Outsourcing and scheduling for two-machine ordered flow shop scheduling problems

This paper considers a two-machine ordered flow shop problem, where each job is processed through the in-house system or outsourced to a subcontractor. For in-house jobs, a schedule is constructed and its 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 the total outsourcing cost. Since this problem is NP-hard, we present an approximation algorithm. Furthermore, we consider three special cases in which job j has a processing time requirement p_j, and machine i a characteristic q_i. The first case assumes the time job j occupies machine i is equal to the processing requirement divided by a characteristic value of machine i, that is, p_j/q_i. The second (third) case assumes that the time job j occupies machine i is equal to the maximum (minimum) of its processing requirement and a characteristic value of the machine, that is, max{p_j, q_i} (min{p_j, q_i}). We show that the first and the second cases are NP-hard and the third case is polynomially solvable.     

Online scheduling of ordered flow shops

We consider online as well as offline scheduling of ordered flow shops with the makespan as objective. In an online flow shop scheduling problem, jobs are revealed to a decisionmaker one by one going down a list. When a job is revealed to the decision maker, its operations have to be scheduled irrevocably without having any information regarding jobs that will be revealed afterwards. We consider for the online setting the so-called Greedy Algorithm which generates permutation schedules in which the jobs on the machines are at all times processed without any unnecessary delays. We focus on ordered flow shops, in particular proportionate flow shops with different speeds and proportionate flow shops with different setup times. We analyze the competitive ratio of the Greedy Algorithm for such flow shops in the online setting. For several cases, we derive lower bounds on the competitive ratios.     

NP-hardness of compact scheduling in simplified open and flow shops

In practical task scheduling it is sometimes required that the components of a system perform consecutively. Such a scheduling is called scheduling without waiting periods or no-wait and/or no-idle. In this article we study the complexity of some simplified scheduling problems of this kind in open shop and flow shop settings. In particular, we show that many trivial questions about the existence of schedule become NP-hard, even if there are only two machines or if the scheduling graph of a system is a path or a cycle.     

Multi-family scheduling in a two-machine reentrant flow shop with setups

This paper considers a two-machine multi-family scheduling problem with reentrant production flows. The problem consists of two machines, M₁ and M₂, and each job has the processing route (M₁, M₂, M₁, M₂). There are identical jobs in the same family and the jobs in the same family are processed in succession. Each machine needs a setup time before the first job in a family is processed. The objective is to minimize the maximum completion time. Examples of such a problem occur in the bridge construction, semiconductor industry and job processing on numerical controlled machines, where they usually require that the jobs are reprocessed once and there are identical jobs in the same family. This problem is shown to be NP-hard. A branch-and-bound algorithm is proposed, and computational experiments are provided.     

Approximating a two-machine flow shop scheduling under discrete scenario uncertainty

This paper deals with the two machine permutation flow shop problem with uncertain data, whose deterministic counterpart is known to be polynomially solvable. In this paper, it is assumed that job processing times are uncertain and they are specified as a discrete scenario set. For this uncertainty representation, the min-max and min-max regret criteria are adopted. The min-max regret version of the problem is known to be weakly NP-hard even for two processing time scenarios. In this paper, it is shown that the min-max and min-max regret versions of the problem are strongly NP-hard even for two scenarios. Furthermore, the min-max version admits a polynomial time approximation scheme if the number of scenarios is constant and it is approximable with performance ratio of 2 and not (4/3 − ?)-approximable for any ? > 0 unless P = NP if the number of scenarios is a part of the input. On the other hand, the min-max regret version is not at all approximable even for two scenarios.     

Minimizing total expected costs in the two-machine,stochastic flow shop

This paper analyzes the n-job, two-machine flowshop sequencing problem with job processing times following exponential distributions. Three sufficient conditions are derived for determining a job sequence which minimizes a total expected linear cost function. Stronger results are obtained for several special cases.     

A heuristic for minimizing the expected makespan in two-machine flow shops with consistent coefficients of variation

The paper deals with the classical problem of minimizing the makespan in a two-machine flow shop. When the job processing times are deterministic, the optimal job sequence can be determined by applying Johnson’s rule. When they are independent and exponential random variables, Talwar’s rule yields a job sequence that minimizes the makespan stochastically.Assuming that the job processing times are independently and Weibull distributed random variables, we present a new job sequencing rule that includes both Johnson’s and Talwar’s rules as special cases. The proposed rule is applicable as a heuristic whenever the job processing times are characterized by their means and the same coefficient of variation. Simulation results show that it leads to very encouraging results when the expected makespan is minimized.     

可转包两台流水作业机排序的近似算法

研究可转包的两台流水作业机排序问题, 目标是极小化最大完工时间和总外包费用之和. 首先给出最坏情况界为2的近似算法, 接着对工件满足有序化约束的情形给出最坏情况界为\frac{3}{2}的改进算法, 以上算法界均为紧界.     

A two-machine flow shop scheduling problem with controllable job processing times

The paper deals with a two-machine flow shop scheduling problem in which both the sequence of jobs and their processing times are decision variables. It is assumed that the cost of performing a job is a linear function of its processing time, and the schedule cost to be minimized is the total processing cost plus maximum completion time cost. In is shown that the decision form of this problem is NP-complete, even when the processing times on one machine only are controllable and all the processing cost units are identical. Two heuristic methods for solving the problem are proposed and their worst-case analysis is presented.     

Approximation algorithms for two-machine flow shop scheduling with batch setup times

In many practical situations, batching of similar jobs to avoid setups is performed while constructing a schedule. This paper addresses the problem of non-preemptively scheduling independent jobs in a two-machine flow shop with the objective of minimizing the makespan. Jobs are grouped into batches. A sequence independent batch setup time on each machine is required before the first job is processed, and when a machine switches from processing a job in some batch to a job of another batch. Besides its practical interest, this problem is a direct generalization of the classical two-machine flow shop problem with no grouping of jobs, which can be solved optimally by Johnson's well-known algorithm. The problem under investigation is known to be NP-hard. We propose two O(n logn) time heuristic algorithms. The first heuristic, which creates a schedule with minimum total setup time by forcing all jobs in the same batch to be sequenced in adjacent positions, has a worst-case performance ratio of 3/2. By allowing each batch to be split into at most two sub-batches, a second heuristic is developed which has an improved worst-case performance ratio of 4/3. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Bicriterion scheduling in the two-machine flowshop

Bicriterion scheduling problems have attracted the attention of many researchers, especially in the past decade. Although more than fifty papers have been published on this topic, most studies done so far focus only on a single machine. In this paper, we extend the development to the two-machine case and present algorithms for the bicriterion of minimising makespan and number of tardy jobs and of makespan and total tardiness. Computational results are also presented.     

Theory and Methodology A bicriteria approach to the two-machine flow shop scheduling problem

In this paper we address the problem of minimizing makespan and sum of completion times simultaneously in a two-machine flow shop environment. We formulate the problem as a bicriteria scheduling problem, and develop a branch-and-bound procedure that iteratively solves restricted single objective scheduling problems until the set of efficient solutions is completely enumerated. We report computational results, and explore certain properties of the set of efficient solutions. We then discuss their implications for the Decision Maker.     

Minimizing the expected weighted number of tardy jobs in stochastic flow shops

This paper is devoted to two types of stochastic scheduling problems, one involving a single machine and the other involving a flow shop consisting of an arbitrary number of machines. In both problem types, all jobs to be processed have due dates, and the objective is to find a job sequence that minimizes the expected weighted number of tardy jobs. For the single-machine case, sufficient optimality conditions for job sequences are derived for various choices of due date and processing time distributions. For the case of a flow shop with an arbitrary number of machines and identically distributed due dates for all jobs, we prove the following intuitively appealing results: (i) when all jobs have the same processing time distributions, the expected weighted number of tardy jobs is minimized by sequencing the jobs in decreasing order of the weights, (ii) when all weights are equal, the jobs should be sequenced according to an increasing stochastic ordering of the processing time distributions.     

Total completion time minimization in two-machine job shops with unit-time operations

We study the problem of minimizing total completion time in two-machine job shop with unit-time operations. We propose an efficient algorithm for the problem. The algorithm is polynomial with respect to a succinct encoding of the problem instances, where the number of bits necessary to encode a job with k operations is O(log(k + 1)). This result answers a long standing open question about the complexity of the problem.     

Bicriteria hierarchical optimization of two-machine flow shop scheduling problem with time-dependent deteriorating jobs

We study a two-machine flowshop scheduling problem with time-dependent deteriorating jobs, i.e. the processing times of jobs are an increasing function of their starting time. The objective is to minimize the total completion time subject to minimum makespan. We propose a mixed integer programming model, and develop two pairwise interchange algorithms and a branch-and-bound procedure to solve the problem while using several dominance conditions to limit the size of the search tree. Several polynomial-time solvable special cases are discussed. Finally, numerical studies are performed to examine the effectiveness and the efficiency of the proposed algorithms.     

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.     

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.     

An improved heuristic for two-machine flow shop scheduling with an availability constraint and nonresumable jobs

In this paper, we deal with the two-machine flow shop scheduling problem having an unavailability interval on the first machine, and nonresumable jobs. We first present an enhancement procedure that, once applied to any arbitrary solution, produces a schedule that is at most equal 2 times the optimal makespan. We then develop an improved heuristic, with a relative worst-case error of 3/2.     

On-line scheduling of small open shops

We investigate the problem of on-line scheduling open shops of two and three machines with an objective of minimizing the schedule makespan. We first propose a 1.848-competitive permutation algorithm for the non-preemptive scheduling problem of two machines and show that no permutation algorithm can be better than 1.754-competitive. Secondly, we develop a (27/19)-competitive algorithm for the preemptive scheduling problem of three machines, which is most competitive.