GROUP-BAY STOWAGE PLANNING PROBLEM FOR CONTAINER SHIP

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1 POLISH MARITIME RESEARCH Specal Issue 2016 S1 (91) 2016 Vol. 23; pp /pomr GROUP-BAY STOWAGE PLANNING PROBLEM FOR CONTAINER SHIP Shen Yfan a Zhao Nng b M Wejan b a Scentfc Research Academy, Shangha Martme Unversty, , Shangha, CHINA b Logstcs Engneerng College, Shangha Martme Unversty, , Shangha, CHINA ABSTRACT Stowage plannng s the core of shp plannng. It drectly nfluences the seaworthness of contaner shp and the handlng effcency of contaner termnal. As the latter step of contaner shp stowage plan, termnal stowage plannng optmzes termnal cost accordng to pre-plan. Group-Bay stowage plannng s the smallest sub problem of termnal stowage plannng problem. A group-bay stowage plannng model s formulated to mnmze relocaton, crane movement and target weght gap satsfyng both shp owner and contaner termnal. A GA-A* hybrd algorthm s desgned to solve ths problem. Numercal experment shown the valdty and the effcency. Keywords: Contaner Termnal, Group-Bay Stowage Plan, Termnal Stowage Plan, Hybrd GA INTRODUCTION Despte the economc contracton of recent years, the contaner trade volume has been sustanng expandng. To meet the expandng demand of martme transportaton, contaner termnals has been focusng on management and decson evoluton to enlarge ther operaton capablty. Now, the research on the contaner termnal manly focuses on two aspects, equpment automaton and ntellgent decsonmakng. There are qute a few scholars who have carred out some research work on these two ponts[3-4, 9-11]. In ntellgent decson, contaner stowage plan has always been a major problem n termnal decson problems. As a core phase of shp plannng, stowage plannng affects equpment effcency and energy consumpton to a large extent. Snce 1970s, stowage plannng has been modeled and analyzed by experts and scholars from many scopes, such as decson makng systems, mathematcal modelng, smulaton, ntellgent algorthms. H. P Wang[5] desgned an automated stowage expert system wth knowledge nference technques, and mplementaton prncples are provded. Kang et al and Km et al[6-7] solved stowage plannng problem wth greedy algorthm and tree search algorthm. Ambrosno and Scomachen[8] proposed a decson support system to solve Master Bay Plan Problem(MBPP), feasble soluton was found by constrant satsfacton method, but objectve functon and optmal soluton was not provded. Wnter et al[12] ntroduced a stowage plannng problem combnng loadng plan, and a decson support system was proposed to optmze the balance of quae cranes. Cho et al[13] and Botter[14] proposed a LP model for contaner shp stowage plannng problem, some assumptons was made to smplfy the problem. The plannng result shows a weak compatblty to actual operaton process. 152 POLISH MARITIME RESEARCH, No S1/2016

2 Avrel et al[15-16] proposed a 0-1 nteger programmng model, a heurstc method was desgned to solve ths problem. Numercal experment wth actual data shows an acceptable soluton qualty and computng effcency. But ths heurstc method s not wthout flaws, t can only solve smplfed stowage plannng problem and lacks flexblty. Ambrosno et al[17] proposed a model mnmzng total loadng tme, wth some operatonal constrants such as contaner type and load lmt. The soluton allocates contaners wth same port of dscharge nto same bay to avod unnecessary restow. Scomachen and Tanfan[18] solved stowage plannng problem wth 3D-BPP, optmzng total loadng tme and quae crane utlzaton. Pacno et al[19] and Delgado et al[20] solved stowage plannng problem wth two steps. The frst step separates contaners nto dfferent categores based on Master Bay Plan, and the second step decdes the specfc stowage locaton for each contaner. These researches focuses manly on optmzng shp stablty, equpment utlzaton and total loadng tme. Other mpact factors such as reshufflng operaton and yard crane shftng are gnored. In practcal applcaton, stowage plannng s a two-step process. In the frst step, shp stablty constrant s clamed and solved by shp lne, and the result of the frst step so called preplan s the nput and constrant of the second step. Ths research focuses on the second step, whch optmzes costs of contaner termnal, on condton that the constrant of the frst step s results are satsfed. Formal researches about the second step often scales down the whole problem nto shp bay stowage plan, for the ndependence of dfferent bays of shp n stowage plannng process. In practce, contaners of the same bay and wth the same key feature(contaner type, contaner sze, port of dscharge) are called a contaner group. In bay-wse stowage plannng, decsons of dfferent contaner groups are relatve ndependent, thus a bay-wse stowage plannng can be separated nto several sub-bay-wse or so called group-bay plannng problem. Therefore, a group-bay stowage plannng model optmzng key cost factors of contaner termnal consderng preplan constrants and operaton constrants s proposed to solve stowage plan problem. A hybrd A*-genetc algorthm s ntroduced to solve ths problem. The stowage locaton s solved by the genetc phase and the loadng sequence s updated by A*. Numercal experments verfes the valdty of the algorthm. Ths model and algorthm can be mplemented to make stowage plans automatcally. phase optmzes specfc slots and loadng sequences for dedcated contaners from the contaner termnal s pont of vew. Constrants are the result of the frst step, and some operaton constrants. Group-bay stowage plannng s a part of the second step. DEFINITION OF GROUP-BAY STOWAGE PLANNING A contaner group s a set of contaners wth same POD, contaner type, sze, status and heght. As s shown n Fg.1, the pre-plan loadng postons of contaner group A are dstrbuted n to shp bays, thus ths contaner group A can be subdvded nto two sets 02H-A and 06H-A. These subsets called group-bays are the smallest decson unt n the termnal plannng problem. In groupbay stowage plannng problem, constrants are stablty of the shp reflected n pre-plan(manly the weght dstrbuton constrants) and operaton constrants. Fg.1 Group-bays of a pre-plan OPTIMIZATION OBJECTIVES IN GROUP-BAY STOWAGE PLANNING (1) Reshuffles: In the loadng process, f the prevous loadng contaner locates rght under the subsequent loadng contaner, the subsequent loadng contaner must be removed from ts row frst to operate the prevous loadng contaner, ths s called reshuffle. GROUP-BAY STOWAGE PLANNING PROBLEM ANALYSIS INTRODUCTION OF STOWAGE PLANNING PROBLEM Plannng a shp s stowage s a two-step process[1-2]. The frst step s executed by the shppng lne, so called pre-plan. Ths pre-plannng phase optmzes general loadng postons for each contaner group (contaners wth same POD, contaner type, sze, status and heght) from the shppng lne s pont of vew to mnmze the shp s utlzaton. Constrants contans manly the stablty of the shp. The second step s executed by contaner termnals, so called termnal-plan. Ths plannng Fg.2 Reshuffles n yard In Fg.2, two contaners are located n the same row of a yard bay. The contaner wth sequence 19 s located rght under the contaner wth sequence 23. When the yard crane carres contaners n sequence, the 19th contaner should be transferred before the 23th contaner, and at that tme, the 23th contaner must be reshuffled to another avalable locaton for the yard crane to get 19th contaner. Reshuffles cost tme POLISH MARITIME RESEARCH, No S1/

3 and resources. (2) Yard crane shfts: refers to yard crane movements between yard bays. A yard crane move from bay A to bay B s a shft. Contaners of same shp bay often locate n dfferent yard bays, the loadng sequence must be optmzed to reduce yard crane shfts. Fg.3 shows that an nferor loadng sequence can cause frequent shfts, whch costs tme and resources. Fg.3 Yard crane shfts between dfferent loadng sequence (3) Shp stablty: In pre-plan optmzaton, contaner weght s a key decson factor. In termnal stowage plannng, the fnal weght dstrbuton must be as close as possble as the optmal weght dstrbuton to ensure a feasble shp stablty and seaworthness. GROUP-BAY STOWAGE PLANNING MODEL Group-Bay stowage plannng problem s a core and dffcult plannng problem of all plannng n contaner termnals. Ths problem s a mult-objectve combnatoral optmzaton problem wth complex constrants. The major decson context s the specfc loadng locaton of each contaner and loadng sequences of these contaners. The constrants manly embodes operaton restrctons and shp stablty control. MODEL ASSUMPTIONS 1. The export contaners to be stowed are no less than shp slots to be allocated; 2. The specfc locaton of each contaner to be stowed s known; 3. The type of each contaner to be stowed s the same as the shp slots to be allocated; 4. Equpment falure s gnored; 5. Enough horzontal transportaton trucks avalable; 6. The quae crane handles only one contaner per move. SYMBOL DEFINITION (1) Dmensons I : set of all contaners to be stowed,, ' I; J : set of all shp slots to be allocated, j, j' J ; S : set of all loadng sequences, s S ; R : set of rows of shp slots to be allocated, rr, ' R; T : set of ters of shp slots to be allocated, t T ; (2) Parameters A jr : Whether shp slot j s n shp row r, equals 1 f shp slot j s n shp row r, otherwse 0; GP : Whether contaner s a GP contaner, equals 1 f contaner s a GP contaner, otherwse 0; HC : Whether contaner s a HC contaner, equals 1 f contaner s a HC contaner, otherwse 0; GP r : Number of GP contaners allowed n shp row r; HC r : Number of HC contaners allowed n shp row r; W j : Mnmum contaner weght of ter of shp slot j accordng to pre-allocaton accordng to pre-plan; M j : Maxmum contaner weght of ter of shp slot j accordng to pre-allocaton accordng to pre-plan; w : Weght of contaner ; δ : Maxmum weght dfference of heaver contaners above lghter contaners allowed n each shp row; Q s : Matrx of seral numbers of loadng sequences; YP ' : 0-1 Matrx,relatve locaton of contaner and, equals 1 f locates rght above n yard, otherwse 0; VP jj 0-1 Matrx,relatve locaton of shp slots j and j, equals 1 f j locates rght above j n yard, otherwse 0; YB : Yard bay of contaner ; F j : Pre-allocaton weght of shp slot j accordng to pre-plan; (3) Decson varables X :0-1 varable, whether contaner s allocated to shp slot j wth loadng sequence s, 1 for true, 0 for false; β :0-1 varable, relatve loadng sequence relaton of contaner and, equals 0 f contaner s to be loaded earler than, otherwse 1; φ s : 0-1 varable, whether contaners wth consecutve loadng sequence locates at the same yard bay, 1 for true, 0 for false; OBJECTIVE FUNCTION (1) Mnmze reshuffles Expresson 3-1 s the relatve loadng sequence of contaner and, f s to be loaded earler than then β equals 0, otherwse 1. 0, XQs Xs jqs < 0 β ' = 1, XQs Xs jqs > 0 Thus the number of reshuffles s shown n expresson 3-2: mn Z1 YP ' β ', I = (2) Mnmze yard crane shfts Expresson 3-3 s whether the yard crane needs to shft from contaner s-1 to s, 1 for true, 0 for false POLISH MARITIME RESEARCH, No S1/2016

4 0, X YB X ( s 1) j YB = 0 I j J I j J φs = 1, X YB X( s 1) j YB 0 I j J I j J Thus, mnmzng total yard crane shfts can be expressed as: mn Z = φs 3-4 s S (3) Mnmzng total weght gap between planned contaners and optmal weght dstrbuton 3-5 mn Z3 = X w Fj j J (4) Overall objectve These three sub objects are all cost objects, the overall objectve should cover the nteracton of sub objects. Accordng to practcal handng cost measures, these sub objects can be treated as costs wthout normalzng. After consulted stowage plannng stuff of Nngbo Daxe Contaner Termnal, a weghted overall functon s gven as follow: CONSTRAINTS Z =0.5Z + 0.2Z + 0.3Z 3-6 total (1) Unqueness constrant A contaner can only be allocated to a shp slot wth only one loadng sequence: 3-12 W X w M j j (4) Maxmum weght dfference of heaver contaners above lghter contaners lmt To ensure the stablty of the shp, the contaners located above others should be lghter or a lttle bt heaver than the beneath contaner. Ths lmtaton can be expressed as: 3-13 ( X w X w ) VP δ ' jj ' (5) Handlng sequence constrant The lower shp slots should be loaded before the upper shp slots, whch means a contaner cannot be loaded to a shp slot above a not yet loaded shp slot. XQ s ( XQs ) VPjj ' 3-14 j' J ALGORITHM FOR GROUP-BAY STOWAGE PLANNING INTRODUCTION OF HYBRID ALGORITHM A hybrd algorthm combnng GA and A* s proposed to solve ths plannng problem. In ths algorthm, the genetc part s dedcated to solve the allocaton of contaners to shp slots, and the A* part solves the most feasble loadng sequence of these contaners to ths allocaton. The work flow of ths algorthm s shown n Fg.4. X = X = X = I j J (2) GP and HC constrant In shp handlng process, GP and HC contaners are mxed n a shp row, whch means n stowage plannng, GP and HC contaners are only restrcted n numbers n quantty n a shp row, not n specfc shp slots. HC quantty restrcton of a shp row can be expressed as: X Ajr HC = HCr 3-10 j J HC quantty restrcton of a shp row can be expressed as: X AjrGP = GPr 3-11 j J (3) Slot Weght lmt Accordng to optmal weght dstrbuton based on pre-plan, slot weght lmt s appled wth a certan extent of relaxaton. Contaner allocated to a slot must weghts n ths rang: Fg.4 Work flow of hybrd algortm POLISH MARITIME RESEARCH, No S1/

5 CHROMOSOME DESIGN Fg.5 shows a Chromosome of the genetc part, a gene s a shp slot for a contaner. A* FOR LOADING SEQUENCE ADJUSTMENT. Ths A* part adjusts loadng sequence to mnmze yard crane shfts after the genetc operators allocates the slots. ( ) ( ) ( ) Z = g + h SELECTION Fg.5 Example of a chromosome GA part selects ndvduals to breed new generaton wth Roulette, the probablty of ndvduals to be chosen to the new generaton s calculated by ftness-based functon: p f f = 1 n = (15) p s the probablty that s chosen to the new generaton, f s the ftness of, n s the populaton quantty. GENETIC OPERATORS Fg.6(a) shows that an order crossover(ox) operator s appled to make sure the generated chromosome has no shp slot repeat conflcts, but the GP and HC quantty may be nfeasble. Therefore the GP and HC quantty constrant s converted nto a penalty functon to elmnate nfeasble chromosomes durng selecton process. Fg.6(b) s the mutaton operator. Ths operator selects two dfferent genes of one chromosome and swtch them. These genetc operators adjusts only the chromosome of slot allocaton, not the loadng sequence. The loadng sequence s optmzed trough the A* part. Expresson 4-16 shows that the cost functon conssts of two parts: g2 ( ) s the actual cost from start node to loadng contaner, whch s total yard crane shfts untl loadng contaner ; h2 ( ) s the estmated mnmum yard crane shfts after contaner s loaded tll the end of the loadng, whch s the total number of yard bays who have contaners to be loaded after contaner. Fg.7 shows the expanson of chld node of A*, a constrant branch prunng s ntroduced to elmnate nfeasble soluton. In the searchng process, these gray slots are slots that can be loaded accordng to handlng sequence constrant, whch means only these gray slots are feasble chld nodes. Ths A* part expands feasble nodes from start node and calculates the cost functon, and selects the node wth the least cost functon as the current node to expand, and the fnal loadng sequence s get through teratons. After loadng sequence searchng for each chromosome, the ftness functon s calculated accordng to the slot allocaton and the loadng sequence to get the selecton probablty of each chromosome. Fg.7 A* expanson and searchng process NUMERICAL EXPERIMENT In ths part, a numercal example from Nngbo Daxe Contaner Termnal s chosen to valdate the feasblty of the proposed model and algorthm. NUMERICAL EXAMPLE Fg.6 Genetc operators Ths numercal s 30H bay of a contaner shp. The 42 contaners to be stowed dstrbutes n 6 yard bays. Fg.8 shows the pre-plan of ths group-bay, blue slots are to be allocated to. Fg.9 shows the contaners and ther yard locatons and weght, yellow contaners are to be allocated. The experment platform s a computer wth a dual core 2.50GHz Intel Core 156 POLISH MARITIME RESEARCH, No S1/2016

6 7 CPU, 4.00GB RAM and a Wn7 64bt OS. RESULTS AND ANALYSIS (1) Preset of Parameters The parameter set s n Table.1: Tab.1 Parameter set of GA Populaton Iteratons Crossover Prob. Mutaton Prob Fg.8 Pre-plan of bay 30H (2) Valdaton of proposed algorthm The stowage plan of proposed algorthm s n Fg.10. Yard crane shfts n yard block 2A accordng to loadng sequence s Total reshuffle s 0, total shfts are 6, wth only one shft back from yard bay 32 to yard bay 62. And all allocated contaners satsfes the weght constrant and maxmum weght dfference of heaver contaners above lghter contaners constrant. (3) Convergence of proposed algorthm The convergence curve s Fg.11: Fg.9 Yard dstrbuton of contaners Fg.11 Convergence curve of proposed algorthm Fg.10 Stowage plannng result by proposed algorthm POLISH MARITIME RESEARCH, No S1/

7 As s shown n Fg.11, proposed algorthm converged to a near-optmal soluton, the curve went flat afterwards, whch shows an acceptable convergence. To go a step further, the algorthm s run 30 tmes. Table.2 s the result. Iteratons to get near-optmal soluton s the convergence speed. NO. Iteratons to get near-optmal soluton Tab.2 30 run tme result of proposed algorthm NO. Iteratons to get near-optmal soluton NO. Iteratons to get nearoptmal soluton For 30 ndependent experments, the max teraton to get near-optmal soluton s 595, the mn teraton s 289, and the average s 448. Ths result shows a relatvely hgh convergence effcency, and an acceptable stablty. For all numercal examples tested, the proposed algorthm converges n 600 teratons, the parameter of 1000 total teratons s feasble, and the convergence of proposed algorthm s valdated. (4) Effectveness of proposed algorthm To verfy effectveness of proposed algorthm, a standard GA s used as control. The two algorthms use both 1000 teratons. Average teratons to get near-optmal soluton, average ftness value, standard devaton of ftness and average soluton tme are lsted n Table.3. Proposed hybrd algorthm Standard GA Tab.3 Comparson of proposed algorthm and GA Average teratons to get near-optmal soluton Average ftness value Standard devaton of ftness Average soluton tme(s) Table.3 shows that proposed algorthm has advantage over standard GA n all four measures, although both algorthms can get near-optmal soluton n gven teratons, and ths shows the performance of the proposed algorthm. Standard GA gets near-optmal soluton n average 729 teratons aganst hybrd algorthm s 448. Ths verfes the convergence speed superorty of proposed hybrd algorthm. The average value shows that proposed algorthm gets a much more global optmzed result. The standard devaton of ftness advantage of proposed algorthm shows the stablty advantage of hybrd algorthm over GA. And the hybrd algorthm has a 21.37s advantage over GA n average soluton tme, whch counts as has a better soluton effcency. Wth better result and less tme consumpton, the propose algorthm can be consdered to be a feasble algorthm for solvng Group-bay stowage plannng problem. CONCLUSIONS Ths study developed stowage plannng problem n three major aspects: (1) Defnes group-bay stowage plannng problem whch s the smallest plannng sub problem n the termnal plannng problem. (2) A MIP model of group-bay stowage plannng problem optmzng yard reshuffles, yard crane shfts and weght dstrbuton s ntroduced wth operatonal prncples. (3) A GA-A* hybrd algorthm s proposed to solve both the stowage locaton allocaton and loadng sequencng of the group-bay stowage plannng problem. (4) Numercal experment shows the convergence and effectveness of the proposed algorthm n solvng group-bay stowage plannng problem. In the synthess, group-bay stowage plannng problem can be solved through proposed algorthm, whch provdes a feasble soluton and an aspect of vew to the termnal stowage plannng problem. REFERENCES 1. Steenken, D., Voß, S., & Stahlbock, R. Contaner termnal operaton and operatons research A classfcaton and lterature revew. OR Spectrum, 26(1), 2004, Alvarez, J. Ambrosno, D., Anghnolf, D., Paolucc, M A heurstc for vessel plannng n a reach stacker termnal[j]. Journal of Martme Research, III(1), 2006, M Chao, Zhang Zhwe, Huang Youfang, Shen Yang, A FAST AUTOMATED VISION SYSTEM FOR CONTAINER CORNER CASTING RECOGNITION,Journal of Marne Scence and Technology-Tawan, 2016, 24(1): DOI: /JMST M Chao, Shen Yang, M Wejan, Huang Youfang, Shp Identfcaton Algorthm Based on 3D Pont Cloud for Automated Shp Loaders, Journal of Coastal Research, 2015, 2015(SI.73): DOI: /SI POLISH MARITIME RESEARCH, No S1/2016

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