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Spatial scheduling for large assembly blocks in shipbuildingAbstract： This paper addresses the spatial scheduling problem （SPP） for large assembly blocks， which arises in a shipyard assembly shop。 The spatial scheduling problem is to schedule a set of jobs, of which each requires its physical space in a restricted space。 This problem is complicated because both the scheduling of assemblies with different due dates and earliest starting times and the spatial allocation of blocks with different sizes and loads must be considered simultaneously。 This problem under consideration aims to the minimization of both the makespan and the load balance and includes various real-world constraints， which includes the possible directional rotation of blocks， the existence of symmetric blocks, and the assignment of some blocks to designated workplaces or work teams. The problem is formulated as a mixed integer programming (MIP） model and solved by a commercially available solver. A two-stage heuristic algorithm has been developed to use dispatching priority rules and a diagonal fill space allocation method， which is a modification of bottom-left-fill space allocation method。 The comparison and computational results shows the proposed MIP model accommodates various constraints and the proposed heuristic algorithm solves the spatial scheduling problems effectively and efficiently。Keywords： Large assembly block； Spatial scheduling; Load balancing； Makespan; Shipbuilding1. IntroductionShipbuilding is a complex production process characterized by heavy and large parts， various equipment, skilled professionals， prolonged lead time, and heterogeneous resource requirements. The shipbuilding process is divided into sub processes in the shipyard, including ship design， cutting and bending operations， block assembly， outfitting， painting, pre-erection and erection。 The assembly blocks are called the minor assembly block, the sub assembly block， and the large assembly block according to their size and progresses in the course of assembly processes。 This paper focuses on the spatial scheduling problem of large assembly blocks in assembly shops。 Fig. 1 shows a snapshot of large assembly blocks in a shipyard assembly shop.Recently, the researchers and practitioners at academia and shipbuilding industries recently got together at “Smart Production Technology Forum in Shipbuilding and Ocean Plant Industries to recognize that there are various spatial scheduling problems in every aspect of shipbuilding due to the limited space, facilities， equipment， labor and time。 The SPPs occur in various working areas such as cutting and blast shops, assembly shops， outfitting shops, preerection yard, and dry docks。 The SPP at different areas has different requirements and constraints to characterize the unique SPPs. In addition， the depletion of energy resources on land put more emphasis on the ocean development。 The shipbuilding industries face the transition of focus from the traditional shipbuilding to ocean plant manufacturing. Therefore, the diversity of assembly blocks， materials， facilities and operations in ship yards increases rapidly。There are some solution providers such as Siemens and Dassult Systems to provide integrated software including product life management, enterprise resource planning system， simulation and etc。 They indicated the needs of efficient algorithms to solve medium- to large-sized SPP problems in 20min, so that the shop can quickly re-optimize the production plan upon the frequent and unexpected changes in shop floors with the ongoing operations on exiting blocks intact。There are many different applications which require efficient scheduling algorithms with various constraints and characteristics (Kim and Moon, 2003, Kim et al。， 2013， Nguyen and Yun, 2014andYan et al.， 2014)。 However， the spatial scheduling problem which considers spatial layout and dynamic job scheduling has not been studied extensively. Until now， spatial scheduling has to be carried out by human schedulers only with their experiences and historical data。 Even when human experts have much experience in spatial scheduling, it takes a long time and intensive effort to produce a satisfactory schedule， due to the complexity of considering blocks geometric shapes, loads， required facilities, etc。 In practice, spatial scheduling for more than a sixmonth period is beyond the human schedulers capacity. Moreover, the space in the working areas tends to be the most critical resource in shipbuilding。 Therefore, the effective management of spatial resources through automation of the spatial scheduling process is a critical issue in the improvement of productivity in shipbuilding plants.A shipyard assembly shop is consisted of pinned workplaces, equipment， and overhang cranes。 Due to the heavy weight of large assembly block, overhang cranes are used to access any areas over other objects without any hindrance in the assembly shop。 The height of cranes can limit the height of blocks that can be assembled in the shop。 The shop can be considered as a twodimensional space。 The blocks are placed on precisely pinned workplaces。Once the block is allocated to a certain area in a workplace， it is desirable not to move the block again to different locations due to the size and weight of the large assembly blocks. Therefore, it is important to allocate the workspace to each block carefully, so that the workspace in an assembly shop can be utilized in a most efficient way。 In addition， since each block has its due date which is predetermined at the stage of ship design, the tardiness of a block assembly can lead to severe delay in the following operations。 Therefore， in the spatial scheduling problem for large assembly blocks， the scheduling of assembly processes for blocks and the allocation of blocks to specific locations in workplaces must be considered at the same time. As the terminology suggests, spatial scheduling pursues the optimal spatial layout and the dynamic schedule which can also satisfy traditional scheduling constraints simultaneously。 In addition, there are many constraints or requirements which are serious concerns on shop floors and these complicate the SPP。 The constraints or requirements this study considered are explained here： （1） Blocks can be put in either directions， horizontal or vertical。 (2) Since the ship is symmetric around the centerline， there exist symmetric blocks. These symmetric blocks are required to be put next to each other on the same workplace。 （3) Some blocks are required to be put on a certain special area of the workplace, because the work teams on that area has special equipment or skills to achieve a certain level of quality or complete the necessary tasks。 (4） Frequently, the production plan may not be implemented as planned， so that frequent modifications in production plans are required to cope with the changes in the shop。 At these modifications， it is required to produce a new modified production plan which does not remove or move the preexisting blocks in the workplace to complete the ongoing operations. （5） If possible at any time, the load balancing over the work teams， i。e., workplaces are desirable in order to keep all task assignments to work teams fair and uniform。Lee， Lee， and Choi （1996） studied a spatial scheduling that considers not only traditional scheduling constraints like resource capacity and due dates， but also dynamic spatial layout of the objects. They used twodimensional arrangement algorithm developed by Lozano-Perez (1983) to determine the spatial layout of blocks in shipbuilding。 Koh, Park, Choi, and Joo （1999) developed a block assembly scheduling system for a shipbuilding company. They proposed a twophase approach that includes a scheduling phase and a spatial layout phase。 Koh, Eom, and Jang (2008） extended their precious works (Koh et al。， 1999) by proposing the largest contact area policy to select a better allocation of blocks。 Cho， Chung， Park， Park, and Kim （2001） proposed a spatial scheduling system for block painting process in shipbuilding， including block scheduling, four arrangement algorithms and block assignment algorithm。 Park et al. (2002) extended Cho et al。 (2001） utilizing strategy simulation in two consecutive operations of blasting and painting。 Shin, Kwon, and Ryu (2008） proposed a bottomleft-fill heuristic method for spatial planning of block assemblies and suggested a placement algorithm for blocks by differential evolution arrangement algorithm. Liu, Chua, and Wee （2011) proposed a simulation model which enabled multiple priority rules to be compared。 Zheng, Jiang， and Chen (2012) proposed a mathematical programming model for spatial scheduling and used several heuristic spatial scheduling strategies (grid searching and genetic algorithm）。 Zhang and Chen （2012） proposed another mathematical programming model and proposed the agglomeration algorithm.This study presents a novel mixed integer programming （MIP） formulation to consider block rotations， symmetrical blocks, pre-existing blocks， load balancing and allocation of certain blocks to predetermined workspace. The proposed MIP models were implemented by commercially available software， LINGO and problems of various sizes are tested. The computational results show that the MIP model is extremely difficult to solve as the size of problems grows。 To efficiently solve the problem, a twostage heuristic algorithm has been proposed。Section 2 describes spatial scheduling problems and assumptions which are used in this study。 Section 3 presents a mixed integer programming formulation. In Section 4, a two-stage heuristic algorithm has been proposed, including block dispatching priority rules and a diagonal fill space allocation heuristic method, which is modified from the bottomleftfill space allocation method。 Computational results are provided in Section 5。 The conclusions are given in Section 6。2. Problem descriptionsThe ship design decides how to divide the ship into many smaller pieces. The metal sheets are cut， blast， bend and weld to build small blocks。 These small blocks are assembled to bigger assembly blocks. During this shipbuilding process， all blocks have their earliest starting times which are determined from the previous operational step and due dates which are required by the next operational step. At each step, the blocks have their own shapes of various sizes and handling requirements. During the assembly, no block can overlap physically with others or overhang the boundary of workplace.The spatial scheduling problem can be defined as a problem to determine the optimal schedule of a given set of blocks and the layout of workplaces by designating the blocks workplace simultaneously。 As the term implies， spatial scheduling pursues the optimal dynamic spatial layout schedule which can also satisfy traditional scheduling constraints. Dynamic spatial layout schedule can be including the spatial allocation issue， temporal allocation issue and resource allocation issue.An example of spatial scheduling is given in Fig. 2。 There are 4 blocks to be allocated and scheduled in a rectangular workplace。 Each block is shaded in different patterns。 Fig。 2 shows the 6day spatial schedule of four large blocks on a given workplace。 Blocks 1 and 2 are preexisted or allocated at day 1. The earliest starting times of blocks 3 and 4 are days 2 and 4, respectively. The processing times of blocks 1， 2 and 3 are 4， 2 and 4days， respectively。The spatial schedule must satisfy the time and space constraints at the same time. There are many objectives in spatial scheduling, including the minimization of makespan， the minimum tardiness， the maximum utilization of spatial and non-spatial resources and etc. The objective in this study is to minimize the makespan and balance the workload over the workspaces。There are many constraints for spatial scheduling problems in shipbuilding, depending on the types of ships built， the operational strategies of the shop， organizational restrictions and etc. Some basic constraints are given as follows; （1） all blocks must be allocated on given workplaces for assembly processes and must not overstep the boundary of the workplace; (2） any block cannot overlap with other blocks; （3） all blocks have their own earliest starting time and due dates; (4） symmetrical blocks needs to be placed side-byside in the same workspace. Fig。 3 shows how symmetrical blocks need to be assigned； (5) some blocks need to be placed in the designated workspace; (6） there can be existing blocks before the planning horizon； (7) workloads for workplaces needs to be balanced as much as possible。In addition to the constraints described above， the following assumptions are made。(1） The shape of blocks and workplaces is rectangular。（2 ）Once a block is placed in a workplace, it cannot be moved or removed from its location until the process is completed.（3 ） Blocks can be rotated at angles of 0 and 90 (see Fig. 4）。（4） The symmetric blocks have the same sizes, are rotated at the same angle and should be placed side-by-side on the same workplace。(5） The nonspatial resources (such as personnel or equipment） are adequate。3。 A mixed integer programming modelA MIP model is formulated and given in this section. The objective function is to minimize makespan and the sum of deviation from average workload per workplace, considering the block rotation, the symmetrical blocks， preexisting blocks， load balancing and the allocation of certain blocks to predetermined workspace。A workspace with the length LENW and the width WIDW is considered two-dimensional rectangular space. Since the rectangular shapes for the blocks have been assumed, a block can be placed on workspace by determining （x， y) coordinates， where 0xLENW and 0yWIDW。 Hence, the dynamic layout of blocks on workplaces is similar to two-dimensional bin packing problem。 In addition to the block allocation, the optimal schedule needs to be considered at the same time in spatial scheduling problems。 Z axis is introduced to describe the time dimension. Then, spatial scheduling problem becomes a threedimensional bin packing problem with various objectives and constraints。The decision variables of spatial scheduling problem are (x， y, z） coordinates of all blocks within a threedimensional space whose sizes are LENW, WIDW and T in x， y and z axes, where T represents the planning horizon. This space is illustrated in Fig。 5。In Fig。 6, the spatial scheduling of two blocks into a workplace is illustrated as an example。 The parameters p1 and p2 indicate the processing times for Blocks 1 and 2， respectively. As shown in z axis， Block 2 is scheduled after Block 1 is completed。4。 A two-stage heuristic algorithmThe computational experiments for the MIP model in Section 3 have been conducted using a commercially a