A Hybrid Model for Decision Making

Utilizing TFNs and Soft Sets as Tools

MICHAEL GR. VOSKOGLOU

Mathematical Sciences, School of Technological Applications

University of Peloponnese (ex T.E.I. of Western Greece)

Meg. Alexandrou 1, 26334 Patras

GREECE

Abstract: - Decision making is the process of evaluating multiple alternatives to choose the one satisfying in the

best way the existing goals. Frequently, however, the boundaries of those goals and/or of the existing

constraints are not sharply defined due to the various forms of uncertainty appearing in the corresponding

problems. Since the classical methods cannot be applied in such cases, several methods for decision making

under fuzzy conditions have been proposed. In this work a new hybrid decision making method is developed

utilizing triangular fuzzy numbers (TFNs) and soft sets as tools, which improves an earlier method of Maji,

Roy and Biswas, which uses soft sets only. The importance of this improvement is illustrated by an application

concerning the decision for the purchase of a house satisfying in the best possible way the goals put by the

candidate buyer.

Key-Words: - Decision making, fuzzy logic, fuzzy set (FS), triangular fuzzy number (TFN), soft sets, intelligent

computing, soft computing

Received: August 15, 2021. Revised: March 23, 2022. Accepted: April 25, 2022. Published: June 3, 2022.

1 Introduction

Decision making is the process of evaluating, with

the help of suitable criteria, multiple alternatives to

choose the one satisfying in the best way the

existing goals. In decision making under fuzzy

conditions the boundaries of the goals and/or of the

existing constraints are not sharply defined. Several

methods for decision making in such cases have

been proposed, e.g. [1-5], etc. Maji et al. [6] used a

tabular form of a soft set in the form of a binary

matrix to introduce a type of parametric decision

making method. Their method, however, replaces

the characterizations (parameters; e.g. beautiful) of

the elements of the set of the discourse (houses in

their example) with the binary elements 0, 1

representing the truth values of those parameters. In

other words, although their method starts from a

fuzzy framework (soft sets), it uses bivalent logic

for making the required decision (e.g. beautiful or

not beautiful)! Consequently, this approach could

lead to a wrong decision, when some (or all) of the

parameters have not a bivalent texture; e.g. the

parameter “wooden” characterizing a house has a

bivalent texture, but not the parameter “beautiful”.

In this work, in order to tackle this problem, we

modify the method of Maji et al. by using triangular

fuzzy numbers (TFNs) instead of the binary elements

0, 1. The rest of the paper is organized as follows: In

section 2 the basic information about TFNs and soft

sets is given which is necessary for the

understanding of the paper. In section 3 the

modified decision making method is presented and

compared with the method of Maji et al. through a

suitable example. The paper closes with the final

conclusions and some hints for future research

included in section 4.

2 Preliminaries

The frequently appearing in real world, in science

and in everyday life uncertainty is due to several

reasons, like randomness, imprecise or incomplete

data, vague information, etc. Probability theory has

been proved sufficient for dealing with the cases of

uncertainty due to randomness. During the last 50-

60 years, however, various mathematical theories

have been introduced for tackling effectively the

other forms of uncertainty, including fuzzy sets [7],

intuitionistic fuzzy sets [8], neutrosophic sets [9],

rough sets [10] and several others [11]. The

combination of two or more of those theories gives

in many cases better results for the solution of the

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Michael Gr. Voskoglou

E-ISSN: 2732-9976

Volume 2, 2022

corresponding problems and that is what we are

going to attempt here for decision making

2.1 Triangular Fuzzy Numbers

Definition 1: Let U be the universal set of the

discourse, then a fuzzy set (FS) Α on U is defined

with the help of its membership function m: U



[0,1] as the set of the ordered pairs

A = {(x, m(x)): x



U} (1)

The real number m(x) is called the membership

degree of x in Α. The greater is m(x), the more x

satisfies the characteristic property of Α. Many

authors, for reasons of simplicity, identify a fuzzy

set with its membership function.

A crisp subset A of U is a fuzzy set on U with

membership function taking the values m(x)=1, if x

belongs to A, and 0 otherwise.

Zadeh [12] has also introduced FNs as follows:

Definition 2: A FN is a FS A on the set R of real

numbers with membership function m: R



[0, 1],

such that:

 A is normal, i.e. there exists x in R such

that m(x) = 1,

 A is convex, i.e. all its a-cuts Aa = {x



mA(x)



α}, α in [0, 1], are closed real

intervals, and

 Its membership function y = m(x) is a

piecewise continuous function.

Arithmetic operations between FNs have been

introduced in two, equivalent to each other,

methods: (i) By applying the Zadeh’s extension

principle [13, Section 1.4, p.20], which provides the

means for any function f mapping a crisp set X to a

crisp set Y to be generalized so that to map fuzzy

subsets of X to fuzzy subsets of Y and (ii) with the

help of the α-cuts of the corresponding FNs and the

representation-decomposition theorem of Ralesscou

- Negoita [14, Theorem 2.1, p.16) for FS. With the

second method the fuzzy arithmetic is turned to the

arithmetic of the closed real intervals.

In practice, however, the previous two general

methods of fuzzy arithmetic requiring laborious

calculations are rarely used in applications, where

the utilization of simpler forms of FNs is preferred.

For general facts on FNs we refer to the book [15].

TFNs is the simplest form of FNs. A TFN is defined

as follows:

Definition 3: Let a, b and c be real numbers with

a < b < c. Then the TFN (a, b, c) is a FN with

membership function:

, [ , ]

( ) [ , ]

xa x a b

y m x x b c

x a or x c







   











Proposition 1: The coordinates (X, Y) of the centre

of gravity (COG) of the graph of the TFN (a, b, c)

are calculated by the formulas X =

abc

, Y =

Proof: The graph of the TFN (a, b, c) is the triangle

ABC of Fig. 1, with A (a, 0), B(b, 1) and C (c, 0).

Figure 1: Graph and COG of the TFN (a, b, c)

Then, the COG, say G, of ABC is the intersection

point of its medians AN and BM. The proof of the

Proposition is easily obtained by writing down the

equations of AN and BM and by solving the linear

system of those two equations.-

The x-coordinate of the COG of a TFN A= (a, b, c)

is usually considered as the crisp representative of A

(defuzzification of A). We write then

D(A)=

abc

(2)

It can be shown [15] that the two previously

mentioned general methods of defining arithmetic

operations between FNs reduce to the following

simple rules for the addition and subtraction of

TFNs:

Let A = (a, b, c) and B = (a1, b1, c1) be two TFNs.

Then A + B = (a+a1, b+b1, c+c1) (3), and

A - B = A + (-B) = (a-c1, b-b1, c-a1) (4),

where –B = (-c1, -b1, -a1) is defined to be the

opposite of B.

On the contrary, the product and the quotient of two

TFNs, although they are FNs, need not be TFNs.

Further the scalar product of a real number k with a

TFN A(a, b, c) is defined by

kA = (ka, kb, kc), if k>0 and kA = (kc, kb, ka), if

k<0 (5)

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2.2 Soft Sets

Molodstov [16] introduced in 1999 the concept of

soft set for tackling the uncertainty created by a set

of parameters characterizing the elements of the set

of the discourse.

Let E be a set of parameters, let Q be a subset of E

and let g: Q → P(V) be a map from Q to the power

set P(V) of the universal set V. Then the soft set on

V defined with the help of g and Q and denoted by

(g, Q) is the set of ordered pairs:

(g, Q) = {(q, g(q)): q∈Q} (6)

The characterization “soft” is related to the fact that

the form of the set (g, Q) depends on the parameters.

For example, let V = {H1, H2, H3, H4, H5, H6} be a

set of houses and let E = {e1, e2, e3, e4, 55} be the set

of the parameters e1=beautiful, e2=wooden, e3=in

the country, e4=modern and e5=cheap. Consider the

subset Q = {e1, e2, e3, e5} of E and assume that H1,

H2, H6 are the beautiful, H2, H3, H5, H6 are the

wooden, H3, H5 are the country houses and H4 is the

unique cheap house. Then a map g: Q → P(V) is

defined by g(e1) = {C1, C2, C6}, g(e2) = { H2, H3, H5,

H6}, g(e3) = {H3, H5}, g(e5) = { H4} and the soft set

(g, Q) = {(e1, {H1, H2, H6}), (e2, {H2, H3, H5, H6}),

(e3, {H3, H5}), (e5, {H4})} (7)

A FS on V, with membership function y=m(x) is a

soft set (g, [0, 1]) on V with g(a)={x ∈ V: m(x)≥a},

for each a in [0, 1].

The use of soft sets instead of FSs has, among

others, the advantage that, by using the parameters,

one overpasses the existing difficulty of defining

properly the membership function of a FS. In fact,

the definition of the membership function is not

unique, depending on the observer’s subjective

criteria [13]. In case of determining the FS of the

cheap houses under sale in a certain area of a city,

for example, the membership function could be

defined in several ways, according to the financial

power of each candidate buyer.

Maji et al. [17] introduced a tabular representation

of soft sets in the form of a binary matrix in order to

be stored easily in a computer’s memory. For

example, the tabular representation of the soft set

(5) is given in Table 1.

Table 1: Tabular representation of the soft set (g, Q)

Soft sets have found important applications to

several sectors of the human activity like decision

making, parameter reduction, data clustering and

data dealing with incompleteness, etc. [18]. One of

the most important steps for the theory of soft sets

was to define mappings on soft sets, which was

achieved by Kharal & Ahmad [19] and was applied

to the problem of medical diagnosis in medical

expert systems. But fuzzy mathematics has also

significantly developed at the theoretical level

providing important insights even into branches of

classical mathematics like algebra, analysis,

geometry, topology etc. For example, one can

extend the concept of topological space, the most

general category of mathematical spaces, to fuzzy

structures and in particular can define soft

topological spaces and generalize the concepts of

convergence, continuity and compactness within

such kind of spaces [20]. The present author has

recently used soft sets as tools in assessment

problems [21].

The Hybrid Decision Making Method

3.1 The Method of Maji et al.

Maji et al. [17] used the tabular form of a soft set

described in section 2.2 as a tool for decision

making in a parametric manner. Here, we use the

example of section 2.2 to highlight their method.

For this, assume that one is interested to buy a

beautiful, wooden, country and cheap house by

choosing among the six houses of the previous

example. Forming the tabular representation of the

soft set (g, Q) (Table 1) the choice value of each

house is calculated by adding the binary elements of

the corresponding row of it. The houses H1 and H4

have, therefore, choice value 1 and all the others

have choice value 2. Consequently, the buyer must

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choose one of the houses H2, H3, H5 or H6, which is

not a very helpful decision.

3.2 The New Hybrid Method

Observe now that the parameters e1 and e5 have not

a bivalent texture. In fact, how beautiful is a house

depends on the subjective opinion of each person,

whereas its low or high price depends on the

financial ability of the candidate buyer. In other

words, the previous decision could not be the best

one.

To tackle this problem, we replace in Table 1 the

binary elements 0, 1 for the parameters e1 and e5 by

the TFNs G1 = (0.85, 0.925, 1), G2 = (0.75, 0.795,

0.84), G3 = (0.6, 0.67, 0.74], G4 = (0.5, 0.545, 0.59)

and G5 = (0, 0.245, 0.49). Assume further that the

candidate buyer, after studying the existing

information about the six houses, decided to use

Table 2 for making the final decision.

Table 2: Revised tabular representation

of the soft set (g, Q)

From Table 2 we calculate the choice value Vi of the

house Hi, i=1, 2, 3, 4, 5, 6 as follows:

V1=D(G1+G3), or by (3) V1= D(1.45, 1.595, 1.74)

and finally by (2) V1=

1.45 1.595 1.74



=1.595.

Similarly V2=1+D(G1+G5)=2.17,

V3=2+D(G3+G3)=3.34, V4=D(G4+G1)=1.47,

V5=2+D(G4+G1)=3.215, V6=1+D(G1+G4)=2.47.

Therefore, the right decision is to buy the house H3.

3.3 Remarks

I) The choice of the extreme entries of the TFNs Gi,

i=1, 2, 2, 4, 5, was made according to the accepted

standards for the linguistic grades excellent (A),

very good (B), good (C), almost good (D) and

unsatisfactory (F) respectively. Their choice

however is not unique and could slightly differ from

case to case according to the goals of the decision

maker. The middle entry of each TFN is equal to the

mean value of its two extreme entries.

II) If a parameter takes the value 0 (or 1) for all

elements of the set of the discourse, then it can be

withdrawn from the tabular form of the

corresponding soft set, since it does not affect the

final decision. The resulting table in those cases is

called a reduct table of the soft set. When all the

parameters having the previous property have been

withdrawn, then the resulting reduct table is called

the core of the soft set [17]. Obviously the core of a

soft set is contained in all reduct tables of it.

III) An analogous to the previous one decision

making method could be introduced by using grey

numbers instead of TFNs [22].

3.4 Weighted Decision Making

Frequently, the goals put by the decision-maker

have not the same importance. In this case, and in

order to make the proper decision, weight

coefficients must be assigned to each parameter.

For instance, assume that in the previous example

the candidate buyer has assigned the weight

coefficients 0.9 to e1, 0.7 to e2, 0.6 to e3 and 0.5 to

the parameter e5. Then, the weighted choice values

of the houses are calculated with the help of Table 2

as follows:

V1=D(0.9G1+0.5G3), or by (2) and (3) V1=1.46.

Similarly V2=0.7+D(0.9G1+0.5G5) =1.655,

V3=0.7+0.6+D(0.9G3+0.5G3)=2.238,

V4=D(0.9G4+0.5G1)=0.953,V5=0.7+0.6

+D(0.9G4+0.5G3)=2.1255,

V6=0.7+W(0.9G1+0.5G4)=1.805.

Consequently, the right decision is again to buy the

house H3.

4. Discussion and Conclusions

In this work a parametric decision making method

was introduced using TFNs and soft sets as tools. As

it was illustrated by the given example, this hybrid

approach is very useful in cases where one or more

parameters have not a bivalent texture (beautiful and

cheap houses in our example), because it enables

one to make the best decision, which is possibly not

possible to be succeeded by using only soft sets, as

the method of Maji et al. suggests.

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Michael Gr. Voskoglou

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The combination of two or more of the theories that

have been developed for tackling efficiently the

various forms of the existing in real world, everyday

life and science uncertainty, appears in general to be

an effective tool for obtaining better results, not

only for decision making, but also for assessment

and for a variety of other human activities.

Consequently this is a promising area for future

research.

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Michael Gr. Voskoglou

E-ISSN: 2732-9976

Volume 2, 2022