Robust single machine scheduling with uncertain release times for minimising the maximum waiting time

We study a single machine scheduling problem (SMSP) with uncertain job release times (JRTs) under the maximum waiting time (MWT) criterion. To deal with the uncertainty, a robust model is established to find an optimal schedule, which minimises the worst-case MWT (W-MWT) when JRTs vary over given time intervals. Although infinite possible scenarios for JRTs exist, we show that only n scenarios are needed for calculating the W-MWT, where n is the number of jobs. Based on this property, the robust (SMSP) with uncertain JRTs to minimise the W-MWT is formulated as a mixed integer linear programming problem. To solve large-size problem instances, an efficient two-stage heuristic (TSH) is proposed. In the first stage, n near-optimal schedules are obtained by solving n deterministic scenario-based SMSPs, and their W-MWTs are evaluated. To speed up the solution and evaluation process, a modified Gusfield’s heuristic is proposed by exploiting the inner connections of these SMSPs. To further improve the schedule obtained in the first stage, the second stage consists of a variable neighbourhood search method by combining both swap neighbourhood search and insert neighbourhood search. We also develop a method to calculate the lower bound of the proposed model so that we can evaluate the performance of the solutions given by the TSH. Experimental results confirm the robustness of schedules produced and advantages of the proposed TSH over other algorithms in terms of solution quality and run time.     

A hybrid particle swarm optimisation for scheduling just-in-time single machine with preemption,machine idle time and unequal release times

This paper addresses preemption in just-in-time (JIT) single–machine-scheduling problem with unequal release times and allowable unforced machine idle time as realistic assumptions occur in the manufacturing environments aiming to minimise the total weighted earliness and tardiness costs. Delay in production systems is a vital item to be focussed to counteract lost sale and back order. Thus, JIT concept is targeted including the elements required such as machine preemption, machine idle time and unequal release times. We proposed a new mathematical model and as the problem is proven to be NP-hard, three meta-heuristic approaches namely hybrid particle swarm optimisation (HPSO), genetic algorithm and imperialist competitive algorithm are employed to solve the problem in larger sizes. In HPSO, cloud theory-based simulated annealing is employed with a certain probability to avoid being trapped in a local optimum. Taguchi method is applied to calibrate the parameters of the proposed algorithms. A number of numerical examples are solved to demonstrate the effectiveness of the proposed approach. The performance of the proposed algorithms is evaluated in terms of relative percent deviation and computational time where the computational results clarify better performance of HPSO than other algorithms in quality of solutions and computational time.     

Exact and approximate methods for parallel multiple-area spatial scheduling with release times

Spatial scheduling problems involve scheduling jobs that each require certain amounts of two-dimensional space within a processing area of limited width and length. Thus, this requires not only assigning time slots to each job but also locations and orientations within the limited physical processing space as well. Such problems, often encountered in shipbuilding and aircraft manufacturing, are generally difficult to solve, and there is a relatively small amount of literature addressing these problems compared to other types of scheduling. In this paper, we consider a particularly complex class of spatial scheduling problems that involve scheduling each job into one of several possible processing areas in parallel to minimize the total amount of tardy time. In addition, each job has a release time before which it may not be processed. We introduce two methods for solving this type of problem: an integer programming (IP) model and a heuristic algorithm. We perform computational tests and comparisons of each method over a large number of generated benchmark problems with varying characteristics, and also compare these to a more naïve heuristic. Solving the IP model was effective for small problems but required excessive amounts of time for larger ones. The heuristic was effective and produced solutions of comparable quality to the IP model for many problems while requiring very little computational time.     

Simulated annealing for parallel machine scheduling with earliness-tardiness penalties and sequence-dependent set-up times

Scheduling problems with earliness and tardiness penalties are commonly encountered in today's manufacturing environment due to the current emphasis on the just-in-time (JIT) production philosophy. The problem studied in this work is the parallel machine earliness-tardiness non-common due date sequence-dependent set-up time scheduling problem (PETNDDSP) for jobs with varying processing times, where the objective is to minimize the sum of the absolute deviations of job completion times from their corresponding due dates. The research presented provides a first step towards obtaining near optimal solutions for this problem using local search heuristics in the framework of a meta-heuristic technique known as simulated annealing (SA). The computational study shows that, using the SA methodology, significant improvements to the local search heuristic solutions can be achieved for problems of this type.     

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.     

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.     

Single CNC machine scheduling with controllable processing times and multiple due dates

In this study, we solve the single CNC machine scheduling problem with controllable processing times. Our objective is to maximize the total profit that is composed of the revenue generated by the set of scheduled jobs minus the sum of total weighted earliness and weighted tardiness, tooling and machining costs. Customers offer multiple due dates to the manufacturer, each coming with a distinct price for the order that is decreasing as the date gets later, and the manufacturer has the flexibility to accept or reject the orders. We propose a number of ranking rules and scheduling algorithms that we employ in a four-stage heuristic algorithm that determines the processing times for each job and a final schedule for the accepted jobs simultaneously, to maximize the overall profit.     

An efficient heuristic approach for parallel machine scheduling with job splitting and sequence-dependent setup times

In this paper, we consider a simplified real-life identical parallel machine scheduling problem with sequence-dependent setup times and job splitting to minimize makespan. We propose a heuristic to solve this problem. Our method is composed of two parts. The problem is first reduced into a single machine scheduling problem with sequence-dependent setup times. This reduced problem can be transformed into a Traveling Salesman Problem (TSP), which can be efficiently solved using Little's method. In the second part, a feasible initial solution to the original problem is obtained by exploiting the results of the first part. This initial solution is then improved in a step by step manner, taking into account the setup times and job splitting. We develop a lower bound and evaluate the performances of our heuristic on a large number of randomly generated instances. The solution given by our heuristic is less than 4.88% from the lower bound.     

Parallel machine scheduling to minimize the sum of quadratic completion times

We consider the parallel machine scheduling problem of minimizing the sum of quadratic job completion times. We first prove that the problem is strongly NP-hard. We then demonstrate by probabilistic analysis that the shortest processing time rule solves the problem asymptotically. The relative error of the rule converges in probability to zero under the assumption that the job processing times are independent random variables uniformly distributed in (0, 1). We finally provide some computational results, which show that the rule is effective in solving the problem in practice.     

Heuristic methods for the single machine scheduling problem with different ready times and a common due date

The single machine scheduling problem with a common due date and non-identical ready times for the jobs is examined in this work. Performance is measured by the minimization of the weighted sum of earliness and tardiness penalties of the jobs. Since this problem is NP-hard, the application of constructive heuristics that exploit specific characteristics of the problem to improve their performance is investigated. The proposed approaches are examined through a computational comparative study on a set of 280 benchmark test problems with up to 1000 jobs.     

A two-stage stochastic programming for single yard crane scheduling with uncertain release times of retrieval tasks

Nowadays maritime transportation has become the mainstream of the global logistics, and the operational efficiency of container terminals plays a critical role in maritime transportation. As one of the most important terminal operational issues, yard crane scheduling that handles both storage and retrieval tasks has caught a lot of interest. However, the uncertainty on the release times of retrieval tasks, as one common phenomenon in daily operations, has been ignored in the literature. This paper investigates single yard crane scheduling to minimise the expected total tardiness of tasks, and focus on the case with uncertain release times of retrieval tasks. A two-stage stochastic programming model is proposed, and the sample average approximation (SAA) approach is applied to solve small instances of the problem. For large-scale instances, a genetic algorithm (GA) and a rule-based heuristic are developed. To evaluate the performances of the solution methods, numerical experiments with 300 instances are implemented. Computational results show that the rule-based heuristic outperforms both GA and SAA in terms of solution quality and running time.     

A particle swarm optimization algorithm for hybrid flow-shop scheduling with multiprocessor tasks

The multistage hybrid flow-shop scheduling problem with multiprocessor tasks has been found in many practical situations. Due to the essential complexity of the problem, many researchers started to apply metaheuristics to solve the problem. In this paper, we address the problem by using particle swarm optimization (PSO), a novel metaheuristic inspired by the flocking behaviour of birds. The proposed PSO algorithm has several features, such as a new encoding scheme, an implementation of the best velocity equation and neighbourhood topology among several different variants, and an effective incorporation of local search. To verify the PSO algorithm, computational experiments are conducted to make a comparison with two existing genetic algorithms (GAs) and an ant colony system (ACS) algorithm based on the same benchmark problems. The results show that the proposed PSO algorithm outperforms all the existing algorithms for the considered problem.     

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.     

Single machine due-window assignment scheduling with resource-dependent processing times to minimise total resource consumption cost

This paper considers a single-machine scheduling problem involving convex resource-dependent processing times and due-window assignment simultaneously. The goal is to minimise the total resource consumption cost under the constraint that the schedule cost involving earliness, tardiness, window location, window size and makespan does not exceed a given limit for two popular due window assignment methods: the common flow allowance (slack) due window assignment method (referred to SLKW) and the common due window assignment method (referred to CONW). We show that the problem can be solved in polynomial time. Some extensions of the problem are also given.     

Parallel-machine scheduling with controllable processing times

We consider a problem of scheduling nindependent and simultaneously available jobs on munrelated parallel machines. The job processing times can be compressed through incurring an additional cost, which is a convex function of the amount of compression. Two problems are formulated as assignment problems, which can be solved inO (n³m + n²m log(nm))time. One is to minimize the total compression cost plus the total flow time. The other is to minimize the total compression cost plus the sum of earliness and tardiness costs.     

Branch and bound methods for scheduling problems with multiprocessor tasks on dedicated processors

Branch and bound methods for the scheduling problem with multiprocessor tasks on dedicated processors and arbitrary precedences are presented. The methods are based on different representations of feasible schedules. Computational results show that the methods surpass each other on different types of problems with multiprocessor tasks.Supported by the Deutsche Forschungsgemeinschaft, Project JoPTAG     

Parallel machine scheduling with step-deteriorating jobs and setup times by a hybrid discrete cuckoo search algorithm

This article considers the parallel machine scheduling problem with step-deteriorating jobs and sequence-dependent setup times. The objective is to minimize the total tardiness by determining the allocation and sequence of jobs on identical parallel machines. In this problem, the processing time of each job is a step function dependent upon its starting time. An individual extended time is penalized when the starting time of a job is later than a specific deterioration date. The possibility of deterioration of a job makes the parallel machine scheduling problem more challenging than ordinary ones. A mixed integer programming model for the optimal solution is derived. Due to its NP-hard nature, a hybrid discrete cuckoo search algorithm is proposed to solve this problem. In order to generate a good initial swarm, a modified Biskup–Hermann–Gupta (BHG) heuristic called MBHG is incorporated into the population initialization. Several discrete operators are proposed in the random walk of Lévy flights and the crossover search. Moreover, a local search procedure based on variable neighbourhood descent is integrated into the algorithm as a hybrid strategy in order to improve the quality of elite solutions. Computational experiments are executed on two sets of randomly generated test instances. The results show that the proposed hybrid algorithm can yield better solutions in comparison with the commercial solver CPLEX^® with a one hour time limit, the discrete cuckoo search algorithm and the existing variable neighbourhood search algorithm.     

Flow-line scheduling problem with controllable processing times

In this paper we address the simultaneous scheduling and optimal-processing-times selection problem in a multi-product deterministic flow line operated under a cyclic scheduling approach. The selection of processing times plays an important role in achieving the desired production rate with the least possible operating cost. We first formulate the important subproblem of optimal-processing-times selection for different objectives, when the sequence of jobs is fixed, and then develop an efficient solution procedure for it. The fast solution of the fixed sequence problem is necessary for the development of efficient approximate solution procedures for the simultaneous scheduling and optimal-processing-times problem. A computational study on the effectiveness of the proposed solution procedure is presented. For the solution of the simultaneous scheduling and optimal-processing-times problem we suggest an iterative solution procedure, and report our computational experience with this procedure. For the solution of large problems we present a genetic algorithm. The effectiveness of the algorithm is demonstrated through computational results and by evaluating the performance of the obtained solutions against lower bounds that we developed for the problem.     

Finite-capacity multi-class production scheduling with set-up times

We treat the scheduling of a single server in a finite-buffer capacity, multi-class, make-to-order production system subject to inventory holding costs, set-up times, and customer rejection costs. We employ theoretical and numerical analysis of a Markov decision process model to investigate the structure of optimal policies and the performance of heuristic policies. We establish the monotonicity of optimal performance with respect to the system parameters. Based on our insights, we provide a heuristic policy called the Capacitated Modified Index Rule (CMIR) for capacitated scheduling with customer loss penalties. The CMIR heuristic can easily be precomputed and stored for real-time control. Numerical benchmarking with respect to the optimal performance as well as an existing heuristic suggests that CMIR is very effective.