Brouer, Berit Dangaard

The Liner Service Network Design Problem (LSN-DP) is the problem of constructing a set of routes for a heterogeneous vessel fleet of a global liner shipping operator. Routes in the liner shipping context are non-simple, cyclic routes constructed for a specific vessel type. The problem is challenging due to the size of a global liner shipping operation and due to the hub-and-spoke network design, where a high percentage of the total cargo is transshipped. We present the first construction heuristic for large scale instances of the LSN-DP. The heuristic is able to find a solution for a real life case with 234 unique ports and 14000 demands in 33 seconds

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Downloaded from orbit.dtu.dk on: Dec 18, 2017A greedy construction heuristic for the liner service network design problemBrouer, Berit DangaardPublication date:2010Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Brouer, B. D. (2010). A greedy construction heuristic for the liner service network design problem. Abstract fromSeventh Triennial Symposium on Transportation Analysis, Tromsø, Norway, .A greedy construction heuristic for the LinerService Network Design ProblemBerit LøfstedtDTU Management EngineeringTechnical University of DenmarkEmail: blof@man.dtu.dk3rd of November 2009The Liner Service Network Design Problem (LSN-DP) is the problem of constructing a setof routes for a heterogeneous vessel fleet of a global liner shipping operator. Routes in the linershipping context are non-simple, cyclic routes constructed for a specific vessel type. The problemis challenging due to the size of a global liner shipping operation and due to the hub-and-spokenetwork design, where a high percentage of the total cargo is transshipped. We present the firstconstruction heuristic for large scale instances of the LSN-DP. The heuristic is able to find asolution for a real life case with 234 unique ports and 14000 demands in 33 seconds.Literature overview: A MIP model of the LSN-DP consist of a highly unconstrained routingproblem subject to a large degree of symmetry and a multicommodity flow problem dominatingthe constraint set and accountable for a large fraction of the cost. Previous work on liner servicenetwork design may be found in [1],[2], [3] and [4]. The models are distinct with regards totransshipments and the vessel fleet. In the early paper of [1] transshipments are not supported,whereas they are supported in [2], [3] and [4]. Transshipment costs are accounted for in theobjective function of [2],[4] as opposed to [3]. The characteristics of the fleet differ as to whetherit is heterogeneous for every vessel [1],[2] or consist of a heterogeneous fleet of homogeneous vesselclasses [3],[4]. Non-linear capacity constraints are found in [1],[2] assuming that a vessel maycomplete its route an integral number of times during the planning horizon. Integrality is notimposed by [4]. The weekly frequency constraint is introduced by [3] assuming a number ofhomogeneous vessels assigned to each service to offer a weekly visit to each port en route with thecapacity of the vessel class in question. Optimal results for smaller instances are presented by [2]and [4]. The approaches of [1],[2] and [3] have focused on solving the liner service network designproblem with traditional decomposition and integer programming methods and fail to produceresults for realistic network sizes of a global liner shipping operator anno 2009. This is addressedby [4] benchmarking a tabu search approach presenting results within 3-5% of the optimal solutionfor up to 7 ports. The multicommodity flow problem is solved in each iteration, which is reportedby [4] to become computationally expensive already for the 7 port instance. A case study of 120ports in [4] show that a heuristic approach may scale to large instances but no execution time isreported and the quality of the solution is hard to evaluate. It is reported to visit important portsinfrequently. A global network connects several hundred ports worldwide and the correspondingforecasted cargo demand comprises a commodity set of 4 orders of magnitude. Methods based onrelaxation of the proposed models or simply evaluating the objective function during a search isnot computationally efficient for large scale problems.Heuristic approach: A solution to the LSN-DP is a set of routes covering the ports servicedby the shipping operator and transporting the forecasted cargo demand. Viewed as a graphpartitioning problem the solution is a set of strongly connected components with a high degree ofinterconnection. The construction heuristic is based on the Multiple Quadratic Knapsack Problem(MQKP) and relies on a graph of the current schedule, which is divided into a set of dense subgraphsrelated by demand, expected transshipment flow and geographical proximity. A solution found bythe construction heuristic is expected to be feasible and realistic, but the quality of the solutioncannot be guaranteed as the heuristic cannot account for the flow problem and the transshipmentcost. In MQKP a set of mutually exclusive items i ∈ V are placed in R knapsacks with differentweight bounds Cr. The objective is to maximise the profit of the knapsacks defined by the profitmatrix P .maximize(MQKP ) =∑r∈R∑i∈V∑j∈Vpijxrixrj +∑r∈R∑j∈Vpjxrj (1)subject to:∑i∈Vwixri ≤ Cr ∀r ∈ R (2)∑r∈Rxri ≤ 1 ∀i ∈ V (3)xri ∈ {0, 1} ∀i ∈ V (4)The variables xri indicate whether item i is included in the r’th knapsack. The knapsack constraint(2) makes the total item weight obey the bound Cr and constraints (3) ensure that items aremutually exclusive to the knapsacks. When the MQKP is applied to the LSN-DP the knapsackitems i ∈ V are the accumulated port visits of each port t ∈ T and the knapsacks r ∈ R representservices, which are a specific vessel class visiting a sequence of ports. Let A be the set of vesselclasses and let Na denote the number of available vessels of class a ∈ A in the fleet. Let Cadenote the capacity in TEU of a single vessel of class a ∈ A. The number of services/knapsacksis dependent on the expected rotation time of a service σ(Ca). σ(Ca) depends on vessel capacityas large vessels are typically assigned to cross regional services and small vessels are assignedto regional services. The number of knapsacks for the LSN-DP is hence |R| = ∑a∈A |Ra| =∑a∈Ad Naσ(Ca)e. The profit matrix P defines each entry pij = f(lij , dij , hij) where lij is the sailingdistance in nautical miles and dij is the demand between ports i, j ∈ V . hij is the potentialhub flow between port i ∈ V and a hub port j ∈ H ⊂ V , where H ⊂ V are ports with a smallpercentage of demand compared to the terminal capacity. A port t ∈ T may be visited multipletimes mt by multiple services according to the capacity and schedule requirements of a port. LetM be a vector of size |T | containing the number of weekly visits to each port t ∈ T . In the MQKPport t ∈ T is duplicated mt times for the knapsack items i ∈ V , V = {T ×M} to represent thecurrent schedule of ports. It is important to observe the weekly frequency constraint of the originalproblem in order to obtain a feasible solution to the LSN-DP using the construction heuristic. Toensure feasibility, each knapsack r ∈ R is required to provide a Hamiltonian cycle of the items inknapsack r. The length of the cycle cannot exceed the mileage coverable by the vessels assignedto knapsack r. Edge variables yrij and enumeration variables uri are introduced in the MQKP toorder the ports in each knapsack into a simple, cyclic route constrained by σ(Ca). taij express thesailing time between ports i and j and tai is the expected time spent in port i for vessel type a.maximize(MQKP ) =∑r∈R∑i∈V∑j∈Vpijxrixrj +∑r∈R∑j∈Vpjxrj (5)subject to: xrixrj ≥ yrij ∀i ∈ V, j ∈ V, r ∈ R (6)∑j∈Vyrij −∑j∈Vyrji = 0 ∀i ∈ V, r ∈ R (7)∑j∈Vyrij ≤ 1 ∀i ∈ V, r ∈ R (8)uri − urj + yrij∑i∈Vxri ≤∑i∈Vxri − 1 ∀i ∈ V, j ∈ V, r ∈ R (9)∑i∈V∑j∈Vyrij(tij + ti) ≤ σ(Ca) ∀r ∈ Ra, a ∈ A (10)∑r∈Rxri = 1 ∀i ∈ V (11)xri ∈ {0, 1} ∀i ∈ V, r ∈ R (12)yrij ∈ {0, 1} ∀i ∈ V, j ∈ V, r ∈ R (13)uri ∈ Z+ ∀i ∈ V, r ∈ R (14)Constraints (6) ensure that an edge variable can only be activated if both endpoints of the arcare included in the knapsack. Constraints (7) ensure a cyclic route. Constraints (8) ensure thatthe cyclic route is simple and constraints (9) that the route is connected. Constraints (10) are theweekly frequency constraint ensuring that the simple, cyclic route has a voyage duration less thanthe expected rotation time.The MQKP is solved using a greedy heuristic, where the knapsacks apply the football teamingprinciple taking turns at picking the best remaining item max ∆f(lij , dij , hij), i ∈ r, j ∈ V¯ whereV¯ are the unassigned items.In a hub-and-spoke network design large vessels are deployed on deep sea services to achieveeconomies of scale[6], while smaller vessels are deployed between spoke and hub. The algorithm ismultilayered to reflect the hub-and-spoke network design of major liner shipping operators. Thefunction f(lij , dij , hij) is adapted to each layer of the network and the ports are correspondinglyassigned to the layers according to their capacity requirements.Computational results: The computational results are based on a real life case from MaerskLine with 234 unique ports and 14000 demands. The MQKP is able to find a solution in 33seconds for the entire network with some ports unplaced. The solutions have been evaluatedby optimization managers at Maersk Line regarding them as realistic with some modifications.Current work is on implementing layer specific seeding to improve the number of unplaced ports.We evaluate the actual flow and the network cost of the solution. We believe that meta heuristicapproaches are needed to optimize liner service networks of global shipping operators and areworking on specializing the Adaptive Large Neighbourhood Search [5] VRP framework towardsthe context of transshipments and cyclic routes. The first step is the construction heuristic forthe LSN-DP presented here. The ALNS is to search for an improved solution according to a moresophisticated objective function and cargo allocation detection.References[1] K. Rana and R.G. Vickson, “Routing container Ships Using Lagrangian Relaxation and De-composition”, Transportation Science 25, 201-214 (1991).[2] L. B. Reinhardt and B. Kallehauge, “Network Design for Container Shipping Using CuttingPlanes”,Conference on Maritime & Intermodal Logistics, Singapore, December 2007.[3] R. Agarwal and O. Ergun,”Ship Scheduling and Network Design for Cargo routing in LinerShipping”,Transportation Science 42, 175-196 (2008).[4] J.F. Alvarez, “ Joint Routing and Deployment of a Fleet of Container Vessels”, MaritimeEconomics and Logistics 11, 186-208 (2009).[5] S. Røpke and D. Pisinger, “A general heuristic for vehicle routing problems”, Computers andOperations Research 34, 2403-2435 (2007).[6] I. Akio and K. Shintani and S.Papadimitriou, ”Multi-port vs. Hub-and-Spoke port calls bycontainerships” Transportation Research Part E 42 , 740-757 (2009).

A greedy construction heuristic for the liner service network design problem

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A greedy construction heuristic for the liner service network design problem

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