AN EXACT METHOD FOR BERTH ALLOCATION AT RAW MATERIAL DOCKS

AN EXACT METHOD FOR BERTH ALLOCATION AT RAW MATERIAL DOCKS Shaohua L, a, Lxn Tang b, Jyn Lu c a Key Laboatoy of Pocess Industy Automaton, Mnsty of Educaton, Chna b Depatment of Systems Engneeng, Notheasten Unvesty, Shenyang, 0004, Chna c Busness School, Loughboough Unvesty, Lecesteshe, LE 3TU, UK Abstact: Ths pape studes a shot-tem beth allocaton poblem encounteed n the Baoshan Ion and Steel complex. A mathematcal model s developed fo the poblem to mnmze the total tadness patculaly consdeng specal ndustal chaactestcs. A lowe bound deved by pefomng a Lagangan elaxaton, along wth appopate banchng ules, s ncopoated nto a banch and bound algothm fo the beth allocaton poblem. Real data collected fom the Baoshan Ion and Steel Complex ae used to test the pefomance of the algothm. Computaton esult ndcates that the optmal beth schedulng can be obtaned fo the ndustal-szed poblem wthn an acceptable unnng tme. Copyght 2005 IFAC Keywods: Optmzaton poblem; Mathematcal models; Lnea pogammng; Boundng method; Algothms. INTRODUCTION Wth the development of moden on and steel ndusty, the opeatons at aw mateal docks ae enomously nceasng so that they become vey congested. Although constuctng moe beths can avod o decease such congeston, t s a cheape and moe feasble method to mpove the poductvty of the exstng beths. So to make an effectve beth allocaton has become a key facto to save fnancal and mpove effcency. L et al. (998) consdeed the beth allocaton poblem wth a "multple-job-onone-pocesso" patten and appled a genealzed Fst-Ft Deceasng heustc to seveal vaatons of the poblem. Chen and Lee (999) pesented a onejob-on-multple-machne model whch can be used to handle a complcated beth allocaton. Ima et al. (2003) modfed the exstng fomulaton of the beth allocaton poblem n ode to teat callng shps at vaous sevce potes. Compaed wth the lteatue n contane temnals, studes often gve moe attentons to stategc and plannng poblems n Coespondng autho. E-mal: qhjytlx@mal.neu.edu.cn the on and steel ndusty. Only vey scace studes focus on the beth allocaton poblem n such an ndusty. Suzuk et al. (996) descbed the coastwse tanspotaton plannng and admnstaton system of Kawasak steel n Japan. An expet system technology was appled to develop the subsystem povdng tanspotaton schedules by shps. Kao et al. (990) expessed the constants of the pot and the wokng ules adopted by the pots of Chna Steel Copoaton as knowledge ules and embedded them nto the famewok of the logc of dock aangement. Kao and Lee (996) egaded the medum-tem shp schedulng poblem as paallel-machne schedulng poblem. Ths s the sole one whch has fomulated a pue zeo-one ntege pogam fo the beth allocaton poblem n the on and steel ndusty. Howeve such a model does not match the chaactestcs of the beth allocaton poblem n the Baosteel. The mateal type s a detemnstc facto to a beth allocaton n the Ion and Steel ndusty fo the geat dffeence n shp types and othe facltes, fo example, conveyos and stoage yads. If Beths ae egaded as machnes n a geneal schedulng poblem, the dffeence n beths and unloades nstalled on them make t mpossble to teat a beth allocaton poblem n the on and steel ndusty as an

dentcal paallel machne poblem. Compaed wth contane temnals, docks n the on and steel ndusty have a smalle sze and dvesfed shps. In ode to mpove the poductvty of beths, moe than one beth may pocess a huge shp and one beth can be shaed by seveal small shps at the same tme. These two specal cases may occu n contane temnals and a vey few eseaches have made the studes on one of them (L, et al., 998, Chen and Lee, 999), wheeas no lteatue nclude both of them. Fom the above dscusson we can see that a beth allocaton n the on and steel ndusty s much dffeent fom that at contane temnals and by now we have not found any models and algothms that clealy fomulate and solve ths specal poblem n the on and steel ndusty. Howeve, besdes the Baosteel, aw mateal docks ae constucted and used n many lage-scale on and Steel Copoatons, fo example, Wuhan Ion and Steel Copoaton n Chna, Chna Steel Copoaton n Tawan, and Kawasak steel plant n Japan. So thee s a pactcal need fo eseaches on ths topc. In the next seveal sectons, we pesent ou model and algothm fo a beth allocaton poblem n the Baosteel. 2. PROBLEM DESCRIPTION AND MODEL 2. Poblem assumptons Lke othe studes n the beth allocaton poblem, we make the followng geneal assumptons: (a) All of the shps ae eady at the begnnng of the plannng hozon. (b) Resouces n docks ae avalable at the begnnng of the plannng hozon. (c) Evey shp must be sevced once and exactly once. (d) No peempton s allowed. Howeve, besdes the geneal chaactestcs, ou poblem has some specal featues that ae summazed as follows. () Beth allocaton has a vey close elaton wth mateals loaded on ncomng shps. (2) Moe than one shp can shae the same beth at the same tme f the total length of such shps does not exceed the quay length of the beth. (3) If the length of a shp s geate than the quay length of one beth, t can occupy seveal beths that belong to the same dock. (4) The pocessng tme of each shp s dependent on the beth t s pocessed. (5) Shps can be allocated to beths wth acceptable physcal condtons such as wate depth and quay length. The geneal and specal featues fom constants of the followng model. 2.2 Model In ode to fully expess such featues, we fst make some classfcaton of shps. Let Ω be the set of all ncomng shps dung the plannng hozon, we can dvde the set nto subsets below, accodng to the mateals loaded on these shps. Ω The set of shps that can only be pocessed at the man mateal dock; Ω 2 The set of shps that can only be pocessed at the auxlay dock; Ω 3 The set of shps that can be sevced at ethe the man mateal dock o the auxlay dock; Ω 4 The set of ol shps. Based on the above defnton, we have Ω = (,, N) =Ω Ω 2 Ω 3 Ω 4, whee N s the total numbe of ncomng shps. In ou poblem, beths do not always sevce one shp each tme. A bulk shp may occupy moe than one beth and one beth may be shaed by seveal small shps. Ths makes the poblem dffcult to be fomulated and solved f we stll use a geneal beth as ou pocesso. Theefoe one concept of beth type s ntoduced to the complcated and pactcal beth allocaton poblem n the Baosteel. Beth type s defned as follows: () Only the beths that ae adjacent wth each othe can be made up of one beth type. (2) Unloades nstalled on the beths n the same beth type ae consdeed as the esouces of ths beth type. (3) One beth type must have the same beths and unloades. (4) If a beth cannot be ncluded n any othe beth type, t s a dependent beth type. Let Ψ be the set of all of beth types n the aw mateal docks, and then we can dvde ths set nto fou subsets accodng to the stuaton of aw mateal docks n the Baosteel. Ψ The set of beth types equpped wth sngleam bdge-type shp unloades at the man mateal dock; Ψ 2 The set of beth types equpped wth doubleam bdge-type unloades at the man aw mateal dock; Ψ 3 The set of beth types equpped wth sngle-am unloades at the auxlay dock; Ψ 4 The set of beth types fo ol shps at the heavy ol dock. Thus fom the above defnton, we have Ψ = (,, M)=Ψ Ψ 2 Ψ 3 Ψ 4, whee M s the total numbe of beth types. To model the poblem, the ente plannng hozon and the total length of each beth type ae dvded nto small unts so that all the tme and length paametes, such as pocessng tmes, due dates and shp lengths, ae of ntege values. The followng addtonal symbols ae used fo defnng the poblem paametes and vaables. T Tme set, T = {,, K}, whee K s the total numbe of plannng peods n the plannng hozon; p The pocessng tme of shp on beth type j; ε The pecentage of the total length of beth type j that s occuped by shp ; m The numbe of unloades that s used to pocess shp on beth type j; d The due date of shp ; l The length of shp (ncludng the hozontal safety length); wd j The wate depth n beth type j;

d The daft depth of shp (ncludng the vetcal safety length); q The pecentage of the total length of beth type j that s avalable at tme k; cn The avalable unloade numbe of beth type j at tme k. Decson vaables: f shp occupes beth type j at tme k x k = =,, N; j =,, M; k =,, K; f shp occupes locaton p on beth type j y p = =,, N; j =,, M; p =,, bl j, whee bl j s the quay length of beth type j; f shp occupes unloade q on beth type j z q = =,, N; j =,, M; q =,, cn j, whee cn j s the numbe of unloades on beth type j; f shp occupes beth type j γ = =,, N; j =,, M. c The depatue tme of shp, =,, N; s The statng bethng locaton of shp when t s sevced on beth type j, =,, N; j =,, M; u The fst unloade allocated to shp when t s sevced on beth type j, =,, N; j =,, M. Objectve functon Mnmze the total tadness can exempt o educe expensve demuage chaged to docks, the man taget of the beth allocaton poblem, whch can be expessed as: mn max{0, c } d Resouce constants ε x q, k Τ () m x k k cn, k Τ (2) ( wd d ) γ 0 Ω (3) j Constants () ensue that the assgned beth length on each beth type must be less than ts total beth length at any tme. Constants (2) ensue that the assgned numbe of unloades on each beth type cannot be moe than the total numbe of unloades. Constants (3) state that each shp can occupy one beth type whose wate depth s deepe than the daft of the shp. Contnuty constants c x k k= c p + s + l yp p= s = p γ (4) = l γ (5) u + m zq q= u = m γ (6) Constants (4) assue that a shp must be sevced at a beth type dung an unnteupted tme peod. Constants (5) ndcate that a shp can be sevced at some beth type only when t has been bethed entely, that s to say, t can not stat dschagng f only pat of ths shp occupes the allocated beth type. Constants (6) state that a shp s pocessed by a gven fxed numbe of consecutve unloades, m, smultaneously n mateal unloadng opeaton. Beth type constants x k k Τ (7) γ = Ω (8) Ψ 2 3 4 γ = Ω (9) Ω2 γ = (0) γ = Ω () Constants (7) guaantee that each shp can occupy at most one beth type at any tme. Constants (8) guaantee that one shp can be sevced at one and only one beth type. Constants (8) and (9) ensue that the shps whch should be sevced at the man aw mateal dock must occupy the beth type at the man aw mateal dock. Constants (8) and (0) estct that the shps whch should be sevced at the auxlay aw mateal dock must occupy the beth type at the auxlay mateal dock. Constants (8) and () guaantee that the shps whch should be sevced at the heavy ol dock must occupy the beth type at the heavy ol dock. Vaable constants x k {0, }, =,, N; j =,, M; k =,, K (2) y p {0, }, =,, N; j =,, M; p =,, bl j (3) z q {0, }, =,, N; j =,, M; q =,, cn j (4) γ {0, }, =,, N; j =,, M (5) c, s, u ae ntege numbes, =,, N; j =,, M(6) 3. LOWER BOUND BASED ON LAGRANGIAN RELAXATION In ths secton we develop a Lowe bound mechansm based on Lagangan elaxaton (Fshe, 98) of esouce constants that wll contbute to ou late banch and bound algothm. The Lagangan elaxaton can be both ndependently used as a nea optmal soluton algothm (Luh, et al. 998) and teated as a method to povde lowe 4

bounds, whch has been successfully used n many poblems. 3. Model of the elaxaton poblem Relaxng constants () and (2) wth the nonnegatve Lagange multples u and v, the elaxed poblem s fomulated as follows. (LR) Mnmze Z L, wth Z L (c, k ) + max{0, c v ( k T d } + m ) ( k T k cn u ε k q ) (7) subject to constants (3)-(6), and u, v 0, j Ψ, k T. Ths poblem can be decomposed nto sub-poblems, each fo one shp. The sub-poblem fo shp, s gven below. (LR ) Mnmze Z L (c, k ) = max{0, c d } + k T v m k k T subject to (3)-(6), and u, v 0, j Ψ, k T. 3.2 Solvng the sub-poblem u ε k + (8) The sub-poblem fo shp can be solved optmally as follows: Step. Fo all the combnatons of beth type j whch can be occuped by shp and tme k, whee j Ψ, k T, pefom the followng pocedue: () Locate the shp on beth type j at tme k; (2) Evaluate the objectve value of the equaton (8). Step2. Fnd the combnaton that has the mnmal objectve value. Step3. Let the stat tme of shp equal to the tme n the combnaton that has the mnmal objectve value. 3.3 Fndng a feasble soluton The decomposed poblems ae easly nfeasble because the constants () and (2) ae elaxed. To fnd a feasble soluton, we popose a heustc fo the lagangan elaxaton poblem. The task of the heustc s to allocate appopate esouce fo each shp ncludng a contnuous beth length and consecutve unloades whch belong to the beth type selected fo t n secton 3.2. We desgn an occuped esouces queue, an avalable contnuous beth length queue and an avalable consecutve unloade queue fo each beth type to descbe the dynamc esouces allocaton pocess. Each of the thee queues s soted n non-deceasng o nceasng ode of the due dates, stat beth locatons and stat unloade numbes, espectvely. In ode to make full use of esouces at each beth type, the fst appopate contnuous beth length and consecutve unloades n the coespondng queues ae chosen. 3.4 Update Lagangan multples We use a subgadent method to obtan the values of the Lagangan multples {u, v }. Let u and v be the multples at teaton, and let Z UB, updated by the heustc algothm n secton 3.3, denote the uppe bound on the mnmum value of the total tadness tme. Afte the elaxaton poblem s solved by the method n secton 3.2, the obtaned soluton Z LB s the lowe bound on the optmal objectve functon. λ s a step length and ntally set to be 2. If fve consecutve teatons fal to mpove the lowe bound, λ wll be havled. Then the multples can be detemned by the followng ecusve fomulaton: u v + = Max{0, u + λ ( Z j =,..., M, k =,, K + = Max{0, v j =,..., M, k =,, K Whee B A = = + λ k ( Z UB LB Z ) A 2 A k K UB Z k K LB B ) B 2 } } (9) (20) ε q, j =,..., M, k =,, K; m cn, j =,..., M, k =,, K. k 4. BRANCH AND BOUND ALGORITHM In ths secton we gve ou banch and bound algothm to obtan an optmal soluton fo the pesented beth allocaton poblem. Specal attenton s gven to the descpton of ou banchng ules that ae elated to the specal chaactestcs of aw mateal docks n the Baosteel and the pocess that ae used to pune the seach tee. Befoe we constuct a seach tee fo the stated poblem, some defntons and ules ae necessay. The seach tee contans two types of nodes, each of whch coesponds wth a patal beth allocaton. If the node s a ccle node, shp j s assgned on the same beth type wth ts fathe node; howeve, f the node s a squae node, shp j s assgned on a new beth type. The followng ae the necessay ules fo the algothm to develop a seach tee fo the consdeed poblem. Rule. At level, thee ae at most N squae nodes whch epesent the cuent beth type s and no ccle nodes can occu n such a level. Rule 2. At othe levels, thee ae at most 2*(N l + ) nodes whch may be squae nodes o ccle nodes. Rule 3. In each banch on the tee, the numbe of the squae nodes must less than o equal to the total

numbe of beth types. Rule 4. One shp can appea exactly once n each banch. Rule 5. At level l, f the last shp n the patal sequence of the cuent actve node s sevced at beth type j, then only those shps whch can occupy beth type j o beth type j+ may geneate both o ethe of one ccle node and one squae node. If thee ae no such shps, ths node s elmnated. Rule 6. Befoe a squae node can be geneated, all of the shps, whose maxmum beth type that can pocess them s smalle than the beth type that s epesented by ths squae node, must have been aanged befoe. All of the above sx ules ae used n ou banch and bound algothm to elmnate the nodes that can not mpove the value of the total tadness. Ou banch and bound has the smla pocedue as the one by Lu and MacCathy (99), n whch a depth-fst seach stategy s used and the mmedate descendants of the actve node ae exploed n nondeceasng ode of the lowe bounds. If the lowe bound at one node s lage than o equal to the mnmal objectve value obtaned by all of the feasble solutons havng been found, ths node and all of ts bothes ae elmnated. 5. COMPUTATION EXPERIMENTS To test the pefomance of the algothm and study the chaactestcs of solutons, computatonal expement has been caed out wth eal poblem nstances, whch wee collected fom the aw mateal management cente n the Baosteel. 5. The selecton of expement data Accodng to the defnton of beth types, aw mateal docks n the Baosteel can be classfed nto fou beth types, whch have been shown n fgue. Shps allocated to beth type must be sevced on the ve sde of the dock. Because the unloades n beth type 2 ae double-am, shps allocated to ths type can be sevced on ethe sde of the man aw mateal dock. Shps allocated to beth type 3 must be dschaged on the ve sde of the auxlay aw mateal dock. Shps allocated to beth type 4 must cay out the dschagng opeaton at heavy ol dock locatng on the land sde of the auxlay aw mateal dock. In the Baosteel, ncomng shps ae dected by the aw mateal day plannng to pull nto docks. Although the day plannng s made once eveyday, t s elated to all about ten ncomng shps on the aveage wthn a thee-day hozon. The pobabltes of the shp type beng oe, auxlay mateals, coal, and heavy ol ae 38.8%, 9.0%, 35.8%, and 6.4% espectvely. Snce the mnmal tme unt of the eal beth plannng n the Baosteel s one hou, the plannng hozon T s selected as 72 fo a thee-day hozon. We select ten examples fom the eal day beth plannng to test the pefomance of the pesented banch and bound algothm. Shp numbe vaes fom 8 to 4 so that they can nclude dffeent dock stuatons. 5.2 Computaton esults In the followng we select an example wth 4 shps to llustate the computaton esult. The esultng beth allocaton s dawn n fgues 2 and 3, n whch fgue 2 shows the stuaton n the man aw mateal dock and fgue 3 shows the one n auxlay aw mateal dock and heavy ol dock. In fgue 2 and 3, the numec value at the left cone of each ectangle ndcates a shp. Numbes enclosed by ccles denote unloades allocated to each shp. And the two ectangles flled wth dots n fgue 3 epesent the tady shps. Table summazes the esults fo all the ten examples. Wthn the tme hozon of thee days, the aveage total numbe of shps s.. Fg.. Beth types n the Baosteel Fg. 2. Beth allocaton n the aw mateal dock

Lagangan Relaxaton of esouce capacty constants s ncopoated nto a banch and bound algothm, along wth new banchng ules, to solve ths beth allocaton poblem. Computaton expement on ten eal beth allocaton poblems n the Baosteel ndcates that the banch and bound algothm can obtan an optmal soluton wthn a easonable tme. ACKNOWLEDGEMENTS Fg. 3. Beth allocaton n auxlay mateal dock and heavy ol dock Table. Ten examples of Baosteel day mateal plannng Poblems Shps Tadness Tadness Runnng numbe tme (s) 8 0 0 0.92 2 9 0 0 2.97 3 0 7 3 59.0 4 0 7 2 6.44 5 0 7 2 5.66 6 2 2 5.25 7 2 3 3 23.7 8 3 8 4 8.3 9 4 5 2 264.23 0 4 2 2 443.38 Fom the data shown n table, we can conclude that: () Wth the numbe of shps nceasng, tadness tme and tadness numbe ae nceasng. Ths can be explaned that the esouce s elatvely scace when the ncomng shps ae congested on one day. Ths makes the beth allocaton moe mpotant to mnmze the tadness and educe demuage cost. (2) Due date has a geat mpact on the unnng tme of the banch and bound algothm. Ths may be because tadness s lage when due date s small so that the objectve value s lage. Ths esults n lage unnng tme as the dffeence of uppe bound and lowe bound s lage. (3) Optmal solutons can be obtaned fo eal theeday beth allocaton poblems n the Baosteel n an acceptable unnng tme. And such a soluton sze s n accodance wth the shp numbe n a geneal day beth plannng of the Baosteel. 6. CONCLUSIONS Beth allocaton s a key facto to mpove the poductvty of the docks. Unlke pevous studes usng Expet System to beth allocaton, ths pape fomulates a novel ntege pogammng model that fully expesses the chaactestcs of eal beth allocaton poblem n the Shangha Baosteel Complex. A lowe bound scheme based on Ths eseach s patly suppoted by Natonal Natual Scence Foundaton fo Dstngushed Young Scholas of Chna (Gant No. 70425003), Natonal Natual Scence Foundaton of Chna (Gant No. 60274049) and (Gant No. 707030), Fok Yng Tung Educaton Foundaton and the Excellent Young Faculty Pogam of the Mnsty of Educaton, Chna. The authos would lke to thank aw mateals cente n Baosteel fo povdng a lot of poducton nfomaton and data. REFERENCES Chen J, Lee C-Y. (999). Geneal Multpocesso Task Schedulng. Naval Reseach Logstcs, 46, 57-74. Fshe ML (98). Lagangan elaxaton method fo solvng ntege pogammng. Management Scence, 27, -8. Ima A, Nshmua E, Papadmtou S (2003). Beth allocaton wth sevce poty. Tanspotaton Reseach Pat B, 37, 437-457. Kao C, L D-C, Wu C, Tsa, C-C(990). Knowledgebased appoach to the optmal dock aangement. Intenatonal Jounal of Systems Scence, 2, 2209-225. Kao C, Lee HT (996). Dscete tme paallelmachne schedulng: a case of shp schedulng. Engneeng Optmzaton, 26, 287-294. L C-L, Ca X, Lee, C-Y (998). Schedulng wth multple-job-on-one-pocesso patten. IIE Tansactons, 30, 433-445. Lu J, MacCathy BL (99). Effectve heustcs fo sngle machne sequencng poblem wth eady tmes. Intenatonal Jounal of Poducton Reseach, 29, 52-533. Luh PB, Gou L, Zhang Y, Nagahoa T, Tsuj M, Yoneda K, Hasegawa T, Kyoya Y, Kano T (998). Job shop schedulng wth goupdependent setups, fnte buffes and long tme hozon. Annals of Opeatons Reseach, 76, 233-259. Suzuk K, Taka Y, Hyama T (996). Coastwse tanspotaton plannng and admnstaton system. Kawasak steel techncal epot, 33, 30-37.