Shape preservation has been the heart of subdivision schemes (SSs) almost from its origin, and several analyses of SSs have been established. Shape preservation properties are commonly used in SSs and various ways have been discovered to connect smooth curves/surfaces generated by SSs to applied geometry. With an eye on connecting the link between SSs and applied geometry, this paper analyzes the geometric properties of a ternary four-point rational interpolating subdivision scheme. These geometric properties include monotonicity-preservation, convexity-preservation, and curvature of the limit curve. Necessary conditions are derived on parameter and initial control points to ensure monotonicity and convexity preservation of the limit curve of the scheme. Furthermore, we analyze the curvature of the limit curve of the scheme for various choices of the parameter. To support our findings, we also present some examples and their graphical representation.

Closed-loop supply chain networks are gaining research popularity due to environmental, economic and social concerns. Such networks are primarily designed to overcome carbon footprints and to retrieve end of life products from customers. This study considers a multi echelon closed-loop supply chain in the presence of machine disruption. A multi-objective model is presented to optimize the total cost, the total time and emissions in a closed-loop supply chain network. The aim is to analyze the trade-off between the objectives of cost, time, and emissions and how these decisions are impacted by the selection of different available machines. A number of solution approaches are tested on a case study from the tire industry. The results suggest the improved performance of the hybrid heuristic and the importance of controlling disruption in a closed-loop supply chain network. Furthermore, there is a trade-off between the different objective functions which can help the decision maker to choose a particular solution according to the preference of an organization. Finally, conclusion and future research avenues are provided.

In this paper, we analyze shape-preserving behavior of a relaxed four-point binary interpolating subdivision scheme. These shape-preserving properties include positivity-preserving, monotonicity-preserving and convexity-preserving. We establish the conditions on the initial control points that allow the generation of shape-preserving limit curves by the four-point scheme. Some numerical examples are given to illustrate the graphical representation of shape-preserving properties of the relaxed scheme.

In this paper, a generalized algorithm to develop a class of approximating binary subdivision schemes is presented. The proposed algorithm is based on three-point approximating binary and four-point interpolating binary subdivision schemes. It contains a parameter which classifies members of the new class of subdivision schemes. A set of efficient properties, for instance, polynomial generation and reproduction, support, continuity, and Hölder continuity, is discussed. Moreover, applications of the proposed subdivision schemes are given in order to demonstrate their variety, flexibility, and visual performance.

The main objective of this research is to analyze some geometrical properties of a quaternary four-point interpolatory subdivision scheme ([ ]-scheme) with a shape parameter and then find the precise range of the shape parameter of the [ ]-scheme to generate fractal and convexity of limit curves. That allows the curve to change easily and be more flexible without altering its control points. Therefore, by taking the various values of tension parameters, the curve still preserves its characteristics and geometrical configuration. These geometric modeling examples show that our techniques can be easily performed, and they can also provide us with an alternative strong strategy for the modeling of complex figures.

This paper introduces a new class of stationary refinement algorithms that combine the advantages of the four-point interpolatory algorithm and the cubic B-spline. This algorithm includes a tension parameter. This analysis identifies the range of this parameter and specifies conditions under which the proposed algorithm maintains shape-preserving properties such as monotonicity and convexity. The algorithm achieves improved smoothness, reaching C3 and fourth-order accuracy, while maintaining the same support length as both parent algorithms. Unlike many high-order algorithms that are nonlinear and complex, this method remains simple and efficient. Numerical examples are provided to demonstrate its practical performance, and the proposed algorithm is particularly well-suited for a discontinuous type of data set.