Minimizing maximum earliness on parallel identical machines

This paper addresses the scheduling problem of minimizing maximum earliness (or more generally — maximizing minimum lateness) on parallel identical machines. We prove that the two-machine case is NP-hard in the ordinary sense, and introduce a pseudo-polynomial dynamic programming algorithm for this case. When the number of machines is arbitrary, the problem is shown to be NP-hard in the strong sense. Then we introduce an efficient heuristic and two simple upper bounds on the optimal minimum lateness value. Finally we provide an extensive numerical study which indicates that the heuristic performs well in various job and machine settings.Scope and purposeIn recent years many researchers have focused on minimizing both earliness and tardiness costs. Only a few studies have considered problems with (maximum or total) earliness as the sole performance measure. We believe that the earliness measure is appropriate for many real-life settings, where the main cost component is the earliness (inventory) cost, and the tardiness (positive lateness) cost component is negligible. Our paper studies the scheduling problem of minmax earliness on parallel identical machines: we analyze the complexity of the problem, and introduce an efficient heuristic and simple bounds on the optimal cost.     

On the equivalence of single machine earliness/tardiness problems with job rejection

We present optimal algorithms for single-machine scheduling problems with earliness criteria and job rejection and compare them with the algorithms for the corresponding problems with tardiness objectives. We present an optimal O(n log n) algorithm for minimizing the maximum earliness on a single machine with job rejection. Our algorithm also solves the bi-criteria scheduling problem is which the objective is to simultaneously minimize the maximum earliness of the scheduled jobs and the total rejection cost of the rejected jobs. We also show that the optimal pseudo-polynomial time algorithm for the total tardiness problem with job rejection can be used to solve the corresponding total earliness problem with job rejection.     

Local and global dominance conditions for the weighted earliness scheduling problem with no idle time

In this paper, we present dominance conditions for the single machine weighted earliness scheduling problem with no idle time. We also propose an algorithm that can be used to improve upper bounds for the weighted earliness criterion and lower bounds for an earliness/tardiness problem. The computational tests show that the algorithm is superior to an initial heuristic schedule and an existing adjacency condition.     

Local and global dominance conditions for the weighted earliness scheduling problem with no idle time

In this paper, we present dominance conditions for the single machine weighted earliness scheduling problem with no idle time. We also propose an algorithm that can be used to improve upper bounds for the weighted earliness criterion and lower bounds for an earliness/tardiness problem. The computational tests show that the algorithm is superior to an initial heuristic schedule and an existing adjacency condition.     

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 approach to early/tardy scheduling with release dates

In this paper, we consider the single machine earliness/tardiness scheduling problem with different release dates and no unforced idle time. The problem is decomposed into weighted earliness and weighted tardiness subproblems. Lower bounding procedures are proposed for each of these subproblems, and the lower bound for the original problem is the sum of the lower bounds for the two subproblems. The lower bounds and several versions of a branch-and-bound algorithm are then tested on a set of randomly generated problems, and instances with up to 30 jobs are solved to optimality. To the best of our knowledge, this is the first exact approach for the early/tardy scheduling problem with release dates and no unforced idle time.     

多品种小批量粮油机械制造业流水车间JIT调度问题研究

粮油机械制造业是为粮油加工企业提供技术装备的重要产业。随着人民群众对粮油的品种、质量的要求越来越高,传统的大批量生产方式逐渐被多品种、小批量现代化大生产模式所取代。作为提供粮油工业装备的行业,生产技术和传统工艺革新迫在眉睫。调度作为保证制造车间有序生产、稳定运行的决定性因素,传统算法很难针对准时制(just-in-time,JIT)要求下流水车间调度问题进行建模和有效求解。在此提出一种基于JIT的遗传算法和模拟退火算法的混合式智能优化算法,采用流水车间10×5 Benchmark进行了调度性能基准测试。结果表明,针对多品种、小批量粮油机械制造流水车间提前/拖期调度问题,搜索效率高,解质量稳定,具有较好地全局优化能力。     

解决并行多机提前/拖后调度问题的混合遗传算法方法

研究了带有公共交货期的并行多机提前/拖后调度问题.提出了一种混合遗传算法 方法,以便于确定公共交货期和每台机器上加工的任务代号及其加工顺序,即找到一个最优 公共交货期和最优调度,使加工完所有任务后交货期安排的成本、提前交货成本和拖后交货 成本的总和最小.数值计算结果表明了该混合遗传算法优于启发式算法,并能适用于较大规 模并行多机提前/拖后调度问题.算法计算量小,鲁棒性强.     

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.     

Benchmarks for scheduling on a single machine against restrictive and unrestrictive common due dates

We consider the NP-hard problem of scheduling jobs on a single machine against common due dates with respect to earliness and tardiness penalties. The paper covers two aspects: Firstly, we develop a problem generator and solve 280 instances with two new heuristics to obtain upper bounds on the optimal objective function value. Secondly, we demonstrate computationally that our heuristics are efficient in obtaining near-optimal solutions for small problem instances. The generated problem instances in combination with the upper bounds can be used as benchmarks for future approaches in the field of common due-date scheduling.Scope and purposeIn connection with just-in-time production and delivery, earliness as well as tardiness penalties are of interest. Thus scheduling against common due dates has received growing attention during the last decade. Many algorithms have been developed to solve the different variants of this problem. But whenever a new algorithm for scheduling against common due dates is proposed, its quality is assessed only on a few self-generated examples. Hence it is difficult to evaluate the various approaches, particularly in comparison with each other. Therefore the goal of this paper is to present numerous benchmark problems together with some upper bounds on the optimal objective function value.     

Earliness–tardiness scheduling with setup considerations

The one-machine scheduling problem with sequence-dependent setup times and costs and earliness–tardiness penalties is addressed. A time-indexed formulation of the problem is presented as well as different relaxations that give lower bounds of the problem. Then, a branch-and-bound procedure based on one of these lower bounds is presented. The efficiency of this algorithm also relies on new dominance rules and on a heuristic to derive good feasible schedules. Computational tests are finally presented.     

Scheduling job classes on uniform machines

We study a scheduling problem with job classes on parallel uniform machines. All the jobs of a given class share a common due-date. General, non-decreasing and class-dependent earliness and tardiness cost functions are assumed. Two objectives are considered: (i) minmax, where the scheduler is required to minimize the maximum earliness/tardiness cost among all the jobs and (ii) minmax-minsum, where the scheduler minimizes the sum of the maximum earliness/tardiness cost in all job classes. The problem is easily shown to be NP-hard, and we focus here on the introduction of simple heuristics. We introduce LPT (Largest Processing Time first)-based heuristics for the allocation of jobs to machines within each class, followed by a solution of an appropriate non-linear program, which produces for this job allocation an optimal schedule of the classes. We also propose a lower bound, based on balancing the load on the machines. Our numerical tests indicate that the heuristics result in very small optimality gaps.     

Minimization of earliness, tardiness and due date penalties on uniform parallel machines with identical jobs

We consider a problem of scheduling n identical nonpreemptive jobs with a common due date on m uniform parallel machines. The objective is to determine an optimal value of the due date and an optimal allocation of jobs onto machines so as to minimize a total cost function, which is the function of earliness, tardiness and due date values. For the problem under study, we establish a set of properties of an optimal solution and suggest a two-phase algorithm to tackle the problem. First, we limit the number of due dates one needs to consider in pursuit of optimality. Next, we provide a polynomial-time algorithm to build an optimal schedule for a fixed due date. The key result is an O(m² log m) algorithm that solves the main problem to optimality.Scope and purpose: To extend the existing research on cost minimization with earliness, tardiness and due date penalties to the case of uniform parallel machines.     

交货期窗口下带有附加惩罚的单机提前/拖期调度问题

交货期窗口下的交货期确定和排序问题是调度领域研究的一个方面,本文对交货期口下的单机作业问题进行了研究,目标函数不仅考虑提前／拖期惩罚,还考虑附加惩罚,假设如果任务在交货期窗口内完工,则不受提前／拖期片罚;如果在交货期窗口外完工,将导致提前／拖期惩罚,本文确定了最优公共交货期,给出了相庆的最优排序,并提出了一个多项式时间算法确定了使目标函数为最小的最优调度,最后的数值例子说明了算法的有效性。     

Notes on “some single-machine scheduling problems with general position-dependent and time-dependent learning effects”

This paper provides a continuation of the idea presented by Yin et al. [Yin et al., Some scheduling problems with general position-dependent and time-dependent learning effects, Inform. Sci. 179 (2009) 2416-2425]. For each of the following three objectives, total weighted completion time, maximum lateness and discounted total weighted completion time, this paper presents an approximation algorithm which is based on the optimal algorithm for the corresponding single-machine scheduling problem and analyzes its worst-case bound. It shows that the single-machine scheduling problems under the proposed model can be solved in polynomial time if the objective is to minimize the total lateness or minimize the sum of earliness penalties. It also shows that the problems of minimizing the total tardiness, discounted total weighted completion time and total weighted earliness penalty are polynomially solvable under some agreeable conditions on the problem parameters.     

用模拟退火法解任务的加工时间为模糊区间数的单机提前/拖期调度问题

研究了单机作业下任务的加工时间为模糊区间数的提前／拖期调度问题。     

A local-best harmony search algorithm with dynamic sub-harmony memories for lot-streaming flow shop scheduling problem

In this paper, a local-best harmony search (HS) algorithm with dynamic sub-harmony memories (HM), namely DLHS algorithm, is proposed to minimize the total weighted earliness and tardiness penalties for a lot-streaming flow shop scheduling problem with equal-size sub-lots. First of all, to make the HS algorithm suitable for solving the problem considered, a rank-of-value (ROV) rule is applied to convert the continuous harmony vectors to discrete job sequences, and a net benefit of movement (NBM) heuristic is utilized to yield the optimal sub-lot allocations for the obtained job sequences. Secondly, an efficient initialization scheme based on the NEH variants is presented to construct an initial HM with certain quality and diversity. Thirdly, during the evolution process, the HM is dynamically divided into many small-sized sub-HMs which evolve independently so as to balance the fast convergence and large diversity. Fourthly, a new improvisation scheme is developed to well inherit good structures from the local-best harmony vector in the sub-HM. Meanwhile, a chaotic sequence to produce decision variables for harmony vectors and a mutation scheme are utilized to enhance the diversity of the HM. In addition, a simple but effective local search approach is presented and embedded in the DLHS algorithm to enhance the local searching ability. Computational experiments and comparisons show that the proposed DLHS algorithm generates better or competitive results than the existing hybrid genetic algorithm (HGA) and hybrid discrete particle swarm optimization (HDPSO) for the lot-streaming flow shop scheduling problem with total weighted earliness and tardiness criterion.     

并行机成组调度问题的启发式算法

研究了优化目标为总拖后/提前时间最小化的并行机成组调度问题,提出了一种三阶段启发式近似求解算法。首先把并行机问题看成单机问题,以最小化总拖后时间为优化目标排列工件的加工次序;然后将工件按第一阶段所求得的次序指派到最先空闲的并行的机器上;最后采用改进的GTW算法对各机器上的工件调度插入适当的空闲时间。计算表明该算法能够在很短的时间内给出大规模调度问题的近似最优解。     

A branch-and-bound algorithm for single machine scheduling with quadratic earliness and tardiness penalties

This paper considers the problem of scheduling a single machine, in which the objective function is to minimize the weighted quadratic earliness and tardiness penalties and no machine idle time is allowed. We develop a branch and bound algorithm involving the implementation of lower and upper bounding procedures as well as some dominance rules. The lower bound is designed based on a lagrangian relaxation method and the upper bound includes two phases, one for constructing initial schedules and the other for improving them. Computational experiments on a set of randomly generated instances show that one of the proposed heuristics, used as an upper bound, has an average gap less than 1.3% for instances optimally solved. The results indicate that both the lower and upper bounds are very tight and the branch-and-bound algorithm is the first algorithm that is able to optimally solve problems with up to 30 jobs in a reasonable amount of time.     

软计算求解分布式多工厂单件制造业的提前/拖期生产计划问题

描述了分布式多工厂单件制造企业准时化生产计划问题, 以实现最小化提前/拖期惩罚费用、生产成本、产品运输费用之和为目标建立了0-1规划数学模型; 设计了基于模糊规则量化的方法求解模糊决策, 并将模糊决策嵌入到遗传算法中的软计算方法求解模型, 使得算法具有比分枝定界法更快速的寻找优解的能力以及更广泛的适应范围. 结果表明了该模型和算法的有效性和应用潜力.