OWA operators in the calculation of the average green-house gases emissions

. This study proposes, through weighted averages and ordered weighted averaging operators, a new aggregation system for the investigation of average gases emissions. We present the ordered weighted averaging operators gases emissions, the induced ordered weighted averaging operators gases emissions, the weighted ordered weighted averaging operators gases emissions and the induced probabilistic weighted ordered weighted averaging operators gases emissions. These operators represent a new way of analyzing the average gases emissions of different variables like countries or regions. The work presents further generalizations by using generalized and quasi-arithmetic means. The article also presents an illustrative example with respect to the calculations of the average gases emissions in the European region.


Introduction
Within the exceptionally later decades, since of an gigantic development of the population and the need to supply nourishment for them from one hand and the other hand an immethodical utilization of fossil fuel, our planet is experiencing an unexampled growth in terms of green-house gases (GHG) emission such as CO2, CH4 and N2O in its atmosphere that cause an ascending amount of global warming year by year and a drastic climate change [13,15,37].
There are many works that study the ways that can lead the GHG emission toward the minimization. [36], evaluate the potential influence of vehicle electrification on grid infrastructure and road-traffic green-house emission. [12] Study the impact of electrical power generation on GHG emission in Europe, [10] analyze green-house gases emission in concrete manufacture while there are some papers that focus on agriculture and farming [4,17,35].
Besides, although these works exist but it seems vital to present a comprehensive forecast about the future of countries based on the experts' opinions to provide a clear plan and make a suitable decision to decrease this emission in any of the studied sectors and under various conditions. Aggregation operators in the related literature with the aim of decision making are diverse and each of them can be used to collect the information [3,[26][27][28][29]. These techniques give importance to the variables according to certain available subjective or objective findings [34,31,38,40,42]. A very popular aggregation operator is the weighted average. This aggregation operator is flexible to use in a wide range of problems. Another popular aggregation operator is the ordered weighted average (OWA) [41,45]. The OWA operator provides a parametrized family of aggregation operators between the minimum and the maximum, weighting the data according to the attitudinal character of the decisionmaker. Based on this operator and with the purpose of expanding it, many authors expand and generalize it [9,16,24,39,46]. There are several types for the concept of expanding and generalizing and the most important item is the form of integrating OWA operator with some key concepts such as, using the induced variables, the probability and the weighted average. [40] propose some new aggregation operators such as the induced ordered weighted geometric averaging (IOWGA) operator, generalized induced ordered weighted averaging (GIOWA) operator, hybrid weighted averaging (HWA) operator.
The purpose of this work is to concentrate on the analysis of the use of the aggregation operators in the calculation of green-house gases (GHG) emission with the aim of developing better decision-making techniques. To this end, the paper studies several aggregation operators including the WA [3], OWA [23,41], OWAWA and IOWAWA [25], IOWA [44], POWAWA and IPOWAWA operator [29]. With the use of each operator, a new operator for GHG emission is produced including the OWA GHG emission (OWAGE), induced OWA GHG emission (IOWAGE), ordered weighted averaging weighted average GHG emission (OWAWAGE), induced OWAWA GHG emission (IOWAWAGE), probabilistic OWAWA GHG emission (POWAWAGE) and induced probabilistic OWAWA GHG emission (IPOWAWAGE).
The work also presents further generalizations by using generalized and quasi-arithmetic means obtaining the generalized OWAGE (GOWAGE). The aim of this approach is to show a more general framework in the analysis of averages by using complex aggregations including with geometric and quadratic averages. The study presents a wide range of particular types of aggregations under this approach.
During the related literature there are several works dedicated to the application of these aggregation operators such as, demand analysis [32], economic growth analysis [33], portfolio selection [18], support vector machines [22] and the average price [30]. On the other hand, many works are dedicated to making decision in different fields to solve the problem. As an example, [7] with mixing induced OWA operators and Minkowski distances, try to present a method to decide in reinsurance. [8] present a new method for handling multi-criteria fuzzy decision-making problems by using FN-IOWA operators or in the other study, [14] analyse the origin and uses of the ordered weighted geometric operator in multicriteria decision making and [21], proposes a model for the best-suited OWA operators and [6] by using bibliometric method review the contribution in fuzzy decision-making area. This work develops OWA operators in the analysis of the average green-house gases emissions.
The work presents an application regarding the calculation of the average gases emissions in Europe. For doing so, the paper considers a multi-expert aggregation problem where four experts analyze the expected average emissions of each European country for the next period. From, the analysis develops several aggregation methods based on the tools developed in the paper including the OWAGE, IOWG and OWAWAGE operators. The main advantage of the OWA operator is the possibility of under or overestimate the information according to the attitudinal character of the decision maker. Thus, depending on the degree of optimism or pessimism of the decision maker, the results may lead to different decisions and interpretations of the information.
This paper is organized as follows. Section 2 briefly reviews some basic OWA operators. Section 3 introduces the use of the OWA operator in the calculation of the average green-house gases emissions. Section 4 develops further generalization with generalized and quasi-arithmetic means. Section 5 presents an illustrative example regarding the calculation of average gases emissions with OWA operators. Section 6 ends the paper summarizing the main findings and conclusions of the paper.

The induced OWA operator (IOWA)
The IOWA operator [44] is an extension of the OWA operator. The main difference between OWA and IOWA is that the reordering step is not developed with the values of the arguments i a . In this case, the reordering step is carried out with order inducing variables. The IOWA operator also includes as particular cases the maximum, the minimum and the average criteria. It can be defined as follows.
where j b is the i a value of the IOWA pair , ii ua having the jth largest i u . i u is the order-ranking variable and i a is the argument variable.

The ordered weighted averaging-weighted average (OWAWA)
The OWAWA operator [25] is a new model that unifies the OWA operator and the weighted average in the same formula. Therefore, both concepts can be seen as a particular case of a more general one. It can be defined as follows.
, that is, according to the jth largest of the i a .

The probabilistic ordered weighted averagingweighted average (POWAWA)
The POWAWA [34] operator uses probabilities, weighted average and OWA in the same formulation. It unifies these three concepts by considering the degree of importance that each concept has in the aggregation, depending on the situation considered. The POWAWA operator is defined as follows.

The induced probabilistic OWAWA operator
The IPOWAWA [29] is an aggregation operator that extends POWAWA operator that uses order-inducing variables that represent complex reordering processes of an aggregation. Thus, it is an aggregation operator that uses induced variables, the probability, the weighted average and the OWA operator. Moreover, it can assess complex reordering processes by using order-inducing variables. Its main advantage is that it provides a more robust formulation than the POWAWA operator because it includes a wide range of cases. It can be defined as follows.

Calculation of the average green-house gases (GHG) emission with OWA operators
The purpose of this paper is to calculate the average GHG emission. The average GHG emission represents a numerical value that reports the information of the GHG emission. To calculate this item, using many aggregation operators is possible likewise normal arithmetic mean. These possible aggregation operators could be WA, OWA, IOWA or a combination of them such as OWAWA, IOWAWA, etc. Through using them we prepare some possibilities for the future of GHG emission in different scenarios in a spectrum from the worst case to the best case based on experts' opinions.
The basic operator for analyzing a set of GHG emission is OWAGE. The OWAGE operator is an aggregation operator that analyses an average GHG emission under uncertainty situation. It can be defined as follows for the set of GHG emission The other significant aggregation operator is the induced OWA (IOWA) that its reordering step is developed with order including variables. So, by using the IOWA operator we obtain IOWA GHG emission (IOWAGE) that can be defined as follows: It is important to mention that this operator is based on considering no extra information. One of the very important aspects of the average GHG emission is the importance of each of them and in other words, their weights in comparison with each other. To this end it is better to use some approaches of information aggregation that combine OWA operators and WA. In the literature there are some aggregation operators with this structure like, the WOWA operator [38], the hybrid average [26] and the OWAWA operators [25]. In this work we apply OWAWA to obtain the OWAWA GHG emission (OWAWAGE) and it is defined as follows for a set of GHG emission To focus more deeply on our contributions, we implement IOWAWA which is a combination of IOWA operators and WA in the same formulation. By using the IOWAWA operator we obtain IOWAWA GHG emission (IOWAWAGE) that can be defined as follows: according to , j f that is, according to the jth largest .
i u Besides, the other aspect that can be considered and leads results to a better form is probabilities in the attitudinal character of the decision-maker. For this reason, we apply POWAWA operator. By applying the Eq. (3) we could obtain the probabilistic OWAWA GHG emission (POWAWAGE). It can be defined as follows: ( ) 12 1, , , • Note that when  increases, we are giving more importance to the IOWAGE operator and when  decreases, we give more importance to the WA. Another group of interesting families are the maximum-WAGE, the minimum-WAGE, the step-IOWAWAGE operator and the usual average.
The arithmetic PWAGE (if 1, j wn = for all j): • The arithmetic POWAGE operator (if 1, i vn = for all i): Many other particular cases can be studied by looking at different expressions of the weighting vectors and the coefficients 12 ,

Generalizations with generalized and quasiarithmetic means
Generalization of the OWA operators is possible to do by generalized and quasi-arithmetic averaging aggregation operators that as the most common one generalized OWA (GOWA) [43] and then quasiarithmetic OWA (Quasi-OWA) [11] are formed. These functions apply a general framework including particular cases. The GOWA operator applied to the analysis of gases emissions is called GOWA gases emissions (GOWAGE) and is defined as follows.
where j b is the jth largest of the i e and g is strictly continuous monotonic function.

Illustrative example
In this section through a numerical example we try to show the applicability of OWA operators. This work concentrates on the calculation of different OWA operators' aggregation on green-house gases emission of European countries and makes a comparison on them to gain a clear decision about their possible future scenarios. To this end and with the purpose of giving a correct overview to solve the problem, a group of four experts analyses the information in seven scenarios. This step by step process can be explained as follows.
Step 1: Four experts analyze the green-house gases emission of European countries in seven possible scenarios in future based on the environmental and economic situation of the mentioned country. Table 1, 2, 3 and 4 represent the opinions of the experts. Table  2,3 and 4 are the same as 1 but to avoid repeating, we summarized them to a short form.
Step 2: The next step belongs to unify the experts' opinions to achieve to a collective result that cover all the information. To this end, it is necessary to assign the degree of importance to each of the experts: (0.4, 0.35, 0.15, 0.1). Z = Table 5 reports the collective results of each country.
Step 3: Based on the objective of this work it is necessary to assign weighting vectors to consider subjective and objective information and an attitudinal character that underestimates the results.
• Step 4: Present the obtained results of the average green-house gases for each country for the OWAGE, WAGE, OWAWAGE, IOWAGE, IOWAWAGE, POWAWAGE and IPOWAWAGE. Table 6 dedicates to the aggregated results.
Step 5: Rank the countries from the lowest to the highest in each of the operators to draw some conclusions. Table 7 presents the results of this ranking based on the abbreviation of the name of each country.

Conclusions
The purpose of this study is to concentrate on the analysis of the use of the aggregation operators in the calculation of GHG emission with the aim of developing better decision-making techniques. In this study we reviewed some of the important operators of the family of OWA. This review started with simple WA and continued with OWA operator. Moreover, we also analyzed some operators that form by combination of two or more aggregation operators. So, these operators are, IOWAGE, OWAWAGE, IOWAWAGE, POWAWAGE and IPOWAWAGE. In addition, through these formulations, we found some particular cases in either IOWAWAGE or POWAWAGE operators such as, olympic-IOWAWAGE, S-IOWAWAGE, centered-IOWAWAGE, maximum, minimum and arithmetic probabilistic weighted average, and arithmetic probabilistic ordered weighted average. Furthermore, some other generalizations are developed by using generalized and quasi-arithmetic means obtaining the GOWAGE and the Quasi-OWAGE operators. The study provides a simple example to review the function of two simple aggregations operators of average green-house gases emission. During this example we review weighted average gases emission (WAGE) and ordered weighted average gases emission (OWAGE) to represent the difference between the result of the calculation based on these operators. We also analyzed the applicability of these approaches for the process of decision-making problem in GHG emission. To achieve to this aim, we implement an illustrative example regarding the calculation of the average of green-house gases emission among European countries. To this end we collect the opinions of the four experts in this area in seven various scenarios in a multi-person analysis. Based on this example, and through five steps we obtain the final table that demonstrate comprehensively the situation of the European countries in a descending trend based on the results of different aggregation operators that can occur according to different scenarios between the minimum and maximum results.
In the future research, by using the different aggregation operators such as logarithmic [1], heavy [19,20], Bonferroni [5] and prioritized [2], we calculate the average GHG emission in a wide range of scenarios among the countries also among different continents.