Makespan minimisation for a parallel machine scheduling problem with preemption and job incompatibility

In this paper, an extension of the graph colouring problem is introduced to model a parallel machine scheduling problem with job incompatibility. To get closer to real-world applications, where the number of machines is limited and jobs have different processing times, each vertex of the graph requires multiple colours and the number of vertices with the same colour is bounded. In addition, several objectives related to scheduling are considered: makespan, number of pre-emptions and summation over the jobs’ throughput times. Different solution methods are proposed, namely, two greedy heuristics, two tabu search methods and an adaptive memory algorithm. The latter uses multiple recombination operators, each one being designed for optimising a subset of objectives. The most appropriate operator is selected dynamically at each iteration, depending on its past performance. Experiments show that the proposed algorithm is effective and robust, while providing high-quality solutions on benchmark instances for the graph multi-colouring problem, a simplification of the considered 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.     

Loading and sequencing heuristics for job scheduling on two unrelated parallel machines with long,sequence-dependent set-up times

The purpose of this paper is to develop and test intelligible heuristics for the scheduling of production orders that can easily be used in practice. Grounded in a case study, this paper examines the combined effects of assignment and sequencing heuristics on commonly used performance indicators. Discrete event simulation is used in the analysis to adequately capture the complexity found in the case study: production orders differing in many aspects, two unrelated parallel machines with varying and product-specific speed, and set-up times that depend on the (dis)similarity of successive orders. Evaluating 108 strategy–scenario combinations including the base case derived from the case study, it is found that a loading heuristic based on order quantity and scheduled capacity in combination with the shortest set-up heuristic performs best. When compared to the heuristic approach used by the case company, this strategy saves about 13.9% of total machine busy time and increases service level by 10.2%. In addition, using a reduced set of 40 production orders we are able to demonstrate that the best heuristic strategies comes close to results generated in a two-stage optimisation. The gap to optimality is only 3.1% in total busy time on average over all scenarios.     

Genetic algorithm with local search for the unrelated parallel machine scheduling problem with sequence-dependent set-up times

In this paper, a genetic algorithm (GA) with local search is proposed for the unrelated parallel machine scheduling problem with the objective of minimising the maximum completion time (makespan). We propose a simple chromosome structure consisting of random key numbers in a hybrid genetic-local search algorithm. Random key numbers are frequently used in GAs but create additional difficulties when hybrid factors are implemented in a local search. The best chromosome of each generation is improved using a local search during the algorithm, but the better job sequence (which might appear during the local search operation) must be adapted to the chromosome that will be used in each successive generation. Determining the genes (and the data in the genes) that would be exchanged is the challenge of using random numbers. We have developed an algorithm that satisfies the adaptation of local search results into the GAs with a minimum relocation operation of the genes’ random key numbers – this is the main contribution of the paper. A new hybrid approach is tested on a set of problems taken from the literature, and the computational results validate the effectiveness of the proposed algorithm.     

Makespan minimisation on parallel batch processing machines with non-identical job sizes and release dates

The problem we study in this paper arises from the washing step of hospital sterilisation services. Washers in the washing step are capable of handling more than one medical device set as long as their capacity is not exceeded. The medical device set sizes and arrival times to the sterilisation service may be different, but they all have the same washing duration. Thus, we model the washing step as a batch scheduling problem where medical device sets are treated as jobs with non-identical sizes and release dates, but equal processing times. The main findings we present in this paper are the following. First, we study two special cases for which polynomial algorithms are presented. We then develop a 2-approximation algorithm for the general problem. Finally, we develop a MILP model and compare it with another MILP model from the literature. Computational results show that our MILP model outperforms the model from the literature.     

Robust makespan minimisation in identical parallel machine scheduling problem with interval data

Parallel machine scheduling problems are commonly encountered in a wide variety of manufacturing environments and have been extensively studied. This paper addresses a makespan minimisation scheduling problem on identical parallel machines, in which the specific processing time of each job is uncertain, and its probability distribution is unknown because of limited information. In this case, the deterministic or stochastic scheduling model may be unsuitable. We propose a robust (min–max regret) scheduling model for identifying a robust schedule with minimal maximal deviation from the corresponding optimal schedule across all possible job-processing times (called scenarios). These scenarios are specified as closed intervals. To solve the robust scheduling problem, which is NP-hard, we first prove that a regret-maximising scenario for any schedule belongs to a finite set of extreme point scenarios. We then derive two exact algorithms to optimise this problem using a general iterative relaxation procedure. Moreover, a good initial solution (optimal schedule under a mid-point scenario) for the aforementioned algorithms is discussed. Several heuristics are developed to solve large-scale problems. Finally, computational experiments are conducted to evaluate the performance of the proposed methods.     

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.     

Fuzzy bi-objective formulation for a parallel machine scheduling problem with machine eligibility restrictions and sequence-dependent setup times

In this paper, a fuzzy bi-objective mixed-integer linear programming (FBOMILP) model is presented. FBOMILP encompasses the minimisation workload imbalance and total tardiness simultaneously as a bi-objective formulation for an unrelated parallel machine scheduling problem. To make the proposed model more practical, sequence-dependent setup times, machine eligibility restrictions and release dates are also considered. Moreover, the inherent uncertainty of processing times, release dates, setup times and due dates are taken into account and modelled by fuzzy numbers. In order to solve the model for small-scale problems, a two-stage fuzzy approach is proposed. Nevertheless, since the problem belongs to the class of NP-hard problems, the proposed model is solved by two meta-heuristic algorithms, namely fuzzy multi-objective particle swarm optimisation (FMOPSO) and fuzzy non-dominated sorting genetic algorithm (FNSGA-II) for solving large-scale instances. Subsequently, through setting up various numerical examples, the performances of the two mentioned algorithms are compared. When α?=?0.5 (α is a level of risk-taking and when it increases the decision-maker’s risk-taking decreases), FNSGA-II is fairly more effective than FMOPSO and has better performance especially in solving large-sized problems. However, when α rises, it can be stated that FMOPSO moderately becomes more appropriate. Finally, directions for future studies are suggested and conclusion remarks are drawn.     

Bi-objective optimisation for integrated scheduling of single machine with setup times and preventive maintenance planning

This paper deals with an integrated bi-objective optimisation problem for production scheduling and preventive maintenance in a single-machine context with sequence-dependent setup times. To model its increasing failure rate, the time to failure of the machine is subject to Weibull distribution. The two objectives are to minimise the total expected completion time of jobs and to minimise the maximum of expected times of failure of the machine at the same time. During the setup times, preventive maintenance activities are supposed to be performed simultaneously. Due to the assumption of non-preemptive job processing, three resolution policies are adapted to deal with the conflicts arising between job processing and maintenance activities. Two decisions are to be taken at the same time: find the permutation of jobs and determine when to perform the preventive maintenance. To solve this integrated problem, two well-known evolutionary genetic algorithms are compared to find an approximation of the Pareto-optimal front, in terms of standard multi-objective metrics. The results of extensive computational experiments show the promising performance of the adapted algorithms.     

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.     

Scheduling policies in multi-product manufacturing systems with sequence-dependent setup times and finite buffers

Multi-product production systems with sequence-dependent setup times are typical in the manufacturing of semiconductor chips and other electronic products. In such systems, the scheduling policies coordinating the production of multiple product types play an important role. In this paper, we study a multi-product manufacturing system with finite buffers, sequence-dependent setup times and various scheduling policies. Using continuous-time Markov chain models, we evaluate the performance of such systems under seven scheduling policies, i.e. cyclic, shortest queue, shortest processing time, shortest overall time (including setup time and processing time), longest queue, longest processing time, and longest overall time. The impacts of these policies on system throughput are compared, and the conditions characterising the superiority of each policy are investigated. The results of this work can provide production engineers and supervisors practical guidance to operate multi-product manufacturing systems with sequence-dependent setups.     

Single machine scheduling problem with stochastic sequence-dependent setup times

In this study, we consider stochastic single machine scheduling problem. We assume that setup times are both sequence dependent and uncertain while processing times and due dates are deterministic. In the literature, most of the studies consider the uncertainty on processing times or due dates. However, in the real-world applications (i.e. plastic moulding industry, appliance assembly, etc.), it is common to see varying setup times due to labour or setup tools availability. In order to cover this fact in machine scheduling, we set our objective as to minimise the total expected tardiness under uncertain sequence-dependent setup times. For the solution of this NP-hard problem, several heuristics and some dynamic programming algorithms have been developed. However, none of these approaches provide an exact solution for the problem. In this study, a two-stage stochastic-programming method is utilised for the optimal solution of the problem. In addition, a Genetic Algorithm approach is proposed to solve the large-size problems approximately. Finally, the results of the stochastic approach are compared with the deterministic one to demonstrate the value of the stochastic solution.     

Minimising makespan in flowshop group scheduling with sequence-dependent family set-up times using inserted idle times

This paper focuses on the permutation flowshop group scheduling problem to minimise makespan, which is typically found in flowline manufacturing cells. In contrast to classical flowshop scheduling, it is characterised by a scheduling task at two levels: on the one hand, jobs within part families and on the other hand, a number of part families have to be sequenced. Integrating sequence-dependent set-up times for every changeover of families, this problem can represent practical cases. By modelling each family as a job with time lags, some specific problem characteristics of the group scheduling problem are pointed out. It is shown that generating job sequences by minimising the sum of inserted machine idle times instead of makespan on the first level of scheduling and the use of the schedule heads on the second level can lead to significant improvements for some test problems. These findings are used for the improvement of existing constructive heuristic algorithms, whose effectiveness is assessed for several test instances with sequence-dependent family set-up times.     

Minimizing makespan on parallel machines with batch arrivals

Most studies in the scheduling literature assume that jobs arrive at time zero, while some studies assume that jobs arrive individually at non-zero times. However, both assumptions may not be valid in practice because jobs usually arrive in batches. In this article, a scheduling model for an identical parallel machine problem with batch arrivals is formulated. Because of the NP-hardness of the problem, a heuristic based on a simplified version of lexicographical search is proposed. To verify the heuristic, two lower bounding schemes are developed, where one lower bound is tight, and the list scheduling heuristic is compared. Extensive computational experiments demonstrate that the proposed heuristic is quite efficient in obtaining near optimal solution with an average error of less than 1.58%. The percentage improvement (from the lower bound) of the heuristic solution on the solution by the list scheduling is as large as 31.68.     

Minimising total tardiness for the identical parallel machine scheduling problem with splitting jobs and sequence-dependent setup times

This paper focuses on an identical parallel machine scheduling problem with minimising total tardiness of jobs. There are two major issues involved in this scheduling problem; (1) jobs which can be split into multiple sub-jobs for being processed on parallel machines independently and (2) sequence-dependent setup times between the jobs with different part types. We present a novel mathematical model with meta-heuristic approaches to solve the problem. We propose two encoding schemes for meta-heuristic solutions and three decoding methods for obtaining a schedule from the meta-heuristic solutions. Six different simulated annealing algorithms and genetic algorithms, respectively, are developed with six combinations of two encoding schemes and three decoding methods. Computational experiments are performed to find the best combination from those encoding schemes and decoding methods. Our findings show that the suggested algorithm provides not only better solution quality, but also less computation time required than the commercial optimisation solvers.     

MIP-based fix-and-optimise algorithms for the parallel machine capacitated lot-sizing and scheduling problem

This paper examines the capacitated lot-sizing and scheduling problem (CLSP) with sequence-dependent setup times, time windows, machine eligibility and preference constraints. Such a problem frequently arises in the semiconductor manufacturing industry by which this paper is motivated. A mixed integer programming (MIP) model is constructed for the problem. Two MIP-based fix-and-optimise algorithms are proposed in which the binary decision variables associated with the assignment of machines are first fixed using the randomised least flexible machine (RLFM) rule and the rest of the decision variables are settled by an MIP solver. Extensive experiments show that the proposed algorithms outperform the state-of-the-art MIP-based fix-and-optimise algorithms in the literature, especially for instances with high machine flexibility and high demand variation.     

Dynamic order acceptance and scheduling problem with sequence-dependent setup time

This paper studies the order acceptance and scheduling problem under a single machine environment when the orders come stochastically during the planning horizon and a sequence-dependent setup time is required between the processing of different types of orders. The objective is to maximise the expected revenue subject to the due date constraints. The problem is formulated as a stochastic dynamic programming model. A rule based on the opportunity cost of the remaining system capacity for the current system state is proposed to make the order acceptance decisions. The remaining system capacity is estimated by a heuristic which generates a good schedule for the accepted orders. Its opportunity cost is estimated by both mathematical programme and greedy heuristic. Computational experiments show that the profit generated by the integrated dynamic programming decision model is much higher than the widely used first-come-first-accept policy in industries and the benefit increases with the length of planning horizon, the arrival rate and the length of lead time. Acceptance decision based on mathematical programming outperforms greedy heuristic by about 7% and its computational time is short. It also shows that the quality of the solutions generated by the opportunity cost based order acceptance rule is satisfactory.     

Parallel machine scheduling with preference of machines

This paper considers a scheduling problem in the manufacturing of anodic electro-etching aluminium foil. To reduce cost and increase efficiency, the manufacturer of aluminium foil usually designs the equipment for electro-etching of aluminium foil into specialised equipment that is dedicated to produce high voltage or medium voltage aluminium foil based on the range the aluminium foil can bear. Nevertheless, high voltage equipment can be used to produce medium voltage aluminium foil with longer processing time, and vice versa. The problem is to schedule jobs on the high and medium voltage equipment, each having several pieces in parallel, with setup times to minimise to the total completion time. In this paper, we propose a three-stage heuristic for this problem and computationally evaluate the performance of the heuristic in comparison to a heuristic for unrelated parallel machines and a branch-and-bound algorithm.     

Integrated maintenance planning and production scheduling with Markovian deteriorating machine conditions

In many industries, production capacity diminishes as machine conditions deteriorate. Maintenance operations improve machine conditions, but also occupy potential production time, possibly delaying the customer orders. Therefore, one challenge is to determine the joint maintenance and production schedule to minimize the combined costs of maintenance and lost production over the long term. In this paper, we address the problem of integrated maintenance and production scheduling in a deteriorating multi-machine production system over multiple periods. Assuming that at the beginning of each period the demand becomes known and machine conditions are observable, we formulate a Markov decision process model to determine the maintenance plan and develop sufficient conditions guaranteeing its monotonicity in both machine condition and demand. We then formulate an integer programming model to find the maintenance and the production schedule in each period. Our computational results show that exploiting online condition monitoring information in maintenance and production decisions leads to 21% cost savings on average compared to a greedy heuristic and that the benefit of incorporating long-term information in making short-term decisions is highest in industries with medium failure rates.