Minimizing the makespan for the two-machine scheduling problem with a single server: Two algorithms for very large instances

This article considers the problem of scheduling a given set of n jobs on two identical parallel machines with a single server. Each job must be processed on one of the machines. Before processing, the server has to set up the relevant machine. The objective is to minimize the makespan. For this unary NP-hard problem, two fast constructive algorithms with a complexity of O(n²) are presented. The performance of these algorithms is evaluated for instances with up to 10,000 jobs. Computational results indicate that the algorithms have an excellent performance for very large instances so that the obtained objective function values are very close to a lower bound, and in many cases even an optimal solution is achieved. Superiority over all existing algorithms is obtained by sequencing the jobs on the two machines so that the machine idle time and the server waiting time are minimized. In doing so, the characteristics of an optimal solution resulting from its relevant lower bound are taken into account.     

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.     

Minimising total weighted earliness and tardiness on parallel machines using a hybrid heuristic

Scheduling jobs with different processing times and distinct known due dates on parallel machines (which may be idle) so as to minimise the total weighted earliness and tardiness is NP-hard. It occurs frequently in industrial settings with a just-in-time production environment. Here, small instances of this problem are tackled via an exact mixed-integer program, whereas larger instances are approximately solved using hybrid algorithms. The computational investigation discerns what makes a difficult instance, and demonstrates the efficiency of the approach.     

Minimising makespan on parallel batch processing machines with non-identical ready time and arbitrary job sizes

This paper considers the parallel batch processing machine scheduling problem which involves the constraints of unequal ready times, non-identical job sizes, and batch dependent processing times in order to sequence batches on identical parallel batch processing machines with capacity restrictions. This scheduling problem is a practical generalisation of the classical parallel batch processing machine scheduling problem, which has many real-world applications, particularly, in the aging test operation of the module assembly stage in the manufacture of thin film transistor liquid crystal displays (TFT-LCD). The objective of this paper is to seek a schedule with a minimum total completion time for the parallel batch processing machine scheduling problem. A mixed integer linear programming (MILP) model is proposed to optimise the scheduling problem. In addition, to solve the MILP model more efficiently, an effective compound algorithm is proposed to determine the number of batches and to apply this number as one parameter in the MILP model in order to reduce the complexity of the problem. Finally, three efficient heuristic algorithms for solving the large-scale parallel batch processing machine scheduling problem are also provided.     

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.     

An effective shuffled frog-leaping algorithm for solving the hybrid flow-shop scheduling problem with identical parallel machines

In this article, an effective shuffled frog-leaping algorithm (SFLA) is proposed to solve the hybrid flow-shop scheduling problem with identical parallel machines (HFSP-IPM). First, some novel heuristic decoding rules for both job order decision and machine assignment are proposed. Then, three hybrid decoding schemes are designed to decode job order sequences to schedules. A special bi-level crossover and multiple local search operators are incorporated in the searching framework of the SFLA to enrich the memetic searching behaviour and to balance the exploration and exploitation capabilities. Meanwhile, some theoretical analysis for the local search operators is provided for guiding the local search. The parameter setting of the algorithm is also investigated based on the Taguchi method of design of experiments. Finally, numerical testing based on well-known benchmarks and comparisons with some existing algorithms are carried out to demonstrate the effectiveness of the proposed algorithm.     

Multi-objective flow-shop scheduling with parallel machines

In this paper, multi-objective flow-shop scheduling with parallel machines is considered, and an algorithm for production-scheduling optimisation is proposed. The objective is to design an algorithm to optimise production scheduling, taking into consideration customer changes to delivery dates and quantities, materials requisitioned from the supplier and inventory management. Manufacturing lead time for the production order is minimised, and, finally, resource use is considered as an auxiliary criterion. Numerical examples, modelled with data from a Mexican enterprise, and computational results are reported. The paper indicates that the mathematical model, as well as the program that solves it, provides flexibility with minimum modifications. Thus it can provide useful information for multiple decision processes.     

A desired load distribution model for scheduling of unrelated parallel machines

Scheduling problems concern the allocation of limited resources over time among both parallel and sequential activities. Load balancing has been adopted as an optimization criterion for several scheduling problems. However, in many practical situations, a load-balanced solution may not be feasible or attainable. To deal with this limitation, this paper presents a generic mathematical model of load distribution for resource allocation, called desired load distribution (DLD). The objective is to develop a DLD model for scheduling of unrelated parallel machines that can be used both in centralized resource management settings and in agent-based distributed scheduling systems. The paper describes the proposed DLD model in details, presents a dynamic programming based optimization algorithm for the proposed model, and then discusses its application to agent-based distributed scheduling.     

Mathematical models and a hunting search algorithm for the no-wait flowshop scheduling with parallel machines

Majority of researches in no-wait flowshop scheduling assume that there is only one machine at each stage. But, factories commonly duplicate machines in parallel for each operation. In this case, they balance the speed of the stages, increase the throughput of the shop floor and reduce the impact of bottleneck stages. Despite their importance, there is no paper to study the general no-wait flowshop with parallel machines. This paper studies this problem where the objective is to minimise makespan. Since there is no mathematical model for the problem, we first mathematically formulate it in form of two mixed integer linear programming models. By the models, the small instances are optimally solved. We then propose a novel hunting search metaheuristic algorithm (HSA) to solve large instances of the problem. HSA is derived based on a model of group hunting of animals when searching for food. A set of experimental instances are carried out to evaluate the algorithm. The algorithm is carefully evaluated for its performance against an available algorithm by means of statistical tools. The related results show that the proposed HSA provides sound performance comparing with other algorithms.     

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.     

Minimising total weighted earliness and tardiness penalties on identical parallel machines using a fast ruin-and-recreate algorithm

This paper studies the scheduling problem of minimising total weighted earliness and tardiness penalties on identical parallel machines against a restrictive common due date. This problem is NP-hard in the strong sense and arises in many just-in-time production environments. A fast ruin-and-recreate (FR&R) algorithm is proposed to obtain high-quality solutions to this complex problem. The proposed FR&R algorithm is tested on a well-known set of benchmark test problems that are taken from the literature. Computational results provide evidence of the efficiency of FR&R, which consistently outperform existing algorithms when applied to benchmark instances. This work provides a viable alternative approach for efficiently solving this practical but complex scheduling problem.     

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 makespan on parallel machines with precedence constraints and machine eligibility restrictions

The block erection problem is defined as a parallel machine scheduling problem with precedence constraints and machine eligibility restrictions. A heuristic algorithm combined of the largest total amount of processing first rule (LTAP) and the enhanced smallest machine load first rule (ESML) is proposed to minimise makespan for the block erection in a shipyard. Finally, four lower bounds and the percentage of the reduced makespan compared with the current solutions are defined to evaluate the performance of the proposed algorithm. The experiments are performed on data selected from a shipbuilding company, and the results demonstrate that the presented algorithm can effectively find a good solution to minimise the makespan of the block erection problem.     

Speeding up a Rollout algorithm for complex parallel machine scheduling

Rollout methodology is a constructive metaheuristic algorithm and its main characteristics are its modularity, the adaptability to different objectives and constraints and the easiness of implementation. Multi-heuristic Rollout extends the Rollout by incorporating several constructive heuristics in the Rollout framework and it is able to easily incorporate human experience inside its research patterns to fulfil complex requirements dictated by the application at hand. However, a drawback for both Rollout and multi-heuristic Rollout is often represented by the required computation time. This paper proposes some alternatives of the full multi-heuristic Rollout algorithm aimed at improving the efficiency by reducing the computational effort while preserving the effectiveness. Namely, we propose dynamic heuristics pruning and candidates reduction strategies. As illustrative case studies, we analyse complex deterministic identical parallel machine scheduling problems showing how Rollout procedures can be used to tackle several additional constraints arising in real contexts. More specifically, we considered both standard (batch production, family set-ups, release, due dates, etc.) and non-standard (machine unavailabilities, maximum campaign size) scheduling constraints. An extensive campaign of computational experiments shows the behaviour of the multi-heuristic Rollout approach and the effectiveness of the different proposed speed-up methods.     

A branch and bound algorithm for scheduling unit size jobs on parallel batching machines to minimize makespan

In this paper, we present a branch and bound algorithm for the parallel batch scheduling of jobs having different processing times, release dates and unit sizes. There are identical machines with a fixed capacity and the number of jobs in a batch cannot exceed the machine capacity. All batched jobs are processed together and the processing time of a batch is given by the greatest processing time of jobs in that batch. We compare our method to a mixed integer program as well as a method from the literature that is capable of optimally solving instances with a single machine. Computational experiments show that our method is much more efficient than the other two methods in terms of solution time for finding the optimal solution.     

An adaptive scheduling algorithm for a parallel machine problem with rework processes

The development of a scheduling methodology for a parallel machine problem with rework processes is presented in this paper. The problem is to make a schedule for parallel machines with rework probabilities, due-dates, and sequence dependent setup times. Two heuristics are developed based on a dispatching algorithm and problem-space-based search method. In order to evaluate the efficacy of the proposed algorithms, six performance indicators are considered: total tardiness, maximum lateness, mean flow-time, mean lateness, the number of tardy jobs, and the number of reworks. This paper shows how these algorithms can adaptively capture the characteristics of manufacturing facilities for enhancing the performance under changing production environments. Extensive experimental results show that the proposed algorithms give very efficient performance in terms of computational time and each objective value.     

Cash management in single and parallel machine scheduling

Traditional scheduling problems are usually focused on minimising time-based objectives. In this paper, we have examined the problems in single and parallel machine scheduling. We have incorporated cash management into the objective function of these problems. It is assumed that processing a given job involves incurring a total cost and receiving a total revenue. Each job also has its own temporal pattern of cost incurring and revenue receiving. Nonetheless, the total cost/revenue of processing a job is assumed to be incurred/received during its processing and not beyond it. We have defined five new objectives: (i) maximising average available cash; (ii) minimising revenue-cost duration; (iii) minimising average cash deficiency; (iv) minimising maximum cash deficiency; and (v) maximising minimum available cash. Based on the results achieved in this paper, the above patterns do not affect the optimal scheduling policy in regards to the first two objectives. For single machine models, all except one of the problems are solved in polynomial time. For parallel machines, all models are proved to be NP-hard. Some of the NP-hard models are convertible to equivalent traditional ones.     

Scheduling two identical parallel machines with preparation constraints

In this paper, we consider the problem of scheduling a set of n jobs on two identical machines with preparation constraints. Each job requires before its execution a set of resources and a non-negligible preparation time. The objective is to minimise the makespan. This problem is NP-hard. We prove the NP-hardness of two specific cases where in the first case preparation times take only three values, whereas in the second case preparation times and the release dates take only two values, respectively. Then, we present some special cases and heuristic algorithms along with an experimental study.     

Scheduling with multi-attribute set-up times on unrelated parallel machines

This paper studies a problem in the knitting process of the textile industry. In such a production system, each job has a number of attributes and each attribute has one or more levels. Because there is at least one different attribute level between two adjacent jobs, it is necessary to make a set-up adjustment whenever there is a switch to a different job. The problem can be formulated as a scheduling problem with multi-attribute set-up times on unrelated parallel machines. The objective of the problem is to assign jobs to different machines to minimise the makespan. A constructive heuristic is developed to obtain a qualified solution. To improve the solution further, a meta-heuristic that uses a genetic algorithm with a new crossover operator and three local searches are proposed. The computational experiments show that the proposed constructive heuristic outperforms two existed heuristics and the current scheduling method used by the case textile plant.