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Punjab University Journal of Mathematics(ISSN 1016-2526)Vol. 52(10)(2020) pp. 55-82

The Fuzzy Cross-Entropy for Picture Hesitant Fuzzy Sets and Their Application inMulti Criteria Decision Making

Tahir Mahmood1, Zeeshan Ali21,2Department of Mathematics and Statistics,

International Islamaic University Islamabad, Pakistan,Email: [email protected],[email protected]

Received: 21 January, 2020 / Accepted: 17 July, 2020 / Published online: 10 October, 2020

Abstract.: In this article, we exploit the cross-entropy of picture hesi-tant fuzzy set is established by distinguishing the cross-entropy of picturefuzzy set and hesitant fuzzy set. First, many measurement concepts areestablished and their basic properties are studied. Further, two measures,which are based on the established picture hesitant fuzzy cross-entropy,are established for evaluating multi-criteria decision making problems inthe environment of picture hesitant fuzzy sets. For both approaches, anoptimization model is pioneered to examine the weight vector for multicriteria decision making problems with incomplete information on criteriaweights. In a last, we give an example to express practical and effective-ness. The comparison of the proposed measures with existing measures isalso discussed.

AMS (MOS) Subject Classification Codes: 03B52; 94D05; 03E72Key Words: Picture fuzzy sets; Hesitant fuzzy sets; Picture hesitant fuzzy sets; Cross

Entropy measures.

1. INTRODUCTION

Multi criteria decision making has been utilized in the establishment of a multi-modalforce recognition system involving multi-criteria with varying weights of importance. Thetheory of a fuzzy set was investigated by Zadeh [1], characterized by only positive grade arerestricted to [0,1]. FS has achieved more success because of its capability of coping withcomplications and troubles. This proves a strong tool for expressing human knowledge[2-5]. However, in some practice cases, the concept of FS cannot cope with complicationsand uncertainty because of the lack of knowledge of the problem. Therefore, Atanassove[6] investigated the intuitionistic fuzzy set contains positive and negative grades, whosesum is bounded to [0,1]. IFS is regarded as a more improved way to cope with complexand awkward information. Although, Garg and Kumar [7] established exponential distancemeasures and TOPSIS methods for the interval-valued intuitionistic fuzzy set. Alcantud et

55

56 Tahir Mahmood and Zeeshan Ali

al. [8] established aggregation of finite chains based on the intuitionistic fuzzy set. VIKORmethod was established by Zhang et al. [9] and Konwar and Debnath [10] established theintuitionistic fuzzy membership metric. Further, Sen and Et [11] explored the intuitionisticfuzzy norm linear space. For more work related to the fuzzy set, we may refer to [12-14].The grade of a neutral cannot be discussed in intuitionistic fuzzy set. However, a neutralgrade is used in many real-life scenarios, such as polling stations, human beings give opin-ions having more answers of a kind: positive, abstinence, negative and refusal/neutral. Forexample, in a democratic voting system, 100 people have appeared in the election and theelection commission issued only 100 ballot paper. One person can take only one ballot pa-per for giving his/her vote and is only one candidate. Basically, the result is dived into fourgroups like vote for candidate 50, abstinence in vote 20, vote negative candidate 20, andrefusal vote is 10. The “abstinence in vote” shows that ballot paper is white which contra-dicts both “vote for a candidate” and “vote negative candidate” but it considers the vote andthe “refusal of voting” means bypassing the vote. Therefore, Cuong [15] investigated thepicture fuzzy set contains positive, abstinence and negative grades, whose sum is boundedto [0,1]. Picture fuzzy set is regarded as a more improved way to cope with complex andawkward information. Although, Zeng et al. [16] established the linguistic picture fuzzyTOPSIS methods and picture fuzzy aggregation operators were established by Garg [17].The notion of picture fuzzy aggregation operators was established by Wei [18]. Wang etal. [19] established the picture fuzzy Muirhead mean operators. After intuitionistic fuzzyset, the picture fuzzy set has achieved more attention from researchers and it is utilized inthe environment of decision making [20-22] problems.Hesitancy occurs in many real-life problems. To handle such types of problems, the hes-itant fuzzy set was explored by Torra [23], as a powerful modification of fuzzy set, thatallows multiple positive grades is associated with each preference information for copingwith an awkward and complicated situation. Keeping the advantages of hesitant fuzzy set,researchers widely explored the concept for pattern recognition [24-26], medical diagnosis[27-30], similarity measures [31], and decision making [32-34] problems.Picture fuzzy set has utilized successfully in various fields but in some practical cases,the picture fuzzy set cannot work effectively. For example, when a decision-maker pro-vides for positive, abstinence, negative grades in the form of a finite subset of [0,1], thenthe condition of existing methods cannot hold. For handling such types of situations, thepicture hesitant fuzzy set was established by Ullah et al. [35], which is more generalizedthan an existing drawback. Picture hesitant fuzzy set is a successful tool for describing theuncertainty and awkward kinds of information in the fuzzy set theory. Basically, picturehesitant fuzzy set contains three degrees is called positive, abstinence, and negative arethe form of a subset of [0,1]. The condition of the picture hesitant fuzzy set is that thesum of the maximum of a positive degree, a maximum of abstinence degree, and maximumof negative degree is bounded to [0,1]. Picture hesitant fuzzy set has achieved more atten-tion like: Wang and Li [36] established the aggregation operators based on picture hesitantfuzzy sets. Jan et al. [37] established the generalized similarity measures based on picturehesitant fuzzy sets. Ahmad et al. [38] established the similarity measures based on picturehesitant fuzzy sets.

Creating an Environment for Learning Mathematics 57

When a decision-maker provides the grade of truth, the grade of abstinence and the grade of

falsity in the form of a subset of unit interval such that

{0.4, 0.3} ,{0.11, 0.01} ,{0.02, 0.01}

, then the exit-

ing notions [1, 6, 15, 23], are cannot deal it effectively, for coping with such kind of issues,the notion of picture hesitant fuzzy set was explored. Although fuzzy set is a valid formto express the uncertain evaluation information, it cannot solve several complex situationsin real life. For more effective expression of the evaluation information, many generalizedforms of fuzzy set were proposed [5–10]. Keeping the advantages of the picture hesitantfuzzy set can express the uncertainty and complexity of human opinions in practice; fur-thermore, the positive, neutral, negative, and refusal membership degrees are representedby several possible values that are given by decision makers, in this article the exploredwork is summarized is follow as:

(1) The notion of cross-entropy of picture hesitant fuzzy set is established by distin-guishing the cross-entropy of picture fuzzy set and hesitant fuzzy set. First, manymeasurement concepts are established and their basic properties are studied. Fur-ther, two measures, which are based on the established picture hesitant fuzzy cross-entropy, are established for evaluating multi criteria decision making problems inthe environment of picture hesitant fuzzy sets.

(2) For both approaches, an optimization model is pioneered in order to examine theweight vector for multi criteria decision making problems with incomplete infor-mation on criteria weights. In last, we give an example to express the practicallyand effectiveness.

(3) The comparison of the proposed measures with existing measures are also dis-cussed in detail.

Besides, the predominance of the picture hesitant fuzz model over a portion of the currentmodels is shown in Table 1.

Table 1. Comparison of picture hesitant fuzzy sets model with existing models in literature.Model Uncertainly falsity hesitations Degrees

in theform ofsubset

StrongCondi-tion

Refusalgrades

FSs √ × × × × ×HFSs √ × × √ × ×IFSs √ √ √ × × ×IHFSs √ √ √ × √ ×PFSs √ √ √ × × √PHFSs √ √ √ √ √ √

The aim of this article is to follow as, In section 2, we recall some concepts of picturefuzzy sets, hesitant fuzzy sets, picture hesitant fuzzy sets, cross-entropy, and the propertiesare discussed. In section 3, the cross-entropy measures based on picture hesitant fuzzy sets


are explored. In section 4, the concept is divided for evaluating multi-criteria decision-making problems using the picture hesitant fuzzy numbers, where the criteria weights arenot completely known. Further, we resolve some numerical examples to show the reliabilityand efficiency of the proposed measures. The compassion o the proposed measures withexisting measures are also conducted. The conclusion is discussed in section 5.

2. NOTATIONS AND PRELIMINARIES

In this section, the ideas of picture fuzzy sets, hesitant fuzzy sets, picture hesitant fuzzysets, cross-entropy, and the properties are discussed. Throughout this article, X representsthe finite fixed set.

Definition 2.1. [15] A picture fuzzy setPPFS onX is an object with the following form:

PPFS = {(MPP F S(x) ,APP F S

(x) ,NPP F S(x)) : x ∈ X}

WhereMPP F S,APP F S

,NPP F S: X → [0, 1] represents the positive, abstinence and nega-

tive grades, with a condition0 ≤MPP F S(x)+APP F S

(x)+NPP F S(x) ≤ 1. The refusal

grade of picture fuzzy set is given by:

πPP F S (x) = 1− (MPP F S (x) +APP F S (x) +NPP F S (x))

The triplet(MPP F S(x) ,APP F S

(x) ,NPP F S(x)) represents the picture fuzzy numbers.

Definition 2.2. [23] A hesitant fuzzy setPHFS onX is an object with the following form:

PHFS = {(x,MPHF S(x)) : x ∈ X}

WhereMPHF S is a set of values in [0,1], presented the positive grade.

Definition 2.3. [35] A picture hesitant fuzzy setPPHFS onX is an object with the followingform:

PPHFS = {(αPP HF S(x) , βPP HF S

(x) , γPP HF S(x)) : x ∈ X}

WhereαPP HF S , βPP HF S andγPP HF S are finite non-empty sets of values in [0, 1], repre-senting the positive, abstinence and negative grades, with a condition0 ≤ max (αPP HF S

(x)) + max (βPP HF S(x)) + max (γPP HF S

(x)) ≤ 1. The refusal gradeof picture hesitant fuzzy set is given by:πPP HF S

(x) = 1 − (MPP HF S(x) +APP HF S

(x) +NPP HF S(x)), whereMPP HF S

∈αPP HF S ,APP HF S ∈ βPP HF S ,NPP HF S ∈ γPP HF S , πPP HF S (x) ∈ δPP HF S (x). The triplet(αPP HF S (x) , βPP HF S (x) , γPP HF S (x)) represents the picture hesitant fuzzy numbers.

Definition 2.4. [35] LetPPHFS−1 =�αPP HF S−1 (x) , βPP HF S−1 (x) , γPP HF S−1 (x)

�andPPHFS−2 =�

αPP HF S−2 (x) , βPP HF S−2 (x) , γPP HF S−2 (x)�

be two picture hesitant fuzzy numbers. Then

(1) PPHFS−1 ⊕ PPHFS−2 =0BBBBBBBBBB@

‘MPP HF S−1 ∈ αPP HF S−1 ,MPP HF S−2 ∈ αPP HF S−2

0@

MPP HF S−1+MPP HF S−2−

MPP HF S−1 .MPP HF S−2

1A,

‘APP HF S−1 ∈ βPP HF S−1 ,APP HF S−2 ∈ βPP HF S−2

�APP HF S−1 .APP HF S−2

�,

‘NPP HF S−1 ∈ γPP HF S−1 ,NPP HF S−2 ∈ γPP HF S−2

�NPP HF S−1 .NPP HF S−2

�

1CCCCCCCCCCA

;


(2) PPHFS−1 ⊗ PPHFS−2 =0BBBBBBBBBBBB@

‘MPP HF S−1 ∈ αPP HF S−1 ,MPP HF S−2 ∈ αPP HF S−2

�MPP HF S−1 .MPP HF S−2

�,

‘APP HF S−1 ∈ βPP HF S−1 ,APP HF S−2 ∈ βPP HF S−2

0@

APP HF S−1+APP HF S−2−

APP HF S−1 .APP HF S−2

1A,

‘NPP HF S−1 ∈ γPP HF S−1 ,NPP HF S−2 ∈ γPP HF S−2

0@

NPP HF S−1+NPP HF S−2−

NPP HF S−1 .NPP HF S−2

1A

1CCCCCCCCCCCCA

;

(3) ∂PPHFS−1 =0BBB@

‘MPP HF S−1

∈αPP HF S−1

�1− �1−MPP HF S−1

�∂�,

‘APP HF S−1

∈βPP HF S−1

�A∂

PP HF S−1

�,

‘NPP HF S−1

∈γPP HF S−1

�N ∂

PP HF S−1

�

1CCCA;

(4) P ∂PHFS−1 =0BBB@

‘MPP HF S−1

∈αPP HF S−1

�M∂

PP HF S−1

�,

‘APP HF S−1

∈βPP HF S−1

�1− �1−APP HF S−1

�∂�,

‘NPP HF S−1

∈γPP HF S−1

�1− �1−NPP HF S−1

�∂�

1CCCA.

3. THE CROSS-ENTROPY MEASURES FORPICTURE HESITANT FUZZY NUMBERS

In this section, we explore the concept of cross-entropy measures based on picture hes-itant fuzzy numbers and their properties. The proposed measures are described below.

Definition 3.1. LetPPHFS−1 =(αPP HF S−1 (x) , βPP HF S−1 (x) , γPP HF S−1 (x)

)and

PPHFS−2 =(αPP HF S−2 (x) , βPP HF S−2 (x) , γPP HF S−2 (x)

)be two picture hesitant fuzzy

numbers. Then the cross-entropyCE1 (PPHFS−1, PPHFS−2) is denoted and defined as:

CE1 : PHFN × PHFN → R+

CE1 (PPHFS−1, PPHFS−2) =

maxMPP HF S−1∈αPP HF S−1

minMPP HF S−2∈αPP HF S−2( MPP HF S−1

log2

(2MPP HF S−1

MPP HF S−1+MPP HF S−2

))

+

maxAPP HF S−1∈βPP HF S−1

minAPP HF S−2∈βPP HF S−2( APP HF S−1

log2

(2APP HF S−1

APP HF S−1+APP HF S−2

))

+

maxNPP HF S−1∈γPP HF S−1

minNPP HF S−2∈γPP HF S−2( NPP HF S−1

log2

(2NPP HF S−1

NPP HF S−1+NPP HF S−2

))


Which holds the following conditions:

(1) CE1 (PPHFS−1, PPHFS−2) ≥ 0;(2) CE1 (PPHFS−1, PPHFS−2) = 0 iff PPHFS−1 = PPHFS−2;(3) CE1

(PC

PHFS−1, PCPHFS−2

)= CE1 (PPHFS−1, PPHFS−2),

wherePCPHFS−1 =

(γPP HF S−1 (x) , βPP HF S−1 (x) , αPP HF S−1 (x)

).


)and



numbers,p ≥ 1. Then the cross-entropyCE2 (PPHFS−1, PPHFS−2) is denoted and de-fined as:



1

|αPP HF S−1 |∑MPP HF S−1∈αPP HF S−1


log2

(2MPP HF S−1


))

p

1p

+

1

|βPP HF S−1 |∑APP HF S−1∈βPP HF S−1


log2

(2APP HF S−1


))

p

1p

+

1

|γPP HF S−1 |∑NPP HF S−1∈γPP HF S−1


log2

(2NPP HF S−1


))

p

1p



(PC



wherePCPHFS−1 =


).

Theorem 3.3. The proposed measuresCE1 (PPHFS−1, PPHFS−2) andCE2 (PPHFS−1, PPHFS−2) are a picture hesitant fuzzy cross-entropy, and satisfy theabove three conditions.Proof: The conditions of def. 3.1 and def.3.2 can be examined and the proof of the otherdefinitions is similar.

(1) It is clear thatCE1 (PPHFS−1, PPHFS−2) ≥ 0 andCE2 (PPHFS−1, PPHFS−2) ≥0.

(2) If PPHFS−1 = PPHFS−2, thenαPP HF S−1 (x) = αPP HF S−2 (x) , βPP HF S−1 (x) =βPP HF S−2 (x) andγPP HF S−1 (x) = γPP HF S−2 (x). Then





log2

(2MPP HF S−1


))

+



log2

(2APP HF S−1


))

+



log2

(2NPP HF S−1


))

= maxMPP HF S−1∈αPP HF S−1(0) +

maxAPP HF S−1∈βPP HF S−1(0) +

maxNPP HF S−1∈γPP HF S−1(0) = 0

and

CE2�PP HF S−1, PP HF S−2

�=

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

0BBBBBBBB@

1��αPP HF S−1

��

PMPP HF S−1

∈αPP HF S−1

0BBBB@

minMPP HF S−2∈αPP HF S−20

B@MPP HF S−1

log2

2MPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@

1��βPP HF S−1

��

PAPP HF S−1

∈βPP HF S−1

0BBBB@

minAPP HF S−2∈βPP HF S−20

B@APP HF S−1

log2

2APP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@

1��γPP HF S−1

��

PNPP HF S−1

∈γPP HF S−1

0BBBB@

minNPP HF S−2∈γPP HF S−20

B@NPP HF S−1

log2

2NPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

= (0)1p + (0)

1p + (0)

1p = 0


(1) LetCE2�PP HF S−1, PP HF S−2

�=


0BBBBBBBB@


��

PMPP HF S−1

∈αPP HF S−1

0BBBB@


B@MPP HF S−1

log2

2MPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@


��

PAPP HF S−1

∈βPP HF S−1

0BBBB@


B@APP HF S−1

log2

2APP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@


��

PNPP HF S−1

∈γPP HF S−1

0BBBB@


B@NPP HF S−1

log2

2NPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p


=


0BBBBBBBB@


��

PMPP HF S−1

∈αPP HF S−1

0BBBB@


B@MPP HF S−1

log2

2MPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@


��

PAPP HF S−1

∈βPP HF S−1

0BBBB@


B@APP HF S−1

log2

2APP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@


��

PNPP HF S−1

∈γPP HF S−1

0BBBB@


B@NPP HF S−1

log2

2NPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p


= CE2

�P

cP HF S−1, P

cP HF S−2

�

Hence the proof of the results is completed.


)and



numbers. Then the cross-entropyCE3 (PPHFS−1, PPHFS−2) is denoted and defined as:






log2

(2MPP HF S−1


))

+



log2

(2APP HF S−1


))

+



log2

(2NPP HF S−1


))

+

maxπPP HF S−1∈δPP HF S−1

minπPP HF S−2∈δPP HF S−2(πPP HF S−1

log2

(2πPP HF S−1

πPP HF S−1+πPP HF S−2

))



(PC



wherePCPHFS−1 =


).


)and



numbers,p ≥ 1. Then the cross-entropyCE4 (PPHFS−1, PPHFS−2) is denoted and de-fined as:




0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

0BBBBBBBB@


��

PMPP HF S−1

∈αPP HF S−1

0BBBB@


B@MPP HF S−1

log2

2MPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@


��

PAPP HF S−1

∈βPP HF S−1

0BBBB@


B@APP HF S−1

log2

2APP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@


��

PNPP HF S−1

∈γPP HF S−1

0BBBB@


B@NPP HF S−1

log2

2NPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@

1��δPP HF S−1

��

PπPP HF S−1

∈δPP HF S−1

0BBBB@

minπPP HF S−2∈δPP HF S−20

B@πPP HF S−1

log2

2πPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA


(1) CE4�PP HF S−1, PP HF S−2

� ≥ 0;


�= 0 iff PP HF S−1 = PP HF S−2 ;

(3) CE4

�P C

P HF S−1, P CP HF S−2

�= CE4

�PP HF S−1, PP HF S−2

�,

whereP CP HF S−1 =

�γPP HF S−1

(x) , βPP HF S−1(x) , αPP HF S−1

(x)�

.

Theorem 3.6. The proposed measuresCE3 (PPHFS−1, PPHFS−2) andCE4 (PPHFS−1, PPHFS−2) are a picture hesitant fuzzy cross-entropy, and satisfy theabove three conditions.Proof: The conditions ofdef.3.1 anddef.3.2 can be examined and the proof of the otherdefinitions is similar.

(1) It is clear thatCE3 (PPHFS−1, PPHFS−2) ≥ 0 andCE4 (PPHFS−1, PPHFS−2) ≥0.

(2) If PPHFS−1 = PPHFS−2, thenαPP HF S−1 (x) = αPP HF S−2 (x) , βPP HF S−1 (x) =βPP HF S−2 (x) andγPP HF S−1 (x) = γPP HF S−2 (x). Then





log2

(2MPP HF S−1


))

+



log2

(2APP HF S−1


))

+



log2

(2NPP HF S−1


))

+

maxπPP HF S−1∈δPP HF S−1

minπPP HF S−2∈δPP HF S−2(πPP HF S−1

log2

(2πPP HF S−1


))

= maxMPP HF S−1∈αPP HF S−1(0) +

maxAPP HF S−1∈βPP HF S−1(0) +

maxNPP HF S−1∈γPP HF S−1(0)+

maxπPP HF S−1∈δPP HF S−1(0) = 0

andCE4 (PPHFS−1, PPHFS−2) =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

0BBBBBBBB@


��

PMPP HF S−1

∈αPP HF S−1

0BBBB@


B@MPP HF S−1

log2

2MPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@


��

PAPP HF S−1

∈βPP HF S−1

0BBBB@


B@APP HF S−1

log2

2APP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@


��

PNPP HF S−1

∈γPP HF S−1

0BBBB@


B@NPP HF S−1

log2

2NPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBB@


��

PπPP HF S−1

∈δPP HF S−1

0BB@


πPP HF S−1log2

2πPP HF S−1


!!1CCA

p

1CCCCCCA

1p

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA


= (0)1p + (0)

1p + (0)

1p + (0)

1p = 0

(1) LetCE4�PP HF S−1, PP HF S−2

�=


0BBBBBBBB@


��

PMPP HF S−1

∈αPP HF S−1

0BBBB@


B@MPP HF S−1

log2

2MPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@


��

PAPP HF S−1

∈βPP HF S−1

0BBBB@


B@APP HF S−1

log2

2APP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@


��

PNPP HF S−1

∈γPP HF S−1

0BBBB@


B@NPP HF S−1

log2

2NPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@


��

PπPP HF S−1

∈δPP HF S−1

0BBBB@


B@πPP HF S−1

log2

2πPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p


=


0BBBBBBBB@


��

PMPP HF S−1

∈αPP HF S−1

0BBBB@


B@MPP HF S−1

log2

2MPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@


��

PAPP HF S−1

∈βPP HF S−1

0BBBB@


B@APP HF S−1

log2

2APP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@


��

PNPP HF S−1

∈γPP HF S−1

0BBBB@


B@NPP HF S−1

log2

2NPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p

+

0BBBBBBBB@


��

PπPP HF S−1

∈δPP HF S−1

0BBBB@


B@πPP HF S−1

log2

2πPP HF S−1


!1CA

1CCCCA

p

1CCCCCCCCA

1p


= CE4

�P

cP HF S−1, P

cP HF S−2

�

Hence the proof of the results is completed.


Definition 3.7. LetPPHFS−1 =�αPP HF S−1 (x) , βPP HF S−1 (x) , γPP HF S−1 (x)

�and

PPHFS−2 =�αPP HF S−2 (x) , βPP HF S−2 (x) , γPP HF S−2 (x)

�be two picture hesitant fuzzy num-

bers. Then the cross-entropyCE5 (PPHFS−1, PPHFS−2) is denoted and defined as:


CE5 (PPHFS−1, PPHFS−2) =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

max0BB@

MPP HF S−1∈ αPP HF S−1

,

APP HF S−1∈ βPP HF S−1

,

NPP HF S−1∈ γPP HF S−1

1CCA

0BBBBBBBBBBBBBBBBBBBBBBBBBBB@

min0BB@


,


,


1CCA

0BBBBBBBBBBBBBBBBBBB@

0@ MPP HF S−1

+ 1−APP HF S−1

−NPP HF S−1

1A

2

log2

0BBBBBBBBBBBBB@

2

0@ MPP HF S−1

+ 1−APP HF S−1

−NPP HF S−1

1A

0BBBBB@

MPP HF S−1+ 1−

APP HF S−1−NPP HF S−1

!+

MPP HF S−2+ 1−


!

1CCCCCA

1CCCCCCCCCCCCCA

1CCCCCCCCCCCCCCCCCCCA

1CCCCCCCCCCCCCCCCCCCCCCCCCCCA

+

max0BB@


,


,


1CCA


min0BB@


,


,


1CCA


0@ 1−MPP HF S−1

+

APP HF S−1+NPP HF S−1

1A

2

log2

0BBBBBBBBBBBBB@

2

0@ 1−MPP HF S−1

+


1A

0BBBBB@

1−MPP HF S−1

+


!+

1−MPP HF S−2

+


!

1CCCCCA

1CCCCCCCCCCCCCA



+

max0BB@


,


,


1CCA


min0BB@


,


,


1CCA


0@ 1−MPP HF S−1

+


1A

2

log2

0BBBBBBBBBBBBB@

2

0@ 1−MPP HF S−1

+


1A

0BBBBB@

1−MPP HF S−1

+


!+

1−MPP HF S−2

+


!

1CCCCCA

1CCCCCCCCCCCCCA



1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA




�P C


�= CE5 (PPHFS−1, PPHFS−2),

whereP CPHFS−1 =

�γPP HF S−1 (x) , βPP HF S−1 (x) , αPP HF S−1 (x)

�.




be two picture hesitant fuzzy numbers,p ≥ 1.Then the cross-entropyCE6 (PPHFS−1, PPHFS−2) is denoted and defined as:


CE6 (PPHFS−1, PPHFS−2) =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@


��.��βPP HF S−1

��.��γPP HF S−1

��

P0BB@


,


,


1CCA


min0BB@


,


,


1CCA


0@ MPP HF S−1

+ 1−APP HF S−1

−NPP HF S−1

1A

2

log2

0BBBBBBBBBBBBB@

2

0@ MPP HF S−1

+ 1−APP HF S−1

−NPP HF S−1

1A

0BBBBB@

MPP HF S−1+ 1−


!+

MPP HF S−2+ 1−


!

1CCCCCA

1CCCCCCCCCCCCCA



p

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

1p

+





��

P0BB@


,


,


1CCA


min0BB@


,


,


1CCA


0@ 1−MPP HF S−1

+


1A

2

log2

0BBBBBBBBBBBBB@

2

0@ 1−MPP HF S−1

+


1A

0BBBBB@

1−MPP HF S−1

+


!+

1−MPP HF S−2

+


!

1CCCCCA

1CCCCCCCCCCCCCA



p


1p

+





��

P0BB@


,


,


1CCA


min0BB@


,


,


1CCA


0@ 1−MPP HF S−1

+


1A

2

log2

0BBBBBBBBBBBBB@

2

0@ 1−MPP HF S−1

+


1A

0BBBBB@

1−MPP HF S−1

+


!+

1−MPP HF S−2

+


!

1CCCCCA

1CCCCCCCCCCCCCA



p


1p

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA




� ≥ 0;



(3) CE6

�P C


�= CE6


�,


�γPP HF S−1


(x)�

.

Theorem 3.9. The proposed measuresCE5 (PPHFS−1, PPHFS−2) andCE6 (PPHFS−1, PPHFS−2) are a picture hesitant fuzzy cross-entropy, and satisfy the above threeconditions.Proof: Straightforward. (The proof of this theorem is similar to the proof of thetheorem3.1)




be two picture hesitant fuzzy numbers. Then thecross-entropyCE7 (PPHFS−1, PPHFS−2) is denoted and defined as:



1

1− 21−q

0BBBBBBBBBBBBBBBBBBBBBBBBBB@


BB@


@Mq

PP HF S−1+Mq

PP HF S−22

−�MPP HF S−1+MPP HF S−22

�q

1A

1CCA+


BB@


@Aq

PP HF S−1+Aq

PP HF S−22

−�APP HF S−1+APP HF S−22

�q

1A

1CCA+


BB@


@Nq

PP HF S−1+Nq

PP HF S−22

−�NPP HF S−1+NPP HF S−22

�q

1A

1CCA

1CCCCCCCCCCCCCCCCCCCCCCCCCCA



�P C



whereP CPHFS−1 =


�.




be two picture hesitant fuzzy numbers,p ≥ 1.Then the cross-entropyCE8 (PPHFS−1, PPHFS−2) is denoted and defined as:




0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

0BBBBB@


��

PMPP HF S−1

∈αPP HF S−1

11−21−q

0BB@


@Mq

PP HF S−1+Mq

PP HF S−22

−�MPP HF S−1+MPP HF S−22

�q

1A

1CCA

p

1CCCCCA

1p

+

0BBBBB@


��

PAPP HF S−1

∈βPP HF S−1

11−21−q

0BB@


@Aq

PP HF S−1+Aq

PP HF S−22

−�APP HF S−1+APP HF S−22

�q

1A

1CCA

p

1CCCCCA

1p

+

0BBBBB@


��

PNPP HF S−1

∈γPP HF S−1

11−21−q

0BB@


@Nq

PP HF S−1+Nq

PP HF S−22

−�NPP HF S−1+NPP HF S−22

�q

1A

1CCA

p

1CCCCCA

1p

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

Which holds the following conditions, for1 < q ≤ 2, p ≥ 1:


�P C



whereP CPHFS−1 =


�.

Theorem 3.12. The proposed measuresCE7 (PPHFS−1, PPHFS−2) andCE8 (PPHFS−1, PPHFS−2) are a picture hesitant fuzzy cross-entropy, and satisfy the above threeconditions.Proof: Straightforward. (The proof of this theorem is similar to the proof of thetheorem3.1)




be two picture hesitant fuzzy numbers. Then thecross-entropyCE9 (PPHFS−1, PPHFS−2) is denoted and defined as:




1

T

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@


BBBBBBBBBBBBBBBBBB@


BBBBBBBBBBBBBBBB@

0BBBBBBBBBB@

0BBBBBB@

�1 + qMPP HF S−1

�ln�1 + qMPP HF S−1

�+�

1 + qMPP HF S−2

�ln�1 + qMPP HF S−2

�

1CCCCCCA

2

1CCCCCCCCCCA

−

0@

2+qMPP HF S−1+qMPP HF S−2

2

ln�


2

�1A

1CCCCCCCCCCCCCCCCA

1CCCCCCCCCCCCCCCCCCA

+


BBBBBBBBBBBBBBBBBB@


BBBBBBBBBBBBBBBB@

0BBBBBBBBBB@

0BBBBBB@

�1 + qAPP HF S−1

�ln�1 + qAPP HF S−1

�+�

1 + qAPP HF S−2

�ln�1 + qAPP HF S−2

�

1CCCCCCA

2

1CCCCCCCCCCA

−

0@

2+qAPP HF S−1+qAPP HF S−2

2

ln�


2

�1A

1CCCCCCCCCCCCCCCCA


+


BBBBBBBBBBBBBBBBBB@


BBBBBBBBBBBBBBBB@

0BBBBBBBBBB@

0BBBBBB@

�1 + qNPP HF S−1

�ln�1 + qNPP HF S−1

�+�

1 + qNPP HF S−2

�ln�1 + qNPP HF S−2

�

1CCCCCCA

2

1CCCCCCCCCCA

−

0@

2+qNPP HF S−1+qNPP HF S−2

2

ln�


2

�1A

1CCCCCCCCCCCCCCCCA


1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA



�P C



whereP CPHFS−1 =


�.




be two picture hesitant fuzzy numbers,p ≥ 1.


Then the cross-entropyCE10 (PPHFS−1, PPHFS−2) is denoted and defined as:


CE10 (PPHFS−1, PPHFS−2) =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@


��

PMPP HF S−1

∈αPP HF S−1

1T

0BBBBBBBBBBBBBBBBBBBBBBBBBBBB@


BBBBBBBBBBBBBBBBBBBBBBBBB@

0BBBBBBBBBBBBBBBBB@

0BBBBBBB@

�1 + qMPP HF S−1

�

ln�1 + qMPP HF S−1

�+�

1 + qMPP HF S−2

�

ln�1 + qMPP HF S−2

�

1CCCCCCCA

2

1CCCCCCCCCCCCCCCCCA

−

0BB@


2

ln

2+qMPP HF S−1

+qMPP HF S−22

!1CCA

1CCCCCCCCCCCCCCCCCCCCCCCCCA

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCA

p

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

1p

+



��

PAPP HF S−1

∈βPP HF S−1

1T




0BBBBBBBBBBBBBBBBB@

0BBBBBBB@

�1 + qAPP HF S−1

�

ln�1 + qAPP HF S−1

�+�

1 + qAPP HF S−2

�

ln�1 + qAPP HF S−2

�

1CCCCCCCA

2

1CCCCCCCCCCCCCCCCCA

−

0BB@


2

ln

2+qAPP HF S−1

+qAPP HF S−22

!1CCA



p


1p

+



��

PNPP HF S−1

∈γPP HF S−1

1T




0BBBBBBBBBBBBBBBBB@

0BBBBBBB@

�1 + qNPP HF S−1

�

ln�1 + qNPP HF S−1

�+�

1 + qNPP HF S−2

�

ln�1 + qNPP HF S−2

�

1CCCCCCCA

2

1CCCCCCCCCCCCCCCCCA

−

0BB@


2

ln

2+qNPP HF S−1

+qNPP HF S−22

!1CCA



p


1p

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

Which holds the following conditions, for1 < q ≤ 2, p ≥ 1:


� ≥ 0;



(3) CE10

�P C


�= CE10


�,


�γPP HF S−1


(x)�

.


We choose the values ofT = (1 + q) ln (1 + q) − (2 + q) (ln (2 + q) − ln2 ). The symmetric discrimination information measure forpicture hesitant fuzzy numbers is follow as:

CE∗i


�

= CEi�PP HF S−1, PP HF S−2

�+ CEi


�

Theorem 3.15. The proposed measuresCE9(PPHFS−1, PPHFS−2) andCE10 (PPHFS−1, PPHFS−2) are a picture hesitant fuzzy cross-entropy, and satisfy the above threeconditions.Proof: Straightforward. (The proof of this theorem is similar to the proof of thetheorem3.1)

4. MULTI CRITERIA DECISION MAKING METHODSBASED ON CROSS-ENTROPY MEASURES

OF PICTURE HESITANT FUZZY NUMBERS

Multi criteria decision making ranking problems with picture hesitant fuzzy information dependnalternatives, represented byPPHFS−i (i = 1, 2, 3, . . . , n) with respect tom criteria expressing byCj (j = 1, 2, 3 . . . , m). The weight vectors forwi ∈ [0, 1] ,

Pni=1 wi = 1 with a conditionw−i ≤

wi ≤ w+i such that

Pni=1 w+

i ≥ 1,Pn

i=1 w−i ≤ 1. The picture hesitant fuzzy numbers is definedbyPPHFS−ij =

�αPP HF S−ij (x) , βPP HF S−ij (x) , γPP HF S−ij (x)

�. The steps of the algorithm is

follow as:Step 1: We construct the decision matrix whose every entities in the form of picture hesitant fuzzynumbers.Step 2: We calculate the weight vector, for this first we choose the positive and negative ideal solutionsuch thatPPositive ideal = (1, 0, 0) andPNegative ideal = (0, 0, 1). Then we use the followingformula such that

CE∗1 (PPHFS−ij , PPHFS−ij)

= CE1 (PPHFS−ij , PPositive ideal) + CE1 (PPositive ideal, PPHFS−ij)

=

0BBBBBBBBBBBBBBBBBBBBBBBBB@

maxMPP HF S−ij∈αPP HF S−ij0

BB@

minMPP ositive ideal∈αPP ositive ideal0

@MPP HF S−ij

log2

�2MPP HF S−ij

MPP HF S−ij+MPP ositive ideal

�1A

1CCA+

maxAPP HF S−ij∈βPP HF S−ij0

BB@

minAPP ositive ideal∈βPP ositive ideal0

@APP HF S−ij

log2

�2APP HF S−ij

APP HF S−ij+APP ositive ideal

�1A

1CCA+

maxNPP HF S−ij∈γPP HF S−ij0

BB@

minNPP ositive ideal∈γPP ositive ideal0

@NPP HF S−ij

log2

�2NPP HF S−ij

NPP HF S−ij+NPP ositive ideal

�1A

1CCA


+


0BBBBBBBBBBBBBBBBBBBBBBBBB@

maxMPP ositive ideal∈αPP ositive ideal0

BB@

minMPP HF S−ij∈αPP HF S−ij0

@MPP ositive ideal

log2

�2MPP ositive ideal

MPP ositive ideal+MPP HF S−ij

�1A

1CCA+

maxAPP ositive ideal∈βPP ositive ideal0

BB@

minAPP HF S−ij∈βPP HF S−ij0

@APP ositive ideal

log2

�2APP ositive ideal

APP ositive ideal+APP HF S−ij

�1A

1CCA+

maxNPP ositive ideal∈γPP ositive ideal0

BB@

minNPP HF S−ij∈γPP HF S−ij0

@NPP ositive ideal

log2

�2NPP ositive ideal

NPP ositive ideal+NPP HF S−ij

�1A

1CCA


and

CE∗1 (PPHFS−ij , PPHFS−ij)

= CE1 (PPHFS−ij , PNegative ideal) + CE1 (PNegative ideal, PPHFS−ij)

=


maxMPP HF S−ij∈αPP HF S−ij0

BB@

minMPNegative ideal∈αPNegative ideal0

@MPP HF S−ij

log2

�2MPP HF S−ij

MPP HF S−ij+MPNegative ideal

�1A

1CCA+

maxAPP HF S−ij∈βPP HF S−ij0

BB@

minAPNegative ideal∈βPNegative ideal0

@APP HF S−ij

log2

�2APP HF S−ij

APP HF S−ij+APNegative ideal

�1A

1CCA+

maxNPP HF S−ij∈γPP HF S−ij0

BB@

minNPNegative ideal∈γPNegative ideal0

@NPP HF S−ij

log2

�2NPP HF S−ij

NPP HF S−ij+NPNegative ideal

�1A

1CCA


+



maxMPNegative ideal∈αPNegative ideal0

BB@

minMPP HF S−ij∈αPP HF S−ij0

@MPNegative ideal

log2

�2MPNegative ideal

MPNegative ideal+MPP HF S−ij

�1A

1CCA+

maxAPNegative ideal∈βPNegative ideal0

BB@

minAPP HF S−ij∈βPP HF S−ij0

@APNegative ideal

log2

�2APNegative ideal

APNegative ideal+APP HF S−ij

�1A

1CCA+

maxNPNegative ideal∈γPNegative ideal0

BB@

minNPP HF S−ij∈γPP HF S−ij0

@NPNegative ideal

log2

�2NPNegative ideal

NPNegative ideal+NPP HF S−ij

�1A

1CCA


Step 3: We examine the closeness of the negative ideal and alternatives by using the following equa-tions, such that

G (PPHFN−j) =

nXi=1

wiGij , Gij =G−ij

G+ij + G−ij

The linear programming model is of the form:

maxG =

mXj=1

G (PPHFN−j) =

mXj=1

nXi=1

wiGij

Step 4: Rank the alternative and examine the best one.Step 5: The end.

(1) (a) An illustrate example

A company has decided to choose a pool of alternatives from several foreign countries based onpreliminary surveys. In this survey, we find a suitable candidate countries, which is representedby a1, a2, a3 anda4. During the assessment, four factors including,c1: Politics and Policy,c2:Infrastructure,c3: Resource andc4: Economy. The weight vector for above alternatives and their

attributes i.e.H =

8>><>>:

0.15 ≤ w1 ≤ 0.3,0.15 ≤ w2 ≤ 0.25,0.25 ≤ w3 ≤ 0.4,0.3 ≤ w4 ≤ 0.45,

9>>=>>;

such thatPn

i=1 wi = 1.

The steps of the algorithm is follow as:Step 1: First we construct the decision matrix whose every entities in the form of picture hesitantfuzzy numbers, which is utilized in the form of table 2


Table 2. Original decision matrix, whose every entries in the form of picture hesitant fuzzy number.Symbols C1 C2 C3 C4

a1

0BB@

{0.3, 0.4} ,{0.1} ,�0.1, 0.2,

0.22

�

1CCA

0@

{0.01, 0.4} ,{0.13} ,

{0.13, 0.21}

1A

0@

{0.03, 0.04} ,{0.23} ,{0.1, 0.22}

1A

0@

{0.3, 0.4} ,{0.1, 0.23} ,{0.23, 0.22}

1A

a2

0BB@

{0.13, 0.14} ,{0.11} ,�

0.11, 0.12,0.122

�

1CCA

0@

{0.23, 0.24} ,{0.11} ,{0.1, 0.12}

1A

0BB@

{0.03, 0.04} ,{0.01} ,�0.1, 0.02,

0.12

�

1CCA

0@

{0.23, 0.21} ,{0.14} ,

{0.21, 0.22}

1A

a3

0@

{0.03, 0.04} ,{0.23} ,{0.1, 0.22}

1A

0@

{0.3, 0.4} ,{0.1, 0.23} ,{0.23, 0.22}

1A

0BB@

�0.11, 0.12

, 0.14

�,

{0.1, 23} ,{0.1, 0.32}

1CCA

0BB@

{0.3, 0.4} ,{0.1} ,�0.1, 0.2,

0.22

�

1CCA

a4

0BB@

{0.03, 0.04} ,{0.01} ,�0.1, 0.02,

0.12

�

1CCA

0@

{0.23, 0.21} ,{0.14} ,

{0.21, 0.22}

1A

0BB@

{0.03, 0.04} ,{0.01} ,�0.1, 0.02,

0.12

�

1CCA

0@

{0.23, 0.21} ,{0.14} ,

{0.21, 0.22}

1A

Step 2: We calculate the weight vector, for this first we choose the positive and negative ideal solutionsuch thatPPositive ideal = (1, 0, 0) andPNegative ideal = (0, 0, 1). Then we use the followingformula such thatCE

∗1�PP HF S−11, PP ositive ideal

�= CE1

�PP HF S−11, PP ositive ideal

�+CE1

�PP ositive ideal, PP HF S−11

�

= max

�min

�0.3log2

�2× 0.3

0.3 + 1

� �, min

�0.4log2

�2× 0.4

0.4 + 1

� � �+

max

�min

�0.1log2

�2× 0.1

0.1 + 0

� � �+

max

8<:

min�0.1log2

�2×0.10.1+0

� �, min

�0.2log2

�2×0.20.2+0

� �,

min�0.22log2

�2×0.220.22+0

� �9=;

+max

�min

�log2

�2× 1

0.3 + 1

� �, min

�log2

�2× 1

0.4 + 1

� � �+ 0 + 0

= max (−0.67,−0.097)− 0.06+

max (−0.06,−0.14, 0.066) + max (0.19, 0.15)

= −0.097− 0.06 + 0.066 + 0.19 = 0.099

And similarly forCE∗1 (PPHFS−11, PNegative ideal) = 1.052

Step 3: We examine the closeness of the negative ideal and alternatives by using the following equa-tions, such that

G11 =CE∗

1 (PPHFS−11, PPositive ideal)�CE∗

1 (PPHFS−11, PPositive ideal)+CE∗

1 (PPHFS−11, PNegative ideal)

�

=0.099

0.099 + 0.952= 0.09420

G12 = 0.023, G13 = 0.0423, G14 = 0.03, G21 = 0.032,

G22 = 0.0243, G23 = 0.0445, G24 = 0.0562, G31 = 0.31,

G32 = 0.024, G33 = 0.0671, G34 = 0.0779,

G41 = 0.012, G42 = 0.23, G43 = 0.03, G44 = 0.651

The linear programming model is of the form:

maxG = 2.9487w1 + 2.6077w2

+2.7233w3 + 3.4084w4

w =

�0.15, 0.15,

0.375, 0.325

�


Step 4: Rank the alternative and examine the best one.

G1 (a1) = 0.04319, G2 (a2) = 0.0434,

G3 (a3) = 0.1, G4 (a4) = 0.259

G4 (a4) ≥ G3 (a3) ≥ G2 (a2) ≥ G1 (a1)

HenceG4 (a4) is the best strategy. Forp = q = 2.Step 5: The end.The comparative analysis of the explored measures with some other existing measures is follow as:the generalized distance and similarity measures based on picture hesitant fuzzy set was explored byJan et al. [37] and some similarity measures based on picture hesitant fuzzy set were presented byAhmad et al. [38]. The comparison between explored measures with existing measures [37, 38] isdiscussed in table 3.Table 3. Comparison between explored work with some existing works.

Methods Similarity Values Ranking ValuesJan et al. [37] G1 (a1) = 0.04353, G2 (a2) =

0.0439,G3 (a3) = 0.98, G4 (a4) =0.246

G4 (a4) ≥ G3 (a3)≥ G2 (a2) ≥ G1 (a1)

Ahmad et al.[38]

G1 (a1) = 0.04820, G2 (a2) =0.0495,G3 (a3) = 0.13, G4 (a4) =0.273

G4 (a4) ≥ G3 (a3)≥ G2 (a2) ≥ G1 (a1)

Proposed Mea-sures

G1 (a1) = 0.04319, G2 (a2) =0.0434,G3 (a3) = 0.1, G4 (a4) =0.259

G4 (a4) ≥ G3 (a3)≥ G2 (a2) ≥ G1 (a1)

From the above analysis, it is clear that the all existing works [37, 38] and the explored measure areprovides the same results, which is discussed in table 3. The best alternative isG4 (a4). When weconsider the positive, abstinence and negative grade in the form of singleton set, then the establishedwork is reduced into picture fuzzy set. The explored work is more general than existing drawbacksbecause of its structure.

5. CONCLUSION

In this article, we exploit the cross-entropy of picture hesitant fuzzy set is established by distin-guishing the cross-entropy of picture fuzzy set and hesitant fuzzy set. First, many measurement con-cepts are established and their basic properties are studied. Further, two measures, which are basedon the established picture hesitant fuzzy cross-entropy, are established for evaluating multi-criteriadecision making problems in the environment of picture hesitant fuzzy sets. For both approaches,an optimization model is pioneered in order to examine the weight vector for multi criteria decisionmaking problems with incomplete information on criteria weights. In last, we give an example toexpress the practically and effectiveness. The comparison of the proposed measures with existingmeasures are also discussed in detail. We aim to broaden our study in the future with the analysisof (1) Complex q-rung orthopair fuzzy sets, (2) Complex bipolar Neutrosophic sets, (3) Fuzzy roughsoft sets and (4) Fuzzy rough Neutrosophic sets. The existing methods are discussed in [39-50] arealso utilized in the environment of proposed approaches.


6. ACKNOWLEDGMENTS

I would like to thank my supervisor, Prof. Tahir Mahmood, for the patient guidance, encourage-ment and advice he has provided throughout my time as his student. I have been extremely luckyto have a supervisor who cared so much about my work, and who responded to my questions andqueries so promptly.

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