A Model for the Stowage Planning of 40 Feet

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					                     International Journal of Information Systems for Logistics and Management
                     Vol. 4, No. 2 (2009) 41-49                     http://www.knu.edu.tw/academe/englishweb/web/ijislmweb/index.html

             A Model for the Stowage Planning of 40 Feet
                 Containers at Container Terminals

                                                   Zhao Ning and Mi Weijian
                          Logistics Engineering School, Shanghai Maritime University, Shanghai, China

                           Received 19 January 2009; received in revised form 20 March 2009; accepted 15 May 2009

                     In container terminals, one of the most important factors driving the logistic efficiency in the yard
              is stowage planning for containers designated to be exported. The stowage plan and operations have to
              meet the requirement of the liner shipping company of ship stability on the one hand, and to ensure the
              smooth and orderly process of handling containers, deployment and movement of the yard cranes on the
              other. These criteria are often in conflict. This paper is concerned with the ship's container stowage
              planning problem which is formulated as a multi-objective integer linear programming. For the sake of
              being practical, the model not only considers the conflict between ship stability and containers' reshuf-
              fling operation but also first takes into account the moving frequency of yard cranes, the probability of
              wait by quay crane and the feasibility of multi-YC feeding one QC during the loading process. A wide
              variety of numerical experiments demonstrated that solutions by this formulation are useful and ap-
              plicable in practice.

              Keywords: multi-objective integer linear programming, container terminal, stowage planning, ship

                  1. INTRODUCTION                                        However, as far as the throughput of container terminals
                                                                         is concerned, the service rate to meet the demand by mega
       Increased regional competition has put further                    vessels has yet to be achieved in the studies that con-
pressure on port operators to stay competitive and relevant              centrated on conventional storage yard, where containers
to their customers. Lower handling charge, shorter tran-                 are stacked on the ground, side by side and one on top
sit time, higher level of service, and wider connectivity                of another.
to the rest of the world have been identified as the main                      The main disadvantage of the conventional stack-
goals to acquire international competitiveness.                          ing scheme is that the reshuffling operations, which incur
       It is therefore not surprising that a number of stud-             additional unproductive moves, have to be performed
ies have been conducted in various aspects for optimiza-                 in order to retrieve a container from a lower tier. The
tion of the container terminal operations. As a result, vari-            vehicle requesting the container will have to wait extra
ous control policies for unloading, storing, and loading                 time which may cause delays in feeding to the quay cranes.
containers have been proposed based on simulation and                    The result could be as serious as lengthening the vessel's
mathematical formulations. One of the most prominent                     turnaround time and consequently a downgrade of ser-
results is the application of object-oriented approach,                  vice level. Therefore, in order to reduce the chances of
in which terminal resources and entities are modeled                     retrieving containers that are not on top of the stacks, and
as individual objects and solutions to the performance                   also due to the weight constraint of containers, the stack-
problem are found via operation research techniques.                     ing height is usually restricted to not more than eight
42          International Journal of the Information Systems for Logistics and Management (IJISLM), Vol. 4, No. 2 (2009)

(in practice even lower than eight). However, this prac-                   Shields (1984) developed a container ship stowage
tice implies a limited utilization of ground space that is           computer-aided preplanning system. Here a small num-
scarce and previous.                                                 ber of stowage plans are created which are then evaluated
       In response to these observations, this work focus            and compared by simulation of the voyage across a
on stowage planning that assigns to each bay position a              number of legs. The order in which loading heuristics
particular outbound container with a type matching the               are applied is determined using a limited number of dif-
preliminary type-based stowage plan provided by shippers.            ferent solutions. Since then further investigations have
It is normally done by assigning outbound containers                 been carried out (Ratcliffe and Sen, 1987; Saginaw and
in inverse order of ports to be visited by the ships and             Parakis, 1989) using expert systems and rule-based tech-
changes may be required from the shipping companies.                 niques to aid the stevedore in finding suitable con-
Stowage plans are prepared a few hours in advance. We                figurations. Rule-based decision systems for dealing with
focus only on the 40 feet full containers stowage prob-              MBPP are presented in Ambrosino and Sciomachen
lem, since different type of containers should obey differ-          (1998), where a constraints satisfaction approach is used
ent rules of stowage, for instance, the empty containers'            for defining and characterising the space of feasible
stowage rules are totally different from the full. However,          solutions without employing an objective function to
it is applicable because the liner shipping company will             optimise, and in Wilson and Roach (2000), where the
designate the bay positions for each type of containers              potential of applying the theory of artificial intelligence
in the PSP (Preliminary Stowage Plan). And the model                 to cargo stowage problems is explored. Todd and Sen
developed in this study is adaptable for the case with 20            (1997) implemented a GA procedure with multiple
feet containers without major modifications.                         criteria such as proximity in terms of container location
       This paper is organized as follows. The next sec-             on board and the minimization of unloading-related
tion reviews the related literature. In the third section            reshuffle, transverse moment and vertical moment. Their
the proposed algorithm is described. In the subsequent               study examined the relationship between the reshuffle
section, a variety of numerical experiments are carried              and the ship stability. Winter (1999) introduced the stow-
out and presented, and the final section reports the paper's         age planning in conjunction with load planning taking
findings and conclusions.                                            into account the equity of quay crane workload. This
                                                                     study also inspired issues of loading-related reshuffle and
              2. LITERATURE REVIEW                                   ship stability. However, although stability is an impor-
                                                                     tant factor of the pre stowage planning for the liners, but
      Management of container terminal operations is                 is not a key factor of the stowage in container terminals.
essentially the allocation and scheduling of the ex-                 Note that all the above studies do not assume that each
pensive resources such as berths, quay cranes, storage               vertical column in holds contains only containers of the
space, yard cranes, and container carriers. Each type                same destination.
of these resources plays an indispensable role in the                      Botter and Brinati (1992), and Cho (1984) explored
interlocking processes in a container terminal. A com-               the application of mathematical models and linear pro-
prehensive review on various decision problems that                  gramming to the problem, whereas too many simplifica-
arise in the planning of export containers’ stowage is               tion hypotheses were incorporated , which have made
given as followed in the literature. Some of the tech-               their approaches unsuitable for practical applications.
niques introduced herein will be extended to evaluate                Avriel and Penn (1993) and Avriel et al. (1998) addressed
the stowage planning.                                                a stowage problem which formulated the problem as a
      Since the 1970s, researchers drawn from academic               0-1 Integer Programming and applied it for loading onto
and commercial shipping organizations, have tried to                 a single hold, but only the unloading-related reshuffles
examine and worked over the problem of stowage plan-                 was taken into consideration.
ning. As a category of the loading problem, stowage                        Some researchers explored the potential of apply-
planning is well recognized in the literature and has                ing the theory of artificial intelligence to cargo stowage
become widely used in a variety of transportation opera-             problems. This class includes the work of Dillingham,
tions. In the early stage, most of the studies were directed         Perakis, Wilson and Roach (1999-2001), and Sato.
at the pre stowage planning in liner shipping companies.                   Ambrosino et al. (2004) addressed a stowage-
Those methods developed have been grouped into the                   planning problem with the objective of minimizing the
following five main classes: (1) simulation based upon               total stowage time where more practical constraints are
probability; (2) heuristic driven; (3) mathematical                  taken into account such as different types of containers
modeling; (4) rule-based expert systems; and (5) decision            in length, weight limit being accepted for securing ship
support systems. None of these approaches have provided              structure, etc. However, they do not explicitly take into
a solution to the complete stowage-planning problem.                 account loading-related reshuffle.
Here follows a brief review of relatively recent research                  Kim et al. analyzed rehandles of transfer crane and
into automating stowage planning.                                    made a evaluation of the number of rehandles in container
               Z. Ning and M. Weijian: A Model for the Stowage Planning of 40 Feet Containers at Container Terminals                                                                43

yards (Kim, 1994, 1997; Kim and Kim, 1994; Kim et al.,
2000, 2004). In 2004 they addressed a load-planning prob-
lem with an objective of proper arrangement of container
stacks on board in light of smooth quay crane operation                     10 08 06 04 02 01 03 05 07 09

and the other of proper container retrieval sequence from                           25              26 23
                                                                              27 28 22 16 24 26 30 26

container stacks in the yard in light of smooth transtainer                   26 26 22 22 23 25 30 30
                                                                                                                                                             6 5 4 3 2 1
                                                                              29 26 21 29 25 16 30 30                                                Truck
operation. For this problem, they developed a beam search                     25 29 27 29 24 27 30 30
                                                                                    28 25 28 24 22 30
       Imai et al. did a series of research on loading busi-
                                                                             Fig. 1. An overview of container terminal port operations
ness in the container terminal (Imai and Miki, 1989; Imai
et al., 2001, 2002, 2006). They formulated a multi-objec-
tive simultaneous stowage planning model for a container                                 06   04   02   00    01     03     05          06     04    02      00    01     03    05
ship with container rehandle in yard stacks. They utilized          Slots
                                                                                              34        73   20     20    22
                                                                                                                                             57     76     76     76    10     10
the estimated number of rehandles in order to take the              Ship-      04
                                                                                              22        22   22     22    22                 10     10     10     10    10     11
rehandle objective into account in the formulation. In this         bays             22       22        22   22     22    22                 10     14     14     15    26     82
                                                                               02                                                  02
study, the rehandle is estimated based on the expected
number of rehandles when retrieving each container in
the block as the first one to be taken.
       Sciomachen and Tanfani (2007) applied the theory             Yard-
of 3D-BPP approach to optimize stowage plans and
terminal productivity. They evaluated how stowage                                A3 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64
                                                                                   01 03 05 07 19 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65

plans can influence the performance of the quay so as                Block
to produce stowage plans that minimize the total loading                         6

time and allow an efficient use of the quay equipment.                               Fig. 2. The stowage between containers and slots
However the performance of the yard crane and other
factors are more important for the stowage planning in
CT. The containership stowage and load-planning prob-               terminal. There is a typical cross-sectional view of a
lem is much more difficult to solve than the three-dimen-           cellular ship. Each cell in the figure represents a slot
sional bin packing problem due to the fact that the ship’s          where a container can be placed and the number in the
stowage plan has to consider the assignment of containers           cell implies the weight of the stowed container.
to a three-dimensional storage space in addition to the                    The stowage planning is to assign a slot to each
restrictions imposed in retrieving containers from the              outbound container stacking in the yard according to
stacks in the field.                                                the preliminary stowage plan provided by the liner ship-
       To sum up the points which we have just indicated,           ping company. The PSP (preliminary stowage plan) is a
three sorts of elementary and crucial factors of this               kind of sketchy plan of slots for each type of containers
stowage problem were barely considered in most of the               classified by size, EF (empty or full), discharging port,
relevant research work. That includes (1) reshuffles in             dangerousness, particularity and so on.
the stacks; (2) overweight stowage; (3) waiting by quay                    As has been elucidated in the foregoing, the stow-
cranes; (4) move frequency of yard cranes; and (5)                  age problem can be attributed to the decision-making
number of feeding blocks. In this paper we do take into             of the relationship between the items of two sets. Set A is
account all the above factors comprehensively.                      the containers in the yard, whereas set B should be
                                                                    the available slots from the PSP. The stowage between
             3. MODEL FORMULATION                                   containers and slots in a ship is shown in Fig. 2.
                                                                           As shown in the following figure, in conventional
3.1 Stowage Planning Description                                    storage yard, containers are stacked by yard cranes side
                                                                    by side and one on top of another to form rectangularly
      As is well known, container terminals play a funda-           shaped heaps called blocks, each of which consists of a
mental role in intercontinental cargo transportation by             number of rows in width, a number of bays in length and
serving as an intermodal interface between the sea and              a number of tiers in height. Similarly, in each ship bay,
the land carriers. Typically, they receive cargos in con-           there are also a number of rows in width and a number
tainers from various transportation devices like vessels            of tiers in height.
or trucks, store them temporarily to account for the dif-
ferences in arrival times of the sea and the land transport,        3.2 Evaluation of Unavoidable Reshuffling
and transfer them to other transportation devices to                    Containers
be delivered to their destinations. Fig. 1 is a schematic
diagram showing the core operations in a container                           Although the loading sequence of containers in a
44           International Journal of the Information Systems for Logistics and Management (IJISLM), Vol. 4, No. 2 (2009)

     1   2    3        4     5     6       04   02    00   01   03
                                                                             Uppos_stowageij =

                                                                              n     m
                                                                              Σ Σ Cudix * STOWAGEij * Sudjy
                                                                             x=1 y=1

                                                                            Based on the analysis above, the unavoidable
                                                                      reshuffle occurs when Uppos_stowage ij >= 1 and
                                                                      STOWAGEij = 1. So the reshuffle can be formulated as
                      Fig. 3. Unavoidable reshuffle                   Uppos_stowageij* STOWAGEij. Since the multiplication
                                                                      of two variables will make the model non-linear, we have
                                                                      to use a trick to make the evaluation of total reshuffles
ship-bay is totally unknown before the stowage planning,              linear.
but in the same row of a ship-bay the slots will be stowed                  In order to make this formulation solvable as a math-
in a certain order. The example illustrated in Fig. 3                 ematical programming, we introduce the binary variable
showed the situation of unavoidable reshuffle. If con-                RESTOWij. By using this definition we may formulate
tainer A is stowed in slot A and container B is stowed                the problem only with the minimization of the Total
in slot B, then container A will definitely result in a re-           Reshuffles as follows:
shuffle of container B because slot A should be loaded
in advance of slot B.                                                 [RS]
       To further calculate the reshuffles caused by a stow-                             n   m
age planning, the following notations should be declared                     Minimize    Σ Σ RESTOWij
                                                                                        i=1 j=1
                                                                      subject to,
Note: the notations with upper characters are unknown
variables to be decided; the notations containing any lower
characters are known parameters.
                                                                              Σ STOWAGEij < = 1∀j

i, x
                    Index of containers to be loaded;
                    Number of containers;
                                                                              Σ STOWAGEij < = 1∀i
j, y              = Index of slots in the ship;
m                 = Number of slots;                                         RESTOWij >= Uppos_stowageij / 10
q                 = Index of blocks in the yard;                             + STOWAGEij – 1∀i, j                            (5)
tq                = Number of blocks;
w                 = Index of yard-bays of the blocks;                        STOWAGEij = {0, 1}∀i, j                         (6)
tw                = Number of yard-bays;
b                 = Index of ship-bays of the ship;                          RESTOWij = {0, 1}∀i, j                          (7)
tb                = Number of ship-bays;
l                 = Index of rows in the ship-bays;                   where STOWAGEij = 1 if a container at position i of yard
tl                = Number of rows in a ship-bay;                     stacks is loaded in slot j of ship; = 0, otherwise and n
STOWAGEij         = Binary variable indicating whether the            is the number of containers to be loaded.
                    container i should be stowed in slot j;                  In the formulation, constraints (3) and (4) ensure
Cudix             = Known binary parameter indicating                 that every container is stowed with one slot and every slot
                    whether container x is stacked above              can only be taken by one container, whereas constraints
                    container i in the same row as container          (5) ensures Uppos_stowageij >= 1 and STOWAGEij = 1
                    i;                                                when RESTOWij = 1.
Sudjy             = Known binary parameter indicating
                    whether slot y is right above slot j in the       3.3 Stability Factor: Overweight Stowage
                    same row as slot j;
RESTOWij          = Binary variable indicating whether the                  The stowage planning has to satisfy the stability re-
                    container i stowed in slot j causes a             quirement of liner shipping company. Hence, it must be
                    reshuffle of other containers;                    approved by the first mate of the ship before the loading
                                                                      process. And the primary concern for the mate is the is-
      To calculate the total number of reshuffles, the vari-          sue of heavy container stowed on top of a lighter one.
able Uppos_stowageij should be introduced to indicate the             Although overweight stowage is not strictly forbidden,
number of containers above container i which are stowed               it should be as fewer as possible. Therefore, we ought
above slot j. We can define this value by                             to formulate it as a stability objective instead of a
                Z. Ning and M. Weijian: A Model for the Stowage Planning of 40 Feet Containers at Container Terminals                  45

constraint, and this objective should focus on the total                        Stowage I                             Stowage II
number of containers that is stowed on a lighter one in-               A A A A              A A A               A A A A A B B B
stead of the GM itself.                                                B B B B              A A A               A A A A A B B B
      The following parameters are declared in order
to formulate the overweight stowage objective.                         A A A B              B B B               A A A A B B B B
                                                                           A A A            B B                   A A A B B B
Vajy         = Known binary parameter indicating                     BLOCK A                                BLOCK A
               whether slot y is on top of slot j.                   BLOCK B                                BLOCK B
Ctn_weighti  = Known parameter, the weight of
               container i;                                               Fig. 4. A ship-bay stowed with containers from two blocks
SLOT_WEIGHTj = Variable, the weight of container that
               is stowed in slot j;
WAVj         = Variable, the weight of container that                ciency of the loading process. Fig. 4 shows two different
               is stowed on top of slot j;                           stowage plans within a ship-bay stowed with containers
UVMBj        = Variable, the weight that remains                     from two blocks. For Stowage I, in each row of the ship-
               after the weight of container stowed                  bay, containers stowed are from different blocks. In
               on top of slot j is subtracted from                   case of this situation, the two yard cranes of Block A and
               SLOT_WEIGHTj;                                         Block B would have a good chance of interacting each
OVERWEIGHTj = Variable, if UVMB j > 0 then                           other's work efficiency and finally result in the wait of
               OVERWEIGHTj = 1, otherwise, 0;                        QC. But for Stowage II, there would be less probability
OVERWEIGHT_STOWAGE = Variable, the total number                      of interaction and conflict.
               of containers that is overweight                            In this regard, we may formulate the problem only
               stowed.                                               with the minimization of the total number of blocks in
                                                                     each row of the ship-bay as follows:
     Based on constraint (3), SLOT_WEIGHTj can be
evaluated by Ctn_weighti and STOWAGEij.                              SLOT_Qjq = Variable indicating whether the container
                                                                                stowed in slot j is from block q;
       SLOT_WEIGHTj =                                                Rrsjl    = Known parameter indicating whether slot
        n                                                                       j is in row l;
        Σ STOWAGEij * Ctn_weighti
                                                           (8)       Rbaysjb  = Known parameter indicating whether slot
                                                                                j is in ship-bay b;
                                                                     Rqciq    = Known parameter indicating whether con-
       WAVj =
                                                                                tainer i is in block q;
        n   m
        Σ Σ Vajy * STOWAGEiy * Ctn_weighti
       i=1 y=1
                                                                     Rwciw    = Known parameter indicating whether con-
                                                                                tainer i is in the yard-bay w;
                                                                     L_Qlq    = Variable, the number of containers from
       UVMBj = WAVj – SLOT_WEIGHTj                        (10)                  block q stowed in ship-row l;
                                                                     L_Q_01lq = Binary variable, whether there are any con-
       OVERWEIGHT_STOWAGE is the count of UVMBj                                 tainers from block q stowed in ship-row l;
which is positive. Therefore, a constraint and the variable
OVERWEIGHTj and are introduced as follows to formu-
late it mathematically, resulting in the formulation [OS].
                                                                           SLOT_Q jq =       Σ STOWAGEij * Rqciq
                  m                                                                   m
       minimize    Σ OVERWEIGHTJ
                                                          (11)             L_Q lq =    Σ SLOT_Qjq * Rrs jl

subject to (3)~(4), (6)~(7) and
                                                                                          tl       tq
       OVERWEIGHTj >= UVMBj / 100                         (12)
                                                                           Minimize        Σ Σ L_Q_01lq
                                                                                          l=1 q=1
       OVERWEIGHTj = {0, 1}∀j                             (13)
                                                                     subject to (3)~(4), (6)~(7) and
where number 100 ensures that UVMBj /100<1.
                                                                                                        n   m

3.4 Minimize the Probability of Wait
                                                                           L_Q_01lq > = (1 /             Σ Σ 1) * L_Qlq
                                                                                                        i=1 n=1

       The wait by quay crane will directly lead to ineffi-                L_Q_01lq = {0, 1}∀l, q                                     (18)
46                International Journal of the Information Systems for Logistics and Management (IJISLM), Vol. 4, No. 2 (2009)

                 Stowage I                                    Stowage II
                                                                                                         X X X X Y Y Y Y
      M M M M M M N N                                M M M M M N N N
                                                                                                         X X X X Y Y Y Y
      N N N M M N N M                                M M M M M N N N
                                                                                                         X X X X Y Y Y Y
      M M M M M N N                                  M M M M N N N N
                  M N N N                                M M M N N N
                                                                                                                X X X Y Y Y
     13    15    17     19        21       23       13   15    17     19   21   23
6                                               6
5                                               5
4                                               4                                                                       QC
3                                               3
2                                               2
1                                               1
      BAY M                       BAY N              BAY M                 BAY N

     Fig. 5. A ship-bay stowed with containers from two yard-bays
                                                                                                    BLOCK X                     BLOCK Y

where L_Q_01lq = 1 if one or more containers from block
q are stowed in ship-row l; = 0 , otherwise, which is en-
sured by constraints (17)~(18).
                                                                                                      6 5 4 3 2 1              1 2 3 4 5 6

3.5 Minimize the Move Frequency of YC
                                                                                           Fig. 6. Two YCs feeding one QC during the loading process

      The move of yard crane from one yard-bay to
another can decrease efficiency with a cost of fuel. As
is shown in Fig. 5, two different stowage plans within a                               yard-bay w are stowed in ship-row l; = 0, otherwise, which
ship-bay stowed with containers from two yard-bays                                     is ensured by constraints (21)~(22).
of one block. In most cases, there is one yard crane
working in one block.                                                                  3.6 Maximize the Number of Feeding Blocks
      For Stowage I, in most rows of the ship-bay con-
tainers stowed are from different bays. As is illustrated                                    The work rate of QC is generally faster than that
in Fig. 5, the loading process is divided into five phases                             of YC, especially during the loading process. One of
according to the four moves of the yard crane from bay                                 the most important factors influencing the loading effi-
to bay like N→M→N→M→N, whereas for Stowage II,                                         ciency is the feeding process in yard. Therefore the feed-
the yard crane only has to move once from bay N to bay                                 ing yard cranes for one ship-bay should be more than
M.                                                                                     one if possible, and this possibility depends on the stow-
      Therefore, the problem can be formulated with the                                age planning.
minimization of the total number of yard-bays in each                                        Fig. 6 is a schematic diagram showing a stowage
row of the ship-bay as follows:                                                        plan with containers from block X and block Y. It is evi-
                                                                                       dent that to maximize the number of feeding blocks is
L_Wlw            = Variable, the number of containers from                             to maximize the number of blocks in each ship-bay.
                   yard-bay w stowed in ship-row l;                                          Accordingly, we may formulate the problem with
L_W_01lw         = Binary variable, whether there are any                              the maximization of the total number of blocks in each
                   containers from yard-bay w stowed in ship-                          ship-bay as follows:
                   row l;
                              n        m
                                                                                       BAY_Qbq        = Variable, the number of containers from
          L_Wlw > =       Σ Σ STOWAGEIij * Rwciw * Rrsjl
                         i=1 n=1
                                                                                                        the block q stowed in ship-bay b;
                                                                                       BAY_Q_01bq     = Binary variable, whether there are any
                                                                                (19)                    containers from the block q stowed in
          [MF]                                                                                          ship-bay b;
                             tl    tw
          Minimize       Σ Σ L_W_01lw                                           (20)                       m
                        l=1 w=1                                                              BAY_Q bq =    Σ SLOT_Qjq * Rbays jb

subject to (3)~(4), (6)~(7) and
                                           n    m                                                          tb    tq
          L_W_01lw > = (1 /                 Σ Σ 1 * L_Wlw
                                           i=1 j=1
                                                                                (21)         Maximize      Σ Σ BAY_Q_01bq
                                                                                                          b=1 q=1

          L_W_01lw = {0, 1}∀l, w                                                (22)   subject to (3)~(4), (6)~(7) and

where L_W_01lw = 1 if one or more containers from                                            BAY_Q_01bq <= BAY_Qbq                                 (25)
                Z. Ning and M. Weijian: A Model for the Stowage Planning of 40 Feet Containers at Container Terminals          47

      BAY_Q_01bq = {0, 1}∀b, q                            (26)       with weights, we obtain the following formulation:

where BAY_Q_01bq = 1 if one or more containers from                        [PA]
block q are stowed in ship-bay b; = 0, otherwise, which                                         n   m

is ensured by constraints (25)~(26).                                       Minimize Z = α Σ         Σ RESTOWij
                                                                                               i=1 j=1
                                                                                m                              tl        tq
3.7 Formulation                                                            +β    Σ OVERWEIGHTj + χ lΣ1 qΣ1 L_Q_01lq
                                                                                j=1                 =   =

      Besides the five objectives above, the port opera-                        tl   tw                   tb        tq

tors have to do the stowage planning under the following                   +δ    Σ Σ L_W_01lw – ε bΣ1 qΣ1 BAY_Q_01bq (30)
                                                                                l=1 w=1            =   =
three constraints.
                                                                     subject to (3)~(7), (12)~(13), (17)~(18), (21)~(22) and
• There is a weight limitation for each row of the ship-             (25)~(29), where α, β, χ, δ and ε are weights for the
  bay. So that the sum of stowed containers’ weight for              objectives of [RS], [OS], [PW], [MF] and [FB], respec-
  each row must be less than or equal to the limitation.             tively. Note that ε is set negative because of the maximi-
• For the containers stowed on deck of the ship, over-               zation of [FB].
  weight stowage is strictly forbidden, which means each
  container on deck must be on the top of a heavier one              3.8 Solution Procedure Using the Genetic Algorithm
  or the first tier of the ship-bay.
• The general height of container is 8.4 feet, while the                   We develop a heuristic algorithm by using the ge-
  high container’s height is 9.6 feet. So there is a limita-         netic algorithm (GA) in Aimms language. The sets we
  tion for the number of high containers in each row                 defined in the model include yard positions, yard blocks,
  of ship-bay.                                                       yard bays, positions in the ship, stacks in the ship and
                                                                     bays of the ship. The relationships described between
      These three constraints can be formulated as                   the sets are also given as constant parameters or decision
follows:                                                             variables. The stowage-planning problem is designed as
                                                                     a minimization mathematical program with all the above
L_weight_limitationl = Known parameter representing the              constraints and the multi-objective formulated in (30).
              weight limitation for each row of the ship-                  GAs are widely applied for plenty of practical prob-
              bay;                                                   lems of mathematical programming, which are difficult
If_ondeckj = Known parameter indicating whether slot                 to solve in terms of polynomially-bounded computational
              j is on deck or not;                                   time. It is solved for each scenario to find the optimal
If_highi   = Known parameter indicating whether                      solution under each scenario, and then the robust model is
              container i is a high container or not;                solved to find the robust solution. Although we can solve
High_container_limit = Known parameter representing                  the model with solver “XA”, but the calculation is too
              the limitation of the number high con-                 much slow when the number of containers in the same
              tainers stowed in a row;                               group is increasing or when the initial feasible solution
                                                                     lags far behind the optimal one. Thus, to accelerate the
       n   m
       Σ Σ STOWAGEij * Ctn_weighti * Rrs jl
      i=1 j=1
                                                                     calculation, we introduced selection operator and muta-
                                                                     tion operators into the solving process. To minimize the
                                                                     objective function, the selection operator and mutation
      <= L_weight_limitationl
                                                                     operators are designed as followed.
      m                                                                    Selection operator:
       Σ OVERWEIGHTj * If_ondeckj = 0
                                                                                           1             n
                                                                           fitness (x) =      ∧ [y (x) / Σ y (x)]             (31)
       n   m                                                                               10           x=1
      Σ Σ STOWAGEij * If_highi * Rrs jl
      i=1 j=1
                                                                     Y(x) denotes the objective function value. Fitness(x)
      <= High_container_limit                             (29)       stands for the probability of Gene X being selected.
                                                                           The mutation operators are designed with a cer-
      Among a number of techniques for generating a                  tain purpose. For example, to minimize the possibility
non-inferior solution set, we employ the weighting method.           of restow, the Gene X (Stowage ij ) mutates when
In this method we define the problem as a mathematical               Max(Uppos_stowageij * Stowageij) >= 1, and the muta-
programming model with a single objective that incorpo-              tion operator is to change the value of Stowageij, Stowagexj,
rates multiple objectives.                                           Stowagexy and Stowageiy into their opposite way (0→1,
      Putting the five objectives into a single objective            1→0).
48                International Journal of the Information Systems for Logistics and Management (IJISLM), Vol. 4, No. 2 (2009)

Table 1. Solution profile for cases of different                                     06 04 02 00 01 03 05                  06 04 02 01 03 05
         container volume                                                         28 06 09 09 09 09 09

Case       Ship       Container         Reshuffle         Over        Time        28 07 21 07 09 09 21                     05 06 07 16 20 21

           size        volume                            weight        (s)        25 09 07 25 26 26 29                     22 22 27 27 28 30

   1          S            50                0               3          142
                                                                                             In a hold                          On deck
   2          S           125                2               4          152
   3          S           150                3               7          176                 Fig. 7. The stowage of bay 18h and 22d in case 1

   4          S           175                2               5          235
   5          S           200                3               5          248
                                                                                                        06 04 03 00 01 03 05
   6          S           200                4               9          268
   7          S           200                4               2          335                             25 22         15 17 18 24
   8          S           250                4              11          503                             26 25 15 22 22 19 24
   9          L           300                2               7          654
  10          L           325                7               9          752                             27 20 24 22 24 24 24
  11          L           345                3               9          809
  12          L           350                2              10          856
  13          L           365                2               1          918
  14          L           375                2              13        1,006
  15          L           400                3              12        1,073
  16          L           400                0              14        1,207
  17          L           450                1               7        1,357
  18          L           450                5               7        1,511      2
  19          L           501                5               9        1,857      4                                                       Block A3
  20          L           553                7              15        2,050
Note that each case of experiment consists of several types of containers                           2
classified by their discharging ports. For each type, it is solved separately.        Block A1      4
So the computation time in the last column of Table 1 is the sum of each                            6
type’s solved time.
                                                                                     Fig. 8. The stowage from two yard-bays to one ship-bay in case 8

                                                                                 bay “A110” of block “A1” and yard-bay “A348” of block
      The solution procedures were coded in “Aimms”                              “A3”
language on the PC with Core(TM) 2, T5600 (1.83GHz)                                    As is illustrated in Fig.8, in each row of the ship-
CPU. Problems used in the experiments were read from                             bay, the stowed containers are from one block which per-
the database of container terminal management informa-                           fectly accord with the objective of [PW] to minimize. The
tion system for Tianjin port. Values of weights α, β, χ,                         fourteen containers stowed on ship-bay “14d” are from
δ and ε in Equation (30) used in the numerical experi-                           only one yard-bay of “A348”, although there are other
ments are respectively 1, 1, 2, 2, 1.                                            bays in block A3, which proves the model can well mini-
      Table 1 demonstrates the numerical cases of differ-                        mize the move frequency of YC (objective of [MF]). And
ent ship size, number of containers to be stowed, the                            according to the above figure 8, we can imagine the
result of reshuffles, overweight stowage and the solving                         deployment of yard cranes during the loading process
time.                                                                            of ship-bay “14d”. Evidently, the containers can be
      The stowage plan demonstrates a balanced stow-                             loaded by two feeding flows, which well meet the re-
age in terms of weight distribution, as the solution for                         quirement of objective [FB].
case 1 is shown in Fig. 7 where the numbers on each slot
represent containers’ weight. There are two bays of                                             5. CONCLUDING REMARKS
containers. As is shown in Fig. 7, the left bay is stowed
in a hold, whereas the right one is stowed on deck.                                   This paper addressed the problem of obtaining a
      And in Fig. 8, it shows the stowage plan of twenty                         non-inferior solution set for the container ship stowage
40” full containers that will be discharged in Singapore.                        planning in the container terminals. The problem was
The stowed ship-bay is “14d” with containers from yard-                          defined as a multi-objective integer programming, for
               Z. Ning and M. Weijian: A Model for the Stowage Planning of 40 Feet Containers at Container Terminals               49

which we obtained a set of non-inferior solutions by us-                  yards. European Journal of Operational Research,
ing the weighting method. For the sake of being prac-                     124(1), 89-101.
tical, the model not only considers the conflict between            Kim, K. H., Kang, J. S. and Ryu, K. R. (2004) A beam search
ship stability and containers' reshuffling operation but                  algorithm for the load sequencing of outbound containers
                                                                          in port container terminals. OR Spectrum, 26(1), 93-116.
also first takes into account the moving frequency of yard
                                                                    Imai, A. and Miki, T. (1989) A heuristic algorithm with
cranes, the probability of wait by quay crane and the
                                                                          expected utility for an optimal sequence of loading
feasibility of multi-YC feeding one QC during the load-                   containers into a containerized ship. Journal of Japan
ing process. A wide variety of experiments demonstrated                   Institute of Navigation, 80, 117-124.
that the solutions by this formulation were acceptable              Imai, A., Nishimura, E., Sasaki, K. and Papadimitriou, S. (2001)
for practical use.                                                        Solution comparisons of algorithms for the container-
                                                                          ship loading problem. Proceedings of the International
                ACKNOWLEDGMENTS                                           Conference on Shipping: Technology and Environment,
                                                                          available on CD-ROM.
      The authors wish to thank the terminal management             Imai, A., Nishimura, E., Papadimitriou, S. and Sasaki, K. (2002)
office of the terminal SMCT for the fruitful support.                     The containership loading problem. International
                                                                          Journal of Maritime Economics, 4(2), 126-148.
Special thanks are to the anonymous referee for the valu-
                                                                    Imai, A., Sasaki, K., Nishimura, E. and Papadimitriou, S. (2006)
able remarks and helpful comments and suggestions.
                                                                          Multi-objective simultaneous stowage and load planning
                                                                          for a container ship with container rehandle in yard stacks.
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