We present an explicit formula for the mask of odd pointsn-ary, for any oddn⩾3, interpolating subdivision schemes. This formula provides the mask of lower and higher arity schemes. The 3-point and 5-pointa-ary schemes introduced by Lian, 2008, and (2m+1)-pointa-ary schemes introduced by, Lian, 2009, are special cases of our explicit formula. Moreover, other well-known existing odd pointn-ary schemes including the schemes introduced by Zheng et al., 2009, can easily be generated by our formula. In addition, error bounds between subdivision curves and control polygons of schemes are computed. It has been noticed that error bounds decrease when the complexity of the scheme decreases and vice versa. Also, as we increase arity of the schemes the error bounds decrease. Furthermore, we present brief comparison of total absolute curvature of subdivision schemes having different arity with different complexity. Convexity preservation property of scheme is also presented.

In this paper, we present the shape-preserving properties of the four-point ternary non-stationary interpolating subdivision scheme (the four-point scheme). This scheme involves a tension parameter. We derive the conditions on the tension parameter and initial control polygon that permit the creation of positivity-and monotonicity-preserving curves after a finite number of subdivision steps. In addition, the outcomes are generalized to determine conditions for positivity-and monotonicity-preservation of the limit curves. Convexity-preservation of the limit curve of the four-point scheme is also analyzed. The shape-preserving behavior of the four-point scheme is also shown through several numerical examples.

No abstract

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.

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.

We apply six-point variant on the Lane-Riesenfeld algorithm to obtain a new family of subdivision schemes. We also determine the support, smoothness, Hölder regularity, magnitude of the artifact, and the shrinkage effect due to the change of integer smoothing parameter that characterizes the members of the family. The degree of polynomial reproduction also has been discussed. It is observed that the proposed schemes have less shrinkage effect and as a result better preserve the shape of control polygon.

In this article, we present a general algorithm to generate a new class of binary approximating subdivision schemes and give derivation of some family members. We discuss important properties of derived schemes such as: convergence, continuity, Hlder regularity, degree of polynomial generation and reproduction, support, limit stencils and artifacts. Furthermore, visual performance of proposed schemes has also been presentedIn this article, we present a general algorithm to generate a new class of binary approximating subdivision schemes and give derivation of some family members. We discuss important properties of derived schemes such as: convergence, continuity, Hlder regularity, degree of polynomial generation and reproduction, support, limit stencils and artifacts. Furthermore, visual performance of proposed schemes has also been presented.