Single two-stage flowshop and parallel two-stage flowshops scheduling with optional job rejection

We study single machine scheduling problems. Generalised due dates are assumed, i.e. job due dates are specified according to the positions of the jobs in the sequence, rather than their identity. Thus, assuming that due dates are numbered in a non-decreasing order, the jth due date refers to the job assigned to the jth position. In addition, we allow the option of job rejection, i.e. not all jobs must be processed. In this case, the scheduler is penalised for each rejected job, and the total rejection cost becomes part of the objective function. Two objective functions are considered: maximum tardiness plus rejection cost, and total tardiness plus rejection cost. Both problems are proved to be NP-hard. Pseudo-polynomial dynamic programmes and efficient heuristics are introduced and tested numerically.

We study a single machine scheduling problem with the total weighted late work and the total rejection cost. The late work of a job is the part of this job executed after its due date, and the rejection cost of a job is the fee of rejecting to process it. We consider a Pareto scheduling and two restricted scheduling problems. The Pareto scheduling problem aims to find all non‐dominated values of the total weighted late work and the total rejection cost. The restricted scheduling problems are dedicated to minimizing the total weighted late work with the total rejection cost not greater than a given threshold, and minimizing the total rejection cost with the total weighted late work not greater than a given threshold, respectively. For the Pareto scheduling problem, we prove that it is binary NP‐hard by providing a pseudo‐polynomial time algorithm, and give a fully polynomial time approximation scheme (FPTAS). For the restricted scheduling problems, we prove that there are no FPTASes unless , which answers an open problem. Moreover, we develop relaxed FPTASes for these two restricted scheduling problems.

An FPTAS for the parallel two-stage flowshop problem

In this paper, we examined single and parallel machine scheduling problems with a learning effect and job rejection simultaneously. In real life, job processing times decrease when there is a learning effect. In some cases, producers cannot process all the jobs and pay the penalty cost for these jobs that they do not process. In our study, learning effect and job rejection are considered at the same time. We examined four different objective functions. Our objectives for single-machine scheduling problems are makespan and rejection cost minimization, total completion time and rejection cost minimization and total absolute deviation of completion times (TADC) and rejection cost minimization. Our objective for parallel machines is makespan and rejection cost minimization. The problems are solved by mathematical models, and four different algorithms are proposed for the problems. From these algorithms, the same results are obtained with single-machine makespan and rejection cost minimization, parallel machine makespan and rejection cost minimization and total completion time and rejection cost minimization. The accuracy for these models is obtained as 100%. The proposed algorithm for TADC and rejection cost minimization yielded close-to-optimal results. Mathematical model and algorithm results for 10 jobs, 20 jobs and 30 jobs are compared and the results are presented. The obtained solutions are obtained in polynomial time.

Two-machine flow-shop scheduling with rejection

A polynomial-time approximation scheme for an arbitrary number of parallel two-stage flow-shops

In this article, we study a parallel machine scheduling problem with inclusive processing set restrictions and the option of job rejection. In the problem, each job is compatible to a subset of machines, and machines are linearly ordered such that a higher-indexed machine can process all those jobs that a lower-indexed machine can process (but not conversely). To achieve a tight production due date, some of the jobs might be rejected at certain penalty. We first study the problem of minimizing the makespan of all accepted jobs plus the total penalty cost of all rejected jobs, where we develop a (5/3+ ε)- approximation algorithm with a time complexity of O(nm/ε2 · log P). We then study two bicriteria variants of the problem. For the variant problem of minimizing the makespan subject to a given bound on the total rejection cost, we develop a (5/3+ ε) -approximation algorithm with a time complexity of O(n2/ε2). For the variant problem of maximizing the total rejection cost of the accepted jobs subject to a given bound on the makespan, we present a 0.5-approximation algorithm with a time complexity of O(n log n).

Single-machine scheduling with total late work and job rejection

Minimizing total late work on a single machine with generalized due-dates

In this paper, we study two scheduling problems on a single machine with rejection and an operator non-availability interval. In the operator non-availability interval, no job can be started or be completed. However, a crossover job is allowed such that it can be started before this interval and completed after this interval. Furthermore, we also assume that job rejection is allowed. That is, each job is either accepted and processed in-house, or is rejected by paying a rejection cost. Our task is to minimize the sum of the makespan (or the total weighted completion time) of accepted jobs and the total rejection cost of rejected jobs. For two scheduling problems with different objective functions, by borrowing the previous algorithms in the literature, we propose a pseudo-polynomial-time algorithm and a fully polynomial-time approximation scheme (FPTAS), respectively.

A number of recent works have studied algorithms for entrywise ep-low rank approximation, namely algorithms which given an n × d matrix A (with n ≥ d), output a rank-k matrix B minimizing ||A − B||pp = ∑i, j |Ai,j − Bi,j|p when p > 0; and ||A − B||0 = ∑i,j [Ai,j ≠ Bi,j] for p = 0, where [·] is the Iverson bracket, that is, ||A − B||0 denotes the number of entries (i, j) for which Ai,j ≠ Bi,j. For p = 1, this is often considered more robust than the SVD, while for p = 0 this corresponds to minimizing the number of disagreements, or robust PCA. This problem is known to be NP-hard for p ∈ {0, 1}, already for k = 1, and while there are polynomial time approximation algorithms, their approximation factor is at best poly(k). It was left open if there was a polynomial-time approximation scheme (PTAS) for ep-approximation for any p ≥ 0. We show the following:1. On the algorithmic side, for p ∈ (0, 2), we give the first npoly(k/e) time (1 + e)-approximation algorithm. For p = 0, there are various problem formulations, a common one being the binary setting in which A ∈ {0, 1}n × d and B = U · V, where U ∈ {0, 1}n × k and V ∈ {0, 1}k × d. There are also various notions of multiplication U · V, such as a matrix product over the reals, over a finite field, or over a Boolean semiring. We give the first almost-linear time approximation scheme for what we call the Generalized Binary e0-Rank-k problem, for which these variants are special cases. Our algorithm computes (1 + e)-approximation in time (1/e)2O (k) / e2 · nd1+o(1), where o(1) hides a factor (log log d)1.1/log d. In addition, for the case of finite fields of constant size, we obtain an alternate PTAS running in time n · dpoly(k/e).2. On the hardness front, for p ∈ (1, 2), we show under the Small Set Expansion Hypothesis and Exponential Time Hypothesis (ETH), there is no constant factor approximation algorithm running in time 2kδ for a constant δ > 0, showing an exponential dependence on k is necessary. For p = 0, we observe that there is no approximation algorithm for the Generalized Binary e0-Rank-k problem running in time 22δk for a constant δ > 0. We also show for finite fields of constant size, under the ETH, that any fixed constant factor approximation algorithm requires 2kδ time for a constant δ > 0.

In this article, we study the method of job rejection in the setting of proportionate flowshop, and focus on minimizing regular performance measures, subject to the constraint that the total rejection cost cannot exceed a given upper bound. In particular, we study total completion time, maximum tardiness, total tardiness, and total weighted number of tardy jobs. All the addressed problems are NP-hard as their single machine counterpart are known to be NP-hard. To the best of our knowledge, there are no detailed solutions in scheduling literature to the first two problems, whereas the last two problems were never addressed to date. For each problem, we provide a pseudo-polynomial dynamic programming solution algorithm, and furthermore, we enhance the reported running time of the first two problems. Our extensive numerical study validates the efficiency of the provided solutions.

Recently, Shabtay considered a scheduling problem on a single serial-batching machine with rejection to minimize the dual criteria of total completion time and total rejection cost, where the number of jobs to be included in each batch is not restricted. He studied four variants of the problem: the first is to minimize the sum of the two criteria; the second and third are to minimize one criterion, subject to the other criterion not exceeding a given value; and the last is to find the Pareto-optimal solutions for the bicriterion problem. Shabtay provided an ${O}$ ( $\textit{n}^{\mathbf {5}}$ ) algorithm for the first variant and an ${O}$ ( ${n} ^{\mathbf {6}}$/$\boldsymbol {\varepsilon }^{\mathbf {2}}$ ) fully polynomial-time approximation scheme (FPTAS) for the fourth variant. In this paper, we provide an alternative ${O}$ ( ${n} ^{\mathbf {4}}$ ) algorithm to solve the first variant and an ${O}$ ( ${n} ^{\mathbf {5}}$/$\boldsymbol {\varepsilon }$ ) FPTAS for the fourth variant, which are more efficient than those developed by Shabtay from a theoretical perspective. However, when the size of each batch is bounded by a given number ${b}~\boldsymbol {>}1$ , the corresponding time complexities of our algorithms for the first and fourth variants reduce to ${O}$ ( bn $^{\mathbf {3}}$ ) and ${O}$ ( bn $^{\mathbf {4}}$/$\boldsymbol {\varepsilon }$ ), respectively.

Single machine scheduling problems have been extensively studied in the literature under the assumption that all jobs have to be processed. However, in many practical cases, one may wish to reject the processing of some jobs in the shop, which results in a rejection cost. A solution for a scheduling problem with rejection is given by partitioning the jobs into a set of accepted and a set of rejected jobs, and by scheduling the set of accepted jobs among the machines. The quality of a solution is measured by two criteria: a scheduling criterion, F1, which is dependent on the completion times of the accepted jobs, and the total rejection cost, F2. Problems of scheduling with rejection have been previously studied, but usually within a narrow framework--focusing on one scheduling criterion at a time. This paper provides a robust unified bicriteria analysis of a large set of single machine problems sharing a common property, namely, all problems can be represented by or reduced to a scheduling problem with a scheduling criterion which includes positional penalties. Among these problems are the minimization of the makespan, the sum of completion times, the sum and variation of completion times, and the total earliness plus tardiness costs where the due dates are assignable. Four different problem variations for dealing with the two criteria are studied. The variation of minimizing F1+F2 is shown to be solvable in polynomial time, while all other three variations are shown to be $\mathcal{NP}$ -hard. For those hard problems we provide a pseudo polynomial time algorithm. An FPTAS for obtaining an approximate efficient schedule is provided as well. In addition, we present some interesting special cases which are solvable in polynomial time.

This chapter focuses on a scheduling problem on identical parallel machines to minimize the total completion time under the assumption that all the jobs are available at time zero. However, before processing begins, some jobs are delayed and become unavailable at time zero, so all the jobs need to be rescheduled with a view to not causing excessive schedule disruption with respect to the planned schedule. To reduce the negative impact of job unavailability and achieve an acceptable service level, one option in rescheduling the jobs is to reject a subset of the jobs at a cost (the rejection cost). Three criteria are thus involved: the total completion time of the accepted jobs in the adjusted schedule, the degree of disruption measured by the maximum completion time disruption to any accepted job between the planned and adjusted schedules, and the total rejection cost. The overall objective is to minimize the former criterion, while keeping the objective values of the latter two criteria to no greater than the given limits.