Citation: Harish Garg, Nancy. Algorithms for single-valued neutrosophic decision making based on TOPSIS and clustering methods with new distance measure[J]. AIMS Mathematics, 2020, 5(3): 2671-2693. doi: 10.3934/math.2020173
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Let A denote the class of functions f which are analytic in the open unit disk Δ={z∈C:|z|<1}, normalized by the conditions f(0)=f′(0)−1=0. So each f∈A has series representation of the form
f(z)=z+∞∑n=2anzn. | (1.1) |
For two analytic functions f and g, f is said to be subordinated to g (written as f≺g) if there exists an analytic function ω with ω(0)=0 and |ω(z)|<1 for z∈Δ such that f(z)=(g∘ω)(z).
A function f∈A is said to be in the class S if f is univalent in Δ. A function f∈S is in class C of normalized convex functions if f(Δ) is a convex domain. For 0≤α≤1, Mocanu [23] introduced the class Mα of functions f∈A such that f(z)f′(z)z≠0 for all z∈Δ and
ℜ((1−α)zf′(z)f(z)+α(zf′(z))′f′(z))>0(z∈Δ). | (1.2) |
Geometrically, f∈Mα maps the circle centred at origin onto α-convex arcs which leads to the condition (1.2). The class Mα was studied extensively by several researchers, see [1,10,11,12,24,25,26,27] and the references cited therein.
A function f∈S is uniformly starlike if f maps every circular arc Γ contained in Δ with center at ζ ∈Δ onto a starlike arc with respect to f(ζ). A function f∈C is uniformly convex if f maps every circular arc Γ contained in Δ with center ζ ∈Δ onto a convex arc. We denote the classes of uniformly starlike and uniformly convex functions by UST and UCV, respectively. For recent study on these function classes, one can refer to [7,9,13,19,20,31].
In 1999, Kanas and Wisniowska [15] introduced the class k-UCV (k≥0) of k-uniformly convex functions. A function f∈A is said to be in the class k-UCV if it satisfies the condition
ℜ(1+zf″(z)f′(z))>k|zf′(z)f′(z)|(z∈Δ). | (1.3) |
In recent years, many researchers investigated interesting properties of this class and its generalizations. For more details, see [2,3,4,14,15,16,17,18,30,32,35] and references cited therein.
In 2015, Sokół and Nunokawa [33] introduced the class MN, a function f∈MN if it satisfies the condition
ℜ(1+zf″(z)f′(z))>|zf′(z)f(z)−1|(z∈Δ). |
In [28], it is proved that if ℜ(f′)>0 in Δ, then f is univalent in Δ. In 1972, MacGregor [21] studied the class B of functions with bounded turning, a function f∈B if it satisfies the condition ℜ(f′)>0 for z∈Δ. A natural generalization of the class B is B(δ1) (0≤δ1<1), a function f∈B(δ1) if it satisfies the condition
ℜ(f′(z))>δ1(z∈Δ;0≤δ1<1), | (1.4) |
for details associated with the class B(δ1) (see [5,6,34]).
Motivated essentially by the above work, we now introduce the following class k-Q(α) of analytic functions.
Definition 1. Let k≥0 and 0≤α≤1. A function f∈A is said to be in the class k-Q(α) if it satisfies the condition
ℜ((zf′(z))′f′(z))>k|(1−α)f′(z)+α(zf′(z))′f′(z)−1|(z∈Δ). | (1.5) |
It is worth mentioning that, for special values of parameters, one can obtain a number of well-known function classes, some of them are listed below:
1. k-Q(1)=k-UCV;
2. 0-Q(α)=C.
In what follows, we give an example for the class k-Q(α).
Example 1. The function f(z)=z1−Az(A≠0) is in the class k-Q(α) with
k≤1−b2b√b(1+α)[b(1+α)+2]+4(b=|A|). | (1.6) |
The main purpose of this paper is to establish several interesting relationships between k-Q(α) and the class B(δ) of functions with bounded turning.
To prove our main results, we need the following lemmas.
Lemma 1. ([8]) Let h be analytic in Δ with h(0)=1, β>0 and 0≤γ1<1. If
h(z)+βzh′(z)h(z)≺1+(1−2γ1)z1−z, |
then
h(z)≺1+(1−2δ)z1−z, |
where
δ=(2γ1−β)+√(2γ1−β)2+8β4. | (2.1) |
Lemma 2. Let h be analytic in Δ and of the form
h(z)=1+∞∑n=mbnzn(bm≠0) |
with h(z)≠0 in Δ. If there exists a point z0(|z0|<1) such that |argh(z)|<πρ2(|z|<|z0|) and |argh(z0)|=πρ2 for some ρ>0, then z0h′(z0)h(z0)=iℓρ, where
ℓ:{ℓ≥n2(c+1c)(argh(z0)=πρ2),ℓ≤−n2(c+1c)(argh(z0)=−πρ2), |
and (h(z0))1/ρ=±ic(c>0).
This result is a generalization of the Nunokawa's lemma [29].
Lemma 3. ([37]) Let ε be a positive measure on [0,1]. Let ϝ be a complex-valued function defined on Δ×[0,1] such that ϝ(.,t) is analytic in Δ for each t∈[0,1] and ϝ(z,.) is ε-integrable on [0,1] for all z∈Δ. In addition, suppose that ℜ(ϝ(z,t))>0, ϝ(−r,t) is real and ℜ(1/ϝ(z,t))≥1/ϝ(−r,t) for |z|≤r<1 and t∈[0,1]. If ϝ(z)=∫10ϝ(z,t)dε(t), then ℜ(1/ϝ(z))≥1/ϝ(−r).
Lemma 4. ([22]) If −1≤D<C≤1, λ1>0 and ℜ(γ2)≥−λ1(1−C)/(1−D), then the differential equation
s(z)+zs′(z)λ1s(z)+γ2=1+Cz1+Dz(z∈Δ) |
has a univalent solution in Δ given by
s(z)={zλ1+γ2(1+Dz)λ1(C−D)/Dλ1∫z0tλ1+γ2−1(1+Dt)λ1(C−D)/Ddt−γ2λ1(D≠0),zλ1+γ2eλ1Czλ1∫z0tλ1+γ2−1eλ1Ctdt−γ2λ1(D=0). |
If r(z)=1+c1z+c2z2+⋯ satisfies the condition
r(z)+zr′(z)λ1r(z)+γ2≺1+Cz1+Dz(z∈Δ), |
then
r(z)≺s(z)≺1+Cz1+Dz, |
and s(z) is the best dominant.
Lemma 5. ([36,Chapter 14]) Let w, x and\ y≠0,−1,−2,… be complex numbers. Then, for ℜ(y)>ℜ(x)>0, one has
1. 2G1(w,x,y;z)=Γ(y)Γ(y−x)Γ(x)∫10sx−1(1−s)y−x−1(1−sz)−wds;
2. 2G1(w,x,y;z)= 2G1(x,w,y;z);
3. 2G1(w,x,y;z)=(1−z)−w2G1(w,y−x,y;zz−1).
Firstly, we derive the following result.
Theorem 1. Let 0≤α<1 and k≥11−α. If f∈k-Q(α), then f∈B(δ), where
δ=(2μ−λ)+√(2μ−λ)2+8λ4(λ=1+αkk(1−α);μ=k−αk−1k(1−α)). | (3.1) |
Proof. Let f′=ℏ, where ℏ is analytic in Δ with ℏ(0)=1. From inequality (1.5) which takes the form
ℜ(1+zℏ′(z)ℏ(z))>k|(1−α)ℏ(z)+α(1+zℏ′(z)ℏ(z))−1|=k|1−α−ℏ(z)+αℏ(z)−αzℏ′(z)ℏ(z)|, |
we find that
ℜ(ℏ(z)+1+αkk(1−α)zℏ(z)ℏ(z))>k−αk−1k(1−α), |
which can be rewritten as
ℜ(ℏ(z)+λzℏ(z)ℏ(z))>μ(λ=1+αkk(1−α);μ=k−αk−1k(1−α)). |
The above relationship can be written as the following Briot-Bouquet differential subordination
ℏ(z)+λzℏ′(z)ℏ(z)≺1+(1−2μ)z1−z. |
Thus, by Lemma 1, we obtain
ℏ≺1+(1−2δ)z1−z, | (3.2) |
where δ is given by (3.1). The relationship (3.2) implies that f∈B(δ). We thus complete the proof of Theorem 3.1.
Theorem 2. Let 0<α≤1, 0<β<1, c>0, k≥1, n≥m+1(m∈ N ), |ℓ|≥n2(c+1c) and
|αβℓ±(1−α)cβsinβπ2|≥1. | (3.3) |
If
f(z)=z+∞∑n=m+1anzn(am+1≠0) |
and f∈k-Q(α), then f∈B(β0), where
β0=min{β:β∈(0,1)} |
such that (3.3) holds.
Proof. By the assumption, we have
f′(z)=ℏ(z)=1+∞∑n=mcnzn(cm≠0). | (3.4) |
In view of (1.5) and (3.4), we get
ℜ(1+zℏ′(z)ℏ(z))>k|(1−α)ℏ(z)+α(1+zℏ′(z)ℏ(z))−1|. |
If there exists a point z0∈Δ such that
|argℏ(z)|<βπ2(|z|<|z0|;0<β<1) |
and
|argℏ(z0)|=βπ2(0<β<1), |
then from Lemma 2, we know that
z0ℏ′(z0)ℏ(z0)=iℓβ, |
where
(ℏ(z0))1/β=±ic(c>0) |
and
ℓ:{ℓ≥n2(c+1c)(argℏ(z0)=βπ2),ℓ≤−n2(c+1c)(argℏ(z0)=−βπ2). |
For the case
argℏ(z0)=βπ2, |
we get
ℜ(1+z0ℏ′(z0)ℏ(z0))=ℜ(1+iℓβ)=1. | (3.5) |
Moreover, we find from (3.3) that
k|(1−α)ℏ(z0)+α(1+z0ℏ′(z0)ℏ(z0))−1|=k|(1−α)(ℏ(z0)−1)+αz0ℏ′(z0)ℏ(z0)|=k|(1−α)[(±ic)β−1]+iαβℓ|=k√(1−α)2(cβcosβπ2−1)2+[αβℓ±(1−α)cβsinβπ2]2≥1. | (3.6) |
By virtue of (3.5) and (3.6), we have
ℜ(1+zℏ′(z0)ℏ(z0))≤k|(1−α)ℏ(z0)+α(1+z0ℏ(z0)ℏ(z0))−1|, |
which is a contradiction to the definition of k-Q(α). Since β0=min{β:β∈(0,1)} such that (3.3) holds, we can deduce that f∈B(β0).
By using the similar method as given above, we can prove the case
argℏ(z0)=−βπ2 |
is true. The proof of Theorem 2 is thus completed.
Theorem 3. If 0<β<1 and 0≤ν<1. If f∈k-Q(α), then
ℜ(f′)>[2G1(2β(1−ν),1;1β+1;12)]−1, |
or equivalently, k-Q(α)⊂B(ν0), where
ν0=[2G1(2β(1−μ),1;1β+1;12)]−1. |
Proof. For
w=2β(1−ν), x=1β, y=1β+1, |
we define
ϝ(z)=(1+Dz)w∫10tx−1(1+Dtz)−wdt=Γ(x)Γ(y) 2G1(1,w,y;zz−1). | (3.7) |
To prove k-Q(α)⊂B(ν0), it suffices to prove that
inf|z|<1{ℜ(q(z))}=q(−1), |
which need to show that
ℜ(1/ϝ(z))≥1/ϝ(−1). |
By Lemma 3 and (3.7), it follows that
ϝ(z)=∫10ϝ(z,t)dε(t), |
where
ϝ(z,t)=1−z1−(1−t)z(0≤t≤1), |
and
dε(t)=Γ(x)Γ(w)Γ(y−w)tw−1(1−t)y−w−1dt, |
which is a positive measure on [0,1].
It is clear that ℜ(ϝ(z,t))>0 and ϝ(−r,t) is real for |z|≤r<1 and t∈[0,1]. Also
ℜ(1ϝ(z,t))=ℜ(1−(1−t)z1−z)≥1+(1−t)r1+r=1ϝ(−r,t) |
for |z|≤r<1. Therefore, by Lemma 3, we get
ℜ(1/ϝ(z))≥1/ϝ(−r). |
If we let r→1−, it follows that
ℜ(1/ϝ(z))≥1/ϝ(−1). |
Thus, we deduce that k-Q(α)⊂B(ν0).
Theorem 4. Let 0≤α<1 and k≥11−α. If f∈k-Q(α), then
f′(z)≺s(z)=1g(z), |
where
g(z)=2G1(2λ,1,1λ+1;zz−1)(λ=1+αkk(1−α)). |
Proof. Suppose that f′=ℏ. From the proof of Theorem 1, we see that
ℏ(z)+zℏ′(z)1λℏ(z)≺1+(1−2μ)z1−z≺1+z1−z(λ=1+αkk(1−α);μ=k−αk−1k(1−α)). |
If we set λ1=1λ, γ2=0, C=1 and D=−1 in Lemma 4, then
ℏ(z)≺s(z)=1g(z)=z1λ(1−z)−2λ1/λ∫z0t(1/λ)−1(1−t)−2/λdt. |
By putting t=uz, and using Lemma 5, we obtain
ℏ(z)≺s(z)=1g(z)=11λ(1−z)2λ∫10u(1/λ)−1(1−uz)−2/λdu=[2G1(2λ,1,1λ+1;zz−1)]−1, |
which is the desired result of Theorem 4.
The present investigation was supported by the Key Project of Education Department of Hunan Province under Grant no. 19A097 of the P. R. China. The authors would like to thank the referees for their valuable comments and suggestions, which was essential to improve the quality of this paper.
The authors declare no conflict of interest.
[1] |
L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X
![]() |
[2] |
K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 87-96. doi: 10.1016/S0165-0114(86)80034-3
![]() |
[3] | F. Smarandache, Neutrosophy. Neutrosophic Probability, Set, and Logic, ProQuest Information & Learning, Ann Arbor, Michigan, USA, 1998. |
[4] | H. Wang, F. Smarandache, Y. Q. Zhang, et al. Single valued neutrosophic sets, Multispace Multistructure, 4 (2010), 410-413. |
[5] |
J. Ye, A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets, J. Intell. Fuzzy Syst., 26 (2014), 2459-2466. doi: 10.3233/IFS-130916
![]() |
[6] |
J. J. Peng, J. Q. Wang, J. Wang, et al. Simplified neutrosophic sets and their applications in multicriteria group decision-making problems, International Journal of System Science, 47 (2016), 2342-2358. doi: 10.1080/00207721.2014.994050
![]() |
[7] |
Nancy, H. Garg, An improved score function for ranking neutrosophic sets and its application to decision-making process, International Journal for Uncertainty Quantification, 6 (2016), 377-385. doi: 10.1615/Int.J.UncertaintyQuantification.2016018441
![]() |
[8] |
D. Rani, H. Garg, Some modified results of the subtraction and division operations on interval neutrosophic sets, J. Exp. Theor. Artif. In., 31 (2019), 677-698. doi: 10.1080/0952813X.2019.1592236
![]() |
[9] | P. Liu, Y. Chu, Y. Li, et al. Some generalized neutrosophic number hamacher aggregation operators and their application to group decision making, Int. J. Fuzzy Syst., 16 (2014), 242-255. |
[10] |
Nancy, H. Garg, Novel single-valued neutrosophic decision making operators under Frank norm operations and its application, Int. J. Uncertain. Quan., 6 (2016), 361-375. doi: 10.1615/Int.J.UncertaintyQuantification.2016018603
![]() |
[11] |
H. Garg, Nancy, New logarithmic operational laws and their applications to multiattribute decision making for single-valued neutrosophic numbers, Cogn. Syst. Res., 52 (2018), 931-946. doi: 10.1016/j.cogsys.2018.09.001
![]() |
[12] |
G. Wei, Z. Zhang, Some single-valued neutrosophic Bonferroni power aggregation operators in multiple attribute decision making, Journal of Ambient Intelligence and Humanized Computing, 10 (2019), 863-882. doi: 10.1007/s12652-018-0738-y
![]() |
[13] | L. Yang, B. Li, A multi-criteria decision-making method using power aggregation operators for single-valued neutrosophic sets, International Journal of Database and Theory and Application, 9 (2016), 23-32. |
[14] | H. Garg, Nancy, Linguistic single-valued neutrosophic power aggregation operators and their applications to group decision-making problems, IEEE/CAA Journal of Automatic Sinica, 7 (2020), 546-558. |
[15] |
P. Ji, J. Q. Wang, H. Y. Zhang, Frank prioritized Bonferroni mean operator with single-valued neutrosophic sets and its application in selecting third-party logistics providers, Neural Comput. Appl., 30 (2018), 799-823. doi: 10.1007/s00521-016-2660-6
![]() |
[16] | H. Garg, Novel neutrality aggregation operators-based multiattribute group decision making method for single-valued neutrosophic numbers, Soft Comput., 2019, 1-23. |
[17] | H. Garg, Nancy, Multiple criteria decision making based on frank choquet heronian mean operator for single-valued neutrosophic sets, Applied and Computational Mathematics, 18, (2019), 163-188. |
[18] | P. Majumdar, Neutrosophic Sets and Its Applications to Decision Making, Computational Intelligence for Big Data Analysis, Springer, Cham, 2015. |
[19] |
H. L. Huang, New distance measure of single-valued neutrosophic sets and its application, Int. J. Intell. Syst., 31 (2016), 1021-1032. doi: 10.1002/int.21815
![]() |
[20] | C. Liu, Y. Luo, The weighted distance measure based method to neutrosophic multiattribute group decision making, Math. Probl. Eng., 2016 (2016), 3145341. |
[21] | H. Garg, Nancy, Some new biparametric distance measures on single-valued neutrosophic sets with applications to pattern recognition and medical diagnosis, Information 8 (2017), 162. |
[22] |
X. H. Wu, J. Q. Wang, J. J. Peng, et al. Cross - entropy and prioritized aggregation operator with simplified neutrosophic sets and their application in multi-criteria decision-making problems, International Journal of Fuzzy Systems, 18 (2016), 1104-1116. doi: 10.1007/s40815-016-0180-2
![]() |
[23] | H. Garg, Nancy, On single-valued neutrosophic entropy of order α, Neutrosophic Sets and Systems, 14 (2016), 21-28. |
[24] | K. Mondal, S. Pramanik, Neutrosophic tangent similarity measure and its application to multiple attribute decision making, Neutrosophic Sets and Systems, 9 (2015), 80-87. |
[25] | K. Mondal, S. Pramanik, B. C. Giri, Hybrid binary logarithm similarity measure for MAGDM problems under SVNS assessments, Neutrosophic Sets and Systems, 20 (2018), 12-25. |
[26] |
F. Liu, G. Aiwu, V. Lukovac, et al. A multicriteria model for the selection of the transport service provider: A single valued neutrosophic DEMATEL multicriteria model, Decision Making: Applications in Management and Engineering, 1 (2018), 121-130. doi: 10.31181/dmame1801121r
![]() |
[27] |
Nancy, H. Garg, A novel divergence measure and its based TOPSIS method for multi criteria decision - making under single - valued neutrosophic environment, J. Intell. Fuzzy Syst., 36 (2019), 101-115. doi: 10.3233/JIFS-18040
![]() |
[28] | C. L. Hwang, K. Yoon, Multiple Attribute Decision Making Methods and Applications A State-ofthe-Art Survey, Springer-Verlag Berlin Heidelberg, 1981. |
[29] |
P. Biswas, S. Pramanik, B. C. Giri, TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment, Neural Computing and Applications, 27 (2016), 727-737. doi: 10.1007/s00521-015-1891-2
![]() |
[30] |
H. Pouresmaeil, E. Shivanian, E. Khorram, et al. An extended method using TOPSIS and VIKOR for multiple attribute decision making with multiple decision makers and single valued neutrosophic numbers, Advances and Applications in Statistics, 50 (2017), 261-292. doi: 10.17654/AS050040261
![]() |
[31] | G. Selvachandran, S. Quek, F. Smarandache, et al. An extended technique for order preference by similarity to an ideal solution (TOPSIS) with maximizing deviation method based on integrated weight measure for single-valued neutrosophic sets, Symmetry, 10 (2018), 236. |
[32] |
X. D. Peng, J. G. Dai, Approaches to single-valued neutrosophic MADM based on MABAC, TOPSIS and new similarity measure with score function, Neural Computing and Applications, 29 (2018), 939-954. doi: 10.1007/s00521-016-2607-y
![]() |
[33] |
I. Mukhametzyanov, D. Pamucar, A sensitivity analysis in MCDM problems: A statistical approach, Decision making: Applications in Management and Engineering, 1 (2018), 51-80. doi: 10.31181/dmame180151b
![]() |
[34] |
E. H. Ruspini, A new approach to clustering, Information and control, 15 (1969), 22-32. doi: 10.1016/S0019-9958(69)90591-9
![]() |
[35] | J. Ye, Single-valued neutrosophic minimum spanning tree and its clustering method, Journal of intelligent Systems, 23 (2014), 311-324. |
[36] | J. Ye, Clustering methods using distance-based similarity measures of single-valued neutrosophic sets, Journal of Intelligent Systems, 23 (2014), 379-389. |
[37] |
Y. Guo, A. Şengür, A novel image segmentation algorithm based on neutrosophic similarity clustering, Applied Soft Computing, 25 (2014), 391-398. doi: 10.1016/j.asoc.2014.08.066
![]() |
[38] | J. Ye, A netting method for clustering-simplified neutrosophic information, Soft Computing, 21 (2016), 7571-7577. |
[39] |
N. D. Thanh, M. Ali, L. H. Son, A novel clustering algorithm in a neutrosophic recommender system for medical diagnosis, Cognitive Computation, 9 (2017), 526-544. doi: 10.1007/s12559-017-9462-8
![]() |
[40] |
A. S. Ashour, Y. Guo, E. Kucukkulahli, et ai. A hybrid dermoscopy images segmentation approach based on neutrosophic clustering and histogram estimation, Applied Soft Computing, 69 (2018), 426-434. doi: 10.1016/j.asoc.2018.05.003
![]() |
[41] |
X. Wang, E. Triantaphyllou, Ranking irregularities when evaluating alternatives by using some ELECTRE methods, Omega - International Journal of Management Science, 36 (2008), 45-63. doi: 10.1016/j.omega.2005.12.003
![]() |
[42] |
Z. S. Xu, J. Chen, J. J. Wu, Cluster algorithm for intuitionistic fuzzy sets, Information Sciences, 178 (2008), 3775-3790. doi: 10.1016/j.ins.2008.06.008
![]() |
[43] |
M. Noureddine, M. Ristic, Route planning for hazardous materials transportation: Multicriteria decision making approach, Decision Making: Applications in Management and Engineering, 2 (2019), 66-85. doi: 10.31181/dmame1901066n
![]() |
[44] | H. Garg, G. Kaur, Quantifying gesture information in brain hemorrhage patients using probabilistic dual hesitant fuzzy sets with unknown probability information, Computers and Industrial Engineering, 140 (2020), 106211. |
[45] | H. Garg, G. Kaur, A robust correlation coefficient for probabilistic dual hesitant fuzzy sets and its applications, Neural Computing & Applications, 2019 (2019), 1-20. |
[46] |
H. Garg, Nancy, Algorithms for possibility linguistic single-valued neutrosophic decision-making based on COPRAS and aggregation operators with new information measures, Measurement, 138 (2019), 278-290. doi: 10.1016/j.measurement.2019.02.031
![]() |
[47] |
H. Garg, Nancy, Non-linear programming method for multi-criteria decision making problems under interval neutrosophic set environment, Applied Intelligence, 48 (2018), 2199-2213. doi: 10.1007/s10489-017-1070-5
![]() |
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