A BICRITERIA APPROACH TO SCHEDULING A SINGLE MACHINE WITH JOB REJECTION AND POSITIONAL PENALTIES Dvir Shabtay and Nufar Gaspar
y
Department of Industrial Engineering and Management Ben-Gurion University of the Negev, Beer-Sheva, Israel Liron Yedidsionz Faculty of Industrial Engineering and Management, Technion - Israel Institute of Technology, Haifa, Israel
Abstract 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, thus resulting in a rejection cost. In such a framework, the scheduler has to decide …rst which jobs will be rejected and which will be accepted. Then he has to schedule the accepted jobs e¢ ciently. Scheduling with job rejection is essentially a problem with two criteria. The …rst is a scheduling criterion, which is dependent on the completion times of the accepted jobs, and the second is the total rejection cost. Problems of scheduling with rejection have been previously studied, but usually in a narrow framework – focusing on one scheduling criterion at a time. This paper provides a uni…ed bicriteria analysis of a large set of single machine problems sharing a common property: all problems can be represented by or reduced to a scheduling problem with a scheduling criterion which includes positional penalties. Among these scheduling criteria 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 di¤erent problem variations for dealing with the two criteria are studied. The variation of minimizing the total integrated cost is shown to be solvable in polynomial time, while all other three variations are shown to be NP-hard. For those hard problems, we provide a pseudo polynomial time algorithm. An FPTAS for obtaining an approximate e¢ cient schedule is provided as well. In addition, we present an interesting special case which is solvable in polynomial time. This research was supported by THE ISRAEL SCIENCE FOUNDATION (grant No. 633/08). Partial support by the Paul Ivanier Center for Robotics and Production Management, Ben-Gurion University of the Negev is also gratefully acknowledged. y e-mails:
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Introduction and Problem Description
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, the manufacturer may wish to reject the processing of some jobs in the shop. The decision to reject jobs may be due to a low machine capacity or high scheduling costs. In such a case, a rejected job may be either outsourced or not get served at all, resulting in a rejection penalty either due to outsourcing cost or loss of income and reputation. In such a framework, the scheduler must …rst decide which jobs will be rejected and which will be accepted. Then the set of accepted jobs has to be e¢ ciently scheduled among the machines as to minimize a given prede…ned scheduling criterion. An instance for a single machine scheduling problem with job rejection is de…ned by the number of jobs to be processed, the jobs’processing times and rejection penalties. A solution is de…ned by a partition of the jobs into a set of accepted, and a set of rejected jobs, and by a job sequence of the set of accepted jobs on the single machine. The quality of a solution is measured by two criteria: The …rst is a scheduling criterion (scheduling cost) depending on the job completion time, and the second one is the total rejection cost. Since scheduling with rejection is essentially a problem with two criteria, four di¤erent problems can arise: The …rst, P1, is to minimize the total integrated cost, i.e., the sum of scheduling and total rejection costs; The second, P2, is to minimize the scheduling criterion subject to an upper bound on the total rejection cost; The third, P3, is to minimize the total rejection cost subject to an upper bound on the scheduling cost; The last one, P4, is to identify the set of Pareto-optimal schedules for both criteria (scheduling and total rejection costs). It should be noted that solving problem P4, also solves problems P1-P3 as a by-product. Note also that the decision version of problems P2 and P3 is identical. The Importance of studying the di¤erent problem variations is derived from di¤erent applications of the problems. For example, one manufacturer may wish to minimize the total integrated cost (P1), while another has a speci…c budget for scheduling or rejection cost and wishes to minimize the other criterion while staying within budget (as described by problems P2 and P3). The idea of scheduling with rejection was …rstly introduced by Bartal et al. [1]. They have studied the problem on a set of identical parallel machines with the makespan scheduling criterion. Other studies related to the makespan scheduling criterion are given in Hoogeveen et al. [7], Cao et al. [2], Cao and Zhang [4], Cheng and Sun [5], Lu et al. [10], Zhang et al. [13], Cao and Yang [3], Zhang et al [12] and Lu et al. [9]. Related works with the maximal lateness and tardiness scheduling criteria were presented by Sengupta [11], Cheng and Sun [5] and Khuller and Mestre [8] and with the weighted sum of completion time scheduling criterion were presented by Engels et al. [6], Cao et al. [2] and Cheng and Sun [5]. 2
When reviewing the existing literature, we noticed that each paper was restricted to a very speci…c type of scheduling criterion (either makespan, lateness or sum of weighted completion time). Moreover, the analysis was usually restricted to the P1 problem variation. Our paper, however, o¤ers a much more general approach towards scheduling with rejection by providing a uni…ed bicriteria analysis for all four problem variations of a large set of scheduling problems in which their scheduling criterion can be represented or reduced to a criterion which has job independent positional penalties. Among those criteria are the makespan, the sum of completion times, the sum and variation of completion times, and the total earliness plus tardiness penalties for several due date assignment problems. The scheduling criteria that are studied have some very important and practical applications for calculating various operational costs. For instance, the sum of completion times is an important element in the calculation of inventory costs, while the well known makespan criterion can be used for calculating machine operational costs.
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Methodology and Main Results Polynomial Time Solution for the P1 Problem
We provide a dynamic programming based algorithm for solving the uni…ed P1 problem in O(n3 ) time. The algorithm is a forward algorithm applied to each number of accepted jobs separately. We also show that if the positional penalties are independent of the number of jobs in the set of accepted jobs and is a monotonous function of the position in the sequence (such as for the sum of completion time criterion) a faster O(n2 ) can be applied to solve the problem. Furthermore, we showed that if the positional penalties are both independent of the number of jobs in the set of accepted jobs and of the position in the sequence (such as for the makespan criterion) the time complexity can be even further reduced to O(n).
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Analysis of Problems P2-P4
For the P2-P4 variations, we show that the problems are N P-hard for any given set of positional penalties which is independent of the number of jobs in the set of accepted jobs and is a monotonous function of the position in the sequence. The proof is done by reducing the even-odd partition problem to the decision version of our uni…ed model. Then we show that all the scheduling criterion mentioned earlier or a special case of them share these characteristics and hence are all N P-hard. This proof as well as being robust, identi…es the complexity of several problems which their complexity was an open question heretofore such as the P2-P4 problem variations with the sum of completion time scheduling criterion. In order to solve the uni…ed P2-P4 problem variations we provide a pseudo polynomial time algorithm which we convert to a fully polynomial approximation scheme (FPTAS) that …nds an approximate e¢ cient solution in the trade o¤ curve. In addition, we show that the P2-P4 problem variations can be solved in a polynomial time if a certain agreeable condition on the job processing times and rejection penalties exists.
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References [1] Bartal, Y., Leonardi, S., Marchetti-Spaccamela, A., Sgall, J., and Stougie, L., 2000, Multiprocessor Scheduling with Rejection, in: Seventh ACM-SIAM Symposium on Discrete Algorithms, 95-103. [2] Cao, Z., Wang, Z., and Shoupeng, L., 2006, On Several Scheduling Problems with Rejection or Discretely Compressible Processing Times, in: Lecture notes in computer science, 3959, 90-98. [3] Cao, Z., Yang, X., 2009, A PTAS for Parallel Batch Scheduling with Rejection and Dynamic Job Arrivals, Theoretical Computer Science, 410, 2732-2745. [4] Cao, Z. and Zhang, Y., 2007, Scheduling with rejection and non-identical job arrivals, Journal of Systems Science and Complexity, 20, 529 535. [5] Cheng, Y., Sun, S., 2007, Scheduling Linear Deteriorating Jobs with Rejection on a Single Machine, European Journal of Operational Research, 194, 18-27. [6] Engels, D.W., Karger, D.R., Kolliopoulos, S.G., Segupta, S., Uma, R.N., and Wein, J., 2003, Techniques for Scheduling with Rejection, Journal of Algorithms, 49, 175-191. [7] Hoogeveen, H., Skutella, M., and Woeginer, G.J., 2003, Preemptive Scheduling with Rejection, Mathematical Programming, 3, 287-326. [8] Khuller, S., and Mestre, J., 2008, An Optimal Incremental Algorithm for Minimizing Lateness with Rejection, in: Lecture notes in computer science, 5193, 601-610. [9] Lu, L., Cheng, T.C.E., Yuan, J., and Zhang, L., 2009, Bounded Single-Machine Parallel-Batch Scheduling with Release Dates and Rejection, Computers & Operations Research, 36, 2748-2751. [10] Lu, L., Zhang, L., and Yuan, J., 2008, The unbounded parallel batch machine scheduling with release dates and rejection to minimize makespan, Theoretical Computer Science, 396, 283–289. [11] Sengupta, S., 2003, Algorithms and Approximation Schemes for Minimum Lateness/ Tardiness Scheduling with Rejection, in: Lecture notes in computer science, 2748, 79-90. [12] Zhang, L., Lu, L., and Yuan, J., 2009, Single Machine Scheduling with Release Dates and Rejection, European Journal of Operational Research, 198, 975-978. [13] Zhang, L., Ren, J., and Wang, C., 2009, Scheduling with Rejection to Minimize the Makespan, in: Lecture notes in computer science, 5573, 411-420.
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