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*Two critical tasks in multi-criteria group decision making MCGDM are to describe criterion values and to aggregate the described information to generate a ranking of alternatives. A flexible and superior tool for the first task is q-rung orthopair fuzzy number qROFN and an effective tool for the second task is aggregation operator. So far, nearly thirty different aggregation operators of qROFNs have been presented.*

## 3. Fuzzy Set Operation (1)

Two critical tasks in multi-criteria group decision making MCGDM are to describe criterion values and to aggregate the described information to generate a ranking of alternatives. A flexible and superior tool for the first task is q-rung orthopair fuzzy number qROFN and an effective tool for the second task is aggregation operator. So far, nearly thirty different aggregation operators of qROFNs have been presented.

Each operator has its distinctive characteristics and can work well for specific purpose. However, there is not yet an operator which can provide desirable generality and flexibility in aggregating criterion values, dealing with the heterogeneous interrelationships among criteria, and reducing the influence of extreme criterion values.

To provide such an aggregation operator, Muirhead mean operator, power average operator, partitioned average operator, and Archimedean T-norm and T-conorm operations are concurrently introduced into q-rung orthopair fuzzy sets, and an Archimedean power partitioned Muirhead mean operator of qROFNs and its weighted form are presented and a MCGDM method based on the weighted operator is proposed in this paper.

The generalised expressions of the two operators are firstly defined. Their properties are explored and proved and their specific expressions are constructed. Finally, the feasibility and effectiveness of the method is demonstrated via a numerical example, a set of experiments, and qualitative and quantitative comparisons.

This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Competing interests: The authors have declared that no competing interests exist. Multi-criteria group decision making MCGDM refers to the process of finding optimal alternatives in complex scenarios via synthetically evaluating the values of multiple criteria of all alternatives provided by a group of domain experts [ 1 ]. In this process, there are two critical tasks.

One critical task is to describe the values of different criteria accurately and effectively. For such description, there are many different kinds of available tools, where fuzzy set is a well-known kind [ 2 — 12 ]. To date, nearly thirty different types of fuzzy sets have been presented [ 13 ].

A qROFS consists of an element and a q-rung orthopair membership grade commonly known as q-rung orthopair fuzzy number qROFN , which is used to quantify the degrees of membership and non-membership of the element to the qROFS. In other words, the rung q in a qROFS is adjustable under the premise of satisfying this condition. In addition, the expressiveness of a qROFS will continue to increase as q increases, which provides enough freedom for the description of fuzzy information. The other critical task in MCGDM is to fuse the described criterion information to generate a ranking of all alternatives.

For such fusion, aggregation operator is regarded as an effective tool. So far, over twenty different aggregation operators of qROFNs have been presented.

Each operator has its own characteristics and can work well for its specific purpose. But there is not yet an operator that has the following three characteristics at the same time: 1 Provide satisfying generality and flexibility in the aggregation of qROFNs; 2 Deal with the situation in which the criteria are divided into several parts and there are interrelationships among different criteria in each part whereas the criteria in different parts are independent of each other; 3 Reduce the negative effect of the unduly high or unduly low criterion values on the aggregation results.

In practical MCGDM problems, aggregation of criterion values is a complex process, in which the preferences of decision makers may change frequently. An ideal aggregation operator should be general and flexible enough to adapt to such change. Moreover, there are usually complex relationships among the different criteria considered in the problems. It is also of importance for an aggregation operator to capture the complex interrelationships of different criteria to generate more reasonable aggregation results [ 28 ].

Further, the values of criteria are generally assessed by domain experts. It is often difficult to ensure the absolute objectivity, which means that a few biased experts will give biased assessment values [ 33 ]. To obtain reasonable aggregation results, it is of necessity to reduce the negative influence of biased criterion values in the aggregation.

Based on these considerations, the motivations of the present paper are explained as follows:. As a result, the presented aggregation operators combine all of their characteristics. The remainder of the paper is organised as follows. A brief introduction of some basic concepts is provided Section 2. Sections 3 explains the details of the presented Archimedean power partitioned MM operators. Section 5 demonstrates and evaluates the presented operators and proposed method via example, experiments, and comparisons.

Section 6 ends the paper with a conclusion. Its formal definition is as follow:. Definition 1 [ 14 ]. To compare two qROFNs, their scores and accuracies are required, which can be calculated according to the following definitions:. Definition 2 [ 26 ]. Definition 3 [ 26 ]. Definition 4 [ 26 ]. Definition 5 [ 19 ]. The formal definition of the rules is as follow:.

Definition 6 [ 28 ]. The PA operator, introduced by Yager [ 56 ], can assign weights to the aggregated arguments via computing the degrees of support between these arguments. This makes it possible to reduce the negative influence of the unduly high or unduly low argument values on the aggregation results.

The formal definition of the PA operator is as follow:. Definition 7 [ 56 ]. 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 [ 51 ]. Definition 8 [ 51 ]. Then the aggregation function 7 is called the partitioned average operator. The MM operator was firstly introduced to aggregate crisp numbers by Muirhead [ 39 ].

It has prominent characteristics in capturing the interrelationships among multiple aggregated arguments and providing a general form of a number of other aggregation operators. The formal definition of the MM operator is as follow:.

Definition 9 [ 39 ]. Then the aggregation function 8 is called the MM operator. The properties of the two operators are explored and their specific cases are discussed. Definition Theorem 1. For the details regarding the proof of this theorem, please refer to Appendix A in S1 File.

Theorem 2. Theorem 3. For the details regarding the proofs of these two theorems, please refer to Appendixes B and C in S1 File , respectively. For example, if the additive generators of Algebraic, Einstein, Hamacher, and Frank T-norms and T-conorms are assigned to f , then four specific operators can be respectively constructed as follows:.

But it does not consider the relative importance of each aggregated qROFN. The formal definition of this operator is as follow:. Theorem 4. For Eq 32 , if the additive generators of Algebraic, Einstein, Hamacher, and Frank T-norms and T-conorms are assigned to f , then four specific operators can be respectively constructed as follows:. In this section, a numerical example is firstly used to illustrate the working process of the proposed MCGDM method.

Then a set of experiments are carried out to explore the influence of different specific operators and parameter values on the aggregation results. Finally, qualitative and quantitative comparisons to the existing methods are made to demonstrate the feasibility and effectiveness of the proposed method.

A numerical example about the determination of the best industry for investment from five possible industries adapted on the basis of Reference [ 31 ] is used to demonstrate the proposed MCGDM method.

To make full use of idle capital, the board of directors of a company decided to invest in a new industry. Five industries were identified as possible industries for investment after preliminary research. The five alternative industries are medical industry A 1 , real estate development industry A 2 , Internet industry A 3 , education and training industry A 4 , and manufacturing industry A 5.

To select the best industry for investment, the board of directors appointed an expert panel, which consists of four different experts E 1 , E 2 , E 3 , and E 4. The four experts were asked to evaluate the five alternative industries on the basis of five criteria, which are the amount of capital profit C 1 , the market potential C 2 , the risk of capital loss C 3 , the growth potential C 4 , and the stability of policy C 5.

To provide enough freedom in the evaluation of the values of the five criteria of each alternative industry, experts were allowed to use qROFNs.

The evaluation results of the four experts are respectively listed in the following four matrices:. With the known conditions above, the determination of the best industry for investment can be carried out leveraging the proposed MCGDM method. Its process consists of the following eight steps:.

To explore the effect of using different specific operators and assigning different parameter values on the aggregation results, the following three experiments were carried out:. The results of the experiment are the calculated scores of Q i and the generated rankings of A i , which are listed in Table 1. As can be seen from the table, there is slight difference among the scores of the same Q i calculated by the four pairs of specific operators, and the rankings of A i also indicate small difference with respect to the four pairs of specific operators.

These indicate that the use of different specific operators has no obvious influence on the aggregation results. The results of the experiment are the calculated scores of Q i and the generated rankings of A i , which are depicted in Fig 1. From the figure, it can be seen that the ranking will change as q changes. The results of the experiment are the calculated scores of Q i and the generated rankings of A i , which are depicted in Fig 2.

As mentioned in the introduction, more than twenty different aggregation operators of qROFNs have been presented within academia. In this subsection, qualitative and quantitative comparisons between the MCGDM methods based on these operators and the proposed MCGDM method are carried out to demonstrate its feasibility and effectiveness.

Generally, a qualitative comparison among different MCGDM methods can be carried out by comparing their characteristics. For the twenty existing methods and the proposed method, the generality and flexibility in the aggregation of qROFNs, the capability to deal with the interrelationships among different criteria, and the capability to reduce the negative influence of the unduly high or unduly low criterion values on the aggregation results are selected as the comparison characteristics.

The results of the comparison are shown in Table 2. The details of the comparison are explained as follows:. Note: Heterogeneous interrelationships refer to the situation in which the criteria are divided into several parts and there are interrelationships among different criteria in each part whereas the criteria in different parts are independent of each other. As can be summarised from the qualitative comparison above, the proposed method has desirable generality and flexibility at both aggregating the q-rung orthopair fuzzy information and dealing with the interrelationships of criteria, and has the capability to reduce the negative influence caused by the deviation of some criterion values.

## Fuzzy Triangular Aggregation Operators

We will study uninorms on the unit square endowed with the natural partial order defined coordinate-wise. We will show that we can choose arbitrary pairs of incomparable elements, a , e and construct a uninorm whose neutral element is e and annihilator is a. As a special case we construct uninorms which are at the same time also nullnorms or, expressed another way, we construct proper nullnorms with neutral element. We will also generalize this result to the direct product of two bounded lattices. This means they are special types of aggregation functions. As such they have proven their importance in various fields of applications, e.

We present a new class of fuzzy aggregation operators that we call fuzzy triangular aggregation operators. To do so, we focus on the situation where the available information cannot be assessed with exact numbers and it is necessary to use another approach to assess uncertain or imprecise information such as fuzzy numbers. We also use the concept of triangular norms t-norms and t-conorms as pseudo-arithmetic operations. Main properties of these operators are discussed as well as their comparison with other existing ones. The fuzzy triangular aggregation operators not only cover a wide range of useful existing fuzzy aggregation operators but also provide new interesting cases. Finally, an illustrative example is also developed regarding the selection of strategies. The available information for the human knowledge is said to be precise crisp information or not fuzzy information.

convex functions, t-norm and t-conorm. 1. Introduction. Type-2 fuzzy sets (T2FSs) were introduced by L.A.. Zadeh in [21], as an extension of type-1 fuzzy.

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Objectives Introduces various operations of fuzzy sets Introduces the concepts of disjunctive sum, distance, difference, conorm and t conorm operators. Example 2 Given two fuzzy sets A and B a. Represent A and B fuzzy sets graphically b. Calculate the of union of the set A and set B c. Calculate the intersection of the set A and set B d.

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Toggle navigation. Help Preferences Sign up Log in. View by Category Toggle navigation. Products Sold on our sister site CrystalGraphics. Continuous Granular t-norm is idempotent The only idempotent t-norm is minimum Tags: absolutely fuzzy granular granularity idempotent logic maximum minimum tconorm tnorm.

g = 1 – x. a s g b = a + b – ab (Archimedean t-norm). ▫ g = e–f(x) multiplicative and additive generators → same t-conorm. Pedrycz and Gomide, FSE