613 research outputs found

    Knapsack Problems with Side Constraints

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    The thesis considers a specific class of resource allocation problems in Combinatorial Optimization: the Knapsack Problems. These are paradigmatic NP-hard problems where a set of items with given profits and weights is available. The aim is to select a subset of the items in order to maximize the total profit without exceeding a known knapsack capacity. In the classical 0-1 Knapsack Problem (KP), each item can be picked at most once. The focus of the thesis is on four generalizations of KP involving side constraints beyond the capacity bound. More precisely, we provide solution approaches and insights for the following problems: The Knapsack Problem with Setups; the Collapsing Knapsack Problem; the Penalized Knapsack Problem; the Incremental Knapsack Problem. These problems reveal challenging research topics with many real-life applications. The scientific contributions we provide are both from a theoretical and a practical perspective. On the one hand, we give insights into structural elements and properties of the problems and derive a series of approximation results for some of them. On the other hand, we offer valuable solution approaches for direct applications of practical interest or when the problems considered arise as sub-problems in broader contexts

    Models and algorithms for decomposition problems

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    This thesis deals with the decomposition both as a solution method and as a problem itself. A decomposition approach can be very effective for mathematical problems presenting a specific structure in which the associated matrix of coefficients is sparse and it is diagonalizable in blocks. But, this kind of structure may not be evident from the most natural formulation of the problem. Thus, its coefficient matrix may be preprocessed by solving a structure detection problem in order to understand if a decomposition method can successfully be applied. So, this thesis deals with the k-Vertex Cut problem, that is the problem of finding the minimum subset of nodes whose removal disconnects a graph into at least k components, and it models relevant applications in matrix decomposition for solving systems of equations by parallel computing. The capacitated k-Vertex Separator problem, instead, asks to find a subset of vertices of minimum cardinality the deletion of which disconnects a given graph in at most k shores and the size of each shore must not be larger than a given capacity value. Also this problem is of great importance for matrix decomposition algorithms. This thesis also addresses the Chance-Constrained Mathematical Program that represents a significant example in which decomposition techniques can be successfully applied. This is a class of stochastic optimization problems in which the feasible region depends on the realization of a random variable and the solution must optimize a given objective function while belonging to the feasible region with a probability that must be above a given value. In this thesis, a decomposition approach for this problem is introduced. The thesis also addresses the Fractional Knapsack Problem with Penalties, a variant of the knapsack problem in which items can be split at the expense of a penalty depending on the fractional quantity

    A decomposition approach for multidimensional knapsacks with family-split penalties

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    The optimization of Multidimensional Knapsacks with Family-Split Penalties has been introduced in the literature as a variant of the more classical Multidimensional Knapsack and Multi-Knapsack problems. This problem deals with a set of items partitioned in families, and when a single item is picked to maximize the utility, then all items in its family must be picked. Items from the same family can be assigned to different knapsacks, and in this situation split penalties are paid. This problem arises in real applications in various fields. This paper proposes a new exact and fast algorithm based on a specific Combinatorial Benders Cuts scheme. An extensive experimental campaign computationally shows the validity of the proposed method and its superior performance compared to both commercial solvers and state-of-the-art approaches. The paper also addresses algorithmic flexibility and scalability issues, investigates challenging cases, and analyzes the impact of problem parameters on the algorithm behavior. Moreover, it shows the applicability of the proposed approach to a wider class of realistic problems, including fixed costs related to each knapsack utilization. Finally, further possible research directions are considered

    Ising formulations of many NP problems

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    We provide Ising formulations for many NP-complete and NP-hard problems, including all of Karp's 21 NP-complete problems. This collects and extends mappings to the Ising model from partitioning, covering and satisfiability. In each case, the required number of spins is at most cubic in the size of the problem. This work may be useful in designing adiabatic quantum optimization algorithms.Comment: 27 pages; v2: substantial revision to intro/conclusion, many more references; v3: substantial revision and extension, to-be-published versio

    Heuristic Solutions for Loading in Flexible Manufacturing Systems

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    Production planning in flexible manufacturing system deals with the efficient organization of the production resources in order to meet a given production schedule. It is a complex problem and typically leads to several hierarchical subproblems that need to be solved sequentially or simultaneously. Loading is one of the planning subproblems that has to addressed. It involves assigning the necessary operations and tools among the various machines in some optimal fashion to achieve the production of all selected part types. In this paper, we first formulate the loading problem as a 0-1 mixed integer program and then propose heuristic procedures based on Lagrangian relaxation and tabu search to solve the problem. Computational results are presented for all the algorithms and finally, conclusions drawn based on the results are discussed
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