Generalized intuitionistic fuzzy soft sets with applications in decision-making

doi:10.1016/j.asoc.2013.03.015

Applied Soft Computing

Volume 13, Issue 8, August 2013, Pages 3552-3566

https://doi.org/10.1016/j.asoc.2013.03.015 Get rights and content

Highlights

•
Intuitionistic fuzzy soft sets are generalized with the inclusion of moderator's opinion about the original evaluation.
•
The properties and operations for Generalized intuitionistic fuzzy soft set (GIFSS) are presented.
•
The notion of generalized intuitionistic fuzzy soft relations is devised.
•
A score function and similarity relation for GIFSS are proposed.
•
We show the application of GIFSS in real supplier selection and medical diagnosis problems.

Abstract

The concept of intuitionistic fuzzy soft set (IFSS) arising from intuitionistic fuzzy set (IFS) is generalized by including a parameter reflecting a moderator's opinion about the validity of the information provided. The resulting generalized intuitionistic fuzzy soft set (GIFSS) finds a special role in the decision making applications. It can evaluate the given criteria along with the moderator's assessment of the furnished data. The properties of GIFSS are investigated and the associated relations called generalized intuitionistic fuzzy soft relations (GIFSR) are given. A similarity measure is given to compare two GIFSSs. As this is not applicable to fuzzy numbers, a new score function is devised to compare two intuitionistic fuzzy numbers (IFNs), the components of IFS. The effectiveness of the proposed GIFSS in decision making is demonstrated on four case studies.

Introduction

The concept of intuitionistic fuzzy set (IFS) [1], [2] extends a fuzzy set by having an additional hesitancy parameter besides the usual membership function. The hesitancy parameter makes IFS useful in modeling many real life activities like evaluation, negotiation and decision making where judgments of human beings play a major role. Molodtsov [3] has pointed out that both the fuzzy set and intuitionistic fuzzy set theories suffer from the inherent difficulty that the magnitude of membership grade for x given by an individual is directly dependent on the knowledge available to the individual; hence prone to variations. Moreover, both the fuzzy and intuitionistic fuzzy theories lack an appropriate parameterization tool. The theory of soft sets addresses these limitations [3].

The soft set theory is based on adequate parameterization. The description of objects is more comprehensive in the soft set theory than in the fuzzy and the intuitionistic fuzzy theories. One can have any number of attributes/parameters in the description of an object in the soft set theory. This makes it easier to compare the objects in terms of the attributes thereby permitting the decision making even with partial information. In view of this, the soft set theory has found several applications in multi-criteria decision making [8], [17].

Maji et al. [4], [5], [6] have combined the soft set theory with the intuitionistic fuzzy set theory to come up with the concept of intuitionistic fuzzy soft set (IFSS). The parameterization and hesitancy accrued to IFSS from this combination facilitate the descriptions of the real-world situations quite accurately. In particular, IFSS is extremely useful in multi-criteria decision making (MCDM) problems where the evaluation of alternatives is done by experts based on the attributes. The suitability of IFSS for the decision making applications is aptly highlighted in [7], [8].

The MCDM model in an IFSS environment takes into account the hesitancy of experts in arriving at the membership grades since the IFS allows an expert to record his hesitancy factor. However, hesitancy is a feature of one's own perception, and it needs to be supported by another independent observer/expert. To address this problem, the notion of generalized intuitionistic fuzzy soft set (GIFSS) is introduced in [25] and developed independently in [26], [27]. The framework of GIFSS requires the moderator's assessment of the credibility of the information in the original IFSS so as to make up for any distortion in the information provided.

In this paper, we further extend the concept of GIFSS by introducing a generalization parameter which itself is intuitionistic fuzzy. We also develop the relations on GIFSS, termed as generalized intuitionistic fuzzy soft relations (GIFSR). A score function and a similarity measure are devised for the comparison of two intuitionistic fuzzy numbers and two GIFSSs respectively. The suitability of GIFSS in supplier selection and medical diagnosis problems is examined by taking up case studies from the real world.

In our view, it is hard to make accurate decisions on the basis of the individuals’ vague conceptions. The GIFSS attempts to minimize the possibility of errors caused by the imprecise information by taking the moderator's opinion on the same. For example, a patient might exaggerate the symptoms which may lead to incorrect diagnosis. The specialist may not have enough time to ascertain the information supplied by probing the patient in detail. It may be more judicious to have a junior doctor moderate the severity of the symptoms of a patient through a generalization parameter.

The GIFSS has a wide scope in the areas like economics, supply chain management, financial accounting and medical expert systems. For an example, in the current medical practice, a specialist diagnoses a patient simply going by the symptoms reported. Similarly the decisions related to the supplier selection problems are based solely upon the evaluation of the suppliers by the domain experts. In such decision making problems, the possibility of error in the judgment (of a patient about the severity of his symptom or of the domain experts in the evaluation of a supplier with respect to a criterion) cannot be ruled out. The potential of the proposed framework in addressing this issue will be demonstrated on a few case-studies. The expert systems deploying the GIFSS would obviate the risk of serious errors in the decision-making.

The paper is organized as follows. Section 2 discusses a few preliminaries and useful definitions related to IFS and IFSS. Section 3 introduces GIFSS and presents its properties. A novel score function and a similarity measure for comparison of two GIFSSs are provided in Section 4. Section 5 is concerned with the concept of generalized intuitionistic fuzzy relations (GIFSR) on GIFSS and its relevant properties. Section 6 presents an approach based on GIFSS to solve the supplier selection problem. A real case study involving supplier selection problem is given in this section. Section 7 shows the utility of the GIFSS in the decision making by solving two supplier selection problems using the similarity measure, proposed in Section 4. An application of generalized intuitionistic fuzzy relations (GIFSR) in solving a medical diagnosis problem is shown in Section 8. Finally, Section 9 gives conclusions of this paper.

Section snippets

Multi-criteria decision making with intuitionistic fuzzy techniques

Fuzzy logic has become an inevitable part of multi criteria decision making because of the need to represent the associated uncertainty. To cater to the increased role of fuzzy logic, fuzzy sets have been extended in the literature leading to type-2 fuzzy sets [28], interval type-2 fuzzy sets [29], vague sets [30], probabilistic fuzzy sets [31], fuzzy soft sets [5] and intuitionistic fuzzy sets [1]. Recently, intuitionistic fuzzy sets have attracted considerable attention in the fuzzy decision

Generalized intuitionistic fuzzy soft set (GIFSS)

With the advent of GIFSS, the evaluation of an alternative against the criteria in an IFSS is moderated with the generalization parameter. The moderated input information can really go a long way in improving the current expert systems by ensuring a higher accuracy in the final decisions. Without moderation, the original evaluation remains uncertified; this means that the veracity of the assessment is doubtful.

In this section, we present the definitions of GIFSS and illustrate it through

Score function and similarity measure in GIFSS

In several areas such as pattern recognition, image processing, region extraction and coding theory, there is a need to compare two fuzzy numbers. A novel score function is devised here to compare two intuitionistic fuzzy numbers (IFNs). A similarity measure to compare two GIFSSs is also proposed.

Relations on GIFSS

The notion of intuitionistic fuzzy soft relation (IFSR) is generalized to GIFSR in the context of GIFSS. The concept of GIFSR is illustrated through simple examples and the properties of GIFSR are investigated in detail.

Definition 5.1

An intuitionistic fuzzy soft relation (IFSR) $\overset{˜;}{R}$ between two IFSSs, $\overset{˜;}{F}$ and $\overset{˜;}{G}$ over soft universes (U, E) and (U, D) respectively, is defined [10] as:

\begin{array}{l} \overset{˜;}{R} (e, d) = \overset{˜;}{F} (e) \overset{˜;}{\cap} \overset{˜;}{G} (d), & \forall e \in E a n d \forall d \in D, \\ where \overset{˜;}{R} : K \to I F^{U} is an IFSS over (U, K), K \subseteq E \times D . \end{array}

where,

\overset{˜;}{\cap}

is a suitable operation.

Definition 5.2

Supplier selection using GIFSS

Selection of the best supplier is of utmost importance for the success of an organization because of the impact of the decisions made on cost, manufacturing, service, efficiency and overall satisfaction of the customers. It is often a difficult decision to select the best supplier as multiple criteria are involved and these are often conflicting. The importance of supplier selection for the businesses is also testified by the numerous studies in the literature [18], [19], [20], [21]. In the

Two case studies based upon similarity measure of GIFSS

The potential of GIFSS is further highlighted through two case-studies that use the similarity measure discussed in Section 4. The case-studies deal with finding the best supplier.

An application of GIFSS and GIFSR for medical diagnosis

GIFSS can be effectively used to reduce the errors encountered in medical diagnosis. These errors often turn out to be quite serious to the victims. The World Health Organization (WHO) recognizes medical errors as having fatal consequences on the patients. As reported by WHO, one in 10 hospital admissions involves an error and one in 300 admissions results in death due to wrong diagnosis. This claim is also reinforced by the British National Health System survey in 2009 [35] which reports that

Conclusions

This paper extends the Intutionisitc Fuzzy Soft Sets (IFSS) to the Generalized IFSS (GIFSS) by introducing the generalization parameter to the pool of the intutionistic fuzzy numbers (IFNs) of IFSS. The generalization parameter is a kind of the moderator's assessment on the presented information as IFNs. The role of moderator is to refine IFNs with his domain specific knowledge. It may be noted that the information of any sort often gets misinterpreted during its presentation. This usually

Acknowledgments

The help rendered by Mr. R. K. Vishwakarma, I.G., CRPF, India in procuring the data for Case-study 1 is gratefully acknowledged. The authors also express their indebtedness to the anonymous reviewers for their suggestions that have led to an improved version of this paper.

References (35)

K. Atanassov
Intuitionistic fuzzy sets
Fuzzy Sets and Systems
(1986)
D. Molodtsov
Soft set theory-First results
Computers & Mathematics with Applications
(1999)
Y. Jiang et al.
An adjustable approach to intuitionistic fuzzy soft sets based decision making
Applied Mathematical Modelling
(2011)
P. Majumdar et al.
Generalised fuzzy soft sets
Computers & Mathematics with Applications
(2010)
E. Sanchez
Inverses of fuzzy relations, application to possibility distributions and medical diagnosis
Fuzzy Sets and Systems
(1979)
J. Ye
Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment
Expert Systems with Applications
(2009)
P.R. Innocent et al.
Computer aided fuzzy medical diagnosis
Information Sciences
(2004)
N. Cagman et al.
Soft set theory and uni-int decision making
European Journal of Operational Research
(2010)
A.H.I. Lee
A fuzzy supplier selection model with the consideration of benefits, opportunities, costs and risks
Expert Systems with Applications
(2009)
A. Sanayei et al.
Group decision making process for supplier selection with VIKOR under fuzzy environment
Expert Systems with Applications
(2010)

T.Y. Wang et al.

A fuzzy model for supplier selection in quantity discount environments

Expert Systems with Applications

(2009)

Z.S. Xu

Intuitionistic preference relations and their application in group decision making

Information Science

(2007)

E. Szmidt et al.

Distances between intuitionistic fuzzy sets

Fuzzy Sets and Systems

(2000)

L.A. Zadeh

The concept of a linguistic variable and its application to approximate reasoning-1

Information Sciences

(1975)

V. Khatibi et al.

Intuitionistic fuzzy set vs. fuzzy set application in medical pattern recognition

Artificial Intelligence in Medicine

(2009)

Z. Zhang et al.

Intuitionistic fuzzy sets with double parameters and its application to multiple attribute decision making of urban planning

Procedia Engineering

(2011)

K. Atanassov

Intuitionistic Fuzzy Sets: Theory and Applications

(1999)

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Generalized intuitionistic fuzzy soft sets with applications in decision-making

Highlights

Abstract

Introduction

Section snippets

Multi-criteria decision making with intuitionistic fuzzy techniques

Generalized intuitionistic fuzzy soft set (GIFSS)

Score function and similarity measure in GIFSS

Relations on GIFSS

Supplier selection using GIFSS

Two case studies based upon similarity measure of GIFSS

An application of GIFSS and GIFSR for medical diagnosis

Conclusions

Acknowledgments

Fuzzy Sets and Systems

Computers & Mathematics with Applications

Applied Mathematical Modelling

Computers & Mathematics with Applications

Fuzzy Sets and Systems

Expert Systems with Applications

Information Sciences

European Journal of Operational Research

Expert Systems with Applications

Expert Systems with Applications

Expert Systems with Applications

Information Science

Fuzzy Sets and Systems

Information Sciences

Artificial Intelligence in Medicine

Procedia Engineering

Intuitionistic Fuzzy Sets: Theory and Applications