Decision algorithm for picture fuzzy sets and Aczel Alsina aggregation operators based on unknown degree of wights.

Aggregation operators (AOs) are well-known and efficient mathematical tools that are utilized to overcome the impact of imprecise and vague information during the aggregation process. The theoretical concepts of Aczel Alsina aggregation expressions are an extension of triangular norms and become a hot research topic in the environment of the fuzzy framework. The power operators provide a smooth approximation and are used to mitigate the influence of redundant or insufficient information on the attributes or criteria. Some robust aggregation approaches are developed by combining two different theories, like power operators and Aczel Alsina aggregation tools. This article aims to explore the theory of picture fuzzy sets (PFSs), an extended version of fuzzy sets, and intuitionistic fuzzy sets. Some robust operations of Aczel Alsina aggregation tools are also present in light of the picture fuzzy environment. We established a class of new methodologies in the light of picture fuzzy information, including picture fuzzy Aczel Alsina power weighted average (PFAAPWA) and picture fuzzy Aczel Alsina power ordered weighted average (PFAAPOWA) operators. We also developed an appropriate approach like picture fuzzy Aczel Alsina power weighted geometric (PFAAPWG) and picture fuzzy Aczel Alsina power ordered weighted geometric (PFAAPOWG) operators. Notable properties and characteristics of proposed methodologies are also demonstrated. Our invented approaches not only aggregate complicated information but can clearly define interrelationships among several arguments. Moreover, we establish an algorithm for the multi-attribute group decision-making (MAGDM) problem to handle the impact of redundant and vague information on human opinions. Finally, we study an experimental case study to evaluate an appropriate optimal option from available options. To reveal consistency and effectiveness of developed approaches, influence study by changing various parametric values and comparative study by comparing results of existing approaches.

Limit properties in the metric semi-linear space of picture fuzzy numbers.

The picture fuzzy set (PFS) just appeared in 2014 and was introduced by Cuong, which is a generalization of intuitionistic fuzzy sets (Atanassov in Fuzzy Sets Syst 20(1):87-96, 1986) and fuzzy sets (Zadeh Inf Control 8(3):338-353, 1965). The picture fuzzy number (PFN) is an ordered value triple, including a membership degree, a neutral-membership degree, a non-membership degree, of a PFS. The PFN is a useful tool to study the problems that have uncertain information in real life. In this paper, the main aim is to develop basic foundations that can become tools for future research related to PFN and picture fuzzy calculus. We first establish a semi-linear space for PFNs by providing two new definitions of two basic operations, addition and scalar multiplication, such that the set of PFNs together with these two operations can form a semi-linear space. Moreover, we also provide some important properties and concepts such as metrics, order relations between two PFNs, geometric difference, multiplication of two PFNs. Next, we introduce picture fuzzy functions with a real domain that is also known as picture fuzzy functions with time-varying values, called geometric picture fuzzy function (GPFFs). In this framework, we give definitions about the limit of GPFFs and sequences of PFN. The important limit properties are also presented in detail. Finally, we prove that the metric semi-linear space of PFNs is complete, which is an important property in the classical mathematical analysis.

Multi-period evaluation and selection of rural wastewater treatment technologies: a case study.

Rapid population growth and agricultural development are generating a considerable amount of effluents, which poses threats to the quality of rural water resources as well as sanitary conditions. However, with a range of rural wastewater treatment (WT) technologies available, one major problem facing the practitioners is which to choose as the most favorable option suited to specific areas. In this study, a novel decision-making framework is proposed to evaluate and select the optimal alternative in rural areas of Xi'an within multiple consecutive time periods. Firstly, an evaluation index system is constructed and picture fuzzy numbers (PFNs) are used to represent both evaluation levels and experts' refusal due to limitation of knowledge. Secondly, fuzzy analytical hierarchy process (FAHP) is applied to derive weights of criteria, which enables experts to assign fuzzy numbers to express their preferences for comparison judgments. Thirdly, evidence theory is utilized to obtain the aggregated values from multiple time periods. Finally, based on the belief intervals obtained, sequencing batch reactor (A4) is determined as the optimal rural WT technology in Xi'an from 2006 to 2020, whereas the membrane bio-reactor (A2) is the last option. The effectiveness of the proposed framework is further validated by comparative analysis. This research can hopefully serve as useful guidance for the assessment of rural WT technologies in various regions.