Two-machine Flow Shop Scheduling Research Articles

Coordinating scheduling and rejection decisions in a two-machine flow shop scheduling problem

We study a two-machine flow shop scheduling problem where any operation can be rejected at a certain cost. A solution for such a problem requires two sets of decisions. The first involves the partition of the set of operations into two subsets: the set of operations that are accepted for scheduling in the shop, and the set of rejected operations. The second decision involves scheduling the set of accepted operations in the shop. The objective is to find a solution that minimizes the sum of the makespan and the total rejection cost. We prove that the problem is NP-hard even if all processing operations have identical processing times and identical rejection costs on either one of the two machines. We show, however, that the problem is fixed parameterized tractable with respect to a parameter that combine the number of different processing times on both machines with the number of different rejection costs on one out of the two machines. We also provide a pseudo-polynomial time algorithm for the problem, which we then convert into a fully polynomial time approximation scheme. This is achieved by dividing the problem into a set of subproblems and deriving a fully polynomial time approximation scheme for each one of them, separately. Finally, we present an integer linear programming formulation of the problem and two simple 2-approximation algorithms.

Two-machine flowshop scheduling with fuzzy processing times and flexible operations

Two-machine flowshop scheduling with fuzzy processing times and flexible operations

Cooperative Game Theory Based Coordinated Scheduling of Two-Machine Flow-Shop and Transportation

Cooperative Game Theory Based Coordinated Scheduling of Two-Machine Flow-Shop and Transportation

Single machine scheduling with resource constraints: Equivalence to two-machine flow-shop scheduling for regular objectives

A number of results have been reported in the literature for a single machine job scheduling problem with resource constraints. We demonstrate that many of these results and some new results follow from an equivalence of this problem and the classical two-machine flow-shop scheduling problem. We further refine computational complexity of the problem with resource constraints by presenting new NP-hardness proofs.

Minimizing the makespan for the two-machine flow shop scheduling problem with random breakdown

AbstractThis paper studies a two-machine flow shop scheduling problem with availability constraints due to a breakdown on the first machine. The starting time of the breakdown is considered stochastic and follows a known probability distribution. A service-level constraint is introduced to model the guarantee with which the obtained schedule takes into account the stochastic nature of the breakdown’s starting time. The objective is to find a solution that minimizes the makespan while satisfying the desired service level. The studied problem is strongly NP-hard. We develop two mixed integer linear models that linearize the non-linear model. Using interval modeling of the breakdown, we propose lower bounds and a valid inequality that are used to strengthen both models. When the lower bounds and the valid inequality are applied, the performance of both models is greatly improved by reducing the gap and reaching optimality for more instances. We also introduced two heuristics that exploit the proposed interval modeling. The computational results indicate that both models are able to reach optimality for the 10-job instances. Moreover, the comparisons results between both models with the two heuristics showed their effectiveness.

Bicriteria two-machine flowshop scheduling: approximation algorithms and their limits

Bicriteria two-machine flowshop scheduling: approximation algorithms and their limits

Bi-criteria two-machine flow shop scheduling problem involving sequence-dependent setup times and pieces multiplicity

Bi-criteria two-machine flow shop scheduling problem involving sequence-dependent setup times and pieces multiplicity

Online scheduling of two-machine flowshop with lookahead and incompatible job families

Online scheduling of two-machine flowshop with lookahead and incompatible job families

A note on valid inequalities for minimizing the total tardiness in a two-machine flow shop

This note reconsiders the valid inequalities designed by Kharbeche and Haouari (2013) for mixed-integer programming formulations of minimizing the total tardiness in a two-machine flow shop scheduling problem. While the majority of their proposed valid inequalities are able to substantially improve the computational time to find an optimal solution, we show that one of them is incorrect, due to the misinterpretation of a dominance criterion proven in Pan and Fan (1997). As a consequence, in some instances, all optimal solutions are excluded from the solution space, resulting in suboptimal solutions. To resolve this issue, we first demonstrate the suboptimality of the proposed valid inequality with a small illustrative example. Second, we formulate a new valid inequality, which correctly captures the dominance criteria of Pan and Fan (1997). Finally, we show the effect of this correction on the solution quality in a numerical study with scheduling instances that have 10 to 20 jobs. Depending on the number of jobs, between 10% and 22% of the supposedly optimal solutions with the incorrect valid inequality are in fact suboptimal. In terms of the objective function value, the error results in a loss of up to 12%.

Two-machine flow shop scheduling with convex resource consumption functions

We consider a two-machine flow shop scheduling problem in which the processing time of each operation is inversely proportional to the power of the amount of resources consumed by it. The objective is to minimize the sum of the makespan and the total resource consumption cost. We show that the problem is NP-hard, and its constrained version remains so. Then, we develop 1.25- and 2-approximation algorithms for the problem and its constrained version, respectively.

Scheduling of two-machine flowshop with outsourcing lead-time

This paper considers a two-machine flowshop scheduling problem with outsourcing lead-time. While the first operation of jobs can be either processed in-house or outsourced to subcontractors, the second operation of jobs must be processed in-house. Outsourcing a job incurs a job-dependent outsourcing cost as well as outsourcing lead-time, which implies that the outsourced job becomes available for the second operation after its outsourcing lead-time. The objective is to minimize the weighted sum of the outsourcing costs and the scheduling costs which is either makespan or total completion time. The problems are shown to be NP-hard, and both exact and heuristic algorithms are developed for the problems. Finally, computational experiments show that the proposed algorithms provide efficient and effective solutions.

두 단계 흐름생산라인에서 가중완료시간을 최소화하기 위해 아웃소싱 전략을 활용하는 일정계획문제

This paper researches a two-machine flowshop scheduling with an outsourcing strategy. In the flowshop, the first processing of jobs is either managed in-house or outsourced to outside suppliers, the second processing is processed in-house only. When a job is outsourced, the company pays outsourcing cost for each job, and considers outsourcing lead-time implying that the second processing can start after the lead-time. The objective cost function of this paper is the sum of the total weighted completion time and the outsourcing costs. This paper derives a branch-and-bound algorithm and some heuristics for the problem. From the computational experiments, we can observe that the performance of the derived algorithms is effective and efficient.

A precedence constrained flow shop scheduling problem with transportation time, breakdown times, and weighted jobs

Job precedence can often be seen in various manufacturing process scenarios. For instance, in the context of flow shop scheduling, certain jobs must be processed before a specific job may be executed. Formally, this scenario is known as precedence constraint, which influences the optimal job sequence. Because of this practical significance, in this study, a two-machine flow shop scheduling problem in which transportation times, breakdown time, and weighted jobs are considered. In addition to that, an ordered precedence constraint is considered that ensures a successor job cannot start on any machine before its predecessor job has been done on all machines. This is the first study that deals with flow shop scheduling problems with transportation times, breakdown time, job weights, and precedence constraints altogether, to the best of the author’s knowledge. To solve this problem, a simple and efficient solution methodology is developed that assures optimal or near-optimal solutions effectively. The developed algorithm is tested on various test instances and results are reported, which will be useful for future comparative studies.

Bi-criteria two-machine flow shop scheduling problem involving sequence-dependent setup times and pieces multiplicity

Bi-criteria two-machine flow shop scheduling problem involving sequence-dependent setup times and pieces multiplicity

Two-machine flow shop scheduling with a common due date to maximize total early work

This paper considers scheduling in a two-machine flow shop to maximize the total early work subject to a common due date. The early work of a job is a parameter defined as the total amount of the job that is processed before the specified due date. We mainly focus on the unweighted model in this paper, and propose a dynamic programming approach running in O(n2d2) time (compared with the previous result in O(n2d4) time for the weighted case discussed in the literature). Then we analyse the problem from an approximation point of view, in which we first show that Johnson’s algorithm, one of the most classical ones in flow shop scheduling, can only achieve the worst performance ratio for the considered problem (although it is an optimal one for makespan minimization). With the motivation of proposing better approximation algorithms, we further design a fully polynomial time approximation scheme (FPTAS). Finally, we point out that the approximation results also work for the weighted model - if a specific constraint is satisfied.

Path cover with minimum nontrivial paths and its application in two-machine flow-shop scheduling with a conflict graph

Path cover with minimum nontrivial paths and its application in two-machine flow-shop scheduling with a conflict graph

흐름 공정의 가중치 조기 종료된 작업 개수와 지체 종료된 작업 개수의 최소화 문제

In this paper, we consider a two-machine flow shop scheduling problem with different job due dates. The objective is to minimize the weighted number of early and tardy jobs. We present a mixed integer programming formulation for the problem and develop efficient cut constraints which are valid inequalities. Extensive experiments are conducted to show the effectiveness of the cut constraints. We propose an O(n²) heuristic algorithm to solve the problem.

A complexity analysis and algorithms for two-machine shop scheduling problems under linear constraints

We study several two-machine shop scheduling problems, namely flow shop, job shop and open shop scheduling problems under linear constraints. In these problems, the processing times of two stages of jobs are also decision variables and satisfy a system of linear constraints. The goal of each problem is to determine the processing time of each job, and to schedule the jobs to the shop machine such that the makespan, i.e., the completion time of all jobs, is minimized. These problems can find application in various areas, such as industrial production, advertising and automotive maintenance. We study the computational complexity and propose polynomial-time optimal or approximation algorithms for them. In particular, we show that although a two-machine flow shop scheduling problem and a two-machine job shop scheduling problem without recirculation can be solved in polynomial time, the problems where processing times satisfy linear constraints are generally NP-hard in the strong sense. Then, we design algorithms for various settings of these problems. We design polynomial-time algorithms for them when there are a fixed number of constraints. For the general case, we first propose a simple 2-approximation algorithm, and then design a polynomial-time approximation schemes. In contrast to the previous two problems, we show that the two-machine open shop scheduling problem under linear constraints can be solved in polynomial time.

Exact and heuristic methods to solve the two-machine cross-docking flow shop scheduling problem

Exact and heuristic methods to solve the two-machine cross-docking flow shop scheduling problem

Robust and Stable Flow Shop Scheduling Problem under Uncertain Processing Times and Machines’ Disruption

This paper presents a predictive robust and stable approach for a two-machine flow shop scheduling problem with machine disruption and uncertain job processing time. Indeed, a general approach is proposed that can be used for robustness and stability optimization in an m-machine flow shop or job shop scheduling problem. The robustness measure is the total expected realized completion time. The expected sum of squared aberration between each jobs’ completion time in the realized and initial schedules is the stability measure. We proposed and compared two methods to deal with such an NP-hard problem; a method based on decomposing the problem into sub-problem and solving each sub-problem, and a theorem-based method. The extensive computational results indicated that the second method has a better performance in terms of robustness and stability, especially in large-sized problems. In other words, the second method is preferable because of the better manufacturer responsiveness to the customer and the production staff satisfaction enhancement.