船舶专业外文文献

Spatial scheduling for large assembly blocks in
shipbuilding
Abstract：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; Shipbuilding
1。

Introduction
Shipbuilding 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, pre—erection 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 20 min，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，2014 and Yan 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 six—month 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 two-dimensional 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 pre-determined 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 pre—existing 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 two-dimensional 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 two-phase 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 bottom—left—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 pre—determined 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 two—stage 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 bottom—left-fill space allocation method. Computational results are provided in Section 5. The conclusions are given in Section 6.
2。

Problem descriptions
The 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 6-day spatial schedule of four large blocks on a given workplace. Blocks 1 and 2 are pre—existed 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 4 days，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-by—side 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 non-spatial resources （such as personnel or equipment）are adequate.
3. A mixed integer programming model
A 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，pre-existing blocks, load balancing and the allocation of certain blocks to pre—determined 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 0 ⩽ x ⩽ LENW and 0 ⩽ y ⩽ WIDW。

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 three-dimensional 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
three-dimensional 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 algorithm
The computational experiments for the MIP model in Section 3 have been conducted using a commercially available solver, LINGO®. Obtaining global optimum solutions is very time consuming，considering the number of variables and constraints。

A ship is consisted of more than 8 hundred large blocks and the size of problem using MIP model is beyond today’s computational ability. A two-stage heuristic algorithm has been proposed using the dispatching priority rules and a diagonal fill method.
4。

1. Stage 1：Load balancing and sequencing
Past research on spatial scheduling problems considers various priority rules。

Lee et al。

(1996) used a
priority rule for the minimum slack time of blocks. Cho et al. （2001) and Park et al。

（2002）used the earliest due date. Shin et al。

（2008) considered three dispatching priority rules for start date, finish date and geometric characteristics (length，breadth, and area）of blocks。

Liu and Teng (1999) compared 9 different dispatching priority rules including first-come first-serve, shortest processing time，least slack，earliest due date，critical ratio，most waiting time multiplied by tonnage，minimal area residue, and random job selection。

Zheng et al。

（2012）used a dispatching rule of longest processing time and earliest start time.
Two priority rules are used in this study to divide all blocks into groups for load balancing and to sequence them considering the due date and earliest starting time. Two priority rules are streamlined to load-balance and sequence the blocks into an algorithm which is illustrated in Fig. 7. The first step of the algorithm in this stage is to group the blocks based on the urgency priority. The urgency priority is calculated by subtracting the earliest starting time and the processing time from the due date for each block. The smaller the urgency priority，the more urgent the block needs to bed scheduled. Then all blocks are grouped into an appropriate number of groups for a reasonable number of levels in urgency priorities. Let g be this discretionary number of groups. There are g groups of blocks based on the urgency of blocks。

The number of blocks in each group does not need to be identical.
Blocks in each group are re-ordered grouped into as many subgroups as workplaces，considering the workload of blocks such as the weight or welding length。

The blocks in each subgroup have the similar urgency and workloads. Then，these blocks in each subgroup are ordered in an ascending order of the earliest starting time. This ordering will be used to block allocations in sequence。

The subgroup corresponds to the workplace.
If block i must be processed at workplace w and is currently allocated to other workplace or subgroup than w，block i is swapped with a block at the same position of block i in an ascending order of the earliest starting time at its workplace (or subgroup）. Since the symmetric blocks must be located on a same workplace, a similar swapping method can be used. One of symmetric blocks which are allocated into different workplace （or subgroups) needs to be selected first。

In this study, we selected one of symmetric blocks whichever has shown up earlier in an ascending order of the earliest starting time at their corresponding workplace （or subgroup）。

Then, the selected block is swapped with a block at the same position of symmetric blocks in an ascending order of the earliest starting time at its workplace (or subgroups)。

4.2。

Stage 2：Spatial allocation
Once the blocks in a workplace （or subgroup) are sequentially ordered in different urgency priority groups, each block can be assigned to workplaces one by one，and allocated to a specific location on a workplace. There has been previous research on heuristic placement methods. The bottom—left (BL) placement method was proposed by Baker, Coffman，and Rivest （1980) and places rectangles sequentially in a bottom—left most position。

Jakobs （1996）used a bottom-left method that is combined with a hybrid genetic algorithm (see Fig。

8）。

Liu and Teng (1999) developed an extended bottom-left heuristic which gives priority to downward movement，where the rectangles is only slide leftwards if no downward movement is possible. Chazele （1983) proposed the bottom-left—fill (BLF）method，which searches for lowest bottom—left point，holes at the lowest bottom—left point and then place the rectangle sequentially in that bottom—left position。

If the rectangle is not overlapped, the rectangle is placed and the point list is updated to indicate new placement positions。

If the rectangle is overlapped，the next point in the point list is selected until the rectangle can be placed without any overlap。

Hopper and Turton (2000) made a comparison between the BL and BLF methods。

They concluded that the BLF method algorithm achieves better assignment patterns than the BL method for Hopper’s example problems。

Spatial allocation in shipbuilding is different from two-dimensional packing problem. Blocks have irregular polygonal shapes in the spatial allocation and blocks continuously appear and disappear since they have their processing times. This frequent placement and removal of blocks makes BLF method less effective in spatial allocation of large assembly block.
In order to solve these drawbacks，we have modified the BLF method appropriate to spatial scheduling for large assembly blocks。

In a workplace，since the blocks are placed and removed continuously, it is more efficient to consider both the bottom-left and top-right points of placed blocks instead of bottom-left points only。

We denote it as diagonal fill placement (see Fig。

9)。

Since the number of potential placement considerations increases，it takes a bit more time to implement diagonal fill but the computational results shows that it is negligible.
The diagonal fill method shows better performances than the BLF method in spatial scheduling problems. When the BLF method is used in spatial allocation，the algorithm makes the allocation of some blocks delayed until the interference by pre-positioned blocks are removed。

It generates a less effective and less efficient spatial schedule. The proposed diagonal fill placement method resolve this delays better by allocating the blocks as soon as possible in a greedy way, as shown in Fig. 10. The potential drawbacks from the greedy approaches is resolved by another placement strategy to minimize the possible dead spaces，which will be explained in the following paragraphs。

The BLF method only focused on two—dimensional bin packing。

Frequent removal and placement of blocks in a workspace may lead to accumulation of dead spaces，which are small and unusable spaces among blocks.
A minimal possible-dead space strategy has been used along with the BLF method. Possible-dead spaces are being generated over the spatial scheduling and they have less chance to be allocated for future blocks。

The minimal possible-dead space strategy minimizes the potential dead space after allocating the following blocks （Chung，2001 and Koh et al。

，2008) by considering the 0° and 90° rotation of the block and allocating the following block for minimal possible-dead space. Fig。

11 shows an example of three possible—dead space calculations using the neighbor block search method。

When a new scheduling block is considered to be allocated, the rectangular boundary of neighboring blocks and the scheduling blocks is searched. This boundary can be calculated by obtaining the smallest and the largest x and y coordinates of neighboring blocks and the scheduling blocks。

Through this procedure, the possible-dead space can be calculated as shown in Fig.
11. Considering the rotation of the scheduling blocks and the placement consideration points from the diagonal fill placement methods, the scheduling blocks will be finally allocated。

In this two-stage algorithm，blocks tend to be placed adjacent to one of the alternative edges and their assignments are done preferentially to minimize fractured spaces.
5. Computational results
To demonstrate the effectiveness and efficiency of the proposed MIP formulation and heuristic algorithm，the actual data about 800+ large assembly blocks from one of major shipbuilding companies has been obtained and used. All test problems are generated from this real—world data。

All computational experiments have been carried out on a personal computer with a Intel® Core™ i3-2100 CPU @ 3。

10 GHz with 2 GB RAM。

The MIP model in Section 3 has been programmed and solved using LINGO® version 10.0，a commercially available software which can solve linear and nonlinear models。

The proposed two-stage heuristic algorithm has been programmed in JAVA programming language。

Because our computational efforts to obtain the optimal solutions for even small problems are more than significant, the complexity of SPP can be recognized as one of most difficult and time consuming problems.
Depending on the scaling factor α in objective function of the proposed MIP formulation, the performance of the MIP model varies significantly。

Setting α less than 0。

01 makes the load balancing capability to be ignored from the optimal solution in the MIP model。

For computational experiments in this study，the results with the scaling factor set to 0。

01 is shown and discussed。

The value needs to be fine—turned to obtain the desirable outcomes。

Table 1 shows a comparison of computational results and performance between the MIP models and two—stage heuristic algorithm。

As shown in Table 1, the proposed two-stage heuristic algorithm finds the near—optimal solutions for medium and large problems very quickly while the optimal MIP models was not able to solve the problems of medium or large sizes due to the memory shortage on computers。

It is observed that the computational times for the MIP problems are rapidly growing as the problem sizes increases. The test problems in Table 1 have 2 workplaces。

Table 1。

Computational results and performance between the MIP models and two-stage heuristic algorithm。

The MIP model Two-stage heuristic algorithm
Number of blocks
Optimal solution Time (s）Best known solution Time （s）
10 12。

360 1014.000 12.360 0.026
20 22。

380a 38250.000 21.380 0.078
30 98。

344a 38255.000 30.740 0.218
50 ––53。

760 0。

719
100 ––133.780 2。

948
200 ––328.860 12.523
300 ––416.060 40。

154
400 ––532。

360 73.214
Best feasible solution after 10 h in Global Solver of LINGO®.
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The optimal solutions for test problems with more than 50 blocks in Table 1 have been not obtained even after 24 h. The best known feasible solutions after 10 h for the test problems with 20 blocks and 30 blocks are reported in Table 1。

It is observed that the LINGO® does not solve the nonlinear constraints very well as shown in Table 1。

For very small problem with 10 blocks，the LINGO® was able to achieve the optimal solutions。

For slightly bigger problems，the LINGO® took significantly more time to find feasible solutions。

From this observation, the approaches to obtain the lower bound through the relaxation method and upper
bounds are significant required in future research。

In contrary, the proposed two—stage heuristic algorithm was able to find the good solutions very quickly. For the smallest test problem with 10 blocks, it was able to find the optimal solution as well. The computational times are 1014 and 0。

026 s，respectively，for the MIP approach and the proposed algorithm。

Interestingly，the proposed heuristic algorithm found significantly better solutions in only 0。

078 and 0.218 s，respectively, for the test problems with 20 and 30 blocks. For these two problems，the LINGO® generates the worse solutions than the heuristics after 10 h of computational times。

The symbol ‘–’ in Table 1 indicates that the Global Solver of LINGO® did not find the feasible solutions。

Another observation on the two-stage heuristic algorithms is the robust computational times。

The computation times does not change much as the problem sizes increase。

It is because the simple priority rules are used without considering many combinatorial configurations.
Fig. 12 shows partial solutions of test problems with 20 and 30 blocks on 2 workplaces. The purpose of Fig。

12 is to show the progress of production planning generated by the two-stage heuristic algorithm. Two workplaces are in different sizes of (40，30) and (35，40), respectively.
6. Conclusions
As global warming is expected to open a new way to transport among continent through North Pole Sea and
to expedite the oceans more aggressively，the needs for more ships and ocean plants are forthcoming. The shipbuilding industries currently face increased diversity of assembly blocks in limited production shipyard. Spatial scheduling for large assembly blocks holds the key role in successful operations of the shipbuilding companies.
The task of spatial scheduling takes place at almost every stage of shipbuilding processes and the large assembly shop is one of the most congested operat ional areas in today’s shipbuilding。

It is also known that the spatial scheduling problem has been the major source of the bottleneck。

The practitioners in shipbuilding industries requires their production planning system to optimize the spatial scheduling and to respond quickly to the changes on the shop floor by re-optimizing the production plan in 20-min time frame. Most companies use a system employing heuristic methods in an ad—hoc manner without knowing how good their planning system is.
To benchmark the performance of the heuristic algorithms，a novel MIP model has been proposed considering various real—world constraints that are raised by field professionals and engineers. Those include block rotations，symmetrical blocks, pre—existing blocks, load balancing and allocation of certain blocks to pre—determined workspace. These constraints have not been considered simultaneously by previous researchers。

The MIP formulation can be used as a target to evaluate their spatial scheduling system in shipbuilding companies。

In addition，the expectation and need of major solution companies such as Siemens™ and Dassult Systems™ is an efficient and effective algorithm to perform spatial scheduling. When a new type of ships or ocean plants is designed and built, there is a higher chance to observe unexpected interruptions in production flows. These interruptions cause significant losses in time，labor, resource and etc. Therefore，the need to re—optimize spatial scheduling quickly in shorter amount of computational times becomes greater。

The proposed two—stage heuristic algorithm uses a simple priority rules and dispatching rules to group the blocks for quick load balancing and suggests the diagonal fill placement method which is fit to spatial scheduling in shipbuilding. The computational results show that the proposed algorithm finds good solutions。