#### Abstract

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 NP-hard 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 parallel-machine 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.

#### 1. Introduction

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 [1] who assumed that the processing times of jobs are nondecreasing and start time dependent linear functions. They derived optimal scheduling policies that minimize the expected makespan. Mosheiov [2] considered simple linear deterioration where jobs have a fixed job-dependent growth rate but no basic processing time. He showed that most commonly applied performance criteria, such as the makespan, the total flow time, the total lateness, the sum of weighted completion time, the maximum lateness, the maximum tardiness, and the number of tardy jobs, remain polynomially solvable. Since then, machine scheduling problems with deteriorating jobs have received increasing attention. An extensive survey of different models and problems was provided by Alidaee and Womer [3]. Cheng et al. [4] presented an updated survey of the results on scheduling problems with time-dependent processing times. Other new results of scheduling with deterioration effect can be found in [5–7].

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. [8]. They studied both the offline and online versions of scheduling problem with rejection on identical parallel machines. The objective is to minimize the sum of the makespan of the accepted jobs and the total rejection penalty of the rejected jobs. After that, the machine scheduling problem with rejection received more and more attention. Hoogeveen et al. [9] considered the offline multiprocessor scheduling problem with rejection where preemption is allowed. Zhang et al. [10] studied the single machine scheduling with release dates and rejection to minimize the sum of the maximum completion times of the scheduled jobs and the total penalty of the rejected jobs. Sengupta [11] considered algorithms and approximation schemes for minimum lateness/tardiness scheduling with rejection. Shabtay et al. [12] recently presented an updated survey on offline scheduling with rejection. In addition, there has been some work on scheduling deteriorating jobs with rejection. For example, Cheng and Sun [13] considered the single machine scheduling problem with deterioration and rejection, in which the processing time of a job is a linear function of its starting time. Li and Yuan [14] studied several parallel-machine scheduling problems in which the processing time of a job is a linear increasing function of its starting time and jobs can be rejected by paying penalties. The objectives are to minimize the makespan and the total weighted completion time plus the total penalty of the rejected jobs. Hsua and Chang [15] considered the unrelated parallel-machine scheduling with deteriorating jobs and rejection. To the best of our knowledge, the problems of scheduling deteriorating jobs with release dates and rejection have not been discussed.

In this paper, we study the single machine (parallel-machine) scheduling of deteriorating jobs with release dates and rejection. The paper is organized as follows. In Section 2, we formulate the problems under consideration and introduce some notations. In Section 3, we first discuss single machine model and show that the problem is strongly NP-hard. Then we propose a fully polynomial time approximation scheme for the case where all jobs have agreeable release dates and due dates. In Section 4, we consider parallel-machine scheduling problem with deterioration and rejection. In Section 5, we conclude the paper and suggest some topics for future research.

#### 2. Problem Description and Notation

The problems considered in this paper can be formally described as follows. There are an independent job set and a single machine ( identical parallel machines ). Each job has a release date , a due date (a delivery time ), a deteriorating rate , and a rejection penalty . Each job is either rejected with a rejection penalty having to be paid or accepted and processed on the single machine (one of parallel machines). The actual processing time of job is given by , where is the starting time of . Denote by and the set of rejected jobs and the set of accepted jobs, respectively. Denote by the completion time of the accepted job . In the single machine model, assume ; the lateness of job is defined as . The objective is to minimize the maximum lateness (i.e., ) plus the total penalty of the rejected jobs. In parallel-machine model, assume . We consider that each job has a delivery time rather than a due date . The delivery of a job begins immediately after it completes processing, and job deliveries are allowed to overlap. We define the delivery completion time as ; the objective is to minimize the maximum delivery completion time (i.e., ) plus the total penalty of the rejected jobs. Using the three-field notation of Graham et al. [16], the problems are denoted by and .

#### 3. The Single Machine Problem

##### 3.1. NP-Hardness Proof

In this subsection, we first discuss the complexity of the problem , where rejection is not considered.

Theorem 1. *The problem , is strongly NP-hard.*

*Proof. *The decision version of the scheduling problem is clearly in NP. We use a reduction from the 4-product problem (4-P), which is strongly NP-hard (see [17]).

An instance of the 4-product problem is formulated as follows. Given , a finite set for some natural number , a size and for every , . Does there exist disjoint subsets such that and the product of sizes of the elements in satisfies for ?

Given an arbitrary instance of 4-product problem, we construct an instance of the scheduling problem as follows.(i)There are jobs.(ii)For , , , and .(iii)For , , , and .(iv)The threshold value is defined by .

The decision version asks whether there is a schedule for problem such that .

It is clear that the reduction can be done in polynomial time. Now we prove that instance has a solution if and only if instance has a schedule with .

Assume that 4-product problem has a solution; that is, there exist disjoint subsets such that and the product of sizes of the elements in satisfies for . For convenience, we denote the set of jobs by . The jobs are scheduled without idle times as follows. First, the set of jobs are scheduled in the interval for . Next, each job is scheduled at time for . Therefore, the maximum lateness of this schedule is .

Conversely, assume that there is a schedule with . We are ready to prove that instance has a solution.

First, each job must start exactly at its release date and finish at its due date , since . Second, the rest jobs are scheduled as the intervals from time to time . Without loss of generality, let be the set of jobs scheduled in the interval .

Therefore, from the above discussion, we know that for . Note that . Thus, we have for . That is, instance has a solution. This completes the proof.

Corollary 2. *The problem is strongly NP-hard, so it has no fully polynomial time approximation scheme unless .*

##### 3.2. An FPTAS for One Special Case

An algorithm is a -approximation algorithm for a minimization problem if it produces a solution that is at most times than the optimal one ( is also referred to as the worst-case ratio). A family of algorithms is called a fully polynomial time approximation scheme (FPTAS), if for each , the algorithm is a approximation algorithm running in polynomial time in the input size and .

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 and . The problem can be denoted by .

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. [18]. The definition of the inflated rejection penalty is stated by Sengupta [11]. The rounding technique and the inflated rejection penalty are stated, respectively, as follows.

For any and , if , then we define . If is an exact power of , then . Note that for any .

For any , we define the modified deteriorating rate of job as , the modified release date of job as and the modified due date of job as , where . Let denote the exponent of ; that is, ; then ; let denote the exponent of ; that is, ; then .

For any , let and be the set of the rejected jobs, where . The inflated rejection penalty of the set is defined as

Lemma 3 (see [11]). *For any set of jobs and any , .*

Lemma 4 (see [19]). *For any and for any integer , holds.*

Theorem 5. *For any , the optimal objective function value for with the modified deteriorating rates, the modified release dates, and the modified due dates is at most times the optimal objective function value for .*

*Proof. *Assume that jobs in the set are scheduled in the order of . Let denote the completion time of job under the modified deteriorating rates and release dates; we have , then
The objective function is
Since the above inequality is valid for any , the results hold.

Assume that the jobs have been reindexed such that and define

Let denote the minimum value of the maximum lateness satisfying the following conditions.(i)The jobs in consideration are .(ii)The maximum completion time of the accepted jobs is .(iii)The inflated rejection penalty of the rejected jobs is .

In any such schedule, there are two possible cases: either job is rejected or is accepted and processed on the machine.

*Case 1 (Job is rejected). *Let be the inflated rejection penalty of the rejected jobs among . From the definition of inflated rejection penalty, the largest value of is given by . Hence, the value of the maximum lateness for the optimal schedule is .

*Case 2 (Job is accepted). *Consider the following.*Case **2**.1* . In this case, the maximum completion time of accepted jobs among is . Therefore, .*Case **2.2* . In this case, let be the maximum completion time of accepted jobs among . Since the starting time of job is , we have . Hence, .

Combining the above two cases, the dynamic programming algorithm is stated as follows.

*Algorithm DP1*

*Step 1 (initialization). *

*Step 2 (recursion). *If , ,

If ,

If ,
where is given by .

*Step 3 (output). *The optimal value is determined by , where is the smallest integer such that .

Let , where , . We have the following theorem.

Theorem 6. *There exists an FPTAS for problem which runs in time.*

*Proof. *We need to compute exactly values and computation of each value takes at most time, where . Hence, the running time of Algorithm DP1 is

#### 4. Parallel-Machine Problem

##### 4.1. An FPTAS

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 has a delivery time rather than a due date . When all jobs have identical release dates, the problem is at least ordinarily NP-hard according to Kononov [20]. We give a fully polynomial time approximation scheme to solve it by considering the modified deteriorating rates, the modified delivery times, and the inflated rejection penalty.

For any , we define the modified deteriorating rate and delivery time of job as , , where . Let denote the exponent of ; that is, ; then .

For any , let and be the set of the rejected jobs, where . The inflated rejection penalty of the set is defined as

Lemma 7. *For any , the optimal objective function value for , , with the modified deteriorating rates and the modified delivery times is at most times the optimal objective function value for
*

Lemma 8. *There exists an optimal job sequence for such that on each machine the accepted jobs are sequenced in nonincreasing order of .*

Based on Lemmas 7 and 8, we design a dynamic programming algorithm for with the modified deteriorating rates and delivery times. Let us index the jobs such that and define

Let denote the optimal value of the maximum delivery completion time satisfying the following conditions.(i)The jobs in consideration are .(ii)The maximum completion time of the accepted jobs on machine is for .(iii)The inflated rejection penalty of the rejected jobs is .

The dynamic programming algorithm is stated as follows.

*Algorithm DP2*

*Step 1 (initialization). *

*Step 2 (recursion). *If job is rejected,
If job is accepted,
Combining the two cases, we have
where is given by .

*Step 3 (output). *The optimal value is determined by
where is the smallest integer such that .

Theorem 9. *There exists an FPTAS for problem , which runs in time.*

*Proof. *We need to compute values . Computation of each such value takes time, where . Therefore the overall time of the algorithm is .

##### 4.2. A Heuristic Algorithm for a Special Case

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 , . We present a heuristic algorithm and give its worst case analysis.

*Heuristic *. Consider the following steps.

*Step 1. *Place all jobs in a list ordered by nonincreasing delivery times. Set for , and denotes the total completion time of jobs assigned to machine .

*Step 2. *Select a machine for which is as small as possible. Choose the first unscheduled job of the list and assign it to machine . Set .

*Step 3. *Repeat Step 2 until all jobs are scheduled.

Theorem 10. *The worst case performance ratio of heuristic is and this bound is tight.*

*Proof. *Let be the maximum delivery completion time according to heuristic and let be the corresponding time in an optimal schedule. Assume that there exists a counterexample with . Let be the first job in the sorted list with , remove all the following jobs from the list, and produce a minimum counterexample with , where is the job with smallest delivery time. Next, we transform the counterexample by subtracting from all delivery times. Obviously, this decreases both and by . Since
holds, the transformation yields another counterexample. Now the last job has delivery times zero and . At the time before job is assigned to its machine ,
This implies

To prove that the bound is tight, consider the following example with jobs, all jobs have delivery time zero, the first of them has deteriorating rates , , and the last job’s deteriorating rate is . Then and . By DLS, . In an optimal schedule, is assigned to one machine and all the previous jobs are equally assigned to the other machines. This yields to . Thus,
This completes the proof.

##### 4.3. Numerical Experiment

This section conducts some numerical experiments to evaluate the performance of heuristic . Without loss of generality, we set . The delivery time and the deteriorating rate are generated from and , respectively. The lower bound of optimal value is . The numerical results are summarized in Table 1. The program was coded in Matlab. The results of numerical experiments show that the algorithm has good performance. The proposed algorithm is acceptable for the -hard problem.

#### 5. Conclusion

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 NP-hard 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 and a heuristic algorithm for a special case. The computational results show that the heuristics is efficient in obtaining near-optimal solutions. For future research, it would be interesting to focus on scheduling deteriorating jobs and rejection with other objectives. Furthermore, investigation of the online version of this scheduling model is also an interesting research direction.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

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 Middle-aged Scientists of Shandong Province (BS2013SF016).