Two-machine Flow Shop Problem Research Articles

An O(n2) algorithm for the variable common due date, minimal tardy jobs bicriteria two-machine flow shop problem with ordered machines

We consider the ordinary NP- hard two-machine flow shop problem with the objective of determining simultaneously a minimal common due date and the minimal number of tardy jobs. We present an O(n2) algorithm for the problem when the machines are ordered, that is, when each job has its smaller processing time on the first (second) machine. We also discuss the applicability of the proposed algorithm to the corresponding single-objective problem in which the common due date is given.

An empirical analysis of heuristics for solving the two-machine flow shop problem with job release times

The theoretical analysis of heuristics for solving intractable optimization problems has many well-known drawbacks. Constructed instances demonstrating an exceptionally poor worst-case performance of heuristics are typically too peculiar to occur in practice. Theoretical results on the average-case performance of most heuristics could not be established due to the difficulty with the use of probabilistic analysis. Moreover, the heuristics for which some type of asymptotic optimality has been proven are likely to perform questionably in practice. The purpose of this paper is to confront known theoretical results with our empirical results concerning heuristics for solving the strongly NP-hard problem of minimizing the makespan in a two-machine flow shop with job release times. The heuristics' performance is examined with respect to their average and maximum relative errors, as well as their optimality rate, that is, the probability of being optimal. In particular, this allows us to observe that the asymptotic optimality rate of so called “almost surely asymptotically optimal” heuristic can be zero. We also present a new heuristic with short worst-case running time and statistically prove that it outperforms all heuristics known so far. However, our empirical experiments reveal that the heuristic is on average slower that its competitors with much longer worst-case running times.

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.

The dominance digraph as a solution to the two-machine flow-shop problem with interval processing times

The flow-shop minimum-length scheduling problem with n jobs processed on two machines is addressed where processing times are uncertain: lower and upper bounds for the random processing time are given before scheduling, but its probability distribution between these bounds is unknown. For such a problem, there often does not exist a dominant schedule that remains optimal for all possible realizations of the job processing times, and we look for a minimal set of schedules that is dominant. Such a minimal dominant set of schedules may be represented by a dominance digraph. We investigate useful properties of such a digraph.

A Truncated Sum of Processing‐Times–Based Learning Model for a Two‐Machine Flowshop Scheduling Problem

AbstractScheduling with learning effects has gained increasing attention in recent years. A well‐known learning model is called “sum‐of‐processing‐times‐based learning” in which the actual processing time of a job is a nonincreasing function of the jobs already processed. However, the actual processing time of a given job drops to zero precipitously when the normal job processing times are large. Moreover, the concept of learning process is relatively unexplored in a flowshop environment. Motivated by these observations, this article addresses a two‐machine flowshop problem with a truncated learning effect. The objective is to find an optimal schedule to minimize the total completion time. First, a branch‐and‐bound algorithm incorporating with a dominance property and four lower bounds is developed to derive the optimal solution. Then three simulated annealing algorithms are also proposed for near‐optimal solution. The experimental results indicated that the branch‐and‐bound algorithm can solve instances up to 18 jobs, and the proposed simulated annealing algorithm performs well in item of CPU time and error percentage. © 2011 Wiley Periodicals, Inc.

An Optimal Algorithm for a Special Flow Shop Problem with Infinite Buffer Capacity

In this paper, a two-machine flow shop problem with infinite buffer capacity is considered. Each of jobs is identical and has two tasks. The first task can be processed on either machine, called flexible task, while the second task must be processed on the second machine and can't be processed unless the first task has been processed. There is infinite buffer capacity between two machines. The problem is to determine the assignment of the flexible tasks to the machines for each job, with the objective of maximizing the makespan. We present an optimal algorithm for this problem.

Quantity-based buffer-constrained two-machine flowshop problem: active and passive prefetch models for multimedia applications

Conventional studies on buffer-constrained flowshop scheduling problems have considered applications with a limitation on the number of jobs that are allowed in the intermediate storage buffer before flowing to the next machine. The study in Lin et al. (Comput. Oper. Res. 36(4):1158---1175, 2008a) considered a two-machine flowshop problem with "processing time-dependent" buffer constraints for multimedia applications. A "passive" prefetch model (the PP-problem), in which the download process is suspended unless the buffer is sufficient for keeping an incoming media object, was applied in Lin et al. (Comput. Oper. Res. 36(4):1158---1175, 2008a). This study further considers an "active" prefetch model (the AP-problem) that exploits the unoccupied buffer space by advancing the download of the incoming object by a computed maximal duration that possibly does not cause a buffer overflow. We obtain new complexity results for both problems. This study also proposes a new lower bound which improves the branch and bound algorithm presented in Lin et al. (Comput. Oper. Res. 36(4):1158---1175, 2008a). For the PP-problem, compared to the lower bounds developed in Lin et al. (Comput. Oper. Res. 36(4):1158---1175, 2008a), on average, the results of the simulation experiments show that the proposed new lower bound cuts about 38% of the nodes and 32% of the execution time for searching the optimal solutions.

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.

Routing open shop and flow shop scheduling problems

We consider a generalization of the classical open shop and flow shop scheduling problems where the jobs are located at the vertices of an undirected graph and the machines, initially located at the same vertex, have to travel along the graph to process the jobs. The objective is to minimize the makespan. In the tour-version the makespan means the time by which each machine has processed all jobs and returned to the initial location. While in the path-version the makespan represents the maximum completion time of the jobs. We present improved approximation algorithms for various cases of the open shop problem on a general graph, and the tour-version of the two-machine flow shop problem on a tree. Also, we prove that both versions of the latter problem are NP-hard, which answers an open question posed in the literature.

Computational complexity and solution algorithms for flowshop scheduling problems with the learning effect

In this paper, we analyze the two-machine flowshop problem with the makespan minimization and the learning effect, which computational complexity was not determined yet. First, we show that an optimal solution of this problem does not have to be the ‘permutation’ schedule if the learning effect is taken into consideration. Furthermore, it is proved that the permutation and non-permutation versions of this problem are NP-hard even if the learning effect, in a form of a step learning curve, characterizes only one machine. However, if both machines have learning ability and the learning curves are stepwise then the permutation version of this problem is strongly NP-hard. Furthermore, we prove the makespan minimization problem in m-machine permutation proportional flowshop environment remains polynomially solvable with identical job processing times on each machine even if they are described by arbitrary functions (learning curves) dependent on a job position in a sequence. Finally, approximation algorithms for the general problem are proposed and analyzed.

Three heuristic procedures for the stochastic, two-machine flow shop problem

Although the deterministic flow shop model is one of the most widely studied problems in scheduling theory, its stochastic analog has remained a challenge. No computationally efficient optimization procedure exists even for the general two-machine version. In this paper, we describe three heuristic procedures for the stochastic, two-machine flow shop problem and report on computational experiments that compare their effectiveness. We focus on heuristic procedures that can be adapted for dispatching without the need for computer simulation or computer-based search. We find that all three procedures are capable of quickly generating solutions close to the best known sequences, which were obtained by extensive search.

A Fuzzy Heuristic Algorithm For The Flow Shop Scheduling Problem

The two-machine flow shop problem with the objective of minimizing makespan that is known as Johnson problem is now standard fundamental in the theory of scheduling. We generalize Johnson’s results for more than two machine problems, using a fuzzy heuristic algorithm. Performance of the new algorithm is analyzed with some numerical examples. To evaluate the performance of the proposed heuristic, we have used it on some small size problems and the results are compared with optimum scheduling. Notice that scheduling the problems with large sizes, is NP hard.

A bicriteria flow-shop scheduling problem with two serial batching machines

Abstract We consider a bicriteria two-machine flow-shop serial-batch problem where the batches have limited size. The first criterion is the number of batches to be minimized. This criterion reflects situations where processing of any batch induces a fixed cost, which leads to a total cost proportional to the number of batches. The second criterion is the makespan. We provide a dynamic programming algorithm in the case where the jobs processing times on the first machine are constant. We also establish a dominance relation allowing us to derive an alternative algorithm with better complexity when the batch size is equal to two.

Polynomial Lower Bounds for the Two-Machine Flowshop Problem with Sequence-Independent Setup Times

In this paper, we address the problem of two-machine flowshop scheduling problem with sequence independent setup times to minimize the total completion time. We propose five new polynomial lower bounds. Computational results based on randomly generated data show that our proposed lower bounds consistently outperform those of the literature.

A two-machine flowshop problem with two agents

The multiple-agent scheduling problems have received increasing attention recently. However, most of the research focuses on deriving feasible/optimal solutions or examining the computational complexity of the intractable cases in a single machine. Often a number of operations have to be done on every job in many manufacturing and assembly facilities (Pinedo, 2002 [1]). In this paper, we consider a two-machine flowshop problem where the objective is to minimize the total completion time of the first agent with no tardy jobs for the second agent. We develop a branch-and-bound algorithm and simulated annealing heuristic algorithms to search for the optimal solution and near-optimal solutions for the problem, respectively.

On-line two-machine job shop scheduling with time lags

We consider the on-line two-machine job shop scheduling problem with time lags so as to minimize the makespan. Each job consists of no more than two operations and time lags exist between the completion time of the first and the start time of the second operation of any two-operation job. We prove that any greedy algorithm is 2-competitive. For the non-clairvoyant variant of the problem, no on-line algorithm can do better. For the clairvoyant variant, no on-line delay algorithm has a competitive ratio better than 5 + 1 2 ≈ 1.618 , and a greedy algorithm is still the best on-line non-delay algorithm. We also show that the same results hold for the two-machine flow shop problem with time lags.

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.

Branch-and-bound and simulated annealing algorithms for a two-agent scheduling problem

We consider a two-agent scheduling problem on a two-machine flowshop setting, where the objective is to minimize the total tardiness of the first agent with the restriction that the number of tardy jobs of the second agent is zero. It is reported in the literature that the complexity of even two-machine flowshop problem, where both agents wish to complete their jobs as soon as possible, is shown NP-hard. Motivated by the limitation and suggestions, this paper proposes a branch-and-bound algorithm and a simulated annealing heuristic algorithm to search for the optimal solution and near-optimal solutions, respectively. Computational results are also presented to evaluate the performance of the proposed algorithms.

Using genetic algorithm in two-machine flexible flow-shop scheduling with setup times

Many studies on scheduling problems assume that each machine is used exclusively for one operation and disregard machine setup time. This exploration studies two-machine flow-shop scheduling problems in which both machines are versatile and setup time is considered. First, a modified optimal algorithm for determining the optimal schedule is developed to minimize the makespan of jobs for these problems. Second, a genetic algorithm is used to quickly find near-optimal schedules for large scale problems. Finally, computational experiments are performed to illustrate the effectiveness and efficiency of the proposed algorithms.

Two-machine flow shop problem with effects of deterioration and learning

In this paper 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 total completion time. Optimal solutions are obtained for some special cases. For the general case, several dominance properties and some lower bounds are derived, which are used to speed up the elimination process of a branch-and-bound algorithm. A heuristic algorithm is also proposed, which is shown by computational experiments to perform effectively and efficiently in obtaining near-optimal solutions.