Int J Adv Manuf Technol (2011) 57:341–342DOI 10.1007/s00170-011-3276-8

ORIGINAL ARTICLE

Notes on “Single-machine schedulingwith past-sequence-dependent setup timesand effects of deterioration and learning”

Yunqiang Yin · Dehua Xu · Xiaokun Huang

Received: 22 January 2011 / Accepted: 13 March 2011 / Published online: 6 April 2011© Springer-Verlag London Limited 2011

Abstract The aim of this paper is to show by a coun-terexample that theorem 6 and corollary 5 in Wanget al. (Int J Adv Manuf Technol 41:1221–1226, 2009)are incorrect.

Keywords Scheduling · Single machine ·Learning effect · Deteriorating jobs · Setup times

1 Introduction

As we observe, theorem 6 and corollary 5 in Wang et al.[1] are incorrect. In this note, we point out these wrongresults by a counterexample.

We shall follow the notations and terminologiesgiven in Wang et al. [1]. There are given a single ma-chine and n independent and non-preemptive jobs thatare available for processing at some time t0 ≥ 0. Themachine can handle one job at a time and preemption is

Y. Yin · D. Xu (B)Key Laboratory of Radioactive Geology and ExplorationTechnology Fundamental Science for National Defense,East China Institute of Technology, Fuzhou,Jiangxi 344000, Chinae-mail: schedulingresearch@gmail.com, dhxu@ecit.edu.cn

Y. Yine-mail: yunqiangyin@gmail.com

Y. Yin · D. XuSchool of Mathematics and Information Sciences,East China Institute of Technology, Fuzhou,Jiangxi 344000, China

X. HuangDepartment of Mathematics, Honghe University,Mengzi, Yunnan 661100, China

not allowed. Let α j be the deterioration rate of job J j ina sequence. In addition, let pA

[k] be the actual processingtime of a job if it is scheduled in the kth position ina sequence. The actual processing time of job J j ifit is started at time t and scheduled in position r isgiven by:

pAjr (t) = α j(b + ct)ra, r, j = 1, 2, · · · , n, (1)

where a ≤ 0 is a constant learning effect. Moreover, thep-s-d setup time of job J[r] if it is scheduled in positionr is given by:

s[1] = 0 and s[r] = dr−1∑

i=1

pA[i],

where d ≥ 0 is a normalizing constant. For convenience,denote by spsd as the p-s-d setup.

Let N denote the set of jobs already scheduled, Ne

be the set of jobs already considered for schedulingbut having been discarded because they will not meettheir due dates in the optimal schedule, and N f denotethe set of jobs not yet considered for scheduling. Theproblem 1|| ∑ U j is known to be solved by Moore’salgorithm [2] as follows:

Wang et al. [1] gave the following results.

Theorem 1′ (Wang et al. [1], theorem 6) For the prob-lem 1|pA

jr (t) = α j(b + ct)ra, spsd| ∑ U j, if the jobs haveagreeable conditions, i.e., α j ≤ αk implies d j ≤ dk forall the jobs J j and Jk, then an optimal schedule can beobtained by Moore’s Algorithm.

Corollary 1′ (Wang et al. [1], Corollary 5) For the prob-lem 1|pA

jr (t) = α j(b + ct)ra, spsd, d j = θα j| ∑ U j, an op-timal schedule can be obtained by Moore’s Algorithm.

342 Int J Adv Manuf Technol (2011) 57:341–342

Algorithm 1 Moore’s AlgorithmStep 1: Order the jobs in non-decreasing order of d j

(the earliest due date (EDD) rule).

Step 2: If no jobs in the sequence are late, stop. Theschedule is optimal.

Step 3: Find the first late job in the schedule. Denotethis job by Ju.

Step 4: Find a job Jv with αv = maxi=1,2,··· ,u αi. Removejob Jv from the schedule and process it after thecompletion of all the jobs that were processed.Go to Step 2.

Table 1 Data of problem set Job J j J1 J2 J3

α j 10 20 40d j 9 18 36

Table 2 The processing information for the sequenceJ1–J2–J3

Job J j J1 J2 J3

pAjr 10.0000 7.0711 6.9464

C j 10.0000 17.0711 24.0175U j 1 0 0

Table 3 The processinginformation for the sequenceJ2–J3

Job J j J2 J3

pAjr 20.0000 21.2132

C j 20.0000 41.2132U j 1 1

Table 4 The processinginformation for the sequenceJ3

Job J j J3

pAjr 40.0000

C j 40.0000U j 1

2 Counterexample

The following counterexample shows that theorem 6and corollary 5 in Wang et al. [1] is incorrect.

Counterexample Let t0 = 0, a = −2.5, b = 1, c = 0.1and d = 0. Consider a problem containing n = 3 jobs,as described in Table 1.

In this counterexample d j = 0.9α j for any job J j, andthe jobs are already indexed by EDD, as required instep 1 of the Moore’s Algorithm. In step 3, job J1 isfound to be the first late job (see Table 2). In step 4, jobJ1 is removed from N and placed in Ne. In the next passat steps 3 and 4, job J2 is removed from N and placed inNe (see Table 3). In the next pass at steps 3 and 4, jobJ3 is removed from N and placed in Ne (see Table 4).

Therefore, Ne = {J1, J2, J3}. According to Moore’sAlgorithm, any processing sequence is an optimalschedule. However, it can be checked that the objectivevalue of the processing sequence J2–J3–J1 is 3 and thatthe objective value of the processing sequence of J1–J2–J3 is 1, a contradiction. It follows that Corollary 5 inWang et al. [1] is incorrect. Since corollary 5 is a specialcase of theorem 6 in Wang et al. [1]. Therefore, theorem6 in Wang et al. [1] is not true as well.

Acknowledgements This research was supported in part bythe Natural Science Foundation of Jiangxi (2010GQS0003), theScience Foundation of Education Committee of Jiangxi forYoung Scholars (GJJ11143, GJJ11144), and the Doctor Founda-tion of East China Institute of Technology.

References

1. Wang J-B, Jiang Y, Wang G (2009) Single-machine schedulingwith past-sequence-dependent setup times and effects of dete-rioration and learning. Int J Adv Manuf Technol 41:1221–1226

2. Moore J (1968) An n job, one machine sequencing algorithmfor minimizing the number of late jobs. Manage Sci 15:102–109