# Hesitant Probabilistic Fuzzy Preference Relations in Decision Making

Authors: Zia Bashir; Tabasam Rashid; Mobashir Iqbal
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Preference of an alternative over another alternative is a useful way to express the opinion of decision maker. In the process of group decision making, preference relations are used in preference modelling of the alternatives under given criteria. The probability is an important tool to deal with uncertainty; in many scenarios of decision making probabilities of different events affect the decision making process directly. In order to deal with this issue, in this paper, hesitant probabilistic fuzzy preference relation (HPFPR) is defined. Furthermore, consistency of HPFPR and consensus among decision makers are studied in the hesitant probabilistic fuzzy environment. In this respect, many novel algorithms are developed to achieve consistency of HPFPRs and reasonable consensus between decision makers and a final algorithm is proposed comprehending all other algorithms, presenting a complete decision support model for group decision making. Lastly, we present a case study with complete illustration of the proposed model and discussed the effects of probabilities on decision making validating the importance of the introduction of probability in hesitant fuzzy preference relation.


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## Body

## 1. Introduction

Fuzzy set theory was initially introduced by Zadeh [1] in 1965 as an extension of the classical set theory. In classical set theory, an element either belongs to or does not belong to the set. In fuzzy set theory, the gradual assessment of elements of set is described by the membership function that is in [0,1]. Fuzzy set theory can be used in which information is vague, incomplete, or imprecise and it is successfully used in decision making problems [2]. After the popularity of this extension in set theory, several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, interval-valued fuzzy sets, intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets, trapezoidal-valued intuitionistic fuzzy sets, type-2 fuzzy sets, and fuzzy multisets. These extensions have been successfully used in several practical applications of real life problems and scientific problems. Applications of these extensions can be found in artificial intelligence, computer science, medicine, control engineering, decision theory, expert systems, logic, management science, operations research, pattern recognition, and robotics. Torra felt that there are some limitations or deficiencies in these extensions. He proposed another extension of fuzzy set theory which is named as hesitant fuzzy set (HFS) theory [3]. This extension permits the several possible membership degrees of an element in [0,1]. HFS provides a much better description than the other extensions of fuzzy sets where the difficulty of establishing the membership degree and there is a specific set of possible values. Many studies on HFS have been conducted, such as extensions of HFS (see [4–9]).Group decision making (GDM) is a procedure to find the best/optimal alternative from a set of alternatives under the basis of certain criteria and the alternatives are evaluated by the group of decision makers for the criteria [10–14]. The opinion of DMs may also be in the form of preference of each pair of alternatives and provides a comparison of one alternative over another and this comparison is a preference relation; for some basics of preference relation see [15]. Preference relations have been developed and investigated in different modes, like multiplicative preference relations [16, 17], fuzzy preference relations [18], multiplicative fuzzy preference relation [10], incomplete fuzzy preference relation [11, 19], linguistic preference relations [20, 21], intuitionistic fuzzy preference relations [22, 23], and interval-valued hesitant preference relations [6]. Due to several external nonfeasible circumstances, like shortage of time, lack of knowledge, and available data resources, DMs provided their preferences opinion over the alternatives in the form of several possible numerical values. All the discussed preference relations do not handle such kind of situations. To overcome this problem, Zhu and Xu [24] introduced hesitant fuzzy preference relation (HFPR). Here the preference of an alternative over another alternative is a HFE; this HFE shows all possible preference values that denote the hesitant degree between two alternatives. HFPRs provide a better framework for the description of the DMs’ hesitation while providing their preferences among the alternatives [25–28].In preference relations based decision making problems, the concept of consistency plays an important role. It is the level or degree of satisfaction among the values in preference relation and these values are given by each DM [29]. Consensus is also an important and valuable concept in decision making problems based on preference relations. Consensus measure is used for the mutual understanding of DMs on the finally obtained alternatives [29]. Until now, several researchers of this area successfully made some progress to convert the preference relations as consistent and generate a certain level of consensus in decision making problems [30–33]. Nowadays, hesitant probabilistic fuzzy sets received good attention in multicriteria decision making. So, it is important to discuss the consistency and consensus measure for these preference relations. In this paper, we target to develop a group decision making model for HPFPRs where the consistency of the model and consensus among the DMs are under consideration. Our proposed model is efficient and practical; furthermore, it is strictly based on theoretical foundations. Pang et al. introduce the idea of probabilistic linguistic term set [34]. They also develop the aggregation operator of this set and proposed the extended TOPSIS version to handle the multicriteria group decision making problems. By getting the motivation from the hesitant fuzzy set and probabilistic linguistic term set, Xu and Zhou proposed the concept of hesitant probabilistic fuzzy set [35]. They investigated several aggregation operators with properties for hesitant probabilistic fuzzy set. A novel algorithm was developed to handle the multicriteria group decision making problems for hesitant probabilistic fuzzy set [36]. This concept was further extended to the continuous form of hesitant probabilistic fuzzy set [37]. Distance measures were also discussed for the continuous form and it was also applied to automotive industry safety evaluation problem. In these no one discusses the preference relations of the hesitant probabilistic fuzzy set. It is worth defining the preference relations for this set and discussing the consistency and consensus measure of the decision model.To accomplish these goals, this paper is structured in the following way. In Section2, some preliminary concepts are discussed to understand our proposal. Section 3 is devoted to design the basic structure of hesitant probabilistic fuzzy preference relations. A distance measured between the preference relations is developed, based on it, consistency measure of preference relations is derived. Also, two novel algorithms are designed to achieve acceptable consistency. In Section 4, consensus measure is defined for the group decision making and a novel algorithm is presented for reaching acceptable consensus among decision makers. Section 5 is dedicated to presenting a complete group decision making model dealing with both issues of consistency and consensus. In Section 6, numerical analysis of the developed model through a case study is performed to understand the importance of our proposal. Section 7 is dedicated for comparison between proposed model and existing ones. Section 8 ends the paper with some concluding remarks.


## 2. Preliminaries

Definition 1 (see [39]).
ForX=x1,x2,…,xn, a fixed set, a preference relation ≽ is a subset of X×X, which satisfies the following two:(1)
(Completeness) for allxi∈X and for all xj∈X, either xi≽xj or xj≽xi; that is, (xi,xj)∈≽ or (xj,xi)∈≽.(2)
(Transitivity) for allxi∈X, for all xj∈X, and for all xk∈X if xi≽xj and xj≽xk, then xi≽xk; that is, (xi,xj)∈≽ and (xj,xk)∈≽⇒(xi,xk)∈≽.Definition 2 (see [1, 40]).
ForX=x1,x2,…,xn, a fixed set, a HFPR is expressed by a matrix A=(aij)n×n⊆X×X, where aij=aijs∣s=1,2,3,…,#aij is a HFE, giving all the possible preference degrees of the alternative xi over xj. Also aij satisfy the following conditions for all i,j∈N: (1)aijσs+aji#aij-σs+1=1,aii=0.5,#aij=#aji,where aijσ(s) is the sth smallest value in aij and also elements of aij are arranged in increasing order.The notion of the hesitant fuzzy set given by Torra [3] is well known and has been successfully used to model vagueness of real life. It allows decision makers to give multiple membership values, but has the deficiency to deal with probabilities of preference degrees. To make hesitant fuzzy sets more compatible with real life, Xu and Zhou [35] defined hesitant probabilistic fuzzy element (HPFE) and hesitant probabilistic fuzzy set (HPFS).Definition 3 (see [35]).
Consider a fixed setR. The HPFS on R is defined as a mathematical symbol:(2)Hp=hγi∣pi∣γi,pi,where h(γi∣pi) is HPFE comprising the elements of the form γi∣pi, expressing the hesitant fuzzy information with probabilities to the set Hp,0≤γi≤1,i=1,2,3,…,#h, where #h is the number of elements in h(γi∣pi), pi∈[0,1] is the respective hesitant probability for γi, and ∑i=1#h‍pi=1.Score function, deviation function, and comparison laws are given to compare different HPFEs.Definition 4.
For a HPFEh(γi∣pi) where i=1,2,…,#h, s(h)=∑i=1#h‍γipi is called the score function of h(γi∣pi), where #h is the number of possible elements in h(γi∣pi).Definition 5.
For a HPFEh(γi∣pi) where i=1,2,…,#h, d(h)=∑i=1#h‍(γi-s(h))2pi is called the deviation function of h(γi∣pi), where s(h)=∑i=1#h‍γipi is the score function of h(γi∣pi) and #h is the number of possible elements in h(γi∣pi).The score and deviation functions are similar to the expectation and variance of the random variable, respectively, and, thus, the comparison laws for two HPFEsh1 and h2 can be presented as follows:Ifs(h1)>s(h2), then h1>h2,Ifs(h1)=s(h2) and d(h1)>d(h2), then h1<h2,Ifs(h1)=s(h2) and d(h1)=d(h2), then h1=h2,Ifs(h1)=s(h2) and d(h1)<d(h2), then h1>h2.


## 3. Hesitant Probabilistic Fuzzy Preference Relation

In order to build a complete model for group decision making first some operations are defined for HPFEs of the same length. Leth(γi∣pi), h1(γi′∣pi′), and h2(γi′′∣pi′′) be HPFEs with #h=#h1=#h2. Then(3)h1⊕h2=⋃γσs′∣pσs′∈h1,γσs′′∣pσs′′∈h2γσs′+γσs′′∣pσs′+pσs′′(4)h1⊖h2=⋃γσs′∣pσs′∈h1,γσs′′∣pσs′′∈h2γσs′-γσs′′∣pσs′+pσs′′(5)maxh1,h2=⋃γσs′∣pσs′∈h1,γσs′′∣pσs′′∈h2max⁡γσs′,γσs′′∣max⁡pσs′+pσs′′(6)minh1,h2=⋃γσs′∣pσs′∈h1,γσs′′∣pσs′′∈h2min⁡γσs′,γσs′′∣min⁡pσs′+pσs′′(7)ωh=⋃γσs∣pσs∈hωγσs∣ωpσs:ω≥0,where γσ(s)′∣pσ(s)′ and γσ(s)′′∣pσ(s)′′ are sth elements of h1(γi′∣pi′) and h2(γi′′∣pi′′), respectively.For simplicityh(γi∣pi) will be written as h throughout this paper.To allow decision makers to provide the preferences in hesitant probabilistic environment, we define hesitant probabilistic fuzzy preference relation (HPFPR).Definition 6 (HPFPR).
LetX=x1,x2,x3,…,xn be the set of alternatives. The HPFPR is a matrix H=(hij)n×n, where hij=hijs∣pijs:s=1,2,3,…,#hij is the HPFE expressing the possible preference degrees of the alternative xi over xj with probabilities and with j>i satisfying the following conditions:(8)hijσs+hjiσs=1,pijσs=pjiσs,#hij=#hji,hiiσs=12,hijσs<hijσs+1,hjiσs+1<hjiσs,where hijσ(s)∣pijσ(s) and hjiσ(s)∣pjiσ(s) are the sth elements in hij and hji, respectively.Remark 7.
The above definition is very much alike to the definition of probabilistic hesitant fuzzy preference relation proposed by Zhou and Xu [41, Definition 4]. They take (9)hii=12∣1:i=1,2,…,n.But if one adopted the technique of β-normalization, that is, making the length of hij the same by adding elements to HPFEs of shorter length, it implicates numerous errors and difficulties when dealing with consistency of HPFPRs and consensus among decision makers based on β-normalization. So fixing the length of diagonal elements to one does not match with the condition ∑s=1#h‍piiσ(s)=1. So by allowing variation, the probabilities of diagonal HPFEs helps us to maintain the spirit of HPFE in discussing consistency and consensus in the context of β-normalization. The diversity in probabilities of diagonal preference degrees does not cause any harm; the net impact remains the same as the sum of all probabilities is 1.Often the length of HPFEs has been different, but to apply the above defined operations (3), (4), and (7), the length is needed to be equal for all HPFEs. Some elements will be added to HPFE who has less elements, but the information it provides that should not be changed. Now, the definition of normalized hesitant probabilistic fuzzy preference relation (NHPFPR) is proposed.Definition 8 (normalized HPFPR).
A HPFPRH=(hij)n×n is called NHPFPR if the length of all hij is the same for all i,j=1,2,…,n.Leth=hi∣pi:i=1,2,3,…,#h be a HPFE. For preference degrees hi Zhu et al. [28] define a way to add elements in HFE; for an optimized parameter 0≤ζ≤1 the preference degree that will be added to hi is ζh++(1-ζ)h-, where h+ is the largest and h- is the smallest among hi. The decision maker can choose the value of ζ according to his risk preferences. The added element will be h+ and h- for ζ=1 and ζ=0, respectively, which demonstrate the optimistic and pessimistic approach of decision maker proposed by Xu and Xia [42]. In hesitant probabilistic fuzzy environment, some way is needed to assign probability to the added preference degree such that the information of HPFPR is not changed. There are many ways to do it; one option is to assign 0 to added preference degree ζh++(1-ζ)h-, but for the extreme cases pessimistic approach, that is, ζ=0, and optimistic approach, that is, ζ=1, the added element in HPFE is h-∣p-/d-#h+1 and h+∣p+/d-#h+1, respectively, where d is the required length of HPFEs and p- and p+ are the probabilities of h- and h+, respectively.For a given HPFPRH=(hij)n×n, we normalize it as follows. Let d=max#hij and i,j=1,2,…,n. For optimized parameter ζ=0,(10)h¯ij=hij,#hij=d;hijσ1∣pijσ1d-#hij+1,…,hijσ1∣pijσ1d-#hij+1︸d-#hij+1,hijσ2∣pijσ2,hijσ3∣pijσ3,…,hijσ#hij∣pijσ#hij︸#hij-1,#hij<d.For ζ=1,(11)h¯ij=hij,#hij=d;hijσ1∣pijσ1,hijσ2∣pijσ2,…,hijσ#hij-1∣pijσ#hij-1︸#hij-1,hijσ#hij∣pijσ#hijd-#hij+1,…,hijσ#hij∣pijσ#hijd-#hij+1︸d-#hij+1,#hij<d.For 0<ζ<1, for i<j(12)h¯ij=hij,#hij=d;hijσ1∣pijσ1,…,hijσtij∣pijσtij︸tij,ζhijσ#hij+1-ζhijσ1∣0,…,ζhijσ#hij+1-ζhijσ1∣0︸d-#hij,hijσtij+1∣pijσtij+1,…,hijσ#hij∣pijσ#hij︸#hij-tij,#hij<d,h¯ji=hji,#hji=d;hjiσ1∣pjiσ1,…,hjiσtji∣pjiσtji︸tji,1-ζhjiσ1+ζhjiσ#hji∣0,…,1-ζhjiσ1+ζhjiσhji∣0︸d-#hji,hjiσtji+1∣pjiσtji+1,…,hjiσ#hji∣pjiσ#hji︸#hji-tji,#hji<d,where tij=tji=maxs∈1,2,…,#hijhijσ(s)≤ζhijσ(#hij)+(1-ζ)hijσ(1).NowH¯=(h¯ij)n×n is NHPFPR; next we deal with consistency.Example 9.
Let(13)H1=.5∣10.1196∣0.4481,0.3567∣0.55190.2898∣10.1702∣0.1906,0.231∣0.4432,0.9392∣0.36620.8804∣0.4481,0.6433∣0.5519.5∣10.0712∣0.0575,0.566∣0.94250.3587∣0.2586,0.7928∣0.74140.7102∣10.9288∣0.0575,0.434∣0.9425.5∣10.5053∣0.9692,0.7557∣0.03080.8298∣0.1906,0.769∣0.4432,0.0608∣0.36620.6413∣0.2586,0.2072∣0.74140.4947∣0.9692,0.2443∣0.0308.5∣1,H2=.5∣10.331∣0.4663,0.4815∣0.53,0.7136∣0.00360.3152∣0.5797,0.3269∣0.1281,0.5828∣0.29210.3593∣0.5865,0.5943∣0.2853,0.8714∣0.12810.669∣0.4663,0.5185∣0.53,0.2864∣0.0036.5∣10.4762∣0.6629,0.7996∣0.33710.2436∣10.6848∣0.5797,0.6731∣0.1281,0.4172∣0.29210.5238∣0.6629,0.2004∣0.3371.5∣10.1829∣0.1629,0.3216∣0.404,0.8871∣0.43310.6407∣0.5865,0.4057∣0.2853,0.1286∣0.12810.7564∣10.8171∣0.1629,0.6784∣0.404,0.1129∣0.4331.5∣1.Then H1 and H2 are HPFPRs. Now by taking optimized parameters 0, 1 for H1 and H2, respectively, the following NHPFPR are obtained. (14)H¯1=.5∣.3333,.5∣.3333,.5∣.33330.1196∣0.2241,0.1196∣0.2241,0.3567∣0.55190.2898∣.3333,0.2898∣.3333,0.2898∣.33330.1702∣0.1906,0.231∣0.4432,0.9392∣0.36620.8804∣0.2241,0.8804∣0.2241,0.6433∣0.5519.5∣.3333,.5∣.3333,.5∣.33330.0712∣0.0287,0.0712∣0.0287,0.566∣0.94250.3587∣0.1293,0.3587∣0.1293,0.7928∣0.74140.7102∣.3333,0.7102∣.3333,0.7102∣.33330.9288∣0.0287,0.9288∣0.0287,0.434∣0.9425.5∣.3333,.5∣.3333,.5∣.33330.5053∣0.4846,0.5053∣0.4846,0.7557∣0.03080.8298∣0.1906,0.769∣0.4432,0.0608∣0.36620.6413∣0.1293,0.6413∣0.1293,0.2072∣0.74140.4947∣0.4846,0.4947∣0.4846,0.2443∣0.0308.5∣.3333,.5∣.3333,.5∣.3333,H¯2=.5∣.3333,.5∣.3333,.5∣.33330.331∣0.4663,0.4815∣0.53,0.7136∣0.00360.3152∣0.5797,0.3269∣0.1281,0.5828∣0.29210.3593∣0.5865,0.5943∣0.2853,0.8714∣0.12810.669∣0.4663,0.5185∣0.53,0.2864∣0.0036.5∣.3333,.5∣.3333,.5∣.33330.4762∣0.6629,0.7996∣0.1686,0.7996∣0.16860.2436∣.3333,0.2436∣.3333,0.2436∣.33330.6848∣0.5797,0.6731∣0.1281,0.4172∣0.29210.5238∣0.6629,0.2004∣0.1686,0.2004∣0.1686.5∣.3333,.5∣.3333,.5∣.33330.1829∣0.1629,0.3216∣0.404,0.8871∣0.43310.6407∣0.5865,0.4057∣0.2853,0.1286∣0.12810.7564∣.3333,0.7564∣.3333,0.7564∣.33330.8171∣0.1629,0.6784∣0.404,0.1129∣0.4331.5∣.3333,.5∣.3333,.5∣.3333.


### 3.1. Consistency Measure of Hesitant Probabilistic Fuzzy Preference Relation

In order to obtain valuable decision from preference relations they should be consistent in a sense that, let us say,x1 is preferable to x2 and x2 is preferable to x3 then x1 must be preferable to x3. Several authors have pursued consistency issues for preference relations [1, 21, 32, 33, 43].Additive consistency for fuzzy preference degrees is well known; actually Tanino [44] defined the additive consistency for fuzzy preference relation based on moderate stochastic transitivity, well known in the probabilistic choice theory [45, page 27]. Furthermore, many kinds of transitivity are proposed and studied for probabilities in comparing the preferences in the choice theory.Considering HPFPRH=(hij)n×n by (10), (11) calculate NHPFPR H¯=(h¯ij)n×n. The weak stochastic transitivity for probability means(15)p¯ikσs≥12∧p¯kjσs≥12⟹p¯ijσs≥12.This will provide a platform to define consistency for HPFPR.Definition 10 (consistency).
For a given HPFPR,H=(hij)n×n and its NHFPR H¯=(h¯ij)n×n with optimized parameter ζ. If(16)h¯ijσs=h¯ikσs-h¯jkσs+12,(17)p¯ikσs≥12∧p¯kjσs≥12⟹p¯ijσs≥12,for all i,j,k=1,2,3,…,n, then H is called consistent HPFPR with optimized parameter ζ.But many times preference relations are not consistent and, for meaningful decision making, some level of consistency is required in the least if it is not fully consistent. For preference degrees, take the summation of (16) for all k(18)nh¯ijσs=∑k=1nh¯ikσs-h¯jkσs+n2;therefore,(19)h¯ijσs=1n∑k=1nh¯ikσs-h¯jkσs+12.Thus, (19) is satisfied by a consistent HPFPR, if not put(20)h~ijσs=1n∑k=1nh¯ikσs-h¯jkσs+12;one can check that the preference degrees h~ijσ(s) obtained from the above equation are consistent. For probabilities, matter is not that simple; some mechanism is needed to make consistent probabilities with all the restrictions of HPFPR like p¯ijσ(s)=p¯jiσ(s) and ∑s=1#h¯ij‍p¯ijσ(s)=1. Let p¯ikσ(s)≥1/2 and p¯kjσ(s)≥1/2. Then define(21)p~ijσs≔p¯ikσs+p¯kjσs2≥12,to keep account for all k=1,2,…,n and keeping in mind ∑s=1#h¯ij‍p¯ijσ(s)=1 we modify (21) as(22)p~ijσs=p¯i1σs+p¯1jσs/2+p¯i2σs+p¯2jσs/2+⋯+p¯inσs+p¯njσs/2n.Hence ∑s=1#h¯ij‍p¯ijσ(s)=1 and p~ijσ(s)=p~jiσ(s) and if p¯ikσ(s)≥1/2 and p¯kjσ(s)≥1/2 for all k=1,2,…,n then surely p~ijσ(s)≥1/2. But, it is possible that p¯ikσ(s)≥1/2 and p¯kjσ(s)≥1/2 are not true for some k that will lead to a situation where p~ikσ(s)≥1/2∧p¯~kjσ(s)≥1/2 and p~ijσ(s)<1/2. Now if another convex combination is calculated by (22) then obtained probability will increase. These observations lead to the following novel algorithm producing a sequence of HPFPRs convergent to fully consistent HPFPR.Algorithm 11 (consistent HPFPR calculator).
Input. HPFPR H and optimized parameter ζ.
Output. NHPFPR H¯, consistent HPFPR H~, and number of iterations t.
Step 1. Compute NHPFPR H¯ by (10) or (11). Let t=0 and H~(0)=hij(0)n×n be defined as(23)h~ij0=1n∑k=1nh¯ikσs-h¯jkσs+12∣p¯ijσs:s=1,2,3,…,d.
Step 2. If the following condition is true, then go to Step 4; otherwise, go to Step 3. (24)p~iktσs≥12∧p~kjtσs≥12⟹p~ijtσs≥12,∀i,j,k=1,2,…,n.
Step 3. H~(t+1)=h~ij(t+1)n×n is defined as(25)h~ijt+1=h~ijtσs∣1n∑k=1np~iktσs+p~kjtσs2:s=1,2,3,…,d;put t=t+1. Go to Step 2.
Step 4. Output NHPFPR H¯, consistent HPFPR H~(t), and number of iterations t.
Step 5. End.Proposition 12.
LetH=(hij)n×n be a HPFPR with its NHFPR H¯=(h¯ij)n×n with optimized parameter ζ. Then H~=(h~ij)n×n output of Algorithm 11 is consistent HPFPR.Proof.
Definition10, (23) and (24) directly imply this proposition.This result also gives the following theorem.Proposition 13.
Consider a HPFPRH=(hij)n×n, its NHFPR H¯=(h¯ij)n×n, and H~=(h~ij)n×n consistent HPFPR with optimized parameter ζ. Then, H is consistent if and only if H¯=H~.The above algorithm is quite efficient; to see this fact, we generate 1000 random HPFPRs with different values ofn, d and apply Algorithm 11 to find their consistent HPFPRs. Table 1 shows the average value of the number of iterations in Algorithm 11.Table 1
Average value of iterations in Algorithm11.
n d ζ = 0 ζ = 1 4 4 1.684 1.679 3 1.357 1.373 2 1.068 1.069  5 5 2.527 2.429 4 1.988 1.958 3 1.405 1.439  6 6 2.924 2.948 5 2.824 2.796 4 2.242 2.25  7 7 2.664 2.839 6 3.14 3.17 5 3.148 3.168  8 8 2.168 2.192 7 2.777 2.732 6 3.254 3.38  9 9 1.65 1.644 8 2.067 2.005 7 2.634 2.653  10 10 1.301 1.345 9 1.56 1.496 8 1.884 1.825Example 14.
TakeH1,H2 and their optimized parameter the same as in Example 9. Then by Algorithm 11, the following consistent HPFPRs H~1 and H~2 are obtained. (26)H~1=.5∣0.2574,.5∣0.2894,.5∣0.45320.3173∣0.2567,0.3325∣0.2882,0.3959∣0.45510.1088∣0.2576,0.124∣0.2891,0.4214∣0.45330.1535∣0.2575,0.1839∣0.2895,0.7684∣0.4530.6827∣0.2567,0.6675∣0.2882,0.6041∣0.4551.5∣0.2559,.5∣0.287,.5∣0.4570.2915∣0.2569,0.2915∣0.2879,0.5256∣0.45520.3361∣0.2568,0.3513∣0.2883,0.8725∣0.45490.8912∣0.2576,0.876∣0.2891,0.5786∣0.45330.7085∣0.2569,0.7085∣0.2879,0.4744∣0.4552.5∣0.2578,.5∣0.2888,.5∣0.45340.5447∣0.2577,0.5599∣0.2892,0.8469∣0.45310.8465∣0.2575,0.8161∣0.2895,0.2316∣0.4530.6639∣0.2568,0.6487∣0.2883,0.1275∣0.45490.4553∣0.2577,0.4401∣0.2892,0.1531∣0.4531.5∣0.2576,.5∣0.2897,.5∣0.4527,H~2=.5∣0.4915,.5∣0.3192,.5∣0.18930.4042∣0.4702,0.4603∣0.3303,0.7095∣0.19950.4035∣0.4631,0.5519∣0.2889,0.6658∣0.2480.1978∣0.4228,0.3906∣0.3291,0.7925∣0.24810.5958∣0.4702,0.5397∣0.3303,0.2905∣0.1995.5∣0.449,.5∣0.3413,.5∣0.20970.4994∣0.4418,0.5916∣0.2999,0.4563∣0.25820.2937∣0.4015,0.4303∣0.3402,0.5829∣0.25830.5965∣0.4631,0.4481∣0.2889,0.3342∣0.2480.5006∣0.4418,0.4084∣0.2999,0.5437∣0.2582.5∣0.4347,.5∣0.2585,.5∣0.30680.2943∣0.3944,0.3387∣0.2987,0.6267∣0.30690.8022∣0.4228,0.6094∣0.3291,0.2075∣0.24810.7063∣0.4015,0.5697∣0.3402,0.4171∣0.25830.7057∣0.3944,0.6613∣0.2987,0.3733∣0.3069.5∣0.354,.5∣0.339,.5∣0.307.Remark 15.
To see the consistency of HPFPR geometrically, three area graphs of fuzzy preference degrees[hijσ(s)], probabilities of preference degrees [pijσ(s)], and score values [hijσ(s)∗pijσ(s)] are made. The procedure to make these graphs is explained for H~1, as follows.
Three matrices ,P, and S of order 4×12 of fuzzy preference values, probability values, and score values are made from HPFPR H~1, (27)F=.5.5.5.3959.3325.3173.4214.124.1088.7684.1839.1535.68270.6675.6041.5.5.5.5256.2915.2915.8725.3513.3361.8912.876.5786.7085.7085.4744.5.5.5.8469.5599.5447.8465.8161.2316.6639.6487.1275.4553.4401.1531.5.5.5,P=.4532.2894.2573.4551.2882.2567.4533.2891.2576.453.2895.2575.4551.2882.2567.457.287.2559.4552.2879.2569.4549.2883.2568.4533.2891.2576.4552.2879.2569.4534.2888.2578.4531.2892.2577.453.2895.2575.4549.2883.2568.4531.2892.2577.4527.2897.2576,S=.2266.1447.1287.1802.0958.0814.191.0359.028.348.0532.0395.2749.1924.1752.2285.1435.128.2392.0839.0749.3969.1013.0863.2623.2533.2295.216.204.0182.2267.1444.1289.3837.1619.1403.2363.218.1049.187.1705.058.1273.1173.0694.2264.1448.1288.The area graphs are made of the above matrices by using Matlab drawing tool bar. Figures 1, 2, and 3 and Figures 4, 5, and 6 show the comparison of area graphs for fuzzy preference degrees, probability values, and score values between H¯1, H~1 and H¯2, H~2, respectively. The areas are more smooth for consistent HPFPRs H~1 and H~2.Figure 1
Area graphs of fuzzy preference degrees forH1.Figure 2
Area graphs of probabilities forH1.Figure 3
Area graphs of score values forH1.Figure 4
Area graphs of fuzzy preference degrees forH2.Figure 5
Area graphs of probabilities forH2.Figure 6
Area graphs of score values forH2.Once the consistent HPFPR is found, we are in a position to make HPFPR acceptably consistent by defining consistency measure. In order to define consistency measure, first distance between two HPFEs is defined. Leth1=γ1s∣p1s:s=1,2,3,…,#h and h2=γ2s∣p2s:s=1,2,3,…,#h2 with #h1=#h2=#h. Then (28)Dh1,h2=∑γ1σs∣p1σs∈h1,γ2σs∣p2σs∈h2maxγ1σs-γ2σs,p1σs-p2σs#h.Based on the above equation the distance between two HPFPRs is defined as follows.Definition 16 (distance).
Consider two HPFPRsH1=(hij,1)n×n and H2=(hij,2)n×n and their NHPFPRs H¯1=(h¯ij,1)n×n and H¯2=(h¯ij,2)n×n. The distance is defined as(29)DH1,H2=2nn+1∑i≤jnDh¯ij,1,h¯ij,2.It is clear that the following properties are satisfied forD(H1,H2):(1)
0≤D(H1,H2)≤1;(2)
D(H1,H2)=D(H2,H1);(3)
H1=H2 if and only if D(H1,H2)=0.While the decision maker provides its preference in form of HPFPR, it should be noted that it can be used for decision making with good results only if it has sufficient consistency.Definition 17 (consistency index).
For a given HPFPRH=(hij)n×n, its NHPFPR H¯=(h¯ij)n×n and consistent HPFPR H~=(h~ij)n×n with optimized parameter 0≤ζ≤1 obtained from Algorithm 11. The consistency index of H is defined to be the distance between H¯ and H~, denoted as (30)CIH=DH¯,H~.It is clear thatCI(H)=0 if and only if H is consistent. Ideally, the decision maker should provide consistent HPFPR so that it can be used for meaningful decision making. However, some margin of error should be provided to decision maker relative to the practical problems.Definition 18 (acceptably consistent HPFPR).
Consider a HPFPRH=(hij)n×n. For a given tolerance value CIr HPFPR H is said to be acceptably consistent HPFPR if(31)CIH≤CIr.If a HPFPR is not even acceptably consistent, then the decision maker should revisit and modify it. A novel algorithm is proposed to make a HPFPR acceptably consistent.Algorithm 19 (acceptably consistent HPFPR calculator).
Input. The HPFPR H=(hij)n×n, the consistency tolerance value CIr, and the parameter α∈(0,1).
Output. The acceptably consistent HPFPR H(t), consistency index CI(H(t)), and number of iterations value t.
Step 1. Compute H¯=(h¯ij)n×n with optimized parameter ζ. Let t=0, H(0)=(hij(0))n×n=H¯=(h¯ij)n×n.
Step 2. Calculate the consistent HPFPR H~(t)=(h~ij(t))n×n by applying Algorithm 11 to H(t) and consistency index CI(H(t)), where(32)CIHt=2nn+1∑i≤jnDhijt,h~ijt.
Step 3. If CI(H(t))≤CIr, then go to Step 5; otherwise, go to Step 4.
Step 4. Make the adjusted HPFPR H(t+1)=hij(t+1), where(33)hijt+1=αhijt⊕1-αh~ijt.Let t=t+1; now return to Step 2.
Step 5. Output , H(t), and CI(H(t)).
Step 6. End.This proposed algorithm will be convergent as the following result shows.Theorem 20.
Consider a HPFPRH, consistency tolerance value CIr, and the sequence Ht of HPFPRs generated by Algorithm 19. Then(34)CIHt+1<CIHt∀t,limt→∞CIHt=0.Proof.
From (23) and (33), it follows that(35)h~ijt+1σs=1n∑k=1nhikt+1σs-hjkt+1σs+12=1n∑k=1nαhiktσs+1-αh~iktσs-αhjktσs-1-αh~jktσs+12=α1n∑k=1nhiktσs-hjktσs+12+1-α1n∑k=1nh~iktσs-h~jktσs+12=h~ijtσs;also (36)hijt+1σs=αhijtσs+1-αh~ijtσs.Thus,(37)hijt+1σs-h~ijt+1σs=αhijtσs+1-αh~ijtσs-h~ijtσs=αhijtσs-h~ijtσs<hijtσs-h~ijtσs;therefore,(38)limt→∞hijtσs-h~ijtσs=limt→∞αhijt-1σs-h~ijt-1σs=⋯=limt→∞αthij0σs-h~ij0σs=0.By (33), (39)pijtσs<pijt+1σs<p~ijtσs,with each iteration pij(t)σ(s), will come closer to p~ij(t)σ(s). As H~(t) is consistent; therefore, each iteration will make H(t) more consistent; that is, if pik(t)σ(s)≥1/2∧pkj(t)σ(s)≥1/2∧pij(t)σ(s)<1/2, then by (33) and (25) (40)pijt+1σs-p~ijt+1σs<pijtσs-p~ijtσs,and each iteration will increase pij(t)σ(s); thus eventually it will be ≥1/2. Hence H(t) will become consistent when t→∞; that is, (41)limt→∞pijtσs-p~ijtσs=0.Finally, (37) and (40) imply (42)CIHt+1<CIHt∀t,limt→∞CIHt=0,following from (38) and (41).The choice of parametersn, d, α directly affects the performance of Algorithm 19. For performance measurement of Algorithm 19, 1000 random HPFPRs are generated and their acceptably consistent HPFPRs are computed by Algorithm 19, the average iteration in Algorithm 19 with respect to different parameters shown in Table 2. It is apparent that Algorithm 19 is quite efficient and, as for the effects of parameters, the increase in value α leads to more iteration in Algorithm 19 to compute consistent HPFPR; therefore, it is suggested to choose α small. Also, the number of iterations in Algorithm 19 is inversely proportional to the consistency index CIr.Table 2
Average iterations values in Algorithm19.
n d C I r α 0.1 0.3 0.6 0.8 4 4 0.05 1.001 1.939 3.226 6.88 0.1 0.996 0.999 1.898 3.659 0.15 0.939 0.959 0.998 1.824 3 0.05 1.001 1.939 3.528 7.326 0.1 0.991 0.997 2 4.084 0.15 0.954 0.95 1.208 2.169 2 0.05 1.05 1.882 3.527 7.415 0.1 0.982 1.054 2.089 4.1 0.15 0.868 0.891 1.297 2.404  5 5 0.05 1 1.976 3.095 6.851 0.1 1 1 1.978 3.71 0.15 0.989 0.994 1.017 1.854 4 0.05 1.004 1.992 3.418 7.266 0.1 1 1 1.989 4.021 0.15 0.999 0.997 1.123 2.214 3 0.05 1.001 1.992 3.767 7.709 0.1 1 1.002 2.123 4.43 0.15 0.994 0.997 1.403 2.575  6 6 0.05 1 1.994 3.028 6.776 0.1 1 1 1.991 3.654 0.15 1 1 1 1.818 5 0.05 1.001 1.999 3.2 7.025 0.1 1 1 1.997 3.942 0.15 1 1 1.016 2.074 4 0.05 1.001 2 3.606 7.446 0.1 1 1 2.011 4.249 0.15 0.999 1 1.178 2.443  7 7 0.05 1 1.993 3.009 6.659 0.1 1 1 1.991 3.487 0.15 0.998 1 0.999 1.732 6 0.05 1 1.998 3.05 6.912 0.1 1 1 1.999 3.816 0.15 1 1 1.001 1.972 5 0.05 1 2 3.261 7.125 0.1 1 1 2 4.023 0.15 1 1 1.022 2.168  8 8 0.05 1 1.993 3.002 6.526 0.1 1 1 1.984 3.317 0.15 1 1 1 1.599 7 0.05 1 2 3.009 6.831 0.1 1 1 1.997 3.658 0.15 1 1 1 1.867 6 0.05 1 2 3.078 7.003 0.1 1 1 2 3.927 0.15 1 1 1 2.026  9 9 0.05 1 1.993 3 6.396 0.1 1 1 1.978 3.229 0.15 1 1 1 1.474 8 0.05 1 2 3 6.676 0.1 1 1 1.998 3.47 0.15 1 1 1 1.74 7 0.05 1 2 3.01 6.92 0.1 1 1 2 3.786 0.15 1 1 1 1.942  10 10 0.05 1 1.986 3 6.326 0.1 1 1 1.973 3.164 0.15 1 1 1 1.367 9 0.05 1 1.994 3 6.543 0.1 1 1 1.995 3.308 0.15 1 1 1 1.58 8 0.05 1 2 3.002 6.798 0.1 1 1 2 3.618 0.15 1 1 1 1.864


## 3.1. Consistency Measure of Hesitant Probabilistic Fuzzy Preference Relation

In order to obtain valuable decision from preference relations they should be consistent in a sense that, let us say,x1 is preferable to x2 and x2 is preferable to x3 then x1 must be preferable to x3. Several authors have pursued consistency issues for preference relations [1, 21, 32, 33, 43].Additive consistency for fuzzy preference degrees is well known; actually Tanino [44] defined the additive consistency for fuzzy preference relation based on moderate stochastic transitivity, well known in the probabilistic choice theory [45, page 27]. Furthermore, many kinds of transitivity are proposed and studied for probabilities in comparing the preferences in the choice theory.Considering HPFPRH=(hij)n×n by (10), (11) calculate NHPFPR H¯=(h¯ij)n×n. The weak stochastic transitivity for probability means(15)p¯ikσs≥12∧p¯kjσs≥12⟹p¯ijσs≥12.This will provide a platform to define consistency for HPFPR.Definition 10 (consistency).
For a given HPFPR,H=(hij)n×n and its NHFPR H¯=(h¯ij)n×n with optimized parameter ζ. If(16)h¯ijσs=h¯ikσs-h¯jkσs+12,(17)p¯ikσs≥12∧p¯kjσs≥12⟹p¯ijσs≥12,for all i,j,k=1,2,3,…,n, then H is called consistent HPFPR with optimized parameter ζ.But many times preference relations are not consistent and, for meaningful decision making, some level of consistency is required in the least if it is not fully consistent. For preference degrees, take the summation of (16) for all k(18)nh¯ijσs=∑k=1nh¯ikσs-h¯jkσs+n2;therefore,(19)h¯ijσs=1n∑k=1nh¯ikσs-h¯jkσs+12.Thus, (19) is satisfied by a consistent HPFPR, if not put(20)h~ijσs=1n∑k=1nh¯ikσs-h¯jkσs+12;one can check that the preference degrees h~ijσ(s) obtained from the above equation are consistent. For probabilities, matter is not that simple; some mechanism is needed to make consistent probabilities with all the restrictions of HPFPR like p¯ijσ(s)=p¯jiσ(s) and ∑s=1#h¯ij‍p¯ijσ(s)=1. Let p¯ikσ(s)≥1/2 and p¯kjσ(s)≥1/2. Then define(21)p~ijσs≔p¯ikσs+p¯kjσs2≥12,to keep account for all k=1,2,…,n and keeping in mind ∑s=1#h¯ij‍p¯ijσ(s)=1 we modify (21) as(22)p~ijσs=p¯i1σs+p¯1jσs/2+p¯i2σs+p¯2jσs/2+⋯+p¯inσs+p¯njσs/2n.Hence ∑s=1#h¯ij‍p¯ijσ(s)=1 and p~ijσ(s)=p~jiσ(s) and if p¯ikσ(s)≥1/2 and p¯kjσ(s)≥1/2 for all k=1,2,…,n then surely p~ijσ(s)≥1/2. But, it is possible that p¯ikσ(s)≥1/2 and p¯kjσ(s)≥1/2 are not true for some k that will lead to a situation where p~ikσ(s)≥1/2∧p¯~kjσ(s)≥1/2 and p~ijσ(s)<1/2. Now if another convex combination is calculated by (22) then obtained probability will increase. These observations lead to the following novel algorithm producing a sequence of HPFPRs convergent to fully consistent HPFPR.Algorithm 11 (consistent HPFPR calculator).
Input. HPFPR H and optimized parameter ζ.
Output. NHPFPR H¯, consistent HPFPR H~, and number of iterations t.
Step 1. Compute NHPFPR H¯ by (10) or (11). Let t=0 and H~(0)=hij(0)n×n be defined as(23)h~ij0=1n∑k=1nh¯ikσs-h¯jkσs+12∣p¯ijσs:s=1,2,3,…,d.
Step 2. If the following condition is true, then go to Step 4; otherwise, go to Step 3. (24)p~iktσs≥12∧p~kjtσs≥12⟹p~ijtσs≥12,∀i,j,k=1,2,…,n.
Step 3. H~(t+1)=h~ij(t+1)n×n is defined as(25)h~ijt+1=h~ijtσs∣1n∑k=1np~iktσs+p~kjtσs2:s=1,2,3,…,d;put t=t+1. Go to Step 2.
Step 4. Output NHPFPR H¯, consistent HPFPR H~(t), and number of iterations t.
Step 5. End.Proposition 12.
LetH=(hij)n×n be a HPFPR with its NHFPR H¯=(h¯ij)n×n with optimized parameter ζ. Then H~=(h~ij)n×n output of Algorithm 11 is consistent HPFPR.Proof.
Definition10, (23) and (24) directly imply this proposition.This result also gives the following theorem.Proposition 13.
Consider a HPFPRH=(hij)n×n, its NHFPR H¯=(h¯ij)n×n, and H~=(h~ij)n×n consistent HPFPR with optimized parameter ζ. Then, H is consistent if and only if H¯=H~.The above algorithm is quite efficient; to see this fact, we generate 1000 random HPFPRs with different values ofn, d and apply Algorithm 11 to find their consistent HPFPRs. Table 1 shows the average value of the number of iterations in Algorithm 11.Table 1
Average value of iterations in Algorithm11.
n d ζ = 0 ζ = 1 4 4 1.684 1.679 3 1.357 1.373 2 1.068 1.069  5 5 2.527 2.429 4 1.988 1.958 3 1.405 1.439  6 6 2.924 2.948 5 2.824 2.796 4 2.242 2.25  7 7 2.664 2.839 6 3.14 3.17 5 3.148 3.168  8 8 2.168 2.192 7 2.777 2.732 6 3.254 3.38  9 9 1.65 1.644 8 2.067 2.005 7 2.634 2.653  10 10 1.301 1.345 9 1.56 1.496 8 1.884 1.825Example 14.
TakeH1,H2 and their optimized parameter the same as in Example 9. Then by Algorithm 11, the following consistent HPFPRs H~1 and H~2 are obtained. (26)H~1=.5∣0.2574,.5∣0.2894,.5∣0.45320.3173∣0.2567,0.3325∣0.2882,0.3959∣0.45510.1088∣0.2576,0.124∣0.2891,0.4214∣0.45330.1535∣0.2575,0.1839∣0.2895,0.7684∣0.4530.6827∣0.2567,0.6675∣0.2882,0.6041∣0.4551.5∣0.2559,.5∣0.287,.5∣0.4570.2915∣0.2569,0.2915∣0.2879,0.5256∣0.45520.3361∣0.2568,0.3513∣0.2883,0.8725∣0.45490.8912∣0.2576,0.876∣0.2891,0.5786∣0.45330.7085∣0.2569,0.7085∣0.2879,0.4744∣0.4552.5∣0.2578,.5∣0.2888,.5∣0.45340.5447∣0.2577,0.5599∣0.2892,0.8469∣0.45310.8465∣0.2575,0.8161∣0.2895,0.2316∣0.4530.6639∣0.2568,0.6487∣0.2883,0.1275∣0.45490.4553∣0.2577,0.4401∣0.2892,0.1531∣0.4531.5∣0.2576,.5∣0.2897,.5∣0.4527,H~2=.5∣0.4915,.5∣0.3192,.5∣0.18930.4042∣0.4702,0.4603∣0.3303,0.7095∣0.19950.4035∣0.4631,0.5519∣0.2889,0.6658∣0.2480.1978∣0.4228,0.3906∣0.3291,0.7925∣0.24810.5958∣0.4702,0.5397∣0.3303,0.2905∣0.1995.5∣0.449,.5∣0.3413,.5∣0.20970.4994∣0.4418,0.5916∣0.2999,0.4563∣0.25820.2937∣0.4015,0.4303∣0.3402,0.5829∣0.25830.5965∣0.4631,0.4481∣0.2889,0.3342∣0.2480.5006∣0.4418,0.4084∣0.2999,0.5437∣0.2582.5∣0.4347,.5∣0.2585,.5∣0.30680.2943∣0.3944,0.3387∣0.2987,0.6267∣0.30690.8022∣0.4228,0.6094∣0.3291,0.2075∣0.24810.7063∣0.4015,0.5697∣0.3402,0.4171∣0.25830.7057∣0.3944,0.6613∣0.2987,0.3733∣0.3069.5∣0.354,.5∣0.339,.5∣0.307.Remark 15.
To see the consistency of HPFPR geometrically, three area graphs of fuzzy preference degrees[hijσ(s)], probabilities of preference degrees [pijσ(s)], and score values [hijσ(s)∗pijσ(s)] are made. The procedure to make these graphs is explained for H~1, as follows.
Three matrices ,P, and S of order 4×12 of fuzzy preference values, probability values, and score values are made from HPFPR H~1, (27)F=.5.5.5.3959.3325.3173.4214.124.1088.7684.1839.1535.68270.6675.6041.5.5.5.5256.2915.2915.8725.3513.3361.8912.876.5786.7085.7085.4744.5.5.5.8469.5599.5447.8465.8161.2316.6639.6487.1275.4553.4401.1531.5.5.5,P=.4532.2894.2573.4551.2882.2567.4533.2891.2576.453.2895.2575.4551.2882.2567.457.287.2559.4552.2879.2569.4549.2883.2568.4533.2891.2576.4552.2879.2569.4534.2888.2578.4531.2892.2577.453.2895.2575.4549.2883.2568.4531.2892.2577.4527.2897.2576,S=.2266.1447.1287.1802.0958.0814.191.0359.028.348.0532.0395.2749.1924.1752.2285.1435.128.2392.0839.0749.3969.1013.0863.2623.2533.2295.216.204.0182.2267.1444.1289.3837.1619.1403.2363.218.1049.187.1705.058.1273.1173.0694.2264.1448.1288.The area graphs are made of the above matrices by using Matlab drawing tool bar. Figures 1, 2, and 3 and Figures 4, 5, and 6 show the comparison of area graphs for fuzzy preference degrees, probability values, and score values between H¯1, H~1 and H¯2, H~2, respectively. The areas are more smooth for consistent HPFPRs H~1 and H~2.Figure 1
Area graphs of fuzzy preference degrees forH1.Figure 2
Area graphs of probabilities forH1.Figure 3
Area graphs of score values forH1.Figure 4
Area graphs of fuzzy preference degrees forH2.Figure 5
Area graphs of probabilities forH2.Figure 6
Area graphs of score values forH2.Once the consistent HPFPR is found, we are in a position to make HPFPR acceptably consistent by defining consistency measure. In order to define consistency measure, first distance between two HPFEs is defined. Leth1=γ1s∣p1s:s=1,2,3,…,#h and h2=γ2s∣p2s:s=1,2,3,…,#h2 with #h1=#h2=#h. Then (28)Dh1,h2=∑γ1σs∣p1σs∈h1,γ2σs∣p2σs∈h2maxγ1σs-γ2σs,p1σs-p2σs#h.Based on the above equation the distance between two HPFPRs is defined as follows.Definition 16 (distance).
Consider two HPFPRsH1=(hij,1)n×n and H2=(hij,2)n×n and their NHPFPRs H¯1=(h¯ij,1)n×n and H¯2=(h¯ij,2)n×n. The distance is defined as(29)DH1,H2=2nn+1∑i≤jnDh¯ij,1,h¯ij,2.It is clear that the following properties are satisfied forD(H1,H2):(1)
0≤D(H1,H2)≤1;(2)
D(H1,H2)=D(H2,H1);(3)
H1=H2 if and only if D(H1,H2)=0.While the decision maker provides its preference in form of HPFPR, it should be noted that it can be used for decision making with good results only if it has sufficient consistency.Definition 17 (consistency index).
For a given HPFPRH=(hij)n×n, its NHPFPR H¯=(h¯ij)n×n and consistent HPFPR H~=(h~ij)n×n with optimized parameter 0≤ζ≤1 obtained from Algorithm 11. The consistency index of H is defined to be the distance between H¯ and H~, denoted as (30)CIH=DH¯,H~.It is clear thatCI(H)=0 if and only if H is consistent. Ideally, the decision maker should provide consistent HPFPR so that it can be used for meaningful decision making. However, some margin of error should be provided to decision maker relative to the practical problems.Definition 18 (acceptably consistent HPFPR).
Consider a HPFPRH=(hij)n×n. For a given tolerance value CIr HPFPR H is said to be acceptably consistent HPFPR if(31)CIH≤CIr.If a HPFPR is not even acceptably consistent, then the decision maker should revisit and modify it. A novel algorithm is proposed to make a HPFPR acceptably consistent.Algorithm 19 (acceptably consistent HPFPR calculator).
Input. The HPFPR H=(hij)n×n, the consistency tolerance value CIr, and the parameter α∈(0,1).
Output. The acceptably consistent HPFPR H(t), consistency index CI(H(t)), and number of iterations value t.
Step 1. Compute H¯=(h¯ij)n×n with optimized parameter ζ. Let t=0, H(0)=(hij(0))n×n=H¯=(h¯ij)n×n.
Step 2. Calculate the consistent HPFPR H~(t)=(h~ij(t))n×n by applying Algorithm 11 to H(t) and consistency index CI(H(t)), where(32)CIHt=2nn+1∑i≤jnDhijt,h~ijt.
Step 3. If CI(H(t))≤CIr, then go to Step 5; otherwise, go to Step 4.
Step 4. Make the adjusted HPFPR H(t+1)=hij(t+1), where(33)hijt+1=αhijt⊕1-αh~ijt.Let t=t+1; now return to Step 2.
Step 5. Output , H(t), and CI(H(t)).
Step 6. End.This proposed algorithm will be convergent as the following result shows.Theorem 20.
Consider a HPFPRH, consistency tolerance value CIr, and the sequence Ht of HPFPRs generated by Algorithm 19. Then(34)CIHt+1<CIHt∀t,limt→∞CIHt=0.Proof.
From (23) and (33), it follows that(35)h~ijt+1σs=1n∑k=1nhikt+1σs-hjkt+1σs+12=1n∑k=1nαhiktσs+1-αh~iktσs-αhjktσs-1-αh~jktσs+12=α1n∑k=1nhiktσs-hjktσs+12+1-α1n∑k=1nh~iktσs-h~jktσs+12=h~ijtσs;also (36)hijt+1σs=αhijtσs+1-αh~ijtσs.Thus,(37)hijt+1σs-h~ijt+1σs=αhijtσs+1-αh~ijtσs-h~ijtσs=αhijtσs-h~ijtσs<hijtσs-h~ijtσs;therefore,(38)limt→∞hijtσs-h~ijtσs=limt→∞αhijt-1σs-h~ijt-1σs=⋯=limt→∞αthij0σs-h~ij0σs=0.By (33), (39)pijtσs<pijt+1σs<p~ijtσs,with each iteration pij(t)σ(s), will come closer to p~ij(t)σ(s). As H~(t) is consistent; therefore, each iteration will make H(t) more consistent; that is, if pik(t)σ(s)≥1/2∧pkj(t)σ(s)≥1/2∧pij(t)σ(s)<1/2, then by (33) and (25) (40)pijt+1σs-p~ijt+1σs<pijtσs-p~ijtσs,and each iteration will increase pij(t)σ(s); thus eventually it will be ≥1/2. Hence H(t) will become consistent when t→∞; that is, (41)limt→∞pijtσs-p~ijtσs=0.Finally, (37) and (40) imply (42)CIHt+1<CIHt∀t,limt→∞CIHt=0,following from (38) and (41).The choice of parametersn, d, α directly affects the performance of Algorithm 19. For performance measurement of Algorithm 19, 1000 random HPFPRs are generated and their acceptably consistent HPFPRs are computed by Algorithm 19, the average iteration in Algorithm 19 with respect to different parameters shown in Table 2. It is apparent that Algorithm 19 is quite efficient and, as for the effects of parameters, the increase in value α leads to more iteration in Algorithm 19 to compute consistent HPFPR; therefore, it is suggested to choose α small. Also, the number of iterations in Algorithm 19 is inversely proportional to the consistency index CIr.Table 2
Average iterations values in Algorithm19.
n d C I r α 0.1 0.3 0.6 0.8 4 4 0.05 1.001 1.939 3.226 6.88 0.1 0.996 0.999 1.898 3.659 0.15 0.939 0.959 0.998 1.824 3 0.05 1.001 1.939 3.528 7.326 0.1 0.991 0.997 2 4.084 0.15 0.954 0.95 1.208 2.169 2 0.05 1.05 1.882 3.527 7.415 0.1 0.982 1.054 2.089 4.1 0.15 0.868 0.891 1.297 2.404  5 5 0.05 1 1.976 3.095 6.851 0.1 1 1 1.978 3.71 0.15 0.989 0.994 1.017 1.854 4 0.05 1.004 1.992 3.418 7.266 0.1 1 1 1.989 4.021 0.15 0.999 0.997 1.123 2.214 3 0.05 1.001 1.992 3.767 7.709 0.1 1 1.002 2.123 4.43 0.15 0.994 0.997 1.403 2.575  6 6 0.05 1 1.994 3.028 6.776 0.1 1 1 1.991 3.654 0.15 1 1 1 1.818 5 0.05 1.001 1.999 3.2 7.025 0.1 1 1 1.997 3.942 0.15 1 1 1.016 2.074 4 0.05 1.001 2 3.606 7.446 0.1 1 1 2.011 4.249 0.15 0.999 1 1.178 2.443  7 7 0.05 1 1.993 3.009 6.659 0.1 1 1 1.991 3.487 0.15 0.998 1 0.999 1.732 6 0.05 1 1.998 3.05 6.912 0.1 1 1 1.999 3.816 0.15 1 1 1.001 1.972 5 0.05 1 2 3.261 7.125 0.1 1 1 2 4.023 0.15 1 1 1.022 2.168  8 8 0.05 1 1.993 3.002 6.526 0.1 1 1 1.984 3.317 0.15 1 1 1 1.599 7 0.05 1 2 3.009 6.831 0.1 1 1 1.997 3.658 0.15 1 1 1 1.867 6 0.05 1 2 3.078 7.003 0.1 1 1 2 3.927 0.15 1 1 1 2.026  9 9 0.05 1 1.993 3 6.396 0.1 1 1 1.978 3.229 0.15 1 1 1 1.474 8 0.05 1 2 3 6.676 0.1 1 1 1.998 3.47 0.15 1 1 1 1.74 7 0.05 1 2 3.01 6.92 0.1 1 1 2 3.786 0.15 1 1 1 1.942  10 10 0.05 1 1.986 3 6.326 0.1 1 1 1.973 3.164 0.15 1 1 1 1.367 9 0.05 1 1.994 3 6.543 0.1 1 1 1.995 3.308 0.15 1 1 1 1.58 8 0.05 1 2 3.002 6.798 0.1 1 1 2 3.618 0.15 1 1 1 1.864


## 4. Consensus Measure in Group Decision Making

For group decision making, letX=x1,x2,…,xn be the set of alternatives and consider E=e1,e2,e3,…,em, the set of decision makers providing their preferences in HPFPRs H1=hij,1,H2=hij,2,…,Hm=(hij,m), respectively. Also, let ω=(ω1,ω2,…,ωm) be the importance/weight vector of decision makers in normalized form; that is, ∑i=1m‍ωi=1. The Algorithm 19 provides the solution of consistency issue. In order to make group decision, the following aggregation operator is defined, which will be used to fuse all respective NHPFPRs of decision makers.Definition 21.
Take a collection of HPFEshi(i=1,2,3,…,n) with equal length and their weight vector λ=(λ1,λ2,…,λn) in normalized form; that is, ∑i=1n‍λi=1. The hesitant probabilistic fuzzy weighted averaging (HPFWA) operator is defined as follows:(43)HPFWAh1,h2,…,hn=⨁i=1nλihi=⋃γ1σs∣p1σs∈h1,γ2σs∣p2σs∈h2,…,γnσs∣pnσs∈hn∑i=1nλiγiσs∣∑i=1nλipiσs.Also, if we take weight vector λ=(1/n,1/n,…,1/n) then the above operator is reduced to hesitant probabilistic fuzzy averaging (HPFA) operator as(44)HPFAh1,h2,…,hn=⨁i=1n1nhi=⋃γ1σs∣p1σs∈h1,γ2σs∣p2σs∈h2,…,γnσs∣pnσs∈hn∑i=1n1nγiσs∣∑i=1n1npiσs.Theorem 22 (boundedness).
If HPFEshi(i=1,2,3,…,n) with equal length and their weight vector λ=(λ1,λ2,…,λn) in normalized form, that is, ∑i=1n‍λi=1, then(45)min⁡λ1h1,λ2h2,…,λnhn≤HPFWAh1,h2,…,hn≤max⁡λ1h1,λ2h2,…,λnhn.Proof.
From (5), (6), and (43), it is easy to yield the desired result.Theorem 23 (commutativity).
If HPFEshi(i=1,2,3,…,n) is a permutation of hi′(i=1,2,3,…,n) with equal length and weight vector λ=(λ1,λ2,…,λn) of hi is a permutation of λ′=(λ1′,λ2′,…,λn′) of hi′ in normalized form, that is, ∑i=1n‍λi=1 and ∑i=1n‍λi′=1, then(46)HPFWAh1,h2,…,hn=HPFWAh1′,h2′,…,hn′.Proof.
Since multiplication is commutative, so it is easy to proof that HPFWA operator is also commutative.Theorem 24 (monotonicity).
If HPFEshi≤hi′ for i=1,2,3,…,n with equal length and weight vector λ=(λ1,λ2,…,λn), that is, ∑i=1n‍λi=1, then(47)HPFWAh1,h2,…,hn≤HPFWAh1′,h2′,…,hn′.Proof.
Sincehi≤hi′ and λi∈0,1, then λihi≤λihi′.
HenceHPFWA(h1,h2,…,hn)≤HPFWA(h1′,h2′,…,hn′).Proposition 25.
TakeHk=(hij,k)n×n(k=1,2,…,m) HPFPRs given by decision makers and ω=(ω1,ω2,…,ωm) is the weight vector of decision makers with ∑i=m‍ωi=1. The NHPFPRs H¯k=(h¯ij,k) are computed with optimized parameters ζk(k=1,2,…,m). Then the HPFPR Hg=(hij,g)n×n calculated as(48)Hg=⨁k=1mωkh¯ij,kn×nis HPFPR.Proof.
It follows from (3), (7), and (43).The next result shows that the aggregated group HPFPRHg obtained by (48) is consistent, provided all the individual HPFPRs are consistent.Theorem 26.
ConsiderHk=(hij,k)n×n(k=1,2,3,…,m) HPFPRs given by decision makers and group HPFPR Hg is computed by (48). Then(49)CIHg≤maxkCIHk.Proof.
LetH~g(t)=h~ij,g(t) and H~k(t)=h~ij,k(t)(k=1,2,…,m) be the sequences of HPFPRs generated by Algorithm 11 applied to Hg and Hk(k=1,2,…,m), respectively. By (28)(50)Dhij,g,h~ij,gt=∑hij,gσs∣pij,gσs∈hij,g,h~ij,gσs∣p~ij,gσs∈h~ij,gmax⁡hij,gσs-h~ij,gσs,pij,gσs-p~ij,gtσsd.Now(51)hij,gσs-h~ij,gtσs=∑k=1mωkh¯ij,kσs-1n∑l=1nhil,gσs-hjl,gσs+12=∑k=1mωkh¯ij,kσs-1n∑l=1n∑k=1mωkh¯il,kσs-∑k=1mωkh¯jl,kσs+12=∑k=1mωkh¯ij,kσs-1n∑l=1nh¯il,kσs-h¯jl,kσs+12hij,gσs-h~ij,gtσs=∑k=1mωkh¯ij,kσs-h~ij,kσs≤maxkh¯ij,kσs-h~ij,kσs≤dmaxk⁡Dh¯ij,k,h~ij,kt.Consider (52)p~ij,g1σs=12n∑l=1npil,gσs+plj,gσs=12n∑l=1n∑k=1mωkp¯il,kσs+∑k=1mωkp¯lj,kσs=∑k=1mωk12n∑l=1np¯il,kσs+p¯lj,kσs=∑k=1mωkp~ij,k1σs.For the inductive step, suppose the following: (53)p~ij,grσs=∑k=1mωkp~ij,krσs,i,j=1,2,…,n.Then (54)p~ij,gr+1σs=12n∑l=1np~il,grσs+p~lj,grσs;by supposition, (55)p~ij,gr+1σs=12n∑l=1n∑k=1mωkp~il,krσs+∑k=1mωkp~lj,krσs=∑k=1mωk12n∑l=1np~il,krσs+p~lj,krσs=∑k=1mωkp~ij,kr+1σs.Hence, by principle of mathematical induction for all t∈N(56)p~ij,gtσs=∑k=1mωkp~ij,ktσs,i,j=1,2,…,n.Therefore,(57)pij,gσs-p~ij,gtσs≤maxkp¯ij,kσs-p~ij,ktσs≤dmaxk⁡Dh¯ij,k,h~ij,kt.Equations (51) and (57) imply (58)Dhij,g,h~ij,gt≤maxk⁡Dh¯ij,k,h~ij,kt;thus by (30) (59)CIHg≤maxkCIHk.Now, the consensus is another very important aspect of group decision making. To deal with the consensus issue among all the individual decision makers, the following consensus index is defined.Definition 27 (consensus index).
LetHk=(hij,k)n×n(k=1,2,3,…,m) be m HPFPRs provided by decision makers and their NHPFPRs H¯k are obtained with optimized parameter ζk(k=1,2,…,m). The group HPFPR Hg is computed by (48). Then the group consensus index (GCI) of HPFPR Hk is defined to be the distance measured between H¯k and Hg; that is,(60)GCIHk=DH¯k,Hg.The agreement between individual decision maker with group decision is measured by the distance between individual HPFPRHk and group HPFPR Hg. Therefore, GCI(Hk)=0 means kth decision maker has full agreement with group decision; otherwise, the smaller the value of GCI(Hk) is, the better the consensus will be. In many real life scenarios, it is important to have consensus among all decision makers, although we have to live with difference of opinion and it is hard to reach complete consensus; for this reason a threshold value can be decided based on practical nature of the problem to allow the difference of opinion to some extent.Definition 28 (acceptably consensus HPFPRs).
LetHk=(hij,k)n×n(k=1,2,3,…,m) be m HPFPRs provided by decision makers and their NHPFPRs H¯k are obtained with optimized ζk(k=1,2,…,m). Furthermore compute group HPFPR Hg and GCI(Hk)(k=1,2,…,m). Consider GCIr a tolerance value of consensus measure; then HPFPR Hk is said to be acceptably consensus with group HPFPR Hg if(61)GCIHk≤GCIr.But it is possible that a decision maker has an unacceptable difference of opinion with group decision; in this regard a novel algorithm is proposed to modify his HPFPR to reach acceptable consensus.Algorithm 29 (consensus improving algorithm).
Input. The tolerance value GCIr of consensus measure, HPFPRs Hk=(hij,k)n×n(k=1,2,3,…,m), and parameter β∈(0,1).
Output. The acceptably consensus HPFPRs Hk(t), group consensus index GCI(Hk(t))(k=1,2,…,m), the group HPFPR Hg(t), and number of iterations value t.
Step 1. Compute H¯k=(h¯ij,k)n×n with optimized parameter ζk. Let t=0, Hk(0)=(hij,k(0))n×n=H¯k=(h¯ij,k)n×n.
Step  2. Compute group HPFPR Hg(t)=(hij,g(t))n×n by using all individual HPFPRs Hk(t)(k=1,2,…,m) according to (48), where(62)hij,gt=⨁k=1mωkhij,kt.
Step 3. Calculate GCI(Hk(t))=D(Hk(t),Hg(t))(k=1,2,…,m). If GCI(Hk(t))≤GCIr for all k=1,2,…,m, then go to Step 5; otherwise go to Step 4.
Step 4. Put Hk(t+1)=(hij,k(t+1))n×n, where(63)hkt+1=βhkt⊕1-βhij,gt.Also let t=t+1 and go to Step 2.
Step 5. Output the adjusted HPFPRs Hk(t)=(hij,k(t))n×n, the group consistency index GCI(Hk(t))(k=1,2,…,m), the group HPFPR Hg(t), and number of iterations value t.
Step 6. End.The convergence of the above algorithm follows from the following result.Theorem 30.
ConsiderHk=(hij,k)n×n(k=1,2,3,…,m) HPFPRs given by decision makers. Let Hkt be the sequence obtained from Algorithm 29. Then (64)GCIHkt+1<GCIHkt,limt→∞GCIHkt=0∀k=1,2,…,m.Proof.
By (62), (3), and (7), (65)hij,gt+1σs∣pij,gt+1σs=∑l=1mωlhij,lt+1σs∣∑l=1mωlpij,lt+1;thus, (66)hij,kt+1σs-hij,gt+1σs=βhij,ktσs+1-βhij,gtσs-∑l=1mωlhij,lt+1σs=βhij,ktσs+1-βhij,gtσs-∑l=1mωlβhij,ltσs+1-βhij,gtσs=βhij,ktσs-∑l=1mωlβhij,ltσs=βhij,ktσs-hij,gtσs;similarly, (67)pij,kt+1σs-pij,gt+1σs=βpij,ktσs-pij,gtσs.Hence, (68)Dhij,kt+1,hij,gt+1=βDhij,kt,hij,gt;now by definition of GCI (60) (69)GCIHkt+1,Hgt+1=2nn-1∑i<jnDhij,kt+1,hij,gt+1=βGCIHkt,Hgt<GCIHkt,Hgt;it also implies (70)limt→∞GCIHkt=limt→∞βGCIHkt-1=limt→∞β2GCIHkt-2=⋯=limt→∞βtGCIHk0=0k=1,2,…,m.Now, 1000 random sets ofm HPFPRs are generated and Algorithm 29 is applied to develop consensus. The average value of iterations of Algorithm 29 is presented in Table 3 for different values of parameters. The readings of Table 3 suggest that the increase in value of parameter β has adverse effects on the number of iterations of Algorithm 29. So, the value of parameter β must be small. The more iterations are needed to develop consensus when GCIr is nearer to 1. Furthermore, Algorithm 29 does not disturb the consistency of HPFPRs; that is, if the individual HPFPRs Hk=(hij,k)n×n(k=1,2,3,…,m) are consistent, then output of Algorithm 29 and the adjusted HPFPRs Hk(t)=(hij,k(t))n×n are also consistent.Table 3
Average values of iterations in Algorithm29.
n m d G C I r β 0.2 0.4 0.7 0.9 4 4 4 0.01 2.002 3.233 3.233 25.301 0.05 1 1.604 3.185 9.672 0.1 0.949 0.956 1.346 3.384 3 3 0.01 2 3.221 7.766 25.102 0.05 1.003 1.701 3.47 10.452 0.1 0.961 0.968 1.542 4.051 2 2 0.01 2.03 3.476 8.166 26.48 0.05 0.991 1.297 2.582 7.754 0.1 0.872 0.911 1.517 3.975  5 4 5 0.01 2 3.137 7.631 24.652 0.05 1 1.294 2.833 8.521 0.1 0.868 0.864 0.974 2.186 3 4 0.01 2 3.234 7.845 25.342 0.05 1 1.223 2.681 7.775 0.1 0.879 0.859 1.114 2.715 3 3 0.01 2.001 3.167 7.816 25.32 0.05 1 1.62 3.197 9.641 0.1 0.802 0.817 0.901 2.024  6 5 6 0.01 2 3.006 7.154 22.888 0.05 1 1.081 2.495 7.288 0.1 0.623 0.606 0.643 1.128 5 5 0.01 2 3.026 7.368 23.74 0.05 1 1.128 2.638 7.784 0.1 0.705 0.706 0.741 1.358 4 4 0.01 2 3.069 7.544 24.335 0.05 1 1.345 2.897 8.539 0.1 0.849 0.838 0.955 1.99  7 7 7 0.01 2 3 6.908 21.984 0.05 1 1.022 2.417 6.912 0.1 0.511 0.485 0.479 0.71 7 6 0.01 2 3.004 7.208 23.336 0.05 1 1.135 2.72 7.903 0.1 0.768 0.777 0.8 1.351 8 4 0.01 2 3.025 7.454 24.147 0.05 1 1.29 2.954 8.655 0.1 0.991 0.992 1.163 2.9  8 9 8 0.01 2 3 6.81 21.664 0.05 1 1.006 2.249 6.615 0.1 0.263 0.256 0.257 0.339 8 7 0.01 2 3 6.907 21.885 0.05 1 1.006 2.232 6.611 0.1 0.329 0.341 0.333 0.435 8 6 0.01 2 3.001 7.022 22.476 0.05 1 1.036 2.453 7.215 0.1 0.309 0.305 0.342 0.356  9 10 9 0.01 2 3 6.633 21.16 0.05 1 1.001 2.052 5.883 0.1 0.083 0.065 0.072 0.1 7 8 0.01 2 3 6.468 20.746 0.05 1 1.003 2.094 6.072 0.1 0.087 0.126 0.105 0.127 9 7 0.01 2 3 6.812 21.546 0.05 1 1.004 2.134 6.298 0.1 0.161 0.162 0.157 0.205  10 9 10 0.01 2 3 6.619 21.049 0.05 1 1 1.955 5.152 0.1 0.052 0.05 0.062 0.062 8 9 0.01 2 3 6.334 20.466 0.05 1 1 2.038 5.746 0.1 0.036 0.038 0.036 0.048 8 8 0.01 2 3 6.454 20.795 0.05 1 1 2.018 5.545 0.1 0.059 0.061 0.047 0.05Proposition 31.
ConsiderHk=(hij,k)n×n(k=1,2,3,…,m) HPFPRs given by decision makers. Let Hkt and Hgt be the sequences obtained from Algorithm 29. If maxkCIHkt≤CIr then(71)maxkCIHkt+1≤maxkCIHkt≤CIr.Proof.
It follows from Theorem20 and (63).


## 5. Decision Support Model for Group Decision Making with HPFPRs

Now, the issues of consistency and consensus are addressed. Algorithms11–29 will provide the consistent individual HPFPRs and the group HPFPR with agreement among decision makers. To comprehend final standing of the alternatives, first for the alternative xi the ith row of group HPFPR is aggregated by HPFA operator (44) and secondly the aggregated HPFEs are ordered according to their score and deviation [35]. In form of the following algorithm a complete decision model is presented.Algorithm 32 (decision support model).
Input. The HPFPRs Hk=(hij,k)n×n(k=1,2,3,…,m) made by decision makers, the weight vector ω=(ω1,ω2,…,ωm) of decision makers in normalized form, consistency tolerance value CIr, group consensus tolerance value GCIr, the maximum number of iterations allowed tmax, and the parameters 0<α,β<1 for modification.
Output. The final standing of all the alternatives.
Step  1. Compute NHPFPRs H¯k=(h¯ij,k)n×n with optimized parameters ζk. Put t=0 and Hk(0)=(hij,k(0))n×n=H¯k=(h¯ij,k)n×n.
Step  2. Calculate the consistent HPFPRs H~k(t)=(h~ij,k(t))n×n by Algorithm 11 and consistency indexes CI(Hk(t)), where(72)CIHkt=2nn-1∑i<jnDhij,kt,h~ij,kt.
Step  3. If CI(Hk(t))≤CIr for all k=1,2,…,m then go to Step 5; otherwise, go to Step 4.
Step  4. Make the adjusted HPFPRs Hk(t+1)=(hij,k(t+1)), where(73)hij,kt+1=hij,kt,CIHkt≤CIrαhij,kt⊕1-αh~ij,kt,CIHkt>CIr.Let t=t+1; now return to Step 2.
Step  5. Apply HPFWA operator (43) to individual HPFPRs Hk(t)=(hij,k(t))n×n(k=1,2,…,m), to get group HPFPR Hg(t)=(hij,g(t))n×n, where(74)hij,gt=⨁k=1mωkhij,kt.
Step  6. Calculate GCI(Hk(t))=D(Hk(t),Hg(t))(k=1,2,…,m). If GCI(Hk(t))≤GCIr for all k=1,2,…,m or t>tmax, then go to Step 8; otherwise, go to Step 7.
Step  7. Let Hk(t+1)=(hij,k(t+1))n×n, where(75)hkt+1=βhkt⊕1-βhij,gt.Also let t=t+1 and go to Step 5.
Step  8. Aggregate each ith-row of HPFPR Hg(t) by HPFA operator (44),(76)hi,g=HPFAhi1,gt,hi2,gt,…,hin,gt=⨁j=1n1nhij,gt,to get collective preference degrees of alternative xi over all other alternatives (i=1,2,…,n).
Step  9. Compute scores s(hi,g) and deviations d(hi,g) [35] as follows: (77)shi,g=∑s=1dhi,gσspi,gσs,dhi,g=∑s=1dhi,gσs-shi,g2pi,gσs,i=1,2,…,n.
Step  10. Determine the final ranking of alternatives xi,i=1,2,3,…,n by comparing the score and deviation values of respective hi,g and output the ranking vector.
Step  11. EndFlowchart of the decision making model is presented in Figure7.Figure 7
Flowchart of the proposed decision model.


## 6. Case Study

The proposed decision making model will be applied to a practical problem of investment in Forex.Example 33.
The Flagship Investment Company (FIC) is known for investment plans in different commodities in Forex. The investors hire FIC for making profitable plans of investments. An investor is interested to invest in four commodities: oil, gold, wheat, and copper denoted asx1,x2,x3, and x4, respectively. He approached FIC to indicate which commodity is more profitable to invest in. FIC makes a committee of four economic experts e1,e2,e3, and e4. Each expert will provide its preferences in terms of HPFPR depending upon many factors like previous market rates, market trends, possible future decisions of different regulatory bodies, economic stability, global peace situation, and so on. The complex nature of the Forex required tools that will model the vagueness and produce results closer to real life. HPFPR will allow the economic experts to express their hesitancy and provide the probabilities of different preference degrees, which are bound to be different because of the factors discussed above. Based on the experience and economic knowledge, weight vector of experts is ω=(0.1,0.4,0.2,0.3)T. The economic expert ek provides his HPFPR Hk(k=1,2,3,4) as follows.(78)H1=.5∣10.1174∣0.2979,0.910∣.70210.4242∣0.6622,0.8341∣0.33780.2625∣0.0636,0.7956∣0.93640.8826∣0.2979,0.0900∣0.7021.5∣10.1921∣0.2236,0.9289∣0.77640.2373∣0.4313,0.5785∣0.56870.5758∣0.6622,0.1659∣0.33780.8079∣0.2236,0.0711∣0.2236.5∣10.2316∣0.7145,0.6211∣0.28550.7375∣0.0636,0.2044∣0.93640.7627∣0.4313,0.4214∣0.56870.7684∣0.7145,0.3789∣0.2855.5∣1,H2=.5∣10.3674∣0.5018,0.3955∣0.3181,0.9880∣0.18000.0987∣0.3339,0.2619∣0.6322,0.5913∣0.03380.3197∣0.2412,0.8691∣0.75880.6326∣0.5018,0.6044∣0.3181,0.0120∣0.1800.5∣10.1340∣0.3627,0.8909∣0.1892,0.9037∣0.44810.7440∣10.9013∣0.3339,0.7381∣0.3339,0.4087∣0.03380.8660∣0.3627,0.1091∣0.1892,0.0963∣0.4481.5∣10.4111∣0.7060,0.9047∣0.29400.6803∣0.2412,0.1309∣0.75880.2559∣10.5889∣0.7060,0.0953∣0.2940.5∣1,H3=.5∣10.1829∣0.1317,0.2399∣0.4745,0.5767∣0.39380.3817∣0.7453,0.9787∣0.25470.0596∣0.2951,0.7219∣0.70490.8171∣0.1317,0.7601∣0.4745,0.4232∣0.3938.5∣10.5∣0.1232,0.5961∣0.87670.1499∣0.2398,0.7224∣0.5387,0.8175∣0.22140.6183∣0.7453,0.0213∣0.25470.5∣0.1233,0.4039∣0.8767.5∣10.4538∣0.5612,0.8003∣0.43880.9404∣0.2951,0.2781∣0.70490.8501∣0.2398,0.2775∣0.5387,0.1825∣0.22150.5462∣0.5612,0.1997∣0.4388.5∣1,H4=.5∣10.0991∣0.629,0.7317∣0.3710.3498∣0.8924,0.9447∣0.10760.1835∣0.2908,0.3685∣0.5157,0.7256∣0.19360.9009∣0.629,0.2683∣0.371.5∣10.2481∣0.7128,0.5∣0.2855,0.9199∣0.00170.3786∣0.6610,0.6443∣0.0808,0.7948∣0.25820.6502∣0.8924,0.0553∣0.10760.7519∣0.7128,0.5∣0.2855,0.0801∣0.0017.5∣10.1349∣0.4378,0.6225∣0.163,0.8759∣0.39920.8165∣0.2908,0.6315∣0.5157,0.2744∣0.19360.6214∣0.6610,0.3557∣0.0808,0.2052∣0.25820.8651∣0.4378,0.3775∣0.163,0.1241∣0.3992.5∣1.Step 1.
NHPFPRsHk(0)=H¯k(k=1,2,3,4) are calculated by (10) as follows. (79)H10=.5∣0.3333,.5∣0.3333,.5∣0.33330.1174∣0.149,0.1174∣0.149,0.91∣0.70210.4242∣0.3311,0.4242∣0.3311,0.8341∣0.33780.2625∣0.0318,0.2625∣0.0318,0.7956∣0.93640.8826∣0.149,0.8826∣0.149,0.09∣0.7020.5∣0.3333,.5∣0.3333,.5∣0.33330.1921∣0.1118,0.1921∣0.1118,0.9289∣0.77640.2373∣0.2156,0.2373∣0.2156,0.5785∣0.56870.5758∣0.3311,0.5758∣0.3311,0.1659∣0.33780.8079∣0.1118,0.8079∣0.1118,0.0711∣0.7764.5∣0.3333,.5∣0.3333,.5∣0.33330.2316∣0.3573,0.2316∣0.3573,0.6211∣0.28550.7375∣0.03180,0.7375∣0.03180,0.2044∣0.93640.7627∣0.2156,0.7627∣0.2156,0.4215∣0.56870.7684∣0.3573,0.7684∣0.3573,0.3789∣0.2855.5∣0.3333,.5∣0.3333,.5∣0.3333,H20=.5∣0.3333,.5∣0.3333,.5∣0.33330.3674∣0.5018,0.3955∣0.3181,0.988∣0.180.0987∣0.3339,0.2619∣0.6322,0.5913∣0.03380.3197∣0.1206,0.3197∣0.1206,0.8691∣0.75880.6326∣0.5018,0.6045∣0.3181,0.012∣0.18.5∣0.3333,.5∣0.3333,.5∣0.33330.134∣0.3627,0.8909∣0.1892,0.9037∣0.44810.7441∣0.3333,0.7441∣0.3333,0.7441∣0.33330.9013∣0.3339,0.7381∣0.6322,0.4087∣0.033830.866∣0.3626,0.1091∣0.1892,0.0963∣0.4481.5∣0.3333,.5∣0.3333,.5∣0.33330.4111∣0.353,0.4111∣0.353,0.9047∣0.2940.6803∣0.1206,0.6803∣0.1206,0.1309∣0.75880.2559∣0.3333,0.2559∣0.3333,0.2559∣0.33330.5889∣0.353,0.5889∣0.353,0.0953∣0.294.5∣0.3333,.5∣0.3333,.5∣0.3333,H30=.5∣0.3333,.5∣0.3333,.5∣0.33330.1829∣0.1317,0.2399∣0.4744,0.5767∣0.39380.3817∣0.3727,0.3817∣0.3727,0.9787∣0.25470.0596∣0.1476,0.0596∣0.1476,0.7219∣0.70490.8171∣0.1317,0.4745∣0.7601,0.4233∣0.3938.5∣0.3333,.5∣0.3333,.5∣0.33330.5∣0.0616,0.5∣0.0616,0.5961∣0.87670.1499∣0.2398,0.7224∣0.5387,0.8175∣0.22150.6183∣0.3727,0.6183∣0.3727,0.0213∣0.25470.5∣0.0616,0.5∣0.0616,0.4039∣0.8767.5∣0.3333,.5∣0.3333,.5∣0.33330.4538∣0.2806,0.4538∣0.2806,0.8003∣0.43880.9404∣0.1476,0.9404∣0.1476,0.2781∣0.70490.8501∣0.239,0.2776∣0.5387,0.1825∣0.22140.5462∣0.2807,0.5462∣0.2807,0.1997∣0.4388.5∣0.3333,.5∣0.3333,.5∣0.3333,H40=.5∣0.3333,.5∣0.3333,.5∣0.33330.0991∣0.3145,0.0991∣0.3145,0.7317∣0.3710.3498∣0.4462,0.3498∣0.4462,0.9447∣0.10760.1835∣0.2908,0.3685∣0.5157,0.7256∣0.19360.9009∣0.3145,0.9009∣0.3145,0.2683∣0.371.5∣0.3333,.5∣0.3333,.5∣0.33330.2481∣0.7128,0.5∣0.2855,0.9199∣0.00170.3786∣0.661,0.6443∣0.0808,0.7948∣0.25810.6502∣0.4462,0.6502∣0.4462,0.0553∣0.10760.7519∣0.7128,0.5∣0.2855,0.0801∣0.0017.5∣0.3333,.5∣0.3333,.5∣0.33330.1349∣0.4378,0.6225∣0.163,0.8759∣0.39920.8165∣0.2908,0.6315∣0.5157,0.2744∣0.19360.6214∣0.661,0.3557∣0.0808,0.2052∣0.25810.8651∣0.4378,0.3775∣0.163,0.1241∣0.3992.5∣0.3333,.5∣0.3333,.5∣0.3333.Step 2.
After calculating consistent HPFPRsH~k(0)=(h~ij,k0))n×n by Algorithm 11 consistency indexes CI(Hk(t)) are computed by (72) as follows: (80)CIH10=.1366,CIH20=.2533,CIH30=.2159,CIH40=.2116.Step 3.
The consistency tolerance valueCIr is decided to be 0.01; therefore, all the HPFPRs Hk(0)(k=1,2,3,4) are needed to be adjusted.
Repeating Steps 2 to 4 in Section5, we get the acceptably consistent HPFPR,(81)H12=0.5∣0.2293,0.5∣0.2293,0.5∣0.54140.3705∣0.2265,0.3705∣0.2265,0.7373∣0.5470.2984∣0.237,0.2984∣0.237,0.9195∣0.52610.1352∣0.2288,0.1352∣0.2288,0.8828∣0.54250.6295∣0.2265,0.6295∣0.2265,0.2627∣0.5470.5∣0.2274,0.5∣0.2274,0.5∣0.54520.4219∣0.2338,0.4219∣0.2338,0.6873∣0.53230.2606∣0.2296,0.2606∣0.2296,0.6475∣0.54070.7015∣0.237,0.7015∣0.237,0.0804∣0.52610.5781∣0.2338,0.5781∣0.2338,0.3127∣0.53230.5∣0.2447,0.5∣0.2447,0.5∣0.51060.3356∣0.2397,0.3356∣0.2397,0.4649∣0.52060.8648∣0.2288,0.8648∣0.2288,0.1171∣0.54250.7394∣0.2296,0.7394∣0.2296,0.3525∣0.54070.6644∣0.2397,0.6644∣0.2397,0.5351∣0.52060.5∣0.2343,0.5∣0.2343,0.5∣0.5315,H22=0.5∣0.3272,0.5∣0.341,0.5∣0.33180.3193∣0.3466,0.1865∣0.324,0.7001∣0.32940.1513∣0.3341,0.428∣0.3516,0.758∣0.31440.3152∣0.3142,0.3626∣0.3196,0.9903∣0.36630.6807∣0.3466,0.8135∣0.324,0.2999∣0.32940.5∣0.3625,0.5∣0.3073,0.5∣0.33010.3311∣0.352,0.7467∣0.3303,0.5659∣0.31770.4989∣0.334,0.6792∣0.3048,0.7939∣0.36120.8487∣0.3341,0.5718∣0.3516,0.242∣0.31440.6689∣0.352,0.2533∣0.3303,0.4341∣0.31770.5∣0.3409,0.5∣0.3561,0.5∣0.3030.6608∣0.3233,0.4331∣0.3295,0.7336∣0.34720.6848∣0.3142,0.6374∣0.3196,0.0097∣0.36630.5011∣0.334,0.3201∣0.3049,0.2061∣0.36120.3392∣0.3233,0.5669∣0.3295,0.2663∣0.34720.5∣0.3054,0.5∣0.3024,0.5∣0.3922,H32=0.5∣0.2437,0.5∣0.3263,0.5∣0.43010.2883∣0.2256,0.1753∣0.3335,0.6098∣0.44080.2642∣0.2486,0.2783∣0.3062,0.7651∣0.44520.07175∣0.243,0.2276∣0.3224,0.9025∣0.43470.7117∣0.2256,0.8247∣0.3335,0.3902∣0.44080.5∣0.2116,0.5∣0.338,0.5∣0.45040.474∣0.2295,0.6016∣0.3089,0.6523∣0.46150.2812∣0.2279,0.5563∣0.3321,0.7944∣0.440.7358∣0.2486,0.7217∣0.3062,0.2349∣0.44520.526∣0.2295,0.3984∣0.3089,0.3477∣0.46150.5∣0.2529,0.5∣0.2853,0.5∣0.46180.3103∣0.2489,0.452∣0.3032,0.6429∣0.44790.9283∣0.243,0.7724∣0.3224,0.0975∣0.43470.7188∣0.227,0.44378∣0.3321,0.2056∣0.440.6897∣0.2489,0.548∣0.3032,0.3571∣0.44790.5∣0.246,0.5∣0.3222,0.5∣0.4318,H42=0.5∣0.2437,0.5∣0.3263,0.5∣0.43010.2883∣0.2256,0.1753∣0.3335,0.6098∣0.44080.2642∣0.2486,0.2783∣0.3062,0.7651∣0.44520.0717∣0.243,0.2276∣0.3224,0.9025∣0.43470.7117∣0.2256,0.8247∣0.3335,0.3902∣0.44080.5∣0.2116,0.5∣0.338,0.5∣0.45040.474∣0.2295,0.60158∣0.3089,0.6523∣0.46150.2812∣0.2279,0.5562∣0.3321,0.7944∣0.440.7358∣0.2486,0.7217∣0.3062,0.2349∣0.44520.5260∣0.2295,0.3984∣0.3089,0.3477∣0.46150.5∣0.2529,0.5∣0.2853,0.5∣0.46180.3103∣0.2489,0.452∣0.3032,0.6429∣0.44790.9283∣0.243,0.7724∣0.3224,0.0975∣0.43470.7188∣0.2279,0.4437∣0.3321,0.2056∣0.440.6897∣0.2489,0.548∣0.3032,0.3571∣0.44790.5∣0.246,0.5∣0.3222,0.5∣0.4318,and consistent HPFPRs, (82)H~12=0.5∣0.2304,0.5∣0.2304,0.5∣0.53920.373∣0.2299,0.373∣0.229,0.7356∣0.54030.2972∣0.2346,0.2972∣0.2346,0.9204∣0.53080.1339∣0.2317,0.1339∣0.2317,0.8837∣0.53650.627∣0.2299,0.627∣0.2299,0.2644∣0.54030.5∣0.2293,0.5∣0.2293,0.5∣0.54130.4242∣0.2341,0.4242∣0.2341,0.6848∣0.53190.2608∣0.2312,0.2608∣0.2312,0.6482∣0.53760.7028∣0.2346,0.7028∣0.2346,0.0796∣0.53080.5758∣0.2341,0.5758∣0.2341,0.3152∣0.53180.5∣0.2388,0.5∣0.2388,0.5∣0.52240.3367∣0.2359,0.3367∣0.2359,0.46337∣0.52810.8661∣0.2317,0.8661∣0.2317,0.1163∣0.53650.7392∣0.2312,0.7392∣0.2312,0.3518∣0.53760.6633∣0.2359,0.6633∣0.2359,0.5367∣0.52810.5∣0.2331,0.5∣0.2331,0.5∣0.5338,H~22=0.5∣0.3305,0.5∣0.3340,0.5∣0.33550.3188∣0.3396,0.18448∣0.3253,0.6971∣0.3350.1519∣0.334,0.4297∣0.338,0.7597∣0.3280.3152∣0.3249,0.363∣0.3241,0.9916∣0.35110.6812∣0.3396,0.8156∣0.3253,0.3029∣0.3350.5∣0.3488,0.5∣0.3166,0.5∣0.33460.3331∣0.3432,0.7453∣0.3292,0.5625∣0.32760.4964∣0.334,0.6786∣0.3153,0.7944∣0.35070.8481∣0.334,0.5703∣0.338,0.2403∣0.3280.6669∣0.3432,0.2547∣0.3292,0.4375∣0.32760.5∣0.3376,0.5∣0.3417,0.5∣0.32060.6633∣0.3284,0.4333∣0.328,0.7319∣0.34360.6848∣0.3249,0.637∣0.3241,0.0084∣0.35110.5036∣0.33,0.3214∣0.3153,0.2056∣0.35070.3367∣0.328,0.5667∣0.328,0.2681∣0.34360.5∣0.3192,0.5∣0.3141,0.5∣0.3667,H~32=0.5∣0.2402,0.5∣0.3221,0.5∣0.43770.2893∣0.2319,0.1747∣0.3251,0.6101∣0.44290.263∣0.2426,0.2771∣0.3115,0.7629∣0.44590.07187∣0.2408,0.2293∣0.321,0.9043∣0.43810.7107∣0.2319,0.8253∣0.3251,0.3899∣0.44290.5∣0.2237,0.5∣0.3281,0.5∣0.44810.4737∣0.2343,0.6026∣0.3145,0.6528∣0.45110.2826∣0.2326,0.5546∣0.3241,0.7942∣0.44340.737∣0.2426,0.7227∣0.3115,0.2371∣0.44590.5263∣0.2343,0.3974∣0.3145,0.3472∣0.45110.5∣0.245,0.5∣0.3009,0.5∣0.45410.3088∣0.2432,0.452∣0.3105,0.6413∣0.44630.9281∣0.2408,0.7707∣0.321,0.0957∣0.43810.7174∣0.2326,0.4454∣0.3241,0.2058∣0.44340.6912∣0.2432,0.548∣0.3105,0.3587∣0.44630.5∣0.2414,0.5∣0.32,0.5∣0.4386,H~42=0.5∣0.4125,0.5∣0.3373,0.5∣0.25020.2762∣0.4365,0.193∣0.3148,0.6047∣0.24870.2739∣0.4331,0.2612∣0.3229,0.8477∣0.24410.0824∣0.4253,0.3632∣0.3178,0.9496∣0.2570.7238∣0.4365,0.806∣0.3148,0.3953∣0.24870.5∣0.4606,0.5∣0.2922,0.5∣0.24710.4977∣0.4572,0.5681∣0.3003,0.7429∣0.24250.3062∣0.4493,0.6701∣0.2952,0.8448∣0.25540.7262∣0.4331,0.7388∣0.3229,0.1523∣0.24410.5023∣0.4572,0.4319∣0.3003,0.2571∣0.24250.5∣0.4537,0.5∣0.3084,0.5∣0.2370.3085∣0.4459,0.602∣0.3033,0.6019∣0.25080.9176∣0.4253,0.6368∣0.3177,0.0504∣0.2570.6938∣0.4493,0.3299∣0.2952,0.1552∣0.25540.6915∣0.4459,0.398∣0.3033,0.3981∣0.25080.5∣0.438,0.5∣0.2982,0.5∣0.2638.Also, Figures 8, 9, 10, and 11 present the comparison between the area graphs of score values for normalized, acceptably consistent, and consistent HPFPRs made from H1, H2, H3, and H4, respectively, by the end of Step 4 in Algorithm 32.Figure 8
Area graphs of score values ofH1.Figure 9
Area graphs of score values ofH2.Figure 10
Area graphs of score values ofH3.Figure 11
Area graphs of score values ofH4.Step 4.
By Applying HPFWA operator (43) to individual HPFPRs Hk(t)=(hij,k(t))n×n(k=1,2,…,m) we get group HPFPR Hg(t)=(hij,g(t))n×n as follows: (83)Hg2=0.5∣0.3081,0.5∣0.3226,0.5∣0.36930.3099∣0.3237,0.2228∣0.3053,0.6612∣0.37110.2403∣0.325,0.3224∣0.3152,0.8189∣0.35980.161∣0.3113,0.2903∣0.3045,0.9384∣0.38430.6901∣0.3238,0.7772∣0.3053,0.3388∣0.37110.5∣0.3399,0.5∣0.2883,0.5∣0.37180.427∣0.3407,0.5989∣0.2952,0.6611∣0.36410.3501∣0.3294,0.5681∣0.2878,0.7799∣0.38270.7597∣0.325,0.6776∣0.3152,0.1811∣0.35980.573∣.3407,0.4011∣0.2952,0.3389∣0.36410.5∣0.3411,0.5∣0.3053,0.5∣0.35370.4195∣0.3298,0.4681∣0.2964,0.623∣0.37380.839∣0.3114,0.7097∣0.3045,0.0616∣0.38430.6499∣0.3294,0.4319∣0.2878,0.2201∣0.38270.5805∣0.3298,0.5319∣0.2964,0.3777∣0.37380.5∣0.3179,0.5∣0.2885,0.5∣0.3936.Step 5.
The group consensus indexes are(84)GCIH1t=0.0823,GCIH2t=0.0709,GCIH3t=0.0899,GCIH4t=0.0743.Now GCIr is decided to be 0.1; therefore all the HPFPRs are acceptably consensus with group HPFPR.Step 6.
Now, all the rows of HPFPRHg are aggregated by HPFA operator (44) as follows:(85)h1,g=0.3028∣0.317,0.3339∣0.3119,0.7296∣0.3711h2,g=0.4918∣0.3334,0.6111∣0.2942,0.57∣0.3724h3,g=0.563∣0.3341,0.5117∣0.303,0.4108∣0.3628h4,g=0.6424∣0.3221,0.5434∣0.2943,0.2897∣0.3836.Step 7.
By (77), scores and deviations of hi,g(i=1,2,3,4) are calculated as follows. (86)sh1,g=.4709,sh2,g=.556,sh3,g=.4922,sh4,g=.4779,dh1,g=.0397,dh2,g=.0023,sh3,g=.0042,sh4,g=.0236.Step 8.
Comparing scores and deviations computed in Step7 provides us the following final preference ranking: (87)x2>x3>x4>x1.Step 9.
End.Remark 34.
Example33 provides the complete illustration of the proposed model dealing with consistency and creating enough consensus among all economic experts. Based on the results it is found that gold is the most profitable commodity, whereas oil is the least preferable for investment.


### 6.1. Effects of Probability on Decision Making

In order to show how important is the role played by the probabilities of preference degrees two new versions are generated from the HPFPRsHk=(hij,k)n×n(k=1,2,3,4) by changing the probabilities only and preserving the preference degrees as follows.(88)H1∗=.5∣10.1174∣0.2,0.91∣0.80.4242∣0.1,0.8341∣0.90.2625∣0.231,0.7956∣0.7690.8826∣0.2,0.0900∣0.8.5∣10.1921∣0.85,0.9289∣0.150.2373∣0.925,0.5785∣0.0750.5758∣0.1,0.1659∣0.90.8079∣0.85,0.0711∣0.15.5∣10.2316∣0.1,0.6211∣0.90.7375∣0.231,0.2044∣0.7690.7627∣0.925,0.4214∣0.0750.7684∣0.1,0.3789∣0.9.5∣1H2∗=.5∣10.3674∣0.1,0.3955∣0.1,0.988∣0.80.0987∣0.2,0.2619∣0.1,0.5913∣0.70.3197∣0.35,0.8691∣0.650.6326∣0.1,0.6044∣0.1,0.0120∣0.8.5∣10.1340∣0.8,0.8909∣0.1,0.9037∣0.10.7440∣10.9013∣0.2,0.7381∣0.1,0.4087∣0.70.8660∣0.8,0.1091∣0.1,0.0963∣0.1.5∣10.4111∣0.11,0.9047∣0.890.6803∣0.35,0.1309∣0.650.2559∣10.5889∣0.11,0.0953∣0.89.5∣1H3∗=.5∣10.1829∣0.01,0.2399∣0.23,0.5767∣0.760.3817∣0.289,0.9787∣0.7110.0596∣0.0586,0.7219∣0.94140.8171∣0.01,0.7601∣0.23,0.4232∣0.76.5∣10.5∣0.7,0.5961∣0.30.1499∣0.8,0.7224∣0.1,0.8175∣0.10.6183∣0.289,0.0213∣0.7110.5∣0.7,0.4039∣0.3.5∣10.4538∣0.15,0.8003∣0.850.9404∣0.0586,0.2781∣0.94140.8501∣0.8,0.2775∣0.1,0.1825∣0.10.5462∣0.15,0.1997∣0.85.5∣1H4∗=.5∣10.0991∣0.05,0.7317∣0.950.3498∣0.15,0.9447∣0.850.1835∣0.1,0.3685∣0.05,0.7256∣0.850.9009∣0.05,0.2683∣0.95.5∣10.2481∣0.75,0.5∣0.1,0.9199∣0.150.3786∣0.75,0.6443∣0.1,0.7948∣0.150.6502∣0.15,0.0553∣0.850.7519∣0.75,0.5∣0.1,0.0801∣0.15.5∣10.1349∣0.15,0.6225∣0.1,0.8759∣0.750.8165∣0.1,0.6315∣0.05,0.2744∣0.850.6214∣0.75,0.3557∣0.1,0.2052∣0.150.8651∣0.15,0.3775∣0.1,0.1241∣0.75.5∣1H1∗∗=.5∣10.1174∣0.2979,0.91∣0.70210.4242∣0.9,0.8341∣0.10.2625∣0.9,0.7956∣0.10.8826∣0.2979,0.0900∣0.7021.5∣10.1921∣0.85,0.9289∣0.150.2373∣0.925,0.5785∣0.0750.5758∣0.9,0.1659∣0.10.8079∣0.85,0.0711∣0.15.5∣10.2316∣0.1,0.6211∣0.90.7375∣0.9,0.2044∣0.10.7627∣0.925,0.4214∣0.0750.7684∣0.1,0.3789∣0.9.5∣1H2∗∗=.5∣10.3674∣0.5018,0.3955∣0.3181,0.9880∣0.180.0987∣0.85,0.2619∣0.1,0.5913∣0.050.3197∣0.75,0.8691∣0.250.6326∣0.5018,0.6044∣0.3181,0.0120∣0.18.5∣10.1340∣0.8,0.8909∣0.1,0.9037∣0.10.7440∣10.9013∣0.85,0.7381∣0.1,0.4087∣0.050.8660∣0.8,0.1091∣0.1,0.0963∣0.1.5∣10.4111∣0.95,0.9047∣0.050.6803∣0.75,0.1309∣0.250.2559∣10.5889∣0.95,0.0953∣0.05.5∣1H3∗∗=.5∣10.1829∣0.1317,0.2399∣0.4745,0.5767∣0.39380.3817∣0.89,0.9787∣0.110.0596∣0.9414,0.7219∣0.05860.8171∣0.1317,0.7601∣0.4745,0.4232∣0.3938.5∣10.5∣0.9,0.5961∣0.10.1499∣0.8,0.7224∣0.1,0.8175∣0.10.6183∣0.89,0.0213∣0.110.5∣0.9,0.4039∣0.1.5∣10.4538∣0.888,0.8003∣0.1120.9404∣0.9414,0.2781∣0.05860.8501∣0.8,0.2775∣0.1,0.1825∣0.10.5462∣0.888,0.1997∣0.112.5∣1H4∗∗=.5∣10.0991∣0.629,0.7317∣0.3710.3498∣0.9,0.9447∣0.10.1835∣0.8,0.3685∣0.05,0.7256∣0.150.9009∣0.629,0.2683∣0.371.5∣10.2481∣0.8,0.5∣0.1,0.9199∣0.10.3786∣0.8,0.6443∣0.1,0.7948∣0.10.6502∣0.9,0.0553∣0.10.7519∣0.8,0.5∣0.1,0.0801∣0.1.5∣10.1349∣0.9,0.6225∣0.05,0.8759∣0.050.8165∣0.8,0.6315∣0.05,0.2744∣0.150.6214∣0.8,0.3557∣0.1,0.2052∣0.10.8651∣0.9,0.3775∣0.05,0.1241∣0.05.5∣1.Remark 35.
Inspection of Table4 will reveal why it is important to consider the probabilities of different preference degrees. Many a time, decision makers are taking decisions related to future scenarios like our case study. The future is a mystery that cannot be predicted completely; probability is a useful tool to deal with uncertainties. In real life scenarios probability of different events can alter the decisions we make today.Table 4
Effects of probability on decision making.
HPFPRs Ranking of alternatives Best commodity H1,H2,H3,H4 x2>x3>x4>x1 Gold H1∗,H2∗,H3∗,H4∗ x1>x2>x3>x4 Oil H1∗∗,H2∗∗,H3∗∗,H4∗∗ x4>x2>x3>x1 Copper


## 6.1. Effects of Probability on Decision Making

In order to show how important is the role played by the probabilities of preference degrees two new versions are generated from the HPFPRsHk=(hij,k)n×n(k=1,2,3,4) by changing the probabilities only and preserving the preference degrees as follows.(88)H1∗=.5∣10.1174∣0.2,0.91∣0.80.4242∣0.1,0.8341∣0.90.2625∣0.231,0.7956∣0.7690.8826∣0.2,0.0900∣0.8.5∣10.1921∣0.85,0.9289∣0.150.2373∣0.925,0.5785∣0.0750.5758∣0.1,0.1659∣0.90.8079∣0.85,0.0711∣0.15.5∣10.2316∣0.1,0.6211∣0.90.7375∣0.231,0.2044∣0.7690.7627∣0.925,0.4214∣0.0750.7684∣0.1,0.3789∣0.9.5∣1H2∗=.5∣10.3674∣0.1,0.3955∣0.1,0.988∣0.80.0987∣0.2,0.2619∣0.1,0.5913∣0.70.3197∣0.35,0.8691∣0.650.6326∣0.1,0.6044∣0.1,0.0120∣0.8.5∣10.1340∣0.8,0.8909∣0.1,0.9037∣0.10.7440∣10.9013∣0.2,0.7381∣0.1,0.4087∣0.70.8660∣0.8,0.1091∣0.1,0.0963∣0.1.5∣10.4111∣0.11,0.9047∣0.890.6803∣0.35,0.1309∣0.650.2559∣10.5889∣0.11,0.0953∣0.89.5∣1H3∗=.5∣10.1829∣0.01,0.2399∣0.23,0.5767∣0.760.3817∣0.289,0.9787∣0.7110.0596∣0.0586,0.7219∣0.94140.8171∣0.01,0.7601∣0.23,0.4232∣0.76.5∣10.5∣0.7,0.5961∣0.30.1499∣0.8,0.7224∣0.1,0.8175∣0.10.6183∣0.289,0.0213∣0.7110.5∣0.7,0.4039∣0.3.5∣10.4538∣0.15,0.8003∣0.850.9404∣0.0586,0.2781∣0.94140.8501∣0.8,0.2775∣0.1,0.1825∣0.10.5462∣0.15,0.1997∣0.85.5∣1H4∗=.5∣10.0991∣0.05,0.7317∣0.950.3498∣0.15,0.9447∣0.850.1835∣0.1,0.3685∣0.05,0.7256∣0.850.9009∣0.05,0.2683∣0.95.5∣10.2481∣0.75,0.5∣0.1,0.9199∣0.150.3786∣0.75,0.6443∣0.1,0.7948∣0.150.6502∣0.15,0.0553∣0.850.7519∣0.75,0.5∣0.1,0.0801∣0.15.5∣10.1349∣0.15,0.6225∣0.1,0.8759∣0.750.8165∣0.1,0.6315∣0.05,0.2744∣0.850.6214∣0.75,0.3557∣0.1,0.2052∣0.150.8651∣0.15,0.3775∣0.1,0.1241∣0.75.5∣1H1∗∗=.5∣10.1174∣0.2979,0.91∣0.70210.4242∣0.9,0.8341∣0.10.2625∣0.9,0.7956∣0.10.8826∣0.2979,0.0900∣0.7021.5∣10.1921∣0.85,0.9289∣0.150.2373∣0.925,0.5785∣0.0750.5758∣0.9,0.1659∣0.10.8079∣0.85,0.0711∣0.15.5∣10.2316∣0.1,0.6211∣0.90.7375∣0.9,0.2044∣0.10.7627∣0.925,0.4214∣0.0750.7684∣0.1,0.3789∣0.9.5∣1H2∗∗=.5∣10.3674∣0.5018,0.3955∣0.3181,0.9880∣0.180.0987∣0.85,0.2619∣0.1,0.5913∣0.050.3197∣0.75,0.8691∣0.250.6326∣0.5018,0.6044∣0.3181,0.0120∣0.18.5∣10.1340∣0.8,0.8909∣0.1,0.9037∣0.10.7440∣10.9013∣0.85,0.7381∣0.1,0.4087∣0.050.8660∣0.8,0.1091∣0.1,0.0963∣0.1.5∣10.4111∣0.95,0.9047∣0.050.6803∣0.75,0.1309∣0.250.2559∣10.5889∣0.95,0.0953∣0.05.5∣1H3∗∗=.5∣10.1829∣0.1317,0.2399∣0.4745,0.5767∣0.39380.3817∣0.89,0.9787∣0.110.0596∣0.9414,0.7219∣0.05860.8171∣0.1317,0.7601∣0.4745,0.4232∣0.3938.5∣10.5∣0.9,0.5961∣0.10.1499∣0.8,0.7224∣0.1,0.8175∣0.10.6183∣0.89,0.0213∣0.110.5∣0.9,0.4039∣0.1.5∣10.4538∣0.888,0.8003∣0.1120.9404∣0.9414,0.2781∣0.05860.8501∣0.8,0.2775∣0.1,0.1825∣0.10.5462∣0.888,0.1997∣0.112.5∣1H4∗∗=.5∣10.0991∣0.629,0.7317∣0.3710.3498∣0.9,0.9447∣0.10.1835∣0.8,0.3685∣0.05,0.7256∣0.150.9009∣0.629,0.2683∣0.371.5∣10.2481∣0.8,0.5∣0.1,0.9199∣0.10.3786∣0.8,0.6443∣0.1,0.7948∣0.10.6502∣0.9,0.0553∣0.10.7519∣0.8,0.5∣0.1,0.0801∣0.1.5∣10.1349∣0.9,0.6225∣0.05,0.8759∣0.050.8165∣0.8,0.6315∣0.05,0.2744∣0.150.6214∣0.8,0.3557∣0.1,0.2052∣0.10.8651∣0.9,0.3775∣0.05,0.1241∣0.05.5∣1.Remark 35.
Inspection of Table4 will reveal why it is important to consider the probabilities of different preference degrees. Many a time, decision makers are taking decisions related to future scenarios like our case study. The future is a mystery that cannot be predicted completely; probability is a useful tool to deal with uncertainties. In real life scenarios probability of different events can alter the decisions we make today.Table 4
Effects of probability on decision making.
HPFPRs Ranking of alternatives Best commodity H1,H2,H3,H4 x2>x3>x4>x1 Gold H1∗,H2∗,H3∗,H4∗ x1>x2>x3>x4 Oil H1∗∗,H2∗∗,H3∗∗,H4∗∗ x4>x2>x3>x1 Copper


## 7. Comparison with Existing Models

Preference relations are one of the important tools that are used for practical decision making in different complex scenarios. A wide number of researchers developed different kinds of preference relations models as fuzzy [10, 11, 32, 40, 43], intuitionistic fuzzy [23, 26, 27], and hesitant fuzzy [6, 24, 28, 33, 38]. Of course fuzzy preference relations are subcase of hesitant fuzzy preference relations, as we have discussed the occurrence probability is considered to be the same which is impractical as Table 4 reflects the impact of probabilities on decision making. So, for this reason the proposed model has provided better modelling of real world problems. In hesitant probabilistic fuzzy environment Zhou and Xu [41] proposed and studied probabilistic fuzzy preference relations. The drawback and difference between their definition of preference relation and proposed Definition 6 have already been discussed in Remark 7. First, they proposed expected consistency based upon multiplicative consistency of fuzzy preference relation [44]; then based on expected consistency multiobjective goal programming models are developed to compute occurrence probability and priority weight vector. Also, an iterative optimization algorithm to improve consistency of nonconsistent preference relations was designed. The comparison is summarized as follows.(i)
Zhou and Xu’s definition [41, Definition 4] is not suitable for decision making based on β-normalization principle, whereas proposed Definition 6 is consistent with β-normalization; in fact HPFPRs are normalized following β-normalization principle.(ii)
They proposed expected consistency, taking a unifying approach to handle both preference degrees and their occurrence probabilities, whereas additive consistency [44] and weak stochastic transitivity [45, Page 27] are adopted for preference degrees and probabilities, respectively, in our proposal.(iii)
They defined expected consistency index based on priority weights computed from multiobjective goal programming model made from expected consistency, whereas for HPFPRH Algorithm 11 (based on convex combination technique) will provide its corresponding NHPFPR H¯ and consistent HPFPR H~ and consistency index is defined to be the distance between H¯ and H~, in our proposal.(iv)
They developed an algorithm which will replace the preference degrees by optimal ones iteratively and occurrence probabilities are computed accordingly to improve consistency, whereas Algorithm19 (based on convex combination technique) will improve consistency, in our proposal.(v)
Both proposals use different techniques and ideas to define and deal with consistency as the above points reflect. However, Zhou and Xu make no effort for group decision making; the main advantage of our proposal is that it is a complete model, not only dealing with group decision making but also providing solution to consensus problem among decision maker, which is a key issue in group decision making.The HFPR is a special case of HPFPR to provide experimental comparison with existing models of group decision making models with HFPRs [33, 38]; we solved their case studies by our proposed model, in a way that preference degrees are the same and their occurrence probabilities are considered to be equal. The comparison is provided in Table 5; clearly our proposed model is quite efficient as it provides the correct results.Table 5
Experimental comparison.
Reference Existing models solution Proposed model solution [38, Case study 5.1] x3>x2>x1>x4 x3>x2>x1>x4 [33, Example  5.1] x3>x2>x1>x4 x3>x2>x1>x4


## 8. Conclusion

In this paper, we have developed a complete group decision support model based on HPFPRs; it consists of three parts: a consistency improving process, a consensus reaching process, and the selection process. The consistency measure of a HPFPR has been defined in the consistency improving process. For HPFPRs that have unacceptable consistency, an optimization method is proposed to improve the consistency until the HPFPRs have an acceptable consistency. In the consensus reaching process, a consensus index is defined to measure the consensus level. For HPFPRs that have an unacceptable consensus, an optimization method is designed to assist them in achieving a predefined consensus level. The proposed model can be used to address GDM problems with HPFPRs. Optimization methods are also developed to help the individual HPFPRs to achieve a predefined consistency level and consensus level with fewer interactions of the DMs. As a consequence, our model is time saving, efficient, and convenient for practical applications. The consistency improving process is performed to ensure that the DMs are neither random nor illogical in their pairwise comparisons. Procedure to reach a consensus level ensures that the adjusted HPFPRs not only achieve the predefined level of consensus but also maintain acceptable consistency. Proposed model also ensures that the consistent HPFPRs do not change at each iteration. This property can retain the DMs’ original decision making information to the greatest extent possible. Here we did not discuss the effects of the application of different controlling parameters and distance functions in the developed model. This model is not useful for the incomplete preference relation. In the future, we will pay attention to addressing these problems. Future research can also focus on the extension of the proposed model to other types of preference relations.



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*Source: 5383487-2018-03-26.xml*