**1. Introduction**

Financial managemen<sup>t</sup> is an important part of strategic managemen<sup>t</sup> research, which investigates how enterprises exploit proper strategies to create and maintain competitive advantages. Presently, research into competition has grown exponentially. Mintzberg et al. [1] criticized overly analytical orientation, upper managemen<sup>t</sup> slant, lack of attention to action and learning, and neglect of the elements that lead to the creation of strategies. Shrivastava [2] focused his research on organizational learning processes; this has the potential to offer insights into these identified drawbacks. Brockman and Morgan [3] showed that organizational knowledge is the basis for gaining and defending competitive advantage and a key variable in the amplification of firm performance. Furthermore, some studies showed evidence of a useful relationship between organizational learning and inflexible performance. For example, Baker and Sinkula [4] proved that learning intention has a direct effect on firm performance. Ussahawanitchakit [5] practiced an advance measure of knowledge and obtained similar issues. Enterprise performance estimation is not only development of market economy within a certain time, but also a scientific method and constructive tool to supervise enterprises for a nation with current market economy. Work for the evaluation of enterprise performance of our country, how to change our economy, social environment, and the trend of internationalization and how to show a system of performance appraisal that fits our economic development plays an especially important operational significance for enhancing the health of our enterprises, improving the managemen<sup>t</sup> level, sharpening the enterprise competitiveness, and further improving the economic growth quality. Many MCDM or multiple attribute decision making (MADM) problems (such as business and strategic financial management, medical diagnosis etc.) have been developed with the use of aggregation [6–8] operators under probabilistic environment. Merigo et al. [9] introduced order weighted averaging operator, induce aggregation operators, weighted averaging operator and then used these operators to develop strategic decision making theory. Li [10] have studied decision and game theory in managemen<sup>t</sup> in the environment of intuitionistic fuzzy information. The study of aggregation functions under different fuzzy environment is an important research instrument in decision science. In next paragraph, we briefly overview of some fuzzy aggregation functions and their corresponding decision making problems.

It is very difficult to take real attribute values, because of complexity presented in serious level in the field of decision environment. In 1965, Zadeh [11] introduced theory of fuzzy sets (FS), a new mathematical notion to handle easily multi-criteria decision making MCDM [12,13] problems and multi-criteria group decision making MCGDM [14,15] problems. It is known to all that intuitionistic fuzzy sets (IFSs) and interval-valued intuitionistic fuzzy sets (IVIFSs) [16] are the generalization of IFS. All these traditional theories are very interesting research topics which have engaged attention of the researchers, because these theories have successful applications in different directions such as decision-making, medical diagnosis, pattern recognition, cluster analysis etc. Then many scholars expressed intuitionistic fuzzy information in their excellent study as follows, Chen and Chou [17] used particle swarm optimization method to develop a MADM problem under the information IVIF numbers. Du and Liu [18] proposed to study a MADM problem based on VIKOR method using intuitionistic trapezoidal fuzzy information. Garg [19] studied an MCDM problem with unknown attribute values by defining a new score and accuracy function under IVIF environment. Kumar and Garg [20] contributed an MADM problem for INVIF environment based on SPA. Li [21] proposed a MADM to develop a closeness coefficient of nonlinear programming method based on IVIF values in which preference values of the attribute are unknown. Lourenzutti and Krohling [22] used IFVs to develop MCDM method using fuzzy-TODIM technique. Wan and Li [23,24] used to develop heterogeneous MAGDM problems with A-IF (A-IVIF) information and then applicability of the proposed method is justified with an example of real supplier selection. Ye [25] motivated to analyze MCDM based on novel accuracy function under IVIF environment. Recently, researchers have drawn attention to model covering-based IF rough sets with their applications to MADM problems [26,27]. Additionally, intuitionistic fuzzy rough graphs are helpful to understand these models and can solved decision-making problems, which is explained in [28].

All the execution of the criteria for alternatives, weighted and order weighted aggregation operators [29,30], have a major role in the course of the document aggregation. In that view, Xu [31,32] introduced some weighted aggregation operators to solve MAGDM problems. Recently, aggregation of information for operators is an interesting research subject, receiving grea<sup>t</sup> attention of the researchers in the light of Hamacher operations. The operation introduced by Hamacher [33], known as "Hamacher operation (HO)", is a combination of algebraic TN and TCN, and Einstein TN and TCN [34]. Huang [35] introduced intuitionistic fuzzy Hamacher aggregation operators and gave application of these operators in MADM. Liu [36] presented some interval-valued intuitionistic fuzzy Hamacher aggregation operators and applied it to MAGDM problems. Xiao [37] proposed order weighted Hamacher geometric operator under IVIF environment. Li [38] gave the attention to the study of Hamacher correlated averaging (HCA) operator with the help of Hamacher sum and the Hamacher product operations based on IVIF information. The readers can ge<sup>t</sup> more information about Hamacher and other aggregation functions and developed decision-making methods from the following references [39–51].

Although IFS and IVIFS have been successfully applied to solve real world problems, in reality there are some conditions which cannot be handled by IFSs. Suppose, that in case of voting, human outlook involved more responses such as yes, abstain, no, refusal, which cannot be perfectly presented

by traditional FS and IFS. To overcome this situation, the notion of picture fuzzy (PFS) set was originated by Cuong [52–54] as a new mathematical tool for computational intelligence problems. A PFS is identified by functions called (positive, neutral, refusal)-membership function during the information analysis. With this point of view, a PFS can be considered as a generalization of IF and the fuzzy sets. Recently, many researchers have studied PFSs and its applications: Sing [55] suggested correlation coefficient of PFS, and gave an application in clustering analysis. Jana et al. [56] used weighted Dombi aggregation operators in picture fuzzy environment, and using these operators developed software selection method. Based on some new fuzzy algorithms on the basis of PFSs environment, Son and others [57,58] provided weather forecasting and time series forecasting. Thong [59] studied a novel fuzzy hybrid model for PF-clustering, and IF-systems for Medical diagnosis and health care support systems. Son [60] investigated generalized PF-distance measure, and applied it to solve clustering analysis problems under PFSs environment. Wei and others [61,62] used PF-information aggregation to find ranking of EPR systems, picture 2-tuple linguistic Bonferroni mean-based model, picture 2-tuple linguistic model for the solution of multi attribute decision making problems. To see more information about PFS applied to risk management, picture preference relation and picture 2-tuple linguistic [63–65] information used to find their corresponding decision making. Recently, Wei [66] introduced Hamacher aggregation operator on picture fuzzy set. He studied different kinds of Hamacher aggregation operators under picture fuzzy environment such as (PFHA) operators, (PFHG) operators, (PFHCA) operators, (IPFHA) operators, (IPFHCA) operators, (PFHPA) operators, (PFHPA) operators, and then provided a MADM problem for the utility and flexibility of proposed method. Therefore, based on Hamacher operation, how to aggregate these PFS is a very useful topic. In this paper, we shall define some PF Hamacher aggregation operators on the basis of traditional arithmetic [29,30,32], geometric operations [31], and Hamacher operations [33,66].

The PFS has a powerful ability to model the ambiguous and imprecise information of the real world. In literature, there are many different works related to applications of fuzzy aggregation method in decision making problem based on Hamacher operations. With this motivation, we used picture fuzzy Hamacher weighted averaging (PFHWA) operator, picture fuzzy Hamacher weighted geometric (PFHWGA) operator to access the best enterprise on the basis of performance evaluation of enterprises.
