The no-wait two-machine flow shop scheduling problem with convex resource-dependent processing times

We extend the classical no-wait two-machine flow shop scheduling problem to the case where job-processing times are controllable through the allocation of a common, limited and nonrenewable resource. Our objective is to simultaneously determine the sequence of the jobs and the resource allocation for each job on both machines in order to minimize the makespan. By using the equivalent load method to obtain the optimal resource allocation on a series-parallel graph, we reduce the problem to a sequencing one and show that it is equivalent to a new special case of the Traveling Salesman Problem (TSP). We prove that although the reduced problem forms a subclass of the TSP on permuted Monge matrices, it is still strongly NP-hard. We provide an approximation result and present three special cases which are polynomially solvable. We have also tested two different subtour-patching heuristics in large-scale computational experiments on randomly generated instances of the problem. Both heuristics produced close-to-optimal solutions in most cases.     

Outsourcing and scheduling for a two-machine flow shop with release times

This article addresses a two-machine flow shop scheduling problem where jobs are released intermittently and outsourcing is allowed. The first operations of outsourced jobs are processed by the first subcontractor, they are transported in batches to the second subcontractor for processing their second operations, and finally they are transported back to the manufacturer. The objective is to select a subset of jobs to be outsourced, to schedule both the in-house and the outsourced jobs, and to determine a transportation plan for the outsourced jobs so as to minimize the sum of the makespan and the outsourcing and transportation costs. Two mathematical models of the problem and several necessary optimality conditions are presented. A solution approach is then proposed by incorporating the dominance properties with an ant colony algorithm. Finally, computational experiments are conducted to evaluate the performance of the models and solution approach.     

Robust scheduling of a two-machine flow shop with uncertain processing times

This paper focuses on manufacturing environments where job processing times are uncertain. In these settings, scheduling decision makers are exposed to the risk that an optimal schedule with respect to a deterministic or stochastic model will perform poorly when evaluated relative to actual processing times. Since the quality of scheduling decisions is frequently judged as if processing times were known a priori, robust scheduling, i.e., determining a schedule whose performance (compared to the associated optimal schedule) is relatively insensitive to the potential realizations of job processing times, provides a reasonable mechanism for hedging against the prevailing processing time uncertainty. In this paper we focus on a two-machine flow shop environment in which the processing times of jobs are uncertain and the performance measure of interest is system makespan. We present a measure of schedule robustness that explicitly considers the risk of poor system performance over all potential realizations of job processing times. We discuss two alternative frameworks for structuring processing time uncertainty. For each case, we define the robust scheduling problem, establish problem complexity, discuss properties of robust schedules, and develop exact and heuristic solution approaches. Computational results indicate that robust schedules provide effective hedges against processing time uncertainty while maintaining excellent expected makespan performance     

An improved approximation algorithm for the two-machine open shop scheduling problem with family setup times

We consider the problem of scheduling families of jobs in a two-machine open shop so as to minimize the makespan. The jobs of each family can be partitioned into batches and a family setup time on each machine is required before the first job is processed, and when a machine switches from processing a job of some family to a job of another family. For this NP-hard problem the literature contains (5/4)-approximation algorithms that cannot be improved on using the class of group technology algorithms in which each family is kept as a single batch. We demonstrate that there is no advantage in splitting a family more than once. We present an algorithm that splits one family at most once on a machine and delivers a worst-case performance ratio of 6/5.     

A two-machine permutation flow shop scheduling problem with buffers

Problems with blocking (limited intermediate storage space) are used frequently for modelling and scheduling just-in-time and flexible manufacturing systems. In this paper, an approximation algorithm is presented for the problem of finding the minimum makespan in a two-machine permutation flow-shop scheduling problem with the mediating buffer of finite capacity. The algorithm is based on the tabu search approach supported by the reduced neighborhood, search accelerator and technique of back jumps on the search trajectory. Due to some special properties, the proposed algorithm provides makespans very close to optimal in a short time. It has been shown that this algorithm outperforms all known approximation algorithms for the problem stated.     

Robust scheduling of a two-stage hybrid flow shop with uncertain interval processing times

This paper studies the makespan minimisation scheduling problem in a two-stage hybrid flow shop. The first stage has one machine and the second stage has m identical parallel machines. Neither the processing time nor probability distribution of the processing time of each job is uncertain. We propose a robust (min–max regret) scheduling model. To solve the robust scheduling problem, which is NP-hard, we first derive some properties of the worst-case scenario for a given schedule. We then propose both exact and heuristic algorithms to solve this problem. In addition, computational experiments are conducted to evaluate the performance of the proposed algorithms.     

No-wait two-machine permutation flow shop scheduling problem with learning effect,common due date and controllable job processing times

We consider a two-machine no-wait permutation flow shop common due date assignment scheduling problem where the processing time of a job is given as a function of its position in the sequence and its amount of resource allocated to this job. The common due date (CON) assignment method means that all the jobs are given a common due date. We need to make a decision on the common due date, resource allocation and the sequence of jobs to minimise total earliness, tardiness, common due date cost and total resource cost. We show that the problem remains polynomially solvable under the proposed model.     

A heuristic model for dynamic flexible job shop scheduling problem considering variable processing times

In real scheduling problems, unexpected changes may occur frequently such as changes in task features. These changes cause deviation from primary scheduling. In this article, a heuristic model, inspired from Artificial Bee Colony algorithm, is proposed for a dynamic flexible job-shop scheduling (DFJSP) problem. This problem consists of n jobs that should be processed by m machines and the processing time of jobs deviates from estimated times. The objective is near-optimal scheduling after any change in tasks in order to minimise the maximal completion time (Makespan). In the proposed model, first, scheduling is done according to the estimated processing times and then re-scheduling is performed after determining the exact ones considering machine set-up. In order to evaluate the performance of the proposed model, some numerical experiments are designed in small, medium and large sizes in different levels of changes in processing times and statistical results illustrate the efficiency of the proposed algorithm.     

Production in a two-machine flowshop scheduling environment with uncertain processing and setup times to minimize makespan

A wide range of uncertainties exists in some real-world production environments which result in uncertain setup and/or processing times. Factors such as crew skills, shortages in equipment and resource breakdowns can be the sources of these uncertainties. This study considers a two-machine production flowshop scheduling problem where both setup and processing times are treated as uncertain variables. The objective is to minimise makespan which is an effective way of resource utilisation. There exists a dominance relation in the literature for the two-machine flowshop scheduling problem with uncertain setup and processing times. However, the dominance relation has not been evaluated. In this study, we evaluate the existing dominance relation. Moreover, a new dominance relation is established and shown to be more effective than the existing one. Furthermore, twenty-five implementations of a polynomial time algorithm are developed. Extensive computational experiments are conducted to evaluate the performance of the implementations of the algorithm. The computational experiments indicate that the overall gap (error) of the best implementation of the algorithm is less than 0.3% when compared to the optimal solution. Moreover, the performance of this implementation of the algorithm is the best one when compared to the remaining implementations for all the considered experimental environments. Additionally, the performance of this implementation of the algorithm is shown to be insensitive to the uncertainty in setup times.     

Heuristics for permutation flow shop scheduling with batch setup times

In this paper we consider permutation flow shop scheduling problems with batch setup times. Each job has to be processed on each machine once and the technological routes are identical for all jobs. The set of jobs is divided into groups. There are given processing timest _ij of jobi on machinej and setup timess _rj on machinej when a job of ther-th group is processed after a job of another group. It is assumed that the same job order has to be chosen on each machine. We consider both the problems of minimizing the makespan and of minimizing the sum of completion times, where batch or item availability of the jobs is assumed. For these problems we give various constructive and iterative algorithms. The constructive algorithms are based on insertion techniques combined with beam search. We introduce suitable neighbourhood structures for such problems with batch setup times and describe iterative algorithms that are based on local search and reinsertion techniques. The developed algorithms have been tested on a large collection of problems with up to 80 jobs.Supported by Deutsche Forschungsgemeinschaft (Project ScheMA) and by the International Association for the Promotion of Cooperation with Scientists from the Independent States of the Former Soviet Union (Project INTAS-93-257)     

Heuristics for the two-machine flowshop scheduling problem to minimise makespan with bounded processing times

The two-machine flowshop scheduling problem of minimising makespan is addressed where jobs have random processing times that are bounded within certain intervals. The probability distributions of job processing times within intervals are not known. The only known information about job processing times is the lower and upper bounds. The decision concerning the solution to the problem, i.e. finding a sequence, has to be made based on these bounds. Different heuristics using the bounds are proposed, and the proposed heuristics are compared based on randomly generated data. Computational analysis has shown that three of the proposed heuristics perform well with an overall average error of less than one percent. Moreover, for symmetric distributions, it is also shown that one of the heuristics, which applies Johnson's algorithm to the average of the lower and upper bounds, performs best with an overall average percentage error of 0.71. The obtained results are also shown to be consistent with recent results reported in the literature.     

A hybrid differential evolution algorithm for a two-stage flow shop on batch processing machines with arbitrary release times and blocking

Batch processing machines (BPMs) have important applications in a variety of industrial systems. This paper considers the problem of scheduling two BPMs in a flow shop with arbitrary release times and blocking such that the makespan is minimised. The problem is formulated as a mixed integer programming model. Subsequently, a hybrid discrete differential evolution (HDDE) algorithm is proposed. In the algorithm, individuals in the population are first represented as discrete job sequences, and mutation and crossover operators are applied based on the representation. Second, after using the first-fit rule to form batches, a novel least idle/blocking time heuristic is developed to schedule the batches in the flow shop. Furthermore, an effective local search technique is embedded in the algorithm to enhance the exploitation ability. The performance of the proposed algorithm is evaluated by comparing its results to a commercial solver (CPLEX), a genetic algorithm and a simulated annealing algorithm. Computational experiments demonstrate the superiority of the HDDE algorithm in terms of solution quality, robustness and run time.     

Open shop scheduling problem to minimize total weighted completion time

A given number of jobs in an open shop scheduling environment must each be processed for given amounts of time on each of a given set of machines in an arbitrary sequence. This study aims to achieve a schedule that minimizes total weighted completion time. Owing to the strong NP-hardness of the problem, the weighted shortest processing time block (WSPTB) heuristic is presented to obtain approximate solutions for large-scale problems. Performance analysis proves the asymptotic optimality of the WSPTB heuristic in the sense of probability limits. The largest weight block rule is provided to seek optimal schedules in polynomial time for a special case. A hybrid discrete differential evolution algorithm is designed to obtain high-quality solutions for moderate-scale problems. Simulation experiments demonstrate the effectiveness of the proposed algorithms.     

An effective discrete harmony search algorithm for flexible job shop scheduling problem with fuzzy processing time

This study addresses flexible job shop scheduling problem (FJSP) with fuzzy processing time. The fuzzy or uncertainty of processing time is one of seven characteristics in remanufacturing. A discrete harmony search (DHS) algorithm is proposed for FJSP with fuzzy processing time. The objective is to minimise maximum fuzzy completion time. A simple and effective heuristic rule is proposed to initialise harmony population. Extensive computational experiments are carried out using five benchmark cases with eight instances from remanufacturing. The proposed heuristic rule is evaluated using five benchmark cases. The proposed DHS algorithm is compared to six metaheuristics. The results and comparisons show the effectiveness and efficiency of DHS for solving FJSP with fuzzy processing time.     

Scheduling of a no-wait two-machine flow shop with sequence-dependent setup times and probable rework using robust meta-heuristics

This paper investigates the scheduling of a no-wait two-machine flow shop considering anticipatory sequence-dependent setup time and a probable rework for both machines to minimise mean completion time (MCT). To tackle the problem, a robust meta-heuristic algorithm, namely the adapted imperialist competitive algorithm (AICA), has been proposed and is compared with two common and popular meta-heuristic algorithms (i.e. genetic algorithm (GA) and population-based simulated annealing (PBSA)). In this study, we have adapted a traditional imperialist competitive algorithm (ICA) with some considerable changes. First of all, a revolution procedure is added to the algorithm for imperialists similar to colonies. Furthermore, the revolution is only performed when the new solution is better than the previous solution, and chief among them for preservation of premature convergence, the concept of global war is applied. However, the performance of AICA is sensitive to the choice of the best parameter values. Thus, to obtain optimal performance, a comprehensive calibration methodology called response surface methodology is employed to obtain the best combination of parameter values. In order to evaluate the effectiveness and efficiency of proposed algorithms, several test problems are generated and the results obtained from algorithms are then compared in terms of relative percentage deviation. Computational experiments indicate that AICA outperforms GA and PBSA in the MCT performance measure, and GA outperforms the others in terms of computational time.     

Asymptotic optimality of shortest processing time-based algorithms for flow shop and open shop problems with nonlinear objective functions

This article investigates the criterion of minimizing total k-power completion time (TKCT) in flow shop and open shop scheduling. For these NP-hard problems, the asymptotic optimality of the shortest processing time-based algorithms is proven for a sufficiently large problem scale. To numerically evaluate the convergence of the algorithms, new lower bounds with performance guarantees are presented for the flow shop TKCT problem. Computational results demonstrate the performance of the proposed algorithms and the effectiveness of the nonlinear objective. In addition, theoretical results on the single-machine TKCT problem are obtained for mathematical deduction.     

Rescheduling strategies for integrating rush orders with preventive maintenance in a two-machine flow shop

Preventive maintenance and rush orders are related. Although preventive maintenance is essential for maximising equipment reliability, it can substantially slow the manufacturing process. Rush order rescheduling involves similar conflicts. Scheduling maintains the robustness of the production schedule, but rush orders require rescheduling. Although preventive maintenance and rush orders are essential manufacturing processes, research on the integration of these functions is insufficient. Unlike recent work that analyses preventive maintenance or rush orders as separate functions, this study proposes an integrated model that analyses both preventive maintenance and rush orders in a two-machine flow shop. The model is then evaluated using two different rescheduling methods. Non-parametric analysis of the models revealed that these two rescheduling methods differ significantly under integrated maintenance and rush order situations.     

A new complexity proof for the two-stage hybrid flow shop scheduling problem with dedicated machines

This paper considers a two-stage hybrid flow shop scheduling problem with dedicated machines at stage 2. The objective is to minimise the makespan. There is one machine at stage 1 and two machines at stage 2. Each job must be processed on the single machine at stage 1 and, depending upon the job type, the job is processed on either of the two machines at stage 2. We first introduce this special type of the two-stage hybrid flow shop scheduling problem and present some preliminary results. We then present a counter example to the known complexity proof of Riane et al. [Riane, F., Artiba, A. and Elmaghraby, S.E., 2002. Sequencing a hybrid two-stage flowshop with dedicated machines. International Journal of Production Research, 40, 4353–4380.] Finally, we re-establish the complexity of the problem.     

An estimation of distribution algorithm for hybrid flow shop scheduling under stochastic processing times

The estimation of distribution algorithm (EDA) has recently emerged as a promising alternative to traditional evolutionary algorithms for solving combinatorial optimisation problems. This paper presents a novel two-phase simulation-based EDA (TPSB-EDA) for minimising the makespan of a hybrid flow shop under stochastic processing times. To address the stochastic scheduling problem efficiently, the proposed TPSB-EDA incorporates a two-phase simulation model to estimate the performance of candidate solutions. In this model, an optimal back propagation network is firstly applied to identify a set of roughly good solutions, and then the selected solutions are further evaluated by a discrete-event simulation algorithm. Moreover, an annealing selection mechanism (ASM) is adopted to preserve the population diversity of EDA. Different from the selection operators of common EDAs, the ASM uses Boltzmann probability in the annealing algorithm to select part of population to establish the probabilistic model. Computation results indicate that the TPSB-EDA provides good solutions in the aspects of solution quality and computational efficiency.