Single machine scheduling to minimize maximum lateness subject to release dates and precedence constraints

This paper considers the problem of finding a nonpreemptive schedule for a single machine to minimize the maximum lateness with release dates and precedence constraints. A branch and bound algorithm is developed. The algorithm uses four different heuristics to find upper bounds at the initial branch node: early release date heuristic, modified Schrage's heuristic, heuristic BLOCK, and a variable neighborhood descent procedure. At each branch node, two branches evolve from a schedule found by heuristic BLOCK using a binary branching rule based on bottleneck and critical jobs, and a lower bound is obtained by optimally solving the relaxed problem with preemption. The algorithm solves 14,984 out of the 15,000 systematically generated instances with up to 1,000 jobs within 1 minute of CPU time.     

A single machine carryover sequence-dependent group scheduling in PCB manufacturing

This paper considers the problem of minimizing the makespan on a single machine with carryover sequence-dependent setup times. A similar problem with multi-machine flow shop usually arises in the assembly of printed circuit boards (PCBs). This research investigates the possibility of processing all components of PCBs using just one machine. By doing so the operational costs of having multi-machines can be reduced, and as a result, finding an optimal solution might be more plausible. The objective is to minimize the maximum completion time of all board groups, commonly known as makespan. The operational constraints are such that all board types within a board group must be completely kitted, as it is traditionally performed by kitting staff, before that board group begins its assembly operation. We introduce the external setup (kitting) time and require that it be performed solely by the machine operator during the run time of the current board group, and thereby completely eliminating the need for kitting staff. The carryover sequence-dependent setup time, namely the internal (machine) setup time, is realized when a new board group is ready for assembly operation and is dependent on all of the previously scheduled board groups and their sequences. To the best of our knowledge, this is the first time the external and internal setup times are integrated in PCB group scheduling research. We develop a branch-and-bound algorithm and a lower-bounding structure. The lower bound consists of two approaches, which enable the algorithm to simultaneously reduce performing unnecessary exploration. In order to test the efficiency of the algorithm, several problem instances with different board groups have been used. The algorithm developed requires a significantly large computation time to optimally solve very large problems. Thus to speak for the efficiency in terms of solving comparable large industry-size problems, we evaluate the deviation of the algorithm from the lower bound which turns out to be very small, with an average of only 6%, in all of the problem instances considered.     

The green vehicle routing problem: A heuristic based exact solution approach

This paper develops a simulated annealing heuristic based exact solution approach to solve the green vehicle routing problem (G-VRP) which extends the classical vehicle routing problem by considering a limited driving range of vehicles in conjunction with limited refueling infrastructure. The problem particularly arises for companies and agencies that employ a fleet of alternative energy powered vehicles on transportation systems for urban areas or for goods distribution. Exact algorithm is based on the branch-and-cut algorithm which combines several valid inequalities derived from the literature to improve lower bounds and introduces a heuristic algorithm based on simulated annealing to obtain upper bounds. Solution approach is evaluated in terms of the number of test instances solved to optimality, bound quality and computation time to reach the best solution of the various test problems. Computational results show that 22 of 40 instances with 20 customers can be solved optimally within reasonable computation time.     

Branch-and-bound algorithm for the maximum triangle packing problem

This work addresses the problem of finding the maximum number of unweighted vertex-disjoint triangles in an undirected graph G. It is a challenging NP-hard combinatorial problem and it is well-known to be APX-hard. A branch-and-bound algorithm which uses a lower bound based on neighborhood degree is presented. A naive upper bound is proposed as well as another one based on a surrogate relaxation of the related integer linear program which is analogous to a multidimensional knapsack problem. Further, a Greedy Search algorithm and a genetic algorithm are described to improve the lower bound. A computational comparison of lower bounds, branch-and-bound algorithm and CPLEX solver is provided using randomly generated benchmarks and well-known DIMACS implementation challenges. The empirical study shows that the branch-and-bound finds the optimal triangle packing solution for small randomly generated MTP instances (up to 100 vertices and 200 triangles) and some DIMACS graphs. For some larger instances and DIMACS challenges graphs, we remark that our lower bound outperforms CPLEX solver regarding the triangle packing solution and the computation time.     

A hybrid genetic and linear programming algorithm for two-agent order acceptance and scheduling problem

In this paper, the simultaneous order acceptance and scheduling problem is developed by considering the variety of customers’ requests. To that end, two agents with different scheduling criteria including the total weighted lateness for the first and the weighted number of tardy orders for the second agent are considered. The objective is to maximize the sum of the total profit of the first and the total revenue of the second agents’ orders when the weighted number of tardy orders of the second agent is bounded by an upper bound value. In this study, it is shown that this problem is NP-hard in the strong sense, and then to optimally solve it, an integer linear programming model is proposed based on the properties of optimal solution. This model is capable of solving problem instances up to 60 orders in size. Also, the LP-relaxation of this model was used to propose a hybrid meta-heuristic algorithm which was developed by employing genetic algorithm and linear programming. Computational results reveal that the proposed meta-heuristic can achieve near optimal solutions so efficiently that for the instances up to 60 orders in size, the average deviation of the model from the optimal solution is lower than 0.2% and for the instances up to 150 orders in size, the average deviation from the problem upper bound is lower than 1.5%.     

Single machine weighted earliness–tardiness penalty problem with a common due date

In this paper, we have considered a class of single machine job scheduling problems where the objective is to minimize the weighted sum of earliness–tardiness penalties of jobs. The weights are job-independent but they depend on whether a job is early or tardy. The restricted version of the problem where the common due date is smaller than a critical value, is known to be NP-complete. While dynamic programming formulation runs out of memory for large problem instances, depth-first branch-and-bound formulation runs slow for large problems since it uses a tree search space. In this paper, we have suggested an algorithm to optimally solve large instances of the restricted version of the problem. The algorithm uses a graph search space. Unlike dynamic programming, the algorithm can output optimal solutions even when available memory is limited. It has been found to run faster than dynamic programming and depth-first branch-and-bound formulations and can solve much larger instances of the problem in reasonable time. New upper and lower bounds have been proposed and used. Experimental findings are given in detail.Scope and purposeA class of single machine problems arising out of scheduling jobs in JIT environment has been considered in this paper. The objective is to minimize the total weighted earliness–tardiness penalties of jobs. In this paper, we have presented a new algorithm and conducted extensive empirical runs to show that the new algorithm performs much better than the existing approaches in solving large instances of the problem.     

An Exact Method to Minimize the Number of Tardy Jobs in Single Machine Scheduling

This paper considers the problem of scheduling n jobs on a single machine to minimize the number of tardy (or late) jobs. Each job has a release date, a processing time and a due date. The general case with non-equal release dates and different due dates is considered. Using new and efficient lower bounds and several dominance rules, a branch and bound scheme is proposed based on the definition of a master sequence, i.e. a sequence containing at least one optimal sequence. With this procedure, 95% of 140-job instances are optimally solved in a maximum of one-hour CPU time.     

Jackson's semi-preemptive scheduling on a single machine

We propose an effective improvement of the well-known Jackson's preemptive schedule lower bound for the single machine scheduling problem with heads and tails. The semi-preemptive scheduling concept roughly consists in reducing the preemption impact by constraining some particular job parts to be processed in reduced time intervals. The impact of semi-preemptive scheduling is twofold: it yields a lower bound which dominates the preemptive one, and enables more effective adjustments of the heads and tails. Our experimental study revealed that suitably embedding our procedure within Carlier's algorithm makes feasible to solve all of the hard instances which could not be solved by its original variant.     

An exact algorithm for the single-machine total weighted tardiness problem with sequence-dependent setup times

This study proposes an exact algorithm for the single-machine total weighted tardiness problem with sequence-dependent setup times. The algorithm is an extension of the authors' previous algorithm for the single-machine scheduling problem without setup times, which is based on the SSDP (Successive Sublimation Dynamic Programming) method. In the first stage of the algorithm, the conjugate subgradient algorithm or the column generation algorithm is applied to a Lagrangian relaxation of the original problem to adjust multipliers. Then, in the second stage, constraints are successively added to the relaxation until the gap between lower and upper bounds becomes zero. The relaxation is solved by dynamic programming and unnecessary dynamic programming states are eliminated to suppress the increase of computation time and memory space. In this study a branching scheme is integrated into the algorithm to manage to solve hard instances. The proposed algorithm is applied to benchmark instances in the literature and almost all of them are optimally solved.     

Minimizing total tardiness in a two-machine re-entrant flowshop with sequence-dependent setup times

We consider a two-machine re-entrant flowshop scheduling problem in which all jobs must be processed twice on each machine and there are sequence-dependent setup times on the second machine. For the problem with the objective of minimizing total tardiness, we develop dominance properties and a lower bound by extending those for a two-machine re-entrant flowshop problem (without sequence-dependent setup times) as well as heuristic algorithms, and present a branch and bound algorithm in which these dominance properties, lower bound, and heuristics are used. For evaluation of the performance of the branch and bound algorithm and heuristics, computational experiments are performed on randomly generated instances, and results are reported.     

单机调度问题对偶集结迭代算法

具有到达时间约束、目标为最小化加权完工时间之和的单机调度问题是一个典型的NP-hard问题,采用时间下标建模的线性规划松弛方法可提供一个很强的下界,但优化求解存在维数困难.为此,本文提出了一种对偶集结优化策略,通过选择一个衰减集结矩阵集结对偶乘子变量,利用对偶理论获得模型的约束集结,从而降低计算复杂度.同时分析了集结模型的结构特性,并提出一种迭代算法来改善下界.仿真结果表明对偶集结迭代算法能够减少计算时间,同时改善下界性能,适用于大规模调度问题.     

A lower bound for the Node,Edge, and Arc Routing Problem

The Node, Edge, and Arc Routing Problem (NEARP) was defined by Prins and Bouchenoua in 2004, although similar problems have been studied before. This problem, also called the Mixed Capacitated General Routing Problem (MCGRP), generalizes the classical Capacitated Vehicle Routing Problem (CVRP), the Capacitated Arc Routing Problem (CARP), and the General Routing Problem. It captures important aspects of real-life routing problems that were not adequately modeled in previous Vehicle Routing Problem (VRP) variants. The authors also proposed a memetic algorithm procedure and defined a set of test instances called the CBMix benchmark. The NEARP definition and investigation contribute to the development of rich VRPs. In this paper we present the first lower bound procedure for the NEARP. It is a further development of lower bounds for the CARP. We also define two novel sets of test instances to complement the CBMix benchmark. The first is based on well-known CARP instances; the second consists of real life cases of newspaper delivery routing. We provide numerical results in the form of lower and best known upper bounds for all instances of all three benchmarks. For three of the instances, the gap between the upper and lower bound is closed. The average gap is 25.1%. As the lower bound procedure is based on a high quality lower bound procedure for the CARP, and there has been limited work on approximate solution methods for the NEARP, we suspect that a main reason for the rather large gaps is the quality of the upper bound. This fact, and the high industrial relevance of the NEARP, should motivate more research on approximate and exact methods for this important problem.     

Optimizing blocking flow shop scheduling problem with total completion time criterion

The blocking flow shop scheduling problem has found many applications in manufacturing systems. There are a few exact methods for solving this problem with different criteria. In this paper, efforts will be made to optimize the total completion time criterion for this problem. We present two mixed binary integer programming models, one of which is based on the departure times of jobs from machines, and the other is based on the idle and blocking times of jobs. An initial upper bound generator and some lower bounds and dominance rules are also developed to be used in a branch and bound algorithm. The algorithm solves 17 instances of the Taillard's benchmark problem set in less than 20 min.     

考虑时间因素的0-1背包调度问题

文中提出考虑时间因素的0-1背包调度问题这一具有NP难度的组合优化问题。给定n个物体(每个物体i的重量为w_i,连续加工时间为t_i),以及一个容量为S的背包,要求给出一个调度方案(物品的放入顺序和放入时间),使得任意时刻放入背包的物品总重量不超过背包容量,每个物体需放入背包连续加工时长t_i后才能取出,该问题是求使所有物体均加工完毕的时间尽可能短的调度方案。提出了3种求解算法:迭代动态规划算法、基于分枝限界的完备算法和遗传进化算法。迭代动态规划算法使用动态规划策略放置尽可能多的未加工物体到背包中,然后每次迭代取出加工完成的物品后再使用动态规划放入尽可能多的剩余未加工物品,直至所有物品被加工完成。基于分枝限界的完备算法通过定义上下界及剪枝操作,有效地降低了算法的计算复杂度。遗传进化算法将一个物品装填序列定义为个体,并定义了相应的适应度、选择、交叉与变异操作。在所设计的3组共计36个算例上的实验结果表明,迭代动态规划算法可以很快求出高质量的解,基于分枝限界的完备算法对小规模算例有很好的效果,遗传算法在处理几百个物体的算例时能在1500s内得到比动态规划算法更好的结果。     

基于TSP方法求解等待时间受限的置换流水车间调度

等待时间受限的置换流水车间调度问题要求工件在连续两个机器间的等待时间满足上限值约束.对此,分析了工件序列中相邻工件的加工持续时间及其上下界关系,并且提出一种启发式方法.首先,建立旅行商间题（TSP）以生成初始调度;然后,采用扩展插入方法优化调度解.为了衡量算法性能,给出问题下界的计算方法和相关评价指标,并通过数据实验验证了该启发式和下界计算方法的可行性和有效性.     

Algorithms for a two‐machine flowshop problem with jobs of two classes

We consider a problem of scheduling jobs of two classes of urgencies in a two‐machine flowshop with the objective of minimizing total tardiness of one class for urgent jobs and the maximum completion time of the other class for non‐urgent jobs. Urgent jobs are an important consideration in the real manufacturing systems, but it has not been studied due to the difficulty of the problem. In this research, a branch‐and‐bound (B&B) algorithm is proposed through developing lower bounds, dominance properties, and heuristic algorithms for obtaining an initial feasible solution. To evaluate the performance of the proposed algorithms, computational experiments on randomly generated instances are performed. Results of the experiments show that the suggested B&B algorithm can solve problems with up to 20 jobs in a reasonable amount of CPU time.     

Cautious Greedy Strategy for Bearing‐only Active Localization: Analysis and Field Experiments

We study the problem of optimally choosing bearing measurement locations for localizing a stationary target in minimum time. The targets are transmitting radio tags, and bearing measurements are acquired from radio signal strength by a robot carrying a direction‐sensitive radio antenna. Actively localizing radio tags has many applications in surveillance, search and rescue, and environmental monitoring. Our work is motivated by the task of monitoring radio‐tagged invasive fish using autonomous vehicles. An active localization algorithm is provided in order to locate a target up to the desired uncertainty. The time required to locate the target includes time spent traveling as well as taking measurements. Since bearing measurements inferred from radio signals have an inherent ambiguity associated with them, the proposed algorithm chooses measurements to minimize the effect of ambiguous measurements on the target estimate. We present a closed‐form bound on the time required to locate a target using the presented active localization strategy. We also present the first known lower bound on the time required by any active localization algorithm (including the unknown optimal). Finally, we bound the ratio of the upper and lower bounds, showing that the expected cost of our algorithm is within a constant factor of the expected cost of the optimal solution. Robust initialization strategies that are motivated by practical sensing limitations are also provided. Our algorithm is shown to reliably locate radio tags to a desired uncertainty in simulations and multiple field experiments.     

The total completion time open shop scheduling problem with a given sequence of jobs on one machine

This paper addresses the open shop scheduling problem to minimize the total completion time, provided that one of the machines has to process the jobs in a given sequence. The problem is NP-hard in the strong sense even for the two-machine case. A lower bound is derived based on the optimal solution of a relaxed problem in which the operations on every machine may overlap except for the machine with a given sequence of jobs. This relaxed problem is NP-hard in the ordinary sense, however it can be quickly solved via a decomposition into subset-sum problems. Both heuristic and branch-and-bound algorithm are proposed. Experimental results show that the heuristic is efficient for solving large-scaled problems, and the branch-and-bound algorithm performs well on small-scaled problems.Scope and purposeShop scheduling problems, widely used in the modeling of industrial production processes, are receiving an increasing amount of attention from researchers. To model practical production processes more closely, additional processing restrictions can be introduced, e.g., the resource constraints, the no-wait in process requirement, the precedence constraints, etc. This paper considers the total completion time open shop scheduling problem with a given sequence of jobs on one machine. This model belongs to a new class of shop scheduling problems under machine-dependent precedence constraints. This problem is NP-hard in the strong sense. A heuristic is proposed to efficiently solve large-scaled problems and a branch-and-bound algorithm is presented to optimally solve small-scaled problems. Computational experience is also reported.     

Minimizing total completion time for re-entrant flow shop scheduling problems

In this paper, we study re-entrant flow shop scheduling problems with the objective of minimizing total completion time. In a re-entrant scheduling problem, jobs may visit some machines more than once for processing. The problem is NP-hard even for machine number m=2. A heuristic algorithm is presented to solve the problem, in which an effective k-insertion technique is introduced as the improvement strategy in iterations. Computational experiments and analyses are performed to give guidelines of choosing parameters in the algorithm. We also provide a lower bound for the total completion time of the optimal solution when there are only two machines. Objective function values of the heuristic solutions are compared with the lower bounds to evaluate the efficiency of the algorithm. For randomly generated instances, the results show that the given heuristic algorithm generates solutions with total completion times within 1.2 times of the lower bounds in most of the cases.