Journal of Applied Mathematics

Journal of Applied Mathematics / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 679702 | https://doi.org/10.1155/2014/679702

G. Moslehi, K. Kianfar, "Approximation Algorithms and an FPTAS for the Single Machine Problem with Biased Tardiness Penalty", Journal of Applied Mathematics, vol. 2014, Article ID 679702, 10 pages, 2014. https://doi.org/10.1155/2014/679702

Approximation Algorithms and an FPTAS for the Single Machine Problem with Biased Tardiness Penalty

Academic Editor: Bernard J. Geurts
Received27 Jan 2014
Accepted13 Mar 2014
Published27 Apr 2014

Abstract

This paper addresses a new performance measure for scheduling problems, entitled “biased tardiness penalty.” We study the approximability of minimum biased tardiness on a single machine, provided that all the due dates are equal. Two heuristic algorithms are developed for this problem, and it is shown that one of them has a worst-case ratio bound of 2. Then, we propose a dynamic programming algorithm and use it to design an FPTAS. The FPTAS is generated by cleaning up some states in the dynamic programming algorithm, and it requires time.

1. Introduction

In this paper, we study a single machine scheduling problem of minimizing total biased tardiness about a common due date. Every job    has a processing time , a weight tardiness factor , and a base tardiness factor . The machine is available at time zero and can process at most one job at a time. The jobs have a common due date . The biased tardiness penalty of job is defined as where is the completion time of job . Figure 1 shows the biased tardiness penalty of job based on its completion time in a sequence. The resulting problem is denoted by , where BTP means “biased tardiness penalty.

Biased tardiness penalty is a kind of performance measures that, according to our observations, has not been studied in the literature in spite of its wide use in practical situations. One of the most common applications of biased tardiness is in designing delivery contracts. In many of delivery contracts, once an order is delivered later than its due date, a fixed penalty must be paid, and when the delivery becomes tardier, the related penalty will increase as well. Many of the practical conditions support this assumption; for example, consider a company where just one day delay in receiving raw materials will break down its production line. In this case, the initial damage caused by late delivery is huge while if the delay increases, the additional damage is relatively small. Another application for biased tardiness penalty is in transportation systems where we must pay extra money for solo-transporting a piece of goods if it is not ready to be carried with other orders.

Problem is NP-hard in the strong sense if the tardiness weights are not all equal [1, 2] and is optimally solvable in pseudo-polynomial time for a fixed number of distinct due dates [3]. Cheng et al. [4] have shown that the schedule that minimizes gives an -approximation for this problem. Kolliopoulos and Steiner [3] design pseudo-polynomial algorithms for the case that there is only a fixed number of different due dates. They also develop an FPTAS if, in addition, the tardiness weights are bounded by a polynomial function of . Karakostas et al. [5] consider the same problem, design a pseudo-polynomial algorithm, and apply a rounding scheme to obtain the desired approximation scheme.

In a special case of problem where the due date is common for all jobs, the resulting problem is proved to be NP-hard in the ordinary sense by Yuan [6], and Lawler and Moore [7] provide a pseudo-polynomial dynamic programming algorithm in time. Fathi and Nuttle [8] develop a 2-approximation algorithm that requires time. Kellerer and Strusevich [9] propose an FPTAS of time complexity, where is the sum of tardiness weights; later, Kacem [10] studies the same problem and develops another approach to obtain a more effective FPTAS in time.

If the tardiness weights are equal, problem is NP-hard in the ordinary sense as proved by Du and Leung [11], and it is solvable by a pseudo-polynomial dynamic programming algorithm proposed by Lawler [1]. For this problem, Lawler [12] proposes a dynamic programming algorithm and converts it into an FPTAS of time complexity. Koulamas [13] provides a faster FPTAS running in time by applying an alternative rounding scheme in conjunction with implementing Kovalyov’s [14] bound improvement procedure. Della Croce et al. [15] consider some popular constructive and decomposition heuristics and conclude that none of them guarantees a constant worst-case ratio bound. Kovalyov and Werner [16] study the approximability of this problem on parallel machines with a common due date.

To examine the complexity of problem , we compare it with the problem of minimizing weighted tardiness on a single machine and common due date. If we set for all jobs , the considered problem transforms to problem that is shown in [1, 2] to be NP-hard in the ordinary sense.

The remainder of this paper is organized as follows. In Sections 2 and 3, we describe two heuristic algorithms for problem and prove their worst-case ratio bounds. Section 4 describes a dynamic programming algorithm that, in Section 5, we convert to an FPTAS using the technique of structuring the execution of an algorithm. Concluding remarks are given in Section 6.

2. SPT Algorithm

In this heuristic algorithm, jobs are sequenced according to a nondecreasing order of processing times, and, hence, it can be implemented in time.

Theorem 1. Let and be, respectively, the smallest and largest weight tardiness factors, and let and be the smallest and largest base tardiness factors. Then, one has where is the penalty created by SPT algorithm and shows the optimal penalty for problem .

Proof. Consider two ordered sets and that include a nondecreasing order of the weight tardiness factors and base tardiness factors, respectively. Suppose that we create dummy jobs by pairing processing times in SPT ordering and tardiness factors according to their reverse order in sets and . It can be easily verified that the associated total penalty, called , is a lower bound on the total penalty of any sequence for real jobs.
Similarly, create another set of dummy jobs by pairing processing times in SPT ordering but tardiness factors consistent with the order of sets and . It can be easily tested that if we sequence these dummy jobs according to SPT ordering, the related total penalty, called , is an upper bound on for the real jobs. Let be the number of tardy jobs under SPT ordering. Also, let and denote the th job in sets and , respectively. Also, let denote the completion time of th job in SPT ordering. Thus, we have
From (3) and the fact that the values are nondecreasingly ordered in sets and , we get And if we signify the term by , then which completes the proof.

The following example illustrates that the worst-case ratio bound obtained by SPT ordering is tight for problem.

Example 2. Suppose that we have two jobs with parameters given in Table 1 and a common due date .
SPT ordering generates the sequence (2-1) with total penalty equal to 1020, while the optimal sequence for this example is (1-2) with the total penalty of 204. So,


Job

110010020
210204

3. Algorithm MPR (Minimum Penalty Rate)

In this section, we present another heuristic algorithm for problem and show that the worst-case ratio bound of this algorithm is 2. Let denote the sequence generated by MPR algorithm, where represents the th job within this sequence. This algorithm requires at most iterations and during its th iteration the th element of , , will be determined. Also, if there are some unscheduled jobs at each iteration filling the whole remaining period through due date (remaining tardiness period), we will choose the one making the minimum penalty and save the related sequence, , beside the main sequence . Finally, the algorithm sorts tardy jobs in the main and secondary sequences according to WSPT (weighted shortest processing time) ordering and returns the sequence with smaller penalty.

Let denote sum penalties related to the jobs from th position through the last job in sequence . So, the algorithm is as follows.

Algorithm MPR.(1)Let be the set of unscheduled jobs and let be the counter index of positions in the sequence. Set and .(2)Define . If is empty, then ; else, consider the job with the minimum penalty in as job that is calculated by and also .(3)For all jobs , calculate values by (7) and select a job such that . If there is a tie, select the job with the smaller processing time; (4)Set , , and .(5)If is not empty, then calculate the value . If , then set , , and   .(6)If , then and go back to step 3.(7)Schedule the remaining jobs at the beginning of sequences and .(8)Sort the tardy jobs in sequences and according to the nonincreasing order of ratios.(9)If , then return the sequence with penalty, and, else, return the sequence with penalty equal to .

It can be easily seen that MPR algorithm runs in time. The following example illustrates the implementation of MPR algorithm on a simple problem.

Example 3. Suppose a problem with four jobs and a common due date . Table 2 shows the jobs’ parameters.
At first, , , and is empty. At the first iteration of running the algorithm , we have , , , and  , where job 2 has the minimum value, and, hence, it is sequenced at the last position of . Also, , , and , and considering , we get . So, and .
At the second iteration , we have , and   , where job 4 has the minimum value, and, hence, . The algorithm calculates , , and , and considering , the value of will remain unchanged. At the last iteration , we have and , and the algorithm sequences job 3 at the second position of sequence . Also, and is empty.
After arbitrary scheduling the remaining jobs at the beginning of sequences and , we get and . These sequences are modified to and after implementing step 8. Also, and , which forces the algorithm to select sequence as the final output.


Job

17818
26416
34224
46517

Here, we present two theorems about problem which are used for proving the worst-case ratio bound of MPR algorithm.

Theorem 4. Consider a problem . Define two sequences and on a common time interval, where the relation holds for all jobs in and jobs in . Then, .

Proof. Consider two sequences and with tardiness penalties shown in Figure 2. Suppose that and denote the slope of the functions related to sequences and , respectively; then, according to the theorem’s assumption, we have for all . It is obvious that for all (especially for point ) the function related to sequence falls under the function related to sequence and, hence, . A similar conclusion can be made for the case where holds.

Theorem 5. In any optimal sequence for problem , the tardy jobs with start time greater than or equal to must be sequenced in WSPT ordering. This means that

Proof. The proof is easily done by swapping each pair of the adjacent tardy jobs.

Theorem 6. Algorithm MPR gives a 2-approximation for problem .

Proof. See the appendix.

The following example illustrates that the worst-case ratio bound obtained by MPR is tight for problem .

Example 7. Suppose that we have jobs with parameters given in Table 3 and a common due date .
Algorithm MPR gives the sequence with total penalty, while the optimal penalty is related to the sequence . Thus,


Job

1n
2 to 1 1

4. Dynamic Programming (DP) Algorithm

Without loss of generality, we consider that jobs are indexed according to the WLPT ordering. For problem , an optimal schedule belongs to the class of schedules in which the early jobs are processed starting at time zero and are followed by a straddling job, called job , that starts no later than time and is completed after time ; in turn, the straddling job is followed by the block of tardy jobs. The early jobs can be processed in any order, while, according to Theorem 5, tardy jobs that start at or after the due date must be processed according to WSPT numbering. Let us introduce the following notations. is a state in the state space, where denotes the total processing time of early jobs and is the total penalty of state. is a set of states for the first jobs, except for job . is the minimum penalty value for problem with a fixed straddling job . is the total processing time of all jobs.

This DP algorithm schedules early jobs starting from time zero and the tardy jobs so that they become complete exactly at time . According to this, the algorithm can be described as in Algorithm 1.

(1) For each
 (1.1) Set
 (1.2) For each
  (i) Consider every state in
   (a) If then add to
   (b) If then add to
  (ii) Delete the state space
 (1.3) Set
(2) Calculate the optimal value from

Let be an upper bound on the optimal penalty, and since and , we can restrict the number of states by . The complexity of substep 1.2 of the DP algorithm is proportional to that leads to time. However, this complexity can be reduced to by selecting a state with the smallest value of at each iteration and for every . Similarly, we can get the complexity of substep 1.3 as , and so, the complexity of step 1 is . Step 2 requires time, and the final complexity of DP algorithm will be calculated as . The following example illustrates the details of DP algorithm.

Example 8. Consider an instance of problem with 3 jobs. The parameters of the jobs are given in Table 4, and the common due date is given as in this example.
Table 5 shows the states generated in each states space regarding the selected straddling jobs as well as subsequences coupled with these states. The optimal value is related to sequence which is obtained by inserting straddling job into the subsequence .


Job

15212
27314
3349


StateSequenceStateSequenceStateSequence

0, 0 (  ,  ,  ) 0, 41 (  ,  ,  2) 3, 41 (3,  ,   2)

0, 0 (  ,  ,  ) 5, 0
0, 30
(1,  ,  ) 
(  ,  ,  1)
5, 45
3, 30
0, 55
(1,  ,  3)  
(3,  ,  1)
(  ,  3,   1)

0, 0 (  ,  ,  ) 5, 0
0, 30
(1,  ,  )
(  ,  ,  1)
5, 41 (1,  ,  2)

5. FPTAS Algorithm

One of the standard approaches to generate an FPTAS is the technique of structuring the execution of an algorithm. Here, the main idea is to take the exact but slow DP algorithm described in Section 4 and to interact with it while it is working. If the algorithm generates a lot of auxiliary states during its execution, then we may remove some of these states and clean up the algorithm’s memory. This method was introduced by Ibarra and Kim [17] for solving the knapsack problem, and in the recent years numerous scheduling problems have applied such an approach (see [1822]). First, let us introduce the following notations. is the error bound of FPTAS algorithm. is a set of states generated by FPTAS for the first jobs, except for job . is the minimum penalty generated by FPTAS for problem with a fixed straddling job . is the minimum penalty generated by FPTAS for problem .

Consider the penalty of algorithm MPR, called , as an upper bound for the problem. To reduce the number of states in each iteration, we split the feasible interval related to the second coordinate of state into equal subintervals of length . For each of the resulting subintervals , we keep at most one state with the smallest value . Given an arbitrary , define

The FPTAS algorithm works on the reduced state space instead of and can be described as in Algorithm 2.

(1) For each
 (1.1) Set
 (1.2) For each
  (i) Consider every state in
   (a) If then add to
   (b) If then add to
  (ii) Delete the state space
  (iii) Let be a state in such that with the smallest (break ties
   by choosing the state of the smallest ). Set
(1.3) Set
(2) Calculate the final solution of FPTAS from

5.1. Worst-Case Analysis of the FPTAS Algorithm

The worst-case analysis is based on comparing the execution of DP and FPTAS algorithms. First, a lemma is provided that will be used to prove the worst-case ratio bound of FPTAS.

Lemma 9. Let be an arbitrary state in . The FPTAS algorithm generates at least one state in such that and .

Proof. The proof is done by induction on . For , obviously we have . Suppose that the lemma is valid up to and we want to show its validity for iteration . Let be a state in generated by the DP algorithm from a feasible state at iteration . Here, two cases can be distinguished. In the first case and in the second case holds. We prove the statement for iteration in these two cases.
Case  1 (). Since , there exists a state such that and . Therefore, the FPTAS algorithm generates the state that may be eliminated when cleaning up the state subset. Let be the remaining state in that is in the same interval as . Thus, we drive that Consequently, the lemma holds for iteration in this case.
Case  2 (). Since , there exists a state such that and . Therefore, the FPTAS algorithm generates the state at iteration that may be eliminated during the cleaning up procedure. Let be the remaining state in that is in the same interval as . So, we have Thus, the lemma is proved for iteration in this case, too.

Theorem 10. Given an arbitrary , the FPTAS algorithm outputs a sequence with penalty such that .

Proof. There exists a straddling job, called , in the optimal sequence for problem . Since the FPTAS algorithm checks all jobs as straddling, then obviously job will be selected in one of its iterations.
By definition, the optimal sequence can be related to a state in . According to Lemma 9, the FPTAS algorithm generates a state in such that and
It is clear that . Let and denote the tardiness of job in the optimal and FPTAS solutions, respectively. From , we have
This will complete the proof.

5.2. Complexity of the FPTAS Algorithm

MPR algorithm runs in time as the initial phase of FPTAS. The state space    is generated at each iteration of substep 1.2 and in time. Since , we have

According to this, substep 1.2 requires time. Noting that step 1 iterates times for every selection of values, the complexity of this step is . Finally, step 2 requires time, and the final complexity of the FPTAS algorithm is computed as .

6. Conclusion

In this paper, we presented a new performance measure for scheduling problems, called biased tardiness penalty. According to this performance measure, two kinds of penalties are assigned to each tardy job: one fixed penalty and the other that linearly increases by the increase in tardiness value. Two approximation algorithms were designed with the polynomial running times. The first approximation algorithm, SPT, gives a worst-case ratio bound linking to size of instances, while the second approximation algorithm, MPR, has a constant worst-case ratio bound of 2. Next, we developed a dynamic programming algorithm and converted it to an FPTAS using the method of structuring the execution of an algorithm. The resulting FPTAS runs in time.

Appendix

Proof of Theorem 6

We consider two main cases for the sequence of tardy jobs and prove the worst-case ratio bound in both cases. Recall that and denote the penalties from MPR and optimal sequences, respectively. Let indicate the sum processing times of jobs in a sequence and let indicate completion time of job in a sequence . Also, shows the penalty of sequence if it ends at time .

Case  1. The first tardy job in MPR sequence before the sorting phase (step 8) is also tardy in optimal sequence.

Algorithm MPR schedules jobs from the end of sequence to the beginning, while some of the selected tardy jobs are also tardy in the optimal sequence and some others are not. According to this, we can show the sequence of tardy jobs before the sorting phase (step 8) as in Figure 3. In this figure, sets contain the jobs that only are tardy in the heuristic sequence and sets contain the jobs that are tardy in both heuristic and optimal sequences. Without loss of generality, suppose that each set contains a single job because sets can be empty. Let sequence begin with a job in from tardy jobs in optimal sequence. Put other tardy jobs in optimal sequence into set . Here, two subcases are identified.

Subcase  1.1 ( is empty). In this case, sets to contain all tardy jobs in the optimal sequence. Figure 4 shows the sequence of tardy jobs after execution of the sorting phase (step 8). From Theorem 5, the jobs included in sets have the same order in both sequences and the optimal sequence. So, we have where the second term shows the decrease in penalty values of to in sequence compared with the related penalties in optimal sequence. The second term indicates penalty related to sets to in the heuristic sequence.

Regarding the fact that all the jobs in sequence which are included in some sets come after the job in , we get From (A.1) and (A.2), Before the sorting phase (step 8), jobs are sequenced according to the nondecreasing order of ’s, and MPR algorithm in step 8 considers all the jobs filling the whole tardiness period in each iteration; thus, Now, from Theorem 4 and (A.5) and the fact that the job in is selected at the last iteration of algorithm, we conclude that From Theorem 5, we get Also, by (A.4), (A.6), and (A.7), it follows that Finally, noting that and we conclude that and (A.8) leads to the proof of in this case.

Subcase  1.2 ( is not empty). Similar to Subcase  1.1, we can show that . Substitute the job in by two dummy jobs, a tardy job having the same tardiness factors as and a processing time and an early job having tardiness factors equal to zero and a processing time . This substitution will not affect the generality of the proof because the penalty of sequence remains unchanged under this substitution, while the optimal penalty cannot increase. So, From Theorem 4 and (A.9) and the fact that the heuristic algorithm has not selected any job in , it follows that According to , we conclude that . According to and , we get which results in the proof of the theorem in this subcase.

Case  2. The first tardy job before sorting phase (step 8) is not tardy in optimal sequence.

Figure 5 shows the sequence of tardy jobs before and after sorting in step 8. Assume that contains tardy jobs in optimal sequence that are not tardy in sequence . cannot be empty because in that condition jobs in to must fill the whole tardiness period from to , and considering , there is no need that the heuristic algorithm selects tardy jobs in . Without loss of generality, the first job in can be substituted by two dummy jobs, a tardy job having the same tardiness factors as the first job in and processing time and an early job having tardiness factors equal to zero and processing time . This substitution gets to exactly fill the tardiness period while it has no effect on the heuristic penalty and will not increase the optimal penalty.

From Figure 5 and (A.2), we can show the relation between the optimal and heuristic sequences as follows: Finally, according to and we conclude the proof of the theorem in this case.

Conflict of Interests

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

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Copyright © 2014 G. Moslehi and K. Kianfar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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