Essential notions and theoretical background

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2. Essential notions and theoretical background

Lotfi A. Zadeh was the first to propose a generalization of the range of values of the membership function {0, 1} to the closed interval [0; 1]. Thus, the value of the membership function can be any real number, starting from zero and ending with unity [3]. Such sets were called“fuzzy”sets. By gradually developing the proposed approach, Zadeh introduced the concept of a fuzzy linguistic variable, which was able to model mathematically linguistic variables [4]. For example, Zadeh made it possible to express mathematically the following linguistic notions:“childhood, young, middle-aged, old.”Zadeh also introduced the concept of fuzzy relations and basic operations on them.

As noted above, in our study we cannot do without expert evaluations. Since subjectivity, vagueness, and imprecision influence the assessments of experts, the use of fuzzy set theory seems to be an effective tool for our research work (see, e.g., [5, 6]).

In [2], to assess credit risks, we used precisely fuzzy relations. In the present work, we propose to use trapezoidal fuzzy numbers as a form of presentation of experts’estimates. The rationale for this choice will be given in the third section.

The chapter consists of six sections. The second section includes all the necessary information to understand the material. The theoretical basis of the offered

approach is laid out, and some theoretical results are given. In the third section, the approach to assessment of credit risk is introduced and discussed. In particular, the rationale for the choice of fuzzy trapezoidal numbers as a form for the presentation of experts’estimates is given. The fourth section looks at the algorithm for realiza- tion of the proposed approach. The fifth section contains toy example of the practi- cal application of the introduced approach. The sixth and the final section

summarize the chapter.

In conclusion, we note that when reading a chapter, the reader is not required to have knowledge of higher mathematics, but only elementary knowledge of arith- metic, algebra, and geometry. Nevertheless, we will try to explain meaningfully mathematical symbols and concepts that may be unfamiliar to the reader.

R~₁¼R~₂⇔a₁ ¼a₂,b₁ ¼b₂,c₁ ¼c₂,d₁ ¼d₂, R~₁,R~₂∈Ψð Þ:X (1) R~1⊕R~2 ¼ða1þa2,b1þb2,c1þc2,d1þd2Þ, R~1,R~2∈Ψð Þ:X (2) α⊗R~ ¼ðαa,αb,αc,αdÞ α>0, R~∈Ψð Þ:X (3) Definition 2.Trapezoidal fuzzy numberR~₁¼ð Þa_i is included in trapezoidal fuzzy numberR~2 ¼ð Þ,bi i ¼1, 4, i:e:R~1⪯R~2, if and only if

a1≤a2,b1≤b2,c1≤c2,d1≤d2 (4) It is known that fuzzy maximum and minimum of two trapezoidal fuzzy numbers is defined as follows [8]:

g

maxR~1,R~2

¼ðmaxfa1,b1g, maxfa2,b2g, maxfa3,b3g, maxfa4,b4gÞ, mingR~1,R~2

¼ðminfa1,b1g, minfa2,b2g, minfa3,b3g, minfa4,b4gÞ:

(

(5) Hence it follows that the above definition is equivalent to those given in the literature (see, e.g., [9, 10]):

R~1⪯ R~2⇔ mingR~₁,R~₂

¼R~₁, g

maxR~₁,R~₂

¼R~₂, (

R~1,R~2∈Ψð Þ:X (6)

Now we are going to introduce a metric onΨ(X) , i.e., define a distance between trapezoidal fuzzy numbers.

We say that the functionv:Ψ(X)!ℜ⁺isisotone valuation onΨ(X) if v maxg R~₁,R~₂

þvmingR~₁,R~₂

¼v R~₁

þv R~₂

(7) and

R~₁⪯R~₂ )v R~₁

≤v R~₂

: (8)

The isotone valuationvdetermines the metriconΨ(X):

ρ R~1,R~2

¼v maxg R~1,R~2

vming R~1,R~2

(9) Ψ(X) with isotone valuationvand metric (Eq. (8)) is called ametric spaceof trapezoidal fuzzy numbers.

Definition 3.In the metric space, the trapezoidal fuzzy numberR~^∗ isthe representa- tiveof the finite collection of trapezoidal fuzzy numbers R~_j

, j¼1,m, m¼2, 3, …if X^m

j¼1

ρR~^∗,R~_j

≤ X^m

j¼1

ρ ~S,R~_j

, ∀~S∈Ψð ÞX (10)

Let us clarify the meaning of this definition. A representative of the given finite collection of trapezoidal fuzzy numbers is a trapezoidal fuzzy number such that the sum of the distances between it and all members of this collection is minimal.

For an accommodation of posterior theoretical constructions, we need to intro- duce a concept ofregulationof finite collection of trapezoidal fuzzy numbers. We begin with an example.

Banking and Finance

Suppose we have the finite collection of trapezoidal fuzzy numbers:

aj bj cj dj

~R1 7 7.5 8 8.8

~R2 6 6.1 7.7 9

~R3 1 3 5 9.5

~R4 7.6 7.9 8.1 8.9

Compare with it the following finite collection of trapezoidal fuzzy numbers:

a^’j b^’j c^’j d^’j

~R1

0 1 3 5 8.8

~R2

0 6 6.1 7.7 8.9

~R3

0 7 7.5 8 9

~R4

0 7.6 7.9 8.1 9.5

We see that the matching columns in both tables consist of equal sets; at the same time the elements of the sets in the second table form nondecreasing

sequences. By the regulation of the finite collection of trapezoidal fuzzy numbers R~_1,R~_2,R~_3,R~₄

, we will mean the finite collection of trapezoidal fuzzy numbers R~⁰_1,R~⁰_2,R~⁰_3,R~⁰₄

n o

. The strict definition of regulation will be given below.

Definition 4.The finite collection of trapezoidal fuzzy numbersn oR~⁰_j is a regulation of the finite collection of trapezoidal fuzzy numbers R~_j

if the finite sets {aj} and {aj0}, {bj} and {bj0}, {cj} and {cj0}, and {dj} and {dj0} are pairwise equal anda1

’≤a20 ≤ … ≤am0,b1

’≤b20≤ … ≤bm

’,c10 ≤c20≤ … ≤cm0,d10≤d20≤ … ≤dm0, andj¼1,m,m¼2, 3, ….

Due to this definition and Eq. (9), it is obvious that the equality X^m

j¼1

ρ ~S,R~_j

¼X^m

j¼1

ρ~S,R~⁰_j

(11)

holds in the metric space for any~S∈Ψð ÞX and the finite collection of trapezoidal fuzzy numbers R~_j

, j¼1,m,m¼2, 3, …. From Eq. (11) it follows that represen- tatives of finite collection of trapezoidal fuzzy numbers and its regulation coincide.

It is obvious that the regulation represents a finite collection of nested trapezoi- dal fuzzy numbers:R~⁰₁⪯R~⁰₂⪯ … ⪯R~⁰_m, m ¼2, 3, …:.

The following theorem yields a formal definition of a representative.

Theorem 1[7].In the metric space of trapezoidal fuzzy numbers, the representative R~^∗ of the finite collection of trapezoidal fuzzy numbers, R~j

, j¼1,m, m¼2, 3,…,is determined as follows:

R~⁰_m=2⪯R~^∗ ⪯R~⁰_m=2þ1 if m is even; (12)

R~^∗ ¼R~⁰_ðmþ1Þ=2 if m is odd: (13)

It follows from the theorem that when the number of members in a finite collection of trapezoidal fuzzy numbers is even, a representative can take on an infinite number of values. Now we introduce the specific aggregation operator that uniquely identifies the representative (here and further on, expression [r], wherer is a real number, denotes the integer part of this number):

R~^∗ ¼

a⁰_½m=2Š,iþa⁰_{½ðmþ3Þ=2Š,i}

=2 if ^½ðmþ1P^Þ=2Š

j¼1

ρR~⁰_j,~R⁰_½m=2Š

¼ P^m

j¼½m=2Šþ1

ρR~⁰_j,R~⁰_{½ðmþ3Þ=2Š} ,

a⁰_½m=2Š,iþ

P½ðmþ1Þ=2Š

j¼1 ρR~⁰_j,R~⁰_½m=2Š P½ðmþ1Þ=2Š

j¼1 ρR~⁰_j,R~⁰_½m=2Š þP_m

j¼½m=2Šþ1ρR~⁰_j,R~⁰_{½ðmþ3Þ=2Š}a⁰_{½ðmþ3Þ=2Š,i} a⁰_½m=2Š,i 0

@

1

Aotherwise 8>

>>

>>

><

>>

>>

>>

:

i¼1, 4

(14) Remark 1.It can be easily shown that the representative determined by Eq. (14) is a trapezoidal fuzzy number.

Summary.In this section, we presented the definitions of fuzzy sets and trape- zoidal fuzzy numbers. Operations on trapezoidal fuzzy numbers are considered, and definitions of concepts that are necessary for constructing the proposed approach are given.