How often are strong forcing constraints generated How long do we kee on generating strong forcing constraints at a node? All nodes Nodes where lower bounds are udated As long as they are violated Nodes where lower bounds are udated No new iterations after seconds or if imrovement < 0.1% Kee strong forcing constraints in roblem after generation? Yes No *In the accelerating technique at most one cycle may be added in each iteration For the three aroaches, we imlemented a dethfirst strategy in branchandrice, and we have set a maximum running time of 10 hours (36000 seconds). In the comutations it aeared to be more efficient to solve all subroblems before reotimizing the RMP, so we do not reort results for the second aroach described in Section 2.2. In solving the RMP, we save the basis before adding columns, and when columns have been added, we reotimize with dual simlex from the saved basis. As ointed out in Sections 2.3 and 2.4, it is comutationally chea to add more than one cycle or ath when the subroblems are solved. There is a tradeoff between the gains of saving iterations as a consequence of adding multile columns, versus the increased comutational time from having more columns in the master roblem. There are a significant amount of commodities in our data sets, and we therefore allow inclusion of at most one ath for each commodity in an iteration of the column generation. However, because of the many commodities, it makes sense to include multile cycles in each iteration if there are several cycles with negative reduced costs. We allow most cycles in solution aroach C, which is targeted at the instances with the largest number of commodities. For branching, the calibration hase indicated that branching on service arcs in the underlying network (BB1) and constraint branching (BB2) was outerformed by branching on slack variables (BB3). The comutational study is therefore limited to branching strategies BB3 and BB4, where BB4 is alied to the largest instances. For solution aroach A, we use branching strategy BB3. For the variable fixing technique described in Section 2.7, the limit for fixing cycles is initially set to 0.7, and sequentially reduced to In each iteration of the column generation, at most 5 cycles may be added. Strong forcing CIRRELT
18 constraints are generated on all nodes, and we kee generating them as long as they are violated. The strong forcing constraints are also ket in the roblem at subsequent nodes. Solution aroach B may be used either with BB3 or BB4. The additional difference from solution aroach A is that strong forcing constraints are only generated at nodes where the lower bounds are udated during branchandbound. This allows exloration of more nodes within the time limit. Solution aroach C is targeted at very large roblem instances, and it utilizes branching strategy BB4. The variable fixing technique starts with a limit for fixing cycles at 0.6, and to facilitate instances with several hundred commodities, we test inclusion of at most 10, 20 and 30 cycles in one iteration of the column generation algorithm. Strong forcing constraints are only generated when the lower bound is udated, and because the reotimizations are timeconsuming, we include the oortunity to sto this rocess if the time exceeds seconds or if the imrovement in objective function value in the last iteration was below 0.1%. Moreover, the strong forcing constraints are removed from the matrix as soon as the bound is udated, to facilitate faster exloration of the search tree. 3.2 Data sets In (Andersen et al., 2007b) the SNDAM roblem with cycle and ath variables was solved directly with the MIPsolver in CPLEX for roblem instances where all design cycles and commodity aths were generated a riori. We test the branchandrice algorithm on a few of the instances from (Andersen et al., 2007b), but the comutational study is focused on instances that are significantly larger than what could be solved with full a riori generation. In Table 2 we resent the dimensions of the roblems that are tested in the comutational study. The first three columns of Table 2 resent number of nodes and arcs in the static network, as well as number of time eriods. In the fourth column we resent the corresonding number of service and holding arcs, and number of commodities aears in the sixth column. In the seventh column a size indicator for roblem dimension is resented, comuted as the roduct of number of service arcs and number of commodities and divided by This measure gives a fair descrition of the relative dimensions of the instances. In the last column we indicate which of the solution aroaches of Section 3.1 we aly to each instance. The instances are insired by a realworld case in rail transortation lanning. Instances 15 are extracted from (Andersen et al., 2007b), while instances 615 are significantly larger than what could be handled with total enumeration of cycles and aths. We observe that there are significant differences in roblem size between the instances. As ointed out in Section 3.1, initial testing has suggested that different aroaches should be alied deending on the sizes of the instances. Instances 15 have size indicators in the range 445, and for these roblems we aly solution aroaches A and B. However, for solution aroach B we only aly the branching on slack variables (BB3). For instances 6 to 12, the size indicator is in the range , which is significantly larger than for instances 15. For these cases, solution aroach A is left out, as it for these instances aeared to be disadvantageous to generate strong forcing constraints on all nodes of the search tree. Finally, for instances 1315, the size indicator exceeds 1000, and these instances are so large that we only aly solution aroach C. CIRRELT
19 Problem id # Static service arcs Table 2.Dimensions of roblem instances. Service + holding arcs Commodities Size indicator Solution aroaches Static nodes Time eriods A and B A and B A and B A and B A and B B B B B B B B C C C 3.3 Results In this section we resent results from model runs for the roblem instances that were defined in Table 2. We allow a maximum CPU time of 10 hours in all cases. In Section we resent results for roblems that have been solved with a riori generation of columns, while we in Section resent results for the roblems that have not been solved earlier Instances solved with a riori generation of columns Results from model runs for instances 15 can be found in Table 3. For each scenario, there is one row with results obtained with solution aroach A, and one row with results obtained with solution aroach B. In each case, we reort lower bound at termination and best MIP solution at termination. Termination either refers to that 10 hours of CPU time has been reached, or that the algorithm has found a roven otimal integer solution. Thereafter the solution time in seconds for finding roven otimal solution is returned, or alternatively the remaining otimality ga after 10 hours of CPU time. In the last two columns of Table 3, we resent the best integer solution that was obtained with a riori enumeration of cycles and aths in (Andersen et al., 2007b) and the corresonding solution time. The a riori enumeration was imlemented on different comuters but with comarable secifications and a maximum running time of 10 hours. The results reorted in Table 3 are very similar for aroaches A and B. For instances 13, the branchandrice algorithm returns roven otimal integer solutions with both solution aroaches. For instances 1 and 3, the solutions are obtained reasonably quickly, while instance 2 requires around and seconds with the two aroaches. The a riori enumeration was more efficient for instances 1 and 2, while instance 3 was solved faster with branchandrice. The a riori enumeration utilized the advantages of a standard MIPsolver with advanced heuristics and more advanced search algorithms than we have imlemented. It is nevertheless good news that the branchandrice algorithm returns roven otimal integer solutions for roblems where these otimal CIRRELT
20 solutions are known. For instances 4 and 5, neither a riori enumeration nor branchandrice returned roven otimal solutions within 10 hours. For instance 4, the integer solutions obtained were slightly better with a riori enumeration, and the ga also smaller in this case due to more advanced bounding. For instance 5, the branchandrice algorithm returns better integer solutions than what was obtained with a riori enumeration. However, the remaining otimality ga after 10 hours is smaller with a riori enumeration; this reflects the stronger focus on bounding in a commercial MIPsolver comared to the branchandrice algorithm resented in this aer. The encouraging news from Table 3 is however that the branchandrice algorithm solves small instances to roven otimality, and that both aroaches A and B return better integer solutions than what was obtained with a riori enumeration for the largest instance. Table 3. Results from model runs for instances that have been comuted with a riori generation of columns. Solution Lower bound at MIP solution at Time use a riori Solution time (sec) / Best MIP with a riori enumeration (sec) / Instance aroach termination termination Otimality ga enumeration Otimality ga 1 A B A B A B A % B % * 0.7% 5 A % B % * 0.5% *For these instances, the a riori aroach did not return a roven otimal solution within 10 hours Larger instances We resent results from model runs for roblem instances 612 in Table 4. In these model runs, all integer solutions have been obtained by use of the accelerating technique resented in Section 2.7. No roblems are solved to roven otimum within 10 hours of CPU time. Again, there are two rows for each instance, reresenting branching on slack variables (BB3) and heuristic branching (BB4). In Table 4, the first column of results resents the lower bounds at termination, which in all cases corresonds to the lower bound obtained at the root node, because the dethfirst search does not return to the root node within 10 hours for any of the instances. Then best integer solutions obtained and otimality gas are reorted. The next two columns elaborate on the best integer solutions that were obtained, and reort node number where the best integer solution was found and the elased time, resectively. In the last two columns we reort number of branchandrice nodes visited during the tree search, and time needed to solve the root node. For each branching strategy, we resent average results from instances 612 in the last two rows. CIRRELT
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