A Choquet integral based fuzzy logic approach to solve uncertain multi-criteria decision making problem

A Choquet Integral based Fuzzy Logic Approach to Solve Uncertain Multi-Criteria Decision Making Problem

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A Choquet Integral based Fuzzy Logic Approach to Solve Uncertain Multi-Criteria Decision Making Problem Chen Li Writing - review and editing, Duan Gang, Wang SuYun, Ma JunFeng PII: DOI: Reference:

S0957-4174(20)30128-7 https://doi.org/10.1016/j.eswa.2020.113303 ESWA 113303

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Expert Systems With Applications

Received date: Revised date: Accepted date:

12 September 2019 8 February 2020 8 February 2020

Please cite this article as: Chen Li Writing - review and editing, Duan Gang, Wang SuYun, Ma JunFeng, A Choquet Integral based Fuzzy Logic Approach to Solve Uncertain Multi-Criteria Decision Making Problem, Expert Systems With Applications (2020), doi: https://doi.org/10.1016/j.eswa.2020.113303

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Highlights • Setup interval valued Choquet integral in the interval valued Sugeno probability space; • Introduce interval valued Sugeno probability measures based on σ − λ rules; • Apply Choquet integrals on interval valued Sugeno probability measure in decision making; • Use refrigerator end-of-life strategy problem to demonstrate proposed approach.

1

A Choquet Integral based Fuzzy Logic Approach to Solve Uncertain Multi-Criteria Decision Making Problem Chen Li1,2 , 1 2

Duan Gang3 , Wang SuYun1 ,

Ma JunFeng2,†

Department of Mathematics, Lanzhou City University, Gansu, 730070, China

Department of Industrial and Systems Engineering, Mississippi State University, Mississippi, 39762 2,3

Abstract:

∗

School of Traffic and Transportation, Lanzhou Jiaotong University, Gansu, 730070, China

Nowadays, the fuzzy measures and fuzzy integrals have been successfully implemented to solve a variety

of uncertain multi-criteria decision-making problems. However, with the growing complexity of the decision-making environment and the diversity of linguistic information in the decision-making process, defining the appropriate and reasonable measures and integrals in the fuzzy logic applications becomes increasingly challenging. As the commonly used interval-valued Sugeno probability space is capable of representing the linguistic information in a more accurate way, in this work, we combine the Choquet integrals with interval-valued Sugeno probability space to develop a new interval-valued function to solve the uncertain multi-criteria decision-making problems. This work first defines the interval-valued Sugeno probability measure based on σ − λ rules and proposes the Choquet integrals in the interval-

valued Sugeno probability space, and then a relevant solution approach is developed to solve the uncertain multi-criteria decision-making problems. The refrigerator components end-of-life strategy determination problem is used as a case study to illustrate the application.

AMS(2000) Subj. Classification: 15A22; 68Q40 Key Words: Fuzzy number; Fuzzy measures; Choquet integrals; Uncertain; Decision making

1. Introduction In today’s highly competitive market, both companies and individuals are always facing complex decisionmaking situations. In order to appropriately support and satisfy customers, people need to not only make decisions in a timely manner, but deal with the uncertainties encountered in decision-making procedure as well. Therefore, the research of decision analysis methods has great theoretical potentials to the management science, and also has significant practical meanings to the real-world complex decision-making environments (Beynon, Cosker & Marshall, 2001; Boran, Genc, Kurt & Akay, 2009; Chen & Wang, 2009; Ma & Kremer, 2014; Ma, Harstvedt, Jaradat & Smith, 2020; Pavan & Todeschini, 2009; Rodolfo & Krohling, 2014). Fuzzy logic, as a widely used approach to handle uncertainty, has received increasingly attentions recently. The uncertainties in decision-making usually come from the experts’ vague or incomplete information-based judgement. Experts are the sources of two types of uncertainties: aleatory and epistemic uncertainties (Ye, Jankovic & Kremer, 2015). Aleatory uncertainty is irreducible because it links to brain processes (Dror & Charlton, 2006); while epistemic uncertainty is resulting from incomplete information of the decision-making environment (Oberkampf, Helton & Sentz, 2001). Two approaches are commonly used to reduce epistemic ∗The author, Li Chen was supported by Scientific Study Project Higher Learning, Ministry Education of Gansu Province (Grant No. 2017B-51), PhD Research Startup Foundation of Lanzhou City University of China (Grant No. LZCU-BS2015-03), and the key constructive discipline of Lanzhou City University of China (Grant No. LZCU-ZDJSXK-201709). Without her contribution, this research will not be possible. † Corresponding author. Tel.:+16623257625. Email Address: [email protected] (J.F. Ma).

uncertainty: probability based approaches and fuzzy logic based methodologies. Probability based approaches model the uncertainty by parameters and utilize relevant distributions to estimate parameters to reduce uncertainty (Ma& Kremer, 2016). Fuzzy logic approaches use ”degree of trust” rather than simple ”true or false” Boolean logic to model and represent uncertainty (Ma, Kremer & Ray, 2018). Because of its flexibility and effectiveness, fuzzy logic is more commonly used in experts’ uncertain judgement based decision-making. The existing fuzzy logic approaches use linguistic assessment instead of exact number to collect experts’ judgements, then convert these linguistic information to fuzzy number, and use membership functions to solve the decision-making problems. Fuzzy logic approaches usually need a step to defuzzify the fuzzy numbers. However, most of defuzzification approaches will either not provide accurate results or processes are too difficulty in calculation. This paper proposes a Choquet integral based fuzzy logic approach to better handle the uncertainty. 2. Literature Review Conventional multi-criteria decision-making (MCDM) approaches do not consider the relevant importance among the decision attributes and cannot deal with inaccurate or incomplete information as well. In the 1980s, Saaty proposed the Analytic Hierarchy Process (AHP) (Saaty, 1980), which is the first work to combine the fuzzy thinking and MCDM. AHP integrates the qualitative and quantitative analysis to make decisions by capturing the judgments from skilled decision-makers (DM), comparing relevant factors and testing the rationality of results layer by layer. Because of many merits, AHP has been widely used in various decision-making processes. However, this approach is suffering several drawbacks, such as too many computational efforts required (Yager, 2004; Chen & Hwang, 1992). In order to identify the relevant importance among multiple attributes in MCDM, Sugeno (1974) proposed a non-additive fuzzy measure to replace additivity fuzzy measures with consideration of weak monotonicity and continuity. According to his interpretation, for any MCDM problem containing n criteria, DMs need to create 2n − 2 fuzzy measures. Sugeno (1974) defined λ fuzzy measure to reduce the

complexity of calculating attributes relevant importance and therefore improve the efficiency of solving MCDM problems. Today, due to the linguistic complexity and vagueness in the decision-making environment, conventional MCDM approaches are difficult to tackle all the uncertainties in the accurate and efficient way, and hence cannot meet the needs of practical implementations. Fuzzy logic, as one of the best approaches to deal with linguistic uncertainties, has been mature for decades. Therefore, integrating fuzzy logic with MCDM becomes a trend (Ma & Kremer, 2015; Wang, Zhang & Zhai, 2007; Wang, Zhai & Lu, 2008). In these fuzzy logic based MCDM approaches, fuzzy linguistics and fuzzy numbers, instead of the traditional precise numbers, are used to describe decision information (Grabisch, 1995; Leung, Wong, Lam, Wang & Xu, 2002; Wang, Klir & Wang, 1996; Wang, Leung, Wong & Fang, 2000; Wang & He, 2011; Xu, Wang, Wong & Leung, 2001). As a classical measure, fuzzy measure (Liu, Jheng, Lin & Chen, 2007; Magdi & Xiao, 2003; Miranda & Grabisch, 2002; Wu, Zhang & Guo, 2000; Yager, 2002) is defined as non-additivity and capable of representing the relevant importance among attributes more accurately. Technically, MCDM requires to have independent criteria, however, it is really difficult to have two completely independent criteria existed in the real practice. Therefore, there is a need to capture the correlation between criteria and take into account in the decisionmaking. The primary contribution of fuzzy measure is to reflect these interactions between criteria, and then provide the accurate evaluation. The Choquet integral based fuzzy measure is a widely used nonlinear function in subjective evaluation (Choquet, 1954; Denneberg, 1994; Murofushi, Sugeno & Machida, 1994; Xu, Wang & Leung, 2003; Yang, Wang, Henga & Leung, 2005). The features of ”super-additive” and ”sub-additive” of fuzzy integral are consistent with the fact that a whole characteristic is not just the simply linear sum of several individual characteristics. This paper uses the Choquet integral as an evaluation tool to support the MCDM problem solving. Extensive Choquet integral fuzzy measure studies have been conducted to use real-valued Choquet integral 3

to solve the problems (Candeloro, Mesiar & Sambucini, 2019; Gong, Chen & Duan, 2014; Huang & Wu, 2014; Jang, 2013; Torra & Narukawa, 2006). However, these studies have limitations in the real practical applications because the real-world complex and uncertain decision-making problems are not appropriate to be modeled and solved by accurate mathematical approaches. Even if some problems can be solved, the results and analysis are not reliable due to the high level of uncertainty. Therefore, in order to remedy this limitation, we will establish interval-valued Choquet integral function based on σ − λ rules. The fuzzy number will be used to represent

the complex linguistic variables, then a Choquet integral fuzzy logic based MCDM model will be formulated. A relevant solution approach will be proposed to solve the decision-making problems under the complex and vague environment. The rest of the paper is organized as follows: Section 3 sets up the interval-valued Choquet integral in the interval-valued Sugeno probability space. The relevant definitions of interval-valued Sugeno probability measures based on σ − λ rules are introduced in section 3.1 and section 3.2 defines the concepts of the Choquet integrals in interval-valued Sugeno probability measure and gives the operational schemes on discrete sets. In section 4, the implementation process of Choquet integrals on interval-valued Sugeno probability measure in MCDM and a relevant solution approach are presented. A refrigerator component end-of-life strategy MCDM problem is used as a case study to illustrate the process and validate the result in Section 5. Section 6 summarizes the work and provides the limitations and future directions. 3 Choquet Integrals in Interval-valued Sugeno Probability Space 3.1 Interval-Valued Sugeno Probability Measure The basic definitions of interval-valued Sugeno probability measure are given below. Definition 3.1.1 (Ha, Li, Li & Tian, 2006) Let X be a nonempty set and A a σ− algebra on the X. A set

function µ is called a fuzzy measure based on σ − λ rules if

µ

∞ [

Ai

i=1

1 where λ ∈ (− supµ , ∞)

S

!

=

)  (∞  1 Y   [1 + λµ(Ai )] − 1 ,    λ i=1 ∞  X     µ(Ai ), 

λ 6= 0, (1) λ = 0,

i=1

{0} , {Ai } ⊂ A, Ai ∩ Aj = ∅ for all i, j = 1, 2, · · · and i 6= j.

Particularly, if λ = 0, then σ − λ rule is σ−additivity.

Remark 3.1.1 In Definition 3.1.1, if n = 2, then   µ(A) + µ(B) + λµ(A)µ(B), µ (A ∪ B) =  µ(A) + µ(B),

λ 6= 0,

(2)

λ = 0.

Definition 3.1.2 (Wu, Zhang & Guo, 1998) Let X be a nonempty set, A a σ−algebra on the X , the set function µ : A → I(R+ ) = {r : [r, r] ⊂ R+ }, that is µ = [µ, µ], where µ : A → R+ , µ : A → R+ , µ and µ satisfying the following conditions: (1) µ(∅) = 0, µ(∅) = 0; (2) if A, B ⊂ X, and A ⊂ B, then µ(A) ≤ µ(B), µ(A) ≤ µ(B); (3) for every A ⊂ X, µ(A) ≤ µ(A),

then µ is called an interval-valued fuzzy measure. Due to the complexity and the vagueness of human thinking, the accurate information is usually hard to be obtained, and such information is often given in the uncertain format, such as interval numbers. Interval number represents a kind of uncertainty and has great potential in various implementations. For example, the interval number has been implemented to mathematical programming to establish the uncertain optimization 4

model. It is meaningful and crucial to handle uncertainty using interval valued fuzzy measure. We give the following definition to support such statement. Definition 3.1.3 If µ and µ satisfy the σ − λ rules in Definition 3.1.2, and µ(X) = 1, µ(X) = 1, then µ = [µ, µ]

is called interval-valued Sugeno probability measure based on σ − λ rules. Or briefly called interval-valued Sugeno probability measure and denoted gλ = [g λ , g λ ].

Below example shows how to handle the joint attribute of interval valued fuzzy measure using Definition 3.1.3 Example 3.1.1 Suppose x1 , x2 , x3 represent three workers engaged in the production process of the same product, respectively. The production efficiency of each worker is given as: µ(x1 ) = [0.5, 0.6], µ(x2 ) = [0.6, 0.7], µ(x3 ) = [0.8, 0.9]. We can obtain µ(x1 ) = 0.5, µ(x2 ) = 0.6, µ(x3 ) = 0.8; µ(x1 ) = 0.6, µ(x2 ) = 0.7, µ(x3 ) = 0.9. Then we can obtain the joint efficiency of three workers by σ − λ rules as shown in Table 1. Table 1 The value of gλ about Example 3.1.1 Set

Value of gλ

E1 = {x1 }

[0.5, 0.6]

E3 = {x3 }

[0.8, 0.9]

E2 = {x2 }

[0.6, 0.7]

E4 = {x1 , x2 }

[1.1+0.3λ1 , 1.3+0.42λ2 ]

E6 = {x2 , x3 }

[1.4+0.48λ1 , 1.6+0.63λ2 ]

E5 = {x1 , x3 } E7 = {x1 , x2 , x3 }

[1.3+0.4λ1 , 1.5+0.54λ2 ] [1.9+1.18λ1 + 0.24λ21 , 2.2+1.59λ2 + 0.378λ22 ]

Remark 3.1.2 In Example 3.1.1, xi can be viewed as an attribute (i = 1, 2, 3), then we can calculate the weight of their joint attributes by σ − λ rules given the weight of individual attribute gλ (xi ), i = 1, 2, 3.

Theorem 3.1.1 Let gλ = [g λ , g λ ] be an interval-valued Sugeno probability measure. If λ ≥ 0, then g λ , g λ has

the monotonicity, respectively.

Proof Let E, F ∈ F, E ⊂ F . Since F = E ∪ (F − E), this implies that g λ (F ) = g λ (E ∪ (F − E)) = g λ (E) + g λ (F − E) + λg λ (E)g λ (F − E) then g λ (F ) ≥ g λ (E) where λ ≥ 0, g λ ≥ 0. That is g λ has the monotonicity. The same theory proves that g λ has the monotonicity. Definition 3.1.4 Let X be a finite set and 2X be the power set of X. Set function µ : 2X −→ [µ, µ] ⊂ [0, 1] is called a regular interval fuzzy measure defined on 2X if the following conditions are satisfied: (1) µ(∅) = 0, µ(∅) = 0, µ(X) = 1, µ(X) = 1; (2) if E ∈ 2X , G ∈ 2X , E ⊂ G, then µ(E) ≤ µ(G), µ(E) ≤ µ(G).

Definition 3.1.5 Let X be a finite set and 2X be the power set of X. Set function µ : 2X −→ [µ, µ] ⊂ [0, 1] is called a regular λ−interval fuzzy measure defined on 2X if the following conditions are satisfied: (1) µ(∅) = 0, µ(∅) = 0, µ(X) = 1, µ(X) = 1; (2) if A ⊂ X, B ⊂ X, A ∩ B = ∅, then µ(A ∪ B) = µ(A) + µ(B) + λµ(A)µ(B) and µ(A ∪ B) = µ(A) + µ(B) +

λµ(A)µ(B), λ ∈ (−1, ∞).

Theorem 3.1.2 Let gλ = [g λ , g λ ] be an interval-valued Sugeno probability measure. Then gλ is a regular λ−interval fuzzy measure defined on A. Proof

Since gλ is an interval-valued Sugeno probability measure, this implies that g λ and g λ are Sugeno

probability measure based on σ −λ rules, respectively, that is for every λ ≥ 0, A ∈ A, B ∈ A, we get g λ (A∪B) = 5

g λ (A) + g λ (B) + λ g λ (A)g λ (B). Furthermore, we can obtain g λ (∅) = 0. Otherwise, g λ (A) = g λ (A ∪ ∅) = g λ (A) + g λ (∅) + λ g λ (A)gλ (∅) and g λ (∅) = −λg λ (A)g λ (∅). for every λ ≥ 0, A ∈ A Since gλ (∅) 6= 0, we have λ g λ (∅) = −1, which contradicts λ g λ (∅) ≥ 0. Then g λ (∅) = 0. Finally, we can conclude that g λ (X) = 1 and g λ have the monotonicity based upon Definition 3.1.3 and

Theorem 3.1.1. Similarly, g λ (∅) = 0, g λ (X) = 1 and g λ have the monotonicity. It can be obtained that gλ is a regular λ−interval fuzzy measure defined on A.

Let X = {x1 , x2 , · · · xn } be a finite set. The value gλi = gλ (xi )(i = 1, 2, · · · , n) is called measure density.

Theorem 3.1.3 The parameter λ = (λ1 , λ2 ) of a regular interval Sugeno probability measure is determined by the equations: n Y

i=1 n Y

(1 + λ1 g λi ) = 1 + λ1 , (3) (1 + λ2 g λi ) = 1 + λ2

i=1

Proof We can prove the above theorem by Theorem 3.1.2 and Wang and Klir (1992). Remark 3.1.3 By Theorem 3.1.1 and Theorem 3.1.2, we can obtain that interval Sugeno probability measure gλ can satisfy the following conditions: (1) gλ (∅) = [0, 0], that is, the empty set has no contribution; (2) gλ (X) = [1, 1], that is, the most weight of the complete set is 1; (3) if A, B ⊂ X, A ⊂ B, then gλ (A) ≤ gλ (B), that is, adding a new attribute does not reduce the weight of

the joint attributes set.

If we know the values of the Sugeno measure based on σ − λ rules, we can use Theorem 3.1.3 to obtain the

values of λ and then use Definition 3.1.1 to obtain the values of the other sets. It implies that a Sugeno measure based on σ − λ can be determined by measure densities.

Example 3.1.2 In Example 3.1.1, we can calculate that λ1 = −0.94 or λ1 = −3.976, λ2 = −0.986 or λ2 = −3.22 using the theorem 3.1.3 for left and right endpoint values, respectively. After screening, we have λ1 = −0.94, λ2 = −0.986. (The left and right endpoints can form an interval and belong to [0, 1], respectively). Furthermore, we can get the value of gλ of the other attributes using σ − λ rules, which are shown in Table 2. Table 2 The value of gλ about Example 3.1.2 Set E1 = {x1 } E2 = {x2 } E3 = {x3 } E4 = {x1 , x2 } E5 = {x1 , x3 } E6 = {x2 , x3 } E7 = {x1 , x2 , x3 }

Value of gλ [0.5, 0.6] [0.6, 0.7] [0.8, 0.9] [0.818, 0.886] [0.924, 0.968] [0.9488, 0.979] [1, 1]

3.2 Choquet Integrals on Interval-valued Sugeno Probability Measure The basic definition of Choquet integral based on interval-valued Sugeno probability measure will be given below. According to Wu, Zhang and Guo (1998), we could assume that R+ = [0, +∞), I(R+ ) = {r : [r, r] ⊂ R+ }

is the subset of interval number and X is the whole set of interval numbers. 6

Interval numbers satisfy the following operations: (1) r ∗ p = [r ∗ p, r ∗ p](∗ denotes + ∨ ∧); (2) k · r =

[kr, kr], (k ∈ R+ ); (3) r ≤ p ⇐⇒ r ≤ p, r ≤ p; (4) d(r, p) = max{|r − p|, |r − p|}; (5) If d(rn , r) → 0,then rn → r.

Definition 3.2.1 (Yang, Wang, Henga & Leung, 2005) An interval-valued function f : X → I(R+ ) is mea-

surable if both f (x) and f (x) are measurable functions, where f (x) = [f (x), f (x)], f (x) is the left end point of interval f (x) and f (x) is the right end point of interval f (x). Interval-valued function f : X → I(R+ ) is C−integrally bounded if there exists a Choquet integrable function h : X → R+ such that |x| ˙ ≤ h(t) for x˙ ∈ f .

Definition 3.2.2 Let (X, A, µ) be a non-additive measure space. Suppose f : X → I(R+ ) is measurable, C−integrally bounded and E ∈ A. f is C−integrable if Z Z (c) f dµ =: {(c) gdµ|g ∈ Sf (x) } E

(4)

E

is a closed interval on I(R+ ), where Sf (x) = {g|g : X → R+ is a measurable selection on f (x)}. Theorem 3.2.1 Let (X, A, µ) be a non-additive measurable space. Suppose µ is a fuzzy measure, E ∈ A, F is nonnegative measurable, C-integrally bounded, then f is C−integrable on E and Z Z Z f dµ]. (c) f dµ = [(c) f dµ, (c) E

(5)

E

E

Proof. Since f is nonnegative measurable and C−integrally bounded, f and f are C−integrable on E. Let R R M ∈ {(c) E gdµ|g ∈ Sf (x) }, then there exists a measurable selection g(x) ∈ f (x) such that (c) E gdµ = M . It satisfies f ≤ g ≤ f so that

(c)

Z

E

that is, {(c)

Z

E

f dµ ≤ (c)

Z

E

gdµ ≤ (c)

gdµ|g ∈ Sf (x) } ⊂ [(c)

Z

Z

f dµ,

E

f dµ, (c)

E

Z

f dµ].

E

On the other hand, since f and f are measurable selections, we have Z Z Z Z gdµ|g ∈ Sf (x) }, (c) f dµ ∈ {(c) gdµ|g ∈ Sf (x) }. (c) f dµ ∈ {(c) E

E

E

E

For measurable selections p(x), m(x) ∈ f (x), we have tp(x) + (1 − t)m(x) ∈ f (x), 0 ≤ t ≤ 1. Hence, tp(x) + (1 − t)m(x) is a measurable selection. Therefore, Z Z (c) [tp(x) + (1 − t)m(x)]dµ ∈ {(c) gdµ|g ∈ Sf (x) }, E

implies that {(c)

R

E

E

gdµ|g ∈ Sf (x) } is a convex set. Z Z Z [(c) f dµ, (c) f dµ] ⊂ {(c) gdµ|g ∈ Sf (x) }. E

Hence, (c)

E

Z

E

That is, F is C−integrable.

f dµ = [(c)

E

Z

E

7

f dgµ , (c)

Z

E

f dµ].

From the aforementioned proof, we know that interval-valued function f is C−integrable on E if (c) R and (c) E f dµ existed and bounded.

R

E

f dµ

Suppose X = {x1 , x2 , · · · xn } be a discrete set. Using Theorem 3.2.1, we can obtain the following theorem.

Theorem 3.2.2 Let f be an interval-valued function on X = {x1 , x2 , · · · , xn }. Then Choquet integral of f with respect to a fuzzy measure µ on X is given by (c)

Z

f dµ =

X

0

0

n X i=1

0

0

0

[µ(Xi ) − µ(Xi+1 )]f (xi ),

0

0

(6) 0

0

0

where x1 , x2 , · · · , xn is a permutation of x1 , x2 , · · · , xn such that f (x0 ) ≤ f (x1 ) ≤ ... ≤ f (xn ), f (x0 ) = 0

0

0

0

0

[0, 0], Xi = {xi , xi+1 , ..., xn }, i = 1, 2, · · · , n and Xn+1 = ∅.

Proof Since f is an interval-valued function on X, by Theorem 3.2.1, we have Z Z Z f dµ]. (c) f dµ = [(c) f dµ, (c) X

X

X

Note that f and f are real-valued functions on X, respectively. Based upon the continuity and the monotonicity of the Choquet integral and the nonnegativity and the monotonicity of the fuzzy measures, we get

(c)

Z

X

n n X X 0 0 0 0 0 0 [µ(Xi ) − µ(Xi+1 )]f (xi )] f dµ = [ [µ(Xi ) − µ(Xi+1 )]f (xi ),

=

i=1

0

0

i=1

i=1

n X

0

0

0

[µ(Xi ) − µ(Xi+1 )]f (xi ). 0

0

0

0

0

where x1 , x2 , · · · , xn is a permutation of x1 , x2 , · · · , xn such that f (x0 ) ≤ f (x1 ) ≤ ... ≤ f (xn ), f (x0 ) = 0

0

0

0

0

[0, 0], Xi = {xi , xi+1 , ..., xn }, i = 1, 2, · · · , n and Xn+1 = ∅.

4. Choquet Integrals on Interval-valued Sugeno Probability Measure for Uncertain MCDM and Relevant Solution Approach The Choquet integral based fuzzy measure is usually used in the information fusion and data mining as a nonlinear aggregation tool. Prior studies have shown the successful applications of the Choquet integral in nonlinear multi-regression, classification and decision-making (Du & Zare, 2019; Fang, Rizzo, Wang, Espy & Wang, 2010; Qin, Liu & Pedrycz, 2016; Tehrani, Cheng & H¨ ullermeier, 2010; Tehrani, Cheng, Dembczynski & H¨ ullermeier, 2012). The fuzzy measure is commonly used to reflect the interaction between criteria so as to obtain more accurate evaluation. Specifically, Choquet integral based fuzzy measure is well-fitted to solve the hierarchical MCDM modeling problems. For instance, Figure 1 shows the three-layer MCDM model based on the Choquet integral. In this three-layer Choquet integral MCDM model, the evaluation of the parent layer attribute is obtained by calculating the Choquet integral of the child layer attributes. Therefore, the parent attribute x1 is assessed by calculating the Choquet integral of all child attributes x1 1 ∼ x1 k1 successively. Furthermore, the comprehensive evaluation value of the whole scheme is obtained by calculating the Choquet integral of the information function relative to all attribute x1 ∼ xn . Then the optimal scheme is is selected based upon the comprehensive evaluation value. For a MCDM problem, suppose there are m objects, each object has n primary criteria(xi ), each primary criteria has k sub-criteria (xji ), and there are n alternatives (options). For each criteria and each alternative, there is an information evaluation table, and we will use it to make the final decision. In order to solve such MCDM problem, we propose a solution approach: firstly, establish language information table and convert linguistic information into triangular fuzzy number; the weight of each criteria is presented as the measure (gλ ) of that criteria, and the language evaluation of the criteria is presented as the 8

Figure 1: The model of 3-layer MCDM based on Choquet integral information function (integral function fi,t ) of that criteria. Then according to Theorem 3.2.2, calculate the j based on sub-criteria x1i -xji , which will obtain the evaluChoquet integral, s value of information function fi,t

ation value of the primary criteria. After that, calculate Choquet integral, s value of information function fi,t based on primary criteria xi - xj , which will obtain the overall evaluation. Later on, rank the evaluation and identify the optimal alternative for the current layer, take the obtained integral value as the information value of the next upper layer, repeat abovementioned process to carry out the integral calculation and rank for this layer, and then repeat the process and get the final optimal selection. Figure 2 summarizes the solution approach of using Choquet integrals on interval-valued Sugeno probability measure to solve uncertain MCDM.

Figure 2: Solution approach

9

5. Case Study 5.1 Refrigerator Component End-of-Life Strategy Determination MCDM Problem In this study, we use a refrigerator component end-of-life strategy determination MCDM problem as a case study to illustrate the proposed Choquet integral based fuzzy approach. This MCDM problem considers 4 main criteria and 14 sub-criteria criteria. Figure 3 presents the hierarchical structure of all criteria. The refrigerator is composed of twenty components, and each component interacts physically with others to generate the primary functions. The refrigerator components only include the main parts and certain small connectors, the fasteners and screw bolts are excluded from this case study. The product data set is adopted from Chung, Okudan & Wysk (2011) and Ma, Kremer & Ray (2018).

Figure 3: The structure of decision attribute

Figure 4: Refrigerator sketch (Adopted from Chung, Okudan &Wysk, 2011)

10

Figure 4 presents the refrigerator dissection sketch. We will need to make the end-of-life strategy decisions for each component of the refrigerator, the decision alternatives include: Reuse, Remanufacture, Primary recycle, Secondary, Incineration and Landfill under 4 main criteria and 14 sub-criteria. The linguistic evaluation and the relevant importance of each attribute are expressed by triangular fuzzy numbers. According to the research results of Chen and Hwang (1992), we can use natural language variables to give the assessment, and furthermore, the natural language variables can be transformed into corresponding triangular fuzzy numbers. Table 3 presents the linguistic evaluation and their corresponding fuzzy numbers. The evaluation needs the involvement of decision makers. Take cabinet frame as an example, Table 4 shows the linguistic evaluation with respect to all criteria and their sub-criteria, and Table 5 presents the corresponding triangular fuzzy numbers. Table 3 Linguistic terms and corresponding fuzzy numbers Evaluation/weighting terms Extra poor/Extra unimportant Very poor/Very unimportant Poor/Unimportant A little Poor/A little unimportant Slightly Poor/Slightly unimportant Fair/Middle Slightly good/Slightly important A little good/A little important Good/Important Very good/Very important Extra good/Extra important

Label EP/EU VP/VU P/U AP/AU SP/SU F/M SG/SI AH/AI G/I VG/VI EG/EI

Triangular fuzzy numbers (0, 0, 0.1) (0, 0.1, 0.2) (0.1, 0.2, 0.3) (0.2, 0.3, 0.4) (0.3, 0.4, 0.5) (0.4, 0.5, 0.6) (0.5, 0.6, 0.7) (0.6, 0.7, 0.8) (0.7, 0.8, 0.9) (0.8, 0.9, 1) (0.9, 1, 1)

Table 4 Criteria importance and EOL options linguistic evaluation with respect to cabinet frame Criteria x1 x2

x3

x4

x11 x21 x12 x22 x32 x42 x52 x13 x23 x33 x43 x14 x24 x34

Weights (gλ )ji VG VG F G G F G VG F VG VG F G P F F F F

j Linguistic evaluation of EOL options (fit ) A1 A2 A3 A4 A5 A6

G G

G G

G VG F F G G P VG F

G G VG G VG P VP F F

G VG VG

G G VG

F VG F G VG G F VG VG G G F VG VG

VG G

VG F

VP VP

F G F G G

VP G VG F P

VP P G F VP

F G G F

VG P VP F

VG G G G

P P VP

F VG VG

F F VG

5.2 Case Study Solution The detailed calculation process of refrigerator EOL determination is presented below following six steps in Figure 2. Before that, we introduce set parameters and variables for calculation . As shown in Figure 3, 4 primary criteria are considered in this example, expressed as xi (i = 1, 2, 3, 4) and 14 sub-criteria are included, expressed as xji , if i = 1 then j = 1, 2; if i = 2 then j = 1, 2, 3, 4, 5; if i = 3 then j = 1, 2, 3, 4; and if i = 4 then j = 1, 2, 3. We represent the six EOL options (Reuse, Remanufacture, Primary recycle, Secondary, Incineration and Landfill) as A1 , A2 , A3 , A4 , A5 and A6 ,. First Step: In this step, the primary activity is to convert the linguistic information into triangular fuzzy numbers. Table 4 shows the weight of criteria and each EOL strategy linguistic evaluation, which were obtained from surveys. According to Table 3, the triangular fuzzy numbers corresponding to the linguistic evaluation are 11

presented in Table 5. Table 5 Triangular fuzzy number evaluation with respect to cabinet frame Criteria x1 x11 x21 x2 x12 x22 x32 x42 x52 x3 x13 x23 x33 x43 x4 x14 x24 x34

Weights (gλ )ji (0.8,0.9,1) (0.8,0.9,1) (0.4,0.5,0.6) (0.7,0.8,0.9) (0.7,0.8,0.9) (0.4,0.5,0.6) (0.7,0.8,0.9) (0.8,0.9,1) (0.4,0.5,0.6) (0.8,0.9,1) (0.8,0.9,1) (0.4,0.5,0.6) (0.7,0.8,0.9) (0.1,0.2,0.3) (0.4,0.5,0.6) (0.4,0.5,0.6) (0.4,0.5,0.6) (0.4,0.5,0.6)

A1

A2

A3

A4

A5

A6

(0.7,0.8,0.9) (0.7,0.8,0.9)

(0.7,0.8,0.9) (0.7,0.8,0.9)

(0.4,0.5,0.6) (0.8,0.9,1)

(0.8,0.9,1) (0.7,0.8,0.9)

(0.8,0.9,1) (0.4,0.5,0.6)

(0,0.1,0.2) (0,0.1,0.2)

(0.7,0.8,0.9) (0.8,0.9,1) (0.4,0.5,0.6) (0.4,0.5,0.6) (0.7,0.8,0.9)

(0.7,0.8,0.9) (0.7,0.8,0.9) (0.8,0.9,1) (0.7,0.8,0.9) (0.8,0.9,1)

(0.4,0.5,0.6) (0.7,0.8,0.9) (0.8,0.9,1) (0.7,0.8,0.9) (0.4,0.5,0.6)

(0.4,0.5,0.6) (0.7,0.8,0.9) (0.4,0.5,0.6) (0.7,0.8,0.9) (0.7,0.8,0.9)

(0,0.1,0.2) (0.7,0.8,0.9) (0.8,0.9,1) (0.4,0.5,0.6) (0.1,0.2,0.3)

(0,0.1,0.2) (0.1,0.2,0.3) (0.7,0.8,0.9) (0.4,0.5,0.6) (0,0.1,0.2)

(0.7,0.8,0.9) (0.1,0.2,0.3) (0.8,0.9,1) (0.4,0.5,0.6)

(0.1,0.2,0.3) (0,0.1,0.2) (0.4,0.5,0.6) (0.4,0.5,0.6)

(0.8,0.9,1) (0.8,0.9,1) (0.7,0.8,0.9) (0.7,0.8,0.9)

(0.4,0.5,0.6) (0.7,0.8,0.9) (0.7,0.8,0.9) (0.4,0.5,0.6)

(0.8,0.9,1) (0.1,0.2,0.3) (0,0.1,0.2) (0.4,0.5,0.6)

(0.8,0.9,1) (0.7,0.8,0.9) (0.7,0.8,0.9) (0.7,0.8,0.9)

(0.7,0.8,0.9) (0.8,0.9,1) (0.8,0.9,1)

(0.7,0.8,0.9) (0.7,0.8,0.9) (0.8,0.9,1)

(0.4,0.5,0.6) (0.8,0.9,1) (0.8,0.9,1)

(0.1,0.2,0.3) (0.1,0.2,0.3) (0,0.1,0.2)

(0.4,0.5,0.6) (0.8,0.9,1) (0.8,0.9,1)

(0.4,0.5,0.6) (0.4,0.5,0.6) (0.8,0.9,1)

Second Step: Carrying out α−level cut set operation using triangular fuzzy numbers in Table 5. Table 6 and Table 7 show the result for α = 0 and α = 1, respectively. Table 6 The value of α−level cut set for α = 0 of Table 5 Criteria x1 x11 x21 x2 x12 x22 x32 x42 x52 x3 x13 x23 x33 x43 x4 x14 x24 x34

Weights (gλ )ji [0.8, 1] [0.8, 1] [0.4, 0.6] [0.7, 0.9] [0.7, 0.9] [0.4, 0.6] [0.7, 0.9] [0.8, 1] [0.4, 0.6] [0.8, 1] [0.8, 1] [0.4, 0.6] [0.7, 0.9] [0.1, 0.3] [0.4, 0.6] [0.4, 0.6] [0.4, 0.6] [0.4, 0.6]

A1

A2

A3

A4

A5

A6

[0.7, 0.9] [0.7, 0.9]

[0.7, 0.9] [0.7, 0.9]

[0.4, 0.6] [0.8, 1]

[0.8, 1] [0.7, 0.9]

[0.8, 1] [0.4, 0.6]

[0, 0.2] [0, 0.2]

[0.7, 0.9] [0.8, 1] [0.4, 0.6] [0.4, 0.6] [0.7, 0.9]

[0.7, 0.9] [0.7, 0.9] [0.8, 1] [0.7, 0.9] [0.8, 1]

[0.4, 0.6] [0.7, 0.9] [0.8, 1] [0.7, 0.9] [0.4, 0.6]

[0.4, [0.7, [0.4, [0.7, [0.7,

0.6] 0.9] 0.6] 0.9] 0.9]

[0, 0.2] [0.7, 0.9] [0.8, 1] [0.4, 0.6] [0.1, 0.3]

[0, 0.2] [0.1, 0.3] [0.7, 0.9] [0.4, 0.6] [0, 0.2]

[0.7, 0.9] [0.1, 0.3] [0.8, 1] [0.4, 0.6]

[0.1, 0.3] [0, 0.2] [0.4, 0.6] [0.4, 0.6]

[0.8, 1] [0.8, 1] [0.7, 0.9] [0.7, 0.9]

[0.4, [0.7, [0.7, [0.4,

0.6] 0.9] 0.9] 0.6]

[0.8, 1] [0.1, 0.3] [0, 0.2] [0.4, 0.6]

[0.8, 1] [0.7, 0.9] [0.7, 0.9] [0.7, 0.9]

[0.7, 0.9] [0.8, 1] [0.8, 1]

[0.7, 0.9] [0.7, 0.9] [0.8, 1]

[0.4, 0.6] [0.8, 1] [0.8, 1]

[0.1, 0.3] [0.1, 0.3] [0, 0.2]

[0.4, 0.6] [0.8, 1] [0.8, 1]

[0.4, 0.6] [0.4, 0.6] [0.8, 1]

Table 7 The value of α−level cut set for α = 1 of Table 5 Criteria x1 x11 x21 x2 x12 x22 x32 x42 x52 x3 x13 x23 x33 x43 x4 x14 x24 x34

Weights (gλ )ji 0.9 0.9 0.5 0.8 0.8 0.5 0.8 0.9 0.5 0.9 0.9 0.5 0.8 0.2 0.5 0.5 0.5 0.5

A1

A2

A3

A4

A5

A6

0.8 0.8

0.8 0.8

0.5 1

0.9 0.8

0.9 0.5

0.1 0.1

0.8 0.9 0.5 0.5 0.8

0.8 0.8 0.9 0.8 0.9

0.5 0.8 0.9 0.8 0.5

0.5 0.8 0.5 0.8 0.8

0.1 0.8 0.9 0.5 0.2

0.1 0.2 0.8 0.5 0.1

0.8 0.2 0.9 0.5

0.2 0.1 0.5 0.5

0.9 0.9 0.8 0.8

0.5 0.8 0.8 0.5

0.9 0.2 0.1 0.5

0.9 0.8 0.8 0.8

0.8 0.9 0.9

0.8 0.8 0.9

0.5 0.9 0.9

0.2 0.2 0.1

0.5 0.9 0.9

0.5 0.5 0.9

12

Third Step: calculate the value of parameter λ and the value of every joint attribute set Eij using the 0

, ...xni }, if i = 1 then single attribute s weight in Table 6, Theorem 3.1.3 and σ − λ rules, where Eij = {xji , xj+1 i

n = 2, if i = 2 then n = 5, if i = 3 then n = 4, if i = 4 then n = 3, 1 ≤ j ≤ n. There are the following marks: g λ (j) = g λ (Eij ), g λ (j) = g λ (Eij ). The results are shown in Table 8 and Table 9.

Table 8 The value of fuzzy measure of sub-criteria for α = 0 A1 g λ (E (j) )

A2 g λ (E (j) )

λ1 = −0.625 λ2 = −1

g λ (2) = 0.4 g λ (2) = 0.6 g λ (1) = 1 g λ (1) = 1 λ1 = −0.993 λ2 = −1

g λ (5) = 0.4 g λ (5) = 0.6

g λ (E (j) )

A3 g λ (E (j) )

λ1 = −0.625 λ2 = −1

g λ (2) = 0.4 g λ (2) = 0.6

g λ (E (j) )

g λ (E (j) )

λ1 = −0.625 λ2 = −1

g λ (2) = 0.4 g λ (2) = 0.6

g λ (1) = 1 g λ (1) = 1

g λ (1) = 1 g λ (1) = 1

λ1 = −0.993 λ2 = −1

λ1 = −0.993 λ2 = −1

g λ (5) = 0.4 g λ (5) = 0.6

g λ (5) = 0.7 g λ (5) = 0.9

g λ (4) = 0.641 g λ (4) = 0.84

g λ (4) = 0.822 g λ (4) = 0.96

g λ (4) = 0.944 g λ (4) = 1

g λ (3) = 0.896 g λ (3) = 0.984

g λ (3) = 0.967 g λ (3) = 1

g λ (3) = 0.969 g λ (3) = 1

g λ (2) = 0.984 g λ (2) = 1

g λ (2) = 0.984 g λ (2) = 1

g λ (2) = 0.984 g λ (2) = 1

g λ (1) = 1 g λ (1) = 1 λ1 = −0.957 λ2 = −1

g λ (1) = 1 g λ (1) = 1

g λ (1) = 1 g λ (1) = 1

λ1 = −0.957 λ2 = −1

λ1 = −0.957 λ2 = −1

g λ (4) = 0.7 g λ (4) = 0.9

g λ (4) = 0.1 g λ (4) = 0.3

g λ (4) = 0.4 g λ (4) = 0.6

g λ (3) = 0.964 g λ (3) = 1

g λ (3) = 0.733 g λ (3) = 0.93

g λ (3) = 0.888 g λ (3) = 1

g λ (2) = 0.972 g λ (2) = 1

g λ (2) = 0.972 g λ (2) = 1

g λ (2) = 0.901 g λ (2) = 1

g λ (1) = 1 g λ (1) = 1 λ1 = −0.443 λ2 = −0.904 g λ (3) = 0.4 g λ (3) = 0.6

g λ (2) = 0.729 g λ (2) = 0.875 g λ (1) = 1 g λ (1) = 1

g λ (1) = 1 g λ (1) = 1

g λ (1) = 1 g λ (1) = 1

λ1 = −0.443 λ2 = −0.904

λ1 = −0.443 λ2 = −0.904

g λ (2) = 0.729 g λ (2) = 0.875

g λ (2) = 0.729 g λ (2) = 0.875

g λ (1) = 1 g λ (1) = 1

g λ (1) = 1 g λ (1) = 1

g λ (3) = 0.4 g λ (3) = 0.6

A4 g λ (E (j) )

g λ (3) = 0.4 g λ (3) = 0.6

A5 g λ (E (j) )

λ1 = −0.625 λ2 = −1

g λ (2) = 0.8 g λ (2) = 0.1 g λ (1) = 1 g λ (1) = 1 λ1 = −0.993 λ2 = −1

g λ (E (j) )

A6 g λ (E (j) )

λ1 = −0.625 λ2 = −1

g λ (2) = 0.8 g λ (2) = 0.1

g λ (E (j) )

g λ (E (j) )

λ1 = −0.625 λ2 = −1

g λ (2) = 0.4 g λ (2) = 0.6

g λ (1) = 1 g λ (1) = 1

g λ (1) = 1 g λ (1) = 1

λ1 = −0.993 λ2 = −1

λ1 = −0.993 λ2 = −1

g λ (5) = 0.7 g λ (5) = 0.9

g λ (5) = 0.7 g λ (5) = 0.9

g λ (5) = 0.7 g λ (5) = 0.9

g λ (4) = 0.822 g λ (4) = 0.96

g λ (4) = 0.944 g λ (4) = 1

g λ (4) = 0.944 g λ (4) = 1

g λ (3) = 0.969 g λ (3) = 1

g λ (3) = 0.969 g λ (3) = 1

g λ (3) = 0.969 g λ (3) = 1

g λ (2) = 0.984 g λ (2) = 1

g λ (2) = 0.984 g λ (2) = 1

g λ (2) = 0.984 g λ (2) = 1

g λ (1) = 1 g λ (1) = 1

g λ (1) = 1 g λ (1) = 1

g λ (1) = 1 g λ (1) = 1

λ1 = −0.957 λ2 = −1

λ1 = −0.957 λ2 = −1

g λ (3) = 0.827 g λ (3) = 0.96

g λ (3) = 0.823 g λ (3) = 1

g λ (3) = 0.823 g λ (3) = 1

g λ (2) = 0.846 g λ (2) = 0.972

g λ (2) = 0.908 g λ (2) = 1

g λ (2) = 0.972 g λ (2) = 1

λ1 = −0.957 λ2 = −1

g λ (4) = 0.7 g λ (4) = 0.9

g λ (1) = 1 g λ (1) = 1 λ1 = −0.443 λ2 = −0.904 g λ (3) = 0.4 g λ (3) = 0.6

g λ (2) = 0.729 g λ (2) = 0.875 g λ (1) = 1 g λ (1) = 1

g λ (4) = 0.8 g λ (4) = 1

g λ (4) = 0.8 g λ (4) = 1

g λ (1) = 1 g λ (1) = 1

g λ (1) = 1 g λ (1) = 1

λ1 = −0.443 λ2 = −0.904

λ1 = −0.443 λ2 = −0.904

g λ (2) = 0.729 g λ (2) = 0.875

g λ (2) = 0.729 g λ (2) = 0.875

g λ (1) = 1 g λ (1) = 1

g λ (1) = 1 g λ (1) = 1

g λ (3) = 0.4 g λ (3) = 0.6

13

g λ (3) = 0.4 g λ (3) = 0.6

Table 9 The value of fuzzy measure sub-criteria for α = 1 A1 gλ (E (j) ) λ = −0.889 gλ (2) = 0.5 gλ (1) = 1 λ = −0.999 gλ (5) = 0.5 gλ (4) = 0.9 gλ (3) = 0.991 gλ (2) = 0.996 gλ (1) = 1 λ = −0.99 gλ (4) = 0.8 gλ (3) = 0.987 gλ (2) = 0.992 gλ (1) = 1 λ = −0.764 gλ (3) = 0.5 gλ (2) = 0.809 gλ (1) = 1

A2 gλ (E (j) ) λ = −0.889 gλ (2) = 0.5 gλ (1) = 1 λ = −0.999 gλ (5) = 0.5 gλ (4) = 0.9 gλ (3) = 0.991 gλ (2) = 0.996 gλ (1) = 1 λ = −0.99 gλ (4) = 0.2 gλ (3) = 0.842 gλ (2) = 0.992 gλ (1) = 1 λ = −0.764 gλ (3) = 0.5 gλ (2) = 0.809 gλ (1) = 1

A3 gλ (E (j) ) λ = −0.889 gλ (2) = 0.5 gλ (1) = 1 λ = −0.999 gλ (5) = 0.8 gλ (4) = 0.981 gλ (3) = 0.991 gλ (2) = 0.996 gλ (1) = 1 λ = −0.99 gλ (4) = 0.5 gλ (3) = 0.955 gλ (2) = 0.966 gλ (1) = 1 λ = −0.764 gλ (3) = 0.5 gλ (2) = 0.809 gλ (1) = 1

A4 gλ (E (j) ) λ = −0.889 gλ (2) = 0.9 gλ (1) = 1 λ = −0.999 gλ (5) = 0.5 gλ (4) = 0.95 gλ (3) = 0.976 gλ (2) = 0.996 gλ (1) = 1 λ = −0.99 gλ (4) = 0.8 gλ (3) = 0.904 gλ (2) = 0.925 gλ (1) = 1 λ = −0.764 gλ (3) = 0.5 gλ (2) = 0.809 gλ (1) = 1

A5 gλ (E (j) ) λ = −0.889 gλ (2) = 0.9 gλ (1) = 1 λ = −0.999 gλ (5) = 0.8 gλ (4) = 0.9 gλ (3) = 0.991 gλ (2) = 0.996 gλ (1) = 1 λ = −0.99 gλ (4) = 0.9 gλ (3) = 0.922 gλ (2) = 0.966 gλ (1) = 1 λ = −0.764 gλ (3) = 0.5 gλ (2) = 0.809 gλ (1) = 1

A6 gλ (E (j) ) λ = −0.889 gλ (2) = 0.5 gλ (1) = 1 λ = −0.999 gλ (5) = 0.8 gλ (4) = 0.981 gλ (3) = 0.991 gλ (2) = 0.996 gλ (1) = 1 λ = −0.99 gλ (4) = 0.9 gλ (3) = 0.922 gλ (2) = 0.992 gλ (1) = 1 λ = −0.764 gλ (3) = 0.5 gλ (2) = 0.809 gλ (1) = 1

Fourth Step: calculate the evaluation value of the primary criteria xi (i = 1.2.3.4) with respect to EOL j options At (t = 1.2.3.4, 5, 6). According to Third step we know that (fi,t )α represents the α−level cut set of the j j j function fi,t . So (fi,t )0 represents the 0−level cut set of the function fi,t .

For EOL options A1 with respect to criteria x2 , α = 0: 1 2 3 4 (1) It can be obtained from Table 6: (f2,1 )0 = [0.7, 0.9], (f2,1 )0 = [0.8, 1], (f2,1 )0 = [0.4, 0.6], (f2,1 )0 = 5 [0.4, 0.6], (f2,1 )0 = [0.7, 0.9];

(2) It can be obtained from Table 6: (gλ )12 = [0.7, 0.9], (gλ )22 = [0.4, 0.6], (gλ )32 = [0.7, 0.9], (gλ )42 = [0.8, 1], (gλ )52 = [0.4, 0.6];

j

(3) It can be obtained from Definition 3.1.2, f ji,t and f i,t represents the left endpoint and right endpoint

j of the α−level cut set of the function fi,t , respectively. Sorting in the order of the size of f j2,1 (j = 1, 2, 3, 4, 5): 0

0

0

0

0

f 32,1 = f 42,1 < f 12,1 = f 52,1 < f 22,1 , then we have x12 , x22 , x32 , x42 , x52 , is a permutation of x12 , x22 , x32 , x42 , x52 , where 0

0

0

0

0

x12 = x32 , x22 = x42 , x32 = x12 , x42 = x52 , x52 = x22 . The value of the parameter λ1 is calculated by Theorem 3.1.3: λ1 = −0.993. Furthermore, we can cal-

culate the weight of joint attributes using σ − λ rules, then, we can obtain the value g λ (j) = g λ (Eij ) = 0

(j+1)0

g λ {xji , xi

0

, · · · , xni , } : g λ (5) = 0.4, g λ (4) = 0.641, g λ (3) = 0.896, g λ (2) = 0.984, g λ (1) = 1. Table 8 and

Table 9 list all measured values and the value of the desired parameter λ.

(4) The value of the primary criteria (xi , i = 1.2.3.4) about EOL options At (t = 1.2.3.4, 5, 6) is calculated by Choquet integral as follows, taking ”primary criteria x2 of EOL options A1 ” for example, α = 0:

(c)

Z

f j2,1 dg λ

= f 32,1 · g λ (1) + (f 42,1 − f 32,1 ) · g λ (2) + (f 12,1 − f 42,1 ) · g λ (3) +(f 52,1 − f 12,1 ) · g λ (4) + (f 22,1 − f 52,1 ) · g λ (5) = 0.4 × 1 + (0.4 − 0.4) × 0.984 + (0.7 − 0.4) × 0.896 +(0.7 − 0.7) × 0.641 + (0.8 − 0.7) × 0.4 = 0.7088.

R j In the same way, (c) f 2,1 dg λ = 0.9552. Therefore, (c)

Z

j fe2,1 dg λ = [(c)

Z

f j2,1 dg λ , (c)

Z

j

f 2,1 dg λ ] = [0.7088, 0.9552] ' [0.709, 0.955].

Similarly, the evaluation value of the remaining primary criteria can be calculated, as shown in Table 10 and Table 11. 14

Table 10 The evaluation value of the primary criteria for α = 0 with respect to the cabinet frame Criteria x1 x11 x21 x2 x12 x22 x32 x42 x52 x3 x13 x23 x33 x43 x4 x14 x24 x34

Weights (gλ )ji [0.8, 1] [0.8, 1] [0.4, 0.6] [0.7, 0.9] [0.7, 0.9] [0.4, 0.6] [0.7, 0.9] [0.8, 1] [0.4, 0.6] [0.8, 1] [0.8, 1] [0.4, 0.6] [0.7, 0.9] [0.1, 0.3] [0.4, 0.6] [0.4, 0.6] [0.4, 0.6] [0.4, 0.6]

A1 [0.7, 0.9] [0.7, 0.9] [0.7, 0.9] [0.709, 0.955] [0.7, 0.9] [0.8, 1] [0.4, 0.6] [0.4, 0.6] [0.7, 0.9] [0.751, 0.99] [0.7, 0.9] [0.1, 0.3] [0.8, 1] [0.4, 0.6] [0.773, 0.988] [0.7, 0.9] [0.8, 1] [0.8, 1]

A2 [0.7, 0.9] [0.7, 0.9] [0.7, 0.9] [0.782, 0.996] [0.7, 0.9] [0.7, 0.9] [0.8, 1] [0.7, 0.9] [0.8, 1] [0.317, 0.579] [0.1, 0.3] [0, 0.2] [0.4, 0.6] [0.4, 0.6] [0.74, 0.96] [0.7, 0.9] [0.7, 0.9] [0.8, 1]

A3 [0.56, 0.84] [0.4, 0.6] [0.8, 1] [0.761, 0.99] [0.4, 0.6] [0.7, 0.9] [0.8, 1] [0.7, 0.9] [0.4, 0.6] [0.788, 1] [0.8, 1] [0.8, 1] [0.7, 0.9] [0.7, 0.9] [0.691, 0.95] [0.4, 0.6] [0.8, 1] [0.8, 1]

A4 [0.78, 1] [0.8, 1] [0.7, 0.9] [0.691, 0.888] [0.4, 0.6] [0.7, 0.9] [0.4, 0.6] [0.7, 0.9] [0.7, 0.9] [0.648, 0.888] [0.4, 0.6] [0.7, 0.9] [0.7, 0.9] [0.4, 0.6] [0.073, 0.287] [0.1, 0.3] [0.1, 0.3] [0, 0.2]

A5 [0.72, 1] [0.8, 1] [0.4, 0.6] [0.742, 0.99] [0, 0.2] [0.7, 0.9] [0.8, 1] [0.4, 0.6] [0.1, 0.3] [0.657, 1] [0.8, 1] [0.1, 0.3] [0, 0.2] [0.4, 0.6] [0.692, 0.95] [0.4, 0.6] [0.8, 1] [0.8, 1]

A6 [0, 0.2] [0, 0.2] [0, 0.2] [0.59, 0.87] [0, 0.2] [0.1, 0.3] [0.7, 0.9] [0.4, 0.6] [0, 0.2] [0.78, 1] [0.8, 1] [0.7, 0.9] [0.7, 0.9] [0.7, 0.9] [0.56, 0.84] [0.4, 0.6] [0.4, 0.6] [0.8, 1]

Table 11 The evaluation value of the primary criteria for α = 1 with respect to the cabinet frame Criteria x1 x11 x21 x2 x12 x22 x32 x42 x52 x3 x13 x23 x33 x43 x4 x14 x24 x34

Weights (gλ )ji 0.9 0.9 0.5 0.8 0.8 0.5 0.8 0.9 0.5 0.9 0.9 0.5 0.8 0.2 0.5 0.5 0.5 0.5

A1 0.8 0.8 0.8 0.847 0.8 0.9 0.5 0.5 0.8 0.874 0.8 0.2 0.9 0.5 0.881 0.8 0.9 0.9

A2 0.8 0.8 0.8 0.89 0.8 0.8 0.9 0.8 0.9 0.452 0.2 0.1 0.5 0.5 0.85 0.8 0.8 0.9

A3 0.7 0.5 0.9 0.877 0.5 0.8 0.9 0.8 0.5 0.895 0.9 0.9 0.8 0.8 0.823 0.5 0.9 0.9

A4 0.89 0.9 0.8 0.793 0.5 0.8 0.5 0.8 0.8 0.771 0.5 0.8 0.8 0.5 0.181 0.2 0.2 0.1

A5 0.86 0.9 0.5 0.847 0.1 0.8 0.9 0.5 0.2 0.833 0.9 0.2 0.1 0.5 0.823 0.5 0.9 0.9

A6 0.1 0.1 0.1 0.733 0.1 0.2 0.8 0.5 0.1 0.899 0.9 0.8 0.8 0.8 0.7 0.5 0.5 0.9

Fifth Step: calculate the evaluation value of EOL options A1 , A2 , A3 , A4 , A5 and A6 , by Choquet integral. For EOL options A1 , α = 0: (1) It can be obtained from Table 10: (f1,1 )0 = [0.7, 0.9], (f2,1 )0 = [0.709, 0.955], (f3,1 )0 = [0.751, 0.99], (f4,1 )0 = [0.773, 0.988]; (2) It can be obtained from Table 10: gλ (x1 ) = [0.8, 1], gλ (x2 ) = [0.7, 0.9], gλ (x3 ) = [0.8, 1], gλ (x4 ) = [0.4, 0.6]; 0

0

0

0

(3) Sorting in the order of the size of f i,1 (i = 1, 2, 3, 4): f 1,1 < f 2,1 < f 3,1 < f 4,1 , then we have x1 , x2 , x3 , x4 0

0

0

0

is a permutation of x12 , x22 , x32 , x42 , where x1 = x1 , x2 = x2 , x3 = x3 , x4 = x4 . Then, using the same method, we can calculate the value of the parameter λ and the weight of joint attributes on the primary criteria. They are shown on Table 12 and Table 13. (4) The value of EOL options At (t = 1.2.3.4, 5, 6) is calculated by Choquet integral as follows, Z (c) f i,1 dg λ = f 1,1 · g λ (1) + (f 2,1 − f 1,1 ) · g λ (2) + (f 3,1 − f 2,1 ) · g λ (3) + (f 4,1 − f 3,1 ) · g λ (4)

= 0.7 × 1 + (0.709 − 0.7) × 0.969 + (0.751 − 0.709) × 0.882 + (0.773 − 0.751) × 0.4 = 0.755.

R In the same way, (c) f i,1 dg λ = 0.99. therefore 15

(c)

Z

fi,1 dg λ = [(c)

Z

f i,1 dg λ , (c)

Z

f i,1 dg λ ] = [0.755, 0.99].

Similarly, the overall evaluation value of the remaining EOL options can be calculated, as shown in Table 14. Table 12 The value of fuzzy measure on the primary criteria for α = 0 with respect to the cabinet frame A1 g λ (E (j) )

A2 g λ (E (j) )

λ1 = −0.992 λ2 = −1 g λ (4) = 0.4 g λ (4) = 1

A3

g λ (E (j) )

g λ (E (j) )

g λ (E (j) )

λ1 = −0.992 λ2 = −1 g λ (4) = 0.7 g λ (4) = 0.9

g λ (E (j) )

λ1 = −0.992 λ2 = −1 g λ (4) = 0.8 g λ (4) = 1

g λ (3) = 0.882 g λ (3) = 1

g λ (3) = 0.822 g λ (3) = 0.96

g λ (3) = 0.944 g λ (3) = 1

g λ (2) = 0.9697 g λ (2) = 1

g λ (2) = 0.9697 g λ (2) = 1

g λ (2) = 0.9697 g λ (2) = 1

g λ (1) = 1 g λ (1) = 1

g λ (1) = 1 g λ (1) = 1

g λ (1) = 1 g λ (1) = 1 A4 g λ (E (j) )

A5 g λ (E (j) )

λ1 = −0.992 λ2 = −1 g λ (4) = 0.8 g λ (4) = 1

A6

g λ (E (j) )

g λ (E (j) )

g λ (E (j) )

λ1 = −0.992 λ2 = −1 g λ (4) = 0.7 g λ (4) = 1

g λ (E (j) )

λ1 = −0.992 λ2 = −1 g λ (4) = 0.8 g λ (4) = 1

g λ (3) = 0.944 g λ (3) = 1

g λ (3) = 0.944 g λ (3) = 1

g λ (3) = 0.944 g λ (3) = 1

g λ (2) = 0.995 g λ (2) = 1

g λ (2) = 0.9697 g λ (2) = 1

g λ (2) = 0.9697 g λ (2) = 1

g λ (1) = 1 g λ (1) = 1

g λ (1) = 1 g λ (1) = 1

g λ (1) = 1 g λ (1) = 1

Table 13 The value of fuzzy measure on the primary criteria for α = 1 λ = −0.998

gλ (4) = 0.5 gλ (3) = 0.95 gλ (2) = 0.992 gλ (1) = 1

λ = −0.998

gλ (4) = 0.8 gλ (3) = 0.9 gλ (2) = 0.992 gλ (1) = 1

λ = −0.998

λ = −0.998

gλ (4) = 0.9 gλ (3) = 0.98 gλ (2) = 0.992 gλ (1) = 1

gλ (4) = 0.9 gλ (3) = 0.98 gλ (2) = 0.999 gλ (1) = 1

λ = −0.998

gλ (4) = 0.9 gλ (3) = 0.98 gλ (2) = 0.999 gλ (1) = 1

λ = −0.998

gλ (4) = 0.9 gλ (3) = 0.98 gλ (2) = 0.992 gλ (1) = 1

Table 14 The comprehensive evaluation value of EOL options with respect to the cabinet frame Main criteria A1 overall EOL options value x1 x2 x3 x4

A3

A1

Crisp number A2 A3

(0.755, 0.876,0.99) (0.75, 0.874,0.989) (0.775, 0.892,1)

0.874 0.871 0.882*

(0.7,0.8,0.9) (0.7,0.8,0.9) (0.56,0.7,0.84) (0.709, 0.847, 0.955) (0.782, 0.89,0.996) (0.761, 0.8773,0.99) (0.751,0.874,0.99) (0.317,0.452,0.579) (0.788,0.896,1) (0.773,0.881,0.988) (0.74,0.85,0.96) (0.692,0.824,0.95)

0.8 0.8 0.7 0.837 0.889* 0.876 0.872 0.449 0.891 0.88* 0.85 0.822

Main criteria A4 overall EOL options value x1 x2 x3 x4

R (c) f dgλ A2

R (c) f dgλ A5

A6

A4

Crisp number A5 A6

(0.757, 0.879,1) (0.733, 0.858,1) (0.723, 0.877,1)

0.879 0.864 0.867

(0.78,0.89,1) (0.72,0.86,1) (0.0, 0.1,0.2) (0.691, 0.793, 0.888) (0.742, 0.847,0.99) (0.59, 0.733,0.87) (0.648,0.771,0.888) (0.657,0.833,1) (0.78,0.899,1) (0.073,0.181,0.287) (0.692,0.824,0.95) (0.56,0.7,0.84)

0.89* 0.86 0.1 0.79 0.859 0.731 0.769 0.83 0.893* 0.18 0.822 0.7

∗ represents the largest value in each group

Sixth Step: In the process of fuzzy data processing, defuzzification (precision) of fuzzy number is a common method due to triangular fuzzy number can not be directly used. There are many ways to defuzzy, in this study,

16

we choose mean value calculation to defuzzy. We use mean value calculation on each triangular fuzzy number in Table 14 to convert the fuzzy number into a crisp number. A similar optimal EOL option for other components can be obtained and be shown in Table 15. Table 15 Refrigerator component relevant closeness RC and appropriate EOL strategy Component Cabinet frame Cabinet Duct in room Fan unit 1 Fan unit 2 Evaporator Rear board Compressor Condenser Base Door 1 Door 2 Gasket 1 Gasket 2 Door liner 1 Door liner 2 Control unit Heater Dryer Shelf set

A1 0.874 0.8859 0.8834 0.8787 0.8847∗ 0.8974∗ 0.8759 0.8873 0.8538 0.8741 0.8573 0.8954∗ 0.8533 0.8532 0.8851 0.8677 0.8981∗ 0.8759 0.8684 0.8566

A2 0.871 0.871 0.8785 0.8685 0.8051 0.8919 0.8382 0.8739 0.8683 0.8856∗ 0.8737 0.8051 0.8685 0.8381 0.8788 0.8774 0.8788 0.8685 0.8736 0.8382

A3 0.882∗ 0.8644 0.8725 0.8981∗ 0.8465 0.891 0.8974∗ 0.8927∗ 0.8944∗ 0.8729 0.8737 0.8196 0.8939∗ 0.8764 0.8726 0.754 0.8779 0.8944∗ 0.8936∗ 0.8953∗

A4 0.879 0.8947∗ 0.8667 0.7563 0.7191 0.8495 0.8473 0.8697 0.7563 0.8493 0.8475 0.7272 0.6727 0.8974∗ 0.8987∗ 0.8886∗ 0.8667 0.7563 0.8382 0.8471

A5 0.864 0.8457 0.8878∗ 0.8889 0.8633 0.8677 0.8954 0.8887 0.8869 0.8326 0.8886∗ 0.8849 0.8868 0.8814 0.8667 0.8422 0.893 0.8889 0.8745 0.8941

A6 0.867 0.88 0.8363 0.7908 0.8074 0.8821 0.879 0.841 0.834 0.881 0.8752 0.8093 0.776 0.888 0.8364 0.7979 0.8364 0.8341 0.867 0.888

Appropriate EOL strategy Primary recycle Secondary recycle Incinerate Primary recycle Reuse Reuse Primary recycle Primary recycle Primary recycle Remanufacturing Incinerate Reuse Primary recycle Secondary recycle Secondary recycle Secondary recycle Reuse Primary recycle Primary recycle Primary recycle

∗ represents the largest value in each group

6. Conclusion and Future Work In this paper, a Choquet integral in interval-valued Sugeno probability space approach is proposed to solve uncertain MCDM problem. In order to describe the rich linguistic information more accurately, we proposed the interval-valued Sugeno probability space. Then the Choquet integrals of interval-valued functions in intervalvalued Sugeno probability space was developed and applied to solve uncertain MCDM problems. A refrigerator component end-of-life strategy determination MCDM problem was employed to illustrate the proposed Choquet integral in interval-valued Sugeno probability space approach. Since the membership degree of the fuzzy set in this proposed approach is an interval, the interval valued fuzzy set is more elastic and easier to be accepted and understood than the general fuzzy set. Therefore, it is more appropriate and reliable to deal with real problems using the interval valued fuzzy set. In this paper, fuzzy probability and fuzzy interval are combined to form fuzzy probability measure of interval value. Based upon this consideration, a new Choquet integral in interval-valued Sugeno probability space is established, which is suitable for solving the decision-making problems with interaction among attributes. Although this paper proposed an innovative Choquet integral based fuzzy logic approach, it still has some limitations. Firstly and foremost, this approach can effectively solve the problem of correlation between decision attributes. However, in many practical decision-making cases, because of the limitations of work experience, knowledge level, and fuzzy human thinking, DMs often show a certain degree of hesitation or uncertainty during the decision making procedure, a new framework that involves three aspects: membership, non-membership and hesitation is needed to remedy this limitation. Furthermore, this approach includes the personal judgments which might be subjective and less professional due to limitation of experience and relevant knowledge. Since group decision making can overcome these drawbacks, it will be integrated with Choquet integral based approach in the near future.

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Acknowledgements The authors would like to thank the referees and editors for providing very helpful comments and suggestions. The author, Li Chen was supported by Scientific Study Project Higher Learning, Ministry Education of Gansu Province (Grant No. 2017B-51), PhD Research Startup Foundation of Lanzhou City University of China (Grant No. LZCU-BS2015-03), and the key constructive discipline of Lanzhou City University of China (Grant No. LZCU-ZDJSXK-201709). Without her contribution, this research will not be possible.

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Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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CRediT author statement Li Chen: Conceptualization, Methodology, Writing-Original, Writing-Review & Editing;Gang Duan: Visualization, Investigation, Writing-Review & Editing;SuYun Wang: Supervision, Investigation; Junfeng Ma: Investigation, Methodology, Supervision, Validation, Writing- Reviewing and Editing.

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