Knapsack-Based Approaches to the Makespan Problem on Multiple Processors

In this paper we study the problem of scheduling n independent jobs available at time zero on m ≥ 2 parallel and identical processors with the objective of minimizing the makespanJWe propose two approaches, both are Knapsack-based. The first is analytical and depends on reducing the set of m occupancy constraints to a single Diophantine equation. We capitalize on the very special structure of the ILP to effect the reduction with small multipliers. The second approach is an iterative heuristic procedure that is based on the observation that a two-machine makespan problem is trivially reduced to a Knapsack problem. Computational experience indicates the superiority of this approach over other existing approaches. Realistic problems of up to 100 jobs on 10 machines are solved in a few seconds on the IBM 370/165.     

A Benders decomposition approach to the weighted number of tardy jobs scheduling problem on unrelated parallel machines with production costs

This paper addresses the problem of scheduling on-time jobs on unrelated parallel machines with machine production costs. The objective is to maximise the net profit which is the sum of the weights of on-time jobs and the cost of using the machines. This scheduling problem is very important and frequent in industrial settings. It is herein solved using an exact approach that applies Benders decomposition to obtain tight upper and lower bounds and uses the bounds within a branch and bound. The computational investigation shows the efficacy of the approach in solving large instances. Most importantly, the proposed approach provides a new venue for solving large-scale scheduling problems.     

A decomposition-based approach to flexible flow shop scheduling under machine breakdown

Manufacturing systems in real-world production are generally dynamic and often subject to a wide range of uncertainties. Recently, research on production scheduling under uncertainty has attracted substantial attention. Although some methods have been developed to address this problem, scheduling under uncertainty remains inherently difficult to solve by any single approach. This article considers makespan optimisation of a flexible flow shop (FFS) scheduling problem under machine breakdown. It proposes a novel decomposition-based approach to decompose an FFS scheduling problem into several cluster scheduling problems which can be solved more easily by different approaches. A neighbouring K-means clustering algorithm is developed to first group the machines of an FFS into an appropriate number of machine clusters, based on a proposed machine allocation algorithm and weighted cluster validity indices. Two optimal back propagation networks, corresponding to the scenarios of simultaneous and non-simultaneous job arrivals, are then selectively adopted to assign either the shortest processing time (SPT) or the genetic algorithm (GA) to each machine cluster to solve cluster scheduling problems. If two neighbouring machine clusters are allocated with the same approach, they are subsequently merged. After machine grouping and approach assignment, an overall schedule is generated by integrating the solutions to the sub-problems. Computation results reveal that the proposed approach is superior to SPT and GA alone for FFS scheduling under machine breakdown.     

Bi-objective scheduling on uniform parallel machines considering electricity cost

This article investigates a bi-objective scheduling problem on uniform parallel machines considering electricity cost under time-dependent or time-of-use electricity tariffs, where electricity price changes with the hours within a day. The aim is to minimize simultaneously the total electricity cost and the number of machines actually used. A bi-objective mixed-integer linear programming model is first formulated for the problem. An insertion algorithm is then proposed for the single-objective scheduling problem of minimizing the total electricity cost for a given number of machines. To obtain the whole Pareto front of the problem, an iterative search framework is developed based on the proposed insertion algorithm. Computational results on real-life and randomly generated instances demonstrate that the proposed approach is quite efficient and can find high-quality Pareto fronts for large-size problems with up to 5000 jobs.     

Worst-case analysis of the LPT algorithm for single processor scheduling with time restrictions

We consider the following scheduling problem. We are given a set S of jobs which are to be scheduled sequentially on a single processor. Each job has an associated processing time which is required for its processing. Given a particular permutation of the jobs in S, the jobs are processed in that order with each job started as soon as possible, subject only to the following constraint: For a fixed integer \(B \ge 2\), no unit time interval \([x, x+1)\) is allowed to intersect more than B jobs for any real x. There are several real world situations for which this restriction is natural. For example, suppose in addition to the jobs being executed sequentially on a single main processor, each job also requires the use of one of B identical subprocessors during its execution. Each time a job is completed, the subprocessor it was using requires one unit of time to reset itself. In this way, it is never possible for more than B jobs to be worked on during any unit interval. In Braun et al. (J Sched 17: 399–403, 2014a) it is shown that this problem is NP-hard when the value B is variable and a classical worst-case analysis of List Scheduling for this situation has been carried out. We prove a tighter bound for List Scheduling for \(B\ge 3\) and we analyze the worst-case behavior of the makespan \(\tau _\mathrm{LPT}(S)\) of LPT (longest processing time first) schedules (where we rearrange the set S of jobs into non-increasing order) in relation to the makespan \(\tau _o(S)\) of optimal schedules. We show that LPT ordered jobs can be processed within a factor of \(2-2/B\) of the optimum (plus 1) and that this factor is best possible.     

Modelling and symmetry breaking in scheduling problems on batch processing machines

Problems of scheduling batch-processing machines to minimise the makespan are widely exploited in the literature, mainly motivated by real-world applications, such as burn-in tests in the semiconductor industry. These problems consist of grouping jobs in batches and scheduling them on machines. We consider problems where jobs have non-identical sizes and processing times, and the total size of each batch cannot exceed the machine capacity. The processing time of a batch is defined as the longest processing time among all jobs assigned to it. Jobs can also have non-identical release times, and in this case, a batch can only be processed when all jobs assigned to it are available. This paper discusses four different versions of batch scheduling problems, considering a single processing machine or parallel processing machines and considering jobs with or without release times. New mixed integer linear programming formulations are proposed as enhancements of formulations proposed in the literature, and symmetry breaking constraints are investigated to reduce the size of the feasible sets. Computational results show that the proposed formulations have a better performance than other models in the literature, being able to solve to optimality instances only considered before to be solved by heuristic procedures.     

含不一致任务重量的同型熔炼炉批调度优化

为求解含不一致任务重量的同型熔炼炉批调度问题,建立了最小化最大任务完工时间优化模型,设计了一种混合粒子群算法（HPSO）。算法使用随机生成的任务序列作为粒子,采用批首次匹配（BFF）规则对任务序列分批,最长加工时间（LPT）规则将批分配到批处理机,并提出了一种最小完工时间差（MCD）规则对LPT调度结果进行优化;为避免早熟,算法引入交叉和变异操作搜索最优解。通过仿真实验与SA、GA算法对比,实验结果表明算法具有良好的性能。     

Parallel machine scheduling with tool loading: a constraint programming approach

This paper presents constraint programming models that aim to solve scheduling and tool assignment problems in parallel machine environments. There are a number of jobs to be processed on parallel machines. Each job requires a set of tools, but limited number of tools are available in the system due to economic restrictions. The problem is to assign the jobs and the required tools to machines and to determine the schedule so that the makespan is minimised. Three constraint programming models are developed and compared with existing methods described in the literature.     

面向节能无关联平行机调度模型及分支定界法

工业企业,特别是高耗能行业,不仅要满足交货期和缩小生产周期的要求,而且不断优化能源配置,降低能耗。研究一类新的以延迟和能源消耗的加权最小为目标的生产调度问题。首先,描述问题并分析问题的复杂性。其次,建立混合整数线性规划模型。进一步,我们提出求解该问题的分支定界算法。最后,通过数值实验和数值试验,验证算法的有效性和高效性。     

Scheduling trucks in container terminals using a genetic algorithm

Trucks are the most popular transport equipment in most mega-terminals, and scheduling them to minimize makespan is a challenge that this article addresses and attempts to resolve. Specifically, the problem of scheduling a fleet of trucks to perform a set of transportation jobs with sequence-dependent processing times and different ready times is investigated, and the use of a genetic algorithm (GA) to address the scheduling problem is proposed. The scheduling problem is formulated as a mixed integer program. It is noted that the scheduling problem is NP-hard and the computational effort required to solve even small-scale test problems is prohibitively large. A crossover scheme has been developed for the proposed GA. Computational experiments are carried out to compare the performance of the proposed GA with that of GAs using six popular crossover schemes. Computational results show that the proposed GA performs best, with its solutions on average 4.05% better than the best solutions found by the other six GAs.     

Meta-heuristics for the distributed two-stage assembly scheduling problem with bi-criteria of makespan and mean completion time

This paper focuses on the distributed two-stage assembly flowshop scheduling problem for minimising a weighted sum of makespan and mean completion time. This problem involves two inter-dependent decision sub-problems: (1) how to allocate jobs among factories and (2) how to schedule the assigned jobs at each factory. A mathematical model is formulated for solving the small-sized instances of the problem. Since the NP-hardness of the problem, we also proposed a variable neighbourhood search (VNS) algorithm and a hybrid genetic algorithm combined with reduced variable neighbourhood search (GA-RVNS) to solve the distributed two-stage assembly flowshop scheduling problems and approximately optimise makespan and mean completion time simultaneously. Computational experiments have been conducted to compare the performances of the model and proposed algorithms. For a set of small-sized instances, both the model and the proposed algorithms are effective. The proposed algorithms are further evaluated on a set of large-sized instances. The results statistically show that both GA-RVNS and VNS obtain much better performances than the GA without RVNS-based local search step (GA-NOV). For the instances with small numbers of jobs, VNS achieves better performances than GA-RVNS. However, for the instances with large numbers of jobs, GA-RVNS yields better performances than the VNS. It is also shown that the overall performances of VNS are very close to GA-RVNS with different numbers of factories, weights given to makespan and numbers of machines at the first stage.     

Bottleneck-based heuristics to minimize tardy jobs in a flexible flow line with unrelated parallel machines

This paper develops new bottleneck-based heuristics with machine selection rules to solve the flexible flow line problem with unrelated parallel machines in each stage and a bottleneck stage in the flow line. The objective is to minimize the number of tardy jobs in the problem. The heuristics consist of three steps: (1) identifying the bottleneck stage; (2) scheduling jobs at the bottleneck stage and the upstream stages ahead of the bottleneck stage; (3) using dispatching rules to schedule jobs at the downstream stages behind the bottleneck stage. A new approach is developed to find the arrival times of the jobs at the bottleneck stage, and two decision rules are developed to schedule the jobs on the bottleneck stage. This new approach neatly overcomes the difficulty of determining feasible arrival times of jobs at the bottleneck stage. In order to evaluate the performance of the proposed heuristics, six well-known dispatching rules are examined for comparison purposes. Six factors are used to design 729 production scenarios, and ten test problems are generated for each scenario. Computational results show that the proposed heuristics significantly outperform all the well-known dispatching rules. An analysis of the experimental factors is also performed and several interesting insights into the heuristics are discovered.     

Scheduling job shop associated with multiple routings with genetic and ant colony heuristics

The general job shop problem is one of the well known machine scheduling problems, in which the operation sequence of jobs are fixed that correspond to their optimal process plans and/or resource availability. Scheduling and sequencing problems, in general, are very difficult to solve to optimality and are well known as combinatorial optimisation problems. The existence of multiple job routings makes such problems more cumbersome and complicated. This paper addresses a job shop scheduling problem associated with multiple job routings, which belongs to the class of NP hard problems. To solve such NP-hard problems, metaheuristics have emerged as a promising alternative to the traditional mathematical approaches. Two metaheuristic approaches, a genetic algorithm and an ant colony algorithm are proposed for the optimal allocation of operations to the machines for minimum makespan time criterion. ILOG Solver, a scheduler package, is used to evaluate the performance of the proposed algorithms. The comparison reveals that both the algorithms are capable of providing solutions better than the solution obtained with ILOG Solver.     

Flexible job-shop scheduling problem under resource constraints

A flexible job-shop-scheduling problem is an extension of classical job-shop problems that permit an operation of each job to be processed by more than one machine. The research methodology is to assign operations to machines (assignment) and determine the processing order of jobs on machines (sequencing) such that the system objectives can be optimized. This problem can explore very well the common nature of many real manufacturing environments under resource constraints. A genetic algorithm-based approach is developed to solve the problem. Using the proposed approach, a resource-constrained operations–machines assignment problem and flexible job-shop scheduling problem can be solved iteratively. In this connection, the flexibility embedded in the flexible shop floor, which is important to today's manufacturers, can be quantified under different levels of resource availability.     

Minimising interference for scheduling two parallel machines with a single server

In this paper, we consider the problem of scheduling a set of jobs on two parallel machines with set-up times. The set-up has to be performed by a single server. The objective is to minimise the forced idle time. The problem of minimising the forced idle time (interference problem) is known to be unary NP-hard for the case of two machines and equal set-up and arbitrary processing times. We propose a mixed integer linear programming model, which describes a special class of schedules where the jobs from a list are scheduled alternatively on the machines, and a heuristic algorithm is tested on instances with up to 100,000 jobs. The computational results indicate that the algorithm has an excellent performance even for very large instances, where mostly an optimal solution is obtained within a very small computational time.     

A bi-objective parallel machine problem with eligibility,release dates and delivery times of the jobs

The scheduling of parallel machines is a well-known problem in many companies. Nevertheless, not always all the jobs can be manufactured in any machine and the eligibility appears. Based on a real-life problem, we present a model which has m parallel machines with different level of quality from the highest level for the first machine till the lowest level for the last machine. The set of jobs to be scheduled on these m parallel machines are also distributed among these m levels: one job from a level can be manufactured in a machine of the same or higher level but a penalty, depending on the level, appears when a job is manufactured in a machine different from the highest level i.e. different from the first machine. Besides, there are release dates and delivery times associated to each job. The tackled problem is bi-objective with the criteria: minimisation of the final date – i.e. the maximum for all the jobs of their completion time plus the delivery time – and the minimisation of the total penalty generated by the jobs. In a first step, we analyse the sub-problem of minimisation of the final date on a single machine for jobs with release dates and delivery times. Four heuristics and an improvement algorithm are proposed and compared on didactic examples and on a large set of instances. In a second step an algorithm is proposed to approximate the set of efficient solutions and the Pareto front of the bi-objective problem. This algorithm contains two phases: the first is a depth search phase and the second is a backtracking phase. The procedure is illustrated in detail on an instance with 20 jobs and 3 machines. Then extensive numerical experiments are realised on two different sets of instances, with 20, 30 and 50 jobs, 3 or 4 machines and various values of penalties. Except for the case of 50 jobs, the results are compared with the exact Pareto front.     

JOB ASSIGNMENT ALGORITHMS ON UNIFORM PARALLEL MACHINES

Uniform parallel machine scheduling problems with a makespan measure cannot generally be solved within polynomial time complexity. This paper considers special problems with a single type of job on the uniform parallel machines, where each machine is available at a given ready time. Also the machine can be restricted on the number of jobs to be processed. The objective is to develop job assignment or batching algorithms which minimize makespan. When all the machines are available at time zero and have no restriction on the number of assignable jobs, a lower bound and optimal solution properties are derived. Based upon these properties, a polynomial algorithm is suggested to find the optimal job assignment on each machine. Three generalized problems are considered under the following situations: (1) some machines have capacity restrictions on the production batch, (2) each machine has its ready time, and (3) the jobs require series-parallel operations. The generalized problems arc also characterized and polynomial algorithms are developed for the same aim of optimal job assignment, except for the case of series-parallel operations. A heuristic algorithm is suggested with numerical tests for the series-parallel operations problem     

Parallel machine scheduling problem with ready times,due times and sequence-dependent setup times using meta-heuristic algorithms

This article considers a parallel machine scheduling problem with ready times, due times and sequence-dependent setup times. The objective of this problem is to determine the allocation policy of jobs and the scheduling policy of machines to minimize the weighted sum of setup times, delay times and tardy times. A mathematical model for optimal solution is derived. An in-depth analysis of the model shows that it is very complicated and difficult to obtain optimal solutions as the problem size becomes large. Therefore, two meta-heuristics, genetic algorithm (GA) and a new population-based evolutionary meta-heuristic called self-evolution algorithm (SEA), are proposed. The performances of the meta-heuristic algorithms are evaluated through comparison with optimal solutions using several randomly generated examples.     

Parallel machine scheduling considering a job-splitting property

This paper focuses on the problem of scheduling jobs on parallel machines considering a job-splitting property. In this problem, it is assumed that a job can be split into a discrete number of subjobs and they are processed on parallel machines independently. A two-phase heuristic algorithm is suggested for the problem with the objective of minimizing total tardiness. In the first phase, an initial sequence is constructed by an existing heuristic method for the parallel-machine scheduling problem. In the second phase, each job is split into subjobs considering possible results of the split, and then jobs and subjobs are rescheduled on the machines using a certain method. To evaluate performance of the suggested algorithm, computational experiments are performed on randomly generated test problems. Results of the experiments show that the suggested algorithm performs better than an existing one.