A computational study of heuristics for two-stage flexible flowshops

We consider the scheduling of two-stage flexible flowshops. This manufacturing environment involves two machine centres representing two consecutive stages of production. Each machine centre is composed of multiple parallel machines. Each job has to be processed serially through the two machine centres. In each machine centre, a job may be processed on any of the machines. There are n independent jobs to be scheduled without preemption. The jobs can wait in between the two machine centres and the intermediate storage is unlimited. Our objective will be to minimize the maximum completion time of the jobs. We formulate the problem as a mixed integer program. Given this problem class is NP-hard in the strong sense, we present three lower bounds to estimate the optimal solution. We then propose a sequence-first, allocate-second heuristic approach for its solution. We heuristically decompose the problem by first creating a priority list to order the jobs and then assign the jobs to the available machines in each machine centre based on this order. We describe seven rules for the sequencing phase. The assignment phase consists of a heuristic which attempts to minimize each partial schedule length while looking ahead at the future assignment of the currently unscheduled jobs. The computational performance of the heuristic approach was evaluated by comparing the value of each heuristic variant to the best among the three lower bounds. Its effectiveness was tested on scenarios pertinent to flexible flowshop environments, such as cellular manufacturing, by conducting a computational study of over 3400 problems. Our computational results indicate that the most effective approach used Johnson's rule to provide the priority list for job assignment. This provided integrality gaps which on the average were less than 0·73%.     

Minimising interference for scheduling two parallel machines with a single server

In this paper, we consider the problem of scheduling a set of jobs on two parallel machines with set-up times. The set-up has to be performed by a single server. The objective is to minimise the forced idle time. The problem of minimising the forced idle time (interference problem) is known to be unary NP-hard for the case of two machines and equal set-up and arbitrary processing times. We propose a mixed integer linear programming model, which describes a special class of schedules where the jobs from a list are scheduled alternatively on the machines, and a heuristic algorithm is tested on instances with up to 100,000 jobs. The computational results indicate that the algorithm has an excellent performance even for very large instances, where mostly an optimal solution is obtained within a very small computational time.     

Job-shop scheduling with limited capacity buffers

In this paper we investigate job-shop problems where limited capacity buffers to store jobs in non-processing periods are present. In such a problem setting, after finishing processing on a machine, a job either directly has to be processed on the following machine or it has to be stored in a prespecified buffer. If the buffer is completely occupied the job may wait on its current machine but blocks this machine for other jobs. Besides a general buffer model, also specific configurations are considered. The aim of this paper is to find a compact representation of solutions for the jobshop problem with buffers. In contrast to the classical job-shop problem, where a solution may be given by the sequences of the jobs on the machines, now also the buffers have to be incorporated in the solution representation. In a first part, two such representations are proposed, one which is achieved by adapting the alternative graph model and a second which is based on the disjunctive graph model. In a second part, it is investigated whether the given solution representation can be simplified for specific buffer configurations. For the general buffer configuration it is shown that an incorporation of the buffers in the solution representation is necessary, whereas for specific buffer configurations possible simplifications are presented.     

Integrated production scheduling and batch delivery with fixed departure times and inventory holding costs

In this paper, we address a model for supply chain coordination. There are m manufacturers modelled as single machines, each of which processes a specific set of jobs (products). After processing is completed, jobs are batched, and batches are shipped to a customer by means of vehicles. The problem consists in concurrently finding a production schedule of the jobs, a partition of jobs into delivery batches and an assignment of delivery batches to vehicles, so that jobs are delivered within their deadlines and total costs are minimised. We focus on a scenario characterised by fixed departure times and inventory holding costs. For each departure time there is a given number of vehicles, possibly having limited capacity. Each job incurs a cost proportional to the time from job completion to delivery departure. In this paper, we show that the problem is NP-hard even for a very restricted case, and report various polynomiality results for two scenarios, namely: (i) when the production sequence of each manufacturer is fixed in advance, and (ii) when there is a single manufacturer and processing times are all equal to 1. We also point out several open problems.     

Spread time considerations in operational fixed job scheduling

In this study, we consider the operational fixed job scheduling problem on identical parallel machines. We assume that the jobs have fixed ready times and deadlines, and spread time constraints are imposed on machines. Our objective is to select a set of jobs for processing so as to maximise the total weight. We show that the problem is strongly NP-hard, and we investigate several special polynomially solvable cases. We propose a branch and bound algorithm that employs size reduction mechanisms, dominance conditions, and powerful lower and upper bounds. The computational results reveal that the branch and bound algorithm returns optimal solutions for problem instances with up to 100 jobs in reasonable solution times.     

A Branch-and-Bound Algorithm for the Continuous-Process Job-Shop Scheduling Problem

The continuous-process job-shop scheduling problem (CPJS) arises typically in the following way: (1) a set of M machines or production facilities are available; (2) a set of N jobs are to be processed through these machines in accordance with a technological matrix; (3) the machines associated with a given job must all be used simultaneously for the completion of this job; (4) a predetermined production time is required for each job; (5) the objective is to determine a production schedule which minimizes the total completion time (makespan) of all jobs. A branch-and-bound type algorithm for the solution of the (CPJS) problem is presented.     

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.     

Scheduling on Parallel Machines to Minimize Two Criteria Related to Job Tardiness

This paper deals with the scheduling of independent, nonpreemptible jobs with due dates on parallel machines. The machines are allowed to have different rates of operation, though a special algorithm is developed for the case in which they are all identical. The principal objective treated is minimizing the number of processors required to meet all due date constraints; in addition, it is shown that the solution methods for this first problem may be used to minimize maximum tardiness on a fixed number of machines. A lower bound on the number of machines required is developed, and several approximation algorithms are presented. Two enumerative optimization algorithms are developed. Computational results are presented for the case of identical machines.     

Parallel machine scheduling with limited controllable machine availability

Most machine scheduling models assume that either the machines are available all the time, or the time of their unavailability is fixed as a constraint. In this paper, we study the problem that neither the unavailability length nor the start time of machine unavailability is fixed. Instead, they would be determined in order to minimise the total cost involved with the completion time and the unavailable time. This model could represent a more realistic and complex situation, in which jobs and machines’ availability operations should be optimised simultaneously. After the model is formulated, some properties of the problem are presented. Then a branch and bound algorithm based on column generation approach is proposed to solve the problem. The computation results show that, within a reasonable computation time, the proposed algorithm can solve medium sized problems optimally.     

Scheduling on uniform parallel machines with periodic unavailability constraints

Scheduling problems under unavailability constraints has become a popular research topic in the last few years. Despite it’s important application in the real world, the uniform parallel machine scheduling problem was the least studied due to its complexity. In this paper, we investigated the uniform parallel machine scheduling problem under deterministic availability constraints. Each machine is subject to one unavailability period. Different versions of the problem regarding the type of jobs (identical and non-identical) and the performance measures (the total completion times and the makespan) were studied. For the case of identical jobs and for both performance measures, we developed linear programming models and optimal algorithms to provide a solution to the problem. For the case of non-identical jobs, we proved that the problem is NP-hard and propose a quadratic program. Because, this later cannot solve problems with very large number of jobs and machines, a heuristic was developed to find near optimal solutions to the problem especially with very large number of jobs and machines. The computational results showed that the heuristic’s performance is very high regardless the dimensions of problem instances.     

Scheduling of machines and automated guided vehicles in FMS using differential evolution

This paper addresses the problem of simultaneous scheduling of machines and two identical automated guided vehicles (AGVs) in a flexible manufacturing system (FMS). For solving this problem, a new meta-heuristic differential evolution (DE) algorithm is proposed. The problem consists of two interrelated problems, scheduling of machines and scheduling of AGVs. A simultaneous scheduling of these, in order to minimise the makespan will result in a FMS being able to complete all the jobs assigned to it at the earliest time possible, thus saving resources. An increase in the performance of the FMS under consideration would be expected as a result of making the scheduling of AGVs as an integral part of the overall scheduling activity. The algorithm is tested by using problems generated by various researchers and the makespan obtained by the algorithm is compared with that obtained by other researchers and analysed.     

Integrated production scheduling and vehicle routing problem with job splitting and delivery time windows

In this paper, we study a production scheduling and vehicle routing problem with job splitting and delivery time windows in a company working in the metal packaging industry. In this problem, a set of jobs has to be processed on unrelated parallel machines with job splitting and sequence-dependent setup time (cost). Then the finished products are delivered in batches to several customers with heterogeneous vehicles, subject to delivery time windows. The objective of production is to minimize the total setup cost and the objective of distribution is to minimize the transportation cost. We propose mathematical models for decentralized scheduling problems, where a production schedule and a distribution plan are built consecutively. We develop a two-phase iterative heuristic to solve the integrated scheduling problem. We evaluate the benefits of coordination through numerical experiments.     

Fast and meta-heuristics for common due-date assignment and scheduling on parallel machines

This study considers common due-date assignment and scheduling on parallel machines. The problem has three decision variables: assigning the common-due-date, allocating jobs to parallel machines, and sequencing the jobs assigned to each machine. The objective is to minimise the sum of due-date assignment, earliness and tardiness penalties. A mathematical programming model is presented, and then two types of heuristics are suggested after characterising the optimal solution properties. The two types of heuristics are: (a) a fast two-stage heuristic with obtaining an initial solution and improvement; and (b) two meta-heuristics, tabu search and simulated annealing, with new neighbourhood generation methods. Computational experiments were conducted on a number of test instances, and the results show that each of the heuristic types outperforms the existing one. In particular, the meta-heuristics suggested in this study are significantly better than the existing genetic algorithm.     

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.     

Unrelated parallel machine scheduling with job splitting

Scheduling jobs on unrelated parallel machines is an activity that is very much a part of industrial scheduling. We report a methodology for minimizing the total weighted tardiness of all jobs intended to be processed on unrelated parallel machines in the presence of dynamic job releases and dynamic machine availability. More importantly, the mixed (binary) integer linear programming model formulated for the problem incorporates a couple of “hard” operational constraints to ensure that just-in-time manufacturing practices are followed by controlling the work-in-process and/or finished goods inventories generated by split jobs mandated by a tight due date, a high priority, and/or a high workload. Four different methods based on simple and composite dispatching rules are used to identify an initial solution, which is then used by the tabu-search-based heuristic solution algorithm to ultimately find the best solution. Incorporating the various tabu search features led to the development of six different heuristics that were tested on eight small problem instances to compare the quality of their solutions to the optimal solutions. The results show that the proposed heuristics are capable of obtaining solutions of good quality in a remarkably short computation time with the best performer among them recording a percentage deviation of only 1.18%. A factorial experiment based on a split-plot design is performed to test the performance of the heuristics on problem structures, ranging from nine jobs and three machines to 60 jobs and 15 machines. The results show that the newly developed composite dispatching heuristic, referred to as the modified apparent tardiness cost, is capable of obtaining initial solutions that significantly accelerate the tabu-search-based heuristics to attain the best solution. The use of a long-term memory function is proven to be advantageous in solving all problem structures. In addition, the variable tabu list size is preferred for solving the small problem structure, while the fixed tabu list size is preferred as the problem size grows from small to medium and then large.     

A simulated annealing procedure for single row layout problems in flexible manufacturing systems

In this paper we address a layout problem in flexible manufacturing systems (FMSs). A layout type that has been extensively implemented in FMSs is the single row machine layout. In such a configuration machines are arranged along a straight track with a material handling device moving jobs from one machine to another. The single row layout problem (SRLP) deals with the optimal arrangement of the machines for the above configuration. We propose a simulated annealing (SA) heuristic for the solution of SRLP. Extensive computational results, and ways to improve the performance of the SA algorithm through parameter fine-tuning procedures, are reported.     

Flexibility-based multi-objective approach for machines selection in reconfigurable manufacturing system (RMS) design under unavailability constraints

The reconfigurable manufacturing system (RMS) is a recent manufacturing paradigm driven by the high responsiveness and performance efficiencies. In such system, machines, material handling units or machines components can be added, modified, removed or interchanged as needed. Hence, the design of RMS is based on reconfigurable machines capabilities and product specification. This paper addresses the problem of machines selections for RMS design under unavailability constraints and aims to develop an approach to ensure the best process plan according to the customised flexibility required to produce all parts of a given product. More specifically, we develop a flexibility-based multi-objective approach using an adapted version of the well-known non-dominated sorting genetic algorithm to select adequate machines from a set of candidate (potential) ones, in order to ensure the best responsiveness of the designed system in case of unavailability of one of the selected machines. The responsiveness is based on the flexibility of the designed system and a generated process plan, which guarantees the management of machines unavailability. It is defined as the ability and the capacity to adapt the process plan in response to machines unavailability. Two objectives are considered, respectively, the maximisation of the flexibility index of the system and the minimisation of the total completion time. To choose the best solution in the Pareto front, a multi-objective decision-making method called technique for order of preference by similarity to ideal solution is used. To demonstrate the applicability of the proposed approach, a simple example is presented and the numerical results are analysed.     

Modelling and optimization of a bi-objective flow shop scheduling with diverse maintenance requirements

In real-world problems, machines cannot continuously operate and have to stop for maintenance before they fail. Lack of maintenance can also affect the performance of machines in processing jobs. In this paper, a permutation flow shop scheduling problem with multiple age-based maintenance requirements is modelled as a novel mixed-integer linear program in which the objectives are conflicting. In modelling the problem, we assume that infrequent maintenance can prolong job processing times. One of the objectives is to minimise the total maintenance cost by planning as few maintenance activities as possible to only meet the minimum requirements, and the other objective tries to minimise the total tardiness by sequencing the jobs and planning the maintenance activities in such a way that the processing times are not prolonged and unnecessary maintenance times are avoided. Because of this conflict, an interactive fuzzy, bi-objective model is introduced. Application of the model is illustrated through a case study for operations and maintenance scheduling of heavy construction machinery. An effective and efficient solution methodology is developed based on the structure of the problem and tested against commercial solvers and a standard GA. Computational results have verified the efficiency of the proposed solution methodology and show that unlike the proposed method, a generic metaheuristic that does not consider the unique structure of the problem can become ineffective for real-world problem sizes.     

Scheduling with agreements: new results

In this paper, the problem of scheduling with agreements (SWA) is considered. In scheduling, this consists of a set of jobs non-preemptively on identical machines subject to constraints that only some specific jobs can be scheduled concurrently on different machines. These constraints are given by an agreement graph and the aim is to minimise the makespan. In the case of two machines we extend two NP-hardness results of SWA with processing times at most three that hold for bipartite agreement graphs to more general agreement graphs. Complexity results of SWA are established in the case of split and complement of bipartite graphs. We also present some approximation results for SWA.     

A meta-heuristic approach for solving the no-wait flow-shop problem

No-wait flow-shop scheduling problems refer to the set of problems in which a number of jobs are available for processing on a number of machines in a flow-shop context with the added constraint that there should be no waiting time between consecutive operations of the jobs. The problem is strongly NP-hard. In this paper, the considered performance measure is the makespan. In order to explore the feasible region of the problem, a hybrid algorithm of Tabu Search and Particle Swarm Optimisation (PSO) is proposed. In the proposed approach, PSO algorithm is used in order to move from one solution to a neighbourhood solution. We first employ a new coding and decoding technique to efficiently map the discrete feasible space to the set of integer numbers. The proposed PSO will further use this coding technique to explore the solution space and move from one solution to a neighbourhood solution. Afterwards, the algorithm decodes the solutions to its respective feasible solution in the discrete feasible space and returns the new solutions to the TS. The algorithm is tested by solving a large number of problems available in the literature. Computational results show that the proposed algorithm is able to outperform competitive methods and improves some of the best-known solutions of the considered test problems.