# A New Method of Multiattribute Decision-Making Based on Interval-Valued Hesitant Fuzzy Soft Sets and Its Application

Authors: Yan Yang; Lei Lang; Liuli Lu; Yangmei Sun
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Combining interval-valued hesitant fuzzy soft sets (IVHFSSs) and a new comparative law, we propose a new method, which can effectively solve multiattribute decision-making (MADM) problems. Firstly, a characteristic function of two interval values and a new comparative law of interval-valued hesitant fuzzy elements (IVHFEs) based on the possibility degree are proposed. Then, we define two important definitions of IVHFSSs including the interval-valued hesitant fuzzy soft quasi subset and soft quasi equal based on the new comparative law. Finally, an algorithm is presented to solve MADM problems. We also use the method proposed in this paper to evaluate the importance of major components of the well drilling mud pump.


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## Body

## 1. Introduction

In 1965, Zadeh [1] introduced the notion of fuzzy set. After that, a variety of extended fuzzy set theories have been proposed such as intuitionistic fuzzy set [2], interval-valued intuitionistic fuzzy set [3], hesitant fuzzy set [4, 5], and interval-valued hesitant fuzzy set [6]. In 1999, Molodtsov [7] firstly proposed soft set theory, which has been widely used in many different areas such as decision analysis [8] and military research [9]. With diversity and complexity of practical problems, the original theories are inapplicable to solve these problems. Maji et al. [10] firstly explored fuzzy soft set, which is a more general notion combining fuzzy set and soft set. Some extensions of fuzzy soft set theory have also been presented such as interval-valued fuzzy soft set [11], intuitionistic fuzzy soft set [12], and interval-valued intuitionistic fuzzy soft set [13]. In 2015, Peng and Yang [14] proposed interval-valued hesitant fuzzy soft set (IVHFSS) combining interval-valued hesitant fuzzy set and soft set.In the aspect of the research works of MADM problems, Roy and Maji [15] presented a method according to a comparison table from fuzzy soft sets. Kong et al. [16] modified Roy and Maji’s algorithm to give a novel one, which is based on the comparison of choice values of different objects. Feng et al. [17] proposed a new approach by using level soft sets to solve MADM problems based on fuzzy soft sets. Yang et al. [18] used Feng et al.’s algorithm to solve MADM problems based on multiple fuzzy soft sets. Xu and Zhang [19] developed a novel approach by TOPSIS with incomplete weight information for solving hesitant fuzzy MADM problems. Wang et al. [20] proposed a MADM method based on hesitant fuzzy soft sets. Wang et al. [21] proposed a MADM approach based on the aggregation operator to aggregate hesitant fuzzy soft information. Peng and Yang [14], by means of TOPSIS and the maximizing deviation method, presented a MADM method based on interval-valued hesitant fuzzy soft sets.The method proposed by Peng and Yang is very effective and allows a decision-maker to quantify his opinion with an interval value within[0,1], since it may be difficult for a decision-maker to exactly quantify his opinion with a crisp number to an attribute due to the insufficiency of available information in practical MADM problems. However, there are two problems with this method. One is that it only considers opinion of a decision-maker and the other is that the comparative law based on the score function is not accurate when it is used to compare two IVHFEs in dealing with MADM problems. Based on the above analysis, we improve Peng and Yang’s method to propose a new one of MADM methods based on IVHFSSs, which not only synthetically considers opinions of many decision-makers but also increases the accuracy of the comparison between two IVHFEs.The remainder of this paper is organized as follows. In Section2, some basic definitions including interval value, possibility degree, interval-valued hesitant fuzzy set, and interval-valued hesitant fuzzy soft set are briefly reviewed. In Section 3, a characteristic function of two interval values and a comparative law of IVHFEs based on the possibility degree are proposed. In Section 4, we define the interval-valued hesitant fuzzy soft quasi subset and soft quasi equal based on the new comparative law presented in Section 3. In Section 5, a MADM method based on IVHFSSs is presented. In Section 6, we apply this method to evaluate the importance of major components of the well drilling mud pump. Analysis and discussion are presented in Section 7. The paper is concluded in Section 8.


## 2. Preliminaries

In this section, we mainly review some definitions including interval value, possibility degree of interval value, interval-valued hesitant fuzzy set, and interval-valued hesitant fuzzy soft set. LetU be an initial universe set of objects and let E be the set of attributes in relation to objects in U; A⊆E.Definition 1 (see [22]).
Letr=rL,rU=x∣0≤rL≤x≤rU; then r is called a nonnegative interval value. Particularly, r is a nonnegative real number, if rL=rU.The comparison of two interval values can be roughly divided into two categories: one is deterministic ordering method [23] and the other is possibility degree method [24]. In this paper, we select the latter method.Definition 2 (see [24]).
Letr1=r1L,r1U and r2=r2L,r2U be two interval values, and λ≥0; then(1)
r1+r2=[r1L+r2L,r1U+r2U];(2)
λr1=[λr1L,λr1U]; particularly, λr1=0, if λ=0.Definition 3 (see [24]).
Letr1=r1L,r1U and r2=r2L,r2U be two interval values; then the possibility degree of r1⪰r2 is defined as follows:(1)Pr1⪰r2=max⁡1-maxr2U-r1Lr1U-r1L+r2U-r2L,0,0.Definition 4 (see [25]).
Letr1=r1L,r1U and r2=r2L,r2U:(1)
IfPr1⪰r2=0.5, thena is quasi-equal tobdenoted by r1≈r2.(2)
IfPr1⪰r2>0.5, thenais more thanb denoted by r1≻r2.(3)
IfPr1⪰r2<0.5, thena is less thanbdenoted by r1≺r2.Definition 5 (see [4, 5]).
LetX be a fixed set. A hesitant fuzzy set onX is in terms of a function that when applied toX returns a subset of [0,1], which can be represented as the following mathematical symbol:(2)H=hHxx∣x∈X,where hHx is a set of values in [0,1], denoting the possible membership degrees of the element x∈X to the setH. For convenience, we call hHx a hesitant fuzzy element.Definition 6 (see [6]).
LetX be a reference set, and let D[0,1] be the set of all closed subintervals of [0,1]. An interval-valued hesitant fuzzy set (IVHFS) onX is (3)H¯=hH¯xx∣x∈X,where hH¯x:X→D0,1 denotes all possible interval-valued membership degrees of the element x∈X to the set H¯. For convenience, we call hH¯x an interval-valued hesitant fuzzy element (IVHFE), which reads hH¯x=r∣r∈hH¯x.Definition 7 (see [6]).
LethH¯1x and hH¯2x be two IVHFEs, where hH¯1x=r1=r1L,r1U∣r1∈hH¯1x and hH¯2x=r2=r2L,r2U∣r2∈hH¯2x; then the operational laws of IVHFEs are defined as follows: (1)
λhH¯1x=1-1-r1Lλ,1-1-r1Uλ∣r1∈hH¯1x, λ>0.(2)
hH¯1x∩hH¯2x=min⁡r1L,r2L,min⁡r1U,r2U∣r1∈hH¯1x,r2∈hH¯2x.(3)
hH¯1x∪hH¯2x=max⁡r1L,r2L,max⁡r1U,r2U∣r1∈hH¯1x,r2∈hH¯2x.Definition 8 (see [7]).
LetPU be the set of all subsets ofU. A pair F¯,A is called a soft set overU, where F¯ is a mapping given by F¯:A→PU.Definition 9 (see [14]).
LetIVHFU be the set of all interval-valued hesitant fuzzy subsets ofU. A pair F¯,A is called an interval-valued hesitant fuzzy soft set (IVHFSS) overU, where F¯ is a mapping given by F¯:A→IVHFU.In [14], some properties of IVHFSSs were given. We give another three properties as follows.Property 10.
Given an IVHFSSF¯,A overU,(1)
ifA=e, then F¯,A is also an interval-valued hesitant fuzzy set,(2)
if there is only one interval value ofhF¯eu for all e∈A and u∈U, then F¯,A is also an interval-valued fuzzy soft set,(3)
if the upper and lower boundaries of each interval value ofhF¯eu are equal for all e∈A and u∈U, then F¯,A is also a hesitant fuzzy soft set.Definition 11 (see [14]).
The union operation on two IVHFSSsF¯,A and G¯,B over U is an IVHFSS H¯,C, where C=A∪B and, for all e∈C,(4)H¯e=F¯e,ife∈A-B,G¯e,ife∈B-A,F¯e∪G¯e,ife∈A∩B.We write F¯,A∪¯G¯,B=H¯,C.Definition 12 (see [14]).
The intersection operation on two IVHFSSsF¯,A and G¯,B with A∩B≠Φ over U is an IVHFSS H¯,C, where C=A∩B and, for all e∈C, H¯e=F¯e∩G¯e. We write F¯,A∩¯G¯,B=H¯,C.


## 3. A New Comparative Law of Interval-Valued Hesitant Fuzzy Elements Based on the Possibility Degree

In this section, a new comparative law of IVHFEs will be proposed which can overcome some defects of the comparative law based on the score function.For convenience, letlhH¯x denote the length of hH¯x and let hH¯σjx denote the jth maximum interval value in hH¯x. lmax is the maximum value of lhH¯1x and lhH¯2x.Definition 13 (see [6]).
The score function ofhH¯x is defined as follows: (5)ShH¯x=1lhH→x∑r∈hH¯xr.Obviously, ShH¯x is still an interval value. Given two IVHFEs hH¯1x and hH¯2x,(1)
ifShH¯1x≈ShH¯2x, then hH¯1x≈hH¯2x;(2)
ifShH¯1x≻ShH¯2x, then hH¯1x≻hH¯2x;(3)
ifShH¯1x≺ShH¯2x, then hH¯1x≺hH¯2x.Example 14.
There arehH¯1x=0.3,0.4,0.4,0.5,0.5,0.6 and hH¯2x=0.3,0.4,0.5,0.6. Obviously, hH¯1x≻hH¯2x. However, by Definition 13, we have ShH¯1x≈ShH¯2x; thus hH¯1x≈hH¯2x.Given two IVHFEshH¯1x and hH¯2x, usually, lhH¯1x≠lhH¯2x. However, the comparative law used in Example 14 ignores this important point. In fact, according to the attitude of decision-makers, there are three different ways to make them equal in length by adding some suitable interval values. When the decision-makers are optimistic, we make them equal in length by adding the maximum interval value. Analogously, we add the minimum interval value when the decision-makers are pessimistic. The decision-makers remain neutral; we add the average of the maximum interval value and the minimum interval value. In this paper, if lhH¯1x≠lhH¯2x, without loss of generality, we take the pessimistic principle to make hH¯1x and hH¯2x equal in length such that lhH¯1x=lhH¯2x=lmax.Example 15.
Two IVHFEs are given in Example14. Obviously, lhH¯1x≠lhH¯2x. Since 0.3,0.4⪰0.5,0.6=0, 0.3,0.4≺0.5,0.6. Based on the pessimistic principle, we can extend hH-2x=0.3,0.4,0.3,0.4,0.5,0.6 such that lhH¯1x=lhH¯2x=3.In 2014, Zhou [9] proposed a characteristic function of two real numbers. Based on this, we give a characteristic function of two interval values.Definition 16.
Assume thatr1=r1L,r1U and r2=r2L,r2U; then the characteristic function fr1,r2 is defined as follows:(6)fr1,r2=1,ifr1≻r2,0.5,ifr1≈r2,0,ifr1≺r2.Definition 17.
LethH¯1x and hH¯2x be two IVHFEs. The possibility degree of hH¯1x⪰hH¯2x is defined as follows:(7)PhH¯1x⪰hH¯2x=1lmax∑j=1lmaxfhH¯1σjx,hH¯2σjx.Definition 18.
LethH¯1x and hH¯2x be two IVHFEs:(1)
IfPhH¯1x⪰hH¯2x=0.5, then IVHFE hH¯1x is quasi-equal to hH¯2x, denoted by hH¯1x≈hH¯2x.(2)
IfPhH¯1x⪰hH¯2x>0.5, then IVHFE hH¯1x is more than hH¯2x, denoted by hH¯1x≻hH¯2x.(3)
IfPhH¯1x⪰hH¯2x<0.5, then IVHFE hH¯1x is less than hH¯2x, denoted by hH¯1x≺hH¯2x.(4)
IfPhH¯1x⪰hH¯2x≥0.5, then IVHFE hH¯1x is more than or equal to hH¯2x, denoted by hH¯1x⪰hH¯2x.(5)
IfPhH¯1x⪰hH¯2x≤0.5, then IVHFE hH¯1x is less than or equal to hH¯2x, denoted by hH¯1x⪯hH¯2x.Example 19.
We consider two IVHFEs given in Example14. Since lhH¯1x>lhH¯2x, we extend hH-2x=0.3,0.4,0.3,0.4,0.5,0.6 based on the pessimistic principle. By Definition 17, we have PhH¯1x⪰hH¯2x=2/3; then hH¯1x≻hH¯2x.Theorem 20.
LethH¯1x and hH¯2x be two IVHFEs. The following conclusions hold:(1)
0≤PhH¯1x⪰hH¯2x≤1.(2)
PhH¯1x⪰hH¯2x=1 if and only if hH¯1σjx≻hH¯2σjx for all j=1,2,…,lmax.(3)
PhH¯1x⪰hH¯2x=0 if and only if hH¯1σjx≺hH¯2σjx for all j=1,2,…,lmax.Proof.
(i) It is easy to prove0≤PhH¯1x⪰hH¯2x≤1 by Definition 17.
(ii) For necessity, ifPhH¯1x⪰hH¯2x=1, then fhH¯1σjx,hH¯2σjx=1 for all j=1,2,…,lmax by Definitions 16 and 17. This implies that hH¯1σjx≻hH¯2σjx for all j=1,2,…,lmax.
For sufficiency, ifhH¯1σjx≻hH¯2σjx, then fhH¯1σjx,hH¯2σjx=1 for all j=1,2,…,lmax by Definition 16; thus, PhH¯1x⪰hH¯2x=1 by Definition 17.
(iii) Similar to(2), we can prove (3).Remark 21.
Formula (7) in Definition 17 can also be defined as follows:(8)PhH¯1x⪰hH¯2x=1lmax∑j=1lmaxPhH¯1σjx⪰hH¯2σjx.However, it does not satisfy Theorem 20. Example 22 can illustrate the point as follows.Example 22.
There arehH¯1x=0.3,0.5,0.3,0.6,0.5,0.7 and hH¯2x=0.3,0.4,0.5,0.6. Based on the pessimistic principle, we extend hH¯2x=0.3,0.4,0.3,0.4,0.5,0.6, and then PhH¯1x⪰hH¯2x=25/36≠1 by Formula (8) in Remark 21. There is a contradiction to Theorem 20, since hH¯1σjx≻hH¯2σjx for all j=1,2,3.Definition 23.
LetH¯1 and H¯2 be two interval-valued hesitant fuzzy sets onX:(1)
IfhH¯1x⪯hH¯2x, for all x∈X, then H¯1 is an interval-valued hesitant fuzzy quasi subset of H¯2, denoted by H¯1⊆H¯2.(2)
IfhH¯1x≈hH¯2x, for all x∈X, then H¯1 is quasi-equal to H¯2, denoted by H¯1≈H¯2.


## 4. Two Important Definitions of Interval-Valued Hesitant Fuzzy Soft Sets Based on the New Comparative Law

In this section, we give definitions of the interval-valued hesitant fuzzy soft quasi subset and soft quasi equal according to Definition23. Let A,B⊆E and let F¯,A and G¯,B be two IVHFSSs overU.Definition 24.
F
¯
,
A is an interval-valued hesitant fuzzy soft quasi subset of G¯,B, if the following two conditions are satisfied:(1)
A⊆B.(2)
F¯e⊆G¯e for all e∈A. That is, hF¯eu⪯hG¯eu, for all e∈A and u∈U, denoted by F¯,A⊆¯G¯,B.Definition 25.
F
¯
,
A is soft quasi-equal to G¯,B, if the following two conditions are satisfied:(1)
A=B.(2)
F¯e≈G¯e for all e∈A. That is, hF¯eu≈hG¯eu, for all e∈A and u∈U, denoted by F¯,A≅G¯,B.Theorem 26.
F
¯
,
A
≅
G
¯
,
B if and only if F-,A⊆¯G-,B and G-,B⊆¯F-,A.Proof.
For necessity, sinceF¯,A≅G¯,B, we have A=B and hF¯eu≈hG¯eu by Definition 25. This implies that A⊆B and F¯e⊆G¯e for all e∈A. Analogously, A⊇B and F¯e⊇G¯e for all e∈A. By Definition 24, we have F-,A⊆¯G-,B and G-,B⊆¯F-,A.
For sufficiency, sinceF-,A⊆¯G-,B and G-,B⊆¯F-,A, we have A⊆B and F¯e⊆G¯e for all e∈A by Definition 24. Analogously, A⊇B and F¯e⊇G¯e for all e∈A. That is, A=B and F¯e≈G¯e for all e∈A. Therefore, F¯,A≅G¯,B by Definition 25.


## 5. A New Method of Multiattribute Decision-Making Based on Interval-Valued Hesitant Fuzzy Soft Sets

In the case of multiple attributes, the MADM is the decision problem to select the optimal alternative or rank the alternatives. LetU=u1,u2,…,un be the set ofn alternatives, let E=e1,e2,…,et be the set oft attributes, and let D=d1,d2,…,ds be the set ofs decision-makers. The decision-makers dk∈Dk=1,2,…,s evaluaten alternatives from Ak=e1k,e2k,…,elAkk, where Ak⊆E and lAk represents the number of attributes in Ak. The attribute weight vector of Ak is ωk=ω1k,ω2k,…,ωlAkkT, where ∑j=1lAkωjk=1.IVHFSSsFk¯,Akk=1,2,…,s can be expressed by a tabular representation shown in Table 1. Firstly, decision-maker dk selects relevant attribute set Ak according to his preference. Next, we collect the historical data of each attribute in the attribute setE and use it as the basis for the evaluation of decision-maker. For example, economy factor is determined by market price, and workforce factor is determined by population, and so forth. Finally, h¯F¯kejkui represents the IVHFE of the alternative ui satisfying the attribute ejk which is given by the decision-maker dk, where j=1,2,…,lAk and i=1,2,…,n.Table 1
Tabular representation ofFk¯,Ak.
u
1
u
2
⋯
u
n
e1k
h
¯
F
¯
k
e
1
k
u
1
h
¯
F
¯
k
e
1
k
u
2
⋯
h
¯
F
¯
k
e
1
k
u
n
e2k
h
¯
F
¯
k
e
2
k
u
1
h
¯
F
¯
k
e
2
k
u
2
⋯
h
¯
F
¯
k
e
2
k
u
n
⋮
⋮
⋮
⋱
⋮
elAkk
h
¯
F
¯
k
e
l
A
k
k
u
1
h
¯
F
¯
k
e
l
A
k
k
u
2
⋯
h
¯
F
¯
k
e
l
A
k
k
u
n
l
kDefinition 27.
Suppose thatA=e1,e2,…,et and F¯,A is an IVHFSS overU. A generalized comparison table (GCT) of F¯,A is a square table in which the number of rows and number of columns are equal. Both rows and columns are named by U=u1,u2,…,un, and elements in GCT are denoted by Cij. Its formula is shown as follows:(9)Cij=∑m=1tPhF¯emui⪰hF¯emuj,i,j=1,2,…,n;m=1,2,…,t,wheret is the number of attributes.Definition 28.
LethH¯x=r=rL,rU∣r∈hH¯x be an IVHFE and let λ be a nonnegative real number; then a new multiplication operational law of IVHFEs is defined as follows:(10)λhH¯x=λrL,λrU∣r∈hH¯x,λ>0.Obviously, it is different from Definition 7.By the IVHFSSs based on the new comparative law of IVHFEs, we propose a new method of the MADM as follows.Algorithm 29.
Step
1. Obtain the tabular representation of Fk¯,Ak by F¯kejk=h¯F¯kejkui/ui∣ui∈Uj=1,2,…,lAk for all decision-makers dkk=1,2,…,s.
Step
2. Compute the weighted IVHFSS F¯k′,Ak for all ejk∈Ak by F′¯kejk=h¯F¯k′ejkui/ui∣h¯F¯k′ejkui=ωjkh¯F¯kejkui,ui∈U.
Step
3. Compute F¯,A=⋃¯k=1sF¯′k,Ak, where A=⋃k=1sAk.
Step
4. Calculate Cij according to Formula (9) and construct the GCT.
Step
5. Compute row sum pi=∑j=1nCij and column sum qj=∑i=1nCij of the GCT, and calculate the score Si=pi-qj of alternative ui.
Step
6. If the score Si is the maximum value, then ui can be used as the optimal alternative.


## 6. Evaluate the Importance of the Major Components of the Well Drilling Mud Pumps

A mud pump, an important part of the well drilling equipment, is a large reciprocating pump, which is used to circulate the mud on a drilling rig. A mud pump is mainly composed of four parts which are pump case, piston, bearing, and impeller. However, the four parts are also most likely to be damaged. Once one of these parts is damaged, the mud pump will not work properly and probably cause huge losses. Aiming at this problem, we will use Algorithm29 to evaluate the importance of the four parts of the well drilling mud pump.Now, we consider four parameters, namely, economic factor, reliability factor, service life factor, and maintenance factor, to evaluate the importance of the four parts. In order to make evaluation by three experts reasonably and comprehensively, we investigated the market price of the four parts in recent years for economic factor, collected the information feedback of the four parts for reliability factor, surveyed the durable years of the four parts for service life factor, and counted the maintenance time of the four parts for maintenance factor. LetU=u1,u2,u3,u4 be a set of the four parts, let D=d1,d2,d3 be the set of the three experts, and let E=e1,e2,e3,e4 be the set of the four attributes.Based on the evaluation of the three experts, we construct three tabular representations ofF¯k,Akk=1,2,3. The evaluations of experts are given as IVHFEs. The information is shown in Tables 2–4.Table 2
Tabular representation ofF¯1,A1.
u
1
u
2
u
3
u
4
e
1
0.5,0.6
,
0.4,0.8
0.4,0.5
,
0.4,0.7
,
0.6,0.8
0.5,0.8
,
0.6,0.8
0.2,0.4
,
0.5,0.6
e
3
0.1,0.2
,
0.4,0.5
,
0.7,0.8
0.4,0.6
,
0.5,0.6
0.1,0.2
,
0.2,0.3
0.2,0.3
,
0.2,0.4Table 3
Tabular representation ofF¯2,A2.
u
1
u
2
u
3
u
4
e
2
0.4,0.5
,
0.7,0.9
0.1,0.3
,
0.2,0.5
0.2,0.5
,
0.3,0.5
,
0.5,0.6
0.3,0.4
,
0.5,0.6
e
3
0.6,0.8
,
0.7,0.8
0.3,0.7
,
0.6,0.8
0.3,0.5
,
0.4,0.7
0.3,0.6
,
0.5,0.6Table 4
Tabular representation ofF¯3,A3.
u
1
u
2
u
3
u
4
e
2
0.3,0.5
,
0.4,0.7
0.2,0.3
,
0.1,0.5
0.6,0.9
,
0.7,0.9
0.6,0.7
,
0.7,0.8
e
4
0.1,0.3
,
0.2,0.5
,
0.4,0.6
0.4,0.7
,
0.5,0.8
0.6,0.8
,
0.7,0.8
0.4,0.5
,
0.5,0.7The weight vectorωk is given by three technical experts dkk=1,2,3, where ω1=0.6,0.4T, ω2=0.7,0.3T, and ω3=0.4,0.6T, respectively. Calculate the weighted IVHFSSs F¯1′,A1, F¯2′,A2, and F¯3′,A3. Their representations are shown in Tables 5–7.Table 5
Tabular representation ofF¯1′,A1.
u
1
u
2
u
3
u
4
e
1
0.3,0.36
,
0.24,0.48
0.24,0.3
,
0.24,0.42
,
0.36,0.48
0.3,0.48
,
0.36,0.48
0.12,0.24
,
0.3,0.36
e
3
0.04,0.08
,
0.16,0.2
,
0.28,0.32
0.16,0.24
,
0.2,0.24
0.04,0.08
,
0.08,0.12
0.08,0.12
,
0.08,0.16Table 6
Tabular representation ofF¯2′,A2.
u
1
u
2
u
3
u
4
e
2
0.28,0.35
,
0.49,0.63
0.07,0.21
,
0.14,0.35
0.14,0.35
,
0.21,0.35
,
0.35,0.42
0.21,0.28
,
0.35,0.42
e
3
0.18,0.24
,
0.21,0.24
0.09,0.21
,
0.18,0.24
0.09,0.15
,
0.12,0.21
0.09,0.18
,
0.15,0.18Table 7
Tabular representation ofF¯3′,A3.
u
1
u
2
u
3
u
4
e
2
0.12,0.2
,
0.16,0.28
0.08,0.12
,
0.04,0.2
0.24,0.36
,
0.28,0.36
0.24,0.28
,
0.28,0.32
e
4
0.06,0.18
,
0.12,0.3
,
0.24,0.36
0.24,0.42
,
0.3,0.48
0.36,0.48
,
0.42,0.48
0.24,0.3
,
0.3,0.42LetF¯,A=⋃¯k=13F¯k′,Ak; then A=A1∪A2∪A3=e1,e2,e3,e4 and (11)F¯e1=F¯1′e1=0.3,0.36,0.24,0.48u1,0.24,0.3,0.24,0.42,0.36,0.48u2,0.3,0.48,0.36,0.48u3,0.12,0.24,0.3,0.36u4,F¯e2=F¯2′e2∪F¯3′e2=0.28,0.35,0.49,0.63u1,0.07,0.21,0.08,0.21,0.14,0.35u2,0.24,0.36,0.28,0.36,0.35,0.42u3,0.24,0.28,0.28,0.32,0.35,0.42u4,F¯e3=F¯1′e3∪F¯2′e3=0.18,0.24,0.21,0.24,0.28,0.32u1,0.16,0.24,0.18,0.24,0.2,0.24u2,0.09,0.15,0.12,0.21u3,0.09,0.18,0.15,0.18u4,F¯e4=F¯3′e4=0.06,0.18,0.12,0.3,0.24,0.36u1,0.24,0.42,0.3,0.48u2,0.36,0.48,0.42,0.48u3,0.24,0.3,0.3,0.42u4.According to Formula (9), we can calculate Ciji,j=1,2,3,4 and get the GCT of the IVHFSS F¯,A shown in Table 8.Table 8
The GCT of IVHFSSF¯,A.
u
1
u
2
u
3
u
4
u1
2
5
/
2
5
/
3
5
/
2
u2
3
/
2
2
7
/
6
3
u3
7
/
3
17
/
6
2
37
/
12
u4
3
/
2
1
11
/
12
2From Table8, we can get the row sum pi, the column sum qj, and the score Si of ui shown in Table 9.Table 9
The scoreSi of ui.
p
i
q
j
S
i
1
2
u1
26
/
3
22
/
3
4
/
3
u2
23
/
3
25
/
3
-
2
/
3
u3
123
/
12
23
/
4
9
/
2
u4
65
/
12
127
/
12
-
31
/
6From Table9, we have S3>S1>S2>S4; thus the part u3, that is, bearing, is the most important component.


## 7. Analysis and Discussion

In the above MADM problem, it is very difficult fordkk=1,2,3 to determine his evaluation on every em(m=1,2,3,4) by a crisp number; thus some interval values in [0,1] are used. Therefore, the existing methods [15–21] are inapplicable to solve this problem. Furthermore, it is necessary to evaluate the mud pump from different aspects, since it is a complex system; thus comprehensive opinions by more than one decision-maker are needed, which implies that the existing method [14] considering a decision-maker is not appropriate to deal with this problem, too.In addition, in the process of evaluating the importance of the four parts of the well drilling mud pump with Algorithm29, we need to calculate Ciji,j=1,2,3,4 by IVHFEs hF¯emui and hF¯emujm=1,2,3,4 according to Formula (9). Therefore, the new comparative law given in Definition 17 is the base for computing Cij.


## 8. Conclusions

Although Peng and Yang [14] have proposed IVHFSSs, the method they gave can only deal with MADM problems with a decision-maker. In most cases, it is necessary to synthesize opinions of more than one decision-maker for complex MADM problems. Based on IVHFSSs and the new comparative law, this paper presents a new method which can effectively solve MADM problems. We also apply this method to evaluate the importance of main parts of the well drilling mud pump, which can help engineers avoid the equipment failure as soon as possible.



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*Source: 9376531-2017-07-30.xml*