Two-machine Flow Shop Scheduling Problem Research Articles

Strong NP-hardness of scheduling problems with learning or aging effect

Proofs of strong NP-hardness of single machine and two-machine flowshop scheduling problems with learning or aging effect given in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c; Applied Mathematical Modelling 37:1523–1536, 2013) contain a common mistake that make them incomplete. We reveal the mistake and provide necessary corrections for the problems in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; Applied Mathematical Modelling 37:1523–1536, 2013). NP-hardness of problems in Rudek (International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c) remains unknown because of another mistake which we are unable to correct.

Solution algorithms for the total weighted completion time minimization flow shop scheduling with decreasing linear deterioration

In this paper, we consider a two-machine flow shop scheduling problem with decreasing linear deterioration. By decreasing linear deterioration, we mean that the processing time is a decreasing function of its execution start time. The objective is to find a sequence that minimizes the total weighted completion time. Several dominance properties and some lower bounds are derived to speed up the elimination process of a branch-and-bound algorithm. Two heuristic algorithms are also proposed to overcome the inefficiency of the branch-and-bound algorithm. Computational results for randomly generated problem instances are presented.

Single-machine and two-machine flowshop scheduling problems with truncated position-based learning functions

Scheduling with learning effects has received growing attention nowadays. A well-known learning model is called ‘position-based learning’ in which the actual processing time of a job is a non-increasing function of its position to be processed. However, the actual processing time of a given job drops to zero precipitously as the number of jobs increases. Motivated by this observation, we propose two truncated learning models in single-machine scheduling problems and two-machine flowshop scheduling problems with ordered job processing times, respectively, where the actual processing time of a job is a function of its position and a control parameter. Under the proposed learning models, we show that some scheduling problems can be solved in polynomial time. In addition, we further analyse the worst-case error bounds for the problems to minimize the total weighted completion time, discounted total weighted completion time and maximum lateness.

Two-machine flow shop scheduling of polyurethane foam production

This paper studies a two-machine flow shop scheduling problem with a supporting precedence relation. The model originates from a real production context of a chemical factory that produces foam-rubber products. We extend the traditional two-machine flow shop by dividing the operations into two categories: supporting tasks and regular jobs. In the model, several different compositions of foam rubber can be mixed at the foam blooming stage, and products are processed at the manufacturing stage. Each job (product) on the second machine cannot start until its supporting tasks (parts) on the first machine are all finished and the second machine is not occupied. The objective is to find a schedule that minimizes the total job completion time. The studied problem is strongly NP-hard. In this paper, we propose a branch-and-bound algorithm incorporating a lower bound and two dominance rules. We also design a simple heuristic and an iterated local search (ILS) algorithm to derive approximate solutions. The performances of the proposed algorithms are examined through computational experiments.

A note on “Two-machine flow-shop scheduling with rejection” and its link with flow-shop scheduling and common due date assignment

In a recent paper by Shabtay and Gasper “Two-machine flow-shop scheduling with rejection, Computers and Operations Research”, forthcoming, doi:10.1016/j.cor.2011.05.023, several complexity and approximation results are proposed for a two-criteria two-machine flow-shop scheduling problem with rejection. The two criteria to be minimized are the makespan the total rejection cost. This note positions the contribution of such results with respect to the contributions of the literature on common due date assignment and flow-shop scheduling not considered in the work of Shabtay and Gasper.

Minimising makespan for a two-machine, flow shop, single-wafer-processing, multiple-jobs-per-carrier scheduling problem

In this paper, we address a two-machine flow shop scheduling problem in which a given set of jobs, each consisting of a number of wafers, is to be configured in the given number of carriers of limited capacity for transferring the wafers, and then, processing the wafers one-at-a-time on the machines, in order to minimise the makespan. First, we consider a relaxation of this problem by converting it to a two-machine flow shop lot streaming problem, in which the lot consists of all the jobs, and it is split into continuous-sized sublots for transferring in uncapacitated carriers. We separately address instances in which all the wafers of a job require either identical or different ratios of processing times on the machines. Several methods are developed for the solution of this problem. Results of our numerical experimentation reveal the efficacy of these methods in obtaining almost-optimal solutions within a few seconds of CPU time for the testbed of problem instances used.

The Two-machine Flow-shop Scheduling Problem with a Single Server and Unit Server Times

We consider the problem of two-machine flow-shop scheduling with a single server and unit server times, we show that this problem is NP -hard in the strong sense and present a simple greedy algorithm for it with worst-case bound $\frac{3}{2}$.

Two-Machine Flow-Shop Scheduling with Job Delivery Coordination

This study considers a class of two-machine flow-shop scheduling problems with job delivery coordination. Both vehicle capacity and transportation times are also investigated. The objective is to minimize the mean arrival time of jobs. Two integer programming models are developed to optimally solve this problem. These two integer programming models are Models 1 and 2. Model 1 adopts the concept of assignment problem to formulate the proposed problems, while Model 2 bases on the dichotomous constraints models. Model 2 is theoretically better than Model 1 in size complexity analysis.

Total completion time minimization in two-machine flow shop scheduling problems with a fixed job sequence

This paper addresses scheduling n jobs in a two-machine flow shop to minimize the total completion time, subject to the condition that the jobs are processed in the same given sequence on both machines. A new concept of optimal schedule block is introduced, and polynomial time dynamic programming algorithms employing this concept are derived for two specific problems. In the first problem, the machine-2 processing time of a job is a step increasing function of its waiting time between the machines, and a decision about machine-1 idle time insertion has to be made. This problem is solved in O(n2) time. In the second problem, the jobs are processed in batches and each batch is preceded by a machine-dependent setup time. An O(n5) algorithm is developed to find an optimal batching decision.

Scheduling in a two-machine flow-shop for earliness/tardiness under learning effect

In the literature, earliness/tardiness (E/T) problem was known as weighted absolute deviation problem, and both tardiness and earliness is very important performance criteria for scheduling problem. While total tardiness criteria provides adaptation for due date (ignoring results of earliness done jobs), it deals with only cost of tardiness. However this phenomenon has been started to change with just-in-time (JIT) production concept. On JIT production, earliness is as important as tardiness. The phenomenon of the learning effect has been extensively studied in many different areas of operational research. However, there have been a few studies in the general context of production scheduling such as flow-shop scheduling. This paper addresses the minimization of the total earliness/tardiness penalties under learning effects in a two-machine flow-shop scheduling problem. Jobs have a common due date. We present mathematical model to obtain an optimal schedule for a given job sequence. We also present heuristics that use genetic algorithm and tabu search, based on proposed properties. Furthermore, random search was used for showing the significance of the study by comparison purpose. A new set of benchmark problems is presented with the purpose of evaluating the heuristics. The experimental results show that the performance of proposed approach is quite well, especially for the instances of large size.

Two-machine flow shop scheduling with deteriorating jobs and chain precedence constraints

Abstract In this paper we investigate two-machine flow shop scheduling problems with deteriorating jobs and chain precedence constraints. We consider two types of precedence constraints. Under the first type constraints, a successor cannot start on any machine before its predecessor has been completed on the same machine. Under the second one, a successor cannot start on any machine before its predecessor has been completed on all machines. We show that the problem with the first type precedence constraints is polynomially solvable, and the problem with the second type precedence constraints is NP-hard in the strong sense.

A bicriteria approach to maximize the weighted number of just-in-time jobs and to minimize the total resource consumption cost in a two-machine flow-shop scheduling system

Abstract We analyze a two-machine flow-shop scheduling problem in which the job processing times are controllable by the allocation of resources to the job operations and the resources can be used in discrete quantities. We provide a bicriteria analysis of the problem where the first criterion is to maximize the weighted number of just-in-time jobs and the second criterion is to minimize the total resource consumption cost. We prove that although the problem is known to be NP - hard even for constant processing times, a pseudo-polynomial time algorithm for its solution exists. In addition, we show how the pseudo-polynomial time algorithm can be converted into a two-dimensional fully polynomial approximation scheme for finding an approximate Pareto solution.

Minimizing makespan in a two-machine flow shop with effects of deterioration and learning

We consider a two-machine flow shop scheduling problem with effects of deterioration and learning. By the effects of deterioration and learning, we mean that the processing time of a job is a function of its execution starting time and its position in a sequence. The objective is to find a sequence that minimizes the makespan. Several dominance properties and two lower bounds are derived, which are used to speed up the elimination process of a branch-and-bound algorithm proposed to solve the problem. Two heuristic algorithms are also proposed to obtain near-optimal solutions. Computational results are presented to evaluate the performance of the proposed algorithms.

Two-machine flow shop scheduling problem with an outsourcing option

We consider a two-machine flow shop problem in which each job is processed through an in-house system or outsourced to a subcontractor. A schedule is established for the in-house jobs, and performance is measured by the makespan. Jobs processed by subcontractors require paying an outsourcing cost. The objective is to minimize the sum of the makespan and total outsourcing costs. We show that the problem is NP-hard in the ordinary sense. We consider a special case in which each job has a processing requirement, and each machine a characteristic value. In this case, the time a job occupies a machine is equal to the job’s processing requirement plus a setup time equal to the characteristic value of that machine. We introduce some optimality conditions and present a polynomial-time algorithm to solve the special case.

A polynomial-time approximation scheme for the two machine flow shop problem with several availability constraints

In this paper, we develop a polynomial-time approximation scheme for the two-machine flow shop scheduling problem with several availability constraints on the first machine, under the resumable scenario.

Minimizing total weighted completion time in a two-machine flow shop scheduling under simple linear deterioration

In this paper, we address a two-machine flow shop scheduling problem under simple linear deterioration. By a simple linear deterioration function, we mean that the processing time of a job is a simple linear function of its execution start time. The objective is to find a sequence that minimizes total weighted completion time. Optimal schedules are obtained for some special cases. For the general case, several dominance properties and two lower bounds are derived to speed up the elimination process of a branch-and-bound algorithm. A heuristic algorithm is also proposed to overcome the inefficiency of the branch-and-bound algorithm. Computational analysis on randomly generated problems is conducted to evaluate the branch-and-bound algorithm and heuristic algorithm.

A note on two-machine no-wait flow shop scheduling with deteriorating jobs and machine availability constraints

This paper is concerned with two-machine no-wait flow shop scheduling problems in which the actual processing time of each job is a proportional function of its starting time and each machine may have non-availability intervals. The objective is to minimize the makespan. We assume that the non-availability intervals are imposed on only one machine. Moreover, the number of non-availability intervals, the start time and end time of each interval are known in advance. We show that the problem with a single non-availability interval is NP-hard in the ordinary sense and the problem with an arbitrary number of non-availability intervals is NP-hard in the strong sense.

Inverse scheduling: two-machine flow-shop problem

We study an inverse counterpart of the two-machine flow-shop scheduling problem that arises in the context of inverse optimization. While in the forward scheduling problem all parameters are given and the objective is to find job sequence(s) for which the value of the makespan is minimum, in the inverse scheduling the exact values of processing times are unknown and they should be selected within given boundaries so that pre-specified job sequence(s) become optimal. We derive necessary and sufficient conditions of optimality of a given solution for the general case of the flow-shop problem when the job sequences on the machines can be different. Based on these conditions we prove that the inverse flow-shop problem is NP-hard even in the case of the same job sequence on both machines and produce a linear programming formulation for a special case which can be solved efficiently.

An improved algorithm for the two machine flow shop problem with several availability constraints

In this paper we deal with the two-machine flow shop scheduling problem with several availability constraints on the first machine, under the resumable scenario. We first develop an improved algorithm with a relative worst-case error bound of 4/3. We then present a polynomially solvable case.

Minimizing Makespan and Mean Flow Time in Two-Versatile-Machine Flow-Shop with Alternative Operations

This study addresses the two-machine flow-shop scheduling problems in which both machines are versatile and thus alternative operations are possible. The performance measures are the mean flow time and makespan of jobs, respectively. The problem is formulated as two integer programming models and two heuristics are developed. Computational results are provided to demonstrate the efficiency of the models and the effectiveness of the heuristics. The integer programming technique is inefficient even for small problems. For the set of problems with known optimal solutions, the average percentage errors of heuristics H1 and H2 are within 10%, respectively. For the set of problems with unknown optimal solutions, the average percentage errors of heuristic H1 for solving the mean flow time is 20.09%, while the average percentage errors of heuristic H2 for solving the makespan is 20.84%.