Introduction to Hesitant Fuzzy Set andIts application in Decision Making

Problems

Laya Aliahmadipour

Department of Computer ScienceFaculty of Mathematics and ComputerShahid Bahonar University of Kerman

21 May 2018

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Outline

1 Introduction

2 Application of HFSsMulti-criteria decision analysis (MCDA)Medical decision makingData mining

3 References

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Outline

1 Introduction

2 Application of HFSsMulti-criteria decision analysis (MCDA)Medical decision makingData mining

3 References

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Basic backgrounds

History

L.A. Zadeh (1965) : Fuzzy Sets,

K. T. Atanassov (1986) : Intuitionistic Fuzzy Sets,

V. Torra (2009) : Hesitant Fuzzy Sets.

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Basic backgrounds

History

L.A. Zadeh (1965) : Fuzzy Sets,

K. T. Atanassov (1986) : Intuitionistic Fuzzy Sets,

V. Torra (2009) : Hesitant Fuzzy Sets.

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Basic backgrounds

History

L.A. Zadeh (1965) : Fuzzy Sets,

K. T. Atanassov (1986) : Intuitionistic Fuzzy Sets,

V. Torra (2009) : Hesitant Fuzzy Sets.

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Basic background

Definition

[5] Let a set X be the universe of discourse. Anintuitionistic fuzzy set is an object having the form:

A = {〈x, µA(x), νA(x)〉|0 ≤ µA(x) + νA(x) ≤ 1, x ∈ X},

where the function µA, νA : X→ [0, 1] define the degree ofmembership and the degree of non-membership of elementsin X to A, respectively. For any A-IFS A and

x ∈ X,πA(x) = 1− (µA(x) + νA(x)) is called the degree ofindeterminacy of x to A.

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Basic background

Definition

[5] Let a set X be the universe of discourse. Anintuitionistic fuzzy set is an object having the form:

A = {〈x, µA(x), νA(x)〉|0 ≤ µA(x) + νA(x) ≤ 1, x ∈ X},

where the function µA, νA : X→ [0, 1] define the degree ofmembership and the degree of non-membership of elementsin X to A, respectively.

For any A-IFS A and

x ∈ X,πA(x) = 1− (µA(x) + νA(x)) is called the degree ofindeterminacy of x to A.

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Basic background

Definition

[5] Let a set X be the universe of discourse. Anintuitionistic fuzzy set is an object having the form:

A = {〈x, µA(x), νA(x)〉|0 ≤ µA(x) + νA(x) ≤ 1, x ∈ X},

where the function µA, νA : X→ [0, 1] define the degree ofmembership and the degree of non-membership of elementsin X to A, respectively. For any A-IFS A and

x ∈ X,πA(x) = 1− (µA(x) + νA(x)) is called the degree ofindeterminacy of x to A.

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Basic backgrounds

Definition

[6] Let X be a fixed set, then a hesitant fuzzy set (HFS) onX in terms of a function h is that when applied to X returnsa subset of [0, 1], i.e.,

h : X→ P([0, 1]).

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Basic backgrounds

Definition

Xia and Xu [7] called h(x) a hesitant fuzzy element (HFE). Ahesitant fuzzy element (HFE) is a set of values in [0, 1], and aHFS is a set of HFEs, for each x ∈ X.

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Basic background

Example

Let X = {x1, x2} be a universal set.

A = {〈x1, 0.3〉, 〈x2, 0.6〉} is a fuzzy set.

B = {〈x1, 0.3, 0.4〉, 〈x2, 0.1, 0.3〉} is an IFS.

hC(x1) = {0.2, 0.4, 0.5}, hC(x2) = {0.3, 0.4, 0.6, 0.7}

C = {〈x1, {0.2, 0.4, 0.5}〉, 〈x2, {0.3, 0.4, 0.6, 0.7}〉} is a HFS.

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Basic background

Example

Let X = {x1, x2} be a universal set.

A = {〈x1, 0.3〉, 〈x2, 0.6〉} is a fuzzy set.

B = {〈x1, 0.3, 0.4〉, 〈x2, 0.1, 0.3〉} is an IFS.

hC(x1) = {0.2, 0.4, 0.5}, hC(x2) = {0.3, 0.4, 0.6, 0.7}

C = {〈x1, {0.2, 0.4, 0.5}〉, 〈x2, {0.3, 0.4, 0.6, 0.7}〉} is a HFS.

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Basic background

Example

Let X = {x1, x2} be a universal set.

A = {〈x1, 0.3〉, 〈x2, 0.6〉} is a fuzzy set.

B = {〈x1, 0.3, 0.4〉, 〈x2, 0.1, 0.3〉} is an IFS.

hC(x1) = {0.2, 0.4, 0.5}, hC(x2) = {0.3, 0.4, 0.6, 0.7}

C = {〈x1, {0.2, 0.4, 0.5}〉, 〈x2, {0.3, 0.4, 0.6, 0.7}〉} is a HFS.

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Basic background

Example

Let X = {x1, x2} be a universal set.

A = {〈x1, 0.3〉, 〈x2, 0.6〉} is a fuzzy set.

B = {〈x1, 0.3, 0.4〉, 〈x2, 0.1, 0.3〉} is an IFS.

hC(x1) = {0.2, 0.4, 0.5},

hC(x2) = {0.3, 0.4, 0.6, 0.7}

C = {〈x1, {0.2, 0.4, 0.5}〉, 〈x2, {0.3, 0.4, 0.6, 0.7}〉} is a HFS.

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Basic background

Example

Let X = {x1, x2} be a universal set.

A = {〈x1, 0.3〉, 〈x2, 0.6〉} is a fuzzy set.

B = {〈x1, 0.3, 0.4〉, 〈x2, 0.1, 0.3〉} is an IFS.

hC(x1) = {0.2, 0.4, 0.5}, hC(x2) = {0.3, 0.4, 0.6, 0.7}

C = {〈x1, {0.2, 0.4, 0.5}〉, 〈x2, {0.3, 0.4, 0.6, 0.7}〉} is a HFS.

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Basic background

Example

Let X = {x1, x2} be a universal set.

A = {〈x1, 0.3〉, 〈x2, 0.6〉} is a fuzzy set.

B = {〈x1, 0.3, 0.4〉, 〈x2, 0.1, 0.3〉} is an IFS.

hC(x1) = {0.2, 0.4, 0.5}, hC(x2) = {0.3, 0.4, 0.6, 0.7}

C = {〈x1, {0.2, 0.4, 0.5}〉, 〈x2, {0.3, 0.4, 0.6, 0.7}〉} is a HFS.

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Basic background

Some Operators

hλ = ∪γ∈h{γλ},λh = ∪γ∈h{1− (1− γ)λ},h1 ⊕ h2 = ∪γ1∈h1,γ2∈h2{γ1 + γ2 − γ1γ2},h1 ⊗ h2 = ∪γ1∈h1,γ2∈h2{γ1γ2},h1 ∪ h2 = ∪γ1∈h1,γ2∈h2 max {γ1, γ2} ,h1 ∩ h2 = ∪γ1∈h1,γ2∈h2 min {γ1, γ2} .

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Outline

1 Introduction

2 Application of HFSsMulti-criteria decision analysis (MCDA)Medical decision makingData mining

3 References

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Application of HFS

Decision making Problems[7],

Medical decision making [4],

Data mining [1, 2, 3].

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Application of HFS

Decision making Problems[7],

Medical decision making [4],

Data mining [1, 2, 3].

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Application of HFS

Decision making Problems[7],

Medical decision making [4],

Data mining [1, 2, 3].

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Multi-criteria decision analysis (MCDA)

MCDA is a formal, structured and transparent decisionmaking methodology. Its aim is to assist groups or individualdecision makers to explore their decisions in the case ofcomplex situations with multiple criteria.

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Multi-criteria decision analysis (MCDA)

Through the use of MCDA a larger number, of perhapsindividually less important, indicators do not get ignored inthe final decision during a simplification that relies tooheavily on a small number of key criteria.

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MCDA

Goal Programming (GP) answers ”how much?”

Analytical Hierarchy Process (AHP) answers ”whichone?”

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MCDA

AHP developed by Saati (1980) Method for ranking decisionalternatives and selecting the best one when the decisionmaker has multiple objectives, or criteria.

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MCDM

Example

Buying a houseCost, proximity of schools, trees, nationhood, publictransportation

Buying a carPrice, interior comfort, mpg, appearance, etc.

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MCDM

In an MCDM problem suppose that there are n alternativesYi(i = 1, 2, . . . . , n) and m attributes Aj(j = 1, 2, · · · ,m)with the attribute weight vector w = (w1, w2, · · · , wn)T suchthat wj ∈ [0, 1], (j = 1, 2, ..., n), and l decision makersEk(k = 1, 2, · · · , l) that provide several values for thealternative Yi under the attribute Aj .

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MCDM

These values can be considered as a hesitant fuzzy elementhij . A basic scheme of a MADM consists of two phases,aggregation and exploitation. In the aggregation phase, theinformation is grouped to reflect a collective value for eachalternative or attribute. In the exploitation phase, the mostdesirable alternative(s) among a set of alternatives isselected. To do so, the HFEs obtained from the previousphase are compared by a score function.

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Example

Suppose there are four possible projects Yi(i = 1, 2, 3, 4) tobe evaluated. It is necessary to compare these projects toselect the most important of them as well as order them fromthe point of view of their importance, taking into accountfour attributes suggested by the Balanced Scorecardmethodology [31] (it should be noted that all of them are ofthe maximization type):

G1: financial perspective,

G2: the customer satisfaction,

G3: internal business process perspective.

And suppose that the weight vector of the attributes isw = (0.2, 0.3, 0.15, 0.35).

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Example

Hesitant fuzzy decision matrixG1 G2 G3

Y1 (0.2, 0.4, 0.7) (0.2, 0.6, 0.8) (0.2, 0.3, 0.6, 0.7, 0.9)Y2 (0.2, 0.4, 0.7, 0.9) (0.1, 0.2, 0.4, 0.5) (0.3, 0.4, 0.6, 0.9)Y3 (0.3, 0.5, 0.6, 0.7) (0.2, 0.4, 0.5, 0.6) (0.3, 0.5, 0.7, 0.8)Y4 (0.3, 0.5, 0.6) (0.2, 0.4) (0.5, 0.6, 0.7)

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Example

Final decision:Y1 Y2 Y3 Y4 Ranking

GHFWA1 0.5634 0.6009 0.5178 0.6524 Y4 > Y2 > Y1 > Y3GHFWA2 0.5847 0.6278 0.5337 0.6781 Y4 > Y2 > Y1 > Y3GHFWG1 0.4783 0.4625 0.4661 0.5130 Y4 > Y1 > Y3 > Y2GHFWG2 0.4546 0.4295 0.4526 0.4755 Y4 > Y1 > Y3 > Y2

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Medical decision making

Motivations

We introduce a new view for applying hesitant fuzzy setin MCDM and data mining problems. In this case, wetry to model hesitancy and uncertainty existing in thenature of some medical application to hesitant fuzzysets.

We attempt to illustrate the advantages of applyinghesitant fuzzy set to specialists and physicians.

we use hesitant fuzzy sets to help each physician tomake the best treatment for similar patients based onthe opinions of specialists.

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Medical decision making

Our approach

In medical data, considering hesitancy in every stage, can beuseful. For instance, in symptoms observation stage:different views about symptoms may lead to completelydifferent diseases or treatments.

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Our approach

collecting patients records by each specialist and saving therecords in a general data base leads to two advantages:

specialists can compare their opinions and hesitation ofprescribed treatment with opinions of the otherspecialists.

Physicians can assess multiple treatments used byspecialists in the similar cases and in vital cases arrangemedical committee.

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Our approach (Steps...)

In the following steps, the approach is explained:

1 Suppose, there is a hesitant fuzzy data base of patientsrecords of specific disease which all specialists confirmrelated symptoms of disease.

2 Once a patient visits a physician for the same disease,based on his/ her observations, the physician recordshis/her opinions for each determined feature of thedisease. There is no limitation for recording his/ heropinions and if there are different views about onespecific feature, he /she can apply set of possible valuesas memberships in [0; 1].

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Our approach (Steps...)

In the following steps, the approach is explained:

1 Suppose, there is a hesitant fuzzy data base of patientsrecords of specific disease which all specialists confirmrelated symptoms of disease.

2 Once a patient visits a physician for the same disease,based on his/ her observations, the physician recordshis/her opinions for each determined feature of thedisease. There is no limitation for recording his/ heropinions and if there are different views about onespecific feature, he /she can apply set of possible valuesas memberships in [0; 1].

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Our approach (Steps...)

3 The physician find the similar records using hesitantfuzzy data mining task an access to the opinions andtreatments of other specialists.

4 If the user is a specialist, he/she should insert his/herobservations of new patient as hesitant fuzzy set andrelated treatments in the database

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Our approach

To illustrate the approach, we apply K-NN method on thePrognostic Breast Cancer data set transferred to hesitantfuzzy data.

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Our approach

Hesitant fuzzy data

The given input information is a set of HFSs, on featurespace X = {x1, x2, ..., xn} in the following form :

S = {D1, D2, . . . , Dm}Dj = {(xi, hDj (xi))|i = 1, 2, . . . , n}, j = 1, 2, . . . ,m,

hDj (xi) = {µk|µk ∈ [0, 1], k = 1, 2, · · · ,K}.

where S is a data set of D,js and hDj (xi) is a set of experts

assessments with arbitrary cardinality K which states thatDj satisfies feature xi. In this paper X is a set of symptoms.

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Our approach

Hesitant fuzzy data

The given input information is a set of HFSs, on featurespace X = {x1, x2, ..., xn} in the following form :

S = {D1, D2, . . . , Dm}Dj = {(xi, hDj (xi))|i = 1, 2, . . . , n}, j = 1, 2, . . . ,m,

hDj (xi) = {µk|µk ∈ [0, 1], k = 1, 2, · · · ,K}.

where S is a data set of D,js and hDj (xi) is a set of experts

assessments with arbitrary cardinality K which states thatDj satisfies feature xi. In this paper X is a set of symptoms.

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Our approach

Hesitant fuzzy data

The given input information is a set of HFSs, on featurespace X = {x1, x2, ..., xn} in the following form :

S = {D1, D2, . . . , Dm}Dj = {(xi, hDj (xi))|i = 1, 2, . . . , n}, j = 1, 2, . . . ,m,

hDj (xi) = {µk|µk ∈ [0, 1], k = 1, 2, · · · ,K}.

where S is a data set of D,js and hDj (xi) is a set of experts

assessments with arbitrary cardinality K which states thatDj satisfies feature xi. In this paper X is a set of symptoms.

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Medical decision making

Example

We apply WPBC to illustrate implementation of theapproach.In WPBC, let X = x1, . . . , xn be a feature space andDj (j = 1, 2, . . . , 699) be a collection of 699 vectors aspatients.

D1 = (5, 1, 1, 1, 2, 1, 3, 1)

D1′=(5,0.1,0.1,0.1,0.2,0.1,0.3,0.1,0.1)

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Medical decision making

Example

We apply WPBC to illustrate implementation of theapproach.In WPBC, let X = x1, . . . , xn be a feature space andDj (j = 1, 2, . . . , 699) be a collection of 699 vectors aspatients.

D1 = (5, 1, 1, 1, 2, 1, 3, 1)

D1′=(5,0.1,0.1,0.1,0.2,0.1,0.3,0.1,0.1)

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Example

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Example

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Data mining

clustering

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Outline

1 Introduction

2 Application of HFSsMulti-criteria decision analysis (MCDA)Medical decision makingData mining

3 References

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References

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Aliahmadipour, L. and Eslami, E., GHFHC : generalizedhesitant fuzzy hierarchical clustering algorithm,International Journal of Intelligent Systems, 31 (2016),855–871.

Aliahmadipour, L., Torra, V., Eslami, E. and Eftekhari,M., A definition for hesitant fuzzy partitions,International Journal of computational Intelligent System, 9 (2016), 497–505.

Aliahmadipour, L., Torra, V. and Eslami, E., On HesitantFuzzy Clustering and Clustering of Hesitant fuzzy Data,Fuzzy sets, Rough set, Multisets and Clustering, Springer2016, under proof.

Aliahmadipour, L., Eslami, E. and Eftekhari, M., Anapplication of hesitant fuzzy sets in medical decisionmaking, 12th International Conference on ModelingDecisions for Artificial Intelligence, Skovde, Sweden,2015, 87–96.

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Atanassov, K.T., Intuitionistic fuzzy sets, Fuzzy Setsand Systems, 20 (1986), 87–96.

Torra, V., Hesitant fuzzy sets, International Journal ofIntelligent System, 25 (2010), 529–539.

Xia, M.M and Xu, Z.S., Hesitant Fuzzy InformationAggregation in Decision Making. International Journal ofApproximate Reasoning, 52 (2011), 395–407.

Xu. Z., Hesitant Fuzzy Sets Theory, Studies in Fuzzinessand Soft Computing. Springer, vol. 314 2014.

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