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1 INTRODUCTION

Transportation safety is one of the most important

problems in shipping. The problem consists of various

tasks one of which is ensuring that ships carrying

cargoes are kept seaworthy at any point of the route.

Container shipping has historically had razor thin

margins and the current pandemic has also cut

terms of capacity went up by 10.9% and the global

shipping fleet has increased by 4.1% which is the

highest since 2014. Cost cutting measures make the

problem of keeping ships seaworthy even more

relevant.

One of key components in keeping a ship

seaworthy is a correctly prepared cargo plan.

Considering the abovementioned factors, relevance of

optimizing such a plan is evident. Container ship

stowage planning problem is known as MBPP (Master

Bay Plan Problem) and its detailed description can be

original work includes a simplified integer

programming model solved by dividing the task in 3

parts. The first part consists of excluding container

positions that don’t satisfy some strict requirements.

Such as excluding reefer containers positions that

aren’t close by electrical sockets from the total set of

solutions. The second part separates containers in

accordance with their destination ports into different

bays; the containers destined to closest ports are

stowed closed to the ship’s center. The resulting

ABSTRACT: One of key components in keeping a ship seaworthy is a correctly prepared cargo plan.

Considering the recent cost cutting measures and reduced transportation volumes relevance of optimizing such

a plan cannot be understated. Though there’s a number of studies addressing the issue none of them covers all

the operational and constructional constraints necessary to factor for. This article presents an integer model that

tries to address some of the constraints missed by other researches. A method for solving the model is designed

and developed based on a steady-state genetic algorithm. A numerical experiment is conducted showing the

method’s feasibility.

Authors of [8] use a simplified containership

In [12] authors propose a method of stowing

Authors of [23] propose four integer models for

In [6] a vessel is simplified to a single m x n bay.

Authors of [15] use the model from [5] and particle

In [9] a genetic algorithm is used for optimizing a

Authors of [19] use a deep reinforcement learning

solution the strictest constraints are shifted towards

the beginning of constraints list which allows to

disregard incorrect solutions faster.

models are proposed for solving the problem.

Finally, authors of [13] use a tabu search algorithm

multi-objective optimization which tries to minimize

the number of re-handles, empty container slots,

horizontal and longitudinal moments and a single

crane discharging time.

The models and methods mentioned above adopt

various simplifying assumptions that make them

applicable to a limited number of cases. The

assumptions include viewing all containers as TEUs

unresolved and requires a more detailed study.

2 MATHEMATICAL MODEL

This study is a continuation of [10]; it tries to improve

the model presented there and solve it using another

approach. A container of size t (0=TEU, 1=FEU),

IMDG class c (c=0 means there’s no dangerous goods),

is stowed in position (i, j, k) (i – bay number, j & k are

coordinates inside the bay, ) if xt,c,i,j,k=1.

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(7)

Constants, sets and variables in the model:

0.. , 1 ;

max max

k k wherek isthemaximumheightof baysinTEU+

t

s

w isanadditionalvariablethatisintroduced foreverys thlimitation−

Inequality (1) limits the total number of containers

to be stowed, (2) makes sure that every FEU container

is stowed in two TEU positions, (3) limits the number

of containers occupying the same position to one, (4)

ensures that no TEUs can be loaded on top of FEUs,

or stability requirements, such checks are planned to

be performed in the next stage of the study.

3 SOLVING THE MODEL

In order to try and solve the abovementioned model a

genetic algorithm was selected. Multiple authors used

variations of this approach [8, 9, 21] and showed its

feasibility. The Genetic Algorithm (GA), often referred

to as genetic algorithms, was invented by John

Holland at the University of Michigan in the 1970s

[14]. It imitates the process of natural selection and is

used to generate solutions based on mutation,

crossover and selection operations. The classic

repeated until the child population is entirely filled.

Children’s fitness is assessed and the algorithm stops

when it has reached a certain value or when a pre-

determined amount of time has passed.

There’s also an alternative approach called a

study. Its main difference consists of updating the

initial population instead of replacing it altogether.

Therefore, the children obtained in crossover are

The fitness function used creates a Boolean

coefficients matrix of the model based on the given

containers set. After that it converts the solution to a

Boolean variables matrix and multiplies the two. The

resulting matrix is compared to the inequalities’

constant matrix. Every equation not satisfied increases

parameters.

For selection a Tournament Selection algorithm is

chosen, which picks n individuals from the initial

population and chooses the fittest of those.

The testing is performed using a simplified ship

model consisting of 4 TEU bays without covers and 25

closed containers to be loaded.

The initial solution is formed via simply filling the

1st bay (Fig. 3).

As we can see two of the requirements are not

The resulting solution (Fig. 4) is obtained by the

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