Journal of Engineering Science and Technology Review 9 (2) (2016) Research Article

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1 Jestr Journal of Engineering Science and Technology Review 9 () (06) Research Article Ship Block Transportation Scheduling Proble Based on Greedy Algorith Chong Wang *,,3, Yun-sheng Mao,, Bing-qiang Hu,, Zhong-jie Deng, and Jong Gye Shin 3,4 Wuhan University of Technology, Key Laboratory of High Perforance Ship Technology, Wuhan, China Wuhan University of Technology, School of Transportation, Wuhan, China 3 Departent of Naval Architecture and Ocean Engineering, Seoul National University, Seoul, Korea 4 Research Institute of Marine Syste Engineering, Seoul National University, Seoul, Korea JOURNAL OF Engineering Science and Technology Review Received 0 January 05 Accepted 0 May 06 Abstract Ship block transportation probles are crucial issues to address in reducing the construction cost and iproving the productivity of shipyards. Shipyards ai to axiize the workload balance of transporters with tie constraint such that all blocks should be transported during the planning horizon. This process leads to three types of penalty tie: epty transporter travel tie, delay tie, and tardy tie. This study ais to iniize the su of the penalty tie. First, this study presents the proble of ship block transportation with the generalization of the block transportation restriction on the ulti-type transporter. Second, the proble is transfored into the classical traveling salesan proble and assignent proble through a reasonable odel siplification and by adding a virtual node to the proposed directed graph. Then, a heuristic algorith based on greedy algorith is proposed to assign blocks to available transporters and sequencing blocks for each transporter siultaneously. Finally, the nuerical experient ethod is used to validate the odel, and its result shows that the proposed algorith is effective in realizing the efficient use of the transporters in shipyards. Nuerical siulation results deonstrate the proising application of the proposed ethod to efficiently iprove the utilization of transporters and to reduce the cost of ship block logistics for shipyards. Keywords: Greedy Algorith Ship Block Transportation Block scheduling.. Introduction In odern shipbuilding, a ship consists of ultiple erection blocks. The block is the ost iportant work-in-process aspect of the shipbuilding process. For exaple, a very large crude oil carrier with deadweight of 350,000 tons coprises around 0 blocks. These blocks undergo a series of construction processes such as assebly, pre-outfitting, painting, and outfitting in the factory or stockyard before erection in the dock. Block weight is generally between 00 and 300 tons, and even the individual block is over 500 tons. Ship block transportation between the factory and the stockyard ainly relies on transporters (Figure ). For a large shipyard, nearly 500 block transportation tasks need to be copleted daily. Thus, the efficiency of shipbuilding ust be iproved through optial block transportation scheduling. Related studies on the optial proble of ship block transportation scheduling in shipyards are liited, although decreasing the construction cost and iproving productivity are possible through efficient transporter operation. Therefore, an efficient ethod is necessary to solve the aforeentioned proble. The present study focuses on how to solve this proble efficiently for shipyards. The rest of this paper is organized as follows. Section explains the ain distinction between the aforeentioned proble and the classical traveling salesan proble (TSP). Section 3 introduces the current research situation regarding this topic. Section 4 derives a atheatical odel to find the optial solution and * E-ail address: chriswang@whut.edu.cn ISSN: Eastern Macedonia and Thrace Institute of Technology. All rights reserved. proposes a heuristic algorith based on greedy algorith. Section 5 presents a coputational nuerical siulation experient to evaluate the perforance of the proposed algorith. Finally, conclusions are suarized and rearks on further study are provided in Section 6. Fig.. Exaple of a transporter for oving blocks in shipyards. Description of the proble When the nuber of transporters is fixed, the type of transporters is the sae, and all blocks are predeterined to be delivered by a specific transporter the scheduling proble is siilar to a ultiple TSP (-TSP) [][][3][4]. In such a proble, each block is considered as a location and transporters are regarded as traveling salesen. However, ship block transport probles in shipyards have a few differences with -TSPTW because of the different nubers and different types of transporter.

2 Journal of Engineering Science and Technology Review 9 () (06) good solution in actual large-sized probles is difficult. Joo et al. (04) [9] expanded the block transportation scheduling proble of Roh and Cha (0)[7] and Ki and Joo (0)[8] by generalizing the block delivery restriction on the ulti-type transporter. Two etaheuristic algoriths based on a genetic algorith and a self-evolution algorith are proposed to avoid the occurrence of infeasible solutions caused by the block delivery restriction. The present study priarily ais to iprove the efficiency of the solution process for assigning blocks to available transporters and sequencing blocks for each transporter siultaneously. The algorith based on greedy algorith is designed to solve the realistic scale proble with an optial feasible solution. 4. Methodology Fig.. Description of ship block transportation proble in shipyards In the exaple shown in Figure, B-B-B5 can only be transported by TP because of the capacity of transporter restrictions. As a result, TP goes through the factory or S3- S4-S-S4-S to coplete the transportation of B-B-B5. In this process, TP fro the workplace S3 transports B to the destination of B (S4) because the destination of B (S4) and the departure of the next transported B (S) are different. Thus, TP ust epty travel to the departure of B (S). The sae TP sequentially goes through workplace S7-S6-S4-S3-S5 to transport B6-B3-B4 to their destinations. The origin and destination of ship block transport probles are different. Therefore, this proble is absolutely different fro that of the classical TSP. 3. State of the art Only a few studies on ship block transportation scheduling proble have been published, although the shipbuilding cost can be reduced through efficient transporter operation. The ain researchers are fro South Korea, which is one of the global industry leaders in shipbuilding. Joo et al. (006)[5] first studied the block transportation scheduling of singletype transporters in shipyards. They proposed a heuristic algorith to iniize the su of total weighted logistic tie of blocks oved by transporters. However, the assuption regarding single-type transporters is inconsistent with the shipyard objective facts. Park et al. (03)[6] addressed the transporter scheduling proble in ship assebly block operation anageent by transforing this proble into parallel achine scheduling with sequencedependent setup tie and precedence constraints with the sae assuption of single-type transporters. Roh et al. (0)[7] expanded the block transportation scheduling proble of Joo et al. (006)[5] when ulti-type transporters are used. Although good solutions were not obtained, this extension preeptively decided the allocation rule of blocks to transporters and deterined the sequence rule of blocks for each transporter. Ki and Joo (0)[8] considered a block transportation scheduling proble with ulti-type transporters as in the work of Roh and Cha (0)[7]. They deterined the blocks assigned to each transporter and the sequence of blocks for each transporter at the sae tie. However, the coputation tie increases significantly when the nuber of blocks and transporters increases. Finding a 4. Matheatical optiization odel To calculate the su of epty transporter travel tie, delay tie, and tardy tie, ti () is set as the tiing to accoplish the transport task of block i. ()If the transport task of block is the first task of the transporter, then oi () =, ti () = 0. () Otherwise, ti () is the tie when the transporter copleted the previous block. Epty transporter travel tie ( ET () i ) is incurred when the current position of the transporter and the origin of the next handling block are different. ()If the transport task of the block i is the first task of the transporter, then ET () i is the tie that transporters ove fro the initial position to the starting position of block i. () Otherwise, ET () i is the tie that transporters ove fro the destination node of the previous block to the start node of the next block. don ( ( j), SNi ( ))/ VE( j) ci ( ) = joj, ( ) = ET ( i) =, ci () = ci (') = j, d( AN( i'), SN( i))/ VE( j) oi () = oi (') + Delay tie ( DT () i ) is incurred when the blocks are collected by transporters after the required preparation tie, which is represented as follows: ( ) ax ( ( ') ( ) ( ),0) ( ') = ( ), ( ') = ( ). DT i = t i + ET i ST i, ci ci oi oi Tardy tie ( LT () i ) is incurred when the block is delivered after the required delivery tie, which is represented as follows: d( SN( i), AN( i) ) ( ) = ( ) + ( ) + ( ) + + ( ) ( ) VF ( c( i) ) d( SN( i), AN( i) ) t( i) = ST( i) + DT( i) + UT( i) + + OT( i). VF c( i) LT i ax ST i DT i UT i OT i AT i,0 ( ) 94

3 Journal of Engineering Science and Technology Review 9 () (06) Block i should be transported by the transporter that has higher weight capacity than the weight of the block, that is, wi () W( ci ()) Decision variables For the i th block: ci () = j: Block i is transported by transporter j. oi () = k: Block i is the kth transportation task of the transporter. The evaluation index of the scheduling plan is as follows: ET () i : epty transporter travel tie DT () i : delay tie LT () i : tardy tie. Paraeters N : Based on the assuption that the workshop or block stockyard yard is siplified as a node, N is the total nuber of nodes. D : Distance atrix aong nodes, D = [ d( i, j )] N N. Based on the assuption that n blocks need to be transported in one day, and the properties of the block transportation task are as follows: SN() i : Start node of block i, SN( i) {,,.., N} AN() i : Destination node of block i, AN( i) {,,.., N} SN AN ) ( i i ST () i : Scheduled start tie of block i AT () i : Scheduled due tie of block i UT () i : Loading tie of block i OT () i : Unloading tie of block i and wi (): Weight of block i. Based on the assuption that transporters exist, each transporter possesses the following attributes: ON( j ): Initial position of transporter j W( j ): Deadweight of transporter j VE( j ) : No-load speed of transporter j and VF( j ): Full-load speed of transporter j. 4. Objective function The odel ais to iniize the su of total ET () i, DT () i, and LT () i, which are represented as follows: n in [ ET( i) + DT ( i) + LT ( i) ] ( ), ( ) ci oi, i= stw.. () i W( c()) i. The optiization variables of this odel are shown as follows: ci (): Assignent variables of block transportation and oi (): Execution sequence variables of block transportation. First, ci () ust satisfy the following constraint: { } ci (),,...,. Then, () ci and ( ) oi were closely related. Based on the assuption that the set of the transportation task of transporter j is C { i: c( i) j} oi (') oi (''). j = =, for any i' i'' C, j The execution sequence variables of block transportation are as follows: oi ( ) starts at and then continues to the consecutive nubers, ( ) { : } {,,..., } O = o i i C = O. j j j 4.3 Optiization algorith In this section, the ain objective is to allow the original optiization proble to approxiately transfor the TSP. Then, an algorith is designed to solve the proble by considering the practical significance of the proble in the shipbuilding industry Model siplification and assuption The optiization odel ais to weigh the su of total ET () i, DT () i, and LT () i. In fact, the delay tie to accoplish the previous block transportation has a significant influence on the delay tie of the next block transportation. In this study, all the previous block transportations are assued to be on tie without the ipact. The concern is how to select the next block transportation task ore effectively. In this case, the su of total ET () i, DT () i, and LT () i for the next block transportation task is called penalty tie. For any block transportation task i and j, the penalty tie is ω (, i j), which eans that block i is transported first and then block j is transported. This process is defined as a transfer aong the block transportation tasks. Therefore, the block transportation tasks were regarded as vertices of a graph. The penalty tie of the transfer was used as edge weights to construct a directed graph. All the blocks should be transported within a reasonable period, and thus, this proble is very close to TSP. Siultaneously, the initial positions of each transporter as a virtual node are added to the directed graph, considering that they had a significant effect on the penalty tie of block transportation tasks. Furtherore, the penalty tie of a transfer is associated with the speed of the transporter, and the dead weight of transporters is liited. Accordingly, this proble was considered in the classification with the sae speed of transporters and according to load capacity to bring this proble closer to the general TSP. The following assuptions were ade: transporters are available with the sae no-load and full-load speeds the noload speed and full speed is VE and VF, respectively the iniu load of the transporters is W n blocks are available with weight wi () less than W. The block transportation tasks n were used as nodes. The penalty tie ω (, i j) of the transfer was used as edge weights to construct a directed graph. The penalty tie is the su of ET () i, DT () i, and LT () i based on the assuption that the block i was transported only in tie and the block j was transported iediately. ω (, i j) = ET(, i j) + DT(, i j) + LT(, i j) ET(, i j) = d( AN(), i SN( j))/ VE DT(, i j) = ax( AT () i + ET (, i j) ST ( j),0) 95

4 Journal of Engineering Science and Technology Review 9 () (06) ( ) + (, ) + ( ) + ( ( ) ( )) + ( ) ( ) ST j DT i j UT j LT(, i j) = ax, /,0 d SN j AN j VF OT j AT j ω ( ii, ) = eans that tasks are not transferred by theselves. The following virtual nodes are available: k = n+, n+,..., n+. The ain reasons for adding the virtual nodes are as follows: a) The penalty tie of transporting other blocks is different because the initial positions of the transporter are different. b) Before the virtual nodes are set up, the path of one transporter in the graph to delivery blocks is only an ordinary path and not a loop. To construct a loop, the virtual node is introduced. After adding virtual nodes, the process of transporter to ove one block are as follows: starting fro the initial position, followed by the copletion of other tasks, and then back to the original node. However, returning to the initial node is in fact unnecessary. The weight of returning to the original node is set as 0 after the other transportation tasks have been accoplished. The constraints of the penalty tie are as follows: () The penalty tie fro the virtual node k to other block transportation tasks is equal to the penalty tie of the transporter oving fro its initial position to other block transportation tasks. ω (,) k j = ET(,) k j + DT(,) k j + LT(,) k j ET(,) k j = d( ON ( k n), SN())/ j VE DT(,) k j = ax( AT () i + ET (,) k j ST (),0) j. ()The penalty tie between virtual nodes is ω ( k, k). ( k, k) k, k n+, n+,..., n+. In other ω =, { } words, the transporter does not ove fro its initial position to the initial position of the other transporters to avoid constituting a loop between virtual nodes. (3)The penalty tie fro the other block transportation j to the virtual node k is ω ( jk, ). ω ( jk, ) = 0, j {,,..., n}, k { n, n,..., n } In particular, the transporter finishes all transportation tasks after copleting any task j. The following decision atrix is set: X = ( x ) ij ( n+ ) ( n+ ) x = eans that the transporter oves block i and then ij oves block j x ij = 0 eans that the transporter does not ove between block i and block j. According to the preceding discussion, the optiization proble is expressed as follows: in ω( i, j) x st.. n+ n+ i= j= n+ xij =, j =,,..., n+, i= n+ xij =, i =,,..., n+ j = ij is where n+ j= n+ xij = indicates that the in-degree of each node i= xij = indicates that the out-degree of each node is. Both of the ensure that the transfer path selection consists of a nuber of loops. For exaple, one transporter and two blocks are constructed in the directed graph (Figure 3) Fig.3. Directed graph of one transporter and two blocks Ordinary nodes and correspond to transportation tasks and, respectively. Virtual node 3 is the initial position of transporter. The weight of the graph edge is the penalty tie of a transfer. The definitions are as follows: () The penalty tie of transporter fro the initial position to the copletion of the block transportation task is. () The penalty tie of transporter fro the initial position to the copletion of the block transportation task is. (3) The penalty tie of transporter to coplete the block transportation tasks and is also. (4) The penalty tie of transporter to coplete the block transportation tasks and is also. This study ais to find the loop in which the penalty tie is the iniu. Clearly, the penalty tie of 3 is, and the penalty tie of 3 is 4. Therefore, the path of 3 is better. This path eans that transporter oves fro the initial position to coplete block transportation task and then copletes block transportation task Greedy algorith optiization In this section, the transporters are classified according to load capacity to design a greedy algorith for obtaining an approxiate solution. In the shipyard, transporters have a variety of different specifications. The reasonable scheduling plan is that the heavyweight blocks are transported by using heavy-load transporters, and the lightweight blocks are transported by using light-load transporters. Based on this classification rule, the approxiation algorith was designed as follows: Based on the assuption that M transporter specifications are present, the load capacity isww,,..., W M. 96

5 Journal of Engineering Science and Technology Review 9 () (06) Step. Set k =, arking all block transportation tasks that are not allocated. Step. If k < M, then go to step 3 otherwise, proceed to step 4. Step 3. Select the block transportation tasks with weight less than W, and construct the assignent proble using k the Hungarian algorith for calculation. To obtain the optial solution as the load of W transporter task, ark the arranged tasks as allocated status then, proceed to step 5. Step 4. Select the block transportation tasks with weight less than in the task without an assigned transporter, and construct TSP using Matlab for calculation. To obtain the k optial solution as the load of transporter task, proceed to step 5. Step 5. If k < M, then update k = k +, and return to step. Otherwise, stop the procedure. 5. Result analysis and discussion 5. Nuerical siulation In this study, the actual production data of D shipyard are used in the nuerical siulation. A total of 5 transporters are used in this shipyard, and the details are shown in Table. Table. Detailed inforation on transporters in D shipyard Load capacity 00 tons 00 tons 300 tons 350 tons 500 tons Nuber No-load speed 50 /in 50 /in 00 /in 80 /in 50 /in Full-load speed 35 /in 0 /in 00 /in 75 /in 50 /in This shipyard has 6 factories and 7 stockyards, that is, 3 nodes are obtained. The distance between the nodes is the sae as the shipyard layout. A total of 50 block transportation tasks are present. The loading and unloading ties of block transportation correspond to noral distribution fro 3 in to 5 in. The scheduled start tie is noral distribution fro 0 in to 400 in. The forula for the scheduled due tie is as follows: AT ()= i ST () i + dij (, ) + R UT() i R is a noral distribution fro 30 in to 50 in. In other words, the scheduled due tie of each block transportation task includes transporter loading travel tie, 0 in loading/unloading block tie, and waiting tie fro approxiately 0 in to 40 in. The nuerical experient environent consists of CPU: Intel (R) Core (TM) i CPU 3.0 GHz RAM: 4.0 G 64-bit operating syste and Matlab R04a.:\ 5. Siulation result and discussion According to the experiental environent, the solving tie is only s by using the greedy algorith. The transportation task of each transporter is shown in Figure 4. The ters in Figure 4 are defined as follows: LIT: No epty travel before the start of the transportation task, and the block arrives at the destination on tie. LDT: No epty travel before the start of the transportation task, but the block is collected by the transporter after the required preparation tie. LDTT: No epty travel before the start of the transportation task, but the block is collected by the transporter after the required preparation tie and is delivered after the required delivery tie. EIT: Epty travel before the start of the transportation task, and the block arrives at the destination on tie. EDT: Epty travel before the start of the transportation task, but the block is collected by the transporter after the required preparation tie. EDTT: Epty travel before the start of the transportation task, but the block is collected by the transporter after the required preparation tie and is delivered after the required delivery tie. In Figure 4, LIT, EIT, and EDT ean that the block arrives at the destination on tie. Figure 4 indicates that ost of the block transportation tasks are copleted on tie, and only a few of the are delivered after the required delivery tie. Furtherore, the load of each transportation is balanced. The entire block transportation task is accoplished before 6: Conclusions This study aied to deterine when the transporter should ove and which transporter should ove the block fro its source factory or stockyard to its destination, with the iniu su of epty transporter travel tie, delay tie, and tardy tie to iniize the cost of ship block logistics. A new atheatical odel was established, which was a hybrid odel based on TSP and AP. A heuristic ethod based on the greedy algorith for the optial solution was derived to deterine the assignent of blocks for available transporters and the transportation sequence of blocks for each transporter. The result indicated that ost of the block transportation tasks were copleted on tie. Moreover, the workload for transporters was balanced appropriately. According to the coputational results, the proposed algorith is a novel solution for ship block transportation scheduling proble in an actual shipyard. The proposed algorith has significant potential to proote transporter efficiency to reduce the cost of ship block logistics for shipyards. In the future, an intelligent scheduling syste for ship block transportation scheduling will be developed based on this ethod. Acknowledgeents This work was supported in part by the scholarship fro China Scholarship Council (CSC) under the Grant CSC NO The authors would like to express their sincere gratitude to the supporter. 97

6 Journal of Engineering Science and Technology Review 9 () (06) Fig.4. Result of block assignent and block transportation t sequence.. Calvo, R.W., A new heuristic for the traveling salesan proble with tie windows. Transportation Science, 34(), 000, pp Baker, E.K., Technical note an exact algorith for the tieconstrained traveling salesan proble. Operations Research, 3(5), 983, pp Bianco, L., Dell Olo, P. and Giordani, S., The Traveling Salesan Proble with Precedence Constraints and Binary Costs. Syste Modelling and Optiization,(7), 996, pp Duas, Y., Desrosiers, J., Gelinas, E. and Soloon, M.M., An optial algorith for the traveling salesan proble with tie windows. Operations research, 43(), 995, pp Joo, C.M., Lee, W.S., Koo, P.H. and Lee, K.B., Transporter scheduling for block transportation in shipbuilding. Journal of the Korea Manageent Engineers Society, (3), 006, pp Park, C. and Seo, J., A GRASP approach to transporter scheduling for ship assebly block operations anageent. European Journal of Industrial Engineering, 7(3), 03, pp Roh, M.I. and Cha, J.H., A block transportation scheduling syste considering a iniisation of travel distance without loading of and interference between ultiple transporters. International Journal of Production Research, 49(), 0, pp References 8. Ki, B.S. and Joo, C.M., Ant colony optiisation with rando selection for block transportation scheduling with heterogeneous transporters in a shipyard. International journal of production research, 50(4), 0, pp Joo, C.M. and Ki, B.S., Block transportation scheduling under delivery restriction in shipyard using eta-heuristic algoriths. Expert Systes with Applications, 4(6), 04, pp Bondy, John Adrian, and Uppaluri Siva Raachandra Murty. Graph theory with applications. Bulletin of the Aerican Matheatical Society, 90(), 976, pp Edonds, J., Matroids and the greedy algorith. Matheatical prograing. Matheatical Prograing,(), 97, pp Pentico. Assignent probles: A golden anniversary survey. European Journal of Operational Research, 76(),007,pp Bednorz, Witold. Advances in greedy algoriths, Wienna: I-Tech Education and Publishing KG, 4(6), 008, pp Tao, N., Jiang, Z. and Qu, S., Assebly block location and sequencing for flat transporters in a planar storage yard of shipyards. International Journal of Production Research, 5(4), 03, pp