Two-machine Flow Shop Problem Research Articles

A hierarchical approach to schedule the two-machine flow shop problem with flexible periodic synchronised maintenance

A hierarchical approach to schedule the two-machine flow shop problem with flexible periodic synchronised maintenance

A hierarchical approach to schedule the two-machine flow shop problem with flexible periodic synchronized maintenance

A hierarchical approach to schedule the two-machine flow shop problem with flexible periodic synchronized maintenance

A Two-Machine Learning Date Flow-Shop Scheduling Problem with Heuristics and Population-Based GA to Minimize the Makespan

This paper delves into the scheduling of the two-machine flow-shop problem with step-learning, a scenario in which job processing times decrease if they commence after their learning dates. The objective is to optimize resource allocation and task sequencing to ensure efficient time utilization and timely completion of all jobs, also known as the makespan. The identified problem is established as NP-hard due to its reduction to a single machine for a common learning date. To address this complexity, this paper introduces an initial integer programming model, followed by the development of a branch-and-bound algorithm augmented with two lemmas and a lower bound to attain an exact optimal solution. Additionally, this paper proposes four straightforward heuristics inspired by the Johnson rule, along with their enhanced counterparts. Furthermore, a population-based genetic algorithm is formulated to offer approximate solutions. The performance of all proposed methods is rigorously evaluated through numerical experimental studies.

Solving the Two-Crane Scheduling Problem in the Pre-Steelmaking Process

This research is motivated by the practical pre-steelmaking stage in large iron and steel companies, which have steady and heavy demands for the steelmaking production process. Our problem studied the pre-steelmaking stage, which consists of two steps that are needed in each convertor before the steelmaking process. During each step, a necessary transportation must be operated by a crane. In contrast to the classical two-machine flowshop problem during which both machines are fixed, these transporting operations are performed by two mounted, removeable cranes. Our problem is scheduling two-crane operations for the sake of minimizing the last convertors’ completion time (makespan); that is, the last finish time among the total operation of the two cranes is minimized. This study was concerned with resolving the interference between two cranes by determining the sequence of loading operations and how each crane avoids the other in order to let it complete its next operation first. A mixed integer linear programming (MILP) model was developed to represent the problem, and we further present the computational complexity of the problem. The result implies that our problem is very difficult to solve, and it is computationally challenging to solve the model. A special case is provided, which can be optimally solved in polynomial time. Furthermore, an evolutionary algorithm cuckoo search (CS) algorithm was attempted to obtain near-optimal solutions for medium- and large-scale problems. Finally, the efficiency and effectiveness of our methods were validated by numerical results in both simulated instances as well as real data from a practical production process.

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.

Exact solution of the two-machine flow shop problem with three operations

We consider the NP-hard two-machine flow shop problem with three operations with the objective of minimizing the makespan. We exploit the links between this problem and another two-machine flow shop problem with common due date and jobs selection. We show how to tailor to this problem the constraint generation approach already successfully applied in the literature to the latter problem. Computational testing indicates that the proposed approach solves to optimality, within 120 s all instances with up to 40000 jobs.

On the complexity of constructing a minmax regret solution for the two-machine flow shop problem under the interval uncertainty

We prove the NP-hardness of constructing a minmax regret solution for the two-machine flow shop problem under the interval uncertainty of the job processing times. The problem complexity status has been an open question for over the past 20 years. We establish the NP-hardness of this problem using a so-called alternative scheme for proving the NP-hardness of optimization problems. Also, we show that the problem is non-approximable in polynomial time.

Algorithms to minimize total completion time in a two-machine flowshop problem with uncertain set-up times

The two-machine flowshop scheduling problem to minimize total completion time with separate set-up times is addressed. Set-up times are modelled as uncertain within an interval where only the lower and upper bounds are known. Eighty-one different versions of a newly developed constructive algorithm are proposed. Computational experiments to evaluate the performance of the proposed algorithm are conducted in two stages. In the first stage, 81 versions of the algorithm are compared with each other and the top seven versions are selected. In the second stage, the performances of the top seven versions are compared with the performance of the best existing known algorithm for the deterministic set-up times solution in the literature. The computational results reveal that errors of the top seven (out of 81) versions of the algorithm are less than 0.005. All computational results are statistically verified. Therefore, the proposed algorithm has excellent performance.

New results for minimising variation of flow time in two-machine proportionate no-wait flow shops

In this paper, the two-machine no-wait proportionate flow shop problem is analysed with the objective of minimising the total absolute deviation of job completion times. The optimality of V-shaped sequences is extended to problems with 12 jobs and it is shown that an optimal sequence may not be V-shaped with 13 jobs. For problems of larger size, the best V-shaped sequence is determined in linear time without resorting to dynamic programming. For larger size problems, it is shown that the number of non-dominated potentially optimal semi-V-shaped sequences increases rapidly. This indicates that derivation of optimal solutions in low order polynomial time is unlikely with a large number of jobs. Finally, the findings are extended to the corresponding problem with the objective of minimising the total absolute differences in waiting times.

Minimizing Makespan in A Two-Machine Flowshop Problem with Processing Time Linearly Dependent on Job Waiting Time

This paper is aimed at studying a two-machine flowshop scheduling where the processing times are linearly dependent on the waiting times of the jobs prior to processing on the second machine. That is, when a job is processed completely on the first machine, a certain delay time is required before its processing on the second machine. If we would like to reduce the actual waiting time, the processing time of the job on the second machine increases. The objective is to minimize the makespan. When the processing time is reduced, it implies that the consumption of energy is reduced. It is beneficial to environmental sustainability. We show that the proposed problem is NP-hard in the strong sense. A 0-1 mixed integer programming and a heuristic algorithm with computational experiment are proposed. Some cases solved in polynomial time are also provided.

A branch and bound algorithm to solve a two-machine no-wait flowshop scheduling problem with truncated learning function

ABSTRACTThis paper addresses a two-machine no-wait flowshop problem with the effect of the truncated learning function of jobs whose actual processing times depend on their positions in the sequence. By considering truncated learning effect, the actual processing time of a job is a function of its position index in the schedule and a control parameter. In a production scenario with truncated learning effect, the actual processing time of a job will not be lower than a threshold when the number of processed jobs increases. We develop a branch and bound algorithm to minimize the makespan. A lower bound and a dominance property have been proposed to increase the speed of elimination process in the branch and bound algorithm.

Multi-scenario scheduling to maximise the weighted number of just-in-time jobs

We study a multi-scenario scheduling problem on a single-machine and a two-machine flow-shop system. The criterion is to maximise the weighted number of just-in-time jobs. We first analyze the case where only processing times are scenario-dependent. For this case, we prove that the single-machine problem is solvable in polynomial time. We also prove that the unit weight two-machine flow-shop problem is solvable in polynomial time if processing times are scenario-dependent only on the second machine, and is ordinary -hard when processing times are scenario-dependent only on the first machine. This ordinary -hard result holds as long as the number of scenarios is fixed. Otherwise, the problem becomes strongly -hard. We then analyze the case where only weights are scenario-dependent. We adopt a multi-criteria approach and define several problem variations. We prove that one of them is polynomial solvable on a single machine and ordinary -hard in a two-machine flow-shop system. We also prove that all other problem variations are ordinary -hard even if there are only two scenarios, and are strongly -hard when the number of scenarios is arbitrary. Finally, we provide two pseudo-polynomial time algorithms for solving all the hard problems when the number of scenarios is fixed.

A branch-and-bound algorithm to minimize total tardiness in a two-machine flowshop problem with different release time

A branch-and-bound algorithm to minimize total tardiness in a two-machine flowshop problem with different release time

Exact method for the two-machine flow-shop problem with time delays

We address the two-machine flow-shop scheduling problem with time delays in order to minimize the makespan, i.e., the maximum completion time. We present a comprehensive theoretical analysis of the different lower bounds of the state of the art and we elucidate dominance relationships between them. We then introduce an exact method based on a branch-and-bound scheme. This method includes a local search-based heuristic and three dominance rules. Finally, a computer simulation of the branch-and-bound method is given on a set of 480 instances. We point out the good performance of our branch-and-bound method that outperforms the state of the art exact method. Precisely, we manage to solve all the state of the art instances except one in a very short time.

Bicriterion Optimization for Flow Shop with a Learning Effect Subject to Release Dates

This paper investigates a two-machine flow shop problem with release dates in which the job processing times are variable according to a learning effect. The bicriterion is to minimize the weighted sum of makespan and total completion time subject to release dates. We develop a branch-and-bound (B&B) algorithm to solve the problem by using a dominance property, several lower bounds, and an upper bound to speed up the elimination process of the search tree. We further propose a multiobjective memetic algorithm (MOMA), enhanced by an initialization strategy and a global search strategy, to obtain the Pareto front of the problem. Computational experiments are also carried out to examine the effectiveness and the efficiency of the B&B algorithm and the MOMA algorithm.

New characteristics of optimal solutions for the two-machine flowshop problem with unlimited buffers

The two-machine flowshop problem with unlimited buffers with the objective of minimising the makespan (F2||Cmax) is addressed. Johnson’s algorithm finds optimal solutions (permutations) to this problem, but are not necessarily the only optimal solutions. We show in this paper that certain jobs that we define as Critical Jobs, must occupy specific positions in any optimal sequence, not only in Johnson’s solutions. We also prove that jobs that precede a critical job cannot be exchanged with jobs that succeed it in an optimal sequence, which reduces the number of enumerations necessary to identify all optimal solutions. The findings of this research can be useful in reducing the search space for optimal enumeration algorithms such as branch-and-bound.

Two-machine flowshop scheduling in a physical examination center

In this paper, we formulate two important operational processes of the physical examination center as a group scheduling problem in two-machine flowshop. The objective is to minimize the makespan. We first analyze the complexity of the problem with a fixed number of release dates and present a pseudo-polynomial-time algorithm. Then we provide an approximation algorithm and analyze its worst-case error bounds for the problem with two release dates and an optimal algorithm for a very practical special case of the problem.

Minimizing total tardiness in a two-machine flowshop scheduling problem with availability constraint on the first machine

Abstract This paper deals with a two-machine flowshop problem in which the machine at the first stage requires preventive maintenance activities that have to be started within a given cumulative working time limit after the previous maintenance. That is, a maintenance activity can be started at any time unless the cumulative working time after the end of the previous maintenance exceeds the given limit. For the problem with the objective of minimizing total tardiness, we develop dominance properties and lower bounds for this scheduling problem as well as a heuristic algorithm, and suggest a branch and bound algorithm in which these properties, lower bounds, and heuristic algorithm are used. Computational experiments are performed to evaluate the algorithm and the results are reported.

On the optimality conditions of the two-machine flow shop problem

This paper tackles the makespan minimization for the well known two-machine flow shop problem. Several optimality conditions are discussed aiming at the characterization of a large set of optimal solutions. We show that our approach dominates some of the results found in the literature. We also establish a number of necessary optimality conditions and discuss the asymptotic behavior of the optimal makespan when the number of jobs goes to infinity.

Properties of two-machine no-wait flow-shop scheduling with a non-resumable unavailable interval

The properties of two-machine no-wait flow-shop scheduling problem, with a non-resumable unavailable interval are studied. This research discusses two cases: a non-resumable unavailable interval in the first and second machines respectively. The problems’ complexities are proved to be NP-hard. The Gilmore and Gomory Algorithm (GGA) is applied to minimize the makespan, and the conditions that GGA yields the optimal solutions are presented. Sixty numerical examples with ten-job and two-machine are generated to test the proposed theorems. Compared to the results obtained by the Johnson algorithm, the GGA algorithm yields an 8.15% improvement on average. Moreover, the worst-case performance ratios of GGA are proved to be 2.