1 Introduction
Multiple criteria decision making (MCDM), or multiple criteria decision analysis, is a process of finding optimal alternatives in complex scenarios via evaluating the values of multiple criteria of all alternatives synthetically (Greco et al.
2016). In this process, the first important task is to express criterion values accurately and effectively. For such expression, there are various kinds of available tools (Abualigah and Hanandeh
2015; Abualigah and Khader
2017; Abualigah et al.
2018a,
b,
c; Abualigah
2019), where fuzzy sets are one of the most representative kinds (Yager
1981). So far, over twenty different types of fuzzy sets have been presented within academia (Bustince et al.
2016). Among them, Zadeh’s fuzzy set (FS) (Zadeh
1965) is a well-known type of fuzzy set that uses a degree of membership
μ (0 ≤
µ ≤ 1) to quantify the degree of satisfaction. In some real applications, FS is enough for the representation of fuzzy information. However, it cannot be used to describe complex fuzzy information, such as the degree of dissatisfaction and the degree of hesitancy. To make up for this deficiency, Atanassov (
1986) presented the theory of intuitionistic fuzzy set (IFS). Compared with a FS, an IFS has a degree of membership
μ and a degree of non-membership
ν (0 ≤
µ ≤ 1; 0 ≤
ν ≤ 1; 0 ≤
µ +
ν ≤ 1), which can, respectively, express the degree of satisfaction and the degree of dissatisfaction, and thus, the degree of hesitancy can be obtained by 1 −
µ −
ν. Due to such characteristic, IFSs have been widely used to describe the values of criteria in MCDM during the past three decades. A number of research topics about IFSs in MCDM, such as operational rules of intuitionistic membership grades (IMGs) (De et al.
2000; Wang and Liu
2012; Jamkhaneh and Garg
2018), aggregation operators of IMGs (Xu and Yager
2011; Xia et al.
2012,
2013; Liu and Chen
2017; He et al.
2017; Liu et al.
2018a), intuitionistic preference relations (Xu
2007; Liao and Xu
2014a; Zhang and Pedrycz
2017), intuitionistic fuzzy calculus (Lei and Xu
2015,
2016; Ai and Xu
2018), and MCDM methods based on IFSs (Wei
2010; Liu and Zhang
2011; Liao and Xu
2014b; Garg
2017), have received extensive attention in this period.
Although IFSs have showed great potential in MCDM, their application range is limited by their capability to express fuzzy information. More specifically, the two components
μ and
ν in an IFS must satisfy the condition that 0 ≤
µ +
ν ≤ 1. In this case, the criterion values whose
µ and
ν do not satisfy this condition cannot be described by IFSs. For example, IFS is not capable of expressing a criterion value whose
µ = 0.8 and
ν = 0.4 because 0.8 + 0.4 > 1. To address this issue, Yager (
2014) proposed the theory of Pythagorean fuzzy set (PFS), which relaxes the condition to 0 ≤
µ2 +
ν2 ≤ 1. For this reason, PFSs can express more fuzzy information than IFSs. For instance, the criterion value whose
µ = 0.8 and
ν = 0.4 can be described by a PFS because 0.8
2 + 0.4
2 < 1. Due to the stronger expressiveness, PFSs have also had a wide range of applications in MCDM. For example, Yager and Abbasov (
2013) investigated the relationships between Pythagorean membership grades (PMGs) and complex numbers; Zhang and Xu (
2014) presented an extension of TOPSIS to MCDM with PFSs; Peng and Yang (
2015) proposed division and subtraction operations on PFSs and developed a Pythagorean fuzzy superiority and inferiority ranking method to address the MCDM problems with uncertainty; Garg (
2016) proposed a new generalised weighted Einstein operator to aggregate PMGs and studied its application in MCDM; Dick et al. (
2016) developed interpretations of complex-valued PMGs; Chen (
2018) developed novel VIKOR-based methods for MCDM involving Pythagorean fuzzy information; Wei and Lu (
2018) presented a set of Pythagorean fuzzy weighted power aggregation operators to resolve MCDM problems; Liang et al. (
2018) proposed a model of three-way decisions and developed the corresponding decision-making process based on PFSs; Khan et al. (
2019) presented a set of Pythagorean hesitant fuzzy Choquet integral aggregation operators for MCDM.
To further improve the expressiveness of PFSs, Yager (
2017) presented the theory of generalised orthopair fuzzy set (GOFS). In a GOFS, the condition of
µ and
ν is further relaxed to 0 ≤
µq +
νq ≤ 1 (
q = 1, 2, 3, …). Obviously, GOFS is the generalisation of FS, IFS, and PFS because: when
q = 1 and 0 <
µq +
νq = 1, GOFS will become FS; when
q = 1 and 0 ≤
µq +
νq ≤ 1, GOFS will become IFS; when
q = 2 and 0 ≤
µq +
νq ≤ 1, GOFS will become PFS. It is also not difficult to find that the greater of the value of the rung
q, the stronger the expressiveness of a GOFS. This provides a mechanism to obtain certain fuzzy information expression range via assigning an appropriate value to
q. As an example, suppose there is a criterion value whose
µ = 0.9 and
ν = 0.5. This value cannot be described by PFS since 0.9
2 + 0.5
2 > 1. However, when
q is assigned at least 3, the value can be expressed by GOFS. From this example, it is no doubt that GOFSs have the strongest expressiveness compared with FSs, IFSs, and PFSs. Due to this, GOFSs have also received extensive attention during the past 2 years. Various research topics regarding GOFSs, which mainly include approximate reasoning in GOFSs (Yager and Alajlan
2017), aspects of GOFSs (Yager et al.
2018), distance measures of GOFSs (Du
2018), correlation and correlation coefficient of GOFSs (Du
2019), MCDM methods based on GOFSs (Liu et al.
2018b; Wang and Li
2018), and aggregation operators of generalised orthopair membership grades (GOMGs) (Liu and Wang
2018a,
b; Liu and Liu
2018; Yang and Pang
2019; Liu et al.
2018c,
d; Wei et al.
2018,
2019; Bai et al.
2018; Wang et al.
2019; Peng et al.
2018; Xing et al.
2019), are gaining importance within academia.
For solving the MCDM problems, there are generally two categories of methods. One category consists of conventional methods (e.g. TOPSIS, VIKOR, PROMETHEE, ELECTRE). The other category includes the methods based on aggregation operators (Grabisch et al.
2009,
2011). Aggregation operators can solve the MCDM problems more effectively because they can provide comprehensive values and rankings of alternatives, while conventional methods can only generate rankings (Liu and Wang
2018b). So far, over twenty different aggregation operators of GOMGs have been presented, which include the weighted averaging (WA) operator and the weighted geometric (WG) operator (Liu and Wang
2018a), the weighted Bonferroni mean (WBM) operator and the weighted geometric Bonferroni mean (WGBM) operator (Liu and Liu
2018), the weighted Archimedean Bonferroni mean (WABM) operators (Liu and Wang
2018b), the weighted partitioned Bonferroni mean (WPBM) operator and the weighted partitioned geometric Bonferroni mean (WPGBM) operator (Yang and Pang
2019), the weighted extended Bonferroni mean (WEBM) operator (Liu et al.
2018b), the weighted Heronian mean (WHM) operator and the weighted geometric Heronian mean (WGHM) operator (Wei et al.
2018), the WHM* operator and the weighted partitioned Heronian mean (WPHM) operator (Liu et al.
2018c), the weighted Maclaurin symmetric mean (WMSM) operator and the weighted geometric Maclaurin symmetric mean (WGMSM) operator (Wei et al.
2019), the weighted power Maclaurin symmetric mean (WPMSM) operator (Liu et al.
2018d), the weighted power partitioned Maclaurin symmetric mean (WPPMSM) operator (Bai et al.
2018), the weighted Muirhead mean (WMM) operator and the weighted geometric Muirhead mean (WGMM) operator (Wang et al.
2019), the weighted exponential (WE) operator (Peng et al.
2018), and the weighted point (WP) operators (Xing et al.
2019). The main characteristics of these operators are listed in Table
1. As can be summarised from the table, among the MCDM methods based on the operators, there is not yet a method that has the following characteristics at the same time: (1) provide desirable generality and flexibility in the aggregation of GOMGs; (2) deal with the case where the criteria are divided into several partitions and there are interrelationships between different criteria in each partition, whereas the criteria in different partitions are independent of each other; (3) reduce the negative effect of the unduly high or unduly low criterion values on the aggregation results; (4) capture the risk attitudes of decision makers.
Table 1
The main characteristics of the existing aggregation operators of GOMGs
WA | | Can provide limited | Independent | Cannot reduce | Cannot capture |
WG | | Can provide limited | Independent | Cannot reduce | Cannot capture |
WBM | | Can provide limited | Between 2 criteria | Cannot reduce | Cannot capture |
WGBM | | Can provide limited | Between 2 criteria | Cannot reduce | Cannot capture |
WABM | | Can provide desirable | Between 2 criteria | Cannot reduce | Can capture |
WPBM | | Can provide limited | Between 2 criteria | Cannot reduce | Cannot capture |
WPGBM | | Can provide limited | Between 2 criteria | Cannot reduce | Cannot capture |
WEBM | | Can provide limited | Between 2 criteria | Cannot reduce | Cannot capture |
WHM | | Can provide limited | Between 2 criteria | Cannot reduce | Cannot capture |
WGHM | | Can provide limited | Between 2 criteria | Cannot reduce | Cannot capture |
WHM* | | Can provide limited | Between 2 criteria | Cannot reduce | Cannot capture |
WPHM | | Can provide limited | Between 2 criteria | Cannot reduce | Cannot capture |
WMSM | | Can provide limited | Among 2 + criteria | Cannot reduce | Cannot capture |
WGMSM | | Can provide limited | Among 2 + criteria | Cannot reduce | Cannot capture |
WPMSM | | Can provide limited | Among 2 + criteria | Can reduce | Cannot capture |
WPPMSM | | Can provide limited | Among 2 + criteria | Can reduce | Cannot capture |
WMM | | Can provide limited | Among 2 + criteria | Cannot reduce | Cannot capture |
WGMM | | Can provide limited | Among 2 + criteria | Cannot reduce | Cannot capture |
WE | | Can provide limited | Independent | Cannot reduce | Cannot capture |
WP | | Can provide moderate | Independent | Cannot reduce | Cannot capture |
In practical MCDM problems, aggregation of values of criteria is a complicated process, in which decision makers’ preferences could vary. A desirable aggregation operator should be general and flexible enough to adapt to such variation. Also, there are generally interrelationships among the criteria considered in the problems. It is also of necessity for an aggregation operator to model such interrelationships to obtain more reasonable aggregation results. Further, criterion values are usually evaluated by domain experts, which are always not absolutely objective. This means that a few domain experts could provide unduly high or unduly low criterion values. To achieve reasonable aggregation results, it is of importance to reduce the effect of such values in the aggregation process. Finally, MCDM problems have certain subjectivity and the preferences of decision makers are their important input. Among various preferences, decision makers’ risk attitudes (e.g. pessimistic, neutral, and optimistic) are an important type. A desirable aggregation operator should have the capability to capture such risk attitudes. Based on these considerations, the motivations of the present paper are explained as follows:
(1)
To develop an aggregation operator of GOMGs that can capture the interrelationships of criteria and the risk attitudes of decision makers, the Bonferroni mean (BM) operator (Bonferroni
1950), geometric BM (GBM) operator (Xia et al.
2013), and partitioned average operator (Dutta and Guha
2015) are introduced to construct partitioned Bonferroni aggregation operators of GOMGs. The BM and GBM operators can capture the interrelationships between the aggregated arguments and were found to, respectively, provide pessimistic and optimistic expectations in MCDM. The partitioned average operator can handle the situation where the aggregated arguments are divided into several partitions and the arguments in different partitions have different interrelations.
(2)
To enable the aggregation operator to reduce the influence of extreme criterion values on the aggregation results, the power average operator (Yager
2001) is combined into the partitioned Bonferroni aggregation operators of GOMGs. The power average operator can assign weights to the aggregated arguments. This makes it possible to reduce the effect of unreasonable arguments values on the aggregation results.
(3)
To improve the generality and flexibility of the combined aggregation operators of GOMGs, the operational rules based on the Archimedean
T-norm and
T-conorm (ATT) (Klement et al.
2000; Deschrijver and Kerre
2002) are used to perform the operations in them. The ATT are important tools that can generate versatile operational rules for membership grades, and the aggregation operators based on them are flexible in the aggregation of fuzzy information.
To sum up, this paper aims to present a set of weighted Archimedean power partitioned BM (WAPPBM) operators and weighted Archimedean power partitioned GBM (WAPPGBM) operators of GOMGs and a MCDM method based on them. This aim is achieved via the combination of the BM, GBM, power average, and partitioned average operators with weights and the operational rules based on ATT in the context of MCDM based on GOMGs. The major contribution of the paper is as follows: A MCDM method based on weighted Archimedean power partitioned Bonferroni aggregation operators of GOMGs is proposed. Compared to the existing MCDM methods based on aggregation operators of GOMGs, the proposed MCDM method simultaneously has the four characteristics above.
The remainder of the paper is organised as follows. A brief introduction of some related basic concepts is given in Sect.
2. Sections
3 and
4, respectively, explain the details of the presented aggregation operators and the proposed MCDM method. A numerical example and qualitative and quantitative comparisons are reported to illustrate and demonstrate the method in Sect.
5. Section
6 ends the paper with a conclusion.
2 Preliminaries
To better understand this paper, some prerequisites in GOFS theory, operational rules of GOMGs based on ATT, BM operator, GBM operator, power average operator, and partitioned average operator are briefly introduced in this section.
2.1 GOFS theory
Yager’s GOFS (Yager
2017) is the generalisation of Zadeh’s FS (Zadeh
1965), Atanassov’s IFS (Atanassov
1986), and Yager’s PFS (Yager
2014). Its formal definition is as follows:
For convenience, a pair 〈µS(x), νS(x)〉 is called as a GOMG, which is usually denoted as G = 〈µ, ν〉. To compare two GOMGs, their scores and accuracies are needed to calculate. The followings are the definitions of the score and the accuracy of a GOMG.
Obviously, −1 ≤ S(G) ≤ 1.
Obviously, 0 ≤ A(G) ≤ 1.
Based on S(G) and A(G), two GOMGs can be compared according to the following definition:
To compute the distance between two GOMGs, a distance measure of GOMGs is required. The following definition provides the Minkowski-type distance measure of GOMGs (Du
2018):
2.2 Operational rules
In mathematics, a
T-norm is a binary operation on the unit interval [0, 1] that satisfies commutativity, associativity, monotonicity, and boundary condition (Klement et al.
2000,
2005; Pap
1997,
2008). The dual notion of a
T-norm is its conorm. Formally, a
T-norm and its conorm can be defined as follows:
A T-norm T is called Archimedean if every sequence xn (where n = 1, 2, …; x1 < 1; and xn+1 = T(xn, xn)) converges to 0. The conorm of an Archimedean T-norm is called as an Archimedean T-conorm.
For an Archimedean T-norm T and its conorm TC: (1) If a function f(t) (t ∈ R) is monotonically decreasing and satisfies the conditions that f(t): (0, 1] → R+; f−1(t): R+ → (0, 1]; limt→∞f−1(t) = 0; and f−1(0) = 1, then f(t) can be used to generate T: T(x, y) = f−1(f(x) + f(y)) and is called as an additive generator of T; (2) If a function g(t) (t ∈ R) is monotonically increasing and satisfies the conditions that g(t): (0, 1] → R+; g−1(t): R+ → (0, 1]; limt→∞g−1(t) = 1; and g−1(0) = 0, then g(t) can be used to generate TC: TC(x, y) = g−1(g(x) + g(y)) and is called as an additive generator of TC. According to the definition of the conorm of a T-norm, f(t) is actually equal to g(1 − t), that is, f(t) = g(1 − t).
During the past few decades, the studies of ATTs and their additive generators have received a lot of attention. Various families of ATTs have been presented in this period. Four well-known families of ATTs and their additive generators are as follows:
(1)
If
f(
t) = − In
t, then
g(
t) = − In(1 −
t),
f−1(
t) = e
−t, and
g−1(
t) = 1 − e
−t. Based on this, the algebraic
T-norm and
T-conorm are obtained as
$$ T_{\text{A}} (x,y) = xy\quad {\text{and}}\quad T_{\text{A}}^{\text{C}} (x,y) = x + y - xy $$
(2)
If
f(
t) = In[(2 −
t)/
t], then
g(
t) = In[(1 +
t)/(1 −
t)],
f−1(
t) = 2/(e
t+ 1), and
g−1(
t) = (e
t− 1)/(e
t+ 1). Based on this, the Einstein
T-norm and
T-conorm are obtained as
$$ T_{\text{E}} (x,y) = \frac{xy}{1 + (1 - x)(1 - y)}\quad {\text{and}}\quad T_{\text{E}}^{\text{C}} (x,y) = \frac{x + y}{1 + xy} $$
(3)
If
f(
t) = In{[
λ + (1 −
λ)
t]/
t} (
λ > 0), then
g(
t) = In{[
λ + (1 −
λ)(1 −
t)]/(1 −
t)},
f−1(
t) =
λ/(e
t+
λ−1), and
g−1(
t) = (e
t− 1)/(e
t+
λ−1). Based on this, the Hamacher
T-norm and
T-conorm are obtained as
$$ T_{\text{H}} (x,y) = \frac{xy}{\lambda + (1 - \lambda )(x + y - xy)}\quad {\text{and}}\quad T_{\text{H}}^{\text{C}} (x,y) = \frac{x + y - xy - (1 - \lambda )xy}{1 - (1 - \lambda )xy} $$
(4)
If
f(
t) = − In[(
ε − 1)/(
εt− 1)] (
ε > 1), then
g(
t) = − In[(
ε − 1)/(
ε1−t−1)],
f−1(
t) = log
ε[(
ε − 1+e
−t)/e
−t], and
g−1(
t) = 1 − log
ε[(
ε − 1 + e
−t)/e
−t]. Based on this, the Frank
T-norm and
T-conorm are obtained as
$$ T_{\text{F}} (x,y) = \log_{\varepsilon } \left( {1 + \frac{{(\varepsilon^{x} - 1)(\varepsilon^{y} - 1)}}{\varepsilon - 1}} \right)\quad {\text{and}}\quad T_{\text{F}}^{\text{C}} (x,y) = 1 - \log_{\varepsilon } \left[ {1 + \frac{{(\varepsilon^{1 - x} - 1)(\varepsilon^{1 - y} - 1)}}{\varepsilon - 1}} \right] $$
Based on ATT, a set of general and versatile operational rules of GOMGs can be established according to the following definition (Liu and Wang
2018b):
2.3 BM operator
The BM operator was presented by Bonferroni (
1950). It is capable of describing the interrelationships between different non-negative real numbers. The formal definition of BM operator is as follows:
2.4 GBM operator
The GBM operator was introduced by Xia et al. (
2013). It was found to obtain more optimistic expectations in MCDM than the BM operator. The GBM operator is actually the dual form of the BM operator. Its formal definition is as follows:
2.5 Power average operator
The power average operator, introduced by Yager (
2001), can assign weights to the aggregated arguments via calculating the degrees of support between these arguments. This makes it possible to reduce the negative effect of the unduly high or unduly low argument values on the aggregation results. The formal definition of the operator is as follows:
2.6 Partitioned average operator
The partitioned average operator can aggregate the arguments in different partitions using the same aggregation operator and aggregate the aggregation results of different partitions using the arithmetic average operator (Dutta and Guha
2015). Its formal definition is as follows: