The paper aims are to propose a new concept called as complex picture fuzzy set (CPFS), as an extension of complex intuitionistic fuzzy set (CIFS). The addition of a neutral membership degree to the definition of CIFS makes CPFS a generalized form of CIFS. The uniqueness of this new theory lies in the capability to attain the wider range with the help of degree of neutral membership, non-membership, and membership. The range of values of membership degrees is broaden to the unit disk in a complex plane. We define elementary operations and properties of CPFSs and explore the MCDM issues with the help of CPFSs, based on Hamacher operations and some aggregation methods. Then, we introduce some operators to aggregate the CPF data, namely complex picture fuzzy Hamacher weighted averaging, ordered weighted averaging, hybrid averaging and complex picture fuzzy Hamacher weighted geometric, ordered weighted geometric and hybrid geometric operators, benefited from the basic Hamacher operations, and averaging, geometric aggregation techniques. We also construct MCDM problem using these operators and perform a calculation for the selection of best ERP system, to demonstrate the authenticity and efficiency of this manuscript. Moreover, we study a comparison to validate the consistency and superiority of our techniques.

中文翻译：

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复杂图片模糊Hamacher聚集算子的决策模型

本文旨在提出一种称为复杂图片模糊集（CPFS）的新概念，作为复杂直觉模糊集（CIFS）的扩展。在CIFS的定义中增加了中性成员资格，使CPFS成为CIFS的广义形式。这种新理论的独特之处在于，在中立成员资格，非成员资格和成员资格的帮助下，能够达到更广泛的范围。隶属度的值的范围扩大到复杂平面中的单位磁盘。我们定义CPFS的基本操作和属性，并基于Hamacher操作和一些聚合方法，借助CPFS探索MCDM问题。然后，我们引入一些运算符来聚合CPF数据，即复杂图片模糊Hamacher加权平均，有序加权平均，混合平均和复杂图片模糊Hamacher加权几何，有序加权几何和混合几何算子，得益于基本Hamacher运算和平均几何聚合技术。我们还使用这些运算符构造MCDM问题，并进行计算以选择最佳ERP系统，以证明此手稿的真实性和效率。此外，我们进行了比较以验证我们技术的一致性和优越性。证明此手稿的真实性和有效性。此外，我们进行了比较以验证我们技术的一致性和优越性。证明此手稿的真实性和有效性。此外，我们进行了比较以验证我们技术的一致性和优越性。