n. mahdavi-amiri

In 1984, Abaffy, Broyden, and Spediacto (ABS) introduced a class of the so-called ABS algorithms to solve systems of real linear equations. Later, the scaled ABS, the extended ABS, the block ABS, and the integer ABS algorithms were introduced leading to various well-known matrix factorizations. Here, we present a generalization of ABS algorithms containing all matrix factorizations such as triangular, W Z, and ZW . We discuss the octant interlocking factorization and make use of the generalized ABS algorithm as a more general approach for producing the octant interlocking factorization.

We study a multi-time, multi-product and multi-echelon supply chain aggregate procurement, production and distribution planning problem and discuss the implications of formulating a tri-level model to integrate procurement, production and distribution, maintaining the existing hierarchy in the decision process. In our model, there are three different decision makers controlling the procurement, production and the distribution processes in the absence of cooperation because of different optimization strategies. First, we present a hierarchical tri-level programming model to deal with decentralized supply chain problems. Then, an algorithm is presented to solve the proposed model. A numerical illustration is provided to show the applicability of the optimization model and the proposed algorithm. In order to evaluate the application of the model and the proposed algorithm, ten sets of small and large problems are randomly generated and tested. The experimental results show that our proposed fuzzy-stochastic simulation based hierarchical interactive particle swarm optimization (Sim-HIPSO) performs well in finding good approximate solutions within reasonable computation times.