The picture fuzzy sets (PFSs) state or model the voting information accurately without information loss. However, their existing operational laws usually generate unreasonable computing results, especially when the agreement degree (AD) or neutrality degree (ND) or opposition degree (OD) is zero. To tackle this issue, we propose the interactional operational laws (IOLs) to compute picture fuzzy numbers (PFNs), which can capture the interaction between the ADs and NDs in two PFNs, as well as the interaction between the ADs and ODs in two PFNs. Based on the proposed novel IOLs, partitioned Heronian mean (PHM) operator, and partitioned geometric Heronian mean (PGHM) operator, some picture fuzzy interactional PHM (PFIPHM), weighted PFIPHM (PFIWPHM), geometric PFIPHM (PFIPGHM), and weighted PFIPGHM (PFIWPGHM) operators are proposed in this paper. Afterwards, we investigate the properties of these operators. Using the PFIWPHM and PFIWPGHM operators, a novel multiple attribute decision-making (MADM) method with PFNs is elaborated. Finally, a study example that involves the service quality ranking of nursing facilities is provided to show the decision procedure of the proposed MADM method and we also give the comparative analysis between the proposed operators and the existing aggregation operators developed for PFNs.

图片模糊集（PFS）准确地陈述或建模投票信息，而不会丢失信息。但是，它们的现有运营法律通常会产生不合理的计算结果，尤其是在一致度（AD）或中立度（ND）或对立度（OD）为零时。为了解决这个问题，我们提出了交互操作法则（IOL）来计算图片模糊数（PFN），它可以捕获两个PFN中的AD和ND之间的交互，以及两个PFN中的AD和OD之间的交互。 。基于提出的新型IOL，分区Heronian均值（PHM）算符和分区几何Heronian均值（PGHM）算子，一些图片模糊交互PHM（PFIPHM），加权PFIPHM（PFIWPHM），几何PFIPHM（PFIPGHM）和加权PFIPGHM（本文提出了PFIWPGHM）运算符。然后，我们研究这些运算符的属性。使用PFIWPHM和PFIWPGHM运算符，阐述了一种带有PFN的新颖多属性决策（MADM）方法。最后，提供了一个涉及护理机构服务质量等级的研究示例，以说明所提出的MADM方法的决策过程，并且还对提议的运营商与为PFN开发的现有聚集运营商进行了比较分析。