Aggregation Operators for Decision Making Based on q-Rung Orthopair Fuzzy Hypersoft Sets: An Application in Real Estate Project

1  Introduction

Decision making [1] is a crucial aspect of our daily lives. Due to the complexity and subjectivity of complex alternatives, decision-makers are frequently confronted with a considerable measure of ambiguity in decision-making problems. To address these problems, Zadeh [2] developed fuzzy set theory. Each element of such set is assigned a membership value, between 0 and 1. The fuzzy set theory has a significant impact on the decision-making process. The fuzzy set theory has numerous applications in numerous domains of engineering, humanities, health, and physical sciences. In recent years, fuzzy set theory has also been utilized in the fields of energy, medicine, materials, economics, and nearly every other scientific discipline. There must be uncertainty if the decision-makers do not have sufficient knowledge to deal with the data necessary to express their preference for the object. Experts mostly consider membership and non-membership values in the decision-making process that the fuzzy set cannot handle. Using the framework of fuzzy multisets, Du et al. [3] developed a multicriteria group decision-making method in 2021, which accounts for varying degrees of risk associated with each criterion.

To deal with this type of circumstance, Atanassov [4,5] proposed intuitionistic fuzzy sets, whose primary components are intuitionistic fuzzy numbers (IFN). Intuitionistic fuzzy numbers are defined by a membership function and a non-membership function, with the condition that the sum of the membership and non-membership grades is less than or equal to one. When the sum is greater than 1, the intuitionistic fuzzy sets do not work under specific conditions. Several operations on intuitionistic fuzzy sets were developed by De et al. [6,7] and incorporated into Sanchez’s method for medical diagnosis. The distance between intuitionistic fuzzy sets was discussed by Szmidt et al. [8]. Using intuitionistic fuzzy sets as a theme, Gulzar et al. [9] introduced the t-intuitionistic fuzzy subgroup with basic algebraic properties and complex intuitionistic fuzzy sets in group theory and its applications. In some instances, when the sum of the degrees of MM and NMM is greater than 1, their sum of squares may be less than or equal to 1. Intuitionistic fuzzy sets fail in this situation.

To address these problems, Yager [10] proposed a novel concept known as the Pythagorean fuzzy set, a generalization of the intuitionistic fuzzy sets. The quadratic sum of membership and non-membership is less than or equal to 1, distinguishing it from other entities. Intuitionistic fuzzy sets and Pythagorean fuzzy sets are incredibly valuable for characterising fuzzy data. It is obvious that the Pythagorean fuzzy set can represent more decision-making information and is more inclusive than the intuitionistic fuzzy set. However, the Pythagorean fuzzy set frequently fails when the decision-maker is able to specify the degree of the membership and non-membership in a particular case where their sum is greater than 1. Some findings for Pythagorean fuzzy sets were developed in 2015 [11]. Pythagorean fuzzy sets, as well as a more refined approach to linguistic, Garg [12] utilized Pythagorean fuzzy sets for MCDM problems.

Yager [13] came up with q-rung orthopair fuzzy sets (q-ROFS), the key feature of which is that the sum of the qth power of the membership and non-membership grade cannot exceed 1. Because of this, q-ROFS is advantageous to both the intuitionistic fuzzy set and the Pythagorean fuzzy set when dealing with ambiguous data. Innovative applications of fuzzy sets include aggregation operators, information measures, and decision-making tools. Madasi et al. [14] recently proposed the topic of N-cubic q-rung orthopair fuzzy sets, along with their properties and the use of mobile apps in the education sector as an application.

In order to address the issue of making group decisions based on several criteria, Zulqarnain et al. [15] developed certain aggregation operators inside the framework of the Interval-Valued Pythagorean fuzzy soft set. After transforming the function F into a multi-valued function, Smarandache [16] introduced the hypersoft set. Hypersoft sets are a fundamental mathematical tool for dealing with ambiguity and uncertainty in problem-based decision-making. Zulqarnain et al. [17,18] developed the Pythagorean fuzzy hypersoft sets, along with fundamental operations and techniques. By combining these concepts, Khan et al. [19] constructed a new structure called q-rung orthopair fuzzy hypersoft sets (q-ROFHSs). In q-ROFHSs, the multi-parameter membership and non-membership grades are limited to the qth power.

It is crucial to understand the aggregation operators in the q-rung orthopair fuzzy environment and other types of fuzzy sets for tackling problems involving decision-making. In order to solve the decision-making techniques, the induced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) and induced interval-valued intuitionistic fuzzy ordered weighted geometric (I-IIFOWG) operators were developed by Wei [20]. The induced generalised intuitionistic fuzzy ordered weighted averaging (I-GIFOWA) and induced generalised interval-valued intuitionistic fuzzy ordered weighted averaging (I-GIIFOWA) operators were introduced by Xu et al. [21]. For the sake of better decision-making, Zulqarnain et al. [22] also explored a variety of aggregation operators in a Pythagorean fuzzy hypersoft environment. Basic aggregation operators, such as the weighted average and weighted geometric operators were designed by Liu et al. [23], and they were applied in a scenario based on q-rung orthopair fuzzy information. Several fundamental operations and exponential aggregation operators were developed by Peng et al. [24] for the q-rung orthopair fuzzy set. Two aggregation operators the complex q-rung orthopair fuzzy weighted average (Cq-ROFWA) and the complex q-rung orthopair fuzzy weighted geometric (Cq-ROFWG) were introduced by Liu et al. [25]. For a complex q-rung orthopair fuzzy (Cq-ROF) environment, Garg et al. [26] developed power aggregation operations. The q-rung orthopair fuzzy hypersoft weighted average (q-ROFHWA) and q-rung orthopair fuzzy hypersoft weighted geometric (q-ROFHWG) aggregation operators were developed by Khan et al. [27] for use in decision-making problems.

Due to the inadequacies of such mathematical modelling problems, we proposed q-ROFHS baesd mathematical model of such daily life problems in decision-making. For these mathematical models in the q-ROFH context, some aggregation operators and decision-making approaches are required. Therefore, we developed ordered weighted average and induced ordered weighted average aggregation operators in the q-ROFH environment.

Here are some of the key aspects of this research:

•   The q-ROFHS process expert data with multi-sub-attributes of uncertainty in order to make decisions.

•   Important aggregation operators in the decision-making process include ordered weighted average and induced ordered weighted average.

•   Using basic operational laws, properties of the ordered weighted average and induced ordered weighted average aggregation operator.

•   A novel decision-making technique was presented by using the proposed aggregation operators.

•   Research is conducted within a real company to determine the efficacy and viability of the proposed method. The interpretations demonstrate that the predefined model is more practical and reliable for understanding the facts.

The remaining study of this article is organized as follows. In Section 2, briefly some basic materials related to the proposed structure are introduced. Section 3 consists of ordered weighted average and induced ordered weighted average operators with properties. In Section 4, the algorithm of the proposed structure is presented based on aggregation operators. A real-life application is also presented in this section. Section 5 describes the comparative analysis of the proposed structure with existing structures. Section 6 presents the conclusion and future work.

2  Preliminaries

In this section, we compile some basic data that will be utilised to construct the structure of the article.

Definition 2.1 (16). The pair (H,A) is called a hypersoft set over Y, where H is the cartesian of n disjoint sets H1,H2,…,Hn having attribute values of n distinct attributes h1,h2,…,hn, respectively, and H:A→P(Y).

Definition 2.2 (19). Let Y={Y1,Y2,…,Yn} be a collection of disjoint attribute valued sets corresponding to n distinct attributes a1,a2,…,an, respectively, with the condition Ai∩Aj=ϕ, for i≠j and i,j∈{1,2,…,n}. A q-rung orthopair fuzzy hypersoft set §A over Y is defined as

§A={(⟨h,ψA(h),ψ~A(h)⟩)|h∈H andq≥1},

where

(a)   q−ROF(Y) is the collection of all q-rung orthopair fuzzy sets over Y.

(b)   H=Y1×Y2×…×Yn

(c)   A is a q-rung orthopair fuzzy over Y with ψA,ψ~A:H→X, and

(d)   δA(h) is a q-rung orthopair fuzzy set for h∈H with δA(h):H→Q(Y).

Here q−ROFHS(Y) represents the collection of all q−ROF subsets of Y, such that 0≤(ψA(h))q+(ψ~A(h))q≤1.

Definition 2.3 (19). The score function of the q-ROFHNs is defined as SA(h)=ψA(h)q−ψ~A(h)q

3  Ordered and Induced Ordered Weighted Average Aggregation Operators

In this section, we use the concept of ordered weighted averaging operator and develop the q-rung orthopair fuzzy hypersoft ordered weighted averaging operator (q-ROFHOWA) and the induced-q-rung orthopair fuzzy hypersoft ordered weighted averaging operator (I-q-ROFHOWA).

Definition 3.1. An ordered weighted averaging q-rung orthopair fuzzy hypersoft (q-ROFHOWA) operator is defined as:

I−q−ROFHOWA(Ea11,Ea12,...,Eanm)=⊕j=1n⁡vj(⊕i=1m⁡wiEσaij)

where wi={1,2,…,n} and vj={1,2,…,m} are weight vectors for experts and sub-attributes of the selected parameters respectively, having the condition that, wi>0,∑i=1mwi=1, and vj>0,∑j=1nvj=1. Then the mapping for the q-ROFHOWA operator is defined as q−ROFHOWA:Δn→Δ is the collection of all q-ROFHNs.

Theorem 3.1. Let Eaij,(i=1,2,…,n)(j=1,2,…,m) be the collection of q-ROFHNs, then their aggregated values by using the q-ROFHOWA operator are also q-rung orhtopair fuzzy hypersoft values, and q−ROFHOWA(Ea11,Ea12,…,Eanm)=(1−∏j=1n(∏1=1m(1−φσaijq)wi)vj,q∏j=1n(∏1=1m(φ~σaij)wi)vj) where wi={1,2,…,n} and vj={1,2,…,m} are weight vectors for experts and sub-attributes of selected parameters, respectively, having the condition that, wi>0,∑i=1mwi=1, and vj>0,∑j=1nvj=1, and σij are permutations of (i=1,2,…,m), (j=1,2,…,n) such that Eσ(i−1)j≥Eσ(i)j and Eσi(j−1)≥Eσi(j)∀i,j.

Proof. The q-ROFHOWA operator can be proved using the principle of mathematical induction as follows. For m=1, we get wi=1. Then, we have,

q−ROFHOWA(Ea11,Ea12,…,Eanm)=⊕j=1n⁡vjEσa1j=(1−∏j=1n⁡(1−φσaijq)vj,q∏j=1n⁡(φ~σaij)vj)=(1−∏j=1n⁡(∏1=11⁡(1−φσaijq)wi)vj,q∏j=1n⁡(∏1=11⁡(φ~σaij)wi)vj)(1)

For n=1, we get vj=1.

q−ROFHOWA(Ea11,Ea12,…,Eanm)=⊕i=1m⁡wiEσai1=(1−∏i=1m⁡(1−φσaijq)wi,q∏i=1m⁡(φ~σaij)wi)=(1−∏j=11⁡(∏i=1m⁡(1−φσaijq)wi)vj,q∏j=11⁡(∏i=1m⁡(φ~σaij)wi)vj)(2)

So, Eq. (2) is true for all m=1 and n=1. Suppose that the equation holds for m=k2,n=k1+1, and m=k2+1,n=k1.

⊕j=1k1+1⁡vj(⊕i=1k2⁡wiEσaij)=(1−∏j=1k1+1⁡(∏1=1k2⁡(1−φσaijq)wi)vj,q∏j=1k1+1⁡(∏1=1k2⁡(φ~σaij)wi)vj)(3)

=⊕j=1k1⁡vj(⊕i=1k2+1⁡wiEσaij)=(1−∏j=1k1⁡(∏1=1k2+1⁡(1−φσaijq)wi)vj,q∏j=1k1⁡(∏1=1k2+1⁡(φ~σaij)wi)vj)(4)

Now we prove the equation for n=k1+1 and m=k2+1.

⊕j=1k1+1⁡vj(⊕i=1k2+1⁡wiEσaij)=⊕j=1k1+1⁡Eσaj(⊕i=1k2⁡wiEσaij⊕wi+1Eσa(k2+1)j)(5)

=(⊕j=1k1+1⁡⊕i=1k2⁡wivjEσaj)(⊕j=1k1+1⁡vjwi+1Eσa(k2+1)(j))=2∏j=1k1+1⁡(∏1=1k2+1⁡(φ~σa(i)(j))wi)vj⊕2∏j=1k1+1⁡(∏1=1k2+1⁡(φ~σa(k2+1)(j))wk2+1)vj=1−∏j=1k1+1⁡(∏1=1k2+1⁡(1−φσaijq)wi)vj,q2∏j=1k1+1⁡(∏i=1k2+1⁡(φ~σa(i)(j))wi)vj=⊕j=1k1+1⁡vj(⊕i=1k2+1⁡wiEσaij)(6)

So, it is valid for n=k1+1 and m=k2+1.

Remark 3.2. If the value of q is fixed, which is q=1, the proposed q-ROFHOWA operator reduces to the IFHOWA operator.

Remark 3.3. If the value of q is fixed, which is q=2, the proposed q-ROFHOWA operator reduces to the PFHOWA operator.

Remark 3.4. If there are single parameters, which is e1,e2,…,en, then the proposed q-ROFHOWA operator reduces to the q-ROFSOWA operator.

Remark 3.5. From the above remarks, it is clear that IFHOWA, PFHOWA and q-ROFSOWA operators are the special cases of the proposed q-ROFHOWA operator.

The I-q-ROFHSOWA operator is idempotent, bounded and monotonic. These can be proved by the following theorem:

(Idempotency) Let Eaij=(φaij,φ~aij) be a collection of q-ROFHNs, (∀i=1,2,…,n)(j=1,2,…,m). If Eσaij=E11 are mathematically identical, then q−ROFHOWA(Ea11,Ea12,…,Eanm)=E.

Proof. As we know,

q−ROFHOWA(Ea11,Ea12,…,Eanm)=(1−∏j=1k1+1⁡(∏1=1k2+1⁡(1−φσaijq)wi)vj,q2∏j=1k1+1⁡(∏i=1k2+1⁡(φ~σa(i)(j))wi)vj)

Eσa(i)(j)=Eij, so

=(1−((1−φσaijq)∑i=1mwi)∑j=1nvj,q2((φ~σa(i)(j))∑i=1mwi)∑j=1nvj)=(1−(1−φσaijq),q2(φ~σa(i)(j)))=(φaij,φ~aij)=E

(Boundedness) Let Eaij=(φaij,φ~aij) be a collection of q-ROFHNs, for all i=1,2,... m,j=1,2,…,n. If Eminmin=minminEa(i)(j) and Emaxmax=maxmaxEa(i)(j), then Eminmin≤q−ROFHOWA(Ea11,Ea12,…,Eanm)≤Emaxmax, where wi and vj are weight vectors for expert’s and sub attributes of selected parameters respectively, such that wi>0,∑i=1mwi=1, and wi>0,∑i=1nvj=1.

Proof. Let f(x)=1−(1−xq),qx∈[0,1]. Then,

ddx(f(x))=−qxq−1((1−xqq)−1q)<0,

so f(x) is a decreasing function on [0, 1]. As φminmin≤φaij≤φmaxmax, f(φminjmini)≤f(φaij)≤f(φmaxjmaxi).

⇔1−maxjmaxiφqq≤1−φσaijqq≤1−minjminiφqq⇔∏j=1n⁡(∏i=1m⁡(1−maxjmaxiφq)wi)vjq≤∏j=1n⁡(∏i=1m⁡(1−φσaijq)wi)vjq≤∏j=1n⁡(∏i=1m⁡(1−minjminiφq)wi)vjq⇔((1−maxjmaxiφq)∑i=1mwi)∑j=1nvjq≤∏j=1n⁡(∏i=1m⁡(1−φσaijq)wi)vjq≤((1−minjminiφq)∑i=1mwi)∑j=1nvjq⇔−1+minjminiφqq≤−∏j=1n⁡(∏i=1m⁡(1−φσaijq)wi)vjq≤−1+maxjmaxiφqq⇔minjmini(φ)≤1−∏j=1n⁡(∏i=1m⁡(1−φσaijq)wi)vjq≤maxjmaxi(φ)(7)

As φ~minmin≤φ~aij≤φ~maxmax, f(φ~minjmini)≤f(φ~aij)≤f(φ~maxjmaxi).

⇔minjminiφ~q≤φ~σaijq≤maxjmaxiφ~q⇔∏j=1n(∏i=1m(minjminiφ~)wi)vjq≤∏j=1n(∏i=1m(φ~σaij)wi)vjq≤∏j=1n(∏i=1m(maxjmaxiφ~)wi)vjq⇔((minjminiφ~)∑i=1mwi)∑j=1nvjq≤∏j=1n(∏i=1m(φ~σaij)wi)vjq≤((maxjmaxiφ~)∑i=1mwi)∑j=1nvjq⇔minjminiφ~q≤∏j=1n(∏i=1m(φ~σaij)wi)vjq≤maxjmaxiφq⇔minjmini(φ~)≤∏j=1n(∏i=1m(φ~σaij)wi)vjq≤maxjmaxi(φ~)(8)

Let q−ROFHOWA(Ea11,Ea12,…,Eanm)=E. Then Eqs. (7) and (8) can be written as minjmini(φq)≤φσaijq≤maxjmaxi(φq) and minjmini(φ~)≤φ~σaij≤maxjmaxi(φ~). Thus,

S(H)=φq−φ~q≤φmaxjmaxiq−φminjminiq=S(Hmax)(9)

S(H)=φq−φ~q≥φminjminiq−φmaxjmaxiq=S(Hmin)(10)

If S(H)<S(Hmax) and S(H)>S(Hmin), then we have,

Hmin<qROFHOWA(Ea11,Ea12,…,Eanm)<Hmax(11)

If S(H)=S(Hmax), then we have φq=φmaxjmaxiq and φ~q=φ~maxjmaxiq. Thus A(H)=φq+φ~. Therefore,

q−ROFHOWA(Ea11,Ea12,…,Eanm)<Hmax(12)

If S(H)=S(Hmin), then we have φq−φ~=φminjminiq and φq=φminjminiq and φ~=φ~minjmini. Thus A(H)=φq+φ~=φminjminiq+φ~minjmini. Therefore,

q−ROFHOWA(Ea11,Ea12,…,Eanm)<Hmin(13)

So by Eqs. (14) and (15), we get

Hmin≤q−ROFHOWA(Ea11,Ea12,…,Eanm)≤Hmax(14)

Definition 3.2. An induced q-rung orthopair fuzzy hypersoft ordered weighted averaging (I-q-ROFHSOWA) operator is defined as:

I−q−ROFHSOWA(⟨Ea11,ua11⟩,⟨Ea12,ua12⟩,…,⟨Eanm,uanm⟩)=⊕j=1n⁡vj(⊕i=1m⁡wiEσaij)=(1−∏j=1n⁡(∏1=1m⁡(1−φσaijq)wi)vj,q∏j=1n⁡(∏1=1m⁡(φ~σaij)wi)vj)(15)

where wi={1,2,…,n} and vj={1,2,…,m} are weight vectors for experts and sub-attributes of selected parameters respectively, having the condition that, wi>0,∑i=1mwi=1, and vj>0,∑j=1nvj=1. Also σaij is the σij value of the q-ROFHSOWA pair ⟨Eaij,uaij⟩ having the ith and jth largest uaij, and uaij in ⟨Eaij,uaij⟩ being referred to as the order inducing variable and Eaij as the q-rung orthopair fuzzy hypersoft argument variable.

Theorem 3.6. Let ⟨Eaij,uaij⟩(i=1,2,…,n)(j=1,2,…,m) be a collection of 2 tuples, then their aggregated values by using the I-q-ROFHOWA operator is also a q-rung orhtopair fuzzy hypersoft value and

I−q−ROFHOWA(⟨Ea11,ua11⟩,⟨Ea12,ua12⟩,…,⟨Eanm,uanm⟩)=(1−∏j=1n⁡(∏1=1m⁡(1−φσaijq)wi)vj,q∏j=1n⁡(∏1=1m⁡(φ~σaij)wi)vj)(16)

where wi={1,2,…,n} and vj={1,2,…,m} are weight vectors for experts and sub-attributes of selected parameters respectively, having the condition that, wi>0,∑i=1mwi=1, and vj>0,∑j=1nvj=1.

Proof. The I-q-ROFHOWA operator can be proved using the principle of mathematical induction as follows. For m=1, we get wi=1. Then we have,

I−q−ROFHOWA(⟨Ea11,ua11⟩,⟨Ea12,ua12⟩,…,⟨Eanm,uanm⟩)=⊕j=1n⁡vjEσa1j=(1−∏j=1n⁡(1−φσaijq)vj,q∏j=1n⁡(φ~σaij)vj)=(1−∏j=1n⁡(∏1=11⁡(1−φσaijq)wi)vj,q∏j=1n⁡(∏1=11⁡(φ~σaij)wi)vj)(17)

For n=1, we get vj=1.

I−q−ROFHOWA(⟨Ea11,ua11⟩,⟨Ea12,ua12⟩,…,⟨Eanm,uanm⟩)=⊕i=1m⁡wiEσai1=(1−∏i=1m⁡(1−φσaijq)wi,q∏i=1m⁡(φ~σaij)wi)=(1−∏j=11⁡(∏i=1m⁡(1−φσaijq)wi)vj,q∏j=11⁡(∏i=1m⁡(φ~σaij)wi)vj)(18)

So, Eq. (2) is true for m=1 and n=1. Suppose that the equation holds for m=k2,n=k1+1, and m=k2+1,n=k1.

=⊕j=1k1+1⁡vj(⊕i=1k2⁡wiEσaij)=(1−∏j=1k1+1⁡(∏1=1k2⁡(1−φσaijq)wi)vj,q∏j=1k1+1⁡(∏1=1k2⁡(φ~σaij)wi)vj)(19)

=⊕j=1k1⁡vj(⊕i=1k2+1⁡wiEσaij)=(1−∏j=1k1⁡(∏1=1k2+1⁡(1−φσaijq)wi)vj,q∏j=1k1⁡(∏1=1k2+1⁡(φ~σaij)wi)vj)(20)

Now we prove the equation for n=k1+1 and m=k2+1.

⊕j=1k1+1⁡vj(⊕i=1k2+1⁡wiEσaij)=⊕j=1k1+1⁡Eσaj(⊕i=1k2⁡wiEσaij⊕i=1k2⁡wi+1Eσa(k2+1)j)=∏j=1k1+1⁡(∏1=1k2+1⁡(φ~σaij)wi)vj⊕∏j=1k1+1⁡(∏1=1k2+1⁡(φ~σa(k2+1)j)wk2+1)vj(21)

=1−∏j=1k1+1⁡(∏1=1k2+1⁡(1−φσaijq)wi)vj,q∏j=1k1+1⁡(∏1=1k2+1⁡(φ~σaij)wi)vj=⊕j=1k1+1⁡vj(⊕i=1k2+1⁡wiEσaij)

So, it is valid for n=k1+1 and m=k2+1.

Example 3.7. Let Y be the set of decision makers to decide the best laptop given as Y={y1,y2,y3}. Also consider the set of attributes as P1 and P2, where P1 represents the laptop type and P2 represents the laptop RAM. Then their corresponding attributive sets can be P1={a11=HP,a12=Dell}, and P2={a21=8GB,a22=16GB,a23=32GB}. Suppose A1={a11,a12},A2={a21,a22} and, a¯11=(a11,a21),a¯12=(a11,a22),a¯21=(a12,a21),a¯22=(a12,a22). Then A¯=A1×A2 will have 2-tuple elements, and we suppose q=8. Then the I-q-ROFHS set (H,A¯) can be written as ⟨Ea¯11,ua¯11⟩=⟨.3,(.4,.2)⟩,⟨Ea¯12,ua¯12⟩=⟨.4,(.5,.3)⟩, ⟨Ea¯21,ua¯21⟩=⟨.6,(.4,.7)⟩,⟨Ea¯22,ua¯22⟩=⟨.5,(.6,.3)⟩. Performing the ordering of q-ROFHS pairs with respect to the first component we have ⟨Ea¯21,ua¯21⟩=⟨.6,(.4,.7)⟩,⟨Ea¯22,ua¯22⟩=⟨.5,(.6,.3)⟩, ⟨Ea¯12,ua¯12⟩=⟨.4,(.5,.3)⟩,⟨Ea¯11,ua¯11⟩=⟨.3,(.4,.2)⟩. This order includes the order of the q-ROFHS arguments: ⟨Eσa¯11,uσa¯11⟩=⟨.6,(.4,.7)⟩,⟨Eσa¯12,uσa¯12⟩=⟨.5,(.6,.3)⟩, ⟨Eσa¯21,uσa¯21⟩=⟨.4,(.5,.3)⟩,⟨Eσa¯22,uσa¯22⟩=⟨.3,(.4,.2)⟩. Let wi=(.2,.1,.4,.3),vj=(.6,.1,.3), and q=8.