International Journal of Industrial Engineering & Production Research

 Abstract : A visual cryptography scheme based on a given graph G is a method to distribute a secret image among the vertices of G, the participants, so that a subset of participants can recover the secret image if they contain an edge of G, by stacking their shares, otherwise they can obtain no information regarding the secret image. In this paper a maximal independent sets of the graph G was applied to propose a lower bound on the pixel expansion of visual cryptography schemes with graph access structure  (G ). In addition a lower bound on the pixel expansion of basis matrices C₅ and Peterson graph access structure were presented .

The eternal need for humans' blood as a critical commodity makes the healthcare systems attempt to provide efficient blood supply chains (BSCs) by which the requirements are satisfied at the maximum level. To have an efficient supply of blood, an appropriate planning for blood supply chain is a challenge which requires more attention. In this paper, we address a mixed integer linear programming model for blood supply chain network design (BSCND) with the need for making both strategic and tactical decisions throughout a multiple planning periods. A robust programming approach is devised to deal with inherent randomness in parameters data of the model. To illustrate the usefulness of the model as well as its solution approach, it is tested into a set of numerical examples, and the sensitivity analyses are conducted. Finally, we employ two criteria: the mean and standard deviation of constraint violations under a number of random realizations to evaluate the performance of both the proposed robust and deterministic models. The results imply the domination of robust approach over the deterministic one.