Fast Generation of Near-Optimal Plans for Eco-Efficient Stowage of Large Container Vessels

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1 Downloaded from orbit.dtu.dk on: Se 06, 2018 Fast Generation of Near-Otimal Plans for Eco-Efficient Stowage of Large Container Vessels Pacino, Dario; Delgado, Alberto; Jensen, Rune Møller; Bebbington, Tom Published in: Proceedings of the International Joint Conference on Artificial Intelligence Publication date: 2011 Document Version Peer reviewed version Link back to DTU Orbit Citation (APA): Pacino, D., Delgado, A., Jensen, R. M., & Bebbington, T. (2011). Fast Generation of Near-Otimal Plans for Eco-Efficient Stowage of Large Container Vessels. Proceedings of the International Joint Conference on Artificial Intelligence, General rights Coyright and moral rights for the ublications made accessible in the ublic ortal are retained by the authors and/or other coyright owners and it is a condition of accessing ublications that users recognise and abide by the legal requirements associated with these rights. Users may download and rint one coy of any ublication from the ublic ortal for the urose of rivate study or research. You may not further distribute the material or use it for any rofit-making activity or commercial gain You may freely distribute the URL identifying the ublication in the ublic ortal If you believe that this document breaches coyright lease contact us roviding details, and we will remove access to the work immediately and investigate your claim.

2 Fast Generation of Near-Otimal Plans for Eco-Efficient Stowage of Large Container Vessels Dario Pacino 1, Alberto Delgado 1, Rune Møller Jensen 1, and Tom Bebbington 2 1 IT-University of Coenhagen, Denmark {dacino,alde,rmj}@itu.dk 2 Maersk Line Oerations, Global Stowage Production, Singaore Tom.Bebbington@maersk.com Abstract. Eco-Efficient stowage lans that are both cometitive and sustainable have become a riority for the shiing industry. Stowage lanning is NP-hard and is a challenging otimization roblem in ractice. We roose a new 2-hase aroach that generates near-otimal stowage lans and fulfills industrial time and quality requirements. Our aroach combines an integer rogramming model for assigning grous of containers to storage areas of the vessel over multile orts, and a constraint rogramming and local search rocedure for stowing individual containers. 1 Introduction Cost-efficiency and sustainability are not oosed objectives for the stowage lans generated daily by liner shiing comanies. A stowage lan assigns containers to load in a ort to vessel slots. An eco-efficient stowage lan aims at achieving a minimum ort stay. In this way, ort fees are minimized and the vessel maximizes its time at sea, which can be used to reduce its seed and thus bunker costs and CO 2 emissions. Eco-efficient stowage lans, however, are hard to roduce in ractice. First, they are made under time ressure by human stowage coordinators just hours before the vessel calls the ort. Second, dee-sea vessels are large and often require thousands of container moves in a ort. Third, comlex interactions between lowlevel stacking rules and high-level stress limits and stability requirements make it difficult to minimize the makesan of cranes and, at the same time, avoid that containers block each other (overstowage). Finally, according to our industrial artner, stowage lanning otimization algorithms must be fast. Runtimes of more than 10 minutes are imractical, since stowage coordinators ossibly need to run several forecast scenarios. This aer introduces a new stowage lanning otimization aroach that, similar to the currently most successful aroaches (e.g, [26, 18, 1]), decomoses Loadlists Vessel Data Port Data Muti Port Master Planning Muti Port Master Plans Current Port Master Plan Slot Planning Stowage Plan Fig. 1. Hierarchical decomosition of stowage lanning into master and slot lanning.

3 the roblem hierarchically as deicted in Figure 1. First the multi-ort master lanning hase decides how many containers of each tye to stow in a set of storage areas of the vessel using an Integer Programming (IP) model. Based on this distribution, a comlete stowage lan is generated in the Slot Planning hase where individual containers are stowed by a combination of Constraint Programming (CP) and Local Search (LS) models imlemented in [8, 21]. To avoid negative imact of the final stowage lan in future orts, the multi-ort master lanning hase otimizes simultaneously master lans for the current ort and a number of downstream orts. We evaluated our aroach exerimentally on 20 real instances rovided by our industrial artner. Desite the NP-hardness of multi-ort master lanning and the revious lack of success solving the roblem otimally for large vessels over multile orts (e.g., [26,18,1]), 11 of the 20 instances can be solved otimally in less than 10 minutes using a standard IP solver. More interestingly, the comutation time can often be reduced several orders of magnitude if we relax the roblem by droing the integrality constraint on the decision variables in a mixed integer rogramming (MIP) model, without affecting solution quality. A 2-hase aroach for stowage lanning, combining our MIP aroach for multi-ort master lanning with our slot lanning algorithms, can generate comlete stowage lans in less than 330 seconds for 16 of the 20 instances. This runtime is well within the time bound of 10 minutes assessed by our industrial artner to be necessary for suorting stowage coordination in ractice. The remainder of the aer is organized as follows. Section 2 describes the roblem. Section 3 introduces related work. Section 4 resents our 2-hase aroach. Section 5 and 6 resent the exeriments and draw conclusions. 2 Background and Problem Statement ISO containers transorted on container shis are normally 8 wide, 8 6 high, and either 20, 40, or 45 long. Due to the lack of suort oints, it is not ossible to stack a 20 container on a 40 or 45 container. High cube containers are 9 6 high and allet wide containers are slightly wider and can only be laced side-by-side in certain atterns. Refrigerated containers (reefers) must be laced near ower lugs. Containers with dangerous goods (IMO containers) must be laced according to a comlex set of searation rules. The caacity of a container shi is given in Twenty-foot Equivalent Units (TEUs). As shown in Figure 2, the cargo sace of a vessel is divided into sections called bays and each bay is divided into an on deck and a below deck art by a number of hatch-covers, which are flat, leak-roof structures. A container is overstowing and causing extra crane moves if it is discharged later than a container stowed below it, either in the same stack or under a hatch-cover (hatchoverstowage). Each sub-section of a bay consists of a row of container stacks divided into slots that can hold a 20 ISO container. Figure 3 (a) and (b) show the container slots of a bay and stack, resectively. Stacks have max height and weight limits. Below deck, cell guides secure containers transversely. Containers

4 on deck are secured by lashing rods and twist locks with limited strength. Thus, container weights must normally decrease uwards in stacks on deck. Moreover, lashing rods of 20 stacks must be accessible and stack heights must be under the vessel s minimum line of sight. 45 containers can normally only be stowed over the lashing bridge on deck Line of sight Hatch cover Bay Lashing bridge Waterline Station Fig. 2. The arrangement of bays in a small container vessel. The vertical arrows show an examle of the resulting forces acting on the shi sections between calculation oints (stations). Single crane work hours for adjacent bays are shown at the to. Stacks On Deck Bow Stack Fore Aft M Tiers On Deck Tiers 3 Below Deck 2 1 Stern Stacks Below Deck 40 High Cube 40 Reefer 20 Reefer 20 Metacentric height G Gravity force (a) (b) (c) θ Z Buoyancy force Fig.3. (a) A vessel bay seen from behind. (b) A side view of a stack of containers. As deicted, ower lugs are normally situated at bottom slots. (c) Transverse stability. A container shi must sail at even keel and have sufficient transverse stability. Figure 3(c) shows a cross section of a shi. For small inclination angles, the volume of the emerged and immersed water wedges (shaded areas) and thus the distance GZ are aroximately roortional with the angle such that the buoyancy force intersects the center line in a fixed osition called the metacenter, M [24]. For an inclination angle θ, the shi s urighting force is roortional to GZ = GM sinθ. GM is called the metacentric height and the center of gravity G must be on the center line and result in sufficient GM for the shi to be stable. Maximum and minimum draft restrictions aly due to ort deths, working height of cranes, and the roeller. The trim is the difference between the aft and fore draft and must be ket within a given san. For a station osition, the shear force is the sum of the resulting vertical forces on vessel sections (see Figure 2) acting aft of, and the bending moment is the sum of these forces

5 times the horizontal distance to them from. Both of these stresses must be within limits. The vessel also has transverse bending moment (torsion) limits. The meta center, draft, trim, and the buoyancy of each section of the vessel can be derived from its hydrostatic tables given its dislacement and longitudinal center of gravity. A container shi transorts containers between orts on a fixed cyclic route. It is the liner shiers and not the ort terminals that are in charge of roducing stowagelans.astowage lanassignsthecontainerstoloadinaterminaltoslots on the vessel and it is often sent to the terminal shortly before calling it. The terminal guarantees a certain roductivity in terms of the number and efficiency of assigned quay cranes, but otherwise the liner shier has no control over its oerations. Tyically the terminal substitutes detailed container information for container tyes in order to otimize the load sequence [16]. In this work we assume the role of the liner shier and focus on the generation of tyed stowage lans for the current ort. The lan is then comleted by the terminal during the load sequencing, where the tyes are substituted with concrete containers. The main objective of stowage lanning is to minimize the ort stay by reducing the total number of moves through overstowage minimization and distributing these moves evenly over the available quay cranes. Since quay cranes are too wide to work on two adjacent bays, a good lower bound for the makesan of quay cranes is the maximum work time of a single crane over airs of adjacent bays (10 hours in the examle shown in Figure 2). Stowage lanning should take cargo in future orts into account (e.g., to ensure enough reefer caacity). But since cargo forecasts are inaccurate, the lan must be robust. Achieving this includes arranging containers with same ort of discharge (POD) in vertical stacks rather than horizontal layers, freeing u as much bottom sace as ossible for unexected reefer and long-haul containers, and to avoid mixing containers with different POD under deck to minimize the risk of hatch-overstows. It is imractical to study large otimization models that include all details of stowage lanning. On the other hand, all major asects of the roblem must be modeled for the results to be valuable. For container tyes this includes 20, 40, and reefer containers because short containers cannot be stowed over long ones which can force overstowage, and reefer slots are scarce and laced at the bottom which may force understowage. In addition, since stability, trim, draft and stress moment limits should not fully be ignored, some weight classes of containers must be introduced. It is also imortant to take containers onboard the vessel when arriving the current ort and containers to load in future orts into account. Finally since ort stay minimization is essential, a realistic estimate of quay crane makesan and overstowage must be included. 3 Literature Survey The number of ublications on stowage lanning has grown substantially within the last few years. Contributions can be divided into two main categories: singlehase and multi-hase aroaches. Multi-hase aroaches decomose the rob-

6 lem hierarchically. They are currently the most successful in terms of model accuracy and scalability. The earliest work can be traced back to a 3-hase heuristic [25], but the first contribution that models several major asects of the roblem is a 2-hase aroach that solves a master lanning hase for multile orts with a branch-and-bound algorithm and uses tabu search for bay lanning [26]. Other aroaches that use a similar decomosition include solving multi-ort master lanning with an iterative imrovement aroach based on the transortation simlex method [18] and a bin-acking heuristic [28]. 3-hase aroaches include combinations of constructive heuristics, 0/1 IP, and metaheuristics (e.g., [1]) and heuristics combined with LS (e.g., [27]). Multi-hase aroaches develoed by the industry include a multi-stage lacement heuristic using a number of lower-bounds [15] and a combination of sequential LP for master lanning and a hierarchy of IPs for slot lanning [14]. Excet this last deloyed industrial system which rovides inut data to our exeriments, none of the revious aroaches include all the major asects of stowage lanning mentioned at the end of Section 2. Single-hase aroaches reresent the stowage lanning roblem (or arts of it) in a monolithic otimization model. The earliest work can be traced back to Aslidis[4], 3 whomakesathoroughstudyofoverstowminimizationalgorithmsfor single bays. The remaining work is categorized according to solution aroach. IP aroaches include a very accurate but intractable model of the comlete stowage roblem [6] and simle models [3, 13, 19] with limited scalability. CP aroaches include an early simle model of the comlete roblem [2] and the combinationofcp[8]andls[21]usedinthisaertogeneratenear-otimalslot lans fast and robustly. A number of metaheuristics have been suggested: genetic algorithms (e.g.,[7, 10]), simulated annealing [11], and lacement heuristics (e.g., [5]). These aroaches use simle models and often only focus on one asect such as overstowage minimization. Additional single-hase aroaches include simulation (e.g., [23]), exert systems (e.g., [9]), 3D-acking (e.g, [22]), and casebased methods (e.g., [20]). Also these aroaches lack reresentative roblem models. 4 Solution Aroach We roose the 2-hase aroach shown in Figure 1. The first hase, multi-ort master lanning, distributes container tyes to sub-sections of bays for the entire route of the vessel. This hase deals with high-level constraints, modeling weight classes of standard 20 and 40 containers, and reefer containers. More recisely, we model GM stability, trim and draft limits, weight distribution and shear forces, and otimize hatch-overstowage and crane makesan. IMO, alletwide, and high-cube containers are not modeled. The two former are often stowed in secific areas, while the latter can be modeled with secialized caacity constraints. We also do not model bending moments and ballast water which are the focus of our future work. The inut to muti-ort master lanning is industrial 3 However, some unublished work from the 1970s using simulation has been cited [6].

7 data from a currently deloyed stowage lanning otimization tool [14], and it includes: 1) vessel data with shi layout and stress limits, 2) current ort loadlist, and future ort loadlists based on historical data, 3) ort data with water deths, crane heights, and crane roductivity. The inut is adjusted such that for examle weight and height constants fulfill quay crane height requirements and line of sight constraints. For that reason, the constants used in the models often change from ort to ort. The master lan for the first ort (the one we are making the stowage lan for) is used as inut for the second hase, slot lanning, to assign the containers of the tyes defined in the master lan to concrete slots. In slot lanning, all major stacking rules aly: containers must form stacks, 20 containers cannot be stowed on to of 40 containers, reefer containers can only be stowed in reefer slots, stack maximum height and weight limits must be fulfilled and cell caacity must be observed. Containers are assigned with the aim of minimizing overstowage, clustering containers with the same POD and freeing stack and reefer slots for robustness. 4.1 Multi-Port Master Planning The multi-ort master lanning hase assigns tyes of containers to sub-sections of bays (locations). Locations are either above or under a hatch-cover and are used as a tool to model hatch-overstowage. Figure 3(a) shows four locations within a bay. Outer locations are symmetrically slit (such as location 2 and 4 in the Figure) to ease transverse stability calculations. For 20 and 40 containers, weconsiderasetoffourmutuallyexclusivecontainertyest = {L,H,RL,RH}, resectively light and heavy containers and light and heavy reefer containers. To roduce a robust lan, our model takes into account the current and a set of downstream orts P. We define transorts TR as the set of airs 1, 2 where 1, 2 P are the loading and discharging ort of a container tye. We define two sets of decision variables x 20τ tl and x 40τ tl reresenting resectively the amount of 20 and 40 containers of tye τ T to be stowed in location l L, where L is the set of all locations, during transort t TR. Although the weight tying of containers used in the model might seem too simlistic, one has to take into account that the average weights, Wt 20τ and Wt 40τ, for each tye are calculated at transort level, making the classification much more refined. Following are the constraints of the roosed IP model: ( x 20τ tl +2x 40τ ) tl C + l P,l L (1) t TR ON t TR ON t TR ON l L τ T τ {RL,RH} τ T ( x 20τ tl +2x 40τ ) tl C R l P,l L (2) x ατ tl C α l P,l L,α {20,40} (3) x ατ tl = LD ατ t α {20,40},τ T,t TR (4)

8 t TR ON G ρ l LG ρ l S s τ T α {20,40} l L Aft s t TR ON t TR ON W ατ t x ατ tl W + l P,l L (5) τ T α {20,40} τ T α {20,40} Wt ατ x ατ tl G +ρ P,ρ {L,V} (6) W ατ t x ατ tl S + s P,s S (7) For each ort P, constraints (1) and (2) define the caacity restrictions of each location l L, resectively for the total number of TEUs allowed (C + l ), and the number of reefer containers that can be stowed (Cl R ), where TRON is the set of all the transorts on the vessel at dearture from ort. Similarly constraint (3) restricts the number of 20 (C 20 ) and 40 (C40) that can be stowed in a location. Constraint (4) forces the loading of all containers in the loadlists. The amount of containers to load is given by the constants LD 20τ t and LD 40τ The average weight of each container tye under a secific transort W ατ t l l t. is used in constraint (5) to limit the load of containers to the max weight allowance W + l reflecting draft limits and vessel caacity. Stability constraints can be calculated w.r.t. the center of gravity of the shi using the hydrostatic data table, and thus be satisfied by limiting its osition using constraint (6). For each ort P, the center of gravity limits, G +ρ and G ρ, have been calculated, where ρ {L, V} reresent the longitudinal and vertical comonents reflecting trim, GM, and draft limits. Due to the symmetrical definition of outer locations, we assumecargotobeequally stowedoneachsideoftheshithusmakingirrelevant the calculations of the transverse comonent (G ρ l indicate the comonents of the center of gravity for each location l L). Given a set of stations s S (calculation oints as shown in Figure 2), constraint (7) calculates the downward force created by the cargo aft of each station s, where L Aft s is the set of locations aft of station s, and S s + and Ss are the maximum and minimum shear limits at station s S for ort P. Since the weight of cargo is constant, buoyancy is included in the calculations behind the constant limits. In the master lanning hase, overstowage minimization focuses on hatchoverstowage. This is modeled by a number of binary variables δ l {0,1}, indicating the resence of containers to load or unload at ort P under on deck locations L O. This is accomlished with the following constraint Ri D + i L U l t TR A τ T ( x 20τ ti +x 40τ ) ti Mδ l P,l L O (8) where L U l is the set of locations under l L O, TR A is the set of transorts that are either loaded or unloaded in ort, and Ri D is the containers already on board the vessel when arriving at the first ort (hereafter referred to as the release) that are discharge from location i in ort. The indicator variable can

9 now be used to calculate hatch-overstowage as follows: Rl OV + ( x 20τ tl +x 40τ ) tl M(1 δl ) yl O P,l L O (9) TR OV t TR OV τ T Constraint (9) defines the cost variable yl O which, given the set of transorts (and release containers Rl OV ) that overstow containers to load or unload at ort P, counts the overstowing containers. In constraints (8-11) we make use of BigM constants M tightened to the uer bounds of the constraints. Due to the fact that slot lanning constraints force 20 containers to be stowed under 40 ones, there is the ossibility of forcing the introduction of overstowage within locations. To alleviate this roblem, we estimate the otential overstowage between 20 and 40 containers within each location. R OV40 l + R D20 l t TR OV τ T + t TR A τ T x 20τ tl Mφ l P,l L (10) x 40τ tl M(1 φ l ) y P l P,l L (11) Constraint (10) introduces a new set of Boolean variables φ l for each ort P and location l L, indicating the resence of 20 containers to load or unload. These indicator variables define the cost variable yl P in constraint (11). This variable holds the number of otential overstows between 40 and 20 containers within a location. Rl D20 and Rl OV40 are the number of containers in the release discharged and otentially overstowing in ort, resectively. Otimization of crane utilization is modeled as the minimization of the makesan of cranes. We calculate a lower bound of the makesan of cranes y T in ort as demonstrated in Figure 2 with the following constraint: C Time ( x 20τ tl +2x 40τ ) tl y T b B, P (12) t TR A l L b τ T where B is the set of adjacent bays, L b is the locations of b B, and C Time is the average crane time needed to load or unload a container. Caacity constraints over reefer containers do not ensure feasibility of a location in the slot lanning hase due to stacking rules. The following constraint ( x 20τ tl +2x 40τ ) tl C R l yl R P,l L (13) t TR ON τ T F τ l alleviate this issue by reducing the maximum caacity of reefer containers within a location by a roortional factor Fl τ, where Fτ l = CR l /C+ l for all non-reefer containers and a factor 1 for all reefer containers. The reduction is then catured in the cost variable yl R. The model minimizes a weighted sum of the cost variables ( C O yl O +C P yl P +C R yl) R + C T y T. (14) P l L P

10 The weights (C O,C P,C R,C T ) have been derived from the deloyed system of our industrial artner and thus reflect a refined adjustment to the economy of stowage lanning and the references of stowage coordinators. Comlexity of Multi-Port Master Planning The Hatch Overstow Problem (HOP) is NP-comlete [17]. This roblem models the same hatch-overstow objective as multi-ort master lanning. Given the otimization version of the HOP, we can reduce it to the multi-ort master lanning roblem by having only one crane and using the reefer caacities to disallow containers below deck. Thus, multi-ort master lanning is NP-hard. 4.2 Slot Planning The master lan for the first ort, which is the ort to generate a stowage lan for, becomes an inut to the slot lanning hase. In this way information is assed in a to-down fashion from the master lanning to the slot lanning hase. There is currently no information flowing in the other direction. The inut defines the number of containers of each tye to stow in each location of the vessel, and by generating a slot lan for each location, a tye-based stowage lan is created. Each slot lan is an indeendent sub-roblem. It must decide which container tye (if any container at all) to stow in each slot of the location. The container tyes here corresond to those defined in the multi-ort master lanning hase (Section 4.1) and must satisfy the constraints and otimize the objectives mentioned in Section 4 for slot lanning. We direct the reader to [8] for a more detailed resentation of the model. The slot lanning hase is solved using a combination of the CP and LS algorithmsdescribedin[8,21].thecpalgorithmisinitiallyrunwithatimelimit of one second. 4 Otimal or near-otimal solutions are often found within this timeframe.insomecases,however,thecparoachisnotabletofindsolutions. When this haens, the LS algorithm is run and the resulting solution is used. Such situations originate in two cases: 1) the roblem is too comlex for CP to be solved in one second (only few cases), and/or 2) the roblem is infeasible. The latter case haens due to the abstraction used in the multi-ort master lanning hase. For instance, when the total weight of containers assigned to a location is within limits, but due to stack weight limits and the arrangement of already onboard containers it is not ossible to stow all containers in the location. Since the loadlist is not strict, and removing a few containers from the locations has small imact on the stability of the vessel, the LS algorithm has been modified to roll out containers that cannot be stowed. 5 Comutational Results To evaluate our aroach, we use 20 instances from a stowage lanning otimization tool [14] deloyed by our industrial artner. Table 1 gives an overview 4 This corresonds to an uer bound of about 100 seconds for a tyical large vessel when solving all the slot lanning sub-roblems sequentially.

11 ID Instances Characteristics Vessel Route Encoding Ca. (TEU) Loc. Ports Util (%) Weight (%) Moves Trans. Bools Table 1. Problem instances overview. Columns under Vessel indicate shi deendent data: Ca. is the maximum nominal caacity of the shi and Loc is the total number of locations. Notice that given the same number of locations, different vessels can have different caacities. Columns under Route show information about the route, Ports indicates the number of calls in the route, Util. and Weight are the maximum utilization during the voyage in terms of TEU and weight. Moves is the total number of crane moves on the route. The Encoding columns resent the number of Boolean variables (Bools) needed by the multi-ort master lanning hase after rerocessing, while Trans. is the total number of active transorts. of the characteristics of the instances. The instances are real stowage roblems that coordinators have solved using the deloyed tool and thus have very high data quality. All exeriments were run on a Linux machine with two Six Core AMD Oteron rocessors at 2.0 Ghz and 32 GB of memory. Multi-Port master and slot lanning models were imlemented in C++ and used resectively CPLEX 12.2, and Gecode 3.5 [12] libraries. Due to the non-deterministic nature of the LS algorithm, results of slot lanning are reorted in average over 10 runs of the algorithm. 5.1 Multi-Port Master Planning Exeriments Given that no revious aroaches have solved the master lanning roblem to otimality, we did not exect our IP model to be efficient. The results, however, did rise attention. Table 2 resents the results of the IP model, and two exeriments where the algorithm is stoed at a 2 and 5% ga from the LP relaxation. A 5% aroximation is accetable since forecasted loadlist data is imrecise. The results are quite interesting for two reasons. First, the IP model was able to calculate otimal solutions for 13 of the 20 instances within a 5 hours limit. Second, taking the results with a 5% ga, it is ossible to generate a comlete stowage lan within 10 minutes for 12 of the instances, suggesting

12 IP Results Otimal 2% Ga 5% Ga ID Obj. Time Ga Time Ga Time Total (10 5 ) (sec.) (%) (sec.) (%) (sec.) (sec.) timeout - timeout - timeout timeout timeout - timeout - timeout timeout timeout timeout - timeout timeout timeout timeout - timeout - timeout timeout 20 - timeout Table 2. Multi-Port Master Planning with IP. The first column is the instance number. The next columns resent groued results of three runs of the model: the first for otimality and the others ending resectively at 2 and 5% ga from the LP relaxation. Column Obj is the otimal value, and column Ga the distance to otimality w.r.t. Obj. Times are reorted in Time, while time to generate a comlete stowage lan is shown in column Total which includes the runtime of the slot lanning hase. Instances that could not be solved within 5 hours are marked with timeout. The bold face shows results obtained within 10 min. the need for further research on IP models. As exected, the objective value is dominated by the overstowage objectives (9) and (11). MIP relaxation To tackle the weaknesses of the IP model, we roose a mixed integer rogramming model where we relax the integrality constraints over the decision variables x 20τ tl and x 40τ tl. We exerimentally evaluate the MIP relaxation by comaring its results to the otimals from the IP model. Table 3 resents results for otimal runs and for 2 and 5% ga from the LP relaxation of the MIP model. In terms of objective value MIP and IP solutions are very similar. It is only in the 5% ga results that it is ossible to notice some significant difference which, however, does not exceed 5.12%. In contrast, runtime results are drastically different. It is now ossible to generate comlete stowage lans for nearly all instances within 10 minutes. Only 4 of the 20 instances could not be solved within this time frame, and 14 could be solved to otimality. Exeriments have shown no difference in objective value w.r.t. overstowage. Also we see a very small difference in crane utilization, clearly due to the increased flexibility of the decision variables. This imortant observation can be used to advocate the use of the MIP model in exchange for IP. The IP model has to face the combinatorial roblem of deciding whether one single container should be stowed in one location or another. Using a MIP we can slit this container and avoid the combinatorial uzzle. Even a single container could, however, cause a large

13 MIP Results Otimal 2% Ga 5% Ga Inst. Ga Time Ga Time Ga Time Total (%) (sec.) (%) (sec.) (%) (sec.) (sec.) timeout - timeout - timeout timeout timeout timeout timeout timeout - timeout - timeout timeout 20 - timeout Table 3. Multi-Port Master Planning with MIP. The second column describes the ga between the IP and MIP otimal solution. For the other columns see Table 2. amount of overstowage, but we do not see this haening in ractice. We thus believe the MIP model can be used in ractice by the industry. Using a MIP also gives the industry the ability to use standard solvers and eases the rocess of adding side constraints. 5.2 Slot lanning exeriments The slot lanning exeriments discussed in this section are based on master lans obtained from the IP and MIP multi-ort master lanning exeriment with 5% otimality ga. Master lans based on MIP may have fractional numbers of containers to stow in locations which is hysically imossible. We attemt to stow a fractioned container in one of the locations where a fraction of it has been assigned by the master lan. If none of these locations have caacity left, the container is rolled out. Based on revious exeriments ([8,21]), we set u a time limit of one second for slot lanning each location of a vessel. The quality of each slot lan is evaluated by comaring it with the best slot lan generated by CP for the same location within twenty minutes. In the cases where the number of containers of one or more tyes were reduced by LS, the slot lan is evaluated against the best lan generated by CP within twenty minutes, stowing the same containers as LS. Table 4 summarizes the results of our exeriments. Vessels are slot lanned fast by our aroach. For the instances with a master lan available, slot lans are generated within 17 seconds in total. There is no time-wise dominance of slot lans generated from MIP and IP master lans, indicating that integrality constraints do not affect the comlexity of slot lanning. When slot lanning IP master lans, a reduction in the number of

14 ID Conts. Slot Planning Current Port Time(s) Locs. Frac.+Odd LS rolled Rolled out(%) Ga(%) Cont. Int. Cont. Int. Cont. Int. Cont. Int. Cont. Int. Cont. Int ,82 7, ,01 9,75 Table 4. Slot Planning MIP (Cont.) and IP (Int.) master lans with 5% ga. The first and second columns are the id of the instance and the number of containers to stow in the first ort. The next columns show groued results of slot lanning based on MIP and IP master lans. The third and fourth columns show the runtime for the slot lans, fifth and sixth columns is the number of locations. The seventh and eighth columns totalize the number of rolled out containers by fractionality and odd number of 20 tyes, the ninth and tenth columns are the containers rolled out by LS, and eleventh and twelfth columns are the ercentage of total containers rolled out. The last two columns show the average ga of the slot lans. A dash indicates that no master lan was rovided. containers rolled out due to fractionality and odd number of 20 tyes in locations is observed in most of the instances. This is, however, a very small (0.4% max) fraction. The number of containers rolled out by LS differs in average only in 1.6 containers (9.7 for the IP and 8.1 for the MIP master lans), and the average ercentage of total containers rolled out is the same (1.2%) for both models. These facts indicate no considerable imact of using MIP master lans. Moreover, it was ossible to generate slot lans for two extra instances using MIP master lans. The maximum roll-out of an instance is 5.99%, a reasonable number given the amount of containers tyically rolled from a loadlist by stowage coordinators. Only two instances have an average otimality ga over 10%, due to the resence of outliers, and the median of all instances is always 0%. CP generated otimal slot lans within one second for 91.8% of the locations of the MIP master lans and 94.8% of the locations of the IP ones. Moreover, CP was able to rove otimality in 83% of the slot lans generated for the locations for both, the MIP and IP master lans.

15 6 Conclusion This aer resented a 2-hase stowage lanning otimization aroach able to solve industry instances to near-otimality within the time limits required for ractical usage. In articular, we introduced an IP model of multi-ort master lanning that includes all major asects of the roblem. Our exeriments have shown that multi-ort master lanning can be solved fast to otimality for a significant number of instances, and that runtime can be drastically reduced by relaxing integrality constraints without comromising solution quality. We have shown that in 16 of our 20 test instances, comlete stowage lans can be comuted in less than 330 seconds. The remaining four instances take longer due to the time needed for master lanning. In future work we lan to develo heuristic master lanning algorithms to handle such hard cases. We are also interested in otimizing ballast water and comare the erformance of our system with human stowage coordinators. Acknowledgments We would like to thank Stehen Barraclough, Jon Adam Freeman, Robert John Milton, and Mikkel Mühldorff Sigurd at Maersk Line and Nicolas Guilbert, Kim Hansen, Esben Mose Hansen, Benoit Paquin, Marc Cromme, and Hans Schou at Ange Otimization for their extensive suort of this work. This research is sonsored in art by the Danish Maritime Fund under the BAYSTOW roject. References 1. Ambrosino, D., Anghinolfi, D., Paolucci, M., Sciomachen, A.: An exerimental comarison of different heuristics for the master bay lan roblem. In: Proceedings of the 9th Int. Symosium on Exerimental Algorithms (2010) 2. Ambrosino, D., Sciomachen, A.: A constraint satisfaction aroach for master bay lans. Maritime Engineering and Ports 36, (1998) 3. Ambrosino, D., Sciomachen, A.: Imact of yard organization on the master bay lanning roblem. Maritime Economics and Logistics (5), (2003) 4. Aslidis, A.H.: Otimal Container Loading. Master s thesis, Massachusetts Institute of Technology (1984) 5. Avriel, M., Penn, M., Shirer, N., Witteboon, S.: Stowage lanning for container shis to reduce the number of shifts. Annals of Oer. Research 76, (1998) 6. Botter, R., Brinati, M.A.: Stowage container lanning: A model for getting an otimal solution. In: Proceedings of the 7th Int. Conf. on Comuter Alications in the Automation of Shiyard Oeration and Shi Design (1992) 7. Davidor, Y., Avihail, M.: A method for determining a vessel stowage lan, Patent Publication WO (1996) 8. Delgado, A., Jensen, R.M., Schulte, C.: Generating otimal stowage lans for container vessel bays. In: Proceedings of the 15th Int. Conf. on Princiles and Practice of Constraint Programming (CP-09). LNCS Series, vol. 5732, (2009) 9. Dillingham, J., Perakis, A.N.: Design of an exert system for container stowage lanning. In: Proceedings of the Fleet Management Technology Conference. Baltimore (1987)

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