SCHEDULING A SINGLE MACHINE TO MINIMIZE TWO CRITERIA: MAXIMUM TARDINESS AND NUMBER OF TARDY JOBS

In this paper we consider a single machine scheduling problem with two criteria; minimizing both maximum tardiness and the number of tardy jobs. We present both heuristic and branch-and-bound algorithms to find the schedule which minimizes the number of tardy jobs among all schedules having minimal maximum tardiness. Computational results show that problems with up to 40 jobs can be solved in less than one minute of computer time, and solution difficulty tends to increase as the range of due dates increases relative to the total processing time. We extend our results to generate all nondominated schedules for the two criteria. Computational experiments indicate that all non-dominated solutions to problems with 40 jobs can be generated. However, solution difficulty for these problems is highly dependent on problem parameters.     

On-line maintenance of optimal machine schedules

Effective and efficient scheduling in a dynamically changing environment is important for real-time control of manufacturing, computer, and telecommunication systems. This paper illustrates the algorithmic and analytical issues associated with developing efficient and effective methods to update schedules on-line. We consider the problem of dynamically scheduling precedence-constrained jobs on a single processor to minimize the maximum completion time penalty. We first develop an efficient technique to reoptimize a rolling schedule when new jobs arrive. The effectiveness of reoptimizing the current schedule as a long-term on-line strategy is measured by bounding its performance relative to oracles that have perfect information about future job arrivals.     

Minimizing earliness–tardiness on a single burn-in oven with a common due date and maximum allowable tardiness constraint

This paper considers a scheduling problem for a single burn-in oven in the semiconductor manufacturing industry where the oven is a batch processing machine with restricted capacity. The batch processing time is set by the longest processing time among those of all the jobs contained in the batch. All jobs are assumed to have the same due date. The objective is to minimize the sum of the absolute deviations of completion times from the due date (earliness–tardiness) of all jobs under the constraint that the maximum tardiness should be less than or equal to the maximum allowable time value. We suggest several two-phase heuristic algorithms for this problem. In the first phase, some heuristic algorithms are developed without maximum allowable tardiness constraint. If the schedule from the first phase violates the maximum tardiness constraint, then the schedule is changed to satisfy maximum allowable tardiness constraint in the second phase. The suggested heuristics are based on genetic algorithms and dominance properties of optimal schedules. We present the results of computational experiments that clearly show the solution quality obtained by the suggested heuristics.     

Scheduling a Single Machine to Minimize the Weighted Number of Tardy Jobs

In this paper the problem of scheduling jobs on a single machine to minimize the weighted number of tardy jobs is examined. It contains the framework for a new branch-and-bound procedure as well as the first extensive computational study of the problem. Results indicate that large problems, e.g. 50 jobs, can be solved in just a few seconds of computer time. Further, the computational results provide insight into how various problem parameters affect the solution difficulty of particular problems.     

Scheduling a tempered glass manufacturing system: a three-stage hybrid flow shop model

Chinese tempered glass has entered a fast and stable growing era. To improve the productivity of tempered glass manufacturers, this paper investigates a scheduling problem in tempered glass production system, originated from a tempered glass manufacturer in China. This problem can be formulated as a three-stage hybrid flow shop (HFS). Single and batch processing machines coexist in this HFS. Besides, a limited buffer, between the first two stages, and machine eligibility requirement are also significant characteristics. To address this complicated scheduling problem, we first establish an integer programming model with the objective of minimising the makespan, i.e. the maximum completion time of jobs in the system. Due to the strong NP-hard nature of the problem, we then propose a constructive heuristic method, a genetic algorithm, as well as a simulated annealing algorithm, to solve practical large-scale problems. Computational results demonstrate the efficiency of the proposed approaches.     

Single-machine earliness–tardiness scheduling with two competing agents and idle time

Two-agent scheduling has gained a lot of research attention recently. Two competing agents who have their own objective functions have to perform their respective set of jobs on one or more shared machines. This study considers a two-agent single-machine earliness and tardiness scheduling problem where jobs have distinct due dates and unforced idleness in between any two consecutive jobs is allowed. The objective is to minimize the total earliness and tardiness of jobs from one agent given that the maximum earliness–tardiness of jobs from the other agent cannot exceed an upper bound. In other words, each job from the second agent has a hard due window, whereas each job from the first agent will incur a penalty if completed either before or after its due date. Two mathematical models of the problem are presented, and several necessary optimality conditions are derived. By exploiting the established dominance properties, heuristic algorithms are developed for the problem. Finally, computational experiments are conducted to assess the models and heuristic procedures.     

Probabilistic asymptotic analysis of stochastic online scheduling problems

In the stochastic online scheduling environment, jobs with unknown release times and weights arrive over time. Upon arrival, the information on the weight of the job is revealed but the processing requirement remains unknown until the job is finished. In this paper we consider the objective of minimizing the total weighted completion time. With the assumptions that job weights are bounded, machine capacity is adequate, and processing requirements are bounded and identical and independently distributed across the machines and jobs, we show that any nondelay algorithm is asymptotically optimal for the stochastic online single machine problem, flow shop problem, and uniform parallel machine problem. Our simulation studies of these stochastic online scheduling problems show that two generic nondelay algorithms perform very well as long as the number of jobs is larger than 100.     

A heuristic model for dynamic flexible job shop scheduling problem considering variable processing times

In real scheduling problems, unexpected changes may occur frequently such as changes in task features. These changes cause deviation from primary scheduling. In this article, a heuristic model, inspired from Artificial Bee Colony algorithm, is proposed for a dynamic flexible job-shop scheduling (DFJSP) problem. This problem consists of n jobs that should be processed by m machines and the processing time of jobs deviates from estimated times. The objective is near-optimal scheduling after any change in tasks in order to minimise the maximal completion time (Makespan). In the proposed model, first, scheduling is done according to the estimated processing times and then re-scheduling is performed after determining the exact ones considering machine set-up. In order to evaluate the performance of the proposed model, some numerical experiments are designed in small, medium and large sizes in different levels of changes in processing times and statistical results illustrate the efficiency of the proposed algorithm.     

Scheduling with generalized batch delivery dates and earliness penalties

In this paper, we study some single machine scheduling problems with generalized batch delivery dates and earliness penalties. The generalized delivery dates are given a-priori before any jobs are processed. They are unrelated to the jobs and the processing order. Each specific delivery batch contains jobs completed but undelivered until the specific delivery date. We consider scheduling problems to minimize two types of earliness penalties: one is the total earliness; the other is the maximum earliness. For these two problems, first we show that they are NP-hard in the strong sense for general cases; then we prove that they are still NP-hard even if there are only two generalized batch dates. We also prove that they are solved in polynomial time for general earliness penalty function if all processing times are equal, and give an O(n log(n)) algorithm to solve the weighted earliness cases.     

Continuous-time algorithms for scheduling a single machine with sequence-dependent setup times and time window constraints in coordinated chains

In this paper we address the problem of selecting and scheduling several jobs on a single machine with sequence-dependent setup times and strictly enforced time window constraints on the start time of each job. We use short-term production targets to coordinate decentralised local schedulers and to make the objectives of specific areas in line with the chain objectives by maintaining a desired work in process profile in manufacturing environments. The existing literature in this domain is based on discrete-time approaches. We depart from prior approaches by considering continuous time. We introduce a two-step mathematical programming model based on disjunctive constraints to solve small problems to optimality, and propose an insertion-based heuristic to solve large-scale instances. We provide several variations of the insertion heuristic based on different score functions. The primary objective of these approaches is to maximise the total defined score for jobs while satisfying production targets for families of jobs in each shift. Further, our models minimise the maximum completion time of all selected jobs. The effectiveness, efficiency, and robustness of the proposed algorithms are analysed and compared with the existing literature.     

Scheduling with alternatives: a link between process planning and scheduling

The objective of this research is to develop and evaluate effective, computationally efficient procedures for scheduling jobs in a large-scale manufacturing system involving, for example, over 1000 jobs and over 100 machines. The main performance measure is maximum lateness; and a useful lower bound on maximum lateness is derived from a relaxed scheduling problem in which preemption of jobs is based on the latest finish time of each job at each machine. To construct a production schedule that minimizes maximum lateness, an iterative simulation-based scheduling algorithm operates as follows: (a) job queuing times observed at each machine in the previous simulation iteration are used to compute a refined estimate of the effective due date (slack) for each job at each machine; and (b) in the current simulation iteration, jobs are dispatched at each machine in order of increasing slack. Iterations of the scheduling algorithm terminate when the lower bound on maximum lateness is achieved or the iteration limit is reached. This scheduling algorithm is implemented in Virtual Factory, a Windows-based software package. The performance of Virtual Factory is demonstrated in a suite of randomly generated test problems as well as in a large furniture manufacturing facility. To further reduce maximum lateness, a second scheduling algorithm also incorporates a tabu search procedure that identifies process plans with alternative operations and routings for jobs. This enhancement yields improved schedules that minimize manufacturing costs while satisfying job due dates. An extensive experimental performance evaluation indicates that in a broad range of industrial settings, the second scheduling algorithm can rapidly identify optimal or nearly optimal schedules.     

A method combining rules with genetic algorithm for minimizing makespan on a batch processing machine with preventive maintenance

This paper considers the problem of minimising makespan on a single batch processing machine with flexible periodic preventive maintenance. This problem combines two sub-problems, scheduling on a batch processing machine with jobs’ release dates considered and arranging the preventive maintenance activities on a batch processing machine. The preventive maintenance activities are flexible but the maximum continuous working time of the machine, which is allowed, is determined. A mathematical model for integrating flexible periodic preventive maintenance into batch processing machine problem is proposed, in which the grouping of jobs with incompatible job families, the starting time of batches and the preventive maintenance activities are optimised simultaneously. A method combining rules with the genetic algorithm is proposed to solve this model, in which a batching rule is proposed to group jobs with incompatible job families into batches and a modified genetic algorithm is proposed to schedule batches and arrange preventive maintenance activities. The computational results indicate the method is effective under practical problem sizes. In addition, the influences of jobs’ parameters on the performance of the method are analyzed, such as the number of jobs, the number of job families, jobs’ processing time and jobs’ release time.     

On bilevel machine scheduling problems

Bilevel scheduling problems constitute a hardly studied area of scheduling theory. In this paper, we summarise the basic concepts of bilevel optimisation, and discuss two problem classes for which we establish various complexity and algorithmic results. The first one is the bilevel total weighted completion time problem in which the leader assigns the jobs to parallel machines and the follower sequences the jobs assigned to each machine. Both the leader and the follower aims to minimise the total weighted completion time objective, but with different job weights. When the leader’s weights are arbitrary, the problem is NP-hard. However, when all the jobs are of unit weight for the leader, we provide a heuristic algorithm based on iterative LP-rounding along with computational results, and provide a sufficient condition when the LP-solution is integral. In addition, if the follower weights induce a monotone (increasing or decreasing) processing time order in any optimal solution, the problem becomes polynomially solvable. As a by-product, we characterise a new polynomially solvable special case of the MAX m-CUT problem, and provide a new linear programming formulation for the P||?_j C_j{P||\sum_j C_j} problem. Finally, we present some results on the bilevel order acceptance problem, where the leader decides on the acceptance of orders and the follower sequences the jobs. Each job has a deadline and if a job is accepted, it cannot be late. The leader’s objective is to maximise the total weight of accepted jobs, whereas the follower aims at minimising the total weighted job completion times. For this problem, we generalise some known single-level machine scheduling algorithms.     

带成组加工的二阶段柔性流水作业问题

本文仔细剖析混杂二阶段流水作业问题,其中第一阶段由m台同型机组成,第二阶段由一台批处理机M组成,并以最大完工时间Cmax为极小化目标函数.我们证明了该类问题除一种情况有多项式时间可解外,其余情况为(强)NP-hatd的.文中对所有(强)NP-hard情况均给出了近似算法并作了性能比分析.     

A novel hybrid meta-heuristic algorithm for a no-wait flexible flow shop scheduling problem with sequence dependent setup times

In this paper, we contemplate the problem of scheduling a set of n jobs in a no-wait flexible flow shop manufacturing system with sequence dependent setup times to minimising the maximum completion time. With respect to NP-hardness of the considered problem, there seems to be no avoiding application of metaheuristic approaches to achieve near-optimal solutions for this problem. For this reason, three novel metaheuristic algorithms, namely population based simulated annealing (PBSA), adapted imperialist competitive algorithm (AICA) and hybridisation of adapted imperialist competitive algorithm and population based simulated annealing (AICA?+?PBSA), are developed to solve the addressed problem. Because of the sensitivity of our proposed algorithm to parameter's values, we employed the Taguchi method as an optimisation technique to extensively tune different parameters of our algorithm to enhance solutions accuracy. These proposed algorithms were coded and tested on randomly generated instances, then to validate the effectiveness of them computational results are examined in terms of relative percentage deviation. Moreover, some sensitive analyses are carried out for appraising the behaviour of algorithms versus different conditions. The computational evaluations manifestly support the high performance of our proposed novel hybrid algorithm against other algorithms which were applied in literature for related production scheduling problems.     

Scheduling a no-wait flow shop containing unbounded batching machines

We study the problem of scheduling n jobs in a no-wait flow shop consisting of m batching machines. Each job has to be processed by all the machines. All jobs visit the machines in the same order. A job completed on an upstream machine should be immediately transferred to the downstream machine. Batching machines can process several jobs simultaneously in a batch so that all jobs of the same batch start and complete together. The processing time of a batch is equal to the maximum processing time of the jobs in this batch. We assume that the capacity of any batch is unbounded. The problem is to find an optimal batch schedule such that the maximum job completion time, that is the makespan, is minimized. For m = 2, we prove that there exists an optimal schedule with at most two batches and construct such a schedule in O(n log n) time. For m = 3, we prove that the number of batches can be limited to nine and give an example where all optimal schedules have seven batches. Furthermore, we prove that the best schedules with at most one, two and three batches are 3-, 2- and 3/2-approximate solutions, respectively. The first two bounds are tight for corresponding schedules. Finally, we suggest an assignment method that solves the problem with m machines and at most r batches in O(n^{m(r-2)+1+[m/r]}) time, if m and r are fixed. The method can be generalized to minimize an arbitrary maximum cost or total cost objective function.     

Scheduling with shutdowns and sequence dependent set-up times

This paper considers a single machine scheduling problem with ready and due times constraints on jobs, shutdown constraints on the machine and sequence dependent set-up times among jobs. The shutdown is a disruptive event such as holiday, breaks or machine maintenance, and has a prespecified period when the machine will be interrupted. If no pre-emption is allowed for jobs, shutdown constraints divide the planning horizon into disconnected time windows. An optimization algorithm based on the branch-and-bound method is developed to minimize the maximum tardiness for solving the problem. This paper further develops the post-processing algorithm that manipulates the starting time of the shutdown period so as to reduce the obtained maximum tardiness. The post-processing algorithm can determine plural schedules to reduce the maximum tardiness, and the production manager will select the objective schedule among them for the interest of overall efficiency. Computational results for the proposed algorithms will indicate that the post-processing algorithm can improve upon the original solution and the problems with multiple shutdowns and with set-up times varying widely can be satisfactorily solved.     

Working time constraints in operational fixed job scheduling

In this study we consider the operational fixed job scheduling problem under working time limitations. The problem has several practical implications in both production and service operations; however the relevant research is scarce. We analyse pre-emptive and non pre-emptive versions of the problem and its special cases. We provide polynomial-time algorithms for some special cases. We show that the non pre-emptive jobs problem is strongly NP-hard, and propose a branch-and-bound algorithm that employs efficient bounding procedures and dominance properties. We conduct a numerical experiment to observe the effects of parameters on the quality of the solution. The results of our computational tests for the branch-and-bound algorithm reveal that our algorithm can solve the instances with up to 100 jobs in reasonable times. To the best of our knowledge our branch-and-bound algorithm is the first optimisation attempt to solve the problem.     

A flowshop scheduling problem with two operations per job

This paper considers a special case of the flowshop scheduling problem where each job requires only two operations on specified machines and shows that this problem is NP-hard in the strong sense even if the first operation of all jobs is processed on the same machine and the number of machines performing the second operation equals two. For the case when the first operation of all jobs is performed on the same machine, it is sufficient to consider only permutation schedules for minimizing any regular measure of performance. Five polynomially bounded heuristic algorithms are described for minimizing makespan for this case and their performance in finding a minimum makespan schedule is theoretically and empirically evaluated.     

Two-machine flow shop problem with unit-time operations and conflict graph

This paper addresses the problem of scheduling, on a two-machine flow shop, a set of unit-time operations subject to the constraints that some conflicting jobs cannot be scheduled simultaneously on different machines. In the context of our study, these conflicts are modelled by general graphs. The problem of minimising the maximum completion time (makespan) is known to be NP-hard in the strong sense. We propose a mixed-integer linear programming (MILP) model. Then, we develop a branch and bound algorithm based on new lower and upper bound procedures. We further provide a computer simulation to measure the performance of the proposed approaches. The computational results show that the branch and bound algorithm outperforms the MILP model and can solve instances of size up to 20,000 jobs.