1. Introduction
While dealing with any real world problems, a decision maker (DM) often feels discomfort when expressing his\her evaluation information by utilizing a single real number in multi-attribute decision making (MADM) or multi-attribute group decision making (MAGDM) problems due to the intellectual fuzziness of DMs. For this cause, Zadeh [
1] developed fuzzy sets (FSs), which are assigned by a truth-membership degree (TMD) in
and are a better tool to present fuzzy information for handling MADM or MAGDM problems. After the introduction of FSs, different fuzzy modelling approaches were developed to deal with uncertainty in various fields [
2,
3,
4]. However, in some situations, it is difficult to express truth-membership degree with an exact number. In order to overcome this defect and to express TMD in a more appropriate way, Turksen [
5] developed interval valued FSs (IVFSs), in which TMD is represented by interval numbers instead of exact numbers. Since only TMD was considered in FSs or IVFSs and the falsity-membership degree (FMD) came automatically by subtracting TMD from one, it is hard to explain some complicated fuzzy information, for example, for the selection of Dean of a faculty, if the results received from five professors are in favor, two are against and three are neither in favor nor against. Then, this type of information cannot be expressed by FSs. So, in order to handle such types of information, Atanassov [
6] developed intuitionistic fuzzy sets (IFSs), which were assigned by TMD and FMD. Atanassov et al. [
7] further enlarged IFSs and developed the interval valued IFS (IVIFSs). However, the shortcoming of FSs, IVFSs, IFSs and IVIFSs are that they cannot deal with unreliable or indefinite information. To solve such problems, Smarandache [
8,
9] developed neutrosophic sets (NSs). In neutrosophic set, every member of the domain set has TMD, an indeterminacy-membership degree (IMD) and FMD, which capture values in ]0
−, 1
+[. Due to the containment of subsets of ]0
−, 1
+[ in NS, it is hard to utilize NS in real world and engineering problems. To make NSs helpful in these cases, some authors developed subclasses of NSs, such as single valued neutrosophic sets (SVNSs) [
10], interval neutrosophic sets (INSs) [
11,
12], simplified neutrosophic sets (SNSs) [
13,
14] and so forth. In recent years, INSs have gained much attention from the researchers and a great number of achievement have been made, such as distance measures [
15,
16,
17], entropies of INS [
18,
19,
20], correlation coefficient [
21,
22,
23]. The theory of NSs has been extensively utilized to handle MADM and MAGDM problems.
For the last many years, information aggregation operators [
24,
25,
26,
27] have stimulated much awareness of authors and have become very dominant research topic of MADM and MAGDM problems. The conventional aggregation operators (AGOs) proposed by Xu, Xu and Yager [
28,
29] can only aggregate a group of real numbers into a single real number. Now these conventional AGOs were further extended by many authors, for example, Sun et al. [
30] proposed the interval neutrosophic number Choquet integral operator for MADM and Liu et al. [
31] developed prioritized ordered weighted AGOs for INSs and applied them to MADM. In addition, some decision-making methods were also developed for MADM problems, for example, Mukhametzyanov et al. [
32] developed a statistically based model for sensitivity analysis in MADM problems. Petrovic et al. [
33] developed a model for the selection of aircrafts based on decision making trial and evaluation laboratory and analytic hierarchy process (DEMATEL-AHP). Roy et al. [
34] proposed a rough relational DEMATEL model to analyze the key success factor of hospital quality. Sarkar et al. [
35] developed an optimization technique for national income determination model with stability analysis of differential equation in discrete and continuous process under uncertain environment. These methods can only give a ranking result, however, AGOs can not only give the ranking result, but also give the comprehensive value of each alternative by aggregating its attribute values.
It is obvious that, different aggregation operators have distinct functions, a few of them can reduce the impact of some awkward data produced by predispose DMs, such as power average (PA) operator proposed by Yager [
36]. The PA operator can aggregate the input data by designating the weight vector based on the support degree among the input arguments, and can attain this function. Now the PA operator was further extended by many researchers into different environments. Liu et al. [
37] proposed some generalized PA operator for INNs, and applied them to MADM. Consequently, some aggregation operators can include the interrelationship between the aggregating parameters, such as the Bonferroni mean (BM) operators developed by Bonferroni [
38], the Heronian mean (HM) operator introduced by Sykora [
39], Muirhead Mean (MM) operator [
40], Maclaurin symmetric mean [
41] operators. In addition, these aggregation operators have also been extended by many authors to deal with fuzzy information [
42,
43,
44,
45,
46].
For aggregating INNs, some AGOs are developed by utilizing different T-norms (TNs) and T-conorms (TCNs), such as algebraic, Einstein and Hamacher. Usually, the Archimedean TN and TCN are the generalizations of various TNs and TCNs such as algebraic, Einstein, Hamacher, Frank, and Dombi [
47] TNs and TCNs. Dombi TN and TCN have the characteristics of general TN and TCN by a general parameter, and this can make the aggregation process more flexible. Recently, several authors defined some operational laws for IFSs [
48], SVNSs [
49], hesitant fuzzy sets (HFSs) [
50,
51] based on Dombi TN and TCN. In practical decision making, we generally need to consider interrelationship among attributes and eliminated the influence of awkward data. For this purpose, some researchers combined BM and PA operators to propose some PBM operators and extended them to various fields [
52,
53,
54,
55]. The PBM operators have two characteristics. Firstly, it can consider the interaction among two input arguments by BM operator, and secondly, it can remove the effect of awkward data by PA operator. The Dombi TN and TCN have a general parameter, which makes the decision-making process more flexible. From the existing literatures, we know that PBM operators are combined with algebraic operations to aggregate IFNs, or IVIFNs, and there is no research on combining PBM operator with Dombi operations to aggregate INNs.
In a word, by considering the following advantages. (1) Since INSs are the more précised class by which one can handle the vague information in a more accurate way when compared with FSs and all other extensions like IVFSs, IFSs, IVIFSs and so forth, they are more suitable to describe the attributes of MADM problems, so in this study, we will select the INSs as information expression; (2) Dombi TN and TCN are more flexible in the decision making process due to general parameter which is regarded as decision makers’ risk attitude; (3) The PBM operators have the properties of considering interaction between two input arguments and vanishes the effect of awkward data at the same time. Hence, the purpose and motivation are that we try to combine these three concepts to take the above defined advantages and proposed some new powerful tools to aggregate INNs. (1) we define some Dombi operational laws for INNs; (2) we propose some new PBM aggregation operators based on these new operational laws; (3) we develop a novel MADM based on these developed aggregation operators.
The following sections of this article are shown as follows. In
Section 2, we review some basic concepts of INSs, PA operators, BM operators, and GBM operators. In
Section 3, we review basic concept of Dombi TN and TCN. After that, we propose some Dombi operations for INNs, and discuss some properties. In
Section 4, we define INDPBM operator, INWDPBM operator, INDPGBM operator and INWDPGBM operator and discuss their properties. In
Section 5, we propose a MADM method based on the proposed aggregation operators with INNs. In
Section 6, we use an illustrative example to show the effectiveness of the proposed MADM method. The conclusion is discussed in
Section 7.
5. MADM Approach Based on the Developed Aggregation Operator
In this section, based upon the developed INWDPBM and INWDPGBM operators, we will propose a novel MADM method, which is defined as follows.
Assume that in a MADM problem, we need to evaluate alternatives with respect to attributes , and the importance degree of the attributes is represented by , satisfying the condition . The decision matrix for this decision problem is denoted by , where is an INN for the alternative with respect to the attribute , Then the main purpose is to rank the alternative and select the best alternative.
In the following, we will use the proposed INWDPBM and INWDPGBM operators to solve this MADM problem, and the detailed decision steps are shown as follows:
- Step 1.
Standardize the attribute values. Normally, in real problems, the attributes are of two types, (1) cost type, (2) benefit type. To get right result, it is necessary to change cost type of attribute values to benefit type using the following formula:
- Step 2.
Calculate the supports
where,
is the distance measure defined in Equation (10).
- Step 3.
- Step 4.
Aggregate all the attribute values
to the comprehensive value
by using INWDPBM or INWDPGBM operators shown as follows.
or
- Step 5.
Determine the score values, accuracy values of using Definition 5.
- Step 6.
Rank all the alternatives according to their score and accuracy values, and select the best alternative using Definition 6.
- Step 7.
End.
This decision steps are also described in
Figure 1.