A discrete artificial bee colony algorithm for single machine scheduling problems

This paper presents a discrete artificial bee colony algorithm for a single machine earliness–tardiness scheduling problem. The objective of single machine earliness–tardiness scheduling problems is to find a job sequence that minimises the total sum of earliness–tardiness penalties. Artificial bee colony (ABC) algorithm is a swarm-based meta-heuristic, which mimics the foraging behaviour of honey bee swarms. In this study, several modifications to the original ABC algorithm are proposed for adapting the algorithm to efficiently solve combinatorial optimisation problems like single machine scheduling. In proposed study, instead of using a single search operator to generate neighbour solutions, random selection from an operator pool is employed. Moreover, novel crossover operators are presented and employed with several parent sets with different characteristics to enhance both exploration and exploitation behaviour of the proposed algorithm. The performance of the presented meta-heuristic is evaluated on several benchmark problems in detail and compared with the state-of-the-art algorithms. Computational results indicate that the algorithm can produce better solutions in terms of solution quality, robustness and computational time when compared to other algorithms.     

An electromagnetism-like mechanism for the single machine total stepwise tardiness problem with release dates

In recent years, a new type of tardiness cost, called stepwise tardiness, has received attention. To the authors’ knowledge, only a few studies have investigated this type of tardiness in the scheduling problem. This study considered the single machine total stepwise tardiness problem with release dates, which is strongly NP hard. Because of the essential complexity of the problem, heuristics were first developed to quickly generate initial solutions. Subsequently, a new electromagnetism-like mechanism (EM), which is a novel metaheuristic, was proposed to improve the solution quality. The new EM includes a natural encoding scheme, a new distance measure between solutions, and effective attraction and repulsion operators. Comparisons with a current EM and other metaheuristics were performed to verify the proposed EM. The computational results show that the proposed EM exhibits good performance for the considered problem.     

Multiple-variable neighbourhood search for the single-machine total weighted tardiness problem

The single-machine total weighted tardiness (SMTWT) problem is a typical discrete combinatorial optimization problem in the scheduling literature. This problem has been proved to be NP hard and thus provides a challenging area for metaheuristics, especially the variable neighbourhood search algorithm. In this article, a multiple variable neighbourhood search (m-VNS) algorithm with multiple neighbourhood structures is proposed to solve the problem. Special mechanisms named matching and strengthening operations are employed in the algorithm, which has an auto-revising local search procedure to explore the solution space beyond local optimality. Two aspects, searching direction and searching depth, are considered, and neighbourhood structures are systematically exchanged. Experimental results show that the proposed m-VNS algorithm outperforms all the compared algorithms in solving the SMTWT problem.     

A branch-and-bound algorithm for a two-stage hybrid flowshop scheduling problem minimizing total tardiness

A two-stage hybrid flowshop-scheduling problem is considered with the objective of minimizing total tardiness of jobs. In the hybrid flowshop, there is one machine at the first stage and multiple identical parallel machines at the second stage. Dominance properties and lower bounds are developed for the problem and a branch-and-bound algorithm is suggested using them. Results of computational experiments show that the suggested algorithm can find optimal solutions for problems with up to 15 jobs in a reasonable amount of central processing unit time.     

A differential evolution algorithm for the no-idle flowshop scheduling problem with total tardiness criterion

In this paper, we investigate the use of a continuous algorithm for the no-idle permutation flowshop scheduling (NIPFS) problem with tardiness criterion. For this purpose, a differential evolution algorithm with variable parameter search (vpsDE) is developed to be compared to a well-known random key genetic algorithm (RKGA) from the literature. The motivation is due to the fact that a continuous DE can be very competitive for the problems where RKGAs are well suited. As an application area, we choose the NIPFS problem with the total tardiness criterion in which there is no literature on it to the best of our knowledge. The NIPFS problem is a variant of the well-known permutation flowshop (PFSP) scheduling problem where idle time is not allowed on machines. In other words, the start time of processing the first job on a given machine must be delayed in order to satisfy the no-idle constraint. The paper presents the following contributions. First of all, a continuous optimisation algorithm is used to solve a combinatorial optimisation problem where some efficient methods of converting a continuous vector to a discrete job permutation and vice versa are presented. These methods are not problem specific and can be employed in any continuous algorithm to tackle the permutation type of optimisation problems. Secondly, a variable parameter search is introduced for the differential evolution algorithm which significantly accelerates the search process for global optimisation and enhances the solution quality. Thirdly, some novel ways of calculating the total tardiness from makespan are introduced for the NIPFS problem. The performance of vpsDE is evaluated against a well-known RKGA from the literature. The computational results show its highly competitive performance when compared to RKGA. It is shown in this paper that the vpsDE performs better than the RKGA, thus providing an alternative solution approach to the literature that the RKGA can be well suited.     

Minimisation of total tardiness for identical parallel machine scheduling using genetic algorithm

In recent years research on parallel machine scheduling has received an increased attention. This paper considers minimisation of total tardiness for scheduling of n jobs on a set of m parallel machines. A spread-sheet-based genetic algorithm (GA) approach is proposed for the problem. The proposed approach is a domain-independent general purpose approach, which has been effectively used to solve this class of problem. The performance of GA is compared with branch and bound and particle swarm optimisation approaches. Two set of problems having 20 and 25 jobs with number of parallel machines equal to 2, 4, 6, 8 and 10 are solved with the proposed approach. Each combination of number of jobs and machines consists of 125 benchmark problems; thus a total for 2250 problems are solved. The results obtained by the proposed approach are comparable with two earlier approaches. It is also demonstrated that a simple GA can be used to produce results that are comparable with problem-specific approach. The proposed approach can also be used to optimise any objective function without changing the basic GA routine.     

Minimising earliness and tardiness penalties in single machine scheduling against common due date using imperialist competitive algorithm

In this paper a novel evolutionary-based approach is utilised for efficiently solving the NP-hard problem of scheduling numerous common-due-date jobs on a single machine. Minimising the sum of earliness and tardiness penalties for all jobs is considered as the target function. The performance of the proposed approach is examined through a computational comparative study with 280 benchmark problems with up to 1000 jobs where the numerical results indicate that it can produce ‘better’ solutions in less computational time when compared to benchmark results and the methods available in the literature, namely genetic algorithm (GA), Tabu search (TS) and differential evolution (DE).     

A vector space model for variance reduction in single machine scheduling

Reducing the variance of part completion times about promised due dates is an important element of Just-In-Time production because it reduces the work-in-process inventory and tardiness simultaneously. Scheduling models and algorithms are developed to minimize the Mean Squared Deviation (MSD) of completion times about due dates on a single machine. A generic model is developed in real vector space for understanding the structural relationship between the optimal schedule and the location of the due dates. Geometric insights gained from this vector space model are used to relate the shortest and longest processing time sequences to the level of difficulty of the MSD optimization problem. The vector space model is used to develop dominance conditions for a branch and bound algorithm and to analytically synthesize parameters for a continuous variable feedback control algorithm for distributed scheduling. The control algorithm lends itself to massively parallel / distributed computation and is found to produce near optimal solutions efficiently, which makes it more scalable and practical compared to the branch and bound algorithm. Computational experiments with both approaches are presented.     

A Meta-RaPS for the early/tardy single machine scheduling problem

This paper investigates a meta-heuristic solution approach to the early/tardy single machine scheduling problem with common due date and sequence-dependent setup times. The objective of this problem is to minimise the total amount of earliness and tardiness of jobs that are assigned to a single machine. The popularity of just-in-time (JIT) and lean manufacturing scheduling approaches makes the minimisation of earliness and tardiness important and relevant. In this research the early/tardy problem is solved by Meta-RaPS (meta-heuristic for randomised priority search). Meta-RaPS is an iterative meta-heuristic which is a generic, high level strategy used to modify greedy algorithms based on the insertion of a random element. In this case a greedy heuristic, the shortest adjusted processing time, is modified by Meta-RaPS and the good solutions are improved by a local search algorithm. A comparison with the existing ETP solution procedures using well-known test problems shows Meta-RaPS produces better solutions in terms of percent difference from optimal. The results provide high quality solutions in reasonable computation time, demonstrating the effectiveness of the simple and practical framework of Meta-RaPS.     

A tabu search algorithm for simultaneous machine/AGV scheduling problem

Machines and automated guided vehicles (AGVs) scheduling problems are two essential issues that need to be addressed for the efficiency of the overall production system. The purpose of this paper is to study the simultaneous scheduling problem of machines and AGVs in a flexible manufacturing system (FMS) since the global optimum cannot be reached by considering each of them individually. In this paper, a mixed integer linear programming (MILP) model is developed with the objective of makespan minimisation. The MILP model consists of the following two constraint sets: machines and AGVs scheduling sub-problems. As both sub-problems are known to be NP-hard, a heuristic algorithm based on tabu search (TS) is proposed to get optimal or near to optimal solution for large-size problems within reasonable computation time. The proposed algorithm includes a novel two-dimensional solution representation and the generation of two neighbour solutions, which are alternately and iteratively applied to improve solutions. Moreover, an improved lower bound calculation method is introduced for the large-size problems. Computational results show the superior performance of the TS algorithm for the simultaneous scheduling problem.     

A hybrid genetic algorithm for the single row layout problem

The arrangement of machines or departments along a straight line is known as single row layout and it is a widely employed configuration in flexible manufacturing systems. In this paper, a hybrid genetic algorithm (HGA) is proposed to solve the single row layout design problem with unequal sized machines and unequal clearances. The algorithm is developed by hybridisation of a genetic algorithm with a local search operator. The proposed HGA is tested on 51 well known data sets from the literature with equal and unequal clearances, and the results are compared with the best known solutions. Finally, algorithm's effectiveness in reaching previously known best solutions is revealed and improvements up to 7% in problems with unequal clearance are obtained.     

An analytical model for service level and tardiness in a single machine MTO production system

In this paper an analytical model to calculate service level, FGI and tardiness for a make-to-order (MTO) production system based on the production leadtime, utilisation and WIP is presented. The distribution of customer required leadtime is linked to the already available equations for an M/M/1 production system from queuing theory. Explicit equations for service level, FGI, FGI leadtime and tardiness are presented for an M/M/1 production system within an MTO environment. For a G/G/1 production system an approximation based extension is provided – discussing the influence of variation in the inter-arrival and processing time distribution in this framework. Moreover, the integration of a work ahead window (WAW) work release policy is discussed. Based on a numerical study, a high potential to decrease FGI (up to 97% FGI reduction) when applying such a WAW strategy is found and it is shown that the higher the targeted service level is, the higher the FGI reduction potential. The paper contributes to a better understanding of the relationship between customer required leadtime distribution and the M/M/1 production system. By applying this model a decision maker can base his capacity investment decisions on the service level and expected tardiness for certain levels of FGI and WIP and can additionally define the optimal WAW policy.     

An efficient iterated local search algorithm for the total tardiness blocking flow shop problem

This paper deals with the blocking flow shop problem and proposes an Iterated Local Search (ILS) procedure combined with a variable neighbourhood search (VNS) for the total tardiness minimisation. The proposed ILS makes use of a NEH-based procedure to generate the initial solution, and uses a local search to intensify the exploration that combines the insertion and swap neighbourhood and uses a perturbation mechanism consisting of three neighbourhood operators to diversify the search. The computational evaluation has shown the effectiveness of combining the insertion and swap neighbourhood during the search despite the insertion neighbourhood being more effective than the swap neighbourhood for this problem. Finally, the computation of this algorithm when evaluated against two other algorithms from the literature shows good performance.     

Assigning jobs to time frames on a single machine to minimize total tardiness

This paper deals with a sequencing problem arising in the management of paced-flowlines, that is production lines where jobs are released at constant time intervals. The problem is to sequence jobs to minimis total tardiness. The problem can be formulated as an assignment problem with a number of knapsack constraints. We prove the strong NP-hardness of the problem and give a number of lower bounds which are used in a branch-and-bound algorithm. Computational results in realistic settings confirm the effectiveness of the procedure developed. The results are particularly interesting with reference to mixed-model assembly lines in which several jobs of few different types are produced periodically.     

Heuristics for minimizing total weighted tardiness in complex job shops

Semi-conductor manufacturing is arguably one of the most complex manufacturing processes in existence today. A semi-conductor wafer fabrication facility is comprised of batching machines, parallel machines, machines with sequence-dependent set-ups, and re-circulating product flow. The individual job release times and due dates combine with the other processing environment characteristics to form a ‘complex’ job shop scheduling problem. We first present a mixed-integer program (MIP) to minimize total weighted tardiness in a complex job shop. Since the problem is NP-hard, we compare a heuristic based on the MIP (MIP heuristic) with both a tuned version of a modified shifting bottleneck heuristic (SB heuristic) and three dispatching rules using random problem instances of a representative model from the literature. While the MIP heuristic typically produces superior schedules for problem instances with a small number of jobs, the SB heuristic consistently outperforms the MIP heuristic for larger problem instances. The SB heuristic's superior performance as compared to additional dispatching rules is also demonstrated for a larger, ‘real world’ dataset from the literature.     

A heuristic to minimize the total weighted tardiness with sequence-dependent setups

We propose a three-phase heuristic for the problem of minimizing the total weighted tardiness on a single machine in the presence of sequence-dependent setup times. In the first phase a number of parameters characterizing the problem instance at hand are calculated. In the second phase we develop a schedule by using a new priority rule whose parameters are calculated based on the results of the first phase. Computational experiments show that this rule significantly outperforms the only other rule so far developed in the literature. The third phase consists of a local improvement procedure to improve the schedule obtained in the second phase. The procedure we suggest has been successfully implemented in an industrial scheduling system.     

Lagrangian relaxation and cut generation for sequence-dependent setup time flowshop scheduling problems to minimise the total weighted tardiness

The Lagrangian relaxation and cut generation technique is applied to solve sequence-dependent setup time flowshop scheduling problems to minimise the total weighted tardiness. The original problem is decomposed into individual job-level subproblems that can be effectively solved by dynamic programming. Two types of additional constraints for the violation of sequence-dependent setup time constraints are imposed on the decomposed subproblems in order to improve the lower bound. The decomposed subproblem with the additional setup time constraints on any subset of jobs is also effectively solved by a novel dynamic programming. Computational results show that the lower bound derived by the proposed method is much better than those of CPLEX and branch and bound for problem instances with 50 jobs and five stages with less computational effort.     

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.