Minimizing total load on a proportionate flowshop with position-dependent processing times and job-rejection

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Highlights

  • We study a scheduling problem on a proportionate flowshop.

  • The objective function is minimum total load.

  • General position-dependent processing times and the option of job-rejection are assumed.

  • The problems studied here are shown to be solved in polynomial time.

Abstract

We study the scheduling problem of minimizing total load on a proportionate flowshop. We consider position-dependent job processing times in the most general way. We show that this problem is solved in O(n4) time, where n is the number of jobs. We then extend the setting to allow job-rejection, where the scheduler may decide to process only a subset of the jobs, and the rejected jobs are penalized. This extension is shown to be solved in O(n5) time.

Introduction

Scheduling problems with the objective function of minimum total load have been rarely studied. This measure becomes relevant usually in multi-machine settings, and is equivalent to the sum of the largest completion times on all the machines. Minimizing total load is justified in settings where the cost is a function of the time that the machines are busy. In such cases the scheduler tries to finish the work on all machines as early as possible. When focusing on the classical setting of parallel identical machines, it is clear that total load is independent of the job allocation to the machines and of the job-sequence on each machine. However, in settings of time-dependent or position-dependent job processing times, minimizing total load becomes less trivial. Mosheiov [5] studied the problem of minimizing total load on parallel identical machines with time-dependent processing times (linear deterioration). He proved that the problem is NP-hard, and introduced and tested a heuristic and a lower bound. Mosheiov [7] studied the same machine setting and objective function with position-dependent processing times (position-based deterioration). This case was shown to be polynomially solvable in the number of jobs and the number of machines. Yu et al. [11] extended the latter model to the case that the job processing times are both position- and machine-dependent. This setting was shown to be polynomially solvable as well. Cheng et al. [3] studied a similar objective function on a two-stage flowshop with a common critical machine in stage one and two independent dedicated machines in stage two. Their goal was to minimize the weighted sum of machine completion times (where, clearly, only stage-two machine completion times are considered in the objective function). They proved that the problem is strongly NP-hard, but the case of a given job-sequence is polynomially solvable.

In this note we study the measure of total load on an m-machine proportionate flowshop. In a proportionate flowshop, the job processing times are assumed to be machine-independent. As above, we focus on the case of position-dependent job-processing time. We allow the processing times to be position-dependent in the most general way. In particular, no monotonicity is required: neither increasing processing time (reflecting deterioration or aging), nor decreasing processing time (reflecting learning). As mentioned, total load refers to the sum of the completion times of the last jobs on all the machines of the flowshop. We first introduce a polynomial (O(n5)) time solution for this problem, where n is the number of jobs. Then, we consider a recently published improved procedure for makespan minimization (Agnetis and Mosheiov [1]). Using this faster procedure, the running time is reduced to O(n4).

In the second part of the paper, we further extend this setting to allow job-rejection, a topic which has become popular among scheduling researchers in the last decade. In many real-life situations, processing all jobs may cause a delay in the completion of orders and may lead to tardiness cost. A decision on rejecting some of the jobs (either outsourcing them or rejecting them all together) becomes reasonable. We refer the reader to the recently published survey on this topic of Shabtay et al. [10]. We show that this problem, i.e., that of minimizing total load plus the rejection cost on a proportionate flowshop with general position-dependent processing times is solved in O(n6) time. Then, based on the improved procedure mentioned above (Agnetis and Mosheiov [1]), the running time is shown to be reduced to O(n5).

Section 2 provides the notation and the formulation. In Section 3 we introduce the solution to the problem of minimum total load. Section 4 contains the solution for the extension in which the objective is minimum total load plus rejection cost. In Section 5 we show how to reduce the above procedures by a factor of n.

Section snippets

Formulation

We consider an n-job m-machine proportionate flowshop problem. The processing time of job j which is performed in position r on machine i is denoted (in a general flowshop) by pijr, i=1,,m, j,r=1,,n. In a proportionate flowshop, job processing times are machine-independent, i.e., pijr=pjr, i=1,,m, j,r=1,,n. The job-position processing times (pjr) are given by a general n×n matrix, and as mentioned, no specific function and no monotonicity in r are assumed.

For a given schedule, Cij denotes

A polynomial-time solution for minimizing total load

In the classical makespan minimization problem on a proportionate flowshop, it is well-known (see e.g., Pinedo [9]) that the optimum is sequence-independent. The makespan value (i.e., the completion time of the last job on machine m) is given by j=1npj+(m1)pmax, where pmax is the largest processing time. Note that the critical path follows the largest job along all m machines, implying that the contribution of this largest job to the makespan is mpmax. When general position-dependent job

Minimizing total load and rejection cost

In this section we consider the option of job rejection, the objective function becomes minimum total load and total rejection cost: i=1mCmax(i)+jRej. We show that this problem remains polynomially solvable. We focus first on the case that the number of the scheduled (and consequently that of the rejected) jobs is given. Let nR denote the number of rejected jobs, i.e., nR=|R|, and nS=|S|=nnR is the number of scheduled or processed jobs. We create an assignment matrix, where the rows

A faster solution

In a recently published paper, Agnetis and Mosheiov [1] introduced a faster solution for the problem of minimizing makespan on a proportionate flowshop with general position-dependent job processing times. A minor modification of this procedure appears to work for the problem of minimum total load studied here. A brief description of the procedure is provided in the following. [We refer the reader to Agnetis and Mosheiov [1] for a full algorithm and a detailed explanation of the complexity

A comment on the case of time-dependent processing times

Scheduling with job processing times which are functions of the jobs' starting times (rather than their position in the sequence) is another very popular research topic in recent years (see Gawiejnowicz, [4]). Browne and Yechiali [2] were the first to study makespan minimization on a single machine, where the job processing times are linear functions of their starting times (also known as linear deterioration). Specifically, they assumed that the processing time of job j is given by the

Conclusion

We studied two scheduling problems on an m-machine proportionate flow shop. In both problems, general position-dependent job processing times were assumed. The objective function in the first problem was minimum total load. Using a recently published algorithm (for the case of makespan minimization), we introduced an O(n4) solution procedure for this problem. The objective function of the second problem is the sum of total load and total rejection cost. This setting was shown to be solved in O(n

Acknowledgements

This research was supported by the Israel Science Foundation (grant No. 1286/14). The second author was supported in part by the Recanati Fund of The School of Business Administration, and by Charles I. Rosen Chair of Management, The Hebrew University of Jerusalem, Israel.

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