# QOE Based Heterogeneous Network Selection Using Pythagorean Fuzzy Methods

Authors: Lei Zhu; Kaixin Cheng; Lei Wang; Wei Tong
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Considering the diversity of demands in heterogeneous network selection problem, we combine Choquet integral with traditional aggregation operators in Pythagorean fuzzy environment, inventing kinds of new operators based on Choquet integral, including PFCOWA (as well as RP-PFOWA) operator, PFCWA operator, PFCOWG (as well as RP-PFOWG) operator, and PFCWG operator, which are capable of solving different attributes relationship prioritized situations. Then, transferring relationship and properties of the operators are further discussed. Additionally, to overcome the weakness of traditional method of evaluation, a novel evaluating system of QOE in fuzzy environment is proposed. Based on these works, an example of heterogeneous network selection is given to illustrate the validity of this method.


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## Body

## 1. Introduction

Pythagorean fuzzy set (PFS) is an extension of intuitionistic fuzzy set (IFS) [1–3] proposed by Yager [4–6]. Intuitionistic fuzzy set [1] describes the imprecise information using two indicators: membership and nonmembership degree. When an expert judges if an alternative is up to standard, the answer can be hard to say by merely yes or no. By using IFS, he can give a membership degree which between 0 and 1 to describe the degree conforms to the standard and vice versa. An important condition is that the total of two membership indicators should not be exceeding one. However, this condition limits the applied range of IFS. Therefore, Yager puts forward PFS to work out the situation when the accumulating of two indicators value is exceeding one while the square sum of them is less than one. Obviously, IFS is the special case of PFS, where the latter one can be more widely used when describing the imprecise and vague information. For the range of membership degree is extended, PFS can express subtler uncertainties than IFS. Since PFS has been developed, the study on it never stopped [7–12].Zhang and Xu [7] defined some operation and the score function of PFSs. Yager [4, 6] proposed some basic aggregation operators including PFWA operator and PFWG operator. Later, various types of operators [8] are developed in Pythagorean environment, including the PFOWA operator, the GPFOWA operator, the PFOWG operator, and the GPFOWG operator. Most of these operators have been used on solving MADM problems. Though many Pythagorean fuzzy aggregation operators had been developed, most of them only consider the elements in fuzzy sets as independent, ignoring the situation when interaction exists between attributes.However, except single attribute performance, correlation between those attributes is more important in some circumstances. Ji and Wang [13] established a new method for selecting TPL providers under single-valued neutrosophic environments, which simultaneously takes into account the interrelationships among criteria. The single-valued neutrosophic hesitant fuzzy geometric weighted Choquet integral Heronian mean operator [14], which is based on the Heronian mean and Choquet integral, is proposed.The application of Choquet integral in fuzzy operators [15–20] is not an emerging idea. The Choquet integral [21] can be used to illustrate the interaction of considered attributes, which is an available tool to measure the intended effect of indefinite event. Yager [15, 16] introduced an inducing factor into Choquet aggregating process, getting a bright new operator: I-COA operator. In intuitionistic fuzzy environment, Xu [17] expended some information aggregating methods by means of the Choquet integral. Tan and Chen [19, 20] strengthened the preference relations of decision-makers by using the I-COA operator in group decision-making. Moreover, Xu and Xia [18] applied Choquet integral to broaden applied scene of former operators. From previous study we can see the feasibility and necessity of transporting this method to Pythagorean environment. So we introduce fuzzy measure into Pythagorean fuzzy operators, acquiring the relationship prioritized Pythagorean aggregation fuzzy operators [22], while fuzzy measure is the basic of Choquet integral.How to evaluate the subjective standard is a tough work. Usually, attributes put into consideration include throughput, delay, jitter, and cost. Those are decisive factors of network QOE (quality of experience), which has arouse the attention of the academic community as well as becoming one of the major evaluation indicators on industry. The concept of QOE is born out of QOS (quality of service), but it focuses on the subjective feelings of users, which meets the need of humanization. Reference [23] introduces the existing QOE models and evaluation methods, and different disciplines have their own emphasis on the structure of the evaluation system. Therefore, this paper proposes a brand-new QOE evaluation system based on Pythagorean fuzzy numbers combined with the heterogeneous network environment and the characteristics of services.This paper further proposes some Pythagorean fuzzy operators, including PFCOWA operator, PFCWA operator, and PFCWG operator, based on our previous work, and unifies their designations and properties, which are capable of solving different attributes relationship prioritized situations. Then, transferring relationship and properties of the operators are further discussed. Additionally, a novel objective evaluating system of QOE in fuzzy environment is proposed. Ultimately, based on the proposed evaluation system, different operators are applied in aggregating information for MADM problems on network selection.


## 2. Basic Knowledge Review

Before formally introducing newly developed operators, it is necessary to recall some fundamental definition and properties.


### 2.1. Pythagorean Fuzzy Numbers

Definition 1 (see [4–7]).
β = P ( μ β , ν β ) is Pythagorean fuzzy number in which μp, νp separately represents the membership degree and the nonmembership degree. μβ,νβ∈[0,1], πβ=1-(μβ)2-(νβ)2 is the degree of uncertainty, and (μβ)2+(νβ)2≤1.Zhang and Xu [7] defined the following operations.Definition 2 (see [7]).
For three PFNsβ=P(μβ,νβ), β1=P(μβ1,νβ1), β2=P(μβ2,νβ2), they have the operations rules as follows:(1)
β1⊕β2=P(μβ12+μβ22-μβ12μβ22,νβ1νβ2).(2)
β1⊗β2=P(μβ1μβ2,νβ12+νβ22-νβ12νβ22)(3)
λβ=P(1-(1-μβ2)λ,(νβ)λ),λ>0.(4)
βλ=P((μβ)λ,1-(1-νβ2)λ),λ>0.
Obviously, the results of operations are still PFNs. They have the following properties:(1)
β1⊕β2=β2⊕β1.(2)
β1⊗β2=β2⊗β1.(3)
λ(β1⊕β2)=λβ1⊕λβ2,λ>0.(4)
(β1⊗β2)λ=β1λ⊗β2λ,λ>0.(5)
(λ1+λ2)β=λ1β⊕λ2β,λ1,λ2>0.(6)
βλ1+λ2=βλ1⊗βλ2,λ1,λ2>0.


### 2.2. Fuzzy Measure

To apply Choquet integral into Pythagorean fuzzy operators, the concept of fuzzy measure [24–26] should be firstly elaborated.Definition 3 (see [24–26]).
Fuzzy measureFm on finite space S is a mapping Fm:ϑ[S]→[0,1], which meets the following conditions:(1)
Fm[⌀]=0,Fm[S]=1.(2)
For allA,B⊆S and A⊆B, Fm(A)≤Fm(B).(3)
Fm(A∪B)=Fm(A)+Fm(B)+ρFm(A)Fm(B), for all A,B⊆S, and A∩B=⌀, ρ∈(-1,∞).Parameterρ has its meaning as follows:ρ = 0 signifies that the attributes under consideration are dependent. ρ>0 shows there is mutual promotion between the attributes, while ρ<0 means the attributes weaken each other when being aggregated.For this reason, this mathematical tool can be used to illustrate relationship between several attributes for multiattribute decision-making problems. The details for how to determine value of the parameters have been discussed in [22]. The application of fuzzy measure as well as Choquet integral in aggregation operators has been proved valid, which brings out a relationship preference strategy distinguished with PROMETHEE. To acquire properties of the operator and further develop some other equipment for different multiattributes problems, we redefine a kind of aggregation operator: Pythagorean fuzzy operators based on Choquet integral. The relationship prioritized operators belong to this category. Besides, it also contains some promotion form operators with more integration traits. Next, we will introduce them in detail.


## 2.1. Pythagorean Fuzzy Numbers

Definition 1 (see [4–7]).
β = P ( μ β , ν β ) is Pythagorean fuzzy number in which μp, νp separately represents the membership degree and the nonmembership degree. μβ,νβ∈[0,1], πβ=1-(μβ)2-(νβ)2 is the degree of uncertainty, and (μβ)2+(νβ)2≤1.Zhang and Xu [7] defined the following operations.Definition 2 (see [7]).
For three PFNsβ=P(μβ,νβ), β1=P(μβ1,νβ1), β2=P(μβ2,νβ2), they have the operations rules as follows:(1)
β1⊕β2=P(μβ12+μβ22-μβ12μβ22,νβ1νβ2).(2)
β1⊗β2=P(μβ1μβ2,νβ12+νβ22-νβ12νβ22)(3)
λβ=P(1-(1-μβ2)λ,(νβ)λ),λ>0.(4)
βλ=P((μβ)λ,1-(1-νβ2)λ),λ>0.
Obviously, the results of operations are still PFNs. They have the following properties:(1)
β1⊕β2=β2⊕β1.(2)
β1⊗β2=β2⊗β1.(3)
λ(β1⊕β2)=λβ1⊕λβ2,λ>0.(4)
(β1⊗β2)λ=β1λ⊗β2λ,λ>0.(5)
(λ1+λ2)β=λ1β⊕λ2β,λ1,λ2>0.(6)
βλ1+λ2=βλ1⊗βλ2,λ1,λ2>0.


## 2.2. Fuzzy Measure

To apply Choquet integral into Pythagorean fuzzy operators, the concept of fuzzy measure [24–26] should be firstly elaborated.Definition 3 (see [24–26]).
Fuzzy measureFm on finite space S is a mapping Fm:ϑ[S]→[0,1], which meets the following conditions:(1)
Fm[⌀]=0,Fm[S]=1.(2)
For allA,B⊆S and A⊆B, Fm(A)≤Fm(B).(3)
Fm(A∪B)=Fm(A)+Fm(B)+ρFm(A)Fm(B), for all A,B⊆S, and A∩B=⌀, ρ∈(-1,∞).Parameterρ has its meaning as follows:ρ = 0 signifies that the attributes under consideration are dependent. ρ>0 shows there is mutual promotion between the attributes, while ρ<0 means the attributes weaken each other when being aggregated.For this reason, this mathematical tool can be used to illustrate relationship between several attributes for multiattribute decision-making problems. The details for how to determine value of the parameters have been discussed in [22]. The application of fuzzy measure as well as Choquet integral in aggregation operators has been proved valid, which brings out a relationship preference strategy distinguished with PROMETHEE. To acquire properties of the operator and further develop some other equipment for different multiattributes problems, we redefine a kind of aggregation operator: Pythagorean fuzzy operators based on Choquet integral. The relationship prioritized operators belong to this category. Besides, it also contains some promotion form operators with more integration traits. Next, we will introduce them in detail.


## 3. Pythagorean Fuzzy Choquet Operators

In this chapter, we utilize the Choquet integral into Pythagorean fuzzy environment, developing a series of neoteric operators. The properties of these operators and the relationship between them are further discussed. Meanwhile, according to different aggregation attributes of these operators, we analyze the using scenes of them, which are their applied environment.


### 3.1. Pythagorean Fuzzy Choquet Averaging (PFCA) Operators

Definition 4.
Pythagorean fuzzy Choquet order weighted averaging (PFCOWA) operator. [22]
β i are PFNs, S:βi=P(μβi,νβi), i=1,2,…,n. si is the ith largest in βi, and set Li as a subset of S:(1)L0=⌀Li=∑k=1isk.The Pythagorean fuzzy Choquet ordered weighted averaging operator is defined as(2)PFCOWAβ1,β2,…,βn=⨁i=1nFmLi-FmLi-1si,Fm is the fuzzy measure on S. According to the operational rules, and the PFCOWA operator can be expressed as below:(3)PFCOWAβ1,β2,…,βn=⨁i=1nFmLi-FmLi-1si=P1-∏i=1n1-μsi2FmLi-FmLi-1,∏i=1nνsiFmLi-FmLi-1.The result is still PFN. And the operator has the following properties.(  1) Commutativity. PFCOWA(β1,β2,…,βn)=PFCOWA(β1,β2,…,βn),(β1,β2,…,βn) is an arbitrary sort of (β1,β2,…,βn).Proof.
This conclusion is easy to prove, for the operator will reorder the set by its value, and the result is the same for one fixed set no matter what the original order is.(  2) Boundedness. Set βmin as the smallest value in (β1,β2,…,βn), while βmax is the largest. We will have the property that(4)βmin≤PFCOWAβ1,β2,…,βn≤βmax.Proof.
(5)PFCOWAβ1,β2,…,βn=⨁i=1n FmLi-FmLi-1si≤⨁i=1n FmLi-FmLi-1βmax, For  ⨁i=1n FmLi-FmLi-1=1,PFCOWAβ1,β2,…,βn≤βmax.
Similarly,βmin≤PFCOWA(β1,β2,…,βn), and the conclusion is successfully proved.(  3) Monotonicity. The PFCOWA operator is monotonically increasing based on value of argument value. If βi≤βi′ is true for each i=1,2,…,n, then(6)PFCOWAβ1,β2,…,βn≤PFCOWAβ1′,β2′,…,βn′.Proof.
Take the derivative of the operator with respect to argument valueβ.(7)∂PFCOWAβ1,β2,…,βn∂βi=FmLi-FmLi-1≥0.
Apparently, the resultFm(Li)-Fm(Li-1) is constantly greater than or equal to zero according to the definition. Therefore, the conclusion is proved.(  4) Idempotency. If βi=β0 is true for each i=1,2,…,n, then(8)PFCOWAβ1,β2,…,βn=β0.This property can be easily proved by the definition and operation rules.Definition 5.
Pythagorean fuzzy Choquet weighted averaging (PFCWA) operator.
S is a collection of PFNs, S:βi=P(μβi,νβi), where i=1,2,…,n. Fm is the fuzzy measure on S, and set Li′ as a subset of S. It meets the following regulations:(9)L0′=⌀Li′=∑k=1iβk.The PFCWA operator is(10)PFCWAβ1,β2,…,βn=P∑i=1nFmLi′-FmLi-1′μβi,∑i=1nFmLi′-FmLi-1′νβi.Furthermore, we take some special occasions into consideration.(1)
WhenFm(S)=∑si∈SFm({si}), which means elements in the set are independent, then, the weighted vector is ωi=Fm(Li)-Fm(Li-1)=Fm(si) and ∑i=1nωi=1, si is the ith largest in βi, and formulas (2) and (3) reduce to(11)PFCOWAβ1,β2,…,βn=⨁i=1nFmsisi=⨁i=1nωisi=P1-∏i=1n1-μsi2ωi,∏i=1nνsiωi=PFOWAβ1,β2,…,βn.The PFCOWA operator is transformed to PFOWA [8]. Especially, when ωi=Fm(si)=1/n, the operator continues to be simplified to PFA operator.(2)
WhenFm(S)=∑si∈SFm({si}), ωi=Fm(Li′)-Fm(Li-1′)=Fm(βi), then, formula (10) reduces to(12)PFCWAβ1,β2,…,βn=P∑i=1nωiμβi,∑i=1nωiνβi=PFWAβ1,β2,…,βn.The PFCWA operator becomes PFWA [8]. Especially, when ωi=1/n, the operator continues to be simplified to PFA operator.Properties of PFCWA operator are quite similar to that of PFCWOA operator except commutativity. Proving process is in accordance with the former operator.(  1)  Boundedness.Set βmin as the smallest value in (β1,β2,…,βn), while βmax is the largest. We will have the property that(13)βmin≤PFCWAβ1,β2,…,βn≤βmax.(  2) Monotonicity. The PFCWA operator is monotonically increasing based on value of argument value. If βi≤βi′, (i=1,2,…,n), then(14)PFCWAβ1,β2,…,βn≤PFCOWAβ1′,β2′,…,βn′.(  3) Idempotency. When βi=β0, for every i=1,2,…,n, then(15)PFCWAβ1,β2,…,βn=β0.


### 3.2. Pythagorean Fuzzy Choquet Geometric (PFCG) Operators

Definition 6.
Pythagorean fuzzy Choquet order weighted geometric (PFCOWG) operator.
S is a collection of PFNs, S:{βi=P(μβi,νβi)}, where i=1,2,…,n. si has the ith largest value in βi, and set Li as a subset of S; it meets the following requirements:(16)L0=⌀Li=∑k=1isk.The Pythagorean fuzzy Choquet order weighted geometric operator is defined as follows:(17)PFCOWGβ1,β2,…,βn=⨂i=1nsiFmLi-FmLi-1,Fm is the fuzzy measure on S; according to the operations of PFNs in Definition 2(3), the PFCOWA operator has form as below:(18)PFCOWGβ1,β2,…,βn=⨂i=1nxiFmLi-FmLi-1=P∏i=1nμsiFmLi-FmLi-1,1-∏i=1n1-νsi2FmLi-FmLi-1.The result of aggregation is still PFNs.Definition 7.
Pythagorean fuzzy Choquet weighted geometric (PFCWG) operator.
S is a collection of PFNs, S:βi=P(μβi,νβi), where i=1,2,…,n. Fm is the fuzzy measure on S, and set Li′ as a subset of S; it meets the following regulations:(19)L0′=⌀Li′=∑k=1iβkThe PFCWG operator is(20)PFCWGβ1,β2,…,βn=P∏i=1nμβiFmLi′-FmLi-1′,∏i=1nνβiFmLi′-FmLi-1′.Also, we take some special occasions into consideration:(1)
When it meets formula (17), which is Fm(S)=∑si∈SFm({si}), then, the weighted vector ωi=Fm(Li)-Fm(Li-1)=Fm(si) and ∑i=1nωi=1, si has the ith largest value in βi, and formulas (17) and (18) reduce to(21)PFCOWGβ1,β2,…,βn=⨂i=1nsiFmsi=⨂i=1nsiωi=P∏i=1nμsiωi,1-∏i=1n1-νsi2ωi=PFOWGβ1,β2,…,βn.The PFCOWG operator is transformed to PFOWG [8]. Especially, when ωi=Fm(si)=1/n, the operator continues to be simplified to PFOWG operator.(2)
When it meets formula (17), Fm(S)=∑si∈SFm({si}), ωi=Fm(Li′)-Fm(Li-1′)=Fm(si), and, then, formula (20) reduces to(22)PFCWGβ1,β2,…,βn=P⨂i=1nμβiωi,⨂i=1nνβiωi.The PFCWG operator becomes PFWG [8].


## 3.1. Pythagorean Fuzzy Choquet Averaging (PFCA) Operators

Definition 4.
Pythagorean fuzzy Choquet order weighted averaging (PFCOWA) operator. [22]
β i are PFNs, S:βi=P(μβi,νβi), i=1,2,…,n. si is the ith largest in βi, and set Li as a subset of S:(1)L0=⌀Li=∑k=1isk.The Pythagorean fuzzy Choquet ordered weighted averaging operator is defined as(2)PFCOWAβ1,β2,…,βn=⨁i=1nFmLi-FmLi-1si,Fm is the fuzzy measure on S. According to the operational rules, and the PFCOWA operator can be expressed as below:(3)PFCOWAβ1,β2,…,βn=⨁i=1nFmLi-FmLi-1si=P1-∏i=1n1-μsi2FmLi-FmLi-1,∏i=1nνsiFmLi-FmLi-1.The result is still PFN. And the operator has the following properties.(  1) Commutativity. PFCOWA(β1,β2,…,βn)=PFCOWA(β1,β2,…,βn),(β1,β2,…,βn) is an arbitrary sort of (β1,β2,…,βn).Proof.
This conclusion is easy to prove, for the operator will reorder the set by its value, and the result is the same for one fixed set no matter what the original order is.(  2) Boundedness. Set βmin as the smallest value in (β1,β2,…,βn), while βmax is the largest. We will have the property that(4)βmin≤PFCOWAβ1,β2,…,βn≤βmax.Proof.
(5)PFCOWAβ1,β2,…,βn=⨁i=1n FmLi-FmLi-1si≤⨁i=1n FmLi-FmLi-1βmax, For  ⨁i=1n FmLi-FmLi-1=1,PFCOWAβ1,β2,…,βn≤βmax.
Similarly,βmin≤PFCOWA(β1,β2,…,βn), and the conclusion is successfully proved.(  3) Monotonicity. The PFCOWA operator is monotonically increasing based on value of argument value. If βi≤βi′ is true for each i=1,2,…,n, then(6)PFCOWAβ1,β2,…,βn≤PFCOWAβ1′,β2′,…,βn′.Proof.
Take the derivative of the operator with respect to argument valueβ.(7)∂PFCOWAβ1,β2,…,βn∂βi=FmLi-FmLi-1≥0.
Apparently, the resultFm(Li)-Fm(Li-1) is constantly greater than or equal to zero according to the definition. Therefore, the conclusion is proved.(  4) Idempotency. If βi=β0 is true for each i=1,2,…,n, then(8)PFCOWAβ1,β2,…,βn=β0.This property can be easily proved by the definition and operation rules.Definition 5.
Pythagorean fuzzy Choquet weighted averaging (PFCWA) operator.
S is a collection of PFNs, S:βi=P(μβi,νβi), where i=1,2,…,n. Fm is the fuzzy measure on S, and set Li′ as a subset of S. It meets the following regulations:(9)L0′=⌀Li′=∑k=1iβk.The PFCWA operator is(10)PFCWAβ1,β2,…,βn=P∑i=1nFmLi′-FmLi-1′μβi,∑i=1nFmLi′-FmLi-1′νβi.Furthermore, we take some special occasions into consideration.(1)
WhenFm(S)=∑si∈SFm({si}), which means elements in the set are independent, then, the weighted vector is ωi=Fm(Li)-Fm(Li-1)=Fm(si) and ∑i=1nωi=1, si is the ith largest in βi, and formulas (2) and (3) reduce to(11)PFCOWAβ1,β2,…,βn=⨁i=1nFmsisi=⨁i=1nωisi=P1-∏i=1n1-μsi2ωi,∏i=1nνsiωi=PFOWAβ1,β2,…,βn.The PFCOWA operator is transformed to PFOWA [8]. Especially, when ωi=Fm(si)=1/n, the operator continues to be simplified to PFA operator.(2)
WhenFm(S)=∑si∈SFm({si}), ωi=Fm(Li′)-Fm(Li-1′)=Fm(βi), then, formula (10) reduces to(12)PFCWAβ1,β2,…,βn=P∑i=1nωiμβi,∑i=1nωiνβi=PFWAβ1,β2,…,βn.The PFCWA operator becomes PFWA [8]. Especially, when ωi=1/n, the operator continues to be simplified to PFA operator.Properties of PFCWA operator are quite similar to that of PFCWOA operator except commutativity. Proving process is in accordance with the former operator.(  1)  Boundedness.Set βmin as the smallest value in (β1,β2,…,βn), while βmax is the largest. We will have the property that(13)βmin≤PFCWAβ1,β2,…,βn≤βmax.(  2) Monotonicity. The PFCWA operator is monotonically increasing based on value of argument value. If βi≤βi′, (i=1,2,…,n), then(14)PFCWAβ1,β2,…,βn≤PFCOWAβ1′,β2′,…,βn′.(  3) Idempotency. When βi=β0, for every i=1,2,…,n, then(15)PFCWAβ1,β2,…,βn=β0.


## 3.2. Pythagorean Fuzzy Choquet Geometric (PFCG) Operators

Definition 6.
Pythagorean fuzzy Choquet order weighted geometric (PFCOWG) operator.
S is a collection of PFNs, S:{βi=P(μβi,νβi)}, where i=1,2,…,n. si has the ith largest value in βi, and set Li as a subset of S; it meets the following requirements:(16)L0=⌀Li=∑k=1isk.The Pythagorean fuzzy Choquet order weighted geometric operator is defined as follows:(17)PFCOWGβ1,β2,…,βn=⨂i=1nsiFmLi-FmLi-1,Fm is the fuzzy measure on S; according to the operations of PFNs in Definition 2(3), the PFCOWA operator has form as below:(18)PFCOWGβ1,β2,…,βn=⨂i=1nxiFmLi-FmLi-1=P∏i=1nμsiFmLi-FmLi-1,1-∏i=1n1-νsi2FmLi-FmLi-1.The result of aggregation is still PFNs.Definition 7.
Pythagorean fuzzy Choquet weighted geometric (PFCWG) operator.
S is a collection of PFNs, S:βi=P(μβi,νβi), where i=1,2,…,n. Fm is the fuzzy measure on S, and set Li′ as a subset of S; it meets the following regulations:(19)L0′=⌀Li′=∑k=1iβkThe PFCWG operator is(20)PFCWGβ1,β2,…,βn=P∏i=1nμβiFmLi′-FmLi-1′,∏i=1nνβiFmLi′-FmLi-1′.Also, we take some special occasions into consideration:(1)
When it meets formula (17), which is Fm(S)=∑si∈SFm({si}), then, the weighted vector ωi=Fm(Li)-Fm(Li-1)=Fm(si) and ∑i=1nωi=1, si has the ith largest value in βi, and formulas (17) and (18) reduce to(21)PFCOWGβ1,β2,…,βn=⨂i=1nsiFmsi=⨂i=1nsiωi=P∏i=1nμsiωi,1-∏i=1n1-νsi2ωi=PFOWGβ1,β2,…,βn.The PFCOWG operator is transformed to PFOWG [8]. Especially, when ωi=Fm(si)=1/n, the operator continues to be simplified to PFOWG operator.(2)
When it meets formula (17), Fm(S)=∑si∈SFm({si}), ωi=Fm(Li′)-Fm(Li-1′)=Fm(si), and, then, formula (20) reduces to(22)PFCWGβ1,β2,…,βn=P⨂i=1nμβiωi,⨂i=1nνβiωi.The PFCWG operator becomes PFWG [8].


## 4. Heterogeneous Network Selection

Operators proposed above are distinct from those of PROMETHEE, taking relationship between attributes into consideration as weighting coefficient, which affects the selection of alternatives. These operators provide fast and effective method with heterogeneous wireless network accessing selection problem, for it is a challenge to decide which network to access when alternatives are multiple, while business needs are various. Therefore, accessing decisions focusing on a particular attribute becomes no longer applicable. According to different business need, the selection standards varying with each other for a business can be sensitive to several attributes at the same time. While satisfying the attribute value requirements, it is also necessary to consider whether attributes needed can be achieved at the same time, which is the relationship between attributes. In brief, the core strategy is to find out the alternative with the best compromise of attribute value and relationship.


### 4.1. QOE Evaluation System

Considering that it is an evaluation index based on user experience, QOE has inherent fuzzy characteristics. Therefore, this paper uses the Pythagorean fuzzy number to evaluate the performance of different networks and their attributes. In the past, experts scoring method is the main source of data. Though it has accuracy, it is also time-consuming and labor-intensive, with low portability and real-time capability, which is unsuitable as a necessary preparation for decision-making problem. So, it is significant to develop an objective attribute data preprocessing system which is highly portable and real-time.(  1) Determination of QOE. There are many ways to evaluate QOE. In this paper, we use the improved type scaling method. Traditional method merely divides QOE into two categories, acceptable and unacceptable. Though it can be easily done, the process is over-rough, which can not express user willing and network performance exactly and scientifically. To overcome this shortcoming, this part we introduce Pythagorean fuzzy number into evaluation. It can use membership and nonmembership of PFNs to describe the degree which the attributes categories conform to. For example, the QOE of a specific business in heterogeneous wireless network is either acceptable or unacceptable in traditional judging method. The improved one gives a PFN (0.82, 0.19) as a result, which means the degree of acceptability of the network for this business is 0.82, while 0.19 is the degree of unacceptability. As a result, the improved method expresses the inherent fuzzy characteristic of QOE, describing users’ subjective feelings in detail at the same time.(  2) Fuzzification. The process of transferring attribute value to fuzzy numbers, as well as the determination of membership and nonmembership degree, is named fuzzification. At present, there are mainly two methods, subjective evaluation and objective evaluation. The subjective evaluation method is to obtain the fuzzy decision matrix for judgment of each network attribute through expert scoring and user feedback. The advantage of this method is its high accuracy, which can directly express the user's feelings. However, the disadvantages are also obvious. The scoring process is inefficient, while portability and real-time performance are both poor. Objective evaluation method is based on attribute characteristics, distinguishing cost-type, and benefit-type attributes and obtains the corresponding user experience quality value by conforming to the actual QOE mapping function. This method is objective and fair, with strong portability and good real-time performance, while the strain capacity is weak, and the accuracy of the results is difficult to determine.In this paper, we adopt these two methods at the same time, contrasting the results to search for the better choice in this experiment circumstance. Figure1 is the QOE evaluation system.Figure 1
QOE evaluation system.


### 4.2. Experiment

Based on the experiment environment in [27], we use proposed preprocessing system to deal with the data and then aggregate values by developed operators to provide selecting strategy. Comparison between different operators will be displayed in chart.(  1) Experiment Environment. The heterogeneous wireless network consists of UMTS, 802.11.a, 802.11.b, and WIMAX. The business includes session, stream, interaction, and background. Different business has different degree of sensitivity toward the same attribute. For example, the business of flow is quite sensitive on packet loss rate and bit error, while it is tolerant on real-time indicators. However, the session business is the opposite. In this experiment, we take session business as example to show the whole process explicitly. Some variable descriptions are illustrated in Table 1.Table 1
Variable description.
Variables Descriptions N i  i = 1,2 , 3,4 set of all networks  Q j  j = 1,2 , 3,4 , 5 set of all attributes  X = x i j original decision matrix  E j information entropy of attributej  R = r i j normalized decision matrix  π j uncertainty of attributej  ω o b j objective weight vector  β i j o b j final decision matrix by objective method  β i j s u b final decision matrix by subjective method(  2) Data Preprocessing. The original data of networks in various attributes in Table 2 is based on [27].Table 2
The original decision matrixX.
Network Delay (ms) Jitter (ms) Throughput (MHz) Packet loss rate (%) Cost UMTS 15 6 0.8 0.02 4  802.11a 29 12 2.5 0.04 1  802.11b 29 12 2 0.04 1  WiMAX 21 9 2 0.03 2.5Because the units of different attributes are not uniform, they need to be normalized firstly. The normalized decision matrix isR=rij, and rij=xij/∑i=1mxij2,i=1,…,m;j=1,…,n.. The result is displayed in Table 3.Table 3
The normalized decision matrixR.
Network Delay Jitter Throughput Packet loss rate Cost UMTS 0.3096 0.2981 0.2073 0.2981 0.8123  802.11a 0.5985 0.5963 0.6479 0.5963 0.2031  802.11b 0.5985 0.5963 0.5183 0.5963 0.2031  WiMAX 0.4334 0.4472 0.5183 0.4472 0.5077Next, the objective weight and uncertainty will be learned through network attributes value. Entropy evaluation method [28] is a regular one in detecting objective weight of network attributes, the principle of which is the statement in information theory about entropy. When the degree of disorder in a system increases, entropy rises when information of system decreases. Applied to weight acquirement, it endows attributes which have low information with low weight and those high information attributes with higher weights.(1)
Calculate information entropypij of network attributes:(23)Ej=∑i=1mpijln⁡pijln⁡m,j=1,2,3,…,npij=rij∑k=1mrkj,i=1,2,3,…,m;j=1,2,3,…,n(2)
Determine the uncertainty of attributesπj:(24)πj=1-Ej,j=1,2,3,…,n(3)
Calculate objective weight:(25)ωobjj=πj∑k=1nπk,j=1,2,3,…,nThe results of E, π, ωobj are shown in Table 4.Table 4
Calculation results.
Network Delay Jitter Throughput Packet loss rate Cost E 0.9762 0.9750 0.9513 0.9750 0.8787  π 0.0238 0.0250 0.0487 0.0250 0.1213  ω o b j 0.0976 0.1024 0.2000 0.1024 0.4975(i) Objective Processing. According to research in [29, 30], use the S-type function in the nonlinear model, combining different characteristics of the network attributes, distinguishing the cost-type and benefit-type attributes, to determine the two mapping models.S m j is the standard value of attribute j in business m, in which m=1,2,3,4; j=1,2,3,4,5. Two mapping models are as follows:Cost-type:μij=1/1+exp⁡(xij-β∙Smj), where β=1.2. The graph of function is shown in Figure 2.Benefit-type:μij=1-1/1+exp⁡(xij-α∙Smj), where α=0.8. The graph of function is shown in Figure 3.Figure 2
Cost-type mapping function.Figure 3
Benefit-type mapping function.The standard values of different business in Table5 are also referred to in [27].Table 5
Standard value matrixSmj.
Business Delay (ms) Jitter (ms) Throughput (MHZ) Packet loss rate (%) Cost Session 15 6 0.02 0.02 1  Stream 30 15 1 0.04 1  Interactive 40 20 0.1 0.04 1  Background 100 30 0.1 0.03 1After acquiring membership degree from mapping function, according toπβ=1-(μβ)2-(νβ)2, the nonmembership degree can be easily resolved. What needs attention is that uncertainty value may be adjusted to πp(x)=1-μp2(x), when μp2(x)+πp2(x)≥1. Meanwhile, the nonmembership degree υp(x) is zero.Table6 shows the final decision matrix results from objective processing. Each member of the matrix is a PFN.Table 6
Final decision matrix 1βijobj.
Network Delay Jitter Throughput Packetloss rate Cost UMTS (0.9526,0.3034) (0.7685, 0.6393) (0.6865, 0.7255) (0.5010, 0.8651) (0.0573, 0.9910)  802.11a (0, 0.9997) (0.0082, 0.9997) (0.9230, 0.3817) (0.4960, 0.8680) (0.5498, 0.8264)  802.11b (0, 0.9997) (0.0082, 0.9997) (0.8791, 0.4741) (0.4960, 0.8680) (0.5498, 0.8264)  WiMAX (0.0474, 0.9986) (0.1419, 0.9896) (0.8791, 0.4741) (0.4985, 0.8665) (0.2142, 0.9692)(ii) Subjective Processing. According subjective evaluation of experts, we have the final decision matrix 2 in Table 7. Users feedback can also be taken into consideration in practical applications.Table 7
Final decision matrix 2βijsub.
Network Delay Jitter Throughput Packet lose rate Cost UMTS (0.71, 0.33) (0.73, 0.31) (0.90, 0.27) (0.74, 0.32) (0.18, 0.84)  802.11a (0.28, 0.77) (0.24, 0.82) (0.79, 0.32) (0.34, 0.62) (0.73, 0.32)  802.11b (0.28, 0.77) (0.24, 0.82) (0.72, 0.38) (0.34, 0.62) (0.73, 0.32)  WiMAX (0.43, 0.65) (0.56, 0.62) (0.73, 0.36) (0.45, 0.68) (0.39, 0.78)As can be seen from Figure4, two pieces of data generally keep a similar trend while the objective is quite fierce when judging a network, for mapping function chosen is strict due to its high slope which is near the critical value. The subjective method shows more tolerance to networks, for experts have put practical situation into consideration. The similar trend proves the validity of developed objective method. Therefore, both methods are valid and can be applied in this experiment.Figure 4
Comparison of sub&obj decision value.It is noteworthy that an objective system established in a specified circumstance needs to be confirmed by traditional methods such as expert scoring, which ensures that it reflects subjective will of people.(  3) Experiment Results. Then, considering different business concerns about network performance distinctively, there exists the possible interaction between these properties. Session business is more sensitive to delay and jitter, while it is tolerant to packet loss. According to the definition in [24], when A is a subset of S, we have(26)
F
m
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Fuzzy measure.
Q 1 Q 2 Q 3 Q 4 Q 5 F m 0.4 0.4 0.2 0.1 0.3According to the above data conditions (Tables6 and 7), we separately use PFCOWA, PFCWA, PFCWOG, PFCWG, and operators to integrate the attributes information. Both subjective fuzzy value and objective fuzzy value are applied to experiment. Table 9 shows the aggregated results by developed operators, using two kinds of evaluation methods. In Figures 5 and 6, the results can be compared to each other more intuitively. Table 10 demonstrates the corresponding decision results of subjective and objective evaluation values in different integration modes, comparing with the result of the improved TOPSIS method used in [27].Table 9
Aggregated results.
subjective PFCOWA PFCWA PFCOWG PFCWG UMTS (0.75051 0.12425) (0.66551 0.38733) (0.3889 0.47469) (0.60936 0.3598)  802.11a (0.53077 0.31589) (0.39058 0.66474) (0.1528 0.65733) (0.34632 0.62287)  802.11b (0.49781 0.32553) (0.38271 0.67149) (0.14769 0.65913) (0.34272 0.63503)  WiMAX (0.53067 0.35575) (0.49818 0.62785) (0.2382 0.64267) (0.48847 0.61634)  objective PFCOWA PFCWA PFCOWG PFCWG  UMTS (0.85391 0.26942) (0.72134 0.57441) (0.31957 0.77158) (0.5653 0.51905)  802.11a (0.77133 0.39978) (0.20675 0.89974) (0 0.92664) (0 0.86773)  802.11b (0.72063 0.47549) (0.20181 0.91013) (0 0.92954) (0 0.88914)  WiMAX (0.70424 0.50083) (0.21456 0.92634) (0.14908 0.917) (0.12652 0.90573)Table 10
Decision strategies by aggregation results by different operators.
Ranking 1 2 3 4 original UMTS WiMAX 802.11a 802.11b PFCOWA(sub) UMTS 802.11a WiMAX 802.11b PFCWA(sub) UMTS WiMAX 802.11a 802.11b PFCOWG(sub) UMTS WiMAX 802.11a 802.11b PFCWG(sub) UMTS WiMAX 802.11a 802.11b PFCOWA(obj) UMTS 802.11a 802.11b WiMAX PFCWA(obj) UMTS 802.11a 802.11b WiMAX PFCOWG(obj) UMTS WiMAX 802.11a 802.11b PFCWG(obj) UMTS 802.11a 802.11b WiMAXFigure 5
Comparison of subjective results.Figure 6
Comparison of objective results.At first, regardless of which algorithm is used in the context of this experiment, the selection of session services is always UMITS. And the priority of 802.11a network is always higher than that of 802.11b. The consistency of these two points confirms the effectiveness of the operators and evaluation system, because the main task in the MADM problems of network selection is to find out the network which has foremost priority.The main difference in ranking results lies in the evaluation of the WIMAX network. Most of the integration results of the subjective evaluation are the same as those of the TOPSIS method [27]. It is considered that the WIMAX has the priority merely after UMITS. However, in the objective evaluation method, its priority is mostly placed at the end. The reason for this is the characteristics of the mapping function in the objective assessment, which has high slope near critical value. Once it exceeds the critical value, it becomes flat rapidly, which no longer has a large discrimination. It brings out a situation when WIMAX performs better than 802.11a and 802.11b in many attributes; however, they all exceed the standard value of session business, so these advantages are no longer considered, whereas, due to the drawback in the cost attribute, the evaluation of WIMAX in the objective evaluation results is the lowest.This kind of evaluation system is similar to the thought that pass rate prioritizes average score. It is undoubtedly a reasonable idea for the users’ quality of experience on network. After comprehensive consideration, the objective evaluation results are scientific and reasonable, which meet the needs of users and can make the same decision as subjective evaluation method does. Moreover, the objective evaluation method has the characteristics of convenience, real-time performance, and high portability. Therefore, in a heterogeneous network environment similar to the experimental environment in this paper, using the proposed QOE objective evaluation system and the proposed operator for MADM problems of network selection is reasonable and effective.


## 4.1. QOE Evaluation System

Considering that it is an evaluation index based on user experience, QOE has inherent fuzzy characteristics. Therefore, this paper uses the Pythagorean fuzzy number to evaluate the performance of different networks and their attributes. In the past, experts scoring method is the main source of data. Though it has accuracy, it is also time-consuming and labor-intensive, with low portability and real-time capability, which is unsuitable as a necessary preparation for decision-making problem. So, it is significant to develop an objective attribute data preprocessing system which is highly portable and real-time.(  1) Determination of QOE. There are many ways to evaluate QOE. In this paper, we use the improved type scaling method. Traditional method merely divides QOE into two categories, acceptable and unacceptable. Though it can be easily done, the process is over-rough, which can not express user willing and network performance exactly and scientifically. To overcome this shortcoming, this part we introduce Pythagorean fuzzy number into evaluation. It can use membership and nonmembership of PFNs to describe the degree which the attributes categories conform to. For example, the QOE of a specific business in heterogeneous wireless network is either acceptable or unacceptable in traditional judging method. The improved one gives a PFN (0.82, 0.19) as a result, which means the degree of acceptability of the network for this business is 0.82, while 0.19 is the degree of unacceptability. As a result, the improved method expresses the inherent fuzzy characteristic of QOE, describing users’ subjective feelings in detail at the same time.(  2) Fuzzification. The process of transferring attribute value to fuzzy numbers, as well as the determination of membership and nonmembership degree, is named fuzzification. At present, there are mainly two methods, subjective evaluation and objective evaluation. The subjective evaluation method is to obtain the fuzzy decision matrix for judgment of each network attribute through expert scoring and user feedback. The advantage of this method is its high accuracy, which can directly express the user's feelings. However, the disadvantages are also obvious. The scoring process is inefficient, while portability and real-time performance are both poor. Objective evaluation method is based on attribute characteristics, distinguishing cost-type, and benefit-type attributes and obtains the corresponding user experience quality value by conforming to the actual QOE mapping function. This method is objective and fair, with strong portability and good real-time performance, while the strain capacity is weak, and the accuracy of the results is difficult to determine.In this paper, we adopt these two methods at the same time, contrasting the results to search for the better choice in this experiment circumstance. Figure1 is the QOE evaluation system.Figure 1
QOE evaluation system.


## 4.2. Experiment

Based on the experiment environment in [27], we use proposed preprocessing system to deal with the data and then aggregate values by developed operators to provide selecting strategy. Comparison between different operators will be displayed in chart.(  1) Experiment Environment. The heterogeneous wireless network consists of UMTS, 802.11.a, 802.11.b, and WIMAX. The business includes session, stream, interaction, and background. Different business has different degree of sensitivity toward the same attribute. For example, the business of flow is quite sensitive on packet loss rate and bit error, while it is tolerant on real-time indicators. However, the session business is the opposite. In this experiment, we take session business as example to show the whole process explicitly. Some variable descriptions are illustrated in Table 1.Table 1
Variable description.
Variables Descriptions N i  i = 1,2 , 3,4 set of all networks  Q j  j = 1,2 , 3,4 , 5 set of all attributes  X = x i j original decision matrix  E j information entropy of attributej  R = r i j normalized decision matrix  π j uncertainty of attributej  ω o b j objective weight vector  β i j o b j final decision matrix by objective method  β i j s u b final decision matrix by subjective method(  2) Data Preprocessing. The original data of networks in various attributes in Table 2 is based on [27].Table 2
The original decision matrixX.
Network Delay (ms) Jitter (ms) Throughput (MHz) Packet loss rate (%) Cost UMTS 15 6 0.8 0.02 4  802.11a 29 12 2.5 0.04 1  802.11b 29 12 2 0.04 1  WiMAX 21 9 2 0.03 2.5Because the units of different attributes are not uniform, they need to be normalized firstly. The normalized decision matrix isR=rij, and rij=xij/∑i=1mxij2,i=1,…,m;j=1,…,n.. The result is displayed in Table 3.Table 3
The normalized decision matrixR.
Network Delay Jitter Throughput Packet loss rate Cost UMTS 0.3096 0.2981 0.2073 0.2981 0.8123  802.11a 0.5985 0.5963 0.6479 0.5963 0.2031  802.11b 0.5985 0.5963 0.5183 0.5963 0.2031  WiMAX 0.4334 0.4472 0.5183 0.4472 0.5077Next, the objective weight and uncertainty will be learned through network attributes value. Entropy evaluation method [28] is a regular one in detecting objective weight of network attributes, the principle of which is the statement in information theory about entropy. When the degree of disorder in a system increases, entropy rises when information of system decreases. Applied to weight acquirement, it endows attributes which have low information with low weight and those high information attributes with higher weights.(1)
Calculate information entropypij of network attributes:(23)Ej=∑i=1mpijln⁡pijln⁡m,j=1,2,3,…,npij=rij∑k=1mrkj,i=1,2,3,…,m;j=1,2,3,…,n(2)
Determine the uncertainty of attributesπj:(24)πj=1-Ej,j=1,2,3,…,n(3)
Calculate objective weight:(25)ωobjj=πj∑k=1nπk,j=1,2,3,…,nThe results of E, π, ωobj are shown in Table 4.Table 4
Calculation results.
Network Delay Jitter Throughput Packet loss rate Cost E 0.9762 0.9750 0.9513 0.9750 0.8787  π 0.0238 0.0250 0.0487 0.0250 0.1213  ω o b j 0.0976 0.1024 0.2000 0.1024 0.4975(i) Objective Processing. According to research in [29, 30], use the S-type function in the nonlinear model, combining different characteristics of the network attributes, distinguishing the cost-type and benefit-type attributes, to determine the two mapping models.S m j is the standard value of attribute j in business m, in which m=1,2,3,4; j=1,2,3,4,5. Two mapping models are as follows:Cost-type:μij=1/1+exp⁡(xij-β∙Smj), where β=1.2. The graph of function is shown in Figure 2.Benefit-type:μij=1-1/1+exp⁡(xij-α∙Smj), where α=0.8. The graph of function is shown in Figure 3.Figure 2
Cost-type mapping function.Figure 3
Benefit-type mapping function.The standard values of different business in Table5 are also referred to in [27].Table 5
Standard value matrixSmj.
Business Delay (ms) Jitter (ms) Throughput (MHZ) Packet loss rate (%) Cost Session 15 6 0.02 0.02 1  Stream 30 15 1 0.04 1  Interactive 40 20 0.1 0.04 1  Background 100 30 0.1 0.03 1After acquiring membership degree from mapping function, according toπβ=1-(μβ)2-(νβ)2, the nonmembership degree can be easily resolved. What needs attention is that uncertainty value may be adjusted to πp(x)=1-μp2(x), when μp2(x)+πp2(x)≥1. Meanwhile, the nonmembership degree υp(x) is zero.Table6 shows the final decision matrix results from objective processing. Each member of the matrix is a PFN.Table 6
Final decision matrix 1βijobj.
Network Delay Jitter Throughput Packetloss rate Cost UMTS (0.9526,0.3034) (0.7685, 0.6393) (0.6865, 0.7255) (0.5010, 0.8651) (0.0573, 0.9910)  802.11a (0, 0.9997) (0.0082, 0.9997) (0.9230, 0.3817) (0.4960, 0.8680) (0.5498, 0.8264)  802.11b (0, 0.9997) (0.0082, 0.9997) (0.8791, 0.4741) (0.4960, 0.8680) (0.5498, 0.8264)  WiMAX (0.0474, 0.9986) (0.1419, 0.9896) (0.8791, 0.4741) (0.4985, 0.8665) (0.2142, 0.9692)(ii) Subjective Processing. According subjective evaluation of experts, we have the final decision matrix 2 in Table 7. Users feedback can also be taken into consideration in practical applications.Table 7
Final decision matrix 2βijsub.
Network Delay Jitter Throughput Packet lose rate Cost UMTS (0.71, 0.33) (0.73, 0.31) (0.90, 0.27) (0.74, 0.32) (0.18, 0.84)  802.11a (0.28, 0.77) (0.24, 0.82) (0.79, 0.32) (0.34, 0.62) (0.73, 0.32)  802.11b (0.28, 0.77) (0.24, 0.82) (0.72, 0.38) (0.34, 0.62) (0.73, 0.32)  WiMAX (0.43, 0.65) (0.56, 0.62) (0.73, 0.36) (0.45, 0.68) (0.39, 0.78)As can be seen from Figure4, two pieces of data generally keep a similar trend while the objective is quite fierce when judging a network, for mapping function chosen is strict due to its high slope which is near the critical value. The subjective method shows more tolerance to networks, for experts have put practical situation into consideration. The similar trend proves the validity of developed objective method. Therefore, both methods are valid and can be applied in this experiment.Figure 4
Comparison of sub&obj decision value.It is noteworthy that an objective system established in a specified circumstance needs to be confirmed by traditional methods such as expert scoring, which ensures that it reflects subjective will of people.(  3) Experiment Results. Then, considering different business concerns about network performance distinctively, there exists the possible interaction between these properties. Session business is more sensitive to delay and jitter, while it is tolerant to packet loss. According to the definition in [24], when A is a subset of S, we have(26)
F
m
S
=
F
m
⋃
i
=
1
n
s
i
=
1
ρ
∏
i
=
1
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1
+
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m
s
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-
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,
ρ
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∑
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=
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,
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(27)
F
m
A
=
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∏
s
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+
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s
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.For Fm(S)=1, we can get ρ+1=∏i=1n(1+ρFm(si)).And when we apply the fuzzy measure value which is determined by attribute relationship in Table8, it is easily to calculate ρ=-0.6253.Table 8
Fuzzy measure.
Q 1 Q 2 Q 3 Q 4 Q 5 F m 0.4 0.4 0.2 0.1 0.3According to the above data conditions (Tables6 and 7), we separately use PFCOWA, PFCWA, PFCWOG, PFCWG, and operators to integrate the attributes information. Both subjective fuzzy value and objective fuzzy value are applied to experiment. Table 9 shows the aggregated results by developed operators, using two kinds of evaluation methods. In Figures 5 and 6, the results can be compared to each other more intuitively. Table 10 demonstrates the corresponding decision results of subjective and objective evaluation values in different integration modes, comparing with the result of the improved TOPSIS method used in [27].Table 9
Aggregated results.
subjective PFCOWA PFCWA PFCOWG PFCWG UMTS (0.75051 0.12425) (0.66551 0.38733) (0.3889 0.47469) (0.60936 0.3598)  802.11a (0.53077 0.31589) (0.39058 0.66474) (0.1528 0.65733) (0.34632 0.62287)  802.11b (0.49781 0.32553) (0.38271 0.67149) (0.14769 0.65913) (0.34272 0.63503)  WiMAX (0.53067 0.35575) (0.49818 0.62785) (0.2382 0.64267) (0.48847 0.61634)  objective PFCOWA PFCWA PFCOWG PFCWG  UMTS (0.85391 0.26942) (0.72134 0.57441) (0.31957 0.77158) (0.5653 0.51905)  802.11a (0.77133 0.39978) (0.20675 0.89974) (0 0.92664) (0 0.86773)  802.11b (0.72063 0.47549) (0.20181 0.91013) (0 0.92954) (0 0.88914)  WiMAX (0.70424 0.50083) (0.21456 0.92634) (0.14908 0.917) (0.12652 0.90573)Table 10
Decision strategies by aggregation results by different operators.
Ranking 1 2 3 4 original UMTS WiMAX 802.11a 802.11b PFCOWA(sub) UMTS 802.11a WiMAX 802.11b PFCWA(sub) UMTS WiMAX 802.11a 802.11b PFCOWG(sub) UMTS WiMAX 802.11a 802.11b PFCWG(sub) UMTS WiMAX 802.11a 802.11b PFCOWA(obj) UMTS 802.11a 802.11b WiMAX PFCWA(obj) UMTS 802.11a 802.11b WiMAX PFCOWG(obj) UMTS WiMAX 802.11a 802.11b PFCWG(obj) UMTS 802.11a 802.11b WiMAXFigure 5
Comparison of subjective results.Figure 6
Comparison of objective results.At first, regardless of which algorithm is used in the context of this experiment, the selection of session services is always UMITS. And the priority of 802.11a network is always higher than that of 802.11b. The consistency of these two points confirms the effectiveness of the operators and evaluation system, because the main task in the MADM problems of network selection is to find out the network which has foremost priority.The main difference in ranking results lies in the evaluation of the WIMAX network. Most of the integration results of the subjective evaluation are the same as those of the TOPSIS method [27]. It is considered that the WIMAX has the priority merely after UMITS. However, in the objective evaluation method, its priority is mostly placed at the end. The reason for this is the characteristics of the mapping function in the objective assessment, which has high slope near critical value. Once it exceeds the critical value, it becomes flat rapidly, which no longer has a large discrimination. It brings out a situation when WIMAX performs better than 802.11a and 802.11b in many attributes; however, they all exceed the standard value of session business, so these advantages are no longer considered, whereas, due to the drawback in the cost attribute, the evaluation of WIMAX in the objective evaluation results is the lowest.This kind of evaluation system is similar to the thought that pass rate prioritizes average score. It is undoubtedly a reasonable idea for the users’ quality of experience on network. After comprehensive consideration, the objective evaluation results are scientific and reasonable, which meet the needs of users and can make the same decision as subjective evaluation method does. Moreover, the objective evaluation method has the characteristics of convenience, real-time performance, and high portability. Therefore, in a heterogeneous network environment similar to the experimental environment in this paper, using the proposed QOE objective evaluation system and the proposed operator for MADM problems of network selection is reasonable and effective.


## 5. Concluding Remarks

In this text, some new kinds of aggregation operators are developed. The common character of them is having the ability to put attributes relationship into consideration of decision, which is important for some special scenarios such as network selection in heterogeneous wireless network. Meanwhile, transfer relationship and properties of the operators are given after definitions. Furthermore, an QOE evaluation system is proposed. Participation of Pythagorean fuzzy number makes exacter description possible for QOE value. The novel concept of objective evaluation process provides effectiveness and portability at the same time. A typical example of network selection has proved the validity of both operators and the system. It is convincing to say the study has a bright potential on practical application. What needs to be improved in the future is how to combine the merits of subjective opinion with objective method, representing a more comprehensive result. Moreover, even more complex application scenarios are worth deeper consideration.



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*Source: 1619623-2018-10-28.xml*