Mathematical models for the cutting stock with limited open stacks problem

This paper describes a biased random‐key genetic algorithm (BRKGA) for the minimization of the open stacks problem (MOSP). The MOSP arises in a production system scenario, and consists of determining a sequence of cutting patterns that minimize the maximum number of open stacks during the cutting process. The proposed approach combines a BRKGA and a local search procedure for generating the sequence of cutting patterns. A novel fitness function for evaluating the quality of the solutions is also developed. Computational tests are presented using available instances taken from the literature. The high quality of the solutions obtained validate the proposed approach.

O problema de geração de padrões de corte (ou problema de corte de estoque) consiste em determinar o conjunto de padrões em que unidades demandadas (itens) são cortadas de unidades maiores (objetos) tal que, por exemplo, o custo ou a perda de material é minimizado. O problema de seqüenciamento de padrões de corte consiste em determinar a seqüência em que os padrões são cortados tal que, por exemplo, o número máximo de pilhas abertas (pilhas de itens com demanda apenas parcialmente produzida, que ainda serão cortados de um ou mais padrões seguintes nessa seqüência) é minimizado. Em geral, uma boa solução para o problema de geração de padrões não corresponde a uma boa solução para o problema de seqüenciamento de padrões e vice-versa. Esses dois problemas são freqüentemente resolvidos, tanto na prática como na literatura, de forma independente e sucessiva. Este trabalho apresenta três abordagens heurísticas para resolver de forma integrada os problemas de geração e seqüenciamento de padrões, considerando o trade-off entre os objetivos envolvidos. Embora essas abordagens possam ser aplicadas para problemas de corte e empacotamento de qualquer dimensão, neste trabalho elas são analisadas e comparadas apenas para o caso de corte unidimensional.

Neste artigo é tratado o clássico problema de corte de estoque bidimensional, cuja solução consiste em um conjunto de padrões de corte que otimiza uma função objetivo, por exemplo, a perda de material. Porém, os padrões de corte podem ser seqüenciados de modo que um outro objetivo também seja otimizado, como, por exemplo, o número máximo de pilhas abertas de itens (uma pilha é aberta quando um tipo de item é cortado pela primeira vez e fechada quando todos os itens deste tipo foram cortados). Uma boa solução para o problema de geração de padrões de corte freqüentemente não resulta numa boa solução para o problema de sequenciamento de padrões de corte, e vice-versa. Em geral, esses dois problemas são abordados, tanto na prática como na literatura, de forma independente e sucessiva. Pileggi et al. (2005) propuseram abordagens heurísticas para resolver esses dois problemas de forma integrada, considerando o trade-off entre os objetivos envolvidos, e analisaram o caso de corte unidimensional (p.e., corte de barras). No presente trabalho estas abordagens são estendidas e aplicadas para analisar o caso de corte bidimensional guilhotinado (p.e., corte de chapas). Resultados computacionais são apresentados para exemplos gerados aleatoriamente e para um exemplo real de uma fábrica de móveis.

Any ultrafast optical spectroscopy experiment is usually accompanied by the necessary routine of ultrashort-pulse characterization. The majority of pulse characterization approaches solve either a one-dimensional (e.g., via interferometry) or a two-dimensional (e.g., via frequency-resolved measurements) problem. Solution of the two-dimensional pulse-retrieval problem is generally more consistent due to the problem's over-determined nature. In contrast, the one-dimensional pulse-retrieval problem, unless constraints are added, is impossible to solve unambiguously as ultimately imposed by the fundamental theorem of algebra. In cases where additional constraints are involved, the one-dimensional problem may be possible to solve, however, existing iterative algorithms lack generality, and often stagnate for complicated pulse shapes. Here we use a deep neural network to unambiguously solve a constrained one-dimensional pulse-retrieval problem and show the potential of fast, reliable and complete pulse characterization using interferometric correlation time traces determined by the pulses with partial spectral overlap.

In this paper, we present a power control algorithm in cognitive radio multiple-input–multiple-output (MIMO) system over time-varying fading channels based on the utility-based framework. In particular, a more general objective function framework is proposed under a long- or short-term transmitting and interference power constraints for secondary users, which results the optimization problem is nonconvex. A stochastic optimization algorithm is proposed based on an improved quantum-behaved particle swarm optimization (IQPSO), which can solve the nonconvex optimization problem efficiently. The acceleration coefficients are adjusted with time-varying equations which can help the particles jump out the local optimum. Due to the characteristics of wave function of quantum Delta potential well model, IQPSO can get better optimal solutions in search space for the stochastic optimization problem. To show the efficiency of the proposed algorithm, the utilities of the MIMO CR system got by IQPSO are compared with other approaches in the literature in different case studies, which show that the proposed algorithm can solve the nonconvex problem efficiently and achieve significant throughput.

This thesis treats different aspects of the class of Mixed-Integer Optimal Control Problems (MIOCPs). These are optimization problems that combine the difficulties of underlying dynamic processes with combinatorial decisions. Typically, these combinatorial decisions are realized as switching decisions between the system’s different operations modes. During the last decades, direct methods emerged as the state-of-the-art solvers for MIOCPs. The formulation of a valid, tight and dependable integral relaxation, i.e., the formulation of a model for fractional values, plays an important role for these direct solution methods. We give detailed insight into several relaxation approaches for MIOCPs and compare them with regard to their respective structures. In particular, these are the typical solution’s structures and properties as convexity, problem size and numerical behavior. From these structural properties, we deduce some required specifications of a solver. Additionally, the modeling and subsequent limitation of the switching process directly tackle the class-specific typical issue of chattering solutions. One of the relaxation methods for MIOCPs is the outer convexification, where the binary variables only enter affinely. For the approximation of this relaxation’s solution, we took up on the control approximation problem in integral sense derived by Sager as part of a decomposition approach for MIOCPs with affine binary controls. This problem describes the optimal approximation of fractional controls with binary controls such that the corresponding dynamic process is changed as little as possible. For the multi-dimensional problem, we developed a new heuristic, which for the first time gives a bound that only depends on the control grid and not anymore on the number of the system’s controls. For the generalization of the control approximation problem with additional constraints, we derived a tailored branch-and-bound algorithm, which is based on the properties of the Lagrangian relaxation of the one-dimensional problem. This algorithm beats state-of-the-art commercial solvers for Mixed-Integer Linear Programs (MILPs) for this special approximation problem by several orders of magnitude. Overall, we present several, partially new modeling approaches for MIOCPs together with the accompanying structural properties. On this basis, we develop new theories for the approximation of certain relaxed solutions. We discuss the efficient implementation of the resulting structure exploiting algorithms. This leads to a deeper and better understanding of MIOCPs. We show the practicability of the theoretical observations with the help of four prototypical problems. The presented methods and algorithms allow on their basis the direct development of decision support and analysis tools in practice.

Orchard Algorithm (OA): A new meta-heuristic algorithm for solving discrete and continuous optimization problems

Range gate pull-off (RGPO) jamming is an electronic countermeasure widely used to fool radar tracking systems. Nevertheless, the research on its strategy optimization has only been tepid. Optimization model, appropriate performance metrics, as well as efficient algorithms are required to further the research in this field. The algebraic description for the objective function of the optimization of the RGPO jamming strategy is difficult to obtain, and the jamming results are not deterministic under the influence of the input noise. To address these issues, this paper models the generation of the RGPO jamming strategy as a stochastic simulation optimization (SSO) problem, and proposes an optimization algorithm of the RGPO jamming strategy with the help of SSO technologies. We propose the committee based active learning (CAL) assisted optimal computing budget allocation based particle swarm optimization (CALPSO-OCBA) to alleviate the competition between solution space search and candidate solution performance evaluation by embedding CAL into PSO-OCBA. To improve the feasibility of the proposed algorithm, a scoring scheme that does not rely on the internal knowledge of the radar tracking system (i.e., tracking model, tracking method and tracking parameters) is designed. In addition, we use four most widely used tracking problems as the benchmarks for testing the proposed optimization algorithm of the RGPO jamming strategy. Experimental results demonstrate that the proposed algorithm is highly competitive in solving the optimization problem of the RGPO jamming strategy.

Genetic algorithms are adaptive methods based on natural evolution that may be used for search and optimization problems. They process a population of search space solutions with three operations: selection, crossover, and mutation. Under their initial formulation, the search space solutions are coded using the binary alphabet, however other coding types have been taken into account for the representation issue, such as real coding. The real-coding approach seems particularly natural when tackling optimization problems of parameters with variables in continuous domains. A problem in the use of genetic algorithms is premature convergence, a premature stagnation of the search caused by the lack of population diversity. The mutation operator is the one responsible for the generation of diversity and therefore may be considered to be an important element in solving this problem. For the case of working under real coding, a solution involves the control, throughout the run, of the strength in which real genes are mutated, i.e., the step size. This paper presents TRAMSS, a Two-loop Real-coded genetic algorithm with Adaptive control of Mutation Step Sizes. It adjusts the step size of a mutation operator applied during the inner loop, for producing efficient local tuning. It also controls the step size of a mutation operator used by a restart operator performed in the outer loop, for reinitializing the population in order to ensure that different promising search zones are focused by the inner loop throughout the run. Experimental results show that the proposal consistently outperforms other mechanisms presented for controlling mutation step sizes, offering two main advantages simultaneously, better reliability and accuracy.

Low-thrust interplanetary spacecraft trajectory optimization poses a uniquely difficult problem to solve because of the inherent nonlinearities of the dynamics and constraints as well as the large size of the search space of possible solutions. Tools currently exist that optimize low-thrust interplanetary trajectories, but these tools are rarely openly available to the public, and when they are available they require multiple interfaces between multiple different packages. The goal of this work is to present a new piece of low-thrust interplanetary spacecraft trajectory optimization software that is open-source and entirely self-contained so that more people can have access to the ability to design interplanetary trajectories. To achieve this goal, a gradient-descent based nonlinear programming method, called the interior point method, was used. The nonlinear programming method was chosen so that results from this work could be compared and contrasted with results from Spacecraft Trajectory Optimization Suite (STOpS), which uses heuristics to iterate towards a solution. Interior point methods are popular because of their ability to handle large amounts of equality and inequality constraints, which is a characteristic that is valuable for low-thrust interplanetary spacecraft trajectories. The software developed, Interior Point Optimizer (IP Optimizer), was then validated against test cases with known solutions to ensure that the software delivered the intended results. Lastly, a constraint satisfaction, a minimum-time, and a maximum-final-mass optimization problem were solved and compared with literature to illustrate the advantages of IP Optimizer and the methods it employs. For the constraint satisfaction problem, IP Optimizer was able to find a solution that exactly satisfied the desired terminal constraints whereas STOpS had an error of 2.29 percent. In this case, IP Optimizer had a reduced runtime of 15 percent compared to STOpS as well. When minimizing time for a spacecraft transfer, IP Optimizer improved upon the solution found by STOpS by 5.3 percent. The speed of convergence for IP Optimizer was almost twice as fast as STOpS for this case. These results show that IP Optimizer is faster than STOpS at converging on a solution and the solution it converges to has a better objective value and more accurately satisfies the terminal constraints than STOpS. Lastly, the maximum-final-mass problem resulted in an objective value that was only 0.5 percent lower than the value found in literature.

In this paper, we present an adaptive coevolutionary memetic algorithm (ACMA) for examination timetabling problems. In our proposed algorithm, the evolutionary search is conducted in two spaces: the heuristic space and the solution space. In the heuristic space, a hyper-heuristic approach is utilised to generate the initial population, and then basic evolutionary operators are used to find the potential heuristic lists. The evolutionary strategy in the heuristic space is regarded as a global search procedure. In the solution space, according to the solution structure, some specific evolutionary operators are designed for expanding the scope of search in solution space. This scheme can be viewed as the local search procedure. By combining two different strategies, the cooperation between them will eventually increase the diversities in the population. In order to determine which space should be selected at each generation, an adaptive parameter is designed based on the proportion of feasible solutions in the current population. Experimental results demonstrated that ACMA obtains competitive results and outperforms the compared approaches on some benchmark instances.

The no-wait flow-shop scheduling problem (NFSSP) with sequence-dependent setup times (SDSTs) and release times (RTs) is applicable in many areas, such as steel production, food processing, and chemical processing. Estimation of the distribution algorithm (EDA) has recently been recognized as a prominent metaheuristic methodology in the field of evolutionary computation due to its excellent performance of global exploration. In this article, an innovative matrix-cube-based (i.e., 3-D) EDA (MCEDA) is first proposed to minimize the total earliness and tardiness (TET) of the NFSSP with SDSTs and RTs. This problem is NP-hard in the strong sense. First, a 3-D matrix cube is devised to learn the valuable information from promising solutions or excellent individuals. Second, an EDA model or probabilistic model based on the matrix cube and a special sampling method is presented to perform effective exploration in solution space and find promising regions. Third, based on a series of newly defined subneighborhoods, a new local search with both a speed-up scanning method and one search strategy is developed to execute exploitation from promising regions. Fourth, a speed-up evaluation method based on the problem’s property is designed to reduce the computational complexity for calculating criterion and accelerate the search process. Owing to the reasonable hybridization of exploration and exploitation, MCEDA can perform very efficient search in solution space. Extensive test results on instances of such a just-in-time problem first show that MCEDA can achieve better solution than state-of-the-art algorithms in obviously less computation time. Additional experiments on instances of various NFSSPs further confirm the efficiency and robustness of MCEDA.

The numerical solution of optimization problems with partial differential equation (PDE) constraints is vital to a growing number of science and engineering applications. The development of robust and efficient algorithms for the solution of these optimization problems presents many challenges that arise out of, e.g., the intricate mathematical structure of these problems, the complicated interactions between numerical methods for PDE and optimization, the large-scale of the optimization problems, and the increasing complexity of applications. To identify and overcome these challenges an integrated approach is needed that builds on a variety of mathematical sub-disciplines, such as theory of PDEs, distributed parameter systems, numerical solution of PDEs, numerical optimization, and numerical linear algebra. This international workshop has brought together some of the leading experts in the fast developing field of optimization problems with PDE constraints to present recent developments in this area as well as to identify open problems and further research needs. Among the themes of this workshop were the design and analysis of approaches for the solution of PDE constrained optimization problems with additional point-wise constraints on controls and states (the solution of the governing PDE). State constrained problems are particularly challenging because of the low regularity properties of the Lagrange multipliers associated with point-wise constraints on the states. A second theme was the development of adaptive methods for the solution of PDE constrained optimization problems and, more generally, the development of optimization level model reduction techniques for these problems. The goal here is to develop models (through, e.g., mesh adaptation or proper orthogonal model reduction) of the PDE constrained optimization problems that capture the relevant features of the optimization problems with a specified accuracy, but involve as few degrees of freedom as possible and, hence, are computationally less expensive to work with. A third theme was The efficient solution of linear systems arising in optimization algorithms for discretized PDE constrained optimization problems represented another theme. Finally, a number of talks presented advances and challenges in the solution of PDE constrained optimization problems arising in important science and industrial applications.

Dividing a Sphere Hierarchically into a Large Number of Spherical Pentagons Using Equal Area or Equal Length Optimization

Particle swarm optimization (PSO) is a swarm intelligence based computer algorithm that is used to find a solution in the search space of an optimization problem. Adding frequency diversity, through subcarrier redundancy, in orthogonal frequency-division multiplexing (OFDM) is a popular approach to improve the robustness of the system. However, frequency redundant OFDM system is prone to high peak-to-average power ratio (PAPR), due to the fact that the same source information is transmitted on multiple subcarriers. Existing schemes such as Selective Mapping (SLM) and partial transmit sequence (PTS) are effective but difficult to implement due to the high computation complexity. In this paper, we propose a two stage PAPR reduction method. We analyze the computational complexity and extensive simulations on the PAPR and show that our scheme considerably reduces the computational complexity while achieving similar PAPR reduction as SLM and better PAPR reduction than PTS.