Modeling and solution for the ship stowage planning problem of coils in the steel industry

Transcription

1 Loughborough Uiversity Istitutioal Repository Modelig ad solutio for the ship stowage plaig problem of coils i the steel idustry This item was submitted to Loughborough Uiversity's Istitutioal Repository by the/a author. Citatio: TANG, L....et al., 05. Modelig ad solutio for the ship stowage plaig problem of coils i the steel idustry. Naval Research Logistics, 6(7), pp Additioal Iformatio: This is the peer reviewed versio of the followig article: TANG, L....et al., 05. Modelig ad solutio for the ship stowage plaig problem of coils i the steel idustry. Naval Research Logistics, 6(7), pp , which has bee published i fial form at This article may be used for ocommercial purposes i accordace with Wiley Terms ad Coditios for Self-Archivig. Metadata Record: Versio: Accepted for publicatio Publisher: c Wiley Rights: This work is made available accordig to the coditios of the Creative Commos Attributio-NoCommercial-NoDerivatives 4.0 Iteratioal (CC BY-NC-ND 4.0) licece. Full details of this licece are available at: Please cite the published versio.

2 Modelig ad solutio for the ship stowage plaig problem of coils i steel idustry Lixi Tag *, Jiyi Liu, Fei Yag, Feg Li, Ku Li The Istitute of Idustrial Egieerig ad Logistics Optimizatio, Liaoig Key Laboratory of Maufacturig System ad Logistics, Northeaster Uiversity, Sheyag, Chia School of Busiess ad Ecoomics, Loughborough Uiversity, Leicestershire LE 3TU, UK This paper has bee published i Naval Research Logistics. It ca be cited as Tag, L., Liu, J., Yag, F., Li, F. ad Li, K. (05) Modelig ad solutio for the ship stowage plaig problem of coils i the steel idustry. Naval Research Logistics, 6(7), pp DOI: 0.00/av.664 * Correspodig author. Tel: ; Fax: ; qhjytlx@mail.eu.edu.c

3 Abstract: We cosider a ship stowage plaig problem (SSPP) where steel coils with kow destiatio ports are to be loaded oto a ship. The coils are to be stowed o the ship i rows. Due to their heavy weight ad cylidrical shape, coils ca be stowed i at most two levels. Differet from stowage problems i previous studies, i this problem there are o fixed positios o the ship for the coils due to their differet sizes. At a destiatio port, if a coil to be uloaded is ot at a top positio, those blockig it eed to be shuffled. stability of ship has to be maitaied after uloadig at each destiatio port. I additio, the The objective for the stowage plaig problem is to miimize a combiatio of ship istability throughout the etire voyage, the shuffles eeded for uloadig at the destiatio ports, ad the dispersio of coils to be uloaded at the same destiatio port. We formulate the problem as a ovel mixed iteger liear programmig model. Several valid iequalities are derived to help reducig solutio time. A tabu search (TS) algorithm is developed for the problem with the iitial solutio geerated usig a costructio heuristic. To evaluate the proposed TS algorithm, umerical experimets are carried out o problem istaces of three differet scales by comparig it with a model-based decompositio heuristic, the classic tabu search algorithm, the particle swarm optimizatio algorithm, ad the maual method used i practice. The results show that for small problems, the proposed algorithm ca geerate optimal solutios. For medium ad large practical problems, the proposed algorithm outperforms other methods. Key Words: stowage plaig, steel coils, iteger programmig, tabu search.

4 . Itroductio Steel products are widely used i costructio ad i producig other products such as automobiles. Due to the large volume ad heavy toage, steel products are ofte trasported by ship wherever possible. From oe of the steel plats of Baosteel where this study is coducted, over three millio tos of steel coils are trasported aually to customers at differet destiatio ports, while sevety percet by ship. Figure shows a ship beig loaded with steel coils. At a destiatio, if a coil to be uloaded is ot o top, those above them must be shuffled. Moreover, the stability of the ship has to be maitaied throughout the whole voyage. To keep ship stability ad to avoid uecessary shufflig, effective ship stowage plaig is critical. Figure Steel coils are beig loaded oto a ship For each arrivig ship, the steel compay eeds to make a loadig pla. The pla is made i two phases. The first phase, called the ship cosolidatio plaig problem (SCPP), is to select steel coils to be loaded oto the ship cosiderig the destiatios of the ship, the quatities ad due dates of customer orders, ad the positios of the coils i the storage yard. The secod phase is to pla the stowage of the coils o the ship, called the ship stowage 3

5 plaig problem (SSPP). I this paper, we study the secod-phase problem, i.e., the ship stowage plaig problem. The study is coducted for a steel factory which ows a port termial to coduct product delivery. The loadig capacities of the smallest ad the biggest ships are about 00 tos ad 000 tos, respectively. The average weight of a coil is about 6 tos. Therefore, the umber of coils that oe ship ca carry rages from to 0. Meawhile, the most commo ships have loadig capacities that rage from 600 tos to 00 tos. Give a set of steel coils to be loaded oto a ship, the problem is to determie the locatio of each coil o the ship, cosiderig the sizes of the coils, stability of the ship ad coveiece of uloadig at the destiatio ports. I curret practice, a rough stowage pla is made maually based o the plaer s experieces. Due to the large problem size i practice, such a maual pla ofte results i the eed for a large umber of shuffles at destiatio ports. A more effective stowage plaig method is eeded. There has bee little research o the ship stowage plaig problem for steel coils. Our literature survey oly foud oe previous study o the problem. Umeda et al. [9] studied a ship stowage plaig problem of steel products icludig coils. The problem was cosidered as a three-dimesioal allocatio problem ad solved usig simulated aealig. I determiig the stowig-positios of the products, the weight balace of the ship, the loadig ratio ad the work efficiecy of stowig were cosidered. The ship cosidered i the study had fixed slots for stowig the products. I practice, the sizes of coils are differet ad there are o fixed positios o the ship. I our problem, the loadig positios o the ship are ufixed ad the coordiates of coils o the ship eed to be determied. The stackig method i coil yards ad warehouses are similar to that for coil stowage o ships. Tag et al. [8] studied a coil shufflig problem (CSP) i a warehouse served by a crae. Cosiderig the practical stackig ad shufflig features of CSP, a liear iteger programmig model was formulated. Miimizig the logistics cost for shufflig operatios is take as the objective. Jag ad Kim [9] studied a pyramid stackig system with multiple groups of cylidrical uits i the warehouse. They established a mathematical model to optimize the re-hadlig cost ad the space cost of the system i three differet cases. I their istaces, the maximum umber of tiers exceeds. 4 However, these problems do ot

6 have ship stability issue, sice the coils are stacked o the groud. Most other papers focus o stowage plaig o cotaier ship ad stackig problems i cotaier termial yards. For example, Avriel et al. [] studied a stowage plaig problem to miimize the umber of cotaier shifts eeded for uloadig them at destiatio port. Some costraits such as stability of the ship were ot cosidered. Ambrosio et al. [] addressed a stowage plaig problem cosiderig costraits such as cotaier dimesios, weights ad destiatios as well as the balace of the ship. The objective was to miimize the total stowage time accordig to the positios of stowed cotaiers o board. The problem was formulated as a biary iteger programmig model ad solved usig a three stage approach. Through preprocessig ad prestowage i the first two stages, the search space was reduced. By further relaxig some costraits the reduced model was the solved i the fial stage. Sciomache ad Tafai [4] related the stowage plaig problem with the three-dimesioal bi packig problem. They cosidered the cotaier stowage ad quay crae assigmet together ad proposed a heuristic solutio method to optimize crae productivity. Imai et al. [8] determied the cotaier stowage ad loadig plas of a ship cosiderig stackig cofiguratios of cotaiers i the yard to maximize ship stability ad miimize the umber of cotaier shuffles durig the loadig process. A geetic algorithm was proposed to solve the problem ad to obtai a set of o-domiated solutios. Álvarez [3] proposed a approach usig tabu search ad multi-start techiques to geerate vessel loadig plas i reach-stacker based termials, cosiderig the vessel stowage requiremets ad the cotaier stackig iformatio i the yard as iputs. Tag et al. [5] ivestigated the cotaier stackig ad reshufflig issues i a cotaier termial. Five effective heuristics were proposed ad a discrete evet simulatio model is developed to aimate the stackig, retrievig, ad reshufflig operatios i both static ad dyamic eviromets. There are sigificat differeces betwee the stowage plaig of steel coils ad that of cotaiers. Cotaiers are of stadard sizes ad the cotaiers of the same size are stacked oe o top of aother o the ship. Steel coils, o the other had, have differet weights, diameters ad widths. Due to the cylidrical shape, the coils caot be stowed i the same way as cotaiers. I additio, ship stability i the cotaier stowage plaig studies was cosidered oly i terms of the total weights i differet parts of the ship rather tha the more 5

7 precise momets. For steel coils, due to the heavy weight ad the differet sizes, oly cosiderig the total weights is far from beig accurate. Oly cosiderig the weight balace of the ship may cause the situatio where some area of the ship is full of coils of the largest weight ad aother area is full of coils of the smallest weight. Although the ship may ot sik durig the voyage, the service life of the ship will be reduced substatially. More recetly, Hvattum et al. [7] studied a tak allocatio problem for bulk cargo shippig. The decisios i the problem icluded ship routig as well as allocatio of loads to taks o the ship with the costrait that each tak could oly hold oe load at ay time. Due to the ature of the cargo cosidered, there were o stackig ad shufflig issues i their problem. They cosidered ship stability ad stregth i a comprehesive way, icludig balaces i fore-ster, left-right as well as diagoal dimesios. Because the positios of the taks were fixed o the ship, it was coveiet to express the momet cotributios of the loads i the taks i each dimesio. Øvstebø et al. [3] studied a stowage problem for RoRo ships. The problem was to decide which cargoes to carry, how may vehicles of each cargo to carry, ad how to stow vehicles o board a RoRo ship with a give voyage route. The objective was to maximize the reveue from optioal cargoes mius pealty costs icurred whe havig to shuffle cargoes. The loadig ad uloadig method of the RoRo ship determied that the shufflig was i horizotal directio rather tha vertical directio. As the vehicles were stowed i laes, the stowage shared some similarities with cotaier stowage, which was differet from the coils stowage. They also cosidered ship stability costraits usig momets, though the costraits i the fore-ster directio was ot explicitly listed. However, while the width of each lae of the vehicles o board was a variable, the distace from each lae to the cetre of the deck was estimated usig a width of equal laes ad a potetial umber of laes. The rest of this paper is orgaized as follows. I the ext sectio, the stowage plaig problem is described i detail ad formulated as a iteger programmig model. Due to the complex loadig costraits, it takes too log to solve the model for practical problems. Therefore, we derive a umber of valid iequalities for the model, ad propose a decompositio heuristic to solve the model approximately i Sectio 3. To further effectively solve the problem, a tabu search (TS) algorithm is developed i Sectio 4 to get a 6

8 efficiet ad superior stowage pla, with the iitial solutio obtaied by a costructed heuristic. Computatioal results are reported i Sectio 5 to demostrate the effectiveess of the proposed TS algorithm. Sectio 6 cocludes the paper.. The ship stowage plaig problem ad model. Problem descriptio A give set of steel coils with kow destiatio ports are to be loaded to a ship. The diameter, width, ad weight of each coil are kow. Cosiderig the practicality of stackig, the coils are to be stored i the ship i the way show i Figure. Due to the heavy weight of the coils ad for stability, at most two layers of coils ca be stored o the ship. Clearly a coil caot be loaded to a upper-layer positio uless the two lower-layer positios udereath it are occupied by other coils. As a result, the maximum umber of coils that ca be loaded to the upper layer of a row is oe fewer tha that to the lower layer of the same row. Due to the differet sizes of the coils, there are o fixed locatios o the ship for the coils. The ship stowage plaig problem is to determie the coordiates of loadig locatios of the coils o the ship. Layer Layer Width Diameter Figure Measuremets ad stackig of coils I practice, steel coils ca be stowed o a ship i the stowage patter described below. Coils are placed i rows across the legth of the ship as show i Figure 3. For coveiece i descriptio, we refer to the directio alog the legth of the ship as the horizotal directio, ad the directio alog the width of the ship as the vertical directio. Accordigly, the way of stowig coils show i Figure 3 is called the horizotal stowage patter. Coils at the lower layer i each row are placed from the ster to the fore, ad o space betwee adjacet coils is allowed i the iitial stowage. Whe the lower layer is full 7

9 of coils, additioal coils ca be loaded to the upper layer. To esure stability ad avoid damage, both the width ad the diameter (ad hece the weight as well) of a coil at the upper layer caot exceed those of ay of the two coils udereath it. The sum of the diameters of the coils at the lower layer of a row caot exceed the legth of the stowage area of the ship. The width of a row is defied as the maximum width of all the coils i the row. I the vertical directio, there must be a safety gap betwee ay two adjacet rows ad betwee a side row ad the edge of the stowage area. The sum of the widths of all rows ad these gaps must ot exceed the width of the stowage area. Right ci Fore Ster Row Left Figure 3 Horizotal stowage patter o a ship The loaded ship is to visit a umber of destiatio ports i a give sequece deliverig the coils. Each coil o board is to be uloaded at a specified destiatio port. Nothig is loaded oto the ship at these destiatio ports. At a destiatio port, if a coil to be uloaded is at the upper layer or at the lower layer without ay coil above it, the the coil ca be uloaded directly. Whe a coil to be uloaded is at the lower layer ad there is ay coil with a later destiatio above it, the the coil above eeds to be shuffled i order to uload the target coil. The shuffled coil will be placed back to the lower layer positio left by the uloaded coil. It ca be see from Figure that there may be two coils blockig the target coil. I this case both blockig coils eed to be shuffled. After the target coil is uloaded, the larger blockig coil (both i terms of width ad i terms of diameter) will be put i the positio left by the uloaded coils ad the other will be placed back to its origial positio. Stability of the ship eeds to be cosidered for the etire voyage. Horizotal ad vertical cetral lies of the stowage area divide the area ito four parts. I most previous studies, ship stability was cosidered by balacig the weights o the two sides of each cetral lie. However eve with perfect weight balace amog the four parts, placig the 8

10 coils differetly withi a part may affect the stability of the ship. A better way to cosider ship stability is to balace the momets i both horizotal ad vertical directios. There has bee oly oe recet study that tried to calculate momets but usig estimated distaces. I this paper we itroduce a ew method for calculatig the momets of the coils with precise distaces. We take horizotal directio as a example to describe the calculatio. The horizotal momet cotributio of a coil is defied as its weight times its distace to the vertical cetral lie. The distace ca be computed by subtractig the half legth of the stowage area from the horizotal coordiate of the coil. Note that the distaces ad thus the momets ca be positive or egative. If the sum of the horizotal momet cotributios of all coils equals zero, the ship is i perfect balace i horizotal directio. Otherwise the absolute value of this sum is defied as horizotal momet imbalace which caot exceed certai limit to esure safety. Similarly, vertical momet imbalace ca be calculated ad should be restricted withi certai limit. Cosiderig the balace of the ship, oce the stowage pla is decided, the order for physically loadig them oto the ship is determied accordig to their positios i the stowage pla ad the stadard loadig procedures. Therefore, we do ot eed to cosider the physical loadig operatios i this paper. Proper stowage plaig of coils helps guarateeig safety durig the voyage as well as ehacig the loadig efficiecy. Cosiderig the stowage requiremet ad characteristics of the problem, the objective to be miimized icludes the followig elemets. ) the momet imbalace of the ship The ship has to keep balace durig the voyage. Therefore, the momet imbalaces of the ship i horizotal ad vertical directios should be miimized for the loaded ship at the origial port as well as for the ship after the uloadig at each destiatio port. As described before, after uloadig a coil, ay blockig coil is either placed back to its origial positio or relocated to the positio left by the uloaded coils. The horizotal coordiate of the blockig coil may chage at most half of a coil diameter while its vertical coordiate will keep uchaged. We igore such small chage, ad therefore the momet cotributios of ay coil will ot be chaged i the voyage util it is uloaded. ) the umber of shuffles 9

11 As described earlier, whe uloadig a lower-layer coil at a destiatio port, if there is ay coil with a later destiatio above it, the the coil above eeds to be shuffled. The umber of coils that eed to be shuffled will be called the umber of shuffles for short. Miimizig the umber of shuffles will ehace the efficiecy of the uloadig operatios ad reduce damages to the coils. 3) the dispersio of coils for the same destiatio Placig the coils with the same destiatio close to each other o the ship ca help to ehace the efficiecy of uloadig. The ship stowage problem is to determie the locatios of a give set of steel coils o a give ship cosiderig the coil iformatio ad the required stowage patter of the ship, ad to miimize the objective fuctio cosistig of the above elemets.. Notatio We defie the followig otatio for modelig our problem. Kow parameters: C the set of coils to be loaded oto the ship, C ={c, c,..., c }, where is the umber of the coils. We will also refer to coil c i as coil i whe it does ot cause cofusio. Q 0 the maximum allowed momet imbalace i horizotal directio. Q 0 the maximum allowed momet imbalace i vertical directio. S iv Parameters of coil c i, v = {,, 3,}. S i, S i, ad S i3 deote the width, diameter ad weight of coil c i, respectively. p i the destiatio umber of coil c i, p i ={,,d,,p}. A smaller umber meas that the ship visits the destiatio earlier. P D the umber of the destiatios of the voyage. the miimum space betwee adjacet rows to allow the quay crae loadig ad uloadig coils. W L M the width of the storage area. the legth of the storage area. a very large positive real umber. the coefficiet of momet imbalace cost. 0

12 the coefficiet of uit shuffle cost. 3 the coefficiet of uit dispersio cost. Because the stowage locatios o the ship are ot fixed, we wat the model to decide the umber of rows that the coils should be stored i ad the umber of coils stored at the lower layer of each row. To build the model we eed to idetify a upper boud for the umber of rows, m, ad a upper boud for the umber of coils at the lower layer of a row, g. These upper bouds ca be calculated as follows. obtai From max,, mi,, m D S i m S i W, we ca i i W max Si i,, mi Si i,, D m ; () D mi{ Si i,, } Re-order the coils such that S[ ] S[] S[ ], the g ca be determied by g i g i S[ i] L, S[ i] L, for each row k =,, m. () Because of the differet sizes of the coils, the actual coordiates of the coils eed to be calculated accordigly i order to calculate their momet cotributios. For this purpose, we defie the left-ster corer as the origi for the coordiates of the locatios of coils. For a coil c i i the jth positio at the lower layer of a row, as the coils are placed from ster to fore without gaps, the horizotal coordiate cx i of this coil ca be expressed as the sum of the diameters of the coils placed i positios to j- at the lower layer i the same row plus half of the diameter of this coil itself. Similarly, for a coil i the jth positio at the upper layer of a row, because it is above the middle poit betwee two coils i positios j ad j+ at the lower layer, its horizotal coordiate ca be cosidered as the sum of the diameters of the coils i positios to j at the lower layer of the same row. Therefore, the distaces dx i (dy i) for calculatig horizotal (vertical) momet cotributio of coil c i ca be obtaied by subtractig L/ (W/) from the horizotal (vertical) coordiate cx i (cy i) as show i Figure 4.

13 Right Y Stowage Area ci Fore dyi dxi cxi Ster X cyi W/ L/ 0 Left Figure 4 Locatio coordiates ad distaces for momet calculatio Decisio variables: x il if coil i is located to locatio j of layer l i row k of the ship 0 otherwise Note that the rows o the ship are umbered from left to right ad the locatios are umbered from ster to fore as show i Figure 3. Therefore, the coil c i is placed at locatio 0 of row 3. The lower layer is defied as layer 0 ad the upper layer is defied as layer. I additio, we also use 0 ad to deote the lower-layer ad upper-layer positio j i the row k, respectively. S max,k = the maximum width of coils i row k, k=,, m. y k = the vertical coordiate of row k, k=,, m. For modelig coveiece, we defie the followig costats: x i,,m+,00, y 00, y m+w-d, S max,0 0, S max,m+ 0. if the upper coil at locatio j of row k is relocated to locatio j at the lower-layer of row k after reachig destiatio d while h=0, uhd or the upper coil at locatio j of row k is relocated to locatio j+ at the lower-layer of row k after reachig destiatio d while h= 0 otherwise u d if there is a coil i locatio j at the upper-layer of row k ad it is still placed at the same positio after reachig destiatio d 0 otherwise if the upper-layer coil i locatio j of row k has a destiatio umber larger tha that of the lower-layer coil i locatio j of row k while h 0, 0 zh or the upper-layer coil i locatio j of row k has a destiatio umber larger tha that of the lower-layer coil i locatio j of row k while h 0 otherwise

14 z if the upper-layer coil i locatio j of row k has a destiatio umber larger tha that of ay coil udereath it 0 otherwise z z 3 If after oe of the coils below the coil at locatio is uloaded, the eighbourig upper-layer coil is first shuffed to this lower-layer positio ad whe that coil is uloaded, the coil at locatio is the shuffled to this positio 0 otherwise If after the two coils below the coil at locatio are uloaded at differet ports, the two eighbourig upper-layer coils are shuffed to these lower-layer positios respectively while the coil at locatio is still placed at its ow positio 0 otherwise If a coil eeds to be shuffled durig the voyage, z idicates the first shuffle eeded. z ad 3 z idicate the secod ad the third shuffles eeded respectively, if there is ay. Note that o coil eeds more three shuffles because of the two-layer stackig structure ad the shuffle rules. if the lower-layer coil at locatio j of row k has a destiatio umber 4 z differet from that of the coil at locatio j+ of the same layer i row k 0 otherwise if the upper-layer coil at locatio j of row k has a destiatio differet from that of the lower-layer coil at locatio j of row k while h=0, 5 zh or the upper-layer coil at locatio j of row k has a destiatio differet from that of the lower-layer coil at locatio j+ of row k while h= 0 otherwise H id the cotributio of coil i to the momet i horizotal directio of the ship just before reachig destiatio d H id the cotributio of coil i to the momet i vertical directio of the ship just before reachig destiatio d Q ( Q ) = the absolute value of the total cotributio of all coils o board to the d d momet i horizotal (vertical) directio of the ship i the voyage lag before destiatio d..3 Optimizatio model Usig the above otatio, the model of the problem ca be preseted. Miimize 3

15 P P m g m g m g m g m g Qd Qd z z z 3 z zh d d k j k j k j k j h0 k j (3) Subject to m gl xil i,..., (4) l0 k j xil, k=,..., m, j,..., g l, l=0, (5) i xi,0 xi, j, k,0 i i, k=,..., m, j,..., g (6) g xi,0si L, =,..., j i k m (7) i, iv i, jl, k,0 iv i i x S x S, k=,..., m, j,..., g, l=0,, v=, (8) Si xi,0 S, maxk =,...,,,..., i y S / D x S / y k max, k i,, k,0 max, k k i k m j g (9), k=,..., m (0) S L L Hid Si3 l xi ' j ' klsi ' l xi ' j ' k, lsi' M xil i' j ' i' j ' j j i, i,...,, k=,..., m, l=0,, j,..., g l, d=,..., P, if p i d () S L L H S ( l) x S l x S M x j j i, id i3 i' j ' kl i' i' j ' k, l i ' il i ' j ' i ' j ' i,...,, k=,..., m, l=0,, j,..., g l, d=,..., P, if p i d () Hid 0,,...,, =,...,, if i i d P p d (3) gl W Hid Si3 yk M ( xil ) l0 j, 4

16 i,...,, k=,..., m, d=,..., P, if p i d (4) gl W Hid Si3 yk M ( xil ) l0 j, i,...,, k=,..., m, d=,..., P, if p i d (5) Hid 0,,...,, =,...,, if i i d P p d (6) Q d Hid i, d=,..., P (7) Q d Hid i, d=,..., P (8) Q d Hid i, d=,..., P (9) Q d Hid i, d=,..., P (0) Q Q, d=,..., P () d 0 Q Q, d=,..., P () d 0 0 h i i, i i, jh, k,0 i i, k=,..., m, h=0,, j,..., g h z ( p x p x ) / P (3) z z, k=,..., m, h=0,, j,..., g h (4) 0 h d pi xi P uhd, i k,..., m, j,..., g, h 0,,, d,..., P (5) pi xi 0 d P u0d,,...,,,...,,,..., i k m j g d P (6) pi xi, j, k 0 d P u d,,...,,,...,,,..., i k m j g d P (7) u u u k,..., m, j,..., g, d,..., P (8) 0 d d d, 5

17 u 0 d u, j, kd, k,..., m, j,..., g, d,..., P (9) i p x d P u u u i i 0 d d d, k,..., m, j,..., g, d,..., P (30) i pi xi d P uh, d uhd, k,..., m, j,..., g, d,..., P, h 0, (3) i pi xi 0 d P u, j, kd u0 d u, j, kd, k,..., m, j,..., g, d,..., P (3) i pi xi 0 d P u, j, kd u0 d u d, k,..., m, j,..., g, d,..., P (33) u, u,0, 0, u0 0, u 0, k,..., m, d,..., P (34),0, kd 0 kd gkd gkd M u u S x S x,, j, kd 0 d iv i, j, k iv i i i k,..., m, j,..., g, d,..., P, v =, (35) M u u S x S x, d, j, kd iv i iv i, j, k i i k,..., m, j,..., g, d,..., P, v =, (36) u u z k,..., m, j,..., g, d,..., P (37), j, kd 0, d, u u z k,..., m, j,..., g, d,..., P (38) 0, j, kd, d, 6

18 3 pi xi 0 pi xi, j, k 0 P u0, j, kd u, j, kd u d 3 z, i i k,..., m, j,..., g, d,..., P (39) 3 pi xi, j, k 0 pi xi 0 P u0, j, kd u, j, kd u d 3 z, i i k,..., m, j,..., g, d,..., P (40) 4 M xi, j, k,0 z ( xi, jh, k,0 pi xi, j h, k,0 pi ) / P i i i, k,..., m, j,..., g, h 0, (4) 5 M xi, zh ( xi, pi xi, jh, k,0 pi ) / P i i i, k,..., m, j,..., g, h 0, (4) 5 M xi, zh ( xi, jh, k,0 pi xi, pi ) / P i i i, k,..., m, j,..., g, h 0, (43) z, z, z, z, z, z, u, u {0, }, h 0,, k,..., m, j,..., g (44) h hp p x {0, }, i,...,, j,..., g, k,..., m, l 0, (45) il The objective of the model is to miimize the weighted sum of three elemets. The three terms i the objective fuctio (3) represet the total cost of ship imbalace durig the voyage, the total cost of shuffles, ad the total cost of the dispersio of coils of each destiatio. Note that z couts the first shuffle eeded for the coil at the upper-level 3 locatio. z ad z cout additioal shuffles eeded for the coil after that. Costraits (4) ad (5) guaratee that each coil ca be placed o oly oe locatio ad each locatio ca be occupied by o more tha oe coil. Costraits (6) require that coils at the lower layer i each row are placed from the ster to the fore with o space betwee adjacet coils. Costraits (7) esure that the sum of diameters of coils at the lower layer i each row caot exceed the legth of the ship. Accordig to the practical operatio 7

19 requiremets, costraits (8) esure that the width ad diameter of a coil at the upper layer caot exceed those of ay of the two coils udereath it. The vertical coordiates of the rows ca be expressed usig costraits (9) ad (0). Costraits (9) esure that the width of ay coil i a row is ot greater tha the maximum width of the row. Costraits (0) set restrictios o the vertical coordiates of the rows so that the gap betwee ay adjacet rows of coils is at least the required miimum D. Whe k= ad k=m+, Costraits (0) esure that the coils i the first ad last rows keep at least the miimum distace D from the left ad right sides of the ship, respectively. Costraits ()-(3) calculate the horizotal momet cotributio of each coil i before each destiatio d, H. If coil i is placed i positio j at layer l of row k (x id il=) ad if it is still o the ship durig the voyage lag before destiatio d (i.e., it is to be uloaded at destiatio d or later (p id), the its horizotal momet is calculated by costraits () ad (). If it is ot placed at that positio, the these two costraits are redudat. If the coil has bee uloaded before this part of the voyage (p i<d), the costraits (3) set H id to be zero. I a similar way, costraits (4)-(6) calculate the momet cotributios of coils i vertical directio. Costraits (7)-(0) calculate the absolute value of the total cotributio of all coils o board to the momet i horizotal ad vertical directios of the ship i the voyage lag before destiatio d. To keep stability, the sums of the momet cotributios i the two directios should be close to zero, ad have to be withi [, Q ] ad [, Q ], respectively, expressed by costraits () ad (). Costraits (3) mea that for the upper-layer coil i locatio j of row k, if ay of the two coils udereath is uloaded at a earlier destiatio tha its ow, the the upper-layer Q 0 0 Q 0 0 coil has to be shuffled ad so z. Costraits (4) cout the first shuffles of the coils. 0 h Costraits (5) esure that coil (the coil allocated to locatio j of the upper-layer i row k) will ot occupy ay positio o the ship after it is uloaded at its destiatio p i. Costraits (6) ad (7) require that coil caot take the positio of ay of the two coils below it before that coil is uloaded. Costraits (8) require that coil may take at most oe of the three positios, its origial positio ad the two positios below it. Costraits (9) guaratee that whe the coil at a lower positio is uloaded, at most oe of the coils above it 8

20 ca be relocated to this positio. Costraits (30) esure that coil must take at least oe of the three positios, its origial positio ad the two positios below it, before it is uloaded. Costraits (3) state that oce coil is shuffled to a lower positio, it will stay there util it is uloaded. Costraits (3) ad (33) state that whe a coil at a lower positio is uloaded, its positio must be take by oe of the coils above it, if there is oe. If there are two coils above it, the oe takes its positio ad the positio of the other is ot chaged. Costraits (34) reflect the fact that there is o coil at locatio 0 or g at the upper layer. This also guaratees that o upper-layer coil will float i the air after a lower-layer coils is uloaded. Safety rules require that the width ad diameter of a lower coil must be larger tha those of ay coil above it. Costraits (35) ad (36) esure that such rules are still satisfied after uloadig ad shufflig at each destiatio. Costraits (37) to (40) calculate additioal shuffles eeded for each upper-layer coil durig the voyage besides the shuffle couted by z. Costraits (4)-(43) calculate the dispersio of coils with the same destiatio over the stowage area. Costraits (4) require that if a coil is placed at locatio j+ of the lower layer i row k ad its destiatio is differet from that of the coil at positio j of the same layer i the same row, the the dispersio of coil i is z 4. Otherwise the costrait is redudat. Costraits (4) ad (43) require that if there is a coil located at locatio j of the upper layer i row k ad its destiatio is differet from ay of the coils udereath it, the dispersio of coil i is z. Otherwise this costrait is redudat. Costraits (44) ad 5 l (45) specify the decisio variables. 3. Valid iequalities ad heuristic solutio for the model The above MILP model ca be solved usig a stadard software package such as CPLEX. However, as the problem size icreases, the computatio time for solvig the model icreases rapidly so that for problem istaces with over 0 coils the model caot be solved withi a reasoable time. To help reducig solutio time, we derive the some valid iequalities for the model, ad propose a decompositio heuristic to solve the model approximately. 9

21 3. Valid iequalities ) Because the umber of coils at the lower level of a row has a upper boud of g, the maximum umber of coils that may be stowed i a row is g-. Based o this, a lower boud for the umber of rows ca be calculated:. I the model the umber g of stowed rows ca be obtaied by checkig whether there is a coil stowed i the first positio of each row. This aalysis gives the followig valid iequality: m xi,, k,0 k i g (46) ) Suppose that the coils are loaded i m 0 rows i the ship. Because i each of these rows the umber of coils o upper layer is at least oe fewer tha that o the lower layer, the total umber of coils o the upper layer ca at most be m 0. We have already kow that m 0 g from ) above. So we have the followig valid iequality: g m g x i, (47) k j i 3) I the situatio where the sizes of the coils are all differet, i.e., Siv Stv for ay i t, v,. Because each upper-layer coils eeds to be supported by two coils at the lower layer, ad the width ad diameter of a upper-layer coil caot exceed of those of ay of its supportig coils, the coils with the maximum ad secod maximum widths (or diameters) caot be placed o the upper layer. Similarly, the coils with the third ad fourth widths (or diameters) caot be both placed o the upper layer. Deote coil i' (i'') as the coils with the ith maximum width (diameter), we have the followig valid iequalities: m g xi, 0,,,, k j i (48) 0

22 m g m g x x (49) 3, 4,, k j k j m g m g x x (50) 3, 4,, k j k j 4) At the iitial port, if the upper-layer coil i locatio j of row k has a destiatio umber equal to or smaller tha that of the lower-layer coils both i locatio j ad j+ of row k, this upper-layer coil will ot be shuffled durig the voyage. Similarly, if there is o coil allocated i positio after a port, the correspodig variable u d of the subsequet ports must be 0. Therefore, we have the followig valid iequalities: z z, j,..., g, k,..., m (5) z z, j,..., g, k,..., m (5) 3 xi u,,...,,,..., i j g k m (53) u d u, d, j,..., g, k,..., m, d,..., P (54) 3. A model-based decompositio heuristic To reduce solutio time, we propose a model-based decompositio heuristic of two steps. I the first step the origial model is simplified by igorig the vertical momet balace requiremet. Hopefully the simplified model ca be solved i much shorter time tha the origial model. The solutio of the simplified model i this step will provide the stackig pla i each row. We refer to each row of coils i this solutio as a lie of coils. However, the vertical momet balace may be poor or eve does ot satisfy the balacig requiremet. Therefore i the secod step we reassig the lies of coils to the rows o the ship ad optimize the vertical coordiates of the rows to miimize the vertical momet imbalace. Note that the decisios of the secod step are for each lie of coils as a whole ad so will ot affect the coil stackig withi the lies. Therefore the objective fuctio value of the first step will ot be chaged, i.e., the ship imbalace i horizotal directio durig the voyage, the umber of shuffles, ad the dispersio of coils of each destiatio will

23 ot be affected by the secod step because the horizotal positio ad the layer for each coil are fixed. The secod step problem ca be solved easily usig a small iteger programmig model. We defie the followig parameters which ca be calculated from the first step results. m' the umber of coil lies. S max,i the largest width of the coils i lie i, i =,, m'. S the total weight of coils i lie i before destiatio d, i =,, m', d =,, P. total i3, d We redefie variables H id ad y k ad defie ew assigmet variables as follows. H id the cotributio of coil lie i to the momet i vertical directio before destiatio d. y k = the vertical coordiate of row k, k=,, m'. ik if lie i of coils is assiged to row k 0 otherwise For modelig coveiece, we defie the followig costats: y0 0, ym' W, Smax,0 = 0, S max,m'+ = 0, 0, 0, i =,, m'. i i m P Miimize Q (55) d Subject to d m' ik, i,..., m' (56) k m' ik, i,..., m' (57) i m' y ( S / ) D ( S / ) y k max, i ik max, i ik k i i m', i,..., m' total W Hid Si3, d yk M ( ik ) total W Hid S i3, d yk M ( ik ) Q m' d Hid i (58), i,..., m', k,..., m', d,..., P (59), i,..., m', k,..., m', d,..., P (60), d =,, P (6)

24 Q m' d Hid i, d =,, P (6) Q Q, d,..., P (63) d 0 {0, }, i,..., m', k,..., m' (64) ik The overall procedure of the heuristic is preseted below. Heuristic H: Step. Solve the simplified model: Miimize P m g m g m g m g m g Qd z z z 3 z zh d k j k j k j k j h0 k j Subject to all costraits i the origial model except costraits (4), (5), (6), (9), (0) ad (). Step. Based o the results of step, calculate parameters m', S max,i ad S i3, total d, the formulate ad solve the model for vertical balace as preseted above. From the descriptio of the heuristic, we ca see clearly that the first step model is a relaxatio of the origial model ad its objective fuctio is part of the origial objective fuctio. Therefore we have the followig property. Property : The objective value of the first step model i H is a lower boud for the objective value of the whole problem. If the secod step model is feasible, the result of the secod step provides a heuristic solutio for the origial problem, with a objective value beig the sum of the objective values of the two steps. I additio, if the objective value of the secod step model is 0, the the solutio obtaied by this algorithm is optimal. Aother possible way to decompose the model is to igore the horizotal balace requiremet first, ad the i step the sequece of coils i the same row ca be adjusted to reduce the horizotal imbalace. This may speed up the solutio process to some extet, but the solutio quality would be affected. I H, though the vertical balace requiremet is igored i step, it ca still be balaced perfectly i step because ot oly the lies (rows) ca be rearraged, their exact positios (ad so the vertical momet) ca be adjusted cotiuously subject to the requiremet of the gap i betwee. Therefore, it gives better solutios. I additio, sice the objective of step oly igores vertical imbalace which 3

25 will be close to zero, it provides a tight lower boud which may be used to evaluate the other heuristics. The model of step is small ad ca be solved very quickly. The solutio also has a excellet vertical balace. Therefore, there is little room for improvig the solutio through iteratively executig the two steps of H. O the other had, step itself takes log to solve. It would be possible to further decompose the problem of this step ad solve it iteratively. For example, we may first divide all coils ito m subsets, each for a row, ad solve the sub-problem for each row; ad the based o the results for differet rows we may re-divide the coils ad start a ew iteratio. This becomes a search process for a good divisio of coils to the rows, ad as the results of the rows may ot provide a good guide o the re-divisio of coils, the search may ot be as effective as metaheuristics such as tabu search. More importatly, the curret step model provides a lower boud for the problem ad we use it to evaluate heuristic solutios for small ad some medium problems. A iterative heuristic solutio of step caot give a lower boud aymore. Therefore, we do ot further decompose the problem i step of H. The computatio time eeded for H is still too log if the umber of coils to be loaded is large. This motivates us to develop faster heuristic algorithms for SSPP. 4. The tabu search algorithm I this sectio we preset a algorithm based o the Tabu Search (TS) techique proposed by Glover [5]. There are may algorithms for various optimizatio problems i steel productio, e.g., Tag et al. [6] ad [7]. However, for sequece-based schedulig problems, TS has bee show to be oe of the most effective local search techiques able to avoid beig trapped at a local optimum ad to fid ear-optimal solutios, e.g., Nowicki ad Smuticki [], Barbarosoglu ad Ozgur [4], Watsoa et al. [0], Norma [], ad Grabowski ad Wodecki [6]. The SSPP is to determie the arragemet of coils o the ship ad the problem shares some features with schedulig problems, which motivates us to adopt TS to solve our problems. The mai compoets of TS iclude iitial solutio, move, eighborhood, tabu list ad stop criteria. Accordig to the features of SSPP, a heuristic is proposed to geerate a iitial 4

26 solutio ad a speed-up strategy is adopted to accelerate the search process of the algorithm. A tabu list with variable legth is proposed for our problems. Each of the compoets is described as follows. 4. Iitial solutio I this sectio, we provide a item stowage heuristic (IS) to geerate a iitial solutio for TS. The IS heuristic has three stages: estimatig the upper boud for the umber of rows, m; sequecig the coils; ad allocatig the coils o the ship. Stage. Calculate the upper boud for the umber of rows, m, ad the upper boud for the umber of coils at the lower layer of a row, g, usig formulae () ad (). Stage. Sequece the coils i reverse order of their destiatios. For coils with the same destiatio, priority is give to those with higher value of Si + Si, where ad are adjustig parameters. Further ties are broke arbitrarily. Stage 3. Determie the stowage positio of the coils i the followig three steps. Step Assig the coils, oe by oe, i the sequece obtaied i stage ito the m rows i the followig way: row, row,, row m; row m, row m-,, row ; row, row,, ad so o. Figure 5 illustrates this method usig a example with 3 rows, where i is the i th coil i the sequece obtaied i Stage. row ' 6' 7' row ' 5' 8' row 3 3' 4' 9' Figure 5. The method of assigig coils ito m=3 rows. Step Calculate the total weights of the coils assiged to the rows. Rearrage the rows o the ship i decreasig order of their weights. If m is a odd umber, place the first row i the middle positio, place the rest rows alterately o the left ad the right sides of the stowage area as expressed i Figure 6(a). If m is a eve umber, the odd umbered ad eve umbered rows are allocated symmetrically as 5

27 show i Figure 6(b) D D 3 4 D D D (a) Figure 6. The rearragemet of rows o board. (b) Step 3 I each row, arrage the stowage of coils, oe by oe i their order i the sequece obtaied i stage, from the middle positio to two sides alterately. Whe the lower-layer of a row is full of coils, the rest coils are assiged o the upper-layer also startig from the middle positio. Figure 7(a) ad 7(b) show situatios where g is odd ad eve, respectively. Fial, shift the positio of whole row backward util there is o gap betwee the rear coil ad the back edge of the stowage area. Upper layer Lower layer Upper layer Lower layer (a) Figure 7. The arragemet of coils i each row. (b) After determiig the positios of coils, the objective fuctio value of the solutio ca be calculated i the same way as i the model. 4. Neighborhood structure Cosiderig the characteristics of the problem, i this algorithm, we geerate a eighbor of a give solutio by feasibly swappig a coil with aother coil or with a empty positio uder the size ad operatio costraits.. A feasible swap of two coils meas that after the swap each of the two coils satisfies the width ad diameter relatioships with coils aroud its ew positio ad that the 6

28 ew legths ad widths of related rows satisfy the size costraits of the storage area.. A feasible swap betwee a coil (i the upper layer) ad a empty positio withi the same row meas that the width ad diameter of the coil must ot exceed, respectively, those of ay coil udereath it after the swap. Because the iitial solutio provides a good vertical balace of the ship, if a upper coil swaps with a empty positio of a differet row, the vertical balace is more likely to become worse. That is why swappig betwee a coil ad a empty positio is limited withi the same row. O the other had, such swappig may geerate more promisig solutio with better horizotal balace. Due to the stowage rules give i the problem descriptio, we kow that there must be o empty positio i the lower layer ad so a coil i the lower layer caot be swapped with a empty positio. We defie the eighborhood of the curret solutio as a set cosistig of N radomly geerated eighbors of the solutio. 4.3 Tabu list I the proposed TS, we costruct a Tabu list to directly record the solutios of most recet moves to avoid the repetitive searchig durig the iteratio process. I the eighborhood of the curret solutio, amog those ot i the tabu list, the oe with the miimum objective value is accepted as the curret solutio for ext iteratio, ad added to the tabu list. Whe the umber of solutios i the tabu list is over a certai legth, the oldest solutio i the list will be dropped. I this paper, the legth of the tabu list (LT) is dyamically chaged accordig to the differece betwee the best objective value foud so far, f best, ad the best objective value obtaied i the last r iteratios, f r. Withi the last r cosecutive iteratios, if the objective value f best is ot improved ad the above metioed differece is above a threshold value u, the tabu list legth will be decreased by. O the cotrary, if f best has ot bee improved ad the differece is below a threshold value l, the tabu list legth will be icreased by. The iitial legth is set to. 7

29 4.4 Stoppig criteria Two commo stoppig criteria are used i the algorithm: ) the maximum umber of iteratios (T) has bee reached; ) the maximum umber of cosecutive iteratios without improvemet o the objective value (TW) has bee reached. 4.5 Calculatio of objective fuctio For ay give solutio, its objective fuctio value ca be calculated usig the expressio i the model. I the tabu search process, each trial solutio is obtaied by a swap o the curret solutio. I order to accelerate the search process, we calculate the objective value of a trial solutio based o the objective value of the curret solutio ad the objective value chage caused by the swap. 4.6 Further improvemet Durig the searchig process of TS, the vertical coordiates of the rows are fixed. Therefore the balace i vertical directio may be further improved by adjustig these coordiates. After the search fiishes, we use step of H to miimize the vertical imbalace based o the solutio obtaied by the TS iteratio. 4.7 The procedure of proposed TS Combiig the compoets itroduced above, the whole procedure of proposed TS ca be described as follows. Step. Geerate a iitial solutio s 0 usig the method preseted i Sectio 4. ad calculate the correspodig objective fuctio value f 0. Set s cur = s best = s 0, ad f cur = f best =f 0. Iitialize N, LT, T, TW, ad r. Set iteratio cout t=0 ad iteratio cout without improvemet tw=0. Step. Iteratio of TS Step.. Create a eighborhood of s cur ad calculate the correspodig objective fuctio values. Step.. Fid the best solutio i the eighborhood but ot i the tabu list, set s cur ad f cur as this solutio ad its correspodig objective value, update the tabu list, ad set t= t +. Step.3. If f cur < f best, set f best = f cur, s best= s cur, ad tw =0, go to Step.5. Otherwise, set tw = tw + ad go to Step.4. 8

30 Step.4. Update LT ad the tabu list accordig to Sectio 4.3. Step.5. If t<t ad tw<tw, go to Step.. Otherwise, go to Step 3. Step 3. Further improve the solutio by step of H. Stop. 5. Computatioal experimets I this sectio, we evaluate the performace of the solutio methods metioed i this paper for the stowage plaig problem through computatioal experimets. We geerate a test istaces accordig to the real data obtaied from Baosteel. These data iclude ship iformatio ad coil iformatio i the stowage plas. The solutios compared i the experimets iclude those obtaied by the origial model, the model with valid iequalities, the model-based decompositio heuristic H, the proposed TS algorithm (pts), the classic TS algorithm (cts), the classic Particle Swarm Optimizatio (PSO), ad the maual method. PSO is a populatio-based meta-heuristic proposed by Keedy ad Eberhart [0], which searches the solutio space based o particles velocity ad positio update mechaisms without ay prior kowledge of the problem ad performs well i solvig may other problems. All the methods are implemeted i Visual Studio C# 0 ad the MILP models are solved usig CPLEX.5. The experimets ru o a persoal computer with a Itel.83GHz CPU, 4 GB memory ad Widows 7 operatig system. 5. Experimetal data Observig the practical data collected, we foud that the diameters ad the widths of most coils are i the rages of [.68m,.78m] ad [0.95m,.5m], respectively. Moreover, based o the data of 50 real coils, we have established the followig relatio fuctio betwee the weights ad the sizes of coils by biary regressio method. S 5.7 S 6.8 S 9.6 (65) i3 i i For all test istaces, the values of S i (width) ad S i (diameter) of each coil c i are geerated radomly from the rages give above, ad correspodig S i3 (weight) value is the calculated usig (65). Accordig to the umber of coils i the problem, the test istaces we geerate ca be classified ito three groups, i.e., the small, the medium, ad the large problems. For the medium ad the large problems, the sizes of ships are similar to those of the real ships as described i sectio. To test our mathematical model, small problem 9