Numerical-analytical algorithm to solve the equation of particle transport in plane geometry

We describe a numerical-analytical algorithm to solve the boundary-value problem for the integrodifferential equation of particle transport in a plane homogeneous medium. The general scheme of approximate solution of this problem is based on its reduction to the solution of some integral equation by summator operators of function approximation theory. Solvability conditions are established for the approximate equations and the algorithm errors are estimated. Working formulas are presented for the algorithm implemented in the form of a computer program. The summator operators in this algorithm are the algebraic interpolation operators with nodes at the extremal points of Chebyshev polynomials of first kind.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 69–76, 1987.     

Simultaneously Applying Multiple Mutation Operators in Genetic Algorithms

The mutation operation is critical to the success of genetic algorithms since it diversifies the search directions and avoids convergence to local optima. The earliest genetic algorithms use only one mutation operator in producing the next generation. Each problem, even each stage of the genetic process in a single problem, may require appropriately different mutation operators for best results. Determining which mutation operators should be used is quite difficult and is usually learned through experience or by trial-and-error. This paper proposes a new genetic algorithm, the dynamic mutation genetic algorithm, to resolve these difficulties. The dynamic mutation genetic algorithm simultaneously uses several mutation operators in producing the next generation. The mutation ratio of each operator changes according to evaluation results from the respective offspring it produces. Thus, the appropriate mutation operators can be expected to have increasingly greater effects on the genetic process. Experiments are reported that show the proposed algorithm performs better than most genetic algorithms with single mutation operators.     

Autonomous operator management for evolutionary algorithms

The performance of an evolutionary algorithm strongly depends on the design of its operators and on the management of these operators along the search; that is, on the ability of the algorithm to balance exploration and exploitation of the search space. Recent approaches automate the tuning and control of the parameters that govern this balance. We propose a new technique to dynamically control the behavior of operators in an EA and to manage a large set of potential operators. The best operators are rewarded by applying them more often. Tests of this technique on instances of 3-SAT return results that are competitive with an algorithm tailored to the problem.     

An inverse problem for Sturm–Liouville differential operators with deviating argument

We consider second-order functional differential operators with a constant delay. Properties of their spectral characteristics are obtained and a nonlinear inverse problem is studied, which consists in recovering the operators from their spectra. We establish the uniqueness and develop a constructive algorithm for solution of the inverse problem.     

A proximal point method for the sum of maximal monotone operators

In this paper, we concentrate on the maximal inclusion problem of locating the zeros of the sum of maximal monotone operators in the framework of proximal point method. Such problems arise widely in several applied mathematical fields such as signal and image processing. We define two new maximal monotone operators and characterize the solutions of the considered problem via the zeros of the new operators. The maximal monotonicity and resolvent of both of the defined operators are proved and calculated, respectively. The traditional proximal point algorithm can be therefore applied to the considered maximal inclusion problem, and the convergence is ensured. Furthermore, by exploring the relationship between the proposed method and the generalized forward‐backward splitting algorithm, we point out that this algorithm is essentially the proximal point algorithm when the operator corresponding to the forward step is the zero operator. Copyright © 2013 John Wiley & Sons, Ltd.     

An inverse problem for operators of a triangular structure

An inverse problem for operators of a triangular structure is studied. An algorithm for the solution and necessary and sufficient conditions for the solvability of this problem are obtained, moreover uniqueness is proved. Applications to difference and differential operators are considered.     

The Split Common Fixed Point Problem for Directed Operators

We propose the split common fixed point problem that requires to find a common fixed point of a family of operators in one space whose image under a linear transformation is a common fixed point of another family of operators in the image space. We formulate and analyze a parallel algorithm for solving this split common fixed point problem for the class of directed operators and note how it unifies and generalizes previously discussed problems and algorithms.     

A genetic algorithm for hybrid flowshops with sequence dependent setup times and machine eligibility

After 50 years of research in the field of flowshop scheduling problems the scientific community still observes a noticeable gap between the theory and the practice of scheduling. In this paper we aim to provide a metaheuristic, in the form of a genetic algorithm, to a complex generalized flowshop scheduling problem that results from the addition of unrelated parallel machines at each stage, sequence dependent setup times and machine eligibility. Such a problem is common in the production of textiles and ceramic tiles. The proposed algorithm incorporates new characteristics and four new crossover operators. We show an extensive calibration of the different parameters and operators by means of experimental designs. To evaluate the proposed algorithm we present several adaptations of other well-known and recent metaheuristics to the problem and conduct several experiments with a set of 1320 random instances as well as with real data taken from companies of the ceramic tile manufacturing sector. The results indicate that the proposed algorithm is more effective than all other adaptations.     

A Study of Permutation Operators for Minimum Span Frequency Assignment Using an Order Based Representation

The genetic algorithm (GA) described in this paper breeds permutations of transmitters for minimum span frequency assignment. The approach hybridizes a GA with a greedy algorithm, and employs a technique called Generalized Saturation Degree to seed the initial population. Several permutation operators from the GA literature are compared, and results indicate that position based operators are more appropriate for this kind of problem than are order based operators. My offspring versus mid-parent correlation studies on crossovers show Pearson's correlation coefficient to be a reliable predictor of performance in most cases. Results presented herein represent improvements over previously published results.     

求解无容量设施选址问题的拉格朗日蝙蝠算法

无容量设施选址问题(Uncapacitated Facility Location Problem,UFLP)是一类经典的组合优化问题,被证明是一种NP-hard问题,易于描述却难于求解.首先根据UFLP的数学模型及其具体特征,重新设计了蝙蝠算法的操作算子,给出了求解UFLP的蝙蝠算法.其次构建出三种可行化方法,并将其与求解UFLP的蝙蝠算法和拉格朗日松弛算法相结合,设计了求解该问题的拉格朗日蝙蝠算法.最后通过仿真实例和与其他算法进行比较的方式,验证了该混合算法用来求解UFLP的可行性,是解决离散型问题的一种有效方式.     

Relaxed two points projection method for solving the multiple-sets split equality problem

The multiple-sets split equality problem, a generalization and extension of the split feasibility problem, has a variety of specific applications in real world, such as medical care, image reconstruction, and signal processing. It can be a model for many inverse problems where constraints are imposed on the solutions in the domains of two linear operators as well as in the operators’ ranges simultaneously. Although, for the split equality problem, there exist many algorithms, there are but few algorithms for the multiple-sets split equality problem. Hence, in this paper, we present a relaxed two points projection method to solve the problem; under some suitable conditions, we show the weak convergence and give a remark for the strong convergence method in the Hilbert space. The interest of our algorithm is that we transfer the problem to an optimization problem, then, based on the model, we present a modified gradient projection algorithm by selecting two different initial points in different sets for the problem (we call the algorithm as two points algorithm). During the process of iteration, we employ subgradient projections, not use the orthogonal projection, which makes the method implementable. Numerical experiments manifest the algorithm is efficient.     

A hybrid generational genetic algorithm for the periodic vehicle routing problem with time windows

We propose a new population-based hybrid meta-heuristic for the periodic vehicle routing problem with time windows. This meta-heuristic is a generational genetic algorithm that uses two neighborhood-based meta-heuristics to optimize offspring. Local search methods have previously been proposed to enhance the fitness of offspring generated by crossover operators. In the proposed method, neighborhood-based meta-heuristics are used for their capacity to escape local optima, and deliver optimized and diversified solutions to the population of the next generation. Furthermore, the search performed by the neighborhood-based meta-heuristics repairs most of the constraint violations that naturally occur after the application of the crossover operators. The genetic algorithm we propose introduces two new crossover operators addressing the periodic vehicle routing problem with time windows. The two crossover operators are seeking the diversification of the exploration in the solution space from solution recombination, while simultaneously aiming not to destroy information about routes in the population as computing routes is NP-hard. Extensive numerical experiments and comparisons with all methods proposed in the literature show that the proposed methodology is highly competitive, providing new best solutions for a number of large instances.     

Bacterial memetic algorithm for offline path planning of mobile robots

The goal of the path planning problem is to determine an optimal collision-free path between a start and a target point for a mobile robot in an environment surrounded by obstacles. This problem belongs to the group of combinatorial optimization problems which are approached by modern optimization techniques such as evolutionary algorithms. In this paper the bacterial memetic algorithm is proposed for path planning of a mobile robot. The objective is to minimize the path length and the number of turns without colliding with an obstacle. The representation used in the paper fits well to the algorithm. Memetic algorithms combine evolutionary algorithms with local search heuristics in order to speed up the evolutionary process. The bacterial memetic algorithm applies the bacterial operators instead of the genetic algorithm??s crossover and mutation operator. One advantage of these operators is that they easily can handle individuals with different length. The method is able to generate a collision-free path for the robot even in complicated search spaces. The proposed algorithm is tested in real environment.     

A multi-criteria large neighbourhood search for the transportation of disabled people

This paper addresses the problem of optimizing the transportation of disabled persons from home to specialized centres or schools. It is modelled as a Dial-a-ride problem (DARP), where several people share the same destination. Particular emphasis is placed on the objective function in order to consider several potentially conflicting interests. We propose a multi-criteria model from Multi-attribute Utility Theory based on the Choquet integral. The resulting multi-criteria DARP is then solved with a large neighbourhood search algorithm. This method includes classical destroy and repair heuristics as well as new operators exploiting the shared destination feature and criterion-specific operators. The algorithm is evaluated on a set of 14 real-world instances in the field of health care logistics, with up to 200 requests and 51 destination points.     

The split common fixed point problem for infinite families of total quasi-asymptotically nonexpansive operators

The main purpose in this paper is to introduce an original technique, namely, a specific way of choosing the indexes of the involved operators, by which a new iterative algorithm is developed for solving the split common fixed point problem for infinite families of total quasi-asymptotically nonexpansive operators and some strong and weak convergence theorems are established for this class of nonlinear operators in the framework of infinite-dimensional Hilbert spaces. The result is more applicable than those of other authors with related interest.     

Iterative algorithm of determining the stability of an equation of oscillations with damping

A problem of studying parametric oscillations with damping is reduced to a spectral problem for a linear bundle of operators in a Hilbert space. This spectral problem is solved by an efficient algorithm. The boundaries of the first stability domain for different values of a damping factor and a special form of a periodic function in the equation are calculated.     

一种改进遗传算法在最大独立子集问题中的应用

最大独立子集问题是组合优化问题中的一个重要问题,该问题是一个NP难题,其目标是在一个环图中找到一个最大的独立子集.提出了一种改进的遗传算法来解决这个问题,用一种基于条件的遗传算子来代替通常的基于概率的遗传算子.实验结果表明提出的算法是有效的.     

Factorisation of Non-negative Fredholm Operators and Inverse Spectral Problems for Bessel Operators

We study the problem of factorisation of non-negative Fredholm operators acting in the Hilbert space L₂(0, 1) and its relation to the inverse spectral problem for Bessel operators. In particular, we derive an algorithm of reconstructing the singular potential of the Bessel operator from its spectrum and the sequence of norming constants.     

A general algorithm for multiple-sets split feasibility problem involving resolvents and Bregman mappings

In this paper, by using Bregman distance, we introduce a new iterative process involving products of resolvents of maximal monotone operators for approximating a common element of the set of common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings and the solution set of the multiple-sets split feasibility problem and common zeros of maximal monotone operators. We derive a strong convergence theorem of the proposed iterative algorithm under appropriate situations. Finally, we mention several corollaries and two applications of our algorithm.     

Optimal Product Line Design: Genetic Algorithm Approach to Mitigate Cannibalization

In this marketing-oriented era where manufacturers maximize profits through customer satisfaction, there is an increasing need to design a product line rather than a single product. By offering a product line, the manufacturer can customize his or her products to the needs of a variety of segments in order to maximize profits by satisfying more customers than a single product would. When the amount of data on customer preferences or possible product configurations is large and no analytical relations can be established, the problem of an optimal product line design becomes very difficult and there are no traditional methods to solve it. In this paper, we show that the usage of genetic algorithms, a mathematical heuristics mimicking the process of biological evolution, can solve efficiently the problem. Special domain operators were developed to help the genetic algorithm mitigate cannibalization and enhance the algorithm’s local search abilities. Using manufacturer’s profits as the criteria for fitness in evaluating chromosomes, the usage of domain specific operators was found to be highly beneficial with better final results. Also, we have hybridized the genetic algorithm with a linear programming postprocessing step to fine tune the prices of products in the product line. Attacking the core difficulty of cannibalization in the algorithm, the operators introduced in this work are unique.