Two-machine Flow Shop Scheduling Problem 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.

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.

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

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.

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

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.

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.

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

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.

Minimizing the weighted sum of maximum earliness and maximum tardiness in a single-agent and two-agent form of a two-machine flow shop scheduling problem

In the past decade, multi-agent scheduling studies have become more widespread. However, the evaluation of these issues in the flow shop scheduling environment has received almost no attention. In this article, we investigate two problems. One problem is a common due date problem of constrained two-agent scheduling of jobs in a two-machine flow shop environment to minimize the weighted sum of maximum earliness and maximum tardiness of first-agent jobs and restrictions of non-eligibility on the tardiness of second-agent jobs. Another problem is a single-agent form of the two-agent problem when the number of second-agent jobs is zero. So, an optimal algorithm with polynomial time complexity is presented for the single-agent problem. For the two-agent problem, after it was shown to have minimum complexity of ordinary NP-hardness, a branch and bound algorithm, based on efficient lower and upper bounds and dominance rules, was developed. The computational results show that the algorithm can solve the large-size instances optimally.

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.

Energy-cost-aware flow-shop scheduling systems with state-dependent energy consumptions

The necessity of managing a production system with less cost and pollution in such a competitive world has enticed the attention of many researchers. This paper deals with a multi-state two-machine flow-shop scheduling problem under Time-Of-Use electricity tariffs. The main issue is to assign a set of jobs to available time slots with different energy costs to minimize the total energy consumption costs. For this purpose, a new linear programming (LP) model is proposed, and several numerical results are solved using CPLEX Studio 12.9.0 solver to evaluate the effciency of the proposed model for the problem. The obtained results demonstrate that CPLEX solver is not able to find the optimal solution for the problems as large as 20 jobs and 90 periods during the 3600 seconds time limitation.

Electricity cost minimisation for optimal makespan solution in flow shop scheduling under time-of-use tariffs

The industrial sector consumes half of the world delivered energy and is responsible for a third of carbon dioxide emissions which cause severe environmental pollution. The industry has to change its behaviour concerning the energy consumption. Since two-machine flow shop scheduling () is one of the typical problems of the manufacturing industry, this paper aims to build an energy-cost-aware scheduling plan. This work tackles the joint optimisation of makespan and electricity cost in two-machine flow shop scheduling problem under electricity pricing. We enhance the financial aspect of the optimal solution of by minimising the electricity cost without increasing the makespan. Firstly, we show the contribution of the generation of several optimal equivalent solutions of . The optimal equivalent solutions have different electricity costs but present the same makespan. Then, we determine the optimal starting time of jobs on several equivalent optimal solutions to get the best production plan. Finally, the numerical tests show that our proposed approach improves the electricity cost significantly under optimal makespan. The results provide good solutions to managers and decision makers to achieve energy cost savings without sacrificing the productivity which can contribute to sustainable development of the manufacturing industry.

Interactive job sequencing system for small make-to-order manufacturers under smart manufacturing environment

Production scheduling is an important research topic widely studied during past few decades. However, many manufacturers still fail to successfully deploy scheduling algorithms and systems, even though information and communication technologies can be used to collect and process data associated with production scheduling under modern smart manufacturing environment. The primary problem is that many scheduling algorithms and systems did not consider diverse variety of scheduling requirements of real production systems. Especially, production schedulers in small make-to-order manufacturers have much trouble in utilizing such algorithms and systems. In order to address this issue, this paper aims to propose a functional architecture of production scheduling system for small make-to-order manufactures under smart manufacturing environment and develop a flexible scheduling algorithm for this system. For illustration, the proposed system and algorithm are applied to a two-machine flow shop scheduling problem, and it is expected that this paper will provide a meaningful insight into the user experiences of production scheduling systems.

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.

A modified hybrid whale optimization algorithm for the scheduling problem in multimedia data objects

SummaryThe scheduling of the Multimedia Data Objects (MDO) is a difficult and extraordinary issue that faces us in the World Wide Web (WWW) to minimize the response time for clients to finish rapidly their tasks. MDO scheduling problem can be modeled as a Two‐Machine Flow Shop Scheduling Problem (T‐MFSSP) that minimizes the makespan. Notwithstanding another objective is to decrease the average lateness time of the clients' jobs. This paper proposes a Modified Hybrid Whale Algorithm (MHWA) to solve a standout among the most imperative applications of flow shop scheduling in the field of MDO. MHWA is hybridized with a local search strategy for solving the scheduling problem of MDO. To cope up with the combinatorial nature of the MDO scheduling, LRV maps the continuous search space into a sequence of jobs. To get better solutions, some operations are applied to the solutions such as swap mutation and reversed block insertion operations. Nawaz‐Enscore‐Ham (NEH) is added to MHWA to upgrade the performance of the algorithm. The solution of Johnson heuristic is added to the initial population of MHWA. It can be inferred that MHWA gives competitive outcomes appeared differently in relation to Johnson and Earliest Due Date (EDD) algorithms. MHWA can obtain the optimal makespan and can minimize the average lateness of the jobs.