A note on “Two-machine flow-shop scheduling with rejection” and its link with flow-shop scheduling and common due date assignment

doi:10.1016/j.cor.2012.04.009

Computers & Operations Research

Volume 39, Issue 12, December 2012, Pages 3244-3246

https://doi.org/10.1016/j.cor.2012.04.009 Get rights and content

Abstract

In a recent paper by Shabtay and Gasper “Two-machine flow-shop scheduling with rejection, Computers and Operations Research”, forthcoming, doi:10.1016/j.cor.2011.05.023, several complexity and approximation results are proposed for a two-criteria two-machine flow-shop scheduling problem with rejection. The two criteria to be minimized are the makespan the total rejection cost. This note positions the contribution of such results with respect to the contributions of the literature on common due date assignment and flow-shop scheduling not considered in the work of Shabtay and Gasper.

Introduction

In [5] Shabtay and Gasper have recently tackled the two-machine flow-shop problem with rejection where two objectives are considered, namely the makespan and the total rejection cost, and four combinations of such objectives are analyzed. Such combinations are the problem of minimizing the weighted sum of such objectives (denoted by P₁), the $ϵ$ constraint problem with respect to the total rejection cost (denoted by P₂), the $ϵ$ constraint problem with respect to the makespan (denoted by P₃) and the more general problem devoted to the search for the Pareto optimal solutions with respect to both objectives (denoted by P₄). Several complexity and approximation results are provided on problems P₁, …, P₄ but some related literature [3], [8] on common due date assignment and flow-shop scheduling is missing. Purpose of this note is to position the contribution of [5] with respect to such literature.

Consider the more general problem P₄. We have n jobs available for processing at time zero to be scheduled on m=2 machines in a flow-shop scheduling system. Each job j has a processing time a_j on the first machine, a processing time b_j on the second machine and a rejection cost w_j. The jobs can be either accepted (belonging then to the set A of accepted jobs) or rejected (belonging then to the set $\bar{A}$ of rejected jobs). The objectives are the makespan of the accepted jobs and the total rejection cost $\sum_{j \in \bar{A}} w_{j}$ and the aim is to search for the Pareto optimal solutions with respect to both objectives.

However, as all the jobs in A will be completed within the makespan $C_{\max}$ , such makespan can be seen as a common due date d to be respected by the jobs in A (hence all jobs $\in A$ will be early with respect to d), while the jobs in $\bar{A}$ can be assumed w.l.o.g. to be completed after d (hence all jobs $\in \bar{A}$ will be tardy with respect to d).

But then, using the extended three-field classification of [7] (which is more common for multi-objective scheduling), problem P₄ can be denoted by $F 2 | d_{j} = d,$ unknown $d | d, \sum_{j} w_{j} U_{j}$ . Correspondingly, P₁ can be denoted by $F 2 | d_{j} = d$ , unknown $d | d + \sum_{j} w_{j} U_{j}$ , P₂ can be denoted by $F 2 | d_{j} = d$ , unknown $d | ϵ (d / \sum_{j} w_{j} U_{j})$ and P₃ can be denoted by $F 2 | d_{j} = d$ , unknown $d | ϵ (\sum_{j} w_{j} U_{j} / d)$ .

It turns out that P₄ is actually the weighted generalization of problem $F 2 | d_{j} = d$ , unknown $d | d, \sum_{j} U_{j}$ considered in [8] (denoted hereafter by P₅) and is also strictly related to problem $F 2 | d_{j} = D | \sum_{j} w_{j} U_{j}$ (minimization of the weighted sum of tardy jobs in a two-machine flow-shop with common due date D—denoted hereafter P₆) considered in [3]. This has several implications with respect to the overall contribution of [5].

Section snippets

NP-hardness of problem P₁—Section 2.1. in [5]

It can be directly derived from the NP-hardness of problem P₆ proved in [3] by showing that P₆ reduces to P₁. Here is a sketch of the proof. Let us denote by $D'$ the makespan obtained by Johnson's algorithm [2] when computing the optimal schedule for the $F 2 ∥ C_{\max}$ problem. Given an instance of P₆ (where we assume w.l.o.g. $w_{i} \geq 1$ integer, i=1, …, n), consider solving several instances of P₁ that keep the same processing times but multiplies the weights by a coefficient $γ$ with $γ \in [0, D']$ . Notice that

Final remarks

In this note we positioned the contribution of [5] with respect to some relevant literature not considered in that paper. We notice, however, that, apparently, in the scheduling literature, there has been little attention in ascertaining the (dis)-similarities between scheduling with rejection and bi-objective scheduling with common due date assignment and this is particularly unfortunate in [5] as one of the authors is the author of several other publications on due date assignment such as,

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Cited by (9)

A constraint generation approach for two-machine shop problems with jobs selection
2017, European Journal of Operational Research
Citation Excerpt :
The literature on shop scheduling problems with job rejection is reviewed by Shabtay, Gasper, and Kaspi (2013). The case of the two-machine flow shop problem has been reviewed by Shabtay and Gasper (2012) while T’kindt and Della Croce (2012) provided links with common due date problems. It is mentioned there that most two-stage shop problems with job rejection are NP-hard even with the equal job rejection cost assumption.
We consider job selection problems in two-stage flow shops and job shops. The aim is to select the best job subset with a given cardinality to minimize the makespan. These problems are known to be ordinary NP-hard and the current state of the art algorithms can solve flow shop problems with up to 3000 jobs. We introduce a constraint generation approach to the integer linear programming (ILP) formulation of these problems according to which the constraints associated with nearly all potential critical paths are relaxed and then only the ones violated by the relaxed solution are sequentially reinstated. The proposed approach is capable of solving problems with up to 100 000 jobs.
Job selection in two-stage shops with ordered machines
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We consider job selection problems in two-stage flow shops, open shops and job shops. The objective is to select the best job subset with a given cardinality to minimize either the total job completion time or the maximum job completion time (makespan). An $O (n^{2})$ algorithm is available for the two-stage flow shop with ordered machines and the minimum makespan objective; we utilize this algorithm to solve the same problem for the total job completion time objective. We then propose an $O (n^{1})$ algorithm for the two-machine open shop job selection problem with ordered machines and the minimum makespan objective. We also consider a two-machine job shop in which the first operation of each job is no longer than the second one. We show that the job selection problem in this job shop with the minimum makespan objective is ordinary NP-hard and that the problem becomes solvable in $O (n \log n)$ time with the additional assumption of ordered jobs.
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Within the wide range of due-date assignment problems, we find three major classes (see e.g. Gordon et al. [1]): (i) assigning a common due-date for all the jobs (also known as CON), (ii) assigning job-dependent due-dates which are (linear) functions of the job processing times (also known as SLK), and (iii) assigning job-dependent due-dates which are penalized if exceed pre-specified deadlines (also known as DIF). For class (i), we refer the readers to the survey papers of Baker and Scudder [2] and of Gordon et al. [1], and to more recent papers such as Shabtay and Gasper [3], Tkindt and Della Croce [4], and Drobouchevitch and Sidney [5]. Relevant references for class (ii) contain Adamopoulos and Pappis [6], Cheng [7–9], Quaddus [10], Alidaee [11], Gordon [12], Karacapilidis and Pappis [13], De et al. [14], Oğuz and Dincer [15], Cheng et al. [16], Gordon and Strusevich [17], Cheng and Kovalyov [18], Wang [19], Shabtay and Steiner [20], Janiak et al. [21], Mosheiov and Oron [22], Wang X.-Y.and Wang M.-Z. [23], and Mor and Mosheiov [24], among others.
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A note on “Two-machine flow-shop scheduling with rejection” and its link with flow-shop scheduling and common due date assignment

Abstract

Introduction

Section snippets

NP-hardness of problem P1—Section 2.1. in [5]

Final remarks

Operations Research Letters

Operations Research Letters

Theoretical Computer Science

Optimal delivery time quotation and order sequencing

Decision Sciences

NP-hardness of problem P₁—Section 2.1. in [5]