三阶段拉格朗日启发式算法求解带同时取送货的绿色车辆路径问题

针对带同时取送货的绿色车辆路径问题,以最小化带碳排放费用的配送成本为优化目标,建立混合整数规划模型,并提出一种结合数学规划方法与启发式算法的三阶段拉格朗日启发式算法进行求解.第1阶段,利用拉格朗日松弛技术得到该问题的拉格朗日对偶模型;第2阶段,设计一种改进的次梯度算法迭代求解该对偶模型,同时引入修复机制,将每次迭代所得下界对应的解修复为原问题较高质量的可行解,并在下次迭代中利用该可行解更新次梯度方向和步长;第3阶段,设计一种启发式局部搜索算法,对第2阶段得到的可行解进行优化,进一步改进解的质量,以得到原问题的近似最优解.实验表明,所提出算法能够获得问题的一个优质解,同时提供一个紧致下界,用以定量评估解的质量.     

装配线物料搬运的拉格朗日松弛算法

为提高汽车制造企业混流装配线的运行效益,提出了基于看板模型的多封闭循环路径多载量小车物料配送调度方法—–装配线物料配送调度的拉格朗日松弛算法.首先对问题域进行了描述并做出了具体假设,以最小化配送系统总成本为目标,建立了混合整数规划模型.在此基础上,针对该模型提出了两种算法—–次梯度和随机步长拉格朗日松弛算法,将松弛问题分解为两个决策子问题分别进行求解.仿真实验表明提出的两种调度算法均适用于该研究问题域,并在求解时间及稳定性上表现出良好的性能.     

可重入混合流水车间调度的拉格朗日松弛算法

为了有效提升多重入车间的生产效率,考虑了实际生产中检查和修复过程对于逐层制造的可重入生产系统的重要性,提出了基于拉格朗日松弛算法的可重入混合流水车间的调度方法.首先进行了问题域的描述,并在此基础上以最小化加权完成时间为调度目标,建立数学规划模型.针对该调度问题提出了基于松弛机器能力约束的拉格朗日松弛算法,使松弛问题分解成工件级子问题,并使用动态规划方法建立递归公式,求解工件级子问题.随后,使用次梯度算法求解拉格朗日对偶问题.最后,对各种不同问题规模进行了仿真实验,结果表明,所提出的调度算法能够在合理的时间内获得满意的近优解.     

基于集结投影次梯度的机组组合算法研究

针对大规模电力系统机组组合问题,提出了基于集结投影次梯度方法的分解协调算法.首先在上层通过拉格朗日松弛方法将原问题分解为多个子问题,从而减小了求解问题的复杂度,避免了维数灾问题,同时显著降低了计算时间,使得原问题可以在多项式时间内求解,随后下层子问题采用动态规划方法很容易求最优解.算例仿真结果表明,所采用的集结投影次梯度方法调整拉格朗日乘子,避免了传统次梯度方法振荡现象严重的缺点,同时加快了收敛速度,得到了令人满意的机组组合方案.     

混合遗传算法求解应急抢修点选址问题

考虑设备应急抢修的时限要求和整个应急抢修系统的服务质量要求,采用0-1整数规划模型描述了应急抢修点选址问题,并针对该问题设计了一种混合遗传算法。在算法中使用启发式算法对种群中的不可行解进行修复,以保持种群在可行域内搜索,并采用近邻搜索算法改善种群中的最佳个体。算例计算的结果表明,该算法求得的结果要优于基于罚函数的遗传算法和采用简单修复算法的遗传算法。     

并行机实时调度问题的LR&CG算法

研究了目标函数为最小化总加权完成时间的并行机实时调度问题.建立该问题混合整数规划模型,并提出融合拉格朗日松弛(LR)和列生成(CG)的 LR & CG 混合算法.该算法包含双重迭代,在内环以次梯度法作为下界求解器和列生成器,在外环通过求解限制主问题来获得影子价格以调节拉格朗日乘子.计算实验结果表明,在相同的计算时间内, LR & CG 能够比常规的 LR 算法获得更好的上界和下界,表明了前者具有更好的收敛性能.     

带最小批量约束的计划问题及其拉格朗日松弛算法

针对一类带最小批量约束的计划问题, 提出了基于拉格朗日松弛策略求解算法. 通过拉格朗日松弛策略, 将原问题转为一系列带最小批量约束的动态经济批量W-W(Wagner-Whitin)子问题. 提出了解决子问题且其时间复杂度O(T³)的最优前向递推算法. 对于拉格朗日对偶问题, 用次梯度算法求解, 获得原问题的下界. 若对偶问题的解是不可行的, 通过固定装设变量, 求解一个剩余的线性规划问题来进行可行化处理. 最后, 数据仿真验证了算法的有效性.     

多学科设计优化中目标层解分析法的研究

目标层解分析是一种层次化、多层系统设计优化方法.为了确保求解多学科设计优化各子问题的可行性,提高求解效率,应用增广拉格朗日惩罚函数松弛化方法,对目标层解分析的内外层嵌套式求解策略进行改进,通过对内层循环的惩罚函数松弛化来减少内层循环病态子问题的求解计算时间,当内层循环获得收敛之后,外层循环更新惩罚权重来获得可行解.并置设计次数由10到1000的具体实例来对比分析各种惩罚函数对求解效率的影响.由实验可得,应用增广拉格朗日惩罚函数松弛化方法求解,计算权重得到减小.迭代次数减少到二次惩罚函数法的2%.     

可延迟供货的冷轧生产库存问题之建模与优化

本文针对可延迟供货的冷轧生产系统,建立了以最小化库存成本、拖期惩罚和启动成本为目标的多阶段生产库存模型,模型中充分考虑了工序不允许停机的情况以及计划与调度之间的一致性问题.同时开发了基于变量分离的有效拉格朗日松弛求解算法,并使用120个基于实际生产数据的算例进行了仿真实验,计算结果显示该算法能够在合理的时间内得到高质量的解.     

奖励收集斯坦利最小树的混合拉格朗日与分散搜索算法

针对PCSTP问题,提出了HLGSS混合算法．通过拉格朗日松弛策略,将PCSTP问题转化为简单的CMST问题;然后由Volume算法求解PCSTP的拉格朗日对偶问题并获得其下界．用SS算法优化原问题的可行解,利用求解拉格朗日对偶问题过程中获得的原始一对偶信息来指导SS算法的搜索．仿真结果表明,HLGSS比SS降低了算法的搜索空间,加速了算法的收敛性．     

车间调度算法的研究和开发

针对车间调度问题,提出了一种改进的拉氏松弛算法,在增加辅助目标函数的基础上,通过对子问题的限制和搜索策略的改变,使拉氏算法的计算量减少,近优解的搜索能力有很大改善,本文还提出了一种基因优化算法,充分利用拉氏算法得到的多个近优解,进一步优化结,仿真结果表明对车间调度问题得到了较好的结果,本方法也可用于其它有约束的规则问题。     

Lagrangian bounds for just-in-time job-shop scheduling

We study the job-shop scheduling problem with earliness and tardiness penalties. We describe two Lagrangian relaxations of the problem. The first one is based on the relaxation of precedence constraints while the second one is based on the relaxation of machine constraints. We introduce dedicated algorithms to solve the corresponding dual problems. The second one is solved by a simple dynamic programming algorithm while the first one requires the resolution of an NP-hard problem by branch and bound. In both cases, the relaxations allow us to derive lower bounds as well as heuristic solutions. We finally introduce a simple local search algorithm to improve the best solution found. Computational results are reported.     

带运输考虑的多阶段动态可重入混合流水车间调度

可重入混合流水车间调度允许一个工件多次进入某些加工阶段,它广泛出现在许多工业制造过程中,如半导体制造、印刷电路板制造等.本文研究了带运输时间的多阶段动态可重入混合流水车间问题,目标是最小化总加权完成时间.针对该问题,建立了整数规划模型,进而基于工件解耦方式提出了两种改进的拉格朗日松弛(LR)算法.在这些算法中,设计了动态规划的改进策略以加速工件级子问题的求解,提出了异步次梯度法以得到有效的乘子更新方向.测试结果说明了所提出的两种改进算法在解的质量和运行时间方面均优于常规LR算法,两种算法都能在可接受的计算时间内得到较好的近优解.     

A heuristic algorithm based on Lagrangian relaxation for the closest string problem

The closest string problem that arises in both computational biology and coding theory is to find a string minimizing the maximum Hamming distance from a given set of strings. This study proposes an efficient heuristic algorithm for this NP-hard problem. The key idea is to apply the Lagrangian relaxation technique to the problem formulated as a mixed-integer programming problem. This enables us to decompose the problem into trivial subproblems corresponding to each position of the strings. Furthermore, a feasible solution can be easily obtained from a solution of the relaxation. Based on this, a heuristic algorithm is constructed by combining a Lagrangian multiplier adjustment procedure and a tabu search. Computational experiments will show that the proposed algorithm can find good approximate solutions very fast.     

A Lagrangian relaxation technique for the general assembly line balancing problem

This paper addresses the problem of balancing assembly or fabrication lines. In order to achieve a given production rate or to optimize the use of workstations, one has to tackle the problem of balancing the production lines. It is well known that this problem belongs to the class of NP-hard problems. In this paper the polyhedron of the feasible solutions of the assembly line balancing problem is first studied. Then a Lagrangian relaxation algorithm that incorporates the set of cycle constraints in the objective function is proposed. These constraints are the complicating restrictions in the model. The relaxed problem has the interesting property that its linear programming relaxation always has integer optimal solutions. The subgradient algorithm is then used to maximize the Lagrangian dual. A heuristic is also used to find primal feasible solutions for the original line balancing integer program. These two bounds are then used to reduce the size of the branch-and-bound tree.     

An integration of Lagrangian split and VNS: The case of the capacitated vehicle routing problem

In this paper, we propose an efficient and novel Lagrangian relaxation method which incorporates a new integer linear programming (ILP) formulation to optimally partition a giant tour in the context of a capacitated vehicle routing problem (CVRP). This approach, which we call Lagrangian split (Ls), is more versatile than the ILP which, in most cases, can be intractable using a conventional solver. An effective repair mechanism followed by a local search are also embedded into the process. The mathematical validity of the repair mechanism and its time complexity are also provided. An integration of Ls into a powerful variable neighbourhood search (VNS) is also presented. Computational experiments are conducted to demonstrate that Ls provides encouraging results when applied on benchmark instances and that the integration of Ls into a metaheuristic scheme produces good results when compared to those found by state-of-the-art methods.     

A new Lagrangian relaxation algorithm for hybrid flowshop scheduling to minimize total weighted completion time

We investigate the problem of scheduling n jobs in s-stage hybrid flowshops with parallel identical machines at each stage. The objective is to find a schedule that minimizes the sum of weighted completion times of the jobs. This problem has been proven to be NP-hard. In this paper, an integer programming formulation is constructed for the problem. A new Lagrangian relaxation algorithm is presented in which precedence constraints are relaxed to the objective function by introducing Lagrangian multipliers, unlike the commonly used method of relaxing capacity constraints. In this way the relaxed problem can be decomposed into machine type subproblems, each of which corresponds to a specific stage. A dynamic programming algorithm is designed for solving parallel identical machine subproblems where jobs may have negative weights. The multipliers are then iteratively updated along a subgradient direction. The new algorithm is computationally compared with the commonly used Lagrangian relaxation algorithms which, after capacity constraints are relaxed, decompose the relaxed problem into job level subproblems and solve the subproblems by using the regular and speed-up dynamic programming algorithms, respectively. Numerical results show that the new Lagrangian relaxation method produces better schedules in much shorter computation time, especially for large-scale problems.     

Lower bounds and exact algorithms for the quadratic minimum spanning tree problem

We address the quadratic minimum spanning tree problem (QMSTP), the problem of finding a spanning tree of a connected and undirected graph such that a quadratic cost function is minimized. We first propose an integer programming formulation based on the reformulation–linearization technique (RLT). We then use the idea of partitioning spanning trees into forests of a given fixed size and obtain a QMSTP reformulation that generalizes the RLT model. The reformulation is such that the larger the size of the forests, the stronger lower bounds provided. Thus, a hierarchy of formulations is obtained. At the lowest hierarchy level, one has precisely the RLT formulation, which is already stronger than previous formulations in the literature. The highest hierarchy level provides the convex hull of integer feasible solutions for the problem. The formulations introduced here are not compact, so the direct evaluation of their linear programming relaxation bounds is not practical. To overcome that, we introduce two lower bounding procedures based on Lagrangian relaxation. These procedures are embedded into two parallel branch-and-bound algorithms. As a result of our study, several instances in the literature were solved to optimality for the first time.