Lower tolerance-based Branch and Bound algorithms for the ATSP

In this paper, we develop a new tolerance-based Branch and Bound algorithm for solving NP-hard problems. In particular, we consider the asymmetric traveling salesman problem (ATSP), an NP-hard problem with large practical relevance. The main algorithmic contribution is our lower bounding strategy that uses the expected costs of including arcs in the solution to the assignment problem relaxation of the ATSP, the so-called lower tolerance values. The computation of the lower bound requires the calculation of a large set of lower tolerances. We apply and adapt a finding from [23] that makes it possible to compute all lower tolerance values efficiently. Computational results show that our Branch and Bound algorithm exhibits very good performance in comparison with state-of-the-art algorithms, in particular for difficult clustered ATSP instances.     

Cut-and-solve: An iterative search strategy for combinatorial optimization problems

Branch-and-bound and branch-and-cut use search trees to identify optimal solutions to combinatorial optimization problems. In this paper, we introduce an iterative search strategy which we refer to as cut-and-solve and prove optimality and termination for this method. This search is different from traditional tree search as there is no branching. At each node in the search path, a relaxed problem and a sparse problem are solved and a constraint is added to the relaxed problem. The sparse problems provide incumbent solutions. When the constraining of the relaxed problem becomes tight enough, its solution value becomes no better than the incumbent solution value. At this point, the incumbent solution is declared to be optimal. This strategy is easily adapted to be an anytime algorithm as an incumbent solution is found at the root node and continuously updated during the search.Cut-and-solve enjoys two favorable properties. Since there is no branching, there are no “wrong” subtrees in which the search may get lost. Furthermore, its memory requirement is negligible. For these reasons, it has potential for problems that are difficult to solve using depth-first or best-first search tree methods.In this paper, we demonstrate the cut-and-solve strategy by implementing a generic version of it for the Asymmetric Traveling Salesman Problem (ATSP). Our unoptimized implementation outperformed state-of-the-art solvers for five out of seven real-world problem classes of the ATSP. For four of these classes, cut-and-solve was able to solve larger (sometimes substantially larger) problems. Our code is available at our websites.     

Multiple asymmetric traveling salesmen problem with and without precedence constraints: Performance comparison of alternative formulations

In this paper, we investigate the performances of 32 formulations for the multiple asymmetric traveling salesman problem (mATSP) from the viewpoint of their tightness and solvability using commercial software. These formulations are either new or are generalizations of those proposed in the literature for the ATSP, including a transformation of the mATSP to an equivalent ATSP formulation. In particular, our results based on the latter strategy reveal that the superiority of the original mATSP formulation over an equivalent transformed ATSP model depends upon the type of formulation used for conducting such a transformation. We also extend our study to the case where precedence relationships exist among the cities. We address two situations in this context. In the first case, a sequential processing order is enforced between a pair of cities only if they both lie on the tour of a particular salesman. The second case involves a more general precedence in which a sequential processing order between certain pairs of cities is enforced regardless of whether the designated pairs are visited by the same or by different salesmen. Our computational experiments demonstrate that the flow-based models afford the tightest LP relaxations. Additionally, the formulations that model the different variants of the mATSP directly, rather than by transforming them to equivalent ATSP problems, are generally faster to solve and are more robust.     

基于反馈校正原理的非对称旅行商问题的自收敛优化算法

针对非对称旅行商问题(ATSP),提出基于反馈校正原理的自收敛求解算法框架.该方法核心是依据ATSP问题松弛模型的对偶关系推断与ATSP最优解无关弧集合的弧排除算法.该算法框架以ATSP问题的初始弧集合作为"参考输入",以ATSP最优解的上下界求解算法作为"控制对象",以弧排除算法作为"反馈校正控制器",其"反馈输入"是"控制对象"的输出差值.算法迭代过程中,上下界差值缩小,排除弧集合增加,算法呈现出自收敛性.该框架集成了数学规划方法和启发式算法的优点,论文从理论证明和仿真分析说明了该自收敛算法的有效性.     

A hybrid genetic algorithm for no-wait job shop scheduling problems

A no-wait job shop (NWJS) describes a situation where every job has its own processing sequence with the constraint that no waiting time is allowed between operations within any job. A NWJS problem with the objective of minimizing total completion time is a NP-hard problem and this paper proposes a hybrid genetic algorithm (HGA) to solve this complex problem. A genetic operation is defined by cutting out a section of genes from a chromosome and treated as a subproblem. This subproblem is then transformed into an asymmetric traveling salesman problem (ATSP) and solved with a heuristic algorithm. Subsequently, this section with new sequence is put back to replace the original section of chromosome. The incorporation of this problem-specific genetic operator is responsible for the hybrid adjective. By doing so, the course of the search of the proposed genetic algorithm is set to more profitable regions in the solution space. The experimental results show that this hybrid genetic algorithm can accelerate the convergence and improve solution quality as well.     

Probabilistic Prediction of the Complexity of Traveling Salesman Problems Based on Approximating the Complexity Distribution from Experimental Data

We show the results of a statistical study of the complexity of the asymmetric traveling salesman problem (ATSP) obtained by processing a specially generated pool of matrices. We show that the normal distribution can serve as an approximation to the distribution of the logarithm of complexity for a fixed problem dimension. We construct a family of probability distributions that represent satisfactory approximations of the complexity distribution with a dimension of the cost matrix from 20 to 49. Our main objective is to make probabilistic predictions of the complexity of individual problems for larger values of the dimension of the cost matrix. We propose a representation of the complexity distribution that makes it possible to predict the complexity. We formulate the unification hypothesis and show directions for further study, in particular proposals on the task of clustering “complex” and “simple” ATSP problems and proposals on the task of directly predicting the complexity of a specific problem instance based on the initial cost matrix.     

A metaheuristic approach for the sequencing by hybridization problem with positive and negative errors

This work introduces a metaheuristic method for the reconstruction of the DNA string from its l-mer content in the presence of large amounts of positive and negative errors. The procedure consists of three parts: the formulation of the problem as an asymmetric traveling salesman problem (ATSP), a technique for handling the positive errors and an optimization algorithm that solves the formulated problem. The optimization algorithm is a variation of the threshold accepting method with intense local search and its function is controlled by a size diminishing shell. The optimization algorithm is used consecutively on ATSPs of continuously decreasing sizes till it reaches a final solution. The proposed method provides solutions of better quality compared to algorithms in the recent bibliography.     

ATSP:时态网络可视化的自适应时间片划分方法

现有的时态网络可视化方法大多采用等量时间片来可视化网络的演变，不利于时态模式的快速挖掘和发现。为此，根据时态网络固有的特征提出自适应时间片划分方法（Adaptive Time Slice Partition method，ATSP）。在时态网络的两种表示方式（基于事件的表示方式和基于快照的表示方式）的基础上，构建了ATSP的基础模型，同时提出了一种改进模型用来描述事件间隔时间服从长尾分布的时态网络。为了实现时间片的不等量划分，针对探索任务的不同提出了基于时态模式的ATSP规则和基于中心节点的ATSP规则，并提出了实现算法--层次划分算法（Hierarchical Partition algorithm，HP）和增量划分算法（Incremental Partition algorithm，IP）。实验结果表明，ATSP方法比传统的时间片划分方法更能准确地表示网络的时态特征，且该方法应用于可视化时，能有效归纳并展示网络的特征，明显提高了视觉分析的效率。     

一种新的求解ATSP问题的PBIL算法

该文将二进制的人口增量学习算法(PBIL)改进为整数(集值)形式(multiplePBIL),并提出了一种新的基于城市间连接关系的非对称旅行商问题(ATSP)的解法。这种解法结合了集值人口增量学习算法和TSP问题的启发式搜索3-opt加强方法。混沌定位,分布式随机遍历构架和判断进化结束条件的可能性分布的熵的确定是该解法的三大创新之处。     

35/44-approximation for Asymmetric Maximum TSP with Triangle Inequality

We describe a new approximation algorithm for the asymmetric maximum traveling salesman problem (ATSP) with triangle inequality. Our algorithm achieves approximation factor 35/44 which improves on the previous 31/40 factor of Bläser, Ram and Sviridenko (Lecture Notes in Computer Science, vol. 3122, pp. 350–359, 2005).     

混沌优化算法及其在组合优化问题中的应用

混沌优化方法(COA)是针对数值优化问题提出的,在解决数值优化问题上具有一定的普遍性,能够很快地搜索到全局最优解,而利用COA解决组合优化问题存在一定的难度,该文提出了混沌优化算法解决组合优化问题的方法,该方法先产生组合优化问题的初始解,再利用混沌变量产生新解或对原解进行混沌扰动,产生新解,然后在解空间中进行最优搜索。将该方法应用到2个典型的组合优化问题(TSP问题,0/1背包问题)的求解中,仿真实验表明了该方法的有效性。     

An approximate programming method based on the simplex method for bilevel programming problem

Many real problems can be modeled to the problems with a hierarchical structure, and bilevel programming is a useful tool to solve the hierarchical optimization problems. So the bilevel programming is widely applied, and numerous methods have been proposed to solve this programming. In this paper, we propose an approximate programming algorithm to solve bilevel nonlinear programming problem. Finally, the example illustrates the feasibility of the proposed algorithm.     

A heuristic algorithm to minimize total weighted tardiness on a single machine with release times

This paper attempts to solve a single machine‐scheduling problem, in which the objective function is to minimize the total weighted tardiness with different release dates of jobs. To address this scheduling problem, a heuristic scheduling algorithm is presented. A mathematical programming formulation is also formulated to validate the performance of the heuristic scheduling algorithm proposed herein. Experimental results show that the proposed heuristic algorithm can solve this problem rapidly and accurately. Overall, this algorithm can find the optimal solutions for 2200 out of 2400 randomly generated problems (91.67%). For the most complicated 20 job cases, it requires less than 0.0016 s to obtain an ultimate or even optimal solution. This heuristic scheduling algorithm can therefore efficiently solve this kind of problem.     

Using evolution strategies to solve DEC-POMDP problems

Decentralized partially observable Markov decision process (DEC-POMDP) is an approach to model multi-robot decision making problems under uncertainty. Since it is NEXP-complete there is no efficient exact algorithm to solve these problems and in spite of the attention it has taken recently, so far only a few approximate solutions that can solve small problems have been proposed. In this study, we offer a novel approximate solution algorithm for DEC-POMDP problems using evolution strategies, and a novel approach to approximately calculate the fitness of the chromosomes which correspond to the expected reward. We also propose a new problem which is a more complex, modified version of the grid meeting problem and solve it. Our results show that our algorithm is scalable and we can solve problems that have more states than the problems attempted in previous studies.     

0/1背包问题的量子算法

近年来针对各种问题提出了许多量子算法,这些量子算法都利用了量子态的可迭加性(Superposition)和纠缠性(Entan-glement),本文在量子环境下对0/1背包问题进行求解,介绍了量子算法的基本思想及相关概念。然后分析并给出求解0/1背包问题的量子算法,在量子物理环境下它能在多项式时间内求出所需要的解。这个量子算法可以推广解决其它NPC问题,如旅行售货员问题等。     

蚁群算法及其在求解旅行商问题中的应用

蚁群算法是一种求解组合优化问题较好的方法。在蚁群算法的基本原理基础上,以旅行商问题为例,介绍了该算法求解TSP的数学模型及具体步骤,并通过仿真实验与粒子群优化算法等方法比较分析,表明了该算法在求解组合优化问题方面具有良好的性能。     

Differential evolution for solving multi-mode resource-constrained project scheduling problems

In this paper we consider the resource-constrained project scheduling problem with multiple execution modes for each activity and minimization of the makespan. To solve this problem, we propose a differential evolution (DE) algorithm. We focus on the performance of this algorithm to solve the problem within small time per activity. Finally, we present the results of our thorough computational study. Results obtained on six classes of test problems and comparison with other algorithms from the literature show that our algorithm gives better solutions.     

A grasp algorithm for m-machine flowshop scheduling problem with bicriteria of makespan and maximum tardiness

In this paper we address the problem of minimizing the weighted sum of makespan and maximum tardiness in an m-machine flow shop environment. This is a NP-hard problem in the strong sense. An attempt has been made to solve this problem using a metaheuristic called Greedy Randomized Adaptive Search Procedure (GRASP). GRASP is a competitive algorithm and is a meta-heuristic for solving combinatorial optimization problems. We have customized the basic concepts of GRASP algorithm to solve a bicriteria flow shop problem and a new algorithm named B-GRASP (Bicriteria GRASP algorithm) is proposed. The new proposed algorithm is evaluated using benchmark problems taken from Taillard and compared with the existing simulated annealing based heuristic developed by Chakravarthy and Rajendran. Computational experiments indicate that the proposed algorithm is much better than the existing one in all cases.     

组合规则启发式算法在模具加工作业调度中的应用研究

由于现行的优化算法在解决模具车间调度问题上存在局限性,目前大部分模具车间由人工编制车间作业计划,导致生产效率较低、物料供应不能同步化等问题.为此,介绍了采用基于组合规则的启发式算法,选择加工时间最短和最小等待时间等规则,解决了现行的优化调度算法在车间调度问题中所遇到的难点.经实例验证获得了较为理想的结果.     

基于递阶遗传算法的多旅行商问题优化*

旅行商问题是一个经典的NP问题,对多人旅行商问题的求解则更具有意义。为了解决所有旅行商路径总和最小为优化标准的多旅行商一类问题,提出了一种递阶遗传算法和矩阵解码方法。该算法根据问题的特点,采用一种递阶编码方案,此编码与多旅行商问题一一对应。用递阶遗传算法优化多旅行商问题无须设计专门的遗传算子,操作简单,并且解码方法适于求解距离对称和距离非对称的多旅行商问题。计算结果表明,递阶遗传算法是有效的,能适用于优化多旅行商问题。