Non-permutation Scheduling Research Articles

Ambiguity of the definition of permutation flow shops in the presence of missing operations

The majority of flow shop studies focused so far on permutation schedules, even though non-permutation schedules might offer better results at the expense however of additional computational effort. It is therefore crucial to determine whether the use of non-permutation schedules is justified by the potential improvements that might be obtained. This question becomes particularly interesting in the case of the flow shop problem with missing operations. But, before investigating this question, we must first clarify the definition of a permutation within this context. The objective of this paper consists in bringing out the ambiguity of the definition of a permutation within the flow shop model with missing operations. To overcome this ambiguity, we propose another way of looking at a permutation that includes the classical understanding. Even though, these two definitions may produce different solutions, we present cases where they are equivalent with respect to the makespan criterion.

Scheduling non-permutation flowshop with finite buffers and two competitive agents

Flowshop scheduling is a popular and valuable optimisation model in academia. Application of the scheduling model for industrial problems has to consider real-world scenarios, such as non-permutation scheduling, finite buffers and individual competition from different customers. These non-negligibly factors can lead to a huge solution space, leaving it barely able to be effectively solved. Therefore, this study investigates a non-permutation bi-agent flowshop scheduling problem with limited buffers, where tasks are released over time and minimisation of a linear criterion (i.e. weighted combination of makespans) improves customer satisfaction. Mixed integer programming model is established for this strong NP-hardness problem that the optimum cannot be found in polynomial time. For small-scale instances, a branch and bound (B&B) algorithm is provided to achieve exact solutions, where effective non-delay pruning rules, well-designed lower and upper bounds are utilised to eliminate invalid nodes during searching. For medium-scale instances, a hybrid particle swarm optimisation (HPSO) algorithm is proposed to seek high-quality solutions, where a two-stage collaborative evolution promotes deep exploration, an efficient sequence-based rule amends infeasible solutions and several validated mechanisms enhances convergence. Evaluation of extensive computational experiments, including components evaluation and nonparametric test, demonstrate the effectiveness of the presented algorithms.

Automatic generation of iterated greedy algorithms for the non-permutation flow shop scheduling problem with total completion time minimization

In this paper we address the non-permutation flow shop scheduling problem, a more general variant of the flow shop problem in which the machines can have different sequences of jobs. We aim to minimize the total completion time. We propose a template to generate iterated greedy algorithms, and use an automatic algorithm configuration to obtain efficient methods. This is the first automated approach in the literature for the non-permutation flow shop scheduling problem. The algorithms start by building a high-quality permutation solution, which is then improved during a second phase that generates non-permutation solutions by changing the job order on some machines. The obtained algorithms are evaluated against two well-known benchmarks from the literature. The results show that they can find better schedules than the state-of-the-art methods for both the permutation and non-permutation flow shop scheduling problems in comparable experimental conditions, as evidenced by comprehensive computational and statistical testing. We conclude that using non-permutation schedules is a viable alternative to reduce the total completion time that production managers should consider.

Scheduling flowline manufacturing cells with inter-cellular moves: non-permutation schedules and material flows in the cell scheduling problem

A major goal of the concept of cellular manufacturing is to form independent cells. However, the formation of entirely independent cells is rarely found in practice, where inter-cellular transports of jobs are often accepted to a certain level. The resulting scheduling task in flowline manufacturing cells is referred to as cell scheduling problem. Usually it is modelled and solved analogously to the group scheduling problem, which arises within a single cell with multiple part families. In this study, we point out several characteristics that specify the distinctiveness of cell scheduling. Furthermore, a new cell-based objective, namely total cell makespan, is introduced. As no appropriate benchmark instances are available so far, new test problems are developed, that integrate the derived characteristics. In an extensive computational study known constructive heuristics as well as simulated annealing algorithms to generate permutation and non-permutation schedules are tested. It is revealed that especially the type of material flows between cells has an immense impact on the algorithms' performance. The findings support a better problem understanding and point out the necessity of developing new heuristics especially for certain types of the cell scheduling problem.

The Dominance Flow Shop Scheduling Problem

We introduce a new line of analysis of Flow Shop scheduling problems, for the case of two jobs and assuming that processing times are unknown. The goal is to determine the domination relations between permutation and non-permutation schedules. We analyze the structural and dominance properties that ensue in this setting, based on the critical paths of schedules.

Non-permutation flowshop scheduling problem with minimal and maximal time lags: theoretical study and heuristic

In this paper, we address the non-permutation flowshop scheduling problem with minimal and maximal time lags between successive operations of each job. For this problem, the set of permutation schedules is not a dominant set but not all non-permutation schedules are feasible because they are not always able to satisfy all time lag constraints. We present a theoretical study, limited to the two-machine case, and related to the change on one machine of the order of two successive jobs of a permutation schedule. This study gives first the necessary conditions to make such move with regard to the feasibility of the schedule; secondly the necessary conditions to make such move interesting with regard to either the makespan or the number of tardy jobs. Through this analysis, we obtain new properties of dominance of permutation schedules. The results of the study are incorporated into a heuristic algorithm which starts the search with optimal permutation schedules and tries to improve them so as to obtain better non-permutation schedules. We also propose a mixed integer linear programming model. The objective function is to minimize lexicographically the number of tardy jobs as primary criterion and the makespan as secondary one. Computational experiments are performed to compare permutation with non-permutation schedules.

Family scheduling with batch availability in flow shops to minimize makespan

This paper addresses a batch scheduling problem in flow shop production systems, where job families are formed based on setup similarities. In order to improve setup efficiency, we consider batching decisions in our solution procedure. Due to its high practical relevance, the batch availability assumption is also adopted in this study. In the presence of sequence-dependent setup times, it is proved that a permutation flow shop is generally not optimal. Therefore, our objective is to determine solutions with inconsistent batches, which essentially lead to non-permutation schedules, to minimize makespan. After examining structural properties, we develop a tabu search algorithm with multiple neighbourhood functions. Computational results confirm the remarkable benefits of batching decisions. Our algorithm also outperforms some well-known and well-performing approaches.

MILP formulation and genetic algorithm for flow shop scheduling problem with missing operations

The purpose of this research is to formulate and solve general flow shop scheduling problems with missing operations in which the objective function is to minimise the maximum completion time (Makespan). Missing operations assumption refers to the fact that at least one job does not visit one machine. We first propose a mixed integer linear programming (MILP) model for the problem that can generates non-permutation schedules. The MILP Model can be used to compute optimal solution for the small-sized problems. In addition, we have presented an adapted genetic algorithm that can be used to generate non-permutation schedule with relatively good solutions in short computational time. Considering of non-permutation schedules is necessary to pass some machine when we have missing operations assumption. To verify the effectiveness of the presented approach, computational experiments are performed on a set of well-known classical flow shop scheduling (without missing operation) benchmark problems. The results show that the performance of the approach is suitable and can reaches good-quality solutions within a reasonable computational time. Thus, we use the genetic algorithm to solve large-sized flow shop scheduling problems with missing operations.

A constructive algorithm and a simulated annealing approach for solving flowshop problems with missing operations

In many practical cases of flowshop environments and especially in flowline manufacturing cells, some or all jobs may not require processing on all machines. Hence, this paper focuses on the flowshop scheduling problem with missing operations. A modification of the constructive NPS-set heuristic is proposed, which generates non-permutation schedules effectively. Furthermore, a two-phase simulated annealing (SA) method is presented that specifically considers missing operations in its procedure. The modified NPS-set heuristic and the two-phase SA are tested and statistically evaluated by an extensive computational study for total flow time criteria. The results show that the modified NPS-set heuristic as well as the specific consideration of missing operations can enhance the algorithms’ performance significantly.

Non-permutation flow shop scheduling with order acceptance and weighted tardiness

This paper studies the non-permutation solution for the problem of flow shop scheduling with order acceptance and weighted tardiness (FSS-OAWT). We formulate the problem as a linear mixed integer programming (LMIP) model that can be optimally solved by AMPL/CPLEX for small-sized problems. In addition, a non-linear integer programming (NIP) model is presented to design heuristic algorithms. A two-phase genetic algorithm (TP-GA) is developed to solve the problem of medium and large sizes based on the NIP model. The properties of FSS-OAWT are investigated and several theorems for permutation and non-permutation optimum are provided. The performance of the TP-GA is studied through rigorous computational experiments using a large number of numeric instances. The LMIP model is used to demonstrate the differences between permutation and non-permutation solutions to the FSS-OAWT problem. The results show that a considerably large portion of the instances have only an optimal non-permutation schedule (e.g., 43.3% for small-sized), and the proposed TP-GA algorithms are effective in solving the FSS-OAWT problems of various scales (small, medium, and large) with both permutation and non-permutation solutions.

Two simple and effective heuristics for minimizing the makespan in non-permutation flow shops

We propose a constructive and an iterated local search heuristic for minimizing the makespan in the non-permutation flow shop scheduling problem. Both heuristics are based on the observation that optimal non-permutation schedules often exhibit a permutation structure with a few local job inversions. In computational experiments we compare our heuristics to the best heuristics for finding non-permutation and permutation flow shop schedules, and evaluate the reduction in makespan and buffer size that can be achieved by non-permutation schedules.

Iterated Local Search Heuristics for Minimizing Total Completion Time in Permutation and Non-permutation Flow Shops

We study the improvement of non-permutation over permutation schedules in flow shops when minimizing the total completion time. We solve both problems by a two-phase heuristic. The first phase uses an iterated local search to find a good permutation schedule. The second phase explores non-permutation schedules using an effective insertion neighborhood, that permits to anticipate or delay a job when passing from one machine to the next. In computational experiments we show that both phases yield state-of-the-art results. We find that allowing non-permutation schedules can reduce the total completion considerably with a moderate extra effort, and without increasing the buffer size needed during processing. We conclude that non-permutation schedules can be viable alternative to permutation schedules in flow shops.

A Non-Permutation Flow Shop Manufacturing Cell Scheduling Problem with Part's Sequence Dependent Family Setup Times

This article presents a new mathematical model for a dynamic flow shop manufacturing cell scheduling problem (DFMCSP) with agreeable job release date for each part family where family setup times are dependent on sequence of parts within families. It means this article considers non-permutation schedules for both sequence of families and sequence of parts within families. The objective is minimizing the Makespan (Cmax). Since, this problem belongs to NP-Hard class. Therefore, reaching an optimal solution in a reasonable computational time by using exact methods is extremely difficult. Thus, this article proposes meta-heuristic methods such as Genetic Algorithms (GA) and Tabu Search (TS). Finally, the computational results compare efficiency of the proposed algorithms under the performance measures.

A general flow shop scheduling problem with consideration of position-based learning effect and multiple availability constraints

In this paper, a more general version of the flow shop scheduling problem with the objective of minimizing the total flow time is investigated. In order to get closer to the actual conditions of the problem, some realistic assumptions including non-permutation scheduling, learning effect, multiple availability constraints, and release times are considered. It is assumed that the real processing time of each job on a machine depends on the position of that job in the sequence, and after processing a specified number of jobs at each machine, an unavailability period is occurring because of maintenance activities. Moreover, it is supposed that each job may not be ready for processing at time zero and may have a release time. According to these assumptions, a new mixed integer linear programming (MILP) model is proposed to formulate the problem. Due to the high complexity of the problem, a heuristic method and a simulated annealing algorithm are presented to find the nearly optimal solutions for medium- and large-sized problems. To obtain better and more robust solutions, the Taguchi method is used in order to calibrate the simulated annealing algorithm parameters. Finally, the computational results are provided for evaluating the performance and effectiveness of the proposed solution methods.

Flow shop batching and scheduling with sequence-dependent setup times

This paper addresses the flow shop batching and scheduling problem where sequence-dependent family setup times are present and the objective is to minimize makespan. We consider violating the group technology assumption by dividing product families into batches. In order to reduce setup times, inconsistent batches are formed on different machines, which lead to non-permutation schedules. To the best of our knowledge, this is the first time that the splitting of job families into inconsistent batches has been considered in a flow shop system. A tabu search algorithm is developed which contains several neighbourhood functions, double tabu lists and a multilevel diversification structure. Compared to the state-of-the-art meta-heuristics for this problem, the proposed tabu search algorithm achieves further improvement when the group scheduling assumption is dropped. Also, various experiments conducted on the benchmark problem instances confirm the benefits of batching. Therefore, it will be prudent for the practitioners to consider adopting inconsistent batches and non-permutation schedules to improve their operational efficiency within a reasonable amount of computational effort.

A Branch-and-Bound Based Heuristic Algorithm for Minimizing Makespan in Machining-Assembly Flowshop Scheduling

This paper proposes a heuristic algorithm, called list-based squeezing branch and bound algorithm, for solving a machine-fixed, machining-assembly flowshop scheduling problem to minimize makespan. The machine-fixed, machining-assembly flowshop consists of some parallel two-machine flow lines at a machining stage and one robot at an assembly stage. Since an optimal schedule for this problem is not always a permutation schedule, the proposed algorithm first finds a promising permutation schedule, and then searches better non-permutation schedules near the promising permutation schedule in an enumerative manner by elaborating a branching procedure in a branch and bound algorithm. The results of numerical experiments show that the proposed algorithm can efficiently provide an optimal or a near-optimal schedule with high accuracy such as mean relative error being less than 0.2% and the maximum relative error being at most 3%.

Multi-objective flow shop scheduling problem with stochastic parameters: fuzzy goal programming approach

Flow shop scheduling problem with stochastic parameters is dealt with in this paper. A multi-objective mixed integer linear programming model is proposed in this concern which can generate non-permutation schedules. To provide a more realistic model, process time and release time are considered stochastic variables with normal distribution. The objective functions are minimising three performance measures including maximum completion time (Makespan), total flow time and total tardiness. Chance constrained programming (CCP) approach and fuzzy goal programming (FGP) are applied to deal with the stochastic parameters and multi-objective function. Due to the complexity of the problem, we have implemented an adapted genetic algorithm to solve large-sized problem. According to the computational experiments, the GA can reach good-quality solutions in reasonable computational time, and can be used to solve large scale problems effectively.

Capacitated production planning problem considering the detailed scheduling constraints in a flow shop environment

In this paper, a multi-product multi-period production system organized as a flow shop is considered. To overcome the shortcomings of hierarchical production planning, detailed production planning scheduling and sequencing constraints are integrated. A new integrated mixed-integer programming (MIP) model, capable of simultaneously performing production planning and nonpermutation scheduling, is proposed for the considered problem. For precise machine workload calculation, detailed scheduling constraints are used to determine a feasible production plan. The objective function includes production, inventory, backlog, set-up costs, and the costs associated with machine idle time and overtime. A hybrid simulated annealing (HSA) algorithm with dispatching rules and an exact solution method is proposed to solve the integrated model. The proposed algorithm searches the solution space for both problems and finds a combination of production planning and scheduling that is feasible and close to optimum. To evaluate the performance of the proposed model and solution method, problems of different scales are studied. The total costs of the proposed solution method are smaller than the results of hierarchical production planning.

Non-permutation flowshop scheduling in a supply chain with sequence-dependent setup times

In this paper, we consider a flowshop scheduling problem with sequence-dependent setup times and a bicriteria objective to minimize the work-in-process inventory for the producer and to maximize the customers' service level. The use of a bicriteria objective is motivated by the fact that successful companies in today's environment not only try to minimize their own cost but also try to fulfill their customers' need. Two main approaches, permutation and non-permutation schedules, are considered in finding the optimal schedule for a flowshop. In permutation schedules the sequence of jobs remains the same on all machines whereas in non-permutation schedule, jobs can have different sequence on different machines. A linear mathematical model for solving the non-permutation flowshop is developed to comply with all of the operational constraints commonly encountered in the industry, including dynamic machine availabilities, dynamic job releases, and the possibility of jobs skipping one or more machines, should their operational requirements deem that it was necessary. As the model is shown to be NP-hard, a metasearch heuristic, employing a newly developed concept known as the Tabu search with embedded progressive perturbation (TSEPP) is developed to solve, in particular, industry-size problems efficiently. The effectiveness and efficiency of the search algorithm are assessed by comparing the search algorithmic solutions with that of the optimal solutions obtained from CPLEX in solvable small problem instances.

Tabu search for non-permutation flowshop scheduling problem with minimizing total tardiness

This paper deals with the non-permutation flowshop problem which means that the job sequences are allowed to be different on machines. The objective function is minimizing the total tardiness. Firstly, three mixed-integer linear programming (MILP) models for non-permutation flowshop problems are described, and then are analyzed and assessed their relative effectiveness. Secondly, two Tabu search based algorithms, denoted by LH1 and LH2, are proposed to solve the complicated non-permutation flowshop problems. The algorithms are constructed on specific neighborhood structures which enable the searching method effective. Finally, the performance is evaluated against Taillard’s famous benchmark. Computational experiments show that the proposed algorithms, LH1 and LH2, are significantly superior to the L_TS algorithm. And the percentages of improved permutation flowshop instances by LH1 and LH2 algorithms are about 87.8% and 98.3% with respect to total tardiness, respectively. The non-permutation schedules also have achieved significant improvement in four different scenarios of due dates. Consequently, average percentage improvement (API) is 14.52% for flowshop problems with low tardiness factors. It is evident that exploring non-permutation schedule is effective and necessary for low tardiness factors.