Permutation Schedules Research Articles

Modeling and solution for hybrid flow-shop scheduling problem by two-stage stochastic programming

To adapt to the uncertainty of the real environment and solve the problem of high efficiency and high quality, this study comprehensively considers a variety of uncertainty factors, focusing on modeling and solving a hybrid flow-shop scheduling problem by two-stage stochastic programming (HFSP-TSP). First, this paper uses a two-stage scenario tree to describe the uncertain factors, the probabilities and values are obtained under different discrete production scenarios, and stochastic programming theory is used to formulate a mixed-integer linear programming (MILP) model for HFSP-TSP. Second, based on a general discrete optimization algorithm framework, that is, pointer-based discrete differential evolution (PDDE) algorithm, a novel variant (H-PDDE) is proposed for effectively solving the HFSP-TSP. It adopts a permutation scheduling decoding method and introduces three new improvement strategies: the strategy of inconsistent code length of individuals in the population, adding local search in discrete mutation operation, using a heuristic algorithm to generate the initial population. Finally, computational experiments were conducted to illustrate the proposed mathematical model and optimization algorithm. The computational results show that the H-PDDE can solve the HFSP-TSP more effectively than existing algorithms and conventional PDDE variants. The stochastic programming model is increasingly superior to the deterministic model in an uncertain environment. The source code of the H-PDDE algorithm and its data files can be found at https://github.com/huangyiping-ai/H-PDDE-algorithm.git, which is publicly available.

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.

On permutation schedules for two-machine flow shops with buffer constraints and constant processing times on one machine

The two-machine flow shop problem with buffers where one of the machines has constant processing times is studied. It is investigated if optimal permutation schedules exist, i.e., schedules which minimize the makespan within the set of all schedules and where the job sequence on both machines is the same. Two standard buffer models (an intermediate buffer that a job occupies between its processing on the two machines and a spanning buffer which is occupied by a job over all of its processing steps) are considered for two types of buffer usage: i) all jobs occupy the same buffer amount and ii) for each job the occupied buffer amount equals its processing time on the machine with non-constant times. For (i) and both buffer models, it is shown that the set of optimal schedules always contains at least one permutation schedule. For (ii) the existence of optimal permutation schedules is only guaranteed for instances with n jobs for all n≤6 (spanning buffer) and for all n≤3 (intermediate buffer). For each n>6 (spanning buffer) and each n>3 (intermediate buffer) exists an instance which does not have an optimal permutation schedule. For (ii) and both buffer models, an upper bound and a lower bound for ratio between the makespan of a best permutation schedule and the makespan of an optimal schedule is given. It is also shown for (ii) and both buffer models that in general it is NP-hard to decide if an optimal permutation schedule exists.

Genetic Algorithm Application for Permutation Flow Shop Scheduling Problems

In this paper, permutation flow shop scheduling problems (PFSS) are investigated with a genetic algorithm. PFSS problem is a special type of flow shop scheduling problem. In a PFSS problem, there are n jobs to be processed on m machines in series. Each job has to follow the same machine order and each machine must process jobs in the same job order. The most common performance criterion in the literature is the makespan for permutation scheduling problems. In this paper, a genetic algorithm is applied to minimize the makespan. Taillard’s instances including 20, 50, and 100 jobs with 5, 10, and 20 machines are used to define the efficiency of the proposed GA by considering lower bounds or optimal makespan values of instances. Furthermore, a sensitivity analysis is made for the parameters of the proposed GA and the sensitivity analysis shows that crossover probability does not affect solution quality and elapsed time. Supplementary to the parameter tuning of the proposed GA, we compare our GA with an existing GA in the literature for PFSS problems and our experimental study reveals that our proposed and well-tuned GA outperforms the existing GA for PFSS problems when the objective is to minimize the makespan.

Solving a flow shop scheduling problem with missing operations in an Industry 4.0 production environment

Industry 4.0 is a modern approach that aims at enhancing the connectivity between the different stages of the production process and the requirements of consumers. This paper addresses a relevant problem for both Industry 4.0 and flow shop literature: the missing operations flow shop scheduling problem. In general, in order to reduce the computational effort required to solve flow shop scheduling problems only permutation schedules (PFS) are considered, i.e., the same job sequence is used for all the machines involved. However, considering only PFS is not a constraint that is based on the real-world conditions of the industrial environments, and it is only a simplification strategy used frequently in the literature. Moreover, non-permutation (NPFS) orderings may be used for most of the real flow shop systems, i.e., different job schedules can be used for different machines in the production line, since NPFS solutions usually outperform the PFS ones. In this work, a novel mathematical formulation to minimize total tardiness and a resolution method, which considers both PFS and (the more computationally expensive) NPFS solutions, are presented to solve the flow shop scheduling problem with missing operations. The solution approach has two stages. First, a Genetic Algorithm, which only considers PFS solutions, is applied to solve the scheduling problem. The resulting solution is then improved in the second stage by means of a Simulated Annealing algorithm that expands the search space by considering NPFS solutions. The experimental tests were performed on a set of instances considering varying proportions of missing operations, as it is usual in the Industry 4.0 production environment. The results show that NPFS solutions clearly outperform PFS solutions for this problem.

Heuristic approaches for scheduling jobs and vehicles in a cyclic flexible manufacturing system

Abstract This paper addresses the scheduling of automated guided vehicles (AGVs) in a cyclic flexible flow shop environment. The vehicles travel along a single loop. All production machines are located alongside the track in the required order, with a possibility for multiple machines per stage. All AGVs are to be scheduled for a specific starting time and will then continuously circle the track. Pickup and delivery times are included in the travel time of a vehicle, stops are forbidden. Jobs may start upon arrival if their predecessor has been started for processing. Therefore, job completion times are dynamic. The considered objectives are the minimization of the number of AGVs and of the total makespan. For the regarded problem, different local search variants are proposed. Optimal results are produced using a brute force enumeration algorithm. Finally, fixed permutation schedules are compared to processing jobs according to a first-come-first-serve rule.

Two-machine flow shop with dynamic storage space

The publications on two-machine flow shop scheduling problems with job dependent storage requirements, where a job seizes a portion of the storage space for the entire duration of its processing, were motivated by various applications ranging from supply chains of mineral resources to multimedia systems. In contrast to the previous publications that assumed that the availability of the storage space remains unchanged, this paper is concerned with a more general case when the availability is a function of time. It strengthens the previously published result concerning the existence of an optimal permutation schedule, shows that the variable storage space availability leads to the NP-hardness in the strong sense even for unit processing times, and presents a polynomial-time approximation scheme together with several heuristic algorithms. The heuristics are evaluated by means of computational experiments.

Robust scheduling for a two-stage assembly shop with scenario-dependent processing times

Recently, finding solutions to assembly flowshop scheduling problems is a topic of extensive discussion in research communities. While existing research assumes that job processing times are constant numbers, in several practical situations, due to several external factors like machine breakdowns, working environment changes, worker performance instabilities, and tool quality variations and unavailability, job processing times may vary. In this study, therefore, we address a two-stage assembly flowshop scheduling problem with two scenario-dependent jobs processing times to minimise the maximum makepsan among both scenarios (called robust makespan) In view of the NP-hard nature, we first derive a dominance property and a lower bound to propose a branch-and-bound algorithm to find a permutation schedule with minimum makespan. Following that, we use Johnson’s rule to propose eight polynomial heuristics for finding near-optimal solutions. Furthermore, we propose four cloud theory-based simulated annealing (CSA) hyper-heuristic algorithms incorporating seven low level heuristics to solve a robust two-stage assembly flowshop problem with scenario-dependent processing times. Finally, we empirically evaluate the effectiveness of all the proposed algorithms in minimising the robust makespan.

Makespan Minimization in Two-Machine Flow-Shop Scheduling under No-wait and Deterministic Unavailable Interval Constraints

This paper systematically studies the two-machine flow-shop scheduling problems with no-wait and deterministic unavailable interval constraints. To minimize the makespan, three integer programming mathematical models are formulated for two-machine flow-shop with no-wait constraint, two-machine flow-shop with resumable unavailable interval constraint, and two-machine flow-shop with no-wait and non-resumable unavailable interval constraints problems, respectively. The optimal conditions of solving the two-machine flow-shop with no-wait constraint problem by the permutation schedules, the two-machine flow-shop with resumable unavailable interval constraint problem by the Johnson algorithm, and two-machine flow-shop with no-wait and non-resumable unavailable interval constraints problem by the Gilmore and Gomory Algorithm (GGA) are presented, respectively. And the tight worst-case performance bounds of Johnson and GGA algorithms for these problems are also proved to be 2. Several instances are generated to demonstrate the proposed theorems. Based on the experimental results, GGA obtains the optimal solution for the two-machine flow-shop with no-wait constraint problem. Although it cannot reach the optimal solution for the two-machine flow-shop with resumable unavailable interval constraint problem, the optimal gap is 0.18% on average when the number of jobs is 100. Moreover, under some special conditions, it yields the optimal solution for the two-machine flow-shop with no-wait and non-resumable unavailable interval constraints problem. Therefore, GGA is an efficient heuristic to solve these problems.

Optimal Makespan for the Coupled Task Ordered Flow-Shop

The ordered flow-shop scheduling problem deals with the case where processing times follow a specific structure. We extend the ordered flow-shop scheduling to the coupled tasks setting, i.e., the coupled task ordered flow-shop, in which an exact delay exists between the consecutive tasks of each job. The aim is to minimize the makespan. The coupled task flow-shop with distinct delays between the consecutive tasks of the jobs is known to be strongly NP-hard, even for two machines and the unit execution time tasks. We investigate the problem when delays follow the ordered structure and show that the optimal permutation schedule has the pyramidal-shaped property. We also introduce two polynomially solvable cases.

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.

Two-machine flow shops with an optimal permutation schedule under a storage constraint

We consider two-machine flow shop scheduling problems with storage to minimize the makespan. Under a storage constraint, each job has its own size, and the total size of the job(s) being processed on the machine(s) and the job(s) waiting for the second operation cannot exceed the given capacity. It is known that the computational complexity is strongly NP-hard and in general an optimal schedule does not exist in the set of schedules with the same job sequences on two machines, referred to as a permutation set. Thus, we consider two cases such that an optimal schedule exists in the permutation set. These two cases have the special structure of processing times. In the first case, the processing times of two machines for each job are identical, while in the second case, machine 2 dominates machine 1, that is, the smallest processing time of the second group of operations is larger than or equal to the largest one of the first group of operations. We show that the first case is strongly NP-hard and admits a polynomial-time approximation scheme (PTAS), while the second case is solved in polynomial time.

Online scheduling of ordered flow shops

We consider online as well as offline scheduling of ordered flow shops with the makespan as objective. In an online flow shop scheduling problem, jobs are revealed to a decisionmaker one by one going down a list. When a job is revealed to the decision maker, its operations have to be scheduled irrevocably without having any information regarding jobs that will be revealed afterwards. We consider for the online setting the so-called Greedy Algorithm which generates permutation schedules in which the jobs on the machines are at all times processed without any unnecessary delays. We focus on ordered flow shops, in particular proportionate flow shops with different speeds and proportionate flow shops with different setup times. We analyze the competitive ratio of the Greedy Algorithm for such flow shops in the online setting. For several cases, we derive lower bounds on the competitive ratios.

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.

Integrated decision making for parts ordering and scheduling of jobs on two-stage assembly problem in three level supply chain

This paper presents an integrated model for two-stage assembly scheduling problem with ordering the required parts in three-level supply chain including parts’ suppliers, components’ manufacturers and an assembler. It considers the quantity and date of parts’ order as decision variables to make the problem seem more realistic. A mixed-integer linear programming model is proposed to minimize sum of the total weighted completion time, parts ordering and holding cost.A genetic algorithm (GA) based heuristic is proposed to solve the problem efficiently since it is NP-hard, whose solutions are compared with those of CPLEX. Computational experiments were conducted through a diverse range of problem instances indicating that the GA performs much better than CPLEX regarding the average percentage of improvement, ranging from 2.96% to 37.70%, and CPU time. The existing study has considered the effect of machine specific and shared parts aligned with integrated decisions. Results indicate that, adding more of both specific and shared parts and increasing the number of jobs lead to dramatic decrease in the performance of CPLEX in comparison with GA. As the results reveal, the integrated model provides enhancements for the supply chain performance up to 8.16%. Furthermore, the permutation and non-permutation types of the problem are analyzed. The obtained Results show that permutation schedules are not sufficient to solve two-stage assembly scheduling problem within inventory constraints.

On the dominance of permutation schedules for some ordered and proportionate flow shop problems

• We consider the ordered and the proportionate flow shop with unequal machine speeds. • We consider both the C max and the ∑ j = 1 n C j objectives. • We show that permutation schedules are dominant for C max with V-shaped speeds. • We show that permutation schedules are dominant for ∑ j = 1 n C j with three stages.

An examination of job interchange relationships and induction-based proofs in single machine scheduling

We provide a generalization of Lawler’s (Mathematical programming the state of the art. Springer, Berlin, pp 202–234, 1983) Theorem on solutions to permutation scheduling problems when the objective function admits a particular job interchange relation. We complete Lawler’s result with a straight-forward proof by induction on n, the number of jobs. A notable application is \(1 ||\varSigma {w}_{j} C_{j}\) where the objective of total weighted completion time admits WSPT (i.e., scheduling jobs in non-decreasing order of \(p_{j}/w_{j}\)). We provide new proofs by induction for the optimality of WSPT as well as for SPT in the unweighted case.

The two-machine flow shop problem with conflict graphs

In this paper, we consider the problem of scheduling jobs on a two-machine flow shop subject to constraints represented by an undirected graph G, in which each edge joins a pair of conflicting jobs that cannot be processed simultaneously on different machines. The problem of minimizing the maximum completion time (makespan) is known to be NP-hard in the strong sense even when all the operations require one unit of processing time. We prove that the permutation schedules are not dominant even for two machines, and we present a special case for which an optimal schedule can be found by a permutation schedule. On the other hand, we propose four Mixed Integer Linear Programming (MILP) models alongside with an experimental study to measure their performance on a wide range of test problems.

Permutation schedules for a two-machine flow shop with storage

This paper considers a two-machine flow shop problem with a buffer, arising in various applications, and addresses a fundamental question of the existence of an optimal permutation schedule. The paper proves that the problem of recognising whether an instance has an optimal permutation schedule is NP-complete in the strong sense, and estimates the deviation from the optimal makespan as a result of the restriction to permutation schedules only.

A modified teaching–learning-based optimisation algorithm for bi-objective re-entrant hybrid flowshop scheduling

In this paper, a modified teaching–learning-based optimisation (mTLBO) algorithm is proposed to solve the re-entrant hybrid flowshop scheduling problem (RHFSP) with the makespan and the total tardiness criteria. Based on the simple job-based representation, a novel decoding method named equivalent due date-based permutation schedule is proposed to transfer an individual to a feasible schedule. At each generation, a number of superior individuals are selected as the teachers by the Pareto-based ranking phase. To enhance the exploitation ability in the promising area, the insertion-based local search is embedded in the search framework as the training phase for the TLBO. Due to the characteristics of the permutation-based discrete optimisation, the linear order crossover operator and the swap operator are adopted to imitate the interactions among the individuals in both the teaching phase and the learning phase. To store the non-dominated solutions explored during the search process, an external archive is used and updated when necessary. The influence of the parameter setting on the mTLBO in solving the RHFSP is investigated, and numerical tests with some benchmarking instances are carried out. The comparative results show that the proposed mTLBO outperforms the existing algorithms significantly.