This paper deals with an identical multiprocessor, multiproduct, lot-size scheduling problem under deterministic conditions. The problem is an extension of the single-processor ELSP problem. The objective is to minimize the average production and inventory costs per unit time subject to constraints of no shortages and no interference among the products. A mathematical formulation for the problem is given. A heuristic procedure, which extends the Common Cycle approach for a single processor, is suggested and computational results are reported. Upper and Lower bounds are developed and used to measure the effectiveness of the procedure.inventory/production, deterministic models, heuristics, multiprocessors, lot-sizing, scheduling

The location of resources in a network satisfying some optimization property is a classical combinatorial problem that can be modeled and solved by using graphs. Key tools in this problem are the domination-type properties, which have been defined and widely studied in different types of graph models, such as undirected and directed graphs, finite and infinite graphs, simple graphs and hypergraphs. When the required optimization property is that every node of the network must have access to exactly one node with the desired resource, the appropriate models are the efficient dominating sets. However, the existence of these vertex sets is not guaranteed in every graph, so relaxing some conditions is necessary to ensure the existence of some kind of dominating sets, as efficient as possible, in a larger number of graphs. In this paper, we study independent [1, 2]-sets, a generalization of efficient dominating sets defined by Chellali et al., in the case of cylindrical networks. It is known that efficient dominating sets exist in very special cases of cylinders, but the particular symmetry of these graphs will allow us to provide regular patterns that guarantee the existence of independent [1, 2]-sets in every cylinder, except in one single case, and to compute exact values of the optimal parameter, the independent [1, 2]-number, in cylinders of selected sizes.

In this paper we present a lower bound for the domination number of the Cartesian product of a path and a cycle, that is tight if the length of the cycle is a multiple of five. This bound improves the natural lower bound obtained by using the domination number of the Cartesian product of two paths, that is the best one known so far.A dominating set in a graph G is a vertex subset S such that every vertex in V (G) \ S has a neighbor in S. The domination number of G is the minimum size of a dominating set of G and it is denoted by γ(G).The domination number is a classical graph parameter that has a challenging development in Cartesian product graphs. As proof of this, the computation of the domination number of the grid graphs, that is, the Cartesian product of two paths, was an open problem for almost thirty years. The final paper concerning this problem [12] closes a list of partial results regarding exact values for particular cases [1,5,7,15] and general upper and lower bounds [4,7,13], among other papers.Contrary to grids, the complete computation of the domination number of cylindrical graphs, that is, the Cartesian product of a path and a cycle, is still open. Currently, there are some partial results, see for instance [8,16,19], showing different techniques to compute exact values of particular cases, where one of the parameters, the path length or the cycle length, is small enough.We devote this paper to computing a general lower bound for the domination number of the cylindrical graphs, that is tight if the length of the cycle is a multiple of five. Our strategy to obtain this bound is similar to the approach in [13], that uses the concept of wasted domination. On the other hand, we found inspiration in [16] to design a (min, +) matrix multiplication algorithm that computes the minimum wasted domination needed for our purpose.

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