# The Neutro-Stability Analysis of Neutrosophic Cubic Sets with Application in Decision Making Problems

Authors: Mohammed A. Al Shumrani; Muhammad Gulistan; Salma Khan
Journal: Journal of Mathematics
 (2020)

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## Abstract

The neutrosophic cubic sets (NCSs) attained attraction of many researchers in the current time, so the need to discuss and study their stability was felt. Thus, in this article, we discuss the three types of stability of NCSs such as truth-stability, indeterminacy-stability, and falsity-stability. We define the left (resp., right) truth-left evaluative set, left (resp., right) indeterminacy-evaluative set, and left (resp., right) falsity-evaluative set. A new notion of stable NCSs, partially stable NCSs, and unstable NCSs is defined. We observe that every NCS needs not to be a stable NCS but each stable NCS must be an NCS, i.e., every internal NCS is a stable NCS but an external NCS may or may not be a stable NCS. We also discuss some conditions under which the left and right evaluative points of an external NCS becomes a neutrosophic bipolar fuzz set. We have provided the condition under which an external NCS becomes stable. Moreover, we discuss the truth-stable degree, indeterminacy-stable degree, and falsity-stable degree of NCSs. We have also defined an almost truth-stable set, almost indeterminacy-stable set, almost falsity-stable set, almost partially stable set, and almost stable set with examples. Application of stable NCSs is given with a numerical example at the end.


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## Body

## 1. Introduction

The crisp set lost the stability as it covers the extremes only, which is not the ideal situation in every problem. To cover this gap, Zadeh [1] presented the idea of the fuzzy set (FS) in 1965 which is stable as compared to the crisp set. But, when there is a case to handle the negative characteristics, the fuzzy set (FS) too lost its stability. To cover this gap, Atanassov [2], in 1986, gave the idea of intutionistic fuzzy sets (IFSs) which are more stable than the fuzzy set. But, the problem with Atanassov’s idea is that indeterminacy is lost and no proper attraction is given to it. Then, Smarandache [3] covered this gap by giving a new idea of a neutrosophic set which is a stable version other than the fuzzy set and intutionistic fuzzy sets. The neutrosophic set NS is the extension of the FS, IVFS, and IFS. In the NS, we deal with its three components, that is, truthfulness, indeterminate, and untruthfulness, and these three functions are independent completely. Neutrosophy gives us a support for a whole family of new mathematical theories with the abstraction of both classical and fuzzy counterparts. In real life and in scientific problems to apply the neutrosophic set, Wang et al. [4] introduced the new idea of a single-valued neutrosophic set SVNS and interval neutrosophic set INS. These are subclasses of the NS, in which truthfulness, indeterminate, and untruthfulness were taken in a closed interval 0,1, see also [5]. On the other side, Zadeh [6] made another extension which is known as the interval-valued fuzzy set (IVFS), in which he described interval membership function. There are many real-life applications of the IVFS, i.e., Sambuc [7] in medical diagnosis in thyroidian, Gorzalczany in approximate reasoning, and Turksen [8, 9] in interval-valued logic. In 2012, the theme of the cubic set CS was used by Jun et al. [10]. CS is the combination of the IVFS and FS in the form of an ordered pair. These all are mathematical tools to determine the complications in our daily life. Jun et al. [11] gave the idea of the NCS. For application of NCSs, we refer to [12–17]. In 2017, the concept of stable cubic sets was introduced by Muhiuddin et al. [18]. In 2019 and 2020, Smarandache [19–21] generalized the classical algebraic structures to neutroalgebraic structures (or neutroalgebras) (whose operations and axioms are partially true, partially indeterminate, and partially false) as extensions of partial algebra and to antialgebraic structures (or antialgebras) (whose operations and axioms are totally false). Also, in general, he extended any classical structure, in no matter what field of knowledge, to a neutrostructure and an antistructure. Similarly, as alternatives to a classical theorem (that is true for all sets’ elements) are the neutrotheorem (partially true, partially indeterminate, and partially false) and antitheorem (false for all sets’ elements), respectively.In this paper, we define different types of the stable neutrosophic cubic set with examples and some basic results. We also define the concept of almost stable neutrosophic cubic sets. At the end, we have provided an application of the presented theory.


## 2. Preliminaries

This section mainly recalls some basic concepts related to fuzzy sets [1], cubic sets [10], neutrosophic sets [3, 4], neutrosophic cubic sets [11], and evaluative structure of cubic sets [18]. For more detail of these sets, we refer the reader to [1, 3, 4, 10, 11, 18].Definition 1 (see [1]).
A mapping p:U⟶0,1 is called an FS, and p˜ (ů) is a membership function and denoted by p.Definition 2 (see [10]).
A structureC=u∘;p˜u∘,pu∘|u∘∈U is a cubic set in U in which p˜(ů) is IVF in U, and p (ů) is an FS in U. This is simply denoted by C=p˜,p. Cu∘ denotes the collection of cubic sets in U.Definition 3 (see [3, 4]).
A neutrosophic set is a structure(1)N=u∘;TNu∘,INu∘,FNu∘|u∘∈U,in U. Here, TNu∘,INu∘,FNu∘∈0,1 are three functions, known as truthfulness, indeterminate, and untruthfulness, respectively, simply denoted by N=TN,IN,FN.Definition 4 (see [11]).
A structure(2)NC=u∘;T˜NCu∘,I˜NCu∘,F˜NCu∘,TNCu∘,INCu∘,FNCu∘|u∘∈U,is an NCS in X. Here,(3)T˜NC=TNCL,TNCU,I˜NC=INCL,INCU,F˜NCFNCL,FNCU,is an interval NS and TNC,INC,FNC is an NS in X simply denoted by(4)NC=T˜NC,I˜NC,F˜NC,TNC,INC,FNC,0,0≤T˜NC+I˜NC+F˜NC≤3,3,0≤TNC+INC+FNC≤1.Definition 5 (see [18]).
A structureC=u∘;p˜u∘,pu∘|u∘∈U is a CS in U in which C (ů) is the evaluative structure defined as follows:(5)EC=u∘;ECu∘|u∘∈U,where ECu∘=lECu∘,rECu∘ with left evaluative point lECu∘=pu∘−p˜u∘ and right evaluative point rECu∘=pu∘+−pu∘ at u∘∈U. We say that ECu∘ is the evaluative point of C=p˜,p at u∘∈U.


## 3. Neutrostable Neutrosophic Cubic Sets

In this section, we provide the concepts of the truth-evaluative set, indeterminacy-evaluative set, falsity-evaluative set, stable truth-element, stable indeterminacy-element, stable falsity-element, and unstable element of the NCS. We also discuss some interesting results.Definition 6.
Letp=Tp,Ip,Fp,tp,ip,fp be an NCS in U. Then,(1)
The truth-evaluative set ofp=Tp,Ip,Fp,tp,ip,fp is represented as(6)ETp=u∘,ETpu∘|u∘∈U=left truth−evaluative point, right truth−evaluative point=lETpu∘,rETpu∘=tu∘−T−u∘,T+u∘−tu∘.(2)
The indeterminacy-evaluative set ofp=Tp,Ip,Fp,tp,ip,fp is represented as(7)EIp=u∘,EIpu∘|u∘∈U=left indeterminacy−evaluative point, right indeterminacy−evaluative point=lEIpu∘,rEIpu∘=iu∘−I−u∘,I+u∘−iu∘.(3)
The falsity-evaluative set ofp=Tp,Ip,Fp,tp,ip,fp is represented as(8)EFp=u∘,EFpu∘|u∘∈U=left falsity−evaluative point, right falsity−evaluative point=lEFpu∘,rEFpu∘=fu∘−F−u∘,F+u∘−fu∘.
The collection(9)ELpu∘=lETpu∘,lEIpu∘,lEFpu∘,is called the left evaluative point and the collection(10)ERpu∘=rETpu∘,rEIpu∘,rEFpu∘,is called the right evaluative point. We say that Eβu∘=ELβu∘,ERβu∘ is the evaluative point.Example 1.
Letβ=u∘,Tu∘,Iu∘,Fu∘,tu∘,iu∘,fu∘|u∘∈I be an NCS in U. If(11)Tu∘,Iu∘,Fu∘,tu∘,iu∘,fu∘=0.2,0.4,0.4,0.6,0.5,0.7,0.3,0.2,0.8,for allu∘∈U,then ETβ=0.1,0.1,EIβ=−0.2,0.4,EFβ=0.3,−0.1. Thus,(12)Eβu∘=ELβu∘,ERβu∘=u∘,0.1,−0.2,0.3,0.1,0.4,−0.1|u∘∈U.Remark 1.
In Example1, we observe that the left or right evaluative point of the NCS is not necessarily an NS. This motivates us to define the following terminologies.Definition 7.
Letβ=Tβ,Iβ,Fβ,tβ,iβ,fβ be an NCS in U with the evaluative set(13)Eβ=u∘;ELβu∘,ERβu∘|u∘∈U.
An element ů∈U is called(1)
Truth stable element of U if(14)ETβ=u∘;lSTβu∘,rSTβu∘=u∘;left truth stable−element, right truth stable−element=u∘;tu∘−T−u∘T+u∘−tu∘≥0.(2)
Indeterminacy stable element of U if(15)EIβ=u∘;lSIβu∘,rSIβu∘=u∘;left indeterminacy stable−element, right indeterminacy stable−element,=u∘;iu∘−I−u∘,I+u∘−iu∘≥0(3)
Falsity stable element of U if(16)EFβ=lSFβu∘,rSFβu∘=u∘;left stable falsity−element, right stable falsity−element=fu∘−F−u∘,F+u∘−fu∘≥0.
An element ů∈U is called stable if it satisfies conditions (1–3). The set of all stable elements of U is called stable cut of β=Tβ,Iβ,Fβ,tβ,iβ,fβ in U and is denoted by Sβ. We say that β=Tβ,Iβ,Fβ,tβ,iβ,fβ is a stable neutrosophic set if Sβ=U.
An element ů∈U is called partially stable if it partially satisfies conditions (1–3). The set of all partially stable elements of U is called partially stable cut of β=Tβ,Iβ,Fβ,tβ,iβ,fβ in U and is denoted by Pβ. We say that β=Tβ,Iβ,Fβ,tβ,iβ,fβ is a partially stable neutrosophic set if Pβ⊂U.
An element ů∈U is called antistable (unstable) if it does not satisfy conditions (1–3). The set of all unstable stable elements of U is called unstable stable cut of β=Tβ,Iβ,Fβ,tβ,iβ,fβ in U and is denoted by Uβ. We say that β=Tβ,Iβ,Fβ,tβ,iβ,fβ is a unstable stable neutrosophic set if U\ss⊆U.
Thus,U=Sβ∪Pβ∪Uβ.Example 2.
Letβ=Tβ,Iβ,Fβ,tβ,iβ,fβ be an NCS in U=0,a,b,c given by Table1.
Clearly,0,a are stable elements of U and b,c are unstable elements of U. Thus,(17)U=a,b,c,d=Sβ=0,a∪Pβ=Φ∪Uβ=b,c.Table 1
Neutrosophic cubic setβ of U.
UTβu∘Iβu∘Fβu∘tβu∘iβu∘fβu∘00.3,0.50.2,0.40.2,0.50.40.30.4a0.3,0.50.3,0.50.3,0.60.40.40.5b0.6,0.80.5,0.60.4,0.50.50.40.3c0.4,0.80.5,0.60.6,0.70.90.70.8Example 3.
Letβ=Tβ,Iβ,Fβ,tβ,iβ,fβ be an NCS in U=a,b given by Table 2.
Clearly,a and b are stable elements of U. Thus,(18)U=a,b=Sβ=a,b∪Pβ=Φ∪Uβ=Φ.Table 2
Neutrosophic cubic setβ of U.
UTβu∘Iβu∘Fβu∘tβu∘iβu∘fβu∘a0.1,0.70.1,0.60.2,0.80.60.50.7b0.6,0.80.6,0.90.5,0.70.70.80.6Remark 2.
Every internal NCS is a stable NCS, as shown in example3. If an NCS is neither internal nor external, then we may have some stable elements with respect to the internal portion and some unstable elements with respect to the external portion as given in the Example 2. Thus, an external NCS may or may not be a stable NCS, as shown in Examples 4 and 5.Example 4.
Letβ=Tβ,Iβ,Fβ,tβ,iβ,fβ be an external NCS in U=a,b given by Table 3.
Then, clearly,a,b are unstable elements of U. Thus,(19)U=a,b=Sβ=Φ∪Pβ=a,b∪Uβ=Φ.Table 3
Neutrosophic cubic setβ of U.
UTβu∘Iβu∘Fβu∘tβu∘iβu∘fβu∘a0.1,0.30.1,0.40.3,0.60.40.50.7b0.5,0.80.6,0.80.4,0.60.40.50.3Example 5.
Letβ=Tβ,Iβ,Fβ,tβ,iβ,fβ be an external NCS in U=a,b given by Table 4.
Then, clearly,a,b are stable elements of U. Thus,(20)U=a,b=Sβ=a,bΦ∪Pβ=Φ∪Uβ=Φ.Table 4
External neutrosophic cubic setβ of U.
UTβu∘Iβu∘Fβu∘tβu∘iβu∘fβu∘a0.2,0.40.3,0.50.3,0.60.20.30.3b0.4,0.80.6,0.70.4,0.50.80.70.5Example 6.
Letβ=Tβ,Iβ,Fβ,tβ,iβ,fβ be an external NCS in U=a given by Table 5.
Clearly,a is an unstable element of U. Thus, Uβ=a=U. Hence, U=Sβ=Φ∪Pβ=Φ∪Uβ=a.Table 5
External neutrosophic cubic setβ of U.
UTβu∘Iβu∘Fβu∘tβu∘iβu∘fβu∘a0.3,0.50.1,0.40.4,0.60.80.50.7Example 7.
Letβ=Tβ,Iβ,Fβ,tβ,iβ,fβ be an external NCS in UU=a given by Table 6.
Clearly,a is an unstable element of U. Thus, Uβ=a=U. Hence, U=Sβ=Φ∪Pβ=Φ∪Uβ=a.Table 6
External neutrosophic cubic setβ of U.
UTβu∘Iβu∘Fβu∘tβu∘iβu∘fβu∘a0.5,0.60.3,0.50.7,0.90.40.20.6Example 8.
Letβ=Tβ,Iβ,Fβ,tβ,iβ,fβ be an NCS in U=a,b,c given by Table 7.
Clearly,a and b are partially stable elements of U, so Pβ=a,b⊂U and c is the only stable element of U, so Sβ=c. Also, there is no element which is unstable, so Uβ=Φ. Hence, U=Sβ∪Pβ∪Uβ.Table 7
Neutrosophic cubic setβ of U.
UTβu∘Iβu∘Fβu∘tβu∘iβu∘fβu∘a0.7,0.80.3,0.50.6,0.90.70.80.2b0.1,0.50.6,0.90.3,0.80.20.70.1c0.1,0.40.2,0.50.3,0.70.30.40.5Remark 3.
(1)
If we have an external NCS which is unstable like in Example6 such that(21)tu∘>T−u∘,T+u∘,iu∘>I−u∘,I+u∘,fu∘>F−u∘,F+u∘,then its right evaluative point becomes a neutrosophic bipolar fuzzy set.(2)
If we have an external NCS which is unstable like in example7 such that(22)tu∘<T−u∘,T+u∘,iu∘<I−u∘,I+u∘,fu∘<F−u∘,F+u∘,then its left evaluative point becomes a neutrosophic bipolar fuzzy set.(3)
Every NCS needs not to be a stable NCS, but each stable NCS must be an NCS.(4)
Observing Example5, we reached at Theorem 1.Theorem 1.
If an external NCSβ=Tβ,Iβ,Fβ,tβ,iβ,fβ in U satisfies the condition(23)∀u∘∈UTβ−u∘=tβu∘,Tβ+u∘=tu∘,I−u∘=iu∘,I+u∘=iu∘,F−u∘=fu∘,F+u∘=fu∘,then β=Tβ,Iβ,Fβ,tβ,iβ,fβ is a stable NCS.Proof.
Straightforward.Remark 4.
We observe that if β is both an internal and external NCS, then β is a stable NCS.Theorem 2.
The complement of a stable NCS is also a stable NCS.Proof.
Letβ=Tβ,Iβ,Fβ,tβ,iβ,fβ be a stable NCS in U. Then,(24)U=Sβ=u∘∈u∘|lEβu∘≥0,rEβu∘≥0.
Hence,(25)tu∘−T−u∘≥0,T+u∘−tu∘≥0,iu∘−I−u∘≥0,I+u∘−iu∘≥0,fu∘−F−u∘≥0,F+u∘−fu∘≥0,∀u∘∈U.
It follows that(26)lEβCu∘=1−tu∘−1−T+u∘=T+u∘−tu∘≥0,lEβCu∘=1−iu∘−1−I+u∘=I+u∘−iu∘≥0,lEβCu∘=1−fu∘−1−F+u∘=F+u∘−fu∘≥0,rEβCu∘=1−T−u∘−1−tu∘=tu∘−T−u∘≥0,rEβCu∘=1−I−u∘−1−iu∘=iu∘−I−u∘≥0,rEβCu∘=1−F−u∘−1−fu∘=fu∘−F−u∘≥0.
Therefore,βC=Tβc,Iβc,Fβc,tβc,iβc,fβc is a stable NCS.Theorem 3.
The complement of an unstable NCS is also an unstable NCS.Proof.
Letβ=Tβ,Iβ,Fβ,tβ,iβ,fβ be an unstable NCS in U. Then,(27)U=Uβ=u∘∈u∘|lEβu∘<0∪u∘∈u∘|rEβu∘<0≠Φ,and so, there exist u∘∈U such that(28)tu∘−T−u∘<0,iu∘−I−u∘<0,fu∘−F−u∘<0,∀u∘∈U,or(29)T+u∘−tu∘<0,I+u∘−iu∘<0,F+u∘−fu∘<0,∀u∘∈U.
It follows that(30)lEβCu∘=1−tu∘−1−T+u∘=T+u∘−tu∘<0,lEβCu∘=1−iu∘−1−I+u∘=I+u∘−iu∘<0,lEβCu∘=1−fu∘−1−F+u∘=F+u∘−fu∘<0,or(31)rEβCu∘=1−T−u∘−1−tu∘=tu∘−T−u∘<0,rEβCu∘=1−I−u∘−1−iu∘=iu∘−I−u∘<0,rEβCu∘=1−F−u∘−1−fu∘=fu∘−F−u∘<0.
Hence,UβC≠Φ, and therefore, βC=Tβc,Iβc,Fβc,tβc,iβc,fβc is an unstable NCS.
Example 9 illustrates Theorem 3.Example 9.
Letβ=Tβ,Iβ,Fβ,tβ,iβ,fβ be an NCS in U=a,b given by Table 8.
Clearly,a and b are unstable elements of U and their complements are represented by Table 9.
Then,βc=Tβc,Iβc,Fβc,tβc,iβc,fβc is unstable since a∈Uβc.Table 8
Neutrosophic cubic setβ of U.
UTβu∘Iβu∘Fβu∘tβu∘iβu∘fβu∘a0.1,0.50.3,0.60.2,0.40.40.50.3b0.6,0.90.1,0.90.1,0.60.70.60.5Table 9
Complement of neutrosophic cubic setβ of U provided in Table 8.
UTβcu∘Iβcu∘Fβcu∘tβcu∘iβcu∘fβcu∘a0.5,0.90.4,0.70.6,0.80.60.50.7b0.1,0.40.1,0.90.4,0.90.30.40.5Theorem 4.
The P-union and P-intersection of two stable NCSs inU are stable cubic sets in U.Proof.
Letβ=Tβ,Iβ,Fβ,tβ,iβ,fβ and β2=Tβ2,Iβ2,Fβ2,tβ2,iβ2,fβ2 be two NCSs in U. Then,(32)Sβ=u∘∈U|lββu∘≥0,rβ5βu∘≥0=U,Sβ2=u∘∈U|lβ5β2u∘≥0,rβ5β2u∘≥0=U.
It follows that(33)tβu∘−Tβ−u∘≥0,Tβ+u∘−tβu∘≥0,iβu∘−Iβ−u∘≥0,Iβ+u∘−iβu∘≥0,fβu∘−Fβ−u∘≥0,Fβ+u∘−fβu∘≥0,∀u∘∈U,tβ2u∘−Tβ2−u∘≥0,Tβ2+u∘−tβ2u∘≥0,iβ2u∘−Iβ2−u∘≥0,Iβ2+u∘−iβ2u∘≥0,fβ2u∘−Fβ2−u∘≥0,Fβ2+u∘−fβ2u∘≥0,∀u∘∈U.
Assume thattβ1u∘≥tβ2u∘,iβ1u∘≥iβ2u∘,fβ1u∘≥fβ2u∘ and consider the following cases:(i)
Tβ1−u∘≥Tβ2−u∘,Tβ1+u∘≥Tβ2+u∘Iβ1−u∘≥Iβ2−u∘,Iβ1+u∘≥Iβ2+u∘Fβ1−u∘≥Fβ2−u∘,Fβ1+u∘≥Fβ2+u∘(ii)
Tβ1−u∘≤Tβ2−u∘,Tβ1+u∘≥Tβ2+u∘Iβ1−u∘≥Iβ2−u∘,Iβ1+u∘≥Iβ2+u∘Fβ1−u∘≥Fβ2−u∘,Fβ1+u∘≥Fβ2+u∘(iii)
Tβ1−u∘≤Tβ2−u∘,Tβ1+u∘≤Tβ2+u∘Iβ1−u∘≥Iβ2−u∘,Iβ1+u∘≥Iβ2+u∘Fβ1−u∘≥Fβ2−u∘,Fβ1+u∘≥Fβ2+u∘(iv)
Tβ1−u∘≤Tβ2−u∘,Tβ1+u∘≤Tβ2+u∘Iβ1−u∘≤Iβ2−u∘,Iβ1+u∘≥Iβ2+u∘Fβ1−u∘≥Fβ2−u∘,Fβ1+u∘≥Fβ2+u∘(v)
Tβ1−u∘≤Tβ2−u∘,Tβ1+u∘≤Tβ2+u∘Iβ1−u∘≤Iβ2−u∘,Iβ1+u∘≤Iβ2+u∘Fβ1−u∘≥Fβ2−u∘,Fβ1+u∘≥Fβ2+u∘(vi)
Tβ1−u∘≤Tβ2−u∘,Tβ1+u∘≤Tβ2+u∘Iβ1−u∘≤Iβ2−u∘,Iβ1+u∘≤Iβ2+u∘Fβ1−u∘≤Fβ2−u∘,Fβ1+u∘≥Fβ2+u∘(vii)
Tβ1−u∘≤Tβ2−u∘,Tβ1+u∘≤Tβ2+u∘Iβ1−u∘≤Iβ2−u∘,Iβ1+u∘≤Iβ2+u∘Fβ1−u∘≤Fβ2−u∘,Fβ1+u∘≤Fβ2+u∘(viii)
Tβ1−u∘≥Tβ2−u∘,Tβ1+u∘≤Tβ2+u∘Iβ1−u∘≤Iβ2−u∘,Iβ1+u∘≤Iβ2+u∘Fβ1−u∘≤Fβ2−u∘,Fβ1+u∘≤Fβ2+u∘(ix)
Tβ1−u∘≥Tβ2−u∘,Tβ1+u∘≥Tβ2+u∘Iβ1−u∘≤Iβ2−u∘,Iβ1+u∘≤Iβ2+u∘Fβ1−u∘≤Fβ2−u∘,Fβ1+u∘≤Fβ2+u∘(x)
Tβ1−u∘≥Tβ2−u∘,Tβ1+u∘≥Tβ2+u∘Iβ1−u∘≥Iβ2−u∘,Iβ1+u∘≤Iβ2+u∘Fβ1−u∘≤Fβ2−u∘,Fβ1+u∘≤Fβ2+u∘(xi)
Tβ1−u∘≥Tβ2−u∘,Tβ1+u∘≥Tβ2+u∘Iβ1−u∘≥Iβ2−u∘,Iβ1+u∘≥Iβ2+u∘Fβ1−u∘≤Fβ2−u∘,Fβ1+u∘≤Fβ2+u∘(xii)
Tβ1−u∘≥Tβ2−u∘,Tβ1+u∘≥Tβ2+u∘Iβ1−u∘≥Iβ2−u∘,Iβ1+u∘≥Iβ2+u∘Fβ1−u∘≥Fβ2−u∘,Fβ1+u∘≤Fβ2+u∘
The first case implies that(34)maxtβ1u∘,iβ1u∘,fβ1u∘,tβ2u∘,iβ2u∘,fβ2u∘=tβ1u∘≥Tβ1−u∘,iβ1u∘≥Iβ1−u∘,iβ1u∘≥Iβ1−u∘=maxTβ1−u∘,Iβ1−u∘,Fβ1−u∘,Tβ2−u∘,Iβ2−u∘,Fβ2−u∘,maxtβ1u∘,iβ1u∘,fβ1u∘,tβ2u∘,iβ2u∘,fβ2u∘=tβ1u∘≥Tβ1+u∘,iβ1u∘≥Iβ1+u∘,iβ1u∘≥Iβ1+u∘=maxTβ1+u∘,Iβ1+u∘,Fβ1+u∘,Tβ2+−u∘,Iβ2+u∘,Fβ2+u∘.
It follows that(35)tβ1u∘−Tβ1−u∘,iβ1u∘−Iβ1−u∘,fβ1u∘−Fβ1−u∘≥0,Tβ1u∘+−tβ1u∘,Iβ1u∘+−iβ1u∘,Fβ1u∘+−fβ1u∘≥0.
The result of the remaining cases can be obtained in the same way. Therefore,β1∪βP2 is a stable CS in U. By the same way, we also know that β1∪βP2 is a stable CS in U.
Example 10 shows that the Ṙ-union and the Ṙ-intersection of two stable NCSs inU may not be a stable NCS in U.Example 10.
Letβ1=Tβ1,Iβ1,Fβ1,tβ1,iβ1,fβ1 and β2=Tβ2,Iβ2,Fβ2,tβ2,iβ2,fβ2 be two NCSs in U=a,b defined by Tables 10 and 11, respectively.
Then,(36)β1∪βR2=a,0.4,0.5,0.3,0.9,0.7,0.9,0.15,0.35,0.6,b,0.6,0.9,0.8,0.9,0.5,0.6,0.6,0.8,0.25,β1∩βR2=a,0.1,0.3,0.1,0.4,0.3,0.7,0.4,0.8,0.80,b,0.6,0.9,0.1,0.9,0.2,0.4,0.7,0.8,0.56.
Hence, we know that(37)Eβ1∪Rβ2a=−0.25,0.35,0.05,0.55,−0.1,0.3,Eβ1∪β2b=0,0.3,0,0.1,−0.25,0.35,Eβ1∩β2a=0.3,−0.1,0.7,−0.4,0.5,−0.1,Eβ1∩β2b=0.1,0.2,0.7,0.1,0.36,−0.16.Table 10
Neutrosophic cubic setβ1 of U.
UTβ1u∘Iβ1u∘Fβ1u∘tβ1u∘iβ1u∘fβ1u∘a0.4,0.50.3,0.40.3,0.70.40.350.60b0.3,0.70.8,0.90.5,0.60.600.80.56Table 11
Neutrosophic cubic setβ2 of U.
UTβ2u∘Iβ2u∘Fβ2u∘tβ2u∘iβ2u∘fβ2u∘a0.1,0.30.1,0.90.7,0.90.150.80.8b0.6,0.90.1,0.90.2,0.40.70.80.25Theorem 5.
Letβ1=Tβ1,Iβ1,Fβ1,tβ1,iβ1,fβ1 and β2=Tβ2,Iβ2,Fβ2,tβ2,iβ2,fβ2 be two internal NCSs in U such that(38)∀u∘∈UmaxTβ1u∘−,Iβ1u∘−,Fβ1u∘−,Tβ2u∘−,Iβ2u∘−,Fβ2u∘−≤tβ1,iβ1,fβ1∧tβ2,iβ2,fβ2u∘.
Then, the Ṙ-union ofβ1 and β2 is a stable NCS in U.Proof.
Letβ1=Tβ1,Iβ1,Fβ1,tβ1,iβ1,fβ1 and β2=Tβ2,Iβ2,Fβ2,tβ2,iβ2,fβ2 be two internal NCSs in U. Then, Tβ1u∘−≤tβ1u∘≤Tβ1u∘+,Iβ1u∘−≤iβ1u∘≤Iβ1u∘+, and Fβ1u∘−≤fβ1u∘≤Fβ1u∘+ and Tβ2u∘−≤tβ2u∘≤Tβ1u∘+,Iβ2u∘−≤iβ2u∘≤Iβ2u∘+, and Fβ2u∘−≤fβ2u∘≤Fβ2u∘+,∀u∘∈U. We know that(39)maxTβ1u∘−,Iβ1−u∘≤Fβ1u∘−,Tβ1u∘−,Iβ1u∘−≤Fβ1u∘−≤tβ1,iβ1,fβ1∧tβ2,iβ2,fβ2u∘≤maxTβ1u∘+,Iβ1u∘+≤Fβ1u∘+,Tβ1u∘+,Iβ1u∘+≤Fβ1u∘+,for all u∘∈U. Hence, the Ṙ-union of β1 and β2 is an internal NCS, and so it is stable by the fact that every internal NCS is stable.Theorem 6.
Letβ1=Tβ1,Iβ1,Fβ1,tβ1,iβ1,fβ1 and β2=Tβ2,Iβ2,Fβ2,tβ2,iβ2,fβ2 be two internal NCSs in U such that(40)∀u∘∈UmaxTβ1u∘+,Iβ1u∘+,Fβ1u∘+,Tβ2u∘+,Iβ2u∘+,Fβ2u∘+≤tβ1,iβ1,fβ1∨tβ2,iβ2,fβ2u∘.
Then, the Ṙ-intersection ofβ1 and β2 is a stable NCS in U.Proof.
Straightforward.


## 4. Neutro-Almost-Stable Neutrosophic Cubic Set

In this section, we introduce a new class of the stable neutrosophic cubic set, namely, the neutro-almost-stable neutrosophic cubic set.Definition 8.
Letβ=Tβ,Iβ,Fβ,tβ,iβ,fβ be an NCS with the evaluative set Eβ=u∘,Eβu∘|u∘∈U in U. Then,(1)
The truth-stable degree ofβ in U is denoted by TruSDβ and is defined as(41)TruSDβ=∑u∘∈UlETβu∘,rETβu∘.(2)
The indeterminacy-stable degree ofβ in U is denoted by IndSDβ and is defined as(42)IndSDβ=∑u∘∈UlEIβu∘,rEIβu∘.(3)
The falsity-stable degree ofβ in U is denoted by FalSDβ and is defined as(43)FalSDβ=∑u∘∈UlEFβu∘,rEFβu∘.(4)
The stable degree ofβ in U is denoted by SDβ and is defined as SDβ=TruSDβ,IndSDβ,FalSDβ.Definition 9.
An NCS with the evaluative setEβ=u∘,Eβu∘|u∘∈U in U is said to be(1)
Almost truth-stable ifTruSDβ≥0(2)
Almost indeterminacy-stable ifIndSDβ≥0(3)
Almost falsity-stable ifFalSDβ≥0(4)
Almost stable if it is almost truth-stable, almost indeterminacy-stable, and almost falsity-stable, i.e.,TruSDβ≥0,IndSDβ≥0,FalSDβ≥0.(5)
Almost partially stable if it is almost partially truth-stable, almost partially indeterminacy-stable, and almost partially falsity-stable.(6)
Almost unstable if it is almost truth-unstable, almost indeterminacy-unstable, and almost falsity-unstable, i.e.,TruSDβ<0,IndSDβ<0,FalSDβ<0.Example 11.
Letβ1=Tβ1,Iβ1,Fβ1,tβ1,iβ1,fβ1 and β2=Tβ2,Iβ2,Fβ2,tβ2,iβ2,fβ2 be two NCSs in U=a,b defined by Tables 12 and 13, respectively,with the evaluative set(44)Eβ1=a;0,0.1,0.05,0.05,0.3,0.1,b;0.3,0.1,0,0.1,0.06,0.04.
Then,TruSDβ1=0.3,0.2≥0, IndSDβ1=0.05,0.15≥0,FalSDβ2=0.36,0.14≥0. Thus,(45)SDβ1=0.3,0.2,0.05,0.15,0.36,0.14≥0,alsowith the evaluative set(46)Eβ2=a0.05,0.15,0.7,0.1,0.1,0.1,b0.1,0.2,0.7,0.1,0.05,0.15.
Then,TruSDβ2=0.15,0.25≥0, IndSDβ2=0.14,0.2≥0,FalSDβ2=0.15,0.25≥0. Thus,(47)SDβ2=0.15,0.35,0.14,0.2,0.15,0.25≥0.
So,β1 and β2 both are almost stable NCSs.Table 12
Neutrosophic cubic setβ1 of U.
UTβ1u∘Iβ1u∘Fβ1u∘tβ1u∘iβ1u∘fβ1u∘a0.4,0.50.3,0.40.3,0.70.40.350.60b0.3,0.70.8,0.90.5,0.60.600.80.56Table 13
Neutrosophic cubic setβ2 of U.
UTβ2u∘Iβ2u∘Fβ2u∘tβ2u∘iβ2u∘fβ2u∘a0.1,0.30.1,0.90.7,0.90.150.80.8b0.6,0.90.1,0.90.2,0.40.70.80.25Example 12.
Letβ3=Tβ3,Iβ3,Fβ3,tβ3,iβ3,fβ3 be an NCS in U=a,b defined by Table 14.
The evaluative set is(48)Eβ3=a−0.1,0.2,0.3,−0.1,−0.1,0.3,b−0.1,0.5,−0.2,0.3,0.2,−0.1.
Then,TruSDβ3=−0.2,0.7<0. Thus, the NCS β3=Tβ3,Iβ3,Fβ3,tβ3,iβ3,fβ3 in U is not almost truth-stable as TruSDβ3<0. Also, IndSDβ3=0.1,0.2≥0. Thus, the NCS β3=Tβ3,Iβ3,Fβ3,tβ3,iβ3,fβ3 in U is almost indeterminacy-stable as IndSDβ3≥0. Similarly β3=Tβ3,Iβ3,Fβ3,tβ3,iβ3,fβ3 in U is almost falsity-stable as FalSDβ3≥0. So, finally, we can say that β3 is an almost partially stable NCS.Table 14
Neutrosophic cubic setβ3 of U.
UTβ3u∘Iβ3u∘Fβ3u∘tβ3u∘iβ3u∘fβ3u∘a0.2,0.30.3,0.50.4,0.60.10.60.3b0.3,0.70.8,0.90.5,0.60.20.60.7Example 13.
Letβ4=Tβ4,Iβ4,Fβ4,tβ4,iβ4,fβ4 be an NCS in U=a,b defined by Table 15
The evaluative set is(49)Eβ4=a;0.2,−0.1,0.3,−0.1,−0.1,0.3,b;−0.1,0.5,−0.2,0.3,0.2,−0.1.
Then,TruSDβ4=0.1,0.4≥0,IndSDβ4=0.1,0.2≥0,FalSDβ4=0.1,0.2≥0. So, β4 is an almost-stable NCS, but it is not a stable NCS, as from Definition 7; Sβ=Φ,Pβ=Φ,Uβ=a,b.Table 15
Neutrosophic cubic setβ4 of U.
UTβ4u∘Iβ4u∘Fβ4u∘tβ4u∘iβ4u∘fβ4u∘a0.2,0.30.3,0.50.4,0.60.40.60.3b0.3,0.70.8,0.90.5,0.60.20.60.7Remark 5.
From Examples11, 12, and 13, we have the following results.Theorem 7.
(1)
Every stable NCSβ=Tβ,Iβ,Fβ,tβ,iβ,fβ in U is an almost-stable NCS, but the converse is not true(2)
Every internal NCS is almost stable(3)
Every external NCS may or may not be stable(4)
The P-union and P-intersection of two stable NCSs are almost stable(5)
The complement of an almost-stable NC is also an almost-stable NCSProof.
Straightforward.


## 5. Application in Decision Making

In this section, we shall define a new approach to multiple attribute group decision making wıth the help of stable neutrosophic cubic sets. We also provide a numerical example. SupposeH=H1,H2,…,Hm. Each alternative Hi respects n criteria Gj=G1,G2,…,Gn which are expressed by a stable NCS qij=q˜Truij,q˜Indij,q˜FalijqTruij,qIndij,qFalij, j=1,2,…n,i=1,2,…,m. The criteria G1,…,Gk are benefit and criteria Gk+1,…,Gn are nonbenefit criteria, and ω=ω1,ω2,…,ωn is the weighted vector of the criteria, where, ωiɛ0,1 and ∑ωi=1. So, the decision matrix is obtained as D=qijm×n. The steps of the decision making based on stable NCSs are given as follows:Step 1: we standardize the decision matrix.Step 2: we construct the normalized decision matrix. Normalize score or data are as follows:(50)rij=u∘ij∑u∘ij2,fori=1,…,m;j=1,…,n.Step 3: we construct the weighted normalized decision matrix:(51)vij=wj⋅rij.Step 4: we determine the ideal and negative ideal solutions. Ideal solutionA∗=v1,…,vn, where(52)vj∗=maxvij,ifj∈J;minvij,ifj∈J′.Negative ideal solution is(53)A′=v1′,…,vn′,where(54)vj′maxvij,ifj∈J;minvij,ifj∈J′.Step 5: we calculate the separation measures for each alternative. Separation from the ideal alternatives is(55)Si∗=∑vj∗−vij2,i=1,…,m.Similarly, separation from negative ideal alternatives is(56)Si′=∑vj′−vij2,i=1,…,m.Step 6: we calculate the relative closeness to the ideal solutionCi∗ where(57)Ci∗=Si′Si∗+Si′,0≤Ci∗≤1.We select the option withCi∗ closest to 1.


### 5.1. Numerical Application

At the end of December 2019 [22], in Wuhan, the China Health Commission reported a cluster of pneumonia cases of unknown etiology. The pathogen was identified as novel coronavirus 2019. Later, the World Health Organization named it Coronavirus Disease 2019 (COVID-19). After the discovery of COVID-19, it spread in more than 200 countries. COVID-19 has zoonotic basis, which was then spread through the human interaction to human population [23]. Common signs of COVID-19 infection are similar to those of common cold and include respiratory symptoms such as dry cough, fever, shortness of breath, and breathing difficulties. Initially its etiology was unknown. Later on, it was studied thoroughly and found that it has an incubation period of 14 days, during which some individuals show all the symptoms while others show mild symptoms. It is sensitive to know that someone have the disease due to the dual nature (same as common flu) of COVID-19 symptoms [24]. In this section, we use the TOPSIS method to rank the COVID-19 in four provinces of Pakistan. A numerical example which is solved using the TOPSIS method is presented to demonstrate the applicability and effectiveness of the proposed method.


### 5.2. Example

Let us consider the decision making problem. Suppose that there is a panel and they selected four possible alternativesH1,H2,H3,H4 to find out the spreading of COVID-19 in provinces of Pakistan: H1is KPK, H2is Sindh, H3is Punjab, and H4is Balochistan. A group of doctors intends to choose one province be the most affected area from four provinces, to be further evaluated according to the four attributes, which are shown as G1 effected people, G2 recovered people, G3 admitted people, and G4 number of deaths. By this method, we can find out which province is more affected. Then, we must take some action to stop the cases in that province. The experts give them advice for quarantine. Also, they suggest them treatment and say that the treatment will be continued until the transmission of virus stops. By using the stable neutrosophic cubic information, the alternatives are evaluated by the decision maker and the results are presented in the decision matrix.The decided steps of the TOPSIS method are presented as follows:Step 1(a)
The decision makers take their analysis of each alternatives based on each criterion and the performance of each alternativeHi with respect to each criterion GjTex translation failed.(58)D=G1G2G3G4H10.1,0.4,0.2,0.6,0.1,0.4,0.2,0.5,0.20.1,0.3,0.1,0.3,0.1,0.3,0.2,0.2,0.20.1,0.4,0.1,0.4,0.1,0.4,0.2,0.2,0.20.1,0.4,0.1,0.3,0.1,0.4,0.2,0.2,0.2H20.1,0.3,0.1,0.4,0.2,0.5,0.2,0.2,0.30.1,0.4,0.1,0.6,0.1,0.4,0.3,0.4,0.20.1,0.4,0.1,0.4,0.1,0.4,0.2,0.2,0.30.1,0.3,0.2,0.6,0.1,0.4,0.2,0.4,0.3H30.2,0.5,0.2,0.5,0.1,0.4,0.3,0.3,0.30.1,0.4,0.1,0.3,0.2,0.6,0.3,0.2,0.40.3,0.6,0.3,0.6,0.1,0.5,0.4,0.4,0.30.1,0.4,0.1,0.4,0.2,0.6,0.2,0.3,0.4H40.1,0.4,0.1,0.4,0.1,0.4,0.2,0.2,0.20.1,0.4,0.3,0.6,0.1,0.5,0.3,0.4,0.40.1,0.3,0.1,0.5,0.1,0.3,0.2,0.3,0.20.1,0.4,0.2,0.4,0.3,0.6,0.3,0.3,0.4(b)
Then, the decision makers present their analysis in the form of a stable neutrosophic cubic set, according to Definitions6 and 7 and Example 3:(59)D=G1G2G3G4H10.1,0.2,0.3,0.1,0.1,0.20.1,0.1,0.1,0.1,0.1,0.10.1,0.2,0.1,0.2,0.1,0.20.1,0.2,0.1,0.1,0.1,0.2H20.1,0.1,0.1,0.2,0.1,0.30.2,0.1,0.3,0.2,0.1,0.20.1,0.2,0.1,0.2,0.2,0.10.1,0.1,0.2,0.2,0.2,0.1H30.1,0.2,0.1,0.2,0.2,0.10.1,0.2,0.1,0.1,0.2,0.20.1,0.2,0.1,0.2,0.2,0.20.1,0.2,0.2,0.1,0.2,0.2H40.1,0.2,0.1,0.2,0.1,0.20.2,0.1,0.1,0.2,0.3,0.10.1,0.1,0.2,0.2,0.1,0.10.2,0.1,0.1,0.1,0.1,0.2.Step 2. The normalized decision matrix is(60)G1G2G3G4H10.25,0.29,0.5,0.143,0.2,0.250.17,0.1,0.17,0.17,0.143,0.170.25,0.29,0.2,0.25,0.17,0.330.20,0.33,0.17,0.20,0.17,0.29H20.25,0.143,0.17,0.29,0.2,0.380.33,0.1,0.5,0.33,0.143,0.330.25,0.29,0.2,0.25,0.33,0.170.20,0.17,0.33,0.4,0.33,0.143H30.25,0.29,0.17,0.29,0.5,0.1250.17,0.2,0.17,0.17,0.29,0.330.25,0.29,0.2,0.25,0.33,0.330.20,0.33,0.33,0.20,0.33,0.29H40.25,0.29,0.17,0.29,0.2,0.250.33,0.1,0.17,0.33,0.43,0.170.25,0.143,0.4,0.25,0.17,0.170.40,0.17,0.17,0.20,0.17,0.29.Step 3. The weighted normalized decision matrix where w=0.3,0.1,0.2,0.4 is(61)G1G2G3G4H10.075,0.087,0.15,0.043,0.06,0.0750.017,0.01,0.017,0.017,0.0143,0.0170.05,0.06,0.04,0.05,0.034,0.0660.08,0.132,0.07,0.08,0.07,0.12H20.075,0.043,0.051,0.087,0.06,0.1140.033,0.01,0.05,0.033,0.0143,0.0330.05,0.06,0.04,0.05,0.066,0.0340.08,0.07,0.132,0.16,0.132,0.06H30.075,0.087,0.051,0.087,0.15,0.0380.017,0.02,0.017,0.017,0.029,0.0330.05,0.06,0.04,0.05,0.066,0.0660.08,0.132,0.132,0.08,0.132,0.12H40.075,0.087,0.051,0.087,0.06,0.0750.033,0.01,0.017,0.033,0.043,0.0170.05,0.143,0.08,0.05,0.034,0.0340.40,0.07,0.07,0.08,0.07,0.12.Step 4. Positive and negative ideal solution: the positive ideal solution A∗=a1,a2,a3,a4 contains the greatest numbers of the first, second, and third column and smallest numbers of the fourth column. The negative ideal solution A′=a1′,a2′,a3′,a4′ contains the smallest numbers of the first, second, and third column and greatest numbers of the fourth column.(62)A∗0.075,0.087,0.15,0.087,0.15,0.1140.033,0.02,0.05,0.033,0.029,0.0330.05,0.143,0.08,0.05,0.066,0.0660.08,0.07,0.07,0.07,0.07,0.06,A′0.075,0.043,0.051,0.043,0.06,0.0380.017,0.01,0.017,0.017,0.0143,0.0170.06,0.05,0.04,0.05,0.034,0.0340.40,0.132,0.132,0.16,0.132,0.12.Step 5. Separation measures for the positive and negative ideal solution are(63)a1∗=0.3694,a2∗=0.2133,a3∗=0.0409,a4∗=0.1292,a1′=0.1308,a2′=0.1206,a3′=0.1236,a4′=0.0349.Step 6. Ranking order of the alternatives is shown by (Figures 1–4). Ranking of COVID-19 is obtained by completing the TOPSIS calculation.(64)H1=0.2615,H2=0.3612,H3=0.7514,H4=0.2127,H3>H2>H1>H4.Figure 1
Total COVID-19 confirmed cases in Punjab till 12 Apr 2020.Figure 2
Total COVID-19 confirmed cases in Sindh till 12 Apr 2020.Figure 3
Total COVID-19 confirmed cases in KPK till 12 Apr 2020.Figure 4
Total COVID-19 confirmed cases in Balochistan till 12 Apr 2020.Thus, we concluded thatH3is the most effected province of Pakistan till April 12, 2020. Here, we used stable neutrosophic cubic sets, but we may use other versions of stable neutrosophic cubic sets.


## 5.1. Numerical Application

At the end of December 2019 [22], in Wuhan, the China Health Commission reported a cluster of pneumonia cases of unknown etiology. The pathogen was identified as novel coronavirus 2019. Later, the World Health Organization named it Coronavirus Disease 2019 (COVID-19). After the discovery of COVID-19, it spread in more than 200 countries. COVID-19 has zoonotic basis, which was then spread through the human interaction to human population [23]. Common signs of COVID-19 infection are similar to those of common cold and include respiratory symptoms such as dry cough, fever, shortness of breath, and breathing difficulties. Initially its etiology was unknown. Later on, it was studied thoroughly and found that it has an incubation period of 14 days, during which some individuals show all the symptoms while others show mild symptoms. It is sensitive to know that someone have the disease due to the dual nature (same as common flu) of COVID-19 symptoms [24]. In this section, we use the TOPSIS method to rank the COVID-19 in four provinces of Pakistan. A numerical example which is solved using the TOPSIS method is presented to demonstrate the applicability and effectiveness of the proposed method.


## 5.2. Example

Let us consider the decision making problem. Suppose that there is a panel and they selected four possible alternativesH1,H2,H3,H4 to find out the spreading of COVID-19 in provinces of Pakistan: H1is KPK, H2is Sindh, H3is Punjab, and H4is Balochistan. A group of doctors intends to choose one province be the most affected area from four provinces, to be further evaluated according to the four attributes, which are shown as G1 effected people, G2 recovered people, G3 admitted people, and G4 number of deaths. By this method, we can find out which province is more affected. Then, we must take some action to stop the cases in that province. The experts give them advice for quarantine. Also, they suggest them treatment and say that the treatment will be continued until the transmission of virus stops. By using the stable neutrosophic cubic information, the alternatives are evaluated by the decision maker and the results are presented in the decision matrix.The decided steps of the TOPSIS method are presented as follows:Step 1(a)
The decision makers take their analysis of each alternatives based on each criterion and the performance of each alternativeHi with respect to each criterion GjTex translation failed.(58)D=G1G2G3G4H10.1,0.4,0.2,0.6,0.1,0.4,0.2,0.5,0.20.1,0.3,0.1,0.3,0.1,0.3,0.2,0.2,0.20.1,0.4,0.1,0.4,0.1,0.4,0.2,0.2,0.20.1,0.4,0.1,0.3,0.1,0.4,0.2,0.2,0.2H20.1,0.3,0.1,0.4,0.2,0.5,0.2,0.2,0.30.1,0.4,0.1,0.6,0.1,0.4,0.3,0.4,0.20.1,0.4,0.1,0.4,0.1,0.4,0.2,0.2,0.30.1,0.3,0.2,0.6,0.1,0.4,0.2,0.4,0.3H30.2,0.5,0.2,0.5,0.1,0.4,0.3,0.3,0.30.1,0.4,0.1,0.3,0.2,0.6,0.3,0.2,0.40.3,0.6,0.3,0.6,0.1,0.5,0.4,0.4,0.30.1,0.4,0.1,0.4,0.2,0.6,0.2,0.3,0.4H40.1,0.4,0.1,0.4,0.1,0.4,0.2,0.2,0.20.1,0.4,0.3,0.6,0.1,0.5,0.3,0.4,0.40.1,0.3,0.1,0.5,0.1,0.3,0.2,0.3,0.20.1,0.4,0.2,0.4,0.3,0.6,0.3,0.3,0.4(b)
Then, the decision makers present their analysis in the form of a stable neutrosophic cubic set, according to Definitions6 and 7 and Example 3:(59)D=G1G2G3G4H10.1,0.2,0.3,0.1,0.1,0.20.1,0.1,0.1,0.1,0.1,0.10.1,0.2,0.1,0.2,0.1,0.20.1,0.2,0.1,0.1,0.1,0.2H20.1,0.1,0.1,0.2,0.1,0.30.2,0.1,0.3,0.2,0.1,0.20.1,0.2,0.1,0.2,0.2,0.10.1,0.1,0.2,0.2,0.2,0.1H30.1,0.2,0.1,0.2,0.2,0.10.1,0.2,0.1,0.1,0.2,0.20.1,0.2,0.1,0.2,0.2,0.20.1,0.2,0.2,0.1,0.2,0.2H40.1,0.2,0.1,0.2,0.1,0.20.2,0.1,0.1,0.2,0.3,0.10.1,0.1,0.2,0.2,0.1,0.10.2,0.1,0.1,0.1,0.1,0.2.Step 2. The normalized decision matrix is(60)G1G2G3G4H10.25,0.29,0.5,0.143,0.2,0.250.17,0.1,0.17,0.17,0.143,0.170.25,0.29,0.2,0.25,0.17,0.330.20,0.33,0.17,0.20,0.17,0.29H20.25,0.143,0.17,0.29,0.2,0.380.33,0.1,0.5,0.33,0.143,0.330.25,0.29,0.2,0.25,0.33,0.170.20,0.17,0.33,0.4,0.33,0.143H30.25,0.29,0.17,0.29,0.5,0.1250.17,0.2,0.17,0.17,0.29,0.330.25,0.29,0.2,0.25,0.33,0.330.20,0.33,0.33,0.20,0.33,0.29H40.25,0.29,0.17,0.29,0.2,0.250.33,0.1,0.17,0.33,0.43,0.170.25,0.143,0.4,0.25,0.17,0.170.40,0.17,0.17,0.20,0.17,0.29.Step 3. The weighted normalized decision matrix where w=0.3,0.1,0.2,0.4 is(61)G1G2G3G4H10.075,0.087,0.15,0.043,0.06,0.0750.017,0.01,0.017,0.017,0.0143,0.0170.05,0.06,0.04,0.05,0.034,0.0660.08,0.132,0.07,0.08,0.07,0.12H20.075,0.043,0.051,0.087,0.06,0.1140.033,0.01,0.05,0.033,0.0143,0.0330.05,0.06,0.04,0.05,0.066,0.0340.08,0.07,0.132,0.16,0.132,0.06H30.075,0.087,0.051,0.087,0.15,0.0380.017,0.02,0.017,0.017,0.029,0.0330.05,0.06,0.04,0.05,0.066,0.0660.08,0.132,0.132,0.08,0.132,0.12H40.075,0.087,0.051,0.087,0.06,0.0750.033,0.01,0.017,0.033,0.043,0.0170.05,0.143,0.08,0.05,0.034,0.0340.40,0.07,0.07,0.08,0.07,0.12.Step 4. Positive and negative ideal solution: the positive ideal solution A∗=a1,a2,a3,a4 contains the greatest numbers of the first, second, and third column and smallest numbers of the fourth column. The negative ideal solution A′=a1′,a2′,a3′,a4′ contains the smallest numbers of the first, second, and third column and greatest numbers of the fourth column.(62)A∗0.075,0.087,0.15,0.087,0.15,0.1140.033,0.02,0.05,0.033,0.029,0.0330.05,0.143,0.08,0.05,0.066,0.0660.08,0.07,0.07,0.07,0.07,0.06,A′0.075,0.043,0.051,0.043,0.06,0.0380.017,0.01,0.017,0.017,0.0143,0.0170.06,0.05,0.04,0.05,0.034,0.0340.40,0.132,0.132,0.16,0.132,0.12.Step 5. Separation measures for the positive and negative ideal solution are(63)a1∗=0.3694,a2∗=0.2133,a3∗=0.0409,a4∗=0.1292,a1′=0.1308,a2′=0.1206,a3′=0.1236,a4′=0.0349.Step 6. Ranking order of the alternatives is shown by (Figures 1–4). Ranking of COVID-19 is obtained by completing the TOPSIS calculation.(64)H1=0.2615,H2=0.3612,H3=0.7514,H4=0.2127,H3>H2>H1>H4.Figure 1
Total COVID-19 confirmed cases in Punjab till 12 Apr 2020.Figure 2
Total COVID-19 confirmed cases in Sindh till 12 Apr 2020.Figure 3
Total COVID-19 confirmed cases in KPK till 12 Apr 2020.Figure 4
Total COVID-19 confirmed cases in Balochistan till 12 Apr 2020.Thus, we concluded thatH3is the most effected province of Pakistan till April 12, 2020. Here, we used stable neutrosophic cubic sets, but we may use other versions of stable neutrosophic cubic sets.


## 6. Conclusions

In this article, we work out with the idea of stable NCSs and internal and external stable NCSs. Also, we define their union, intersection, and complement with examples. After that, we demonstrate the application of the TOPSIS method to find out the ranking of COVID-19. For this purpose, we used a numerical example to find out the most affected area. We reached at the following key points:Every stable NCSβ=Tβ,Iβ,Fβ,tβ,iβ,fβ in U is an almost-stable NCS, which is, of course, an NCS which turns into a cubic set with three different parts as truth, indeterminacy, and falsity, but the converse of this chain is not true always.If we have an external NCS which is unstable such that(65)tu∘>T−u∘,T+u∘,iu∘>I−u∘,I+u∘,fu∘>F−u∘,F+u∘,then its right evaluative point becomes a neutrosophic bipolar fuzzy set.If we have an external NCS which is unstable such that(66)tu∘<T−u∘,T+u∘,iu∘<I−u∘,I+u∘,fu∘<F−u∘,F+u∘,then its left evaluative point becomes a neutrosophic bipolar fuzzy set.We used the idea of stable neutrosophic cubic sets in the application section, so results are within the range; otherwise, we may have results which lie outside the domain of neutrosophic cubic sets. This is the main advantage of stable neutrosophic cubic sets.



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*Source: 8835019-2020-12-01.xml*