In this paper we study and solve two different variants of static knapsack problems with random weights: The stochastic knapsack problem with simple recourse as well as the stochastic knapsack problem with probabilistic constraint. Special interest is given to the corresponding continuous problems and three different problem solving methods are presented. The resolution of the continuous problems allows to provide upper bounds in a branch-and-bound framework in order to solve the original problems. Numerical results on a dataset from the literature as well as a set of randomly generated instances are given.
Chapter the two symmetric subcases that either e lies in some cycle space element, and when we delete e from that, it becomes a cut (this is the statement in (ii)), or else e lies in some cut, and when we delete e from it, this edge set becomes an element of the cycle space (which corresponds to (iii)). If we now naively use the finite version of the cycle space on an infinite, locally finite graph, every element of the cycle space will necessarily be finite, and we will see that this theorem fails for locally finite graphs. One In Chapter 6 we will show that this is also true for locally finite graphs. See also Richter and Shank [22] and Lins, Richter and Shank [19]. In finite graphs, Archdeacon, Bonnington and Little [1] used ladders (which are certain substructures involving left-right tours and bicycles in an unusual way) to give a criterion for planarity. Planarity criteria are very important tools, and this one in particular is purely algebraical and requires
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