Group multi-criteria decision making using intuitionistic multi-fuzzy sets
- Sujit Das^{1},
- Mohuya B Kar^{2} and
- Samarjit Kar^{3}Email author
https://doi.org/10.1186/2195-5468-1-10
© Das et al.; licensee Springer. 2013
Received: 18 April 2013
Accepted: 10 October 2013
Published: 7 November 2013
Abstract
In this paper we propose an efficient approach for group multi-criteria decision making (MCDM) based on intuitionistic multi-fuzzy set (IMFS). First we construct intuitionistic multi-fuzzy matrices for decision makers with respect to the criteria (attributes) of the alternatives. Based on intuitionistic multi-fuzzy matrices, we construct the aggregated intuitionistic multi-fuzzy matrix using the proposed intuitionistic multi-fuzzy weighted averaging (IMFWA) operator. Then we use Hamming distance and Euclidean distance measurements in the context of IMFS between the aggregated matrix and the specified sample matrix to reach the optimal decision. This paper also presents score function and accuracy function of IMFS with an application to MCDM. Finally, a real-life case study related to heart disease diagnosis problem is provided to illustrate the advantage of the proposed approach.
Keywords
Introduction
Intuitionistic fuzzy set (IFS) was introduced by Atanassov [1] and can be shown as a generalization and extension of Zadeh's fuzzy set theory [2]. IFS has emerged as an active research area mainly for solving multiple-criteria decision making problems [3–9] and group decision making (GDM) problems [10–12] where the values of local criteria (attributes) of alternatives and/or their weights are intuitionistic fuzzy values (IFVs). Intuitionistic fuzzy set covers such kind of situations where a human being can express the degree of belonging of an element to a set as well as degree of non-belonging of an element to a set.
A complete account of the development of multi-set theory has been seen in [13]. As a generalization of multi-set, Yager [14] introduced the concept of multi-fuzzy set (MFS). An element of a MFS can occur more than once with possibly the same or different membership values. MFS theory is an extension of theories of fuzzy sets [2], L-fuzzy sets [15], and intuitionistic fuzzy sets [16]. Group multi-criteria decision making requires certain methods to aggregate the opinions provided by different experts. Yager [17] proposed an interesting and well-grounded approach, namely the ordered weighted averaging (OWA) [17], which enabled aggregation of the variables in terms of their order in the set.
Most of the human reasoning involves linguistic variables whose values are words rather than numbers to express a situation in a better way in an uncertain environment. Fuzzy sets act as the basis of these linguistic variables. But in some situations like decision making problems (such as medical diagnosis, sales analysis, marketing, etc.), the description by a linguistic variable in terms of membership function only is not adequate. In many cases, there is an obvious chance of an appearance of a non-null complement. IFS can be used in this context as a proper tool for representing both membership and non-membership of an element to a concept [18]. Again there are situations where each element has different membership values. In such situations, an intuitionistic multi-fuzzy set (IMFS) is more adequate. Here we present IMFS as a tool for representing such situations linguistically.
The aim of this paper is to combine the concept of intuitionistic fuzzy set and multi-fuzzy set to produce an intuitionistic multi-fuzzy set and apply this to group MCDM problems. In group MCDM, a group of decision makers provide their opinions based on their own observations and intuitions over a set of attributes. The individual opinions are finally converted to a collective opinion using some suitable procedure. This article introduces IMFSs, basic operations on IMFSs, and application of IMFSs to MCDM problems by constructing a suitable algorithmic approach. Here we also present score function and accuracy function in the context of IMFS with an application to MCDM. In the proposed approach, we first construct intuitionistic multi-fuzzy matrices (IMFMs) for decision makers. Decision makers provide their opinions in terms of linguistic scales at different time instances which are represented by IMFMs. Next we construct the aggregated intuitionistic multi-fuzzy matrix which reflects the combined opinions of all decision makers using our proposed intuitionistic multi-fuzzy weighted averaging (IMFWA) operator. IMFWA can be treated as an extension of intuitionistic weighted arithmetic mean (IWAM) [19]. Then this study uses IMFS-based normalized Hamming distance and normalized Euclidean distance to find out the final outcome. Finally, this study degenerates the individual experts' opinions (IMFS) into IFS and compares the final outcome. This approach is illustrated using a suitable case study concerned with medical disease diagnosis.
The rest of this article is organized as follows: The second section presents the preliminaries about IFS and some operations on IFS. The third section includes a brief description about MFS, IMFS, and few operations on IMFS including IMFWA operator and distance measurements. The fourth section includes the proposed algorithmic approach, followed by a case study in the fifth section. Finally, the paper is concluded in the sixth section.
Intuitionistic fuzzy sets and some operations
In this section, we briefly review the definition of IFS, its operations, and comparisons.
Definition of IFS and IFV with necessary conditions
The function π _{ A } : X → [0, 1], given by π _{ A }(x) = 1 − μ _{ A }(x) − ν _{ A }(x), x ∈ X, defines the degree of uncertainty of the membership of the element x to set A known as hesitation function. For a fixed x ∈ X, an object {μ _{ A }(x),v _{ A }(x)} is usually called IFV or intuitionistic fuzzy number (IFN).
Description of two basic operations using intuitionistic fuzzy sets
Addition ⊕ and multiplication ⊗ of IFSs
These operations were constructed in such a way that they produce IFVs since it is easy to prove that 0 ≤ μ _{ A } + μ _{ B } − μ _{ A }. μ _{ B } + ν _{ A }. ν _{ B } ≤ 1 and 0 ≤ μ _{ A }. μ _{ B } + ν _{ A } + ν _{ B } − ν _{ A }. ν _{ B } ≤ 1.
Multiplication ⊗ of IFS with real values
where 0 ≤ wi ≤ 1 and $\sum _{i=1}^{n}{w}_{i}}=1$.
This aggregating operator provides IFVs and is currently the most popular in the solution of MCDM problems in the intuitionistic fuzzy setting.
Comparison of IFSs
Intuitionistic multi-fuzzy set and some operations
This section briefly describes multi-fuzzy set, intuitionistic multi-fuzzy set, and a few operations on IMFS including IMFWA operator and distance measurements.
MFS and IMFS
Definition 3 Let k be a positive integer and U be a universal set. A multi-fuzzy set A over U is a set of ordered sequences $A=\left\{x/\left({\mu}_{A}^{1}\left(x\right),{\mu}_{A}^{2}\left(x\right),\dots ,{\mu}_{A}^{i}\left(x\right),\dots ,{\mu}_{A}^{k}\left(x\right)\right):x\in U\right\}$ where ${\mu}_{A}^{i}\in P\left(U\right),\phantom{\rule{1em}{0ex}}i=1,2,\dots ,k$.
The function ${\mu}_{A}=\left({\mu}_{A}^{1},{\mu}_{A}^{2},\dots ,{\mu}_{A}^{k}\right)$ is called a multi-membership function of multi-fuzzy set A, and k is called the dimension of multi-fuzzy set A.
Definition 4 An intuitionistic multi-fuzzy set A is defined by $A=\left\{x/\left({\mu}_{A}^{1}\left(x\right),{\mu}_{A}^{2}\left(x\right),\dots ,{\mu}_{A}^{k}\left(x\right)\right),\left({\nu}_{A}^{1}\left(x\right),{\nu}_{A}^{2}\left(x\right),\dots ,{\nu}_{A}^{k}\left(x\right)\right):x\in U\right\},$ where $0\le {\mu}_{A}^{i}\left(x\right)+{\nu}_{A}^{i}\left(x\right)\le 1$, ${\pi}_{A}^{i}\left(x\right)=1-\left({\mu}_{A}^{i}\left(x\right)+{\nu}_{A}^{i}\left(x\right)\right)$ for every x ∈ U and i = 1,2,…,k where k is a positive integer and U is a universal set.
For a fixed x ∈ U, an object $\left\{{\mu}_{A}^{i}\left(x\right),{\nu}_{A}^{i}\left(x\right)\right\}$ is usually called an intuitionistic multi-fuzzy value (IMFV) or intuitionistic multi-fuzzy number (IMFN) where i = 1,2,…,k and k > 0.
Remark 1 If we arrange the membership sequences in decreasing order, then the corresponding non-membership sequence may not be in decreasing or increasing order.
Remark 2 The sequences of membership functions and non-membership functions only have k terms (finite number of terms), and k is known as the dimension of A.
Some useful operations on IMFS
Addition of IMFS (A ⊕ B)
Multiplication of IMFS (A ⊗ B)
Multiplication of IMFS A with real value
Now λA = (0.45, 0.37, 0.29)(0.32, 0.45, 0.63), A^{ λ } = (0.84, 0.77, 0.71)(0.05, 0.11, 0.23).
Proposed IMFWA operator
where A _{1},A _{2},…,A _{ n } are IMFS of dimension k and w _{1},w _{2},…,w _{ n } are their weights where $\sum _{i=1}^{n}{w}_{i}}=1$ and w _{ i } ∈ [0, 1].
Definition of score function S(x), x∈ U and accuracy function H(x), x∈ U for IMFSs
Comparison of IMFS
Since S(x) > S(y), therefore x > y.
IMFVs and choice values
U/E | e _{1} | e _{2} | e _{3} | e _{4} | e _{5} | Choice value |
---|---|---|---|---|---|---|
h _{1} | (0.3,0.7,0.5) | (0.4,0.3,0.4) | (0.1,0.2,0.0) | (0.5,0.6,0.7) | (0.4,0.3,0.4) | (1.7,2.1,2.0) |
(0.2,0.1,0.4) | (0.3,0.6,0.4) | (0.7,0.7,0.8) | (0.4,0.3,0.2) | (0.6,0.4,0.4) | (2.2,2.1,2.2) | |
h _{2} | (0.4,0.3,0.5) | (0.7,0.6,0.8) | (0.6,0.5,0.4) | (0.3,0.6,0.2) | (0.4,0.7,0.5) | (2.4,2.7,2.4) |
(0.5,0.4,0.4) | (0.2,0.2,0.1) | (0.3,0.3,0.4) | (0.7,0.3,0.7) | (0.1,0.2,0.3) | (1.8,1.4,1.9) | |
h _{3} | (0.1,0.2,0.1) | (0.3,0.2,0.1) | (0.8,0.7,0.8) | (0.3,0.2,0.2) | (0.4,0.3,0.2) | (1.9,1.6,1.4) |
(0.7,0.6,0.9) | (0.6,0.0,0.7) | (0.0,0.1,0.1) | (0.6,0.7,0.6) | (0.4,0.7,0.7) | (2.3,2.1,3.0) | |
h _{4} | (0.5,0.4,0.5) | (0.4,0.3,0.4) | (0.2,0.1,0.0) | (0.5,0.6,0.3) | (0.4,0.5,0.4) | (2.0,1.9,1.6) |
(0.4,0.4,0.3) | (0.5,0.3,0.5) | (0.7,0.6,0.7) | (0.4,0.3,0.6) | (0.6,0.4,0.3) | (2.6,2.0,2.4) | |
h _{5} | (0.6,0.2,0.7) | (0.8,0.3,0.4) | (0.3,0.3,0.3) | (0.3,0.4,0.1) | (0.5,0.3,0.4) | (2.5,1.5,1.9) |
(0.3,0.4,0.2) | (0.1,0.5,0.4) | (0.6,0.4,0.3) | (0.5,0.5,0.8) | (0.3,0.5,0.5) | (1.8,2.3,2.2) | |
h _{6} | (0.6,0.6,0.8) | (0.4,0.3,0.3) | (0.6,0.4,0.3) | (0.6,0.3,0.2) | (0.3,0.4,0.2) | (2.5,2.0,1.8) |
(0.2,0.3,0.1) | (0.5,0.1,0.2) | (0.2,0.4,0.4) | (0.3,0.6,0.4) | (0.3,0.4,0.6) | (1.5,1.8,1.7) |
Score function and choice values
U | Choice value | Score value | Accuracy value |
---|---|---|---|
h _{1} | (1.7,2.1,2.0) | 0.7 | 12.4 |
(2.2,2.1,2.2) | |||
h _{2} | (2.4,2.7,2.4) | 2.4 | 12.6 |
(1.8,1.4,1.9) | |||
h _{3} | (1.9,1.6,1.4) | 2.5 | 12.3 |
(2.3,2.1,3.0) | |||
h _{4} | (2.0,1.9,1.6) | 1.5 | 12.5 |
(2.6,2.0,2.4) | |||
h _{5} | (2.5,1.5,1.9) | 1.8 | 12.2 |
(1.8,2.3,2.2) | |||
h _{6} | (2.5,2.0,1.8) | 1.3 | 11.3 |
(1.5,1.8,1.7) |
Distance measurements over IMFSs
Presentation of IMFS to IFS degeneration
An intuitionistic multi-fuzzy set $A=\left\{x/\left({\mu}_{A}^{1}\left(x\right),{\mu}_{A}^{2}\left(x\right),\dots ,{\mu}_{A}^{k}\left(x\right)\right)\left({\nu}_{A}^{1}\left(x\right),{\nu}_{A}^{2}\left(x\right),\dots ,{\nu}_{A}^{k}\left(x\right)\right):x\in U\right\}$ can be degenerated to an IFS A^{ G } where ${A}^{G}=\left\{x/\left({\mu}_{A}^{G}\left(x\right),\left({\nu}_{A}^{G}\right.\left(x\right)\right):x\in U\right\},{\mu}_{A}^{G}\left(x\right)=\frac{1}{k}{\displaystyle \sum _{i=1}^{k}{\mu}_{A}^{i}},{\nu}_{A}^{G}\left(x\right)=\frac{1}{k}{\displaystyle \sum _{i=1}^{k}{\nu}_{A}^{i}},$ and $0\le {\mu}_{A}^{G}\left(x\right)+{\nu}_{A}^{G}\left(x\right)\le 1$, ${\pi}_{A}^{G}\left(x\right)=1-\left({\mu}_{A}^{G}\left(x\right)+{\nu}_{A}^{G}\left(x\right)\right)$ for every x ∈ U and i = 1,2,…,k where k is a positive integer and U is a universal set.
Example 8 Let A be an IMFS of dimension 3. If A = {(0.7, 0.6, 0.5), (0.1, 0.2, 0.4)}, then IFS A^{ G } can be expressed as A^{ G } = {(0.7 + 0.6 + 0.5)/3, (0.1 + 0.2 + 0.4)/3} = (0.6, 0.2).
An algorithm for IMFS-based GDM
In this section, we develop an algorithm for an intuitionistic multi-fuzzy set-based group decision making method using Hamming distance measurement technique.
Step 2: A weight is provided to each decision maker d _{ l }(l ∈ L) which can be expressed as w _{ i } ∈ [0, 1] ∀ i ∈ L where $\sum _{i=1}^{L}{w}_{i}=1$.
Step 5: Distance matrices are searched to find out minimal distance for each alternative.
Case study
Statistics have consistently shown that heart disease is one of the leading causes of deaths in the USA and all over the world (CDC's report) [22]. Prompt and accurate diagnosis followed by proper treatment is necessary for life saving. Unfortunately, accurate diagnosis of heart diseases has never been an easy task. We illustrate the approach by using a medical diagnosis-based case study specifically for heart disease diagnosis. In this research work, we have categorized heart disease into four different stages based on a common set of symptoms like chest pain, palpitations, dizziness, fainting, and fatigue. Stage ‘I’ indicates that the patient is in the initial stages of heart disease and is fully curable if diagnosed at early stages and follows proper treatment. Patients who belong to stage ‘II’ are in more unsafe conditions and critical to cure. Stage ‘III’ is more unsafe in comparison to stage ‘II’ , and the last stage is stage ‘IV’ wherein the disease is assumed to be not recoverable.
Observation of expert I
Patients/symptoms | Chest pain | Palpitations | Dizziness | Fainting | Fatigue |
---|---|---|---|---|---|
P _{1} | (0.3,0.7,0.5) | (0.4,0.3,0.4) | (0.1,0.2,0.0) | (0.5,0.6,0.7) | (0.4,0.3,0.4) |
(0.2,0.1,0.4) | (0.3,0.6,0.4) | (0.7,0.7,0.8) | (0.4,0.3,0.2) | (0.6,0.4,0.4) | |
(0.5,0.2,0.1) | (0.3,0.1,0.2) | (0.2,0.1,0.2) | (0.1,0.1,0.1) | (0.0,0.3,0.2) | |
P _{2} | (0.4,0.3,0.5) | (0.7,0.6,0.8) | (0.6,0.5,0.4) | (0.3,0.6,0.2) | (0.4,0.7,0.5) |
(0.5,0.4,0.4) | (0.2,0.2,0.1) | (0.3,0.3,0.4) | (0.7,0.3,0.7) | (0.1,0.2,0.3) | |
(0.1,0.3,0.1) | (0.1,0.2,0.1) | (0.1,0.2,0.2) | (0.0,0.1,0.1) | (0.5,0.1,0.2) | |
P _{3} | (0.1,0.2,0.1) | (0.3,0.2,0.1) | (0.8,0.7,0.8) | (0.3,0.2,0.2) | (0.4,0.3,0.2) |
(0.7,0.6,0.9) | (0.6,0.0,0.7) | (0.0,0.1,0.1) | (0.6,0.7,0.6) | (0.4,0.7,0.7) | |
(0.2,0.2,0.0) | (0.1,0.8,0.2) | (0.2,0.2,0.1) | (0.1,0.1,0.2) | (0.2,0.0,0.1) | |
P _{4} | (0.5,0.4,0.5) | (0.4,0.3,0.4) | (0.2,0.1,0.0) | (0.5,0.6,0.3) | (0.4,0.5,0.4) |
(0.4,0.4,0.3) | (0.5,0.3,0.5) | (0.7,0.6,0.7) | (0.4,0.3,0.6) | (0.6,0.4,0.3) | |
(0.1,0.2,0.2) | (0.1,0.4,0.1) | (0.1,0.3,0.3) | (0.1,0.1,0.1) | (0.0,0.1,0.3) |
Observation of expert II
Patients/symptoms | Chest pain | Palpitations | Dizziness | Fainting | Fatigue |
---|---|---|---|---|---|
P _{1} | (0.5,0.6,0.4) | (0.3,0.5,0.3) | (0.4,0.6,0.6) | (0.5,0.2,0.8) | (0.4,0.7,0.8) |
(0.2,0.2,0.5) | (0.3,0.2,0.5) | (0.2,0.3,0.2) | (0.3,0.6,0.1) | (0.4,0.0,0.1) | |
(0.3,0.2,0.1) | (0.4,0.3,0.2) | (0.2,0.1,0.2) | (0.2,0.2,0.1) | (0.2,0.3,0.1) | |
P _{2} | (0.6,0.2,0.7) | (0.8,0.3,0.4) | (0.3,0.3,0.3) | (0.3,0.4,0.1) | (0.5,0.3,0.4) |
(0.3,0.4,0.2) | (0.1,0.5,0.4) | (0.6,0.4,0.3) | (0.5,0.5,0.8) | (0.3,0.5,0.5) | |
(0.1,0.4,0.1) | (0.1,0.2,0.2) | (0.1,0.3,0.4) | (0.2,0.1,0.1) | (0.2,0.2,0.1) | |
P _{3} | (0.6,0.6,0.8) | (0.4,0.3,0.3) | (0.6,0.4,0.3) | (0.6,0.3,0.2) | (0.3,0.4,0.2) |
(0.2,0.3,0.1) | (0.5,0.1,0.2) | (0.2,0.4,0.4) | (0.3,0.6,0.4) | (0.3,0.4,0.6) | |
(0.2,0.1,0.1) | (0.1,0.6,0.5) | (0.2,0.2,0.3) | (0.1,0.1,0.4) | (0.4,0.2,0.2) | |
P _{4} | (0.5,0.6,0.7) | (0.3,0.7,0.2) | (0.2,0.3,0.2) | (0.4,0.3,0.5) | (0.2,0.4,0.3) |
(0.3,0.2,0.1) | (0.6,0.3,0.6) | (0.5,0.3,0.5) | (0.4,0.3,0.3) | (0.5,0.4,0.4) | |
(0.2,0.2,0.2) | (0.1,0.0,0.2) | (0.3,0.4,0.3) | (0.2,0.4,0.2) | (0.3,0.2,0.3) |
Observation of expert III
Patients/symptoms | Chest pain | Palpitations | Dizziness | Fainting | Fatigue |
---|---|---|---|---|---|
P _{1} | (0.3,0.4,0.7) | (0.3,0.8,0.7) | (0.4,0.1,0.2) | (0.3,0.4,0.4) | (0.1,0.3,0.7) |
(0.5,0.5,0.3) | (0.6,0.1,0.2) | (0.3,0.5,0.8) | (0.4,0.5,0.2) | (0.7,0.3,0.3) | |
(0.2,0.1,0.0) | (0.1,0.1,0.1) | (0.3,0.4,0.0) | (0.3,0.1,0.4) | (0.2,0.4,0.0) | |
P _{2} | (0.6,0.4,0.5) | (0.4,0.3,0.6) | (0.4,0.6,0.5) | (0.2,0.5,0.3) | (0.7,0.5,0.3) |
(0.2,0.3,0.4) | (0.5,0.2,0.3) | (0.3,0.3,0.4) | (0.4,0.4,0.7) | (0.2,0.2,0.3) | |
(0.2,0.3,0.1) | (0.1,0.5,0.1) | (0.3,0.1,0.1) | (0.4,0.1,0.0) | (0.1,0.3,0.4) | |
P _{3} | (0.4,0.2,0.6) | (0.4,0.6,0.4) | (0.6,0.5,0.3) | (0.6,0.4,0.3) | (0.3,0.3,0.3) |
(0.3,0.6,0.2) | (0.5,0.3,0.5) | (0.3,0.4,0.6) | (0.3,0.5,0.5) | (0.4,0.6,0.5) | |
(0.3,0.2,0.2) | (0.1,0.1,0.1) | (0.1,0.1,0.1) | (0.1,0.1,0.2) | (0.3,0.1,0.2) | |
P _{4} | (0.8,0.6,0.7) | (0.4,0.7,0.2) | (0.6,0.4,0.4) | (0.6,0.5,0.5) | (0.3,0.4,0.4) |
(0.1,0.1,0.3) | (0.5,0.3,0.5) | (0.3,0.4,0.3) | (0.4,0.3,0.4) | (0.5,0.4,0.3) | |
(0.1,0.3,0.0) | (0.1,0.0,0.3) | (0.1,0.2,0.3) | (0.0,0.2,0.1) | (0.2,0.2,0.3) |
Observation of expert IV
Patients/symptoms | Chest pain | Palpitations | Dizziness | Fainting | Fatigue |
---|---|---|---|---|---|
P _{1} | (0.3,0.3,0.5) | (0.6,0.4,0.4) | (0.4,0.3,0.4) | (0.6,0.3,0.4) | (0.2,0.3,0.3) |
(0.2,0.1,0.4) | (0.3,0.6,0.5) | (0.4,0.7,0.4) | (0.3,0.3,0.2) | (0.7,0.5,0.3) | |
(0.5,0.6,0.1) | (0.1,0.0,0.1) | (0.2,0.0,0.2) | (0.1,0.4,0.4) | (0.1,0.2,0.4) | |
P _{2} | (0.3,0.3,0.5) | (0.4,0.3,0.6) | (0.4,0.5,0.6) | (0.3,0.6,0.7) | (0.3,0.2,0.4) |
(0.5,0.5,0.4) | (0.5,0.2,0.3) | (0.3,0.3,0.3) | (0.5,0.3,0.1) | (0.5,0.2,0.2) | |
(0.2,0.2,0.1) | (0.1,0.5,0.1) | (0.3,0.2,0.1) | (0.2,0.1,0.2) | (0.2,0.6,0.4) | |
P _{3} | (0.3,0.2,0.4) | (0.3,0.2,0.4) | (0.8,0.7,0.3) | (0.3,0.7,0.1) | (0.2,0.3,0.1) |
(0.2,0.6,0.1) | (0.6,0.0,0.2) | (0.0,0.1,0.1) | (0.6,0.3,0.3) | (0.2,0.2,0.1) | |
(0.5,0.2,0.5) | (0.1,0.8,0.4) | (0.2,0.2,0.6) | (0.1,0.0,0.6) | (0.4,0.5,0.8) | |
P _{4} | (0.2,0.4,0.3) | (0.1,0.3,0.3) | (0.2,0.1,0.1) | (0.5,0.3,0.3) | (0.4,0.3,0.0) |
(0.2,0.4,0.3) | (0.5,0.3,0.2) | (0.2,0.6,0.1) | (0.4,0.3,0.1) | (0.0,0.1,0.7) | |
(0.6,0.2,0.4) | (0.4,0.4,0.5) | (0.6,0.3,0.8) | (0.1,0.4,0.6) | (0.6,0.6,0.3) |
Symptoms vs diseases
Symptoms/stages | Stage I | Stage II | Stage III | Stage IV |
---|---|---|---|---|
Chest pain | (0.8,0.1,0.1) | (0.7,0.2,0.1) | (0.5,0.3,0.2) | (0.1,0.7,0.2) |
Palpitations | (0.2,0.7,0.1) | (0.5,0.1,0.4) | (0.3,0.5,0.2) | (0.3,0.6,0.1) |
Dizziness | (0.3,0.5,0.2) | (0.4,0.2,0.4) | (0.8,0.1,0.1) | (0.5,0.1,0.4) |
Fainting | (0.5,0.3,0.2) | (0.4,0.1,0.5) | (0.3,0.2,0.5) | (0.2,0.6,0.2) |
Fatigue | (0.5,0.4,0.1) | (0.1,0.2,0.7) | (0.4,0.4,0.2) | (0.3,0.3,0.4) |
Aggregated intuitionistic multi-fuzzy matrix
Patients/symptoms | Chest pain | Palpitations | Dizziness | Fainting | Fatigue |
---|---|---|---|---|---|
P _{1} | (0.36,0.53,0.54) | (0.41,0.55,0.48) | (0.34,0.33,0.34) | (0.49,0.39,0.62) | (0.29,0.43,0.60) |
(0.25,0.18,0.39) | (0.36,0.29,0.38) | (0.36,0.52,0.48) | (0.35,0.41,0.17) | (0.59,0.00,0.24) | |
(0.39,0.30,0.07) | (0.23,0.16,0.15) | (0.30,0.15,0.19) | (0.17,0.20,0.22) | (0.13,0.57,0.15) | |
P _{2} | (0.49,0.30,0.56) | (0.62,0.39,0.63) | (0.44,0.49,0.46) | (0.28,0.53,0.38) | (0.50,0.46,0.40) |
(0.35,0.39,0.34) | (0.27,0.25,0.24) | (0.36,0.32,0.35) | (0.51,0.37,0.44) | (0.23,0.25,0.31) | |
(0.16,0.30,0.10) | (0.12,0.36,0.13) | (0.21,0.19,0.19) | (0.21,0.10,0.18) | (0.27,0.29,0.29) | |
P _{3} | (0.38,0.33,0.54) | (0.35,0.35,0.31) | (0.72,0.59,0.49) | (0.47,0.44,0.20) | (0.30,0.33,0.20) |
(0.30,0.50,0.21) | (0.55,0.00,0.34) | (0.00,0.20,0.22) | (0.42,0.50,0.44) | (0.31,0.43,0.38) | |
(0.32,0.17,0.25) | (0.10,0.65,0.35) | (0.28,0.21,0.29) | (0.10,0.06,0.36) | (0.38,0.25,0.42) | |
P _{4} | (0.55,0.51,0.58) | (0.31,0.54,0.28) | (0.33,0.24,0.19) | (0.51,0.44,0.41) | (0.33,0.40,0.29) |
(0.22,0.24,0.23) | (0.52,0.30,0.42) | (0.38,0.46,0.32) | (0.40,0.30,0.29) | (0.00,0.38,0.40) | |
(0.23,0.25,0.19) | (0.17,0.16,0.30) | (0.29,0.31,0.49) | (0.09,0.26,0.30) | (0.67,0.31,0.31) |
As the patient's condition and variations of symptoms differ with time, for an expert, it might not be feasible to make a decision based on one inspection of the patient. A probable solution might be to inspect the patient at regular time intervals and record them accordingly. This paper has taken the sample readings at three different time intervals of a day. So each symptom is described by three membership functions (μ), three non-membership functions (v), and three hesitation margins (π).
In accordance with the case study, the various steps of the algorithm are described as follows:
Step 1: Intuitionistic multi-fuzzy matrix R = {R^{1}, R^{2}, R^{3}, R^{4}} for each expert E = {e _{1}, e _{2}, e _{3}, e _{4}} is constructed by considering the three consecutive observations of those experts. Tables 3, 4, 5, and 6 shows the intuitionistic multi-fuzzy decision matrices (R^{1}, R^{2}, R^{3}, R^{4}) for expert I (e _{1}), expert II (e _{2}), expert III (e _{3}), and expert IV (e _{4}), respectively.
Step 2: In the case study, we have considered two real cases. Initially, the weights of each expert are taken to be equal. In the second case, we assume that the weights of each expert are different. Due to experts' variation in experience and domain of expertise, different priorities can be assigned to different experts, so variations in their weights during the decision making process is practical.
Step 3: An aggregated intuitionistic multi-fuzzy matrix is constructed based on the opinions of the decision makers. When the weights of each expert are equal, Table 8 shows the corresponding aggregated intuitionistic multi-fuzzy matrix.
Distance matrix with equal weight (w _{1} = w _{2} = w _{3} = w _{4} = 1/4)
Patients/diseases | Hamming distance | Euclidean distance | Diagnosis result | ||||||
---|---|---|---|---|---|---|---|---|---|
Stage I | Stage II | Stage III | Stage IV | Stage I | Stage II | Stage III | Stage IV | ||
P _{1} | 0.23 | 0.31 | 0.28 | 0.34 | 0.25 | 0.30 | 0.28 | 0.32 | Stage I |
P _{2} | 0.26 | 0.30 | 0.25 | 0.27 | 0.27 | 0.29 | 0.24 | 0.26 | Stage III |
P _{3} | 0.32 | 0.31 | 0.23 | 0.24 | 0.32 | 0.29 | 0.24 | 0.26 | Stage III |
P _{4} | 0.23 | 0.26 | 0.25 | 0.29 | 0.23 | 0.25 | 0.28 | 0.30 | Stage I |
Distance matrix with different weights (w _{1} = 0.1, w _{2} = 0.2, w _{3} = 0.3, w _{4} = 0.4)
Patients/diseases | Hamming distance | Euclidean distance | Diagnosis result | ||||||
---|---|---|---|---|---|---|---|---|---|
Stage I | Stage II | Stage III | Stage IV | Stage I | Stage II | Stage III | Stage IV | ||
P _{1} | 0.24 | 0.31 | 0.27 | 0.34 | 0.26 | 0.30 | 0.27 | 0.31 | Stage I |
P _{2} | 0.26 | 0.29 | 0.23 | 0.26 | 0.26 | 0.28 | 0.23 | 0.26 | Stage III |
P _{3} | 0.34 | 0.29 | 0.26 | 0.25 | 0.34 | 0.28 | 0.26 | 0.25 | Stage IV |
P _{4} | 0.26 | 0.24 | 0.26 | 0.30 | 0.26 | 0.24 | 0.28 | 0.30 | Stage II |
Distance matrix with different weight (w _{1} = 0.3, w _{2} = 0.2, w _{3} = 0.4, w _{4} = 0.1)
Patients/diseases | Hamming distance | Euclidean distance | Diagnosis result | ||||||
---|---|---|---|---|---|---|---|---|---|
Stage I | Stage II | Stage III | Stage IV | Stage I | Stage II | Stage III | Stage IV | ||
P _{1} | 0.23 | 0.32 | 0.29 | 0.35 | 0.25 | 0.31 | 0.30 | 0.33 | Stage I |
P _{2} | 0.27 | 0.31 | 0.25 | 0.27 | 0.27 | 0.30 | 0.25 | 0.27 | Stage III |
P _{3} | 0.31 | 0.34 | 0.22 | 0.24 | 0.31 | 0.32 | 0.23 | 0.25 | Stage III |
P _{4} | 0.19 | 0.27 | 0.26 | 0.30 | 0.21 | 0.27 | 0.28 | 0.31 | Stage I |
Distance matrix with different weight (w _{1} = 0.2, w _{2} = 0.1, w _{3} = 0.3, w _{4} = 0.4)
Patients/diseases | Hamming distance | Euclidean distance | Diagnosis result | ||||||
---|---|---|---|---|---|---|---|---|---|
Stage I | Stage II | Stage III | Stage IV | Stage I | Stage II | Stage III | Stage IV | ||
P _{1} | 0.24 | 0.31 | 0.29 | 0.35 | 0.26 | 0.31 | 0.29 | 0.32 | Stage I |
P _{2} | 0.26 | 0.29 | 0.24 | 0.26 | 0.27 | 0.28 | 0.23 | 0.26 | Stage III |
P _{3} | 0.35 | 0.31 | 0.25 | 0.23 | 0.35 | 0.29 | 0.25 | 0.25 | Stage IV |
P _{4} | 0.26 | 0.25 | 0.26 | 0.30 | 0.25 | 0.24 | 0.28 | 0.30 | Stage II |
Diagnostic results from Table 9 indicates that patient P _{1} and P _{4} are in stage I, while P _{2} and P _{3} are in stage III which is similar to the diagnostic result of Table 11. Other diagnostic results from Table 10 specifies that patient P _{1} is in stage I, P _{2} is in stage III, P _{3} is in stage IV, and P _{4} is in stage II which is similar to Table 12. Hamming distance and Euclidean distance measurements are observed to produce similar diagnostic outcome in all the cases.
Distance matrix with equal weight (w _{1} = w _{2} = w _{3} = w _{4} = 1/4)
Patients/diseases | Hamming distance | Euclidean distance | Diagnosis result | ||||||
---|---|---|---|---|---|---|---|---|---|
Stage I | Stage II | Stage III | Stage IV | Stage I | Stage II | Stage III | Stage IV | ||
P _{1} | 0.17 | 0.31 | 0.20 | 0.29 | 0.33 | 0.50 | 0.41 | 0.49 | Stage I |
P _{2} | 0.26 | 0.29 | 0.22 | 0.25 | 0.44 | 0.49 | 0.40 | 0.42 | Stage III |
P _{3} | 0.30 | 0.29 | 0.18 | 0.20 | 0.49 | 0.46 | 0.32 | 0.36 | Stage III |
P _{4} | 0.18 | 0.28 | 0.21 | 0.26 | 0.31 | 0.45 | 0.43 | 0.48 | Stage I |
Distance matrix with different weights (w _{1} = 0.1, w _{2} = 0.2, w _{3} = 0.3, w _{4} = 0.4)
Patients/diseases | Hamming distance | Euclidean distance | Diagnosis result | ||||||
---|---|---|---|---|---|---|---|---|---|
Stage I | Stage II | Stage III | Stage IV | Stage I | Stage II | Stage III | Stage IV | ||
P _{1} | 0.18 | 0.31 | 0.20 | 0.29 | 0.35 | 0.50 | 0.39 | 0.48 | Stage I |
P _{2} | 0.25 | 0.28 | 0.21 | 0.24 | 0.42 | 0.47 | 0.37 | 0.41 | Stage III |
P _{3} | 0.31 | 0.26 | 0.20 | 0.21 | 0.50 | 0.41 | 0.34 | 0.37 | Stage III |
P _{4} | 0.21 | 0.25 | 0.22 | 0.25 | 0.35 | 0.40 | 0.42 | 0.47 | Stage I |
Conclusion
In this paper, we first present an intuitionistic multi-fuzzy set which is a combination of an intuitionistic fuzzy set and a multi-fuzzy set. We then introduce some basic operations on intuitionistic multi-fuzzy set and their properties. Especially, we introduce the concept of intuitionistic multi-fuzzy weighted averaging operator which can be treated as an extension of intuitionistic weighted arithmetic mean operator. We have proposed an algorithmic approach which combines the opinions of individual decision makers to produce an aggregated matrix by using a group multi-criteria decision making method. In the proposed approach, we measure the Hamming distance and Euclidean distance of each patient for the set of diseases by considering the symptoms of that particular disease where both type of distance measurements yield similar diagnostic result. The lowest distance shows proper diagnosis for both Hamming distance and Euclidean distance measurements. The observed case study illustrates our approach in two phases: in the first phase, all decision makers are assumed to have equal importance or weight, and the next phase considers different weights for each individual decision makers. Our observation proves that changes in experts' weight might have several impacts on disease diagnosis, i.e., diagnosis might vary for different patients. In this study, we also present score function and accuracy function in the context of IMFS with an application to MCDM. Finally, this study degenerates the individual experts' opinions (IMFS) into IFS and compares the final outcome which is found to be different. The concept of multiplicity is incorporated by taking samples from the same patient at different times. A future scope of this research work might be enhancing the study of intuitionistic multi-fuzzy set for uncertain group decision making problems where GDM is crucial due to the lack of information, expertise of the experts, risk amendment, etc. An example of these type of problems might be critical disease diagnosis, risk management, natural disaster forecasting, typical financial decision making, etc. Researchers may introduce the advantages of machine learning techniques to allow updating of information for decision makers which would be more adaptable in the real life environments.
Declarations
Acknowledgements
The authors wish to express sincere gratitude to the anonymous reviewers for their constructive comments and helpful suggestions, which led to substantial improvements of this paper.
Authors’ Affiliations
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