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TONGMYONG UNIVERSITY

FINAL REPORT
DECISION ANALYIS
TOPIC: SELECTING SHIP REGISTRY: CLOSED OR OPEN

Students:

레레레레 – 19330110

레레레레레레레레 – 19330123
레레레레레레레 – 19330109

Instructor: PhD. Enrio Dagostini

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Submission date: 16 June 2022


ABTRACT
Ship registration is an official procedure that puts on a flag for the ship and allows it to trade
commercially. This study is aimed at figuring out whether a ship owner should register his ship in a
closed registry or an opened registry and which factors are the most important in the deliberation
process. There are 2 alternatives which are open registry and closed registry and 8 criteria that were
selected by researching previous studies in the ship registration field, by using Best – Worst Method
(BWM), we have found that ships should be registered in opened registries. Shipbuilding standard is
the most important factor to be deliberated followed by taxation.

Key words: ship registration, open registry, closed registry, best – worst method.

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Table of Contents
ABTRACT..........................................................................................................................................1
LIST OF FIGURE.............................................................................................................................3
LIST OF TABLE................................................................................................................................4
LIST OF ABBREVIATIONS...........................................................................................................5
I.

INTRODUCTION......................................................................................................................6

II. LITERATURE REVIEW..........................................................................................................9
III. METHODOLOGY...................................................................................................................14
IV. DATA COLLECTION..............................................................................................................16
V. DATA ANALYSIS.....................................................................................................................17
STEP 1. Formulation of the problem.........................................................................................17
STEP 2. Find the best and the worst criterion........................................................................... 18
STEP 3. Find the preference of the best criterion over all other criteria...................................20
STEP 4. Find the preference of all other criteria over the worst criterion.................................21
STEP 5. Estimate optimal weights............................................................................................ 21
STEP 6. Final scores of alternatives..........................................................................................23
VI. RESULTS...................................................................................................................................27
VII. CONCLUSION........................................................................................................................28
REFERENCES.................................................................................................................................30

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LIST OF FIGURE
Figure 1. Top 10 flag States in 2019................................................................................................8

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LIST OF TABLE
Table 1. Overview of respondents................................................................................................. 16
Table 2. Other to Most Vector........................................................................................................19
Table 3. Other to Worst Vector...................................................................................................... 20
Table 4. Optimal weights...............................................................................................................22
Table 5. Importance of alternatives under each criterion (respondent 7)...................................... 23
Table 6. Normalized Values for respondent 7................................................................................24
Table 7. Priority of alternatives (respondent 7)............................................................................. 24
Table 8. Priorities for ship registration.......................................................................................... 25
Table 9. Priority of alternatives under each criterion (full sample aggregate level)......................26

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LIST OF ABBREVIATIONS
IMO
UNCLOS
BWM
FOC
ILO
AHP
STCW
SOLAS

MARPOL
IOPP
MCDM
ANP
SWOT


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I.

INTRODUCTION

The concept of freedom of navigation prevails in the high seas, according to the UNCLOS,
which implies that the high seas are accessible to all states. There are state-specific issues
addressing the concept of freedom under this agreement, which involves the development of
topics such as freedom of navigation and overflight. Within the high seas, States cannot assert
sovereignty or authority. This unfettered access to the high seas applies to all vessels. However,
in terms of freedom of navigation, the UNCLOS of the Sea and public international law establish
a framework that forbids conditions aboard vessels.
That is a reason why every vessel should have a nationality because of its registration under a
state, which can allow the vessel to be registered under its regulations and utilize the high seas.
Significantly, a national flag is an official signature proving that a country assigns and authorizes
a vessel or vehicle, which is under its jurisdiction and control in the administrative, technical,
and social spheres of that country. With the increasing demand of economic development, the
purpose of our country's ships flying foreign flags for transportation and circulation purposes has
led to the fact that many goods and exploitation at sea have also become bustling and developed.
Therefore, the registration and flying foreign flags is only a form as well as a measure to make
circulation and anchoring much more convenient at international ports.

The laws of States regarding the registration of ships are divided into two categories include
open registry and closed registry. For the closed registry, it is administered by an individual
country as a national registry for the registration of their own ships flying their own flag, owned,
operated and manned by nationals of that country. In this field, the owner of the ship should
necessarily be from the country of registration and the place of business should be in the country
of registration. This registration also means that these ships may be requisitioned at time of war
for the transportation of goods and people in the service of the nation.
On the other hand, it is common for maritime registries to display open registries in certain countries,
although the owners of vessels can register their vessels under that country's flag despite the fact that
there is no genuine relationship between the flag state and the vessel. Open registers allow ship
owners of other nationalities to flag and operate ships under their flag. It is generally the case that the
flag of that country will appear on the vessel. Moreover, the laws of that country

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will generally apply to the vessel and its crew while on the high seas. To be specific, a ship could
be traded more easily than the others if it registered with any registry mentioned above. The
second one is that registering the vessel in a different country than the one where the owner lives
help them save money and reduce operating costs. Moreover, it helps achieving lower
administrative costs; in some cases, it also allows the ship to operate under less stringent
regulations of the owner’s country. As importantly, the ship owners are beneficial from tax
incentives (tonnage tax), certification and security. To be clearer, tonnage tax is a special tax
regime for shipping companies under which a notional profit is calculated based on the number
and size of ships operating and in which a standard corporate income tax rate is applied. It is
sometimes argued that open registers are associated with comparatively liberal tax regulations,
labor rules, and environmental protection legislation, which are often lax and easily
circumvented. Panama and Liberia are the most popular open registry countries because they do
not require citizenship or residency requirements for ship owners. The registration process in
these countries is comparatively simple, largely online based, and offers ship owners tax-free

money and inexpensive labor. Registration providing ownership for a ship is very important for it
to be able to participate in international trade relations, bringing benefits to the nation.
However, besides the benefits, there are still disadvantages that need to be mentioned to choose the
best option for ship owners. On the contrary, sailing a ship under the flag of convenience could have
serious repercussions for seafarers working on the vessel. Initially, ships sailed under open registry
have been claimed to have more catastrophic working conditions than the others since such kinds of
ships are operating under less stringent regulation. There is a disagreement existing between the
jurisdiction and the appropriate regulations of ship operations under the flag of convenience. The
next one is that seafarers are more at risk of being forced to work in severely hazardous situations
with inadequate insurance and compensation in money. Moreover, the disastrous drawback of sailing
a ship is that it does not mandate fundamental seafarers' rights, such as the right of free creations or
trade union organizations, the demand of achieving adequate salaries, health and safety workplace
conditions, wage scales and so on. The privileges mentioned above are reserved for sailors and may
not be available on ships. Finally, seafarers may not be allowed to take a nap between consecutive
transportation duties as per standard guidelines.

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Figure 1. Top 10 flag States in 2019
This figure above illustrates that this is top 10 flag states in 2019. It is not surprising that Panama
remains at the top of the list by far, with the highest tonnage and boats registered under the flag,
followed by Liberia and Marshall Islands. According to the (ListLloyd's, 2019) Panama has
9,367 vessels flying its flag, some 4,486 ships ahead of China, the next-biggest in terms of
number of vessels. On the other hand, only 3 countries including China, Hong Kong and Greece
still apply closed registry, the remaining are open registry.
Although there have been a lot of research on selecting ship registry ships, our research is one of the
first to use the best-worst method to find out which criteria are most important and least important. It
is highlight that some researchers applied various methodologies, e.g. fuzzy, SWOT, and AHP, to
investigate ship flag choice problems in the past. Few used ANP method to study ship flag selection

problems. (Celik & Kandakoglu, 2012) used a fuzzy quantified Strength, Weakness, Opportunity and
Threat (SWOT) to analyze the case of the flagging-out dilemma in Turkey. Perepelkin, et.al (2010)
proposed an improved methodology to measure flag performances for the shipping industries of
various maritime countries in the world. Kandakoglu, et.al (2009) introduced a multi-methodological
approach for shipping registry selection in maritime transportation industry. Lua, Fan & Li (2013)
used individual ship registration data to analyze flag choice

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behavior, used a binary choice model to analyze flagging-out decision, and used a nested logit
model to analyze final flag choice.
From a ship owner’s point of view, they need to understand what factors influence the decision to
register a ship. Because when competing in the international shipping market, it is imperative for
ship owners to come up with cost-effective strategies to operate their ships, choosing a flag is a
key step for a shipping operation in success. That is a reason why two questions arise here:
“Which registry should the ship owner register their ship in?” and “How important are the
factors that affect the ship registration decision?”
Having researched previous studies, our team identified 8 criteria for 2 alternatives of open and
closed registry: ship building standards, standard living and working conditions, crew’s wage,
availability of well – trained seafarers, ship’s type, trading routes, tax, and bureaucratic control.
This study solves the above problems by applying the Best-Worst Method to find out what are the
factors affecting the ship owner’s decision to register a ship. A key advantage of this study is that it
uses data from leading professors in the shipping industry with highly specialized knowledge.

In the next section, we will first discuss the existing documents on closed registry and open
registry, thereby identifying the factors that influence the decision to register ships as well as the
purpose of the ship. Section 3 will introduce our methodology, including an explanation of the
methods, who developed them, and the areas in which it has been applied. This study design
process will also be detailed in this section. Section 4 describes data processing. Section 5

reports on the estimated outcome model and data analysis, followed by section 6 is result of the
research and section 7 concludes this paper.
AI.

LITERATURE REVIEW

Ship registration is the first step for the ship to be able to trade commercially. Without a flag
hanging on top of the ship, she is not accepted by any ports around the world. So that, choosing
the type of registry is the first and foremost strategy in operating a ship.
There are 2 main types of ship registry: closed registry and open registry. Closed registration is the
fundamental type of ship registration in the maritime industry. After that, the open registration

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appeared with a new term of “flag – out” which means that ships are registered in another
country rather than the national one.
So why do ship owners flag out their ships?
Ship operator’s main goals are to operate vessels as efficiently as possible with the minimum
costs. In developed countries, open registry is an option to outsource the cheap labor market and
“circumvent safety regulations” (Gregory, 2012). For example, once an American – owned ship
is registered in Panama, an open registry or to be more specific, a flag of convenience, the ship
does not constraint on complying with the America’s rules and regulations but that of Panama.
Open registry or flag of convenience usually has similar characteristics: low operation cost and
cheap labor. Ship owner can deliver goods at a much lower cost than registering the ship in
developed countries by decreasing the crew's standard of living and working conditions
(ITFseafarers). According to Judith Swan, open registries offer several benefits to ship owners:
“the low or no vessel restrictions; favorable tax environment; low administration and
registration fees; no or easy to meet nationality requirements; quick and efficient registration
process; flexible manning requirements” (Swan, 2002). Levy no or low local taxes on income is

one of the most significant benefits of open registries.
However, there have been several controversial issues surrounding the open registration. From the
regulation perspective, 2 kinds of registration apply opposite rules: closed registries require high
shipbuilding standards, crew qualification, or proper crew welfare while open registries enforce
maritime laws and regulations rigorously, others do so only infrequently. The front one considers
safety as the most important concern for governments that enforce rigorous standards, whereas the
latter cares about revenue from ship registration (Gianni, 2008). However, because of the emphasis
on profit earning, open registries and ship operators in those countries ignore a vital element:
maritime security, which can lead to “sub-standard shipping, poor performance on safety,
maltreatment of crew” (Gianni, 2008). According to Bye and Aaberg (2018), flag state can be used
as a risk measurement and ships fly under FOC (Flag of Convenience – a type of open registry) tend
to be involved in maritime accidents. “The rise of the flags of convenience created an international,
deregulated, and non-unionized segment of merchant shipping, which as a competitor on the global
market had far-reaching consequences for the nation-state-based

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organization of the maritime transport sector, for the national fleets of the industrial countries,
and for central national trade union activities (from economic and wage bargaining policies to
organizational policies)” (Koch-BaumgartenSigrid, 1998). By sailing under open registry’s flag,
the ship can avoid the unions which set up basic rules for safety, security, or seafarers’ right and
wage so that they can cut down on facility and compensation cost.
Selecting between two types of registries is similar to scaling the benefits and drawbacks of them
in terms of costs, compulsory regulations, and maritime security.
We looked at some previous studies which share the same topic in order to identify the criteria of
selecting ship registry.
In 2009, Celik, Er and Ozok sorted the factors considered in shipping registry selection into 3 main
categories: economic, social, and political. By using the methodology of Fuzzy AHP, they found that
bank finance and tax related expenses in the economic considerations are the most important

elements in the decision-making process (Application of fuzzy extended AHP methodology on
shipping, 2008). Kyriaki and Michael (2016) have a similar result in category of economic that
“country’s tax system, the ease of starting new business and the country’s corruption/ transparency
level” can affect the flag-out ratio. Flags must not only have an investor-friendly tax structure in
place, but also demonstrate skill in running it smoothly and efficiently, resulting in a well-organized
and accommodating tax environment for ship owners. The study indicates that ship owner can decide
whether to flag – out by considering 5 factors of quantitative: “age of the vessel, the trade on which
it is engaged, basic wage costs, national insurance payments, and training costs”. To be more
specific, although basic wage expenses may have a negative impact on the choice, it is the overall
employment costs that are important. In qualitative part, crew salaries, “availability of skilled
labor, and fiscal reasons” are among the qualitative variables influencing the usage of a foreign flag
(1998) In Angela and Peter’ point of view, crew expenses were cited as the most prevalent reason
given by firms that elected not to select national flag. Other elements that had impacted the choices
were: the desire to avoid bureaucratic supervision; the high expenses of complying with national
flag regulations; the unavailability of competent labor and fiscal considerations. Those firms who
decided to fly under closed registry indicated that

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their decision was influenced by a variety of factors, including ship’s type, trading routes,
public relations concerns, marketing, and historical considerations.
Overall, main factors that influence ship registration can be listed as follows: crew’s wage,
seafarers’ living and working conditions, availability of well – trained seafarers, ship’s type, ship
building standards, trading routes, tax, and bureaucratic control.
Crew’s wage relates to the salary of the seafarers that is calculated by the days staying on board
of the ship. Generally, there is a huge gap between the salaries of developed - country seafarers
and developing – country seafarers. For instance, the average salary of a seaman ordinary in
Philippines is about $5,258 while that in Japan is $32,544 ( ERI's Salary Expert Database). So
that ship owners that base in Japan must carefully consider whether to register their ships in

Japan, a closed registry, or an open registry which allow them to outsource the seafarers.
Standard living and working conditions refer to the quality of life on board of the ship: food,
drink, working and resting hour, medical treatment, accommodation, etc.
Those two criteria, which related to seafarers, are protected by International Labour Organization
(ILO) by Maritime Labour Convention (2006) so that “the world's seafarers will be given the
freedom of movement necessary for their well-being and for their professional activities and, in
general, to facilitate international commerce” (ILO).
There are two main kinds of seafarers: national and non – national seafarers. Availability of well
– trained seafarers can affect the decision of out – sourcing the crew. Nowadays, most seafarers are
required to have some conventions in other to work on board of the ships, one of the basic
conventions is STCW - International Convention on Standards of Training, Certification and
Watchkeeping for Seafarers (STCW). The goal of STCW is to maintain safe working environment
for the seafarers and enhance their professional performance in the duties on – board. STCW
includes several certificates for master, chief mate, radio operator and other positions in terms of
basic safety training, medical fitness, firefighting, etc. (International Transport Workers’

Federation).

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Depend on the type of the ship, the ship owner can register in national or opened registry that
brings them the most practical benefit. For instance, if the ship is a fishing vessel which mostly
trades within that national trade water, closed registry is a more appropriate choice.
Trading routes is the shipping networks on the sea between continents, countries, or ports. It is a
matter in ship registration when vessels only transport goods from one port to another port in the
same country.
Tax is one of the most important factors as it directly effects on the ship owner’s profits. Tax rates
also depend on the conditions of the vessels such as age or type. Each country has a different tax
system for ship registration. For instance, when a new built or a secondhand ship is registered in

Panama, the ship owner must pay for 3 basic fees: consular fee, registration fee and annual tonnage
tax for the first year. With each type of ship, the formula for tax calculations is different but overall,
there are 5 basic fees: annual tax, annual consular tax, annual inspection fee, annual investigation fee
and 3% per Net Tonnage. Vessels which make efforts to environmental preservation is offered annual
tax discounted up to 50% for 3 years (Services Merchant Marine).

Shipbuilding standards refer to the required equipment or technology of the ship. For some
registries, like Hong Kong, complying with an IMO’s convention is deemed to have met the
countries’ standards (Mayer Brown, 2008). “The main objective of the SOLAS Convention is to
specify minimum standards for the construction, equipment and operation of ships, compatible
with their safety” (IMO, 1974). Here some SOLAS’s certificates relate to shipbuilding standards:
Navigation License
Passenger Ship Safety Certificate
Cargo Ship Safety Construction Certificate
Cargo Ship Safety Equipment Certificate
Cargo Ship Safety Radio Certificate
Cargo Ship Safety Certificate

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Class Certificates (Ports and Maritime Affairs, 2016).
Complying with the International Convention for the Prevention of Pollution from Ships
(MARPOL) is another requirement, when a new built vessel follows MARPOL, it is then issued
a certificate called IOPP (Wartsila).
“Bureaucratic control which is defined as an approach that emphasizes organizational authority
and relies on administrative rules, regulations, procedures, policies, standardization of activities,
well-defined job descriptions, and other administrative mechanisms to ensure that employees
exhibit appropriate behaviors and meet performance standards. Parties in maritime industry are
expected to adhere closely to the international conventions and stay within the guidelines”

(Asyali, 2005). An example of bureaucratic control is Port State Control. Port State Control “the
inspection of foreign ships in national ports to verify that the condition of the ship and its
equipment comply with the requirements of international regulations and that the ship is manned
and operated in compliance with these rules” (IMO, Port State Control).
By using Best – worst method, this study aims to identify the importance level of each criterion
which affect the ship registration choice between open and closed registries. Our team planned
on collecting data on a survey using Google form and sending the questionnaire to 10
correspondents who are professors or having maritime related academic knowledge in 2
countries Vietnam and Korea.
The results can be used by ship owner, ship operator or any other stakeholders in maritime industry
that are still wondering about ship registration, especially the two mentioned ship registries.

Review studies from a long time ago so some criteria might not be such a matter for ship
registration nowadays. For example, availability of well – trained seafarers are no longer a
problem as all seafarers are required to have basic certificates to work on board of the ship such
as STCW (basic training) and other kinds of certification for each position.
BI.

METHODOLOGY

When a problem is so complicated that it can only be modeled as a hierarchy or network, most
“Best-worst method” fall to arrive at a solution. In a specific, ship registration is a multi-criteria
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concept, therefore a multi-criteria decision-making method (MCDM) could be used to assess the
relevance of the various criteria. Specifically, MCDM methods have been applied to solve many
problems in various fields including sustainable supply chain management, green port and so on.
In this study we use the phrase “BWM” which is known as “best worst method”. This kind of
method brings unique advantages for this paper, which has not been used in this area before. The

method’s description is elaborated in the next section.
With regards to decision-making, this is generally defined as the cognitive process of selecting an
alternative from a set of alternatives. Besides, A MCDM problem is a problem where a decisionmaker has to find one of the most suitable alternatives from a set of those considering a set of
criteria. In order to address this problem, the BWM which means “Best worst method” has been
applied. To be more clearly, this is a multi-criteria decision-making method which was developed by
Dr. Jafar Rezaei from Delft University of Technology in 2015. This kind of method is based on a
systematic pairwise comparison of the decision criteria used to evaluate a set of alternatives.

The most significantly unique of BMW is that it requires less data than other MCDM methods.
To be more specific, BMW requires less pairwise comparisons than AHP, with 2n3 and n(n1)/2
comparisons, respectively. “The pairwise comparison method works by each alternative being
compared against every other alternative in. Typically, all possible pairs of alternatives are
pairwise compared and pairwise ranked against each other” (Pairwise Comparison).
Moreover, the results of the data analysis using BMW are more reliable than that using AHP as it
provides more consistent comparisons. Consistent ratio refers to the reliability of the data.
According to Chu and Kuang (2002), the consistency in research should be <0.1, especially
when analytic hierarchy process is used as the results might be different if the consistency ratio is
>0.1. The creator of BWM stated that consistency ratio in this method is used to measure the
level of reliability because results from BMW are always consistent (Rezaei, 2015) Similar to
other BWM studies, the method develops in the six steps:
(1) Formulation of the problem;
(2) Find the best and the worst criterion;
(3) Find the preference of the best criterion over all other criteria;

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(4) Find the preference of all other criteria over the worst criterion;
(5) Estimate optimal weights;
(6) Final scores of alternatives.


In order to find out what is the best or the worst criteria by using this kind of method, this has
been already utilized in several real situation problems. For instance, (Kusi-Sarpong, Rezaei,
Ahmadi, 2017) have used BWM to determine the most important social sustainability criterion in
supply chains management. (Wankhele & SVinodh, 2021) developed a framework for the
automotive sector and analyzed the identified challenges to derive rank for systematic
implementation in Indian automotive industries. Another example of BWM application include
identifying the appropriate governance model for green port management to ports in the Indian
Ocean Rim relying on social systems engineering principles. (Ziaul, Henrik, Mariia, 2020)
IV.

DATA COLLECTION

We collect data by sending questionnaires in Google Form via email, Zalo.vn or Facebook for
our professors, teachers and people who are working in relation to Maritime and Shipping
Industry. There are 20 questionnaires that were sent but we only received 10 responses. 1 of them
was rejected as the respondent rated only 1 for the question how much he knows about maritime
st

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industry or ship registration. The data was collected in 3 weeks from 1 May to 20 May 2022
and here is an overview of respondents:

Respondent
1
2
3

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4
5
6
7
8
9

Overall, respondents from Vietnam and Korea share nearly equal proportions with 5 and 4,
respectively. So, we expect if there is any different in the result between people from the two
countries. Most of them are from Port and Shipping industries and have the affiliation in
Academic. Only 1 respondent is from the industrial – commercial with a bachelor’s degree in
Chartering Department. All respondents have the year of experience more than 11 years.
V.

DATA ANALYSIS

Empirical application of the BWM
The topic of our group is to evaluate the appropriate kind of ship registry. There are 2 main types of
ship registry: close registry and open registry. Close registration the fundamental type of ship
registration in the maritime industry. After that, the open registration appeared with a new term of
"flag-out" which means that ships are registered in another country rather than the national one.

STEP 1. Formulation of the problem
In this first step, our group identified a set of criteria to describe the subject matter. This section
throws more light on the developmental and refinement processes of the framework proposed in
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this assignment. The criteria were identified through a combination of a literature review and
previous studies, we identified 8 criteria which are important in ship registry selection. They are:
(1) Ship building standards: refer to the required equipment or technology of the ship.
(2) Standard of living and working conditions: refer to the quality of life on board of the

ship: food, drink, working and resting hour, medical treatment, accommodation, etc.
(3) Bureaucratic control: Control of an organization and the individuals that make up the

organization through systems of standardized rules, methods, and verification procedures.
(4) Crew's wage: relates to the salary of the seafarers that is calculated by the days staying on

board of the ship.
(5) Tax: the amount to be paid to the country of registration of the ship.
(6) Trading routes: routes are the shipping networks on the sea between continents, countries,

or ports.
(7) Availability of well – trained seafarers: affect the decision of out – sourcing the crew.
(8) Ship's type: Type of the ship which is registered.

STEP 2. Find the best and the worst criterion
In step 2, respondents were asked to identify best and worst criteria for ship registration. In
particular, respondents were asked to answer the following: “which of the eight criteria is the
most important for ship registration?” and “which of the eight criteria is the least important for
ship registration?”.
The most important means best and the least important means worst.

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Table 2. Other to Most Vector


Respondent

Best

1

Tax

2

Shi
build
standa

3

Tax

4

Shi
build
standa

5

Shi
build
standa


6

Tradi

rout

7

Tradi

rout

8

Bureauc

contr

9

In this table, the respondents were asked to specify the best criterion’s preference over all other
criteria, using 1 to 9 measurement scale. The most important criterion is scaled as 1. We can see
that ship building standards and tax are chosen the most so they are the 2 most common best.

Shi
build
standa



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Table 3. Other to Worst Vector

Respondent

1
2
3

4

5
6

7
8
9

In this step, the respondents were asked to determine the preference ratio of all criteria over the
least important criterion via a questionnaire, again using a measurement scale of 1 to 9. The least
important criteria are scaled as 1. The 2 most common worst are ship’s type and crew’s wage
STEP 3. Find the preference of the best criterion over all other criteria
To obtain the scores in table 2, respondents ranked the importance of the best criterion over all
other criteria using a 1–9 scale. Hence, we can express the best to others vector as:


=(


)

1, 2,….,

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Here,

indicates the preference of the best criterion B over the criterion j.

STEP 4. Find the preference of all other criteria over the worst criterion
In the next step, respondents were asked to rank the importance of all other criteria over the
worst criterion on a scale from 1 to 9. Hence, we can express the others to worst vector as:
=(

)

1 , 2,….,

Here,

indicates the preference of the criterion j over the worst criterion W.

STEP 5. Estimate optimal weights
In this stage, we minimize the maximum absolute differences (| − |, |
The minimization problem can be expressed as follow:
min [max(|

∑


=1

−

|, | −

−

|) for all j to find the optimal weights of a criterion.

|)]

(3)

s.t.
≥ 0,

The, we can solve (3) as a liner optimization model shown in (4).
min
min
s.t.
| −

|

|≤

,


−

|≤ ,

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