599

1 INTRODUCTION

One can examine statements like; “student X is a good

one” or “point (x, y) is a true location of the ship”.

Evaluating the first, we should consult the student’s

teacher for his subjective opinion. Belief, uncertainty

and plausibility measures are expected as an educator

answer. Second hypothesis is of the same type but

appears not so easy. Although one can ask nautical

experts for their help, nonetheless this way of conduct

seems inadequate. Primarily one has to point at

reference data. Statement changes a bit and takes the

form “is the point (x, y) a true location of the ship given

a set of indications delivered by various accompanied

navigational observations”. Two-dimensional problem

can be solved considering each coordinate separately.

In navigation problem of assessing the belief,

uncertainty and presumption of the x coordinate as the

true location in proximity of a given abscissa. The

assessment here is not subjective. The opinion should

be based on the indications of a certain positioning

system. The necessary knowledge available to an

experienced bridge officer is usually based on available

samples of observed, random indications of such a

system – see Figure 1 for illustration. It should be

added that the interesting location here is

approximately the point, not exactly in it. In the latter

case, having density of distributions, inference in

relation to a specific point is not possible.

Proposed way of verifying the statement exploits

collected sets of instances generated by the

navigational aids. Uses fuzzy systems and methods of

multi-attribute decision-making. Contribution from

Nautical Knowledge Extraction and Decision Making

W. Filipowicz

Gdynia Maritime University, Gdynia, Poland

ABSTRACT: One frequently encountered decision-making problem is the evaluation that boils down to judging

hypotheses. Typically, we determine whether they are true or false, although we may also have doubts.

Hypotheses can be statements of various kinds. For example, we may wish to classify a given student as belonging

to the category of good students. Mentioned hypotheses are related to different disciplines, quite often seemingly

uncorrelated. To confirm this hypothesis, we would most often refer to the subjective opinions of their teachers.

A similar issue arises in nautical science; for instance, consider the problem of identifying a particular location as

the most probable one where an observer is situated. Accompanied establishing ranges of the true, false and

uncertain statement might be subjective. Objectivity could be also considered provided stored sets of instances

are available. Expected are adequate functionalities of software tools at hand. Functional aspects tends to increase

nowadays. Random observations are usually accompanied by methods rectifying knowledge regarding their

behaviour and quality. Available data are explored in order to extract necessary parameters required within the

inference schemes of evaluating the hypothesis truth.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 19

Number 2

June 2025

DOI: 10.12716/1001.19.02.32

600

belief theory results in ability of the data informative

context enrichment. The proposed approach refers to

belief structures and the mechanism of evidence

aggregation. The association of encoded fragments of

evidence is a little-known and often undervalued

mechanism. It allows for the enrichment of the

informational context of the components. It provides a

formalized framework through which we can obtain an

answer regarding the cumulative assessment of a given

hypothesis in light of opinions originating from

various sources, often differing in reliability.

2 UNCERTAINTY MODEL

Binary logic rules for the two-state distinction of values

[true, false]. Fuzzy systems of diversification switches

in understanding these values. The statement can be

true/false to some extent. A widely possible interval

model for introducing belief, plausibility and

uncertainty of the hypotheses being true. Suppose we

ask a teacher for an opinion about a certain student.

Only in a few cases will we hear an opinion that clearly

states a high grade for a student. The evaluator will

more likely state that based on the exams conducted,

the student is good, but while direct contacts and

meetings during practical classes, he believes that, the

opinion expressed is not entirely true. The teacher has

doubts reaching a certain value that allows him to

assess the student as good. Above this level, he would

not say that the student belongs to this category. In the

example given, we have three ranges of values:

conviction or belief, uncertainty and presumption or

plausibility, above which we have impossibility. The

state where the truth of the statement is not allowed. In

practice, interval notation [a, b] is used. The value of a

denotes the level of conviction, 1-b is the impossibility

interval, and b-a is the range of uncertainty.

Determining the appropriate interval involves

objectively assessed exams, as well as a subjective

evaluation of the entirety of achievements. From an

axiological point of view, the most important thing is

conviction. The range of uncertainty, although

significant, is of lesser importance. The construction of

a hierarchy in a set of elements should be based on the

values of belief. This problem is of particular

importance during the aggregation of assessments or

opinions.

Figure 1 shows two sets of instances distributions,

their conventional and modern histograms with

examples of selected abscissas. Considering each

coordinate of a location separately thus solving two

single-dimensional problems is an approach enabling

verification nautical hypothesis referring to the

observer location. Moreover relying on density

functions we should modify the hypothesis, finally it

takes the form of: “is the neighbourhood of given

abscissa a range where the true x-coordinate of the

observer position is located given a set of indications

delivered by various accompanied navigational

observations”.

It should be stressed that the area of interest is the

neighbourhood of a point, not the point itself. In the

latter case, if we rely on histograms representing

distribution densities, reasoning about a specific point

is not feasible.

Data for a certain navigation system are shown in

Figure 1. The history of the system's readings is shown

as a set of points located around the identified location.

The distribution of the x-values of these locations are

shown as histogram, although modern continuous

density graph is also included. The figures included

next to the drawing represent vertical and horizontal

assessments of the distribution.

Figure 1. The set of instances distribution, its conventional

and modern histogram with examples of selected abscissas

and quality parameters

Included figures refer to assessment of the

distribution. The first indicator reflects the variation in

the heights of the histogram bars. A higher value is

preferred, as it indicates a better-formed structure. The

second value refers to the width of the histogram,

which in the presented case is the width of a single bin.

Systems with a smaller range of dispersion are

favoured, as they indicate lower uncertainty in

readings. The two indicators for evaluating a given

system are of different types: the first is qualitative—

higher values indicate better assessments. The second

is cost-type—lower values are preferred.

MADM (Multi-Attribute Decision Making) is a field

of knowledge where such contrasting cases are

recognized. Unified methods for treating such values

are proposed [9]. It is often that block of the

characteristics of observations made at the same time is

available. Thus upgrading hierarchy among its

elements is of primary importance. Deferring further

detail for now, it should be known what the given

system indication quality is. At the same time, the

relative belief in the system’s reliability is assessed. It is

at this point that the fundamental difference appears

between the problem of evaluating a student and

determining location based on readings from various

systems. The second case involves geometric

relationships. What is sought is belief and plausibility

regarding the neighbourhood of a given x-coordinate

as the area of the most likely observer location. These

attributes result from the analysed reading but also

depend on geometry, the relative position of the

reading and the considered coordinate [1]. Belief is

therefore variable. Figure 1 also contains two lines

associated with two example x-coordinates. The

proposed method for calculating plausibility refers to

continuous functions. Since histograms do not satisfy

this condition, they require transformation—a concept

previously proposed by the author [2][3].

601

For a system, the maxima of belief and plausibility

regarding the location of the coordinate around which

the observer's position is most likely are identified with

converted histograms. As the distance from the point

of maximal values, increases, both of these indicators

decrease. A reduction in belief has been assumed with

increasing distance from the maximum of the

continuous histogram. The value of uncertainty is a

characteristic feature of the distribution and remains

constant, regardless of the considered abscissa. The

sum of belief and uncertainty gives the value of

plausibility, which therefore decreases. In contrast, the

value describing the impossibility of supporting the

considered hypothesis increases. This implies that the

true location coordinate should be sought elsewhere.

2.1 Simple Belief Structures and Their Aggregation

A practical problem belonging to the decision-making

category may look as follows. Two sources provide

evaluations of the same student. One of the student’s

teachers claims to be convinced that the student is

good. Based on exam results, he assesses his belief level

on a [0, 1] scale at 0.45. Based on personal interactions,

he concludes that the student could be considered good

up to a level of 0.85. However, beyond that threshold,

the student should no longer be regarded as this

category. The second evaluator is more sceptical. He

ultimately sets his belief at 0.20 and suggests a wider

uncertainty interval, which in this case should be 0.65.

In practice, a single assessment is required—one

that consolidates all available opinions. Such a

cumulative evaluation provides a more accurate and

complete picture of the student being assessed. The

parameters of this resulting evaluation might be

calculated using non-null generating conjunctive-like

aggregation of the available belief structures.

Each opinion allows for the definition of an

individual, simple belief distribution. The two simple

belief structures, based on the available evaluations of

the student, are shown in Table 1. The column labelled

{T} contains the belief values for the statement; the

student is good. The second column {¬T} includes data

regarding the impossibility that the student is good.

The last column {T, ¬T} contains the uncertainty, i.e.,

doubts regarding the assessment.

Table 1. Two belief structures on statements regarding

student evaluations

Structure

{T}

{¬T}

{T, ¬T}

0,450

0.400

0,150

0,200

0,150

0,650

{T} belief values for the statement; the student is good

{¬T} impossibility that the student is good

{T, ¬T} uncertainty that the student is good

Two belief structures of the presented forms,

referring to the same domain of discourse, can be

combined. [6][7]. This leads to an enrichment—

compared to the original assignments—of

informational content. The aggregation (association) of

two structures results in a belief distribution with

cumulative content, characterized by a richer

informational context.

The elements of the resulting structure are

calculated from the intersections of each pair of subsets

from the aggregated structures. The intersection of two

sets may be empty. The subsets {T} and {¬T} share no

common elements; in such a case, the aggregation

operation yields the union of the arguments—i.e., the

set {T, ¬T}. In the context of this application, this

corresponds to uncertainty. All of this is consistent

with a method proposed by Hau and Kashyap [5]. This

type of association can be described as

conjunctive/disjunctive, characterized by the absence

of empty values. The masses associated with the

resulting sets are obtained as products of the masses of

each paired element being combined. This basic

scheme of conjunctive association is most easily

implemented using a two-dimensional

representation—an example is shown in Table 2.

Table 2. Conjunctive-like combination of two simple belief

structures on statements regarding student evaluations

Structure II

set mass

{T}

{¬T}

{T, ¬T}

result

0,200

0,150

0,650

Structure I

{T}

{T, ¬T}

{T}

0,45

0,090

0,068

0,293

0,413

{¬T}

{T, ¬T}

{¬T}

0,4

0,080

0,060

0,260

0,343

{T, ¬T}

{T}

{¬T}

{T, ¬T}

0,15

0,030

0,023

0,098

0,245

The number of columns in such a table corresponds

to the number of elements in one of the structures, and

the number of rows equals the number of elements in

the second assignment. Each element corresponds to

an event or option with a non-zero mass. Elements

assigned a mass of zero do not affect the aggregation

results, and for this reason, they should not appear in

the respective structure [4][7].

Table 2 presents the aggregation of the belief

structures based on the student evaluations. The first

few columns show data from the first evaluator. The

second evaluator’s opinions are shown in the first

rows. The table's interior contains the intermediate

results of the conjunctive/disjunctive aggregation for

each pair of elements. The final result, obtained by

summing the values assigned to identical sets, is

presented in the last column. The probabilistic-

possibilistic diagrams of the student assessments and

the result of their assembly are shown in Figure 2.

Figure 2. Diagrams of the student’s evaluations and the

combination result

Compared to the more favourable evaluation, we

observe a slight decrease in the belief that the student

is good (relative to the first evaluator’s value). The

uncertainty range increases (compared to the smaller

value), while the indicator suggesting that the student

should not be considered good decreases (relative to

the higher original value). It is worth noting that the

result confirms the subjective, common-sense

assessment of the student being evaluated.

602

2.2 Nautical belief assignments

Let us return to the nautical example. The problem lies

in seeking support for the hypothesis that the

coordinate of the observer’s true position lies in the

vicinity of a given x-value based on density diagrams.

Thus two sort of models are used. Student’s assessment

rely on possibilistic-probabilistic approach while

reasoning on position fixing exploits possibilistic-

density one. The agreement between the two

approaches is obtained by assuming a unit width of the

neighbourhood of a given abscissa. What is sought is

the total, cumulative support, derived from two

independent observations.

Figure 3. Distributions of the x-coordinates of the instances of

two systems, their densities in the form of histograms and

continuous functions, and an exemplary abscissa in the

vicinity of which the location of the true position of the

observer is determined

The situation is illustrated in Figure 3, which

presents density distributions for two systems of

differing precision. It can be observed that the first

system (I) dominates the second (II) in terms of the

narrower spread of its distribution area.

The goal is to determine the support for the

hypothesis that the true coordinate of the observer's

position lies near the marked x-value. The dataset

characterizing the two simple belief structures is

shown in Table 3. The characteristic values of the

respective belief structures were calculated for the

abscissa marked in Figure 3, based on the continuous

graphs shown there. The distributions are labelled I

and II, and they differ primarily in the spread of the

observed instances. Vertical shape of the step-wise

histogram (I) also dominates over the second one (II).

Table 3 Two simple belief structures defined from the

histories of two independent systems

Structure

{T}

{¬T}

{T, ¬T}

0,456

0,430

0,114

0,138

0,524

0,338

Structure I corresponds to a distribution with less

dispersion. Furthermore, the distance between the

considered coordinate and the centre of dispersion of

this system is very small. This results in significantly

higher belief—derived from this observation—in

support of the hypothesis that the neighbourhood of

the given x-value represents the true coordinate

location, compared to the belief derived from the

second observation. Let us compare the relevant belief

values: 0.456 versus 0.138. At the same time, the

uncertainty ranges for each case display opposite

magnitudes. These are shown in the last column: the

uncertainty induced by the first observation is

significantly lower than that from the second, namely:

0.114 versus 0.338. The method for calculating such

values in the case of simultaneous indications from

systems of varying accuracy will be presented in the

following section.

Table 3. Conjunctive-like combination of two simple belief

structures from Table 2

Structure II

set mass

{T}

{¬T}

{T, ¬T}

result

0,138

0,524

0,338

Structure I

{T}

{T, ¬T}

{T}

0,456

0,063

0,239

0,154

0,233

{¬T}

{T, ¬T}

{¬T}

0,43

0,059

0,225

0,145

0,430

{T, ¬T}

{T}

{¬T}

{T, ¬T}

0,114

0,016

0,060

0,039

0,337

Possibilistic-density diagrams of supporting the

hypothesis of the true coordinate location

approximately the considered x-value (Figure 3) are

shown in Figure 4.

3 DETERMINING THE COMPONENTS OF BELIEF

STRUCTURES FOR DATA SETS

In nautical science, we often work with so-called

simultaneous observations—data sets whose element

distributions indicate different degrees of uncertainty.

Their practical application requires the definition of

universal rules of procedure. Evaluating quality and

building a hierarchy within the available data sets are

fundamental tasks.

Figure 4. The graphs of the support for the hypothesis of

locating the true coordinate of the observer's position in the

vicinity of the abscissa marked in Figure 3

Figure 5 presents the density distributions of

instances for four observations, shown as both

histograms and continuous functions, along with

sample x-values near which support is calculated for

the hypothesis of the observer's true location. Each

system is described using the data gathered in Table 5,

which defines a multi-criteria decision-making

problem (shaded columns), as well as its solution using

the SAW (Simple Additive Weighting) method [9].

The first of the marked columns contains data

labelled di1/vi1. The values before the slash di1

603

correspond to the horizontal spreads of the x-values of

instance sets for systems I, II, III, and IV. These

represent the bin widths [in pixels] of the individual

histograms. Smaller values are favoured, as they

indicate systems with greater precision. For this

reason, this attribute is considered cost-type. The

importance weight for this attribute has been

arbitrarily set to 0.75. The second marked column

contains data labelled di2/vi2. The di2 values reflect the

vertical spreads of the histograms. These are calculated

based on the differences in histogram bin heights (i.e.,

the number of cases) across each system’s structure.

Larger values are favoured, as they indicate histograms

with better shaping. Therefore, this attribute is

considered qualitative. Its importance weight was

arbitrarily set to 0.25. This means that the horizontal

spread of the instances (i.e., the system’s precision) is

considered more important than the vertical shaping of

the histogram.

The vij values placed after the slashes are the

normalized values corresponding to each attribute.

The calculation methods for these values are shown in

Table 6. The appropriate formula is applied depending

on the attribute type: cost or qualitative.

Table 5. Example characteristic data of the x-coordinate

distributions of the instances observed for the four

positioning systems

System

horizontal

expansion

di1/vi1 (cost

0.75)

vertical

expansion

di2/vi2

(quality 0.25)

ranking

value

maximum

density/belief

20.9/1.00

6.5/1.00

1.00

0.46/0.28

44.3/0.00

6.4/0.89

0.22

0.66/0.06

III

22.7/0.92

5.9/0.33

0.78

0.50/0.21

33.2/0.47

5.6/0.00

0.36

0.60/0.10

Table 6. Methods of transforming attributes from the

decision table of a multi-criteria problem

qualitative attribute

min

max min

ij j

−

(1)

cost attribute

min

max min

j ij

−

(2)

Table 7. Set of basic factors and proposed formulae for their

estimation

factor

formula of

evaluation

meaning

fdmax=plmax

read from the

graph

plausibility for the centre of the

graph, maximum of the continuous

density function (see line 1 at

Figure 5)

belmax

C*fdmax

maximum belief obtained from the

maximum of the continuous density

function (see line 2 at Figure 5)

uncrtmin

fdmax - belmax

minimum uncertainty of the system

fdi

read from the

graph

the value of the continuous density

function for a given abscissa (see

line 3 at Figure 5)

beli

belmax*fdi/fdmax

the belief measure evaluated from the

continuous density function for a

given abscissa (see line 4 at Figure 5)

uncrti

fdmax – beli

the uncertainty measure for i-th

abscissa

C constant ratio, assumed as reduced system’s ranking value (see

Table 5)

Figure 5. The set of instances distribution, its conventional

and modern histogram with examples of data for selected

abscissas

Ranking values are obtained by multiplying matrix

V by the transposed weight vector. The resulting

values are placed in the fourth column of Table 5. The

best system turned out to be System I, with a ranking

value of 1.00. The worst is System II, with the value of

0.22. The ranking values determine the quality

hierarchy of the evaluated systems. They define the

membership function shapes when using fuzzy

systems, and consequently, they determine the shape

of the transformed density distributions [4]. The

ranking-based transformations influence the belief

values regarding the truth of hypotheses in the central

regions of the distributions. The dataset, ordered

according to the ranking list, reveals an increasing

trend in uncertainty across the systems. In such

conditions, the belief values at the central coordinates

of the distributions tend to decrease progressively.

The last column of Table 5 shows the maximum

value of the continuous density function of each

distribution. It also lists the belief value associated with

the likelihood that the observer's coordinate lies near

the x-value of the highest distribution density. The

highest belief corresponds to the best system. This

value depends on the ranking function and reaches a

minimum for the lowest-rated distribution. In this way,

an important practical principle is realized: that the

estimated position should depend primarily on the

most reliable data sources.

The set of calculated coefficients necessary for the

construction of belief structures is shown in Table 7. It

shows parameters characterizing the distributions of

instances of individual navigation aids, but also ways

of evaluating the measures of belief and presumption

for any given abscissa.

604

Figure 6. Density distributions of instances for four

observations in the form of histograms and continuous

functions, and sample abscissas, in whose neighbourhoods

the support for the hypothesis about the location of the true

position of the observer is determined

To illustrate the above-mentioned issues, the

following section presents distributions reflecting the

levels of support for the hypothesis that the true

coordinate lies near the specified x-values shown in

Figure 6. The dataset provides the foundation for

defining the relevant belief structures.

Table 8. The ordered belief structures for the abscissas

shown in Figure 6 along with the partial results of their

sequential association

Structure

{T}

{¬T}

{T, T¬}

{T}C

{¬T}C

{T, T¬}C

Abscissa

451

0.388

0.540

0.072

intermediate results 1

III

0.244

0.501

0.255

0.211

0.444

0.345

0.042

0.402

0.556

0.141

0.564

0.295

0.015

0.344

0.641

0.097

0.657

0.246

Abscissa

484

0.308

0.54

0.152

intermediate results 2

III

0.323

0.501

0.176

0.203

0.442

0.355

0.069

0.402

0.529

0.146

0.554

0.300

0.022

0.344

0.634

0.102

0.645

0.253

Abscissa

529

0.081

0.54

0.379

intermediate results 3

III

0.248

0.501

0.251

0.134

0.596

0.270

0.099

0.402

0.499

0.107

0.645

0.248

0.031

0.344

0.625

0.078

0.711

0.212

Table 8 shows three sets of belief structures for each

of the x-values illustrated in Figure 6. Each set is sorted

according to decreasing system-ranking value. The

calculations of the presented elements are based on the

maximum values for each system, taking into account

the offset of the x-values from the peak density location

of the respective system. It is assumed that as the

distance between the x-value and the peak increases,

the corresponding belief decrease. In the last three

columns of Table 8, the results of successive

associations of the four distributions shown alongside

are presented. The outcome of this operation is a

sequence of three belief structures, labelled as

"intermediate result x". The first structure in each trio

is the result of aggregating the two best distributions

from the respective set of four. Initially, in the first two

cases, these are structures I and III. The following

results involve aggregating the previous result with

structure IV, and then combining that new result with

the last distribution. Within each trio of results, the

belief values decrease, the uncertainty levels tends to

increase, while the indicator corresponding to the

rejection of the hypothesis—that the correct coordinate

lies near the x-value—remains the same.

An interesting case is that of a belief distribution in

which the uncertainty reaches a value of 1. This

structure then takes the form m

∅

= {0, 0, 1}, and it serves

as a neutral element in the aggregation process. This

means that the aggregation operation ⊗ in the form

∅

⊗ mi results in mi.

4 SUMMARY

This paper presents a method for encoding opinions

that include an element of doubt. The proposed

approach uses an interval model, which allows for the

creation of belief structures that are elements of the

Mathematical Theory of Evidence (MTE), also known

as belief theory. This scheme provides a mechanism for

aggregation that enriches the informational context of

arguments. Two sort of models were used;

possibilistic-probabilistic and possibilistic-density one.

The agreement between the two approaches is

obtained by assuming the neighbourhood of a given

abscissa of unit width.

The available opinions come from various sources

and may be more or less objective. From a nautical

perspective, an important task is the exploration of

data regarding instances and observations, leading to

the extraction of opinions on hypotheses about the

observer's correct location. The available sets of

observations represent different levels of reliability,

making it necessary to define a hierarchy among these

elements so that the implementation of the concept

produces solutions that meet quality standards.

Preparing a ranking list is a multi-criteria decision-

making problem. This field provides a range of

methods for building hierarchies within a set of

alternatives. One of the simplest is the additive method

known by the acronym SAW (Simple Additive

Weighting). Determining a ranking for a set of

simultaneous observations is a challenge addressed in

this work. The proposed approach utilizes sets of

instances observed for each system used to determine

the observer’s position.

The computed utility values define the hierarchy of

relative quality among the systems under

consideration. These values determine the forms of

membership functions in the context of using fuzzy

sets. Consequently, they allow for defining the shapes

of density distribution functions, usually perceived in

the form of "step-like" histograms. The transformed,

continuous form of the density distribution employs an

approach assuming that histogram bins are fuzzy sets

[2][3].

The ranking function values form the basis for

calculating belief in the truth of the considered

hypotheses, especially in the central regions of the

distributions. The data set, sorted in descending order

according to the ranking list, organizes the systems

from best to worst. The differences in ranking values

make it possible to determine the degree of qualitative

dominance of one system over another.

605

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