Perfect codes in direct products of cycles—a complete characterization

AbstractLet be a direct product of cycles. It is known that for any r⩾1, and any n⩾2, each connected component of G contains a so-called canonical r-perfect code provided that each ℓi is a multiple of rn+(r+1)n. Here we prove that up to a reasonably defined equivalence, these are the only perfect codes that exist

Improving approximation by switching between two error functions

The key to designing a LED luminaire is the choice of secondary optics. The design process can be improved using analytical models and optimization tools. A modified approximation error (evaluation function) is introduced in the standard analytical model for spatial light distribution. It is shown that the new evaluation function provides better quality and consistency compared to the standard evaluation function, and is a better practical approach to the optimization problem

Computing the Weighted Wiener and Szeged Number on Weighted Cactus Graphs in Linear Time

Cactus is a graph in which every edge lies on at most one cycle. Linear algorithms for computing the weighted Wiener and Szeged numbers on weighted cactus graphs are given. Graphs with weighted vertices and edges correspond to molecular graphs with heteroatoms

Coordination of a Retail Supply Chain Distribution Flow

Retail supply chains are very sensitive by their nature and need to adapt to several situations with the aim to increase their reliability, flexibility and convenience. There are many factors affecting the effectiveness of a distribution flow, from perishability, capacities of storage areas, lead times, untimely deliveries and others. Because the latter heavily depend on the planned and realized distribution and not on the demand side perspective, we partially neglect them in the initial study. We focus only on the demand satisfaction, without considering any pricing policies, perishability factors, etc. Beside stochastic demand modelling we introduce the multi-objective optimization approach to cope with the minimization of transport and warehouse costs, minimization of over stock effects and the maximization of customers’ service level. Methodology used produces a set of solutions and its quality estimations in order to find the desired distribution plan that is near optimal. The paper further explains the integration of management decisions with respect to the obtained results of the modelling approach. The applicability of the model will be explained using a numerical example

The Hosoya-Wiener Polynomial of Weighted Trees

Formulas for the Wiener number and the Hosoya-Wiener polynomial of edge and vertex weighted graphs are given in terms of edge and path contributions. For a rooted tree, the Hosoya- Wiener polynomial is expressed as a sum of vertex contributions. Finally, a recursive formula for computing the Hosoya-Wiener polynomial of a weighted tree is given