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The picture fuzzy set, characterized by three membership degrees, is ahelpful tool for multi-criteria decision making (MCDM). This paper investigatesthe structure of the closed operational laws in the picture fuzzy numbers(PFNs) and proposes efficient picture fuzzy MCDM methods. We first introduce anadmissible order for PFNs and prove that all PFNs form a complete lattice underthis order. Then, we give some specific examples to show the non-closeness ofsome existing picture fuzzy aggregation operators. To ensure the closeness ofthe operational laws in PFNs, we construct a new class of picture fuzzyoperators based on strict triangular norms, which consider the interactionbetween the positive degrees (negative degrees) and the neutral degrees. Basedon these new operators, we obtain the picture fuzzy interactional weightedaverage (PFIWA) operator and the picture fuzzy interactional weighted geometric(PFIWG) operator. They are proved to be monotonous, idempotent, bounded,shift-invariant, and homogeneous. We also establish a novel MCDM method underthe picture fuzzy environment applying PFIWA and PFIWG operators. Furthermore,we present an illustrative example for a clear understanding of our method. Wealso give the comparative analysis among the operators induced by six classesof famous triangular norms.

Picture Fuzzy Interactional Aggregation Operators via Strict Triangular Norms and Applications to Multi-Criteria Decision Making