Abstract The objective of this paper is to develop Pythagorean fuzzy (PF) multi‐objective optimization by ratio analysis (PF‐MOORA) plus full MULTIplicative form (PF‐MULTIMOORA) method for solving multicriteria decision making (MCDM) problems with completely unknown information of criteria weights. In the model formulation process, a new distance measure is defined to quantify the difference between PF sets by combining Hamming distance and Hausdorff metric. This distance measure is, subsequently, implemented in entropy weight model for determining unknown weights of criteria, and also in reference point approach for obtaining preference indices of alternatives. To overcome the deficiencies occurred in existing MULTIMOORA method, like multiple comparisons, circular reasoning, and so on, an aggregation‐based approach is recommended in the proposed PF‐MULTIMOORA. To demonstrate the feasibility and practicality of the proposed method, an example concerning strategy prioritization of a tiles manufacturing company is presented. In the strategy evaluation process, at first, the judgement values provided by the decision maker are expressed in linguistic terms, and then those are converted into PF numbers through a PF weighting scale. The sensitivity of the proposed model is validated by changing of weights of criteria which impact on the ranks of the strategies. To show robustness of the developed method, the result attained by applying PF‐MULTIMOORA is compared with existing techniques, not only in crisp and fuzzy quantitative strategic planning matrix context, but also using four other MCDM methods, namely, modified PF‐MOORA, as a particular case of the proposed PF‐MULTIMOORA technique, PF weighted sum, PF‐TOPSIS and PF‐VIKOR.

This article presents a new family of linguistic Pythagorean fuzzy aggregation operations based on Einstein t-norms and t-conorms. Some of their necessary properties are also discussed. In the proposed method, a generalized weighted distance measure is developed using a linguistic-scale function to evaluate differences among linguistic Pythagorean fuzzy sets (LPFSs). An entropy measure is also introduced for LPFS to measure fuzziness associated with linguistic decision information. Moreover, a technique for ordering preference by similarity to ideal solution (TOPSIS)-based methodology is constructed through the newly defined concepts to address linguistic Pythagorean fuzzy multicriteria group decision-making problems, where the information about weight is either completely known or unknown. Subsequently, weights of decision makers are computed through a TOPSIS-based algorithm, and weights of criteria are evaluated by introducing a linguistic Pythagorean fuzzy entropy weight model. To minimize information loss throughout the decision-making procedure, aggregation is done at the final stage for obtaining the final ranking of alternatives. Finally, several practical examples are considered, solved, and compared with existing methods to exhibit the robustness of the proposed methodology. Also, a sensitivity analysis is performed to establish the dynamic nature of the proposed model.

In this study, a new technique for order preference by similarity to ideal solution (TOPSIS)-based methodology is proposed to solve multicriteria group decision-making problems within Pythagorean fuzzy environment, where the information about weights of both the decision makers (DMs) and criteria are completely unknown. Initially, generalized distance measure for Pythagorean fuzzy sets (PFSs) is defined and used to initiate a new Pythagorean fuzzy entropy measure for computing weights of the criteria. In the decision-making process, at first, weights of DMs are computed using TOPSIS through the geometric distance model. Then, weights of the criteria are determined using the entropy weight model through the newly defined entropy measure for PFSs. Based on the evaluated criteria weights, TOPSIS is further applied to obtain the score value of alternatives corresponding to each decision matrix. Finally, the score values of the alternatives are aggregated with the calculated DMs’ weights to obtain the final ranking of the alternatives to avoid the loss of information, unlike other existing methods. Several numerical examples are considered, solved, and compared with the existing methods.

In this paper, a series of similarity measures based on point operators for Pythagorean fuzzy sets are proposed. Using the proposed similarity measures, two new aggregation operators, viz., Pythagorean fuzzy-dependent averaging operator and Pythagorean fuzzy-dependent geometric operator, are developed. The advantage of using these operators is that the influence of unfair arguments of aggregated results could be eliminated, since the associated weights are taken from the aggregated Pythagorean fuzzy arguments. Also, the proposed operators have the capability to adjust the degree of aggregated arguments with the controlling parameters. To establish the application potentiality of those operators, a methodology for solving multicriteria group decision-making problems having Pythagorean fuzzy arguments is developed. A numerical example is provided to demonstrate the proficiency of the proposed method. The achieved results are compared with the results of other existing technique.

Polygonal approximation of digitized curves is an important step in shape representation and analysis. One of the approaches to obtain a good polygonal approximation is that of formulating the approximation task as an optimization problem and then solving it using the available tools. Adopting this strategy, in this article, we present a differential evolution-based method for polygonal approximation. We show that our method not only yields near-optimal results, but also does so very efficiently. The proposed method is also a demonstration of the successful application of differential evolution for solving a combinatorial optimization problem.

Abstract Almost all entities—animate or inanimate—that we see around us change with time. The changes are brought about by changes in the values of their attributes. By using a set of parameters to represent the variable attributes of an entity, and by suitably manipulating their values at run time, the behaviour of an entity can be broadly mimicked in animation. The majority of entities, however, are all too complex to animate directly. They are better described in terms of nested layers of smaller and simpler entities, which we call components. Each component is structurally and behaviourally complete and can be described independent of its application. In the present paper, we propose a scheme for 3D animation that broadly follows this line. The keystone of this scheme is a language, nicknamed ‘V’, which defines the structural and visual attributes of each component of the scene and associates a parameterized behaviour with it, if necessary, in the form of a program script. Thereafter, wherever such a component appears, it does so with a built‐in behaviour, which can nevertheless be regulated by its higher‐level component through its parameters. The advantage is that an entire animation can be modelled in a declarative fashion in terms of nested components with embedded behaviour. Besides, each component is easy to write, alter and reuse. The effort for development, debugging and maintenance of animation modelled in this way is much less as the concerns are almost always local. Copyright © 2002 John Wiley & Sons, Ltd.