We consider the problems of scheduling deteriorating jobs with release dates on a single machine (parallel machines) and jobs can be rejected by paying penalties. The processing time of a job is a simple linear increasing function of its starting time. For a single machine model, the objective is to minimize the maximum lateness of the accepted jobs plus the total penalty of the rejected jobs. We show that the problem is NPhard in the strong sense and presents a fully polynomial time approximation scheme to solve it when all jobs have agreeable release dates and due dates. For parallelmachine model, the objective is to minimize the maximum delivery completion time of the accepted jobs plus the total penalty of the rejected jobs. When the jobs have identical release dates, we first propose a fully polynomial time approximation scheme to solve it. Then, we present a heuristic algorithm for the case where all jobs have to be accepted and evaluate its efficiency by computational experiments.
For most classical scheduling problems, the processing times of jobs are considered to be constant and independent of their starting time. However, this assumption is not appropriate for the modelling of many modern industrial processes; we often encounter situations in which processing time increases over time, when the machines gradually lose efficiency. Such problems are generally known as scheduling with deterioration effects. Scheduling with linear deterioration was first considered by Browne and Yechiali [
At the same time, it is always assumed in traditional research that all the jobs have to be processed. In the real world, however, things may be more complicated. For example, due to limited resources or limited available capacity, the decider of a factory can choose only a subset of these tasks to be scheduled, while perhaps incurring some penalty for the rejected jobs. The machine scheduling problem with rejection was first introduced by Bartal et al. [
In this paper, we study the single machine (parallelmachine) scheduling of deteriorating jobs with release dates and rejection. The paper is organized as follows. In Section
The problems considered in this paper can be formally described as follows. There are an independent job set
In this subsection, we first discuss the complexity of the problem
The problem
The decision version of the scheduling problem is clearly in NP. We use a reduction from the 4product problem (4P), which is strongly NPhard (see [
An instance of the 4product problem is formulated as follows. Given
Given an arbitrary instance
There are
For
For
The threshold value is defined by
The decision version asks whether there is a schedule
It is clear that the reduction can be done in polynomial time. Now we prove that instance
Assume that 4product problem has a solution; that is, there exist disjoint subsets
Conversely, assume that there is a schedule
First, each job
Therefore, from the above discussion, we know that
The problem
An algorithm
In this subsection, we consider the special case where all jobs have agreeable release dates and due dates and assume that the jobs have been indexed such that
We design an FPTAS for this problem by considering the modified deteriorating rates, the modified release dates, the modified due dates, and the inflated rejection penalty. The definition of the modified deteriorating rates and the modified release dates involves a geometric rounding technique developed by Afrati et al. [
For any
For any
For any
For any set
For any
For any
Assume that jobs in the set
Assume that the jobs have been reindexed such that
Let
The jobs in consideration are
The maximum completion time of the accepted jobs is
The inflated rejection penalty of the rejected jobs is
In any such schedule, there are two possible cases: either job
Let
Consider the following.
Combining the above two cases, the dynamic programming algorithm is stated as follows.
If
If
If
The optimal value is determined by
Let
There exists an FPTAS for problem
We need to compute exactly
In this subsection, we focus our attention on identical parallel machines problem with deterioration and rejection. For convenience of discussion, we consider a scheduling model in which each job
For any
For any
For any
There exists an optimal job sequence for
Based on Lemmas
Let
The jobs in consideration are
The maximum completion time of the accepted jobs on machine
The inflated rejection penalty of the rejected jobs is
The dynamic programming algorithm is stated as follows.
If job
The optimal value is determined by
There exists an FPTAS for problem
We need to compute
In this subsection, it is assumed that all jobs have to be processed; that is, rejection is not allowed. This problem can be denoted by
Place all jobs in a list ordered by nonincreasing delivery times. Set
Select a machine
Repeat Step
The worst case performance ratio of heuristic
Let
To prove that the bound is tight, consider the following example with
This section conducts some numerical experiments to evaluate the performance of heuristic
Summary of the computational results.







3  10  240  559  2.329  1.329 
4  12  208  577  2.774  1.774 
4  12  168  485  2.887  1.887 
4  15  380  520  1.368  0.368 
4  15  477  931  1.952  0.952 
5  10  33  60  1.818  0.818 
5  10  57  115  2.018  1.018 
5  18  611  1688  2.762  1.763 
In this paper, we focus on several scheduling problems with deterioration, release date, and rejection. For the single machine model, we show that this problem is strongly NPhard and provides a fully polynomial time approximation scheme when all jobs have agreeable release dates and due dates. For parallel machines model, we propose a fully polynomial time approximation scheme for
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper is supported by the National Natural Science Foundation of China (11201259), the Doctoral Fund of the Ministry of Education (20123705120001 and 20123705110003), and Promotive Research Fund for Young and Middleaged Scientists of Shandong Province (BS2013SF016).