Int J Adv Manuf Technol (2012) 58:763–764DOI 10.1007/s00170-011-3432-1

ORIGINAL ARTICLE

Note to: Deteriorating jobs and learningeffects on a single-machine schedulingwith past-sequence-dependent setup times

Xue Huang

Received: 1 February 2011 / Accepted: 16 March 2011 / Published online: 30 June 2011© Springer-Verlag London Limited 2011

Abstract Some results in a recent paper (Yin et al., IntJ Adv Manuf Technol 46:707–714, 2010) are incorrect.In this note, we show by a counterexample that theresults are incorrect.

Keywords Scheduling · Single machine ·Learning effect · Deteriorating jobs

1 Introduction

The recent paper “Single-machine scheduling withgeneral learning functions” [1] addresses the single-machine past-sequence-dependent (p-s-d) setup timesscheduling problems with deteriorating jobs and learn-ing effects. The authors showed studied single-machinescheduling problems with p-s-d setup times and the ef-fects of learning and deterioration. We proved that themakespan minimization problem, the total completiontime minimization problem, and the sum of the δth(δ ≥ 0) power of job completion times minimizationproblem can be solved by the SPT rule, respectively.We also proved that some special cases of the to-tal weighted completion time minimization problem,the maximum lateness minimization problem, and thenumber of tardy jobs minimization problem can besolved in polynomial time. In this note, we will givea counterexample to show the incorrectness of someresults in Yin et al. [1].

X. Huang (B)School of Science, Shenyang Aerospace University,Shenyang 110136, Chinae-mail: huangxuebj@126.com

We shall follow the notations and terminologiesgiven in Yin et al. [1]. There are given a single machineand n independent and non-preemptive jobs that areimmediately available for processing. The machine canhandle one job at a time and preemption is not allowed.Let pj be the normal processing time of job J j in asequence. In addition, let p[k] and pA

[k] be the normalprocessing time and actual processing time of a jobif it is scheduled in the kth position in a sequence,respectively. The actual processing time of job J j if itis scheduled in position r is given by:

pAjr = pj

(p0 + ∑r−1

l=1 p[l]p0 + ∑n

l=1 pl

)a1

ra2 , r, j = 1, 2, . . . , n, (1)

where p0 is a given parameter, a1 < 0 is a constantdeterioration rate, a2 < 0 is a constant learning rate,and

∑0i=1 p[i] := 0. On the other hand, we assume that

the p-s-d setup time of job J[r] if it is scheduled inposition r is given by:

s[1] = 0 and s[r] = br−1∑i=1

pA[i], (2)

where b ≥ 0 is a normalizing constant. For conve-nience, we denote by spsd the p-s-d setup given by Eq. 2[2–4].

For a given schedule, π = [J1, J2, . . . , Jn], C j =C j(π) represents the completion time of job J j. Let∑

U j, where U j = 1 if C j > d j (i.e., the job is late) andU j = 0 otherwise, j = 1, 2, . . . , n, represent the numberof tardy jobs of a given permutation, where d j is the duedate of job J j.

764 Int J Adv Manuf Technol (2012) 58:763–764

2 A counterexample

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 [5] as follows:

Moore’s Algorithm

Step 1 Order the jobs in non-decreasing order of d j

(the earliest due date rule).Step 2 If no jobs in the sequence are late, stop. The

schedule is optimal.Step 3 Find the first late job in the schedule. Denote

this job by Jμ.Step 4 Find a job Jν with pν = maxi=1,2,...,μ pi. Re-

move job Jν from the schedule and process itafter the completion of all the jobs that wereprocessed. Go to Step 2.

Yin et al. [1] gave the following result.

Theorem 1′ (Theorem 6, [1]) For the problem 1|pAjr =

pj

(p0+∑r−1

l=1 p[l]p0+∑n

l=1 pl

)a1

ra2 , spsd| ∑ U j, if the jobs have agree-

able conditions, i.e., p j ≤ pk implies d j ≤ dk for all thejobs J j and Jk, then an optimal schedule can be obtainedby Moore’s algorithm.

Corollary 1′ (Corollary 6, [1]) For the problem 1|pAjr =

pj

(p0+∑r−1

l=1 p[l]p0+∑n

l=1 pl

)a1

ra2 , spsd, d j = θpj| ∑ U j, an optimal

schedule can be obtained by Moore’s Algorithm.

The following example shows that Theorem 1′ andCorollary 1′ are incorrect.

Counterexample 1 n = 3, p0 = 1, p1 = 10, p2 = 21,

p3 = 32, d1 = 7, d2 = 18, d3 = 29. When a1 = −0.01,

a2 = −1.5 and b = 0. By the Moore’s Algorithm,we know that Jd = {J1, J2, J3}, hence any sequenceis an optimal schedule. However, if the sequence is[J1, J2, J3], ∑

U j = 1 and if the sequence is [J3, J2, J1],∑U j = 3. Therefore, Theorem 6 in Yin et al. [1] is not

correct.Note that the problem 1|pA

jr = pj

(p0+∑r−1

l=1 p[l]p0+∑n

l=1 pl

)a1

ra2 ,

spsd, d j = θpj| ∑ U j is a special case of the problem

1|pAjr = pj

(p0+∑r−1

l=1 p[l]p0+∑n

l=1 pl

)a1

ra2 , spsd, pi ≤ pj ⇒ di ≤d j| ∑ U j. Therefore, Corollary 6 in Yin et al. [1] is alsonot correct.

Acknowledgments This research was supported by the Na-tional Natural Science Foundation of China (grant no. 11001181)and the Science Research Foundation of Shenyang AerospaceUniversity (grant no. 201011Y).

References

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2. Koulamas C, Kyparisis GJ (2008) Single-machine schedulingproblems with past-sequence-dependent setup times. Eur JOper Res 187:68–72

3. Wang J-B (2008) Single-machine scheduling with past-sequence-dependent setup times and time-dependent learningeffect. Comput Ind Eng 55:584–591

4. Wang J-B, Jiang Y, Wang G (2009) Single-machine schedul-ing with past-sequence-dependent setup times and effects ofdeterioration and learning. Int J Adv Manuf Technol 41:1221–1226

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