On Guillotine Cutting Sequences

Color-spanning localized query

Many cutting problems on two- or three-dimensional objects require that the cuts be orthogonal and of guillotine type. However, there are applications in which the cuts must be orthogonal but need not be of guillotine type. In this paper we focus on the latter type of cuts on rectangular bins. We investigate the so-called \(L\)-approach, introduced by Lins et al. (J Oper Res Soc 54:777–789, 2003), which has been used to tackle, among others, the pallet loading problem and the two-dimensional unconstrained knapsack problem. This approach concerns a method to generate two \(L\)-shaped pieces from an \(L\)-shaped piece. More recently, Birgin et al. (J Oper Res Soc 63(2):183–200, 2012), raised two questions concerning this approach. The first question is whether one may restrict only to cuts on raster points (rather than on all discretization points), without loss in the quality of the solution. We prove that the answer to this question is positive. The second question is whether the \(L\)-approach is an optimal method to solve the unconstrained knapsack problem. We show an instance of the problem for which this approach fails to find an optimum solution.

The two-dimensional knapsack problem with splittable items in stacks

The Floating-Cuts model: a general and flexible mixed-integer programming model for non-guillotine and guillotine rectangular cutting problems

The area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+epsilon)-approximations in f(k,epsilon)n^O(1) time where k is some parameter of the input. The goal is to overcome lower bounds from either of the areas. We obtain the following results on parameterized approximability: - In the maximum independent set of rectangles problem (MISR) we are given a collection of n axis parallel rectangles in the plane. Our goal is to select a maximum-cardinality subset of pairwise non-overlapping rectangles. This problem is NP-hard and also W[1]-hard [Marx, ESA'05]. The best-known polynomial-time approximation factor is O(log log n) [Chalermsook and Chuzhoy, SODA'09] and it admits a QPTAS [Adamaszek and Wiese, FOCS'13; Chuzhoy and Ene, FOCS'16]. Here we present a parameterized approximation scheme (PAS) for MISR, i.e. an algorithm that, for any given constant epsilon>0 and integer k>0, in time f(k,epsilon)n^g(epsilon), either outputs a solution of size at least k/(1+epsilon), or declares that the optimum solution has size less than k. - In the (2-dimensional) geometric knapsack problem (2DK) we are given an axis-aligned square knapsack and a collection of axis-aligned rectangles in the plane (items). Our goal is to translate a maximum cardinality subset of items into the knapsack so that the selected items do not overlap. In the version of 2DK with rotations (2DKR), we are allowed to rotate items by 90 degrees. Both variants are NP-hard, and the best-known polynomial-time approximation factor is 2+epsilon [Jansen and Zhang, SODA'04]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA'15]. We show that both variants are W[1]-hard. Furthermore, we present a PAS for 2DKR. For all considered problems, getting time f(k,epsilon)n^O(1), rather than f(k,epsilon)n^g(epsilon), would give FPT time f'(k)n^O(1) exact algorithms by setting epsilon=1/(k+1), contradicting W[1]-hardness. Instead, for each fixed epsilon>0, our PASs give (1+epsilon)-approximate solutions in FPT time. For both MISR and 2DKR our techniques also give rise to preprocessing algorithms that take n^g(epsilon) time and return a subset of at most k^g(epsilon) rectangles/items that contains a solution of size at least k/(1+epsilon) if a solution of size k exists. This is a special case of the recently introduced notion of a polynomial-size approximate kernelization scheme [Lokshtanov et al., STOC'17].

Greedy algorithms for a class of knapsack problems with binary weights

This paper presents the results of an empirical study of the effects of coefficient correlation structure and constraint slackness settings on the performance of solution procedures on synthetic two-dimensional knapsack problems (2KP). The population correlation structure among 2KP coefficients, the level of constraint slackness, and the type of correlation (product moment or rank) are varied in this study. Representative branch-and-bound and heuristic solution procedures are used to investigate the influence of these problem parameters on solution procedure performance. Population correlation structure, and in particular the interconstraint component of the correlation structure, is found to be a significant factor influencing the performance of both the algorithm and the heuristic. In addition, the interaction between constraint slackness and population correlation structure is found to influence solution procedure performance.

Temporal upscaling of instantaneous evapotranspiration: An intercomparison of four methods using eddy covariance measurements and MODIS data

In 1960, Asplund and Grünbaum proved that every intersection graph of axis-parallel rectangles in the plane admits an O(ω2)-coloring, where ω is the maximum size of a clique. We present the first asymptotic improvement over this six-decade-old bound, proving that every such graph is O(ω log ω)-colorable and presenting a polynomial-time algorithm that finds such a coloring. This improvement leads to a polynomial-time O(log log n)-approximation algorithm for the maximum weight independent set problem in axis-parallel rectangles, which improves on the previous approximation ratio of .

This chapter begins with an explanation about the importance of modeling and optimization of manufacturing processes not only from the scientific and researching point of view but also for practical industrial applications. Then it introduces the hybrid approach which combines artificial intelligence tools and finite element method for these modeling and optimization tasks. The advantages and shortcomings of each of these techniques are exposed, highlighting the convenience of combining both methods, increasing the robustness and flexibility. Furthermore, the different approaches for combining artificial intelligence and finite element method in modeling and optimization of manufactured processes are outlined and preliminarily evaluated. 1.1 Relevance and Convenience of Hybrid Modeling and Optimization of Manufacturing Processes Modeling of the physical phenomena involved in manufacturing processes (machining, forming, foundry, etc.) has been recognized as one of the most important tasks in manufacturing research. Accurate and realistic mathematical models not only allow understanding how these phenomena take place, but also facilitate the development of new manufacturing processes. Knowledge about the quantitative relationships between the different parameters involved in manufacturing processes also permits the implementation of effective monitoring and control system, which are indispensable in the high automated modern industry. Moreover, the use of highly optimized manufacturing processes is widely accepted as a necessary condition for achieving effectiveness, efficiency and economic competitiveness in manufacturing workshops. R. Quiza et al., Hybrid Modeling and Optimization of Manufacturing, SpringerBriefs in Computational Mechanics, DOI: 10.1007/978-3-642-28085-6_1, The Author(s) 2012 1 Unfortunately, the physical nature of the phenomena underlying the manufacturing processes is not easy to understand, as they involve complex nonlinear relationships which are no completely explained up to date. This situation worsens with the use of modern materials for parts and tools. For example, for many years, the tool life of high speed steels cutting tools, working at relatively low speeds, was described by the well-known Taylor’s law (Childs et al. 2000) with an acceptable error level. However, the introduction of multi-coated carbides and PCBN tools, at high cutting speeds and hard conditions has made the Taylor’s law useless or, at less, extremely limited (Dolinsek et al. 2001). Two main approaches can be used in modeling manufacturing processes (see Fig. 1.1). On one hand, the phenomenological modeling is based on the identification and mathematical description of the physical phenomena. It has the advantage of being more realistic and accurate. Also they help to understand the mechanisms of these phenomena. This mathematical description comes in the form of expressions, usually differential equations which can be solve analytically only for a limited set of simple problems. For most of the real problems, these equations (especially partial differential equations) have not analytical solutions and approximated numerical solutions must be obtained instead. Between the numerical methods used for solving partial differential equation, the finite element method (FEM) has reach the higher application levels, because of its ability for being applied to problems defined over complex spatial domains and the relative simplicity of its computational implementation (Dixit and Dixit 2008). FEM has been successfully applied to a wide variety of manufacturing processes, such as machining (Arrazola and Ozel 2010; Mamalis et al. 2008), forming (Gudur and Dixit 2008; Shahani et al. 2009), welding (Anca et al. 2011) and foundry (Lewis et al. 2005). Sometimes, the phenomenological models are too much idealized for giving results accurate enough for being used in practical applications. In these cases, empirical models, based on correlation of experimental data, play a crucial role. The main drawback of this kind of models is its incapability for identifying or explaining the physical relationship between the involved variables. However, they usually offer accurate outcomes for industrial applications and had been widely used in optimization of manufacturing processes. Traditionally, the mathematical techniques used for correlating data in empirical models were those based on statistics. These tools are quite simple and are Fig. 1.1 Classification of modeling techniques 2

Online auction is the essence of many modern markets, particularly networked markets, in which information about goods, agents, and outcomes is revealed over a period of time, and the agents must make irrevocable decisions without knowing future information. Optimal stopping theory, especially the classic secretary problem, is a powerful tool for analyzing such online scenarios which generally require optimizing an objective function over the input. The secretary problem and its generalization the multiple-choice secretary problem were under a thorough study in the literature. In this paper, we consider a very general setting of the latter problem called the submodular secretary problem, in which the goal is to select k secretaries so as to maximize the expectation of a (not necessarily monotone) submodular function which defines efficiency of the selected secretarial group based on their overlapping skills. We present the first constant-competitive algorithm for this case. In a more general setting in which selected secretaries should form an independent (feasible) set in each of l given matroids as well, we obtain an O(l log2 r)-competitive algorithm generalizing several previous results, where r is the maximum rank of the matroids. Another generalization is to consider l knapsack constraints (i.e., a knapsack constraint assigns a nonnegative cost to each secretary, and requires that the total cost of all the secretaries employed be no more than a budget value) instead of the matroid constraints, for which we present an O(l)-competitive algorithm. In a sharp contrast, we show for a more general setting of subadditive secretary problem, there is no O(√n)-competitive algorithm and thus submodular functions are the most general functions to consider for constant-competitiveness in our setting. We complement this result by giving a matching O(√n)-competitive algorithm for the subadditive case.

Effective solution of the fundamental quasi-periodic problems of the theory of elasticity for a plane with cuts distributed along a straight line

In this paper, a hybrid algorithm by combining 15 task assignment rules and ant colony optimization (ACO) algorithm to solve straight and U-type assembly line balancing problem to optimize line efficiency and smoothness index is proposed. The algorithm is designed to solve assembly line balancing problems of all sizes. The proposed multi-rule multiobjective ant colony optimization algorithm for straight and U-type assembly line balancing problems is evaluated with various set of benchmark problems and compared with a multi-objective simulated annealing algorithm reported in the literature. The results indicate the better performance of the proposed hybrid algorithm.

The prize-collecting dominating set (PCDS) problem is the generalization of the minimum dominating set (MDS) problem. The MDS problem requires a dominating set in a given graph, namely a subset of vertexes. Any vertex in the graph belongs to the closed neighborhood of the subset of vertexes, where the subset of vertexes with the smallest cardinality is the MDS. In the PCDS problem: given an undirected graph, its vertex setV has nonnegative weighting functionω and nonnegative penalty functionπ . Finding a vertex subset D⊂V, for vertexes do not belong to the closed neighborhood of D , we need to pay penalties for them. The objective of this issue is to minimize the sum of weights and penalties. As we all know that the MDS problem is NP-hard in general graph, obviously, PCDS problem is NP-hard, too., In certain cases, the PCDS problem can achieve better results than the MDS problem, which has a certain value in practical applications such as facility location and logistics management. Whether the problem is limited to a special graph will get useful structural characterization and corresponding algorithm results is considered, designed polynomial time exact algorithm on the star and according to the r -LMP approximation algorithm, designed the 2-LMP approximation algorithm on the path and cycle.

Covering and packing of triangles intersecting a straight line