makespan minimization flowshop with position dependent job processing times—computational...
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Computers & Operations Research 40 (2013) 2071–2082
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Computers & Operations Research
0305-05http://d
n CorrE-m
radoslaw
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Makespan minimization flowshop with position dependent jobprocessing times—computational complexity and solution algorithms
Agnieszka Rudek a, Radosław Rudek b,n
a Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Polandb Wrocław University of Economics, Komandorska 118/120, 53-345 Wrocław, Poland
a r t i c l e i n f o
Available online 15 March 2013
Keywords:SchedulingAging effectLearning effectComputational complexityHeuristic
48/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.cor.2013.02.024
esponding author. Tel.: þ48 71 368 0378; faxail addresses: [email protected]@ue.wroc.pl, [email protected]
a b s t r a c t
This paper considers flowshop scheduling problems where job processing times are described byposition dependent functions, i.e., dependent on the number of processed jobs, that model learning oraging effects. We prove that the two-machine flowshop problem to minimize the maximum completiontime (makespan) is NP-hard if job processing times are described by non-decreasing position dependentfunctions (aging effect) on at least one machine and strongly NP-hard if job processing times are varyingon both machines. Furthermore, we construct fast NEH, tabu search with a fast neighborhood search andsimulated annealing algorithms that solve the problem with processing times described by arbitraryposition dependent functions that model both learning and aging effects. The efficiency of the proposedmethods is numerically analyzed.
& 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Although over 50 years have passed since Johnson [17] publishedthe first paper on flowshop scheduling, it is still topical issue amongresearchers. Nevertheless, due to rigorous constraints, classical flow-shop scheduling problems are perceived to be more interesting in atheoretical context than as a practical research (see [13]). It followsfrom observations that algorithms constructed on the basis of theclassical models usually provide unsatisfactory (unstable) solutionsfor real-life flowshop problems, since these models do not take intoconsideration additional factors that are significant in practice, e.g.,variable parameters [4,6,10,14,22,23,36,43]. Recently, a particularattention has been attracted by a group of problems, where jobprocessing times change with the number of previously processedjobs (e.g., [24,26,41,46]).
If the time required to process a job decreases with the numberof processed jobs, then such phenomenon is called the learningeffect. It is typical for human activity environments or for auto-matized manufacturing, where a human support for machines isneeded during activities such as operating, controlling, setup,cleaning, maintaining, failure removal, etc. Although learning cancease with time, it is often aroused by such factors as newinexperienced employees, extension of an assortment, newmachines, more refined equipment, software update or general
ll rights reserved.
: þ48 71 368 0376.(A. Rudek),m (R. Rudek).
changes of the production environment [1]. However, highlyautomatized manufacturing systems may also benefit on the factthat if a machine does the same task repetitively, then theknowledge from the previous iterations can be used to improvethe performance of a system when the task is processed the nexttime (e.g., [2]). Furthermore, machine learning methods, forinstance reinforcement learning algorithms, that learn and operateon-line (see [39]), improve their efficiency on the basis of inter-actions with an environments (learning-by-doing). Thus, theperformances of the systems optimized by such algorithms areimproved with the number of their iterations (e.g., [3,16]). For asurvey on scheduling problems with learning see [1,15].
On the other hand, if the times required to process jobsincrease with the number of processed jobs, then it is called theaging or position dependent deteriorating effect. The problemswith this phenomenon are present in many manufacturing sys-tems, in which tiredness of human workers (e.g., [7]), tool wear oflathe machines (e.g., [32]) or decreasing activity of some chemicalsubstances (e.g., [25]) affect the production output. More detailsconcerning scheduling problems with the aging effect can befound in [9,21,44,45,47].
Although the makespan minimization flowshop problem withposition dependent processing times (modelling learning or agingeffects) was discussed (e.g., [23,38]), its computational complexitystatus for the two-machine case is still an open issue even forarbitrary aging characteristics. Therefore, the first main contribu-tion of this paper is to prove that the two-machine flowshopproblem to minimize makespan that is known to be polynomiallysolvable (see [17]) becomes NP-hard if job processing times are
A. Rudek, R. Rudek / Computers & Operations Research 40 (2013) 2071–20822072
position dependent non-decreasing functions (aging effect) on onemachine and strongly NP-hard if the aging effect concerns twomachines.
Furthermore, there is a lack of efficient approximation algo-rithms for the makespan minimization flowshop problem withlearning or aging effects (see [6,37,40,42]). Therefore, we contri-bute the field by providing fast heuristic and metaheuristicmethods that solve the general version of the problem, i.e., witharbitrary position dependent job processing times that candescribe both learning and aging effects. Namely, we constructthe fast NEH algorithm that is significantly faster than thestandard NEH (e.g., [6]). Moreover, we propose the fast neighbor-hood search that is much faster than the standard approach andcan be used for tabu search and other metaheuristic algorithmssolving similar problems.
To the best of our knowledge, the results presented in thispaper have never been investigated in the scheduling domain.
This paper is organized as follows. Section 2 contains problemformulation and the computational status is established in Section 3.Subsequently, description of approximation algorithms is provided,followed by an experimental verification of their efficiency. The lastsection concludes the paper.
2. Problem formulation
In this paper, we investigate flowshop problems with the agingeffect. These problems can be defined as follows. There are given aset J ¼ f1,…,ng of n jobs and m machines, namely M¼ fM1,…,Mmg.Each job j consists of a set O¼ fO1,j,…,Om,jg of m operations. Eachoperation Oz,j has to be processed on machine Mz (z¼ 1,…,m).Moreover, operation Ozþ1,j may start only if Oz,j is completed. If itis assumed that machines have to process jobs in the same order,then the problem is called a permutation flowshop and suchproblems are considered in this paper. It is also assumed that eachmachine can process one operation at a time, and there areno precedence constraints between jobs. Operations are non-preemptive and available for processing at time 0 on M1. Further,instead of operation Oz,j, we say job j on machine Mz.
The processing time pðzÞj ðvÞ of job j that is scheduled in positionv in a sequence on machine Mz (z¼ 1,…,m) is a positive functiondependent on its position v (i.e., the number of previouslymachined products, v−1). The function pj(v) models learning oraging effects if it is non-increasing or non-decreasing, respectively.Moreover, each job j is characterized by its normal processing timeaðzÞj on machine Mz that is defined as the time required to process ajob if the machine is not influenced by aging/learning, i.e.,aðzÞj ≜pðzÞj ð1Þ for z¼ 1,…,m and j¼1,…,n.
For the m-machine permutation flowshop problems the sche-dule of jobs on the machines can be unambiguously defined bytheir sequence (permutation) π. Thus, for each job πðiÞ, i.e.,scheduled in the ith position in π, we can determine its completiontime CðzÞ
πðiÞ on machine Mz
CðzÞπðiÞ ¼maxfCðz−1Þ
πðiÞ ,CðzÞπði−1ÞgþpðzÞπðiÞðiÞ, ð1Þ
where Cð0Þπð1Þ ¼ CðzÞ
πð0Þ ¼ 0 for z¼ 1,…,m and Cð1ÞπðiÞ ¼∑i
l ¼ 1pð1ÞπðlÞðlÞ is the
completion time of a job placed in position i in the permutation πon M1.
The objective is to find such a schedule π of jobs on themachines that minimizes the maximum completion time (make-span) Cmax≜maxj∈JfCjg (in the considered problems Cmax ¼ CðmÞ
πðnÞ).The m-machine flowshop problem with non-decreasing (onlyaging), non-increasing (only learning) and arbitrary (learning/aging) processing times, according to the three-field notationscheme XjY jZ, will be denoted as FmjAEjCmax, FmjLEjCmax andFmjpðzÞj ðvÞjCmax, respectively.
3. Computational complexity
In this section, we will prove the strong NP-hardness of theconsidered problem with the aging effect F2jAEjCmax and also NP-hardness of its special case. The proofs are based on similarobservations as in [29].
3.1. NP-hardness
In this part, we will prove that the makespan minimizationproblem with the aging effect F2jAE,pð1Þj ðvÞ ¼ að1Þj jCmax is at leastNP-hard even if job processing times on the first machine areconstant (i.e., it does not deteriorate).
Theorem 1. The problem F2jAE,pð1Þj ðvÞ ¼ að1Þj jCmax is NP-hard even ifjob processing times are constant on the first machine (i.e., thismachine does not deteriorate).
Proof. To prove this theorem, we transform EQUAL CARDINALITY
PARTITION problem, that is known to be NP-complete, to the decisionversion of the considered scheduling problem with aging. At firstthe definition of EQUAL CARDINALITY PARTITION problem is given.EQUAL CARDINALITY PARTITION (ECP) [8]: There are given positive
integers t, B and x1,…,x2t of 2t positive integers satisfying∑2t
q ¼ 1xq ¼ 2B for q¼ 1,…,2t. Does there exist a partition of the setX ¼ f1,…,2tg into two disjoint subsets X1 and X2 such that∑q∈X1xq ¼∑q∈X2xq ¼ B and jX1j ¼ jX2j ¼ t?The decision version of the problem F2jAE,pð1Þj ðvÞ ¼ að1Þj jCmax
(Decision problem with the Aging Effect on one machine, DAE1)is defined as follows: Does there exist such a schedule π of jobs onthe machines for which the criterion value Cmax is not greater thanthe given value y?The transformation from ECP to DAE1 is constructed as follows.
The instances of DAE1 contain 2t partition jobs (j¼ 1,…,2t)constructed on the basis of the elements from the set X of ECPand one enforcer job e that is an auxiliary (additional) job, whichforces the optimal schedules such that a sequence of jobs isimportant (i.e., jobs are scheduled before and after e). The para-meters of the partition jobs (j¼ 1,…,2t) are given as follows:
pð1Þj ðvÞ ¼ að1Þj ¼ 1 for v¼ 1,…,2tþ1,
pð2Þj ðvÞ ¼tBþ2txj for v¼ 1,…,t,3tBþtxj for v¼ tþ1,…,2tþ1,
(
where j¼ 1,…,2t, and the parameters of the enforcer job e are
pð1Þe ðvÞ ¼ að1Þe ¼ 1þtBðtþ2Þ−t for v¼ 1,…,2tþ1,
pð2Þe ðvÞ ¼tB for v¼ 1,…,tþ1,4tBðtþ2Þ for v¼ tþ2,…,2tþ1,
(
and y¼ 1þ4tBðtþ1Þ and n¼ 2tþ1. Thus, we analyze step functionsof job processing times. It is trivial to show that DAE1 belongs to theNP class and the given transformation is polynomial.Now we will establish the position of the enforcer job e in an
optimal schedule π for the constructed instances of DAE1.Observed that if the enforcer job e is scheduled in positionve > tþ1, then pð2Þe ðveÞ ¼ 4tBðtþ2Þ > y. On the other hand, ifve < tþ1, then
Cmax > ∑ve−1
i ¼ 1pð1ÞπðiÞ það1Þe þpð2Þe ðveÞþ ∑
n
i ¼ ve þ1pð2ÞπðiÞðiÞ
> tBðtþ2ÞþtBþð2tþ1−veÞ3tB¼ tBð7tþ6−3veÞ > tBð4tþ6Þ > y:
A. Rudek, R. Rudek / Computers & Operations Research 40 (2013) 2071–2082 2073
Therefore, the enforcer job e occupies position tþ1, otherwiseCmax > y.Let J1 and J2 denote the subsets containing the jobs of DAE1 that
are constructed on the basis of the elements of ECP from thesubsets X1 and X2, respectively. Thus, from the above considera-tions follows that in the optimal solution π of the constructedinstances of DAE1 the machines execute at first the jobs from thesubset J1 in an arbitrary order, next the enforcer job e and finallythe jobs from the subset J2 in an arbitrary order as well. On thisbasis, the criterion value for DAE1 and given π can be calculated asfollows:
Cmax ¼max Cð1Þe ,pð1Þπð1Þ þ ∑
t
i ¼ 1pð2ÞπðiÞðiÞ
( )þpð2Þe ðtþ1Þþ ∑
2tþ1
i ¼ tþ2pð2ÞπðiÞðiÞ, ð2Þ
where Cð1Þe ¼ 1þtBðtþ2Þ.
Now we will show the answer for the constructed instances ofDAE1 is yes (i.e., Cmax≤y) if and only if it is yes for ECP (i.e.,∑q∈X1xq ¼∑q∈X2xq ¼ B and jX1j ¼ jX2j ¼mÞ.“Only if.” Assume that the answer for ECP is yes, thus, there
exists such a partition of the set X into two disjoint subsets X1
and X2 for which ∑q∈X1xq ¼∑q∈X2xq ¼ B and jX1j ¼ jX2j. Therefore,the makespan Cmax is equal to y as it is required, Cmax ¼ 1þ4tBðtþ1Þ ¼ y.“If.” Assume now that the answer for ECP is no, i.e., any partition
of the set X into two disjoint subsets X1 and X2 gives∑q∈X1xq≠∑q∈X2xq or jX1j≠jX2j. If jX1j≠jX2j, then from the propertyof an optimal solution π of DAE1 follows that Cmax > y.Let ∑q∈X1xq ¼ B−λ and ∑q∈X2xq ¼ Bþλ, where 0 < λ≤B is a non-
zero integer. Based on (2) we have
Cmax ¼maxf1þtBðtþ2Þ,1þtBðtþ2Þ−2tλgþtBð3tþ2Þþtλ¼ yþtλ > y,
otherwise (i.e., ∑q∈X1xq ¼ Bþλ and ∑q∈X2xq ¼ B−λ)
Cmax ¼maxf1þtBðtþ2Þ,1þtBðtþ2Þþ2tλgþtBð3tþ2Þ−tλ¼ yþtλ > y:
Thus, for both cases Cmax > y, thereby the answer for DAE1 is no.Therefore, DAE1 has an answer yes if and only if it is yes for
ECP. It means that DAE1 is at least NP-complete, therebyF2jAE,pð1Þj ðvÞ ¼ að1Þj jCmax is at least NP-hard. □
3.2. Strong NP-hardness
Now we will show that if job processing times on all machinesare described by non-decreasing position dependent functions(both machines deteriorate), then the considered problem,F2jAEjCmax, is strongly NP-hard.
Theorem 2. The problem F2jAEjCmax is strongly NP-hard.
Proof. At first 3-PARTITION problem, which is known to be stronglyNP-complete, is defined.3-PARTITION (3PP) [8]: There are given positive integers t, B and
x1,…,x3t of 3t positive integers satisfying ∑3tq ¼ 1xq ¼ tB and B=4 <
xq < B=2 for q¼ 1,…,3t. Does there exist a partition of the setX ¼ f1,…,tg into t disjoint subsets X1,…,Xt such that ∑q∈Xi xq ¼ Bfor i¼ 1,…,t?The decision version of the problem F2jAEjCmax (Decision
problem with the Aging Effect on two machines, DAE2) is definedas follows: Does there exist such a schedule π of jobs on the machinesfor which Cmax≤y?
On this basis, a pseudopolynomial time transformation from 3PPto DAE2 is constructed such that the instances of DAE2 contain 3tpartition jobs (constructed on the basis of the elements from theset X of 3PP) and t enforcer jobs (which force the optimal schedulessuch that a sequence of jobs is important, i.e., the partition jobs arescheduled between the enforcer jobs). The parameters of thepartition jobs are defined as follows:
pð1Þj ðvÞ ¼ að1Þj ¼ A¼ 1 for v¼ 1,…,4t,
pð2Þj ðvÞ ¼
Bþtxj for v¼ 1,…,4⋮iBþðt−iþ1Þxj for v¼ 4ði−1Þþ1,…,4i⋮tBþxj for v¼ 4ðt−1Þþ1,…,4t,
8>>>>>><>>>>>>:
where j¼ 1,…,3t, i¼ 1,…,t and for the enforcer jobs:
pð1ÞeiðvÞ ¼
3ði−1ÞBþðt−iþ2ÞB for v¼ 1,…,4ði−1Þþ12y for v¼ 4ði−1Þþ2,…,4t,
(
pð2Þei ðvÞ ¼ að2Þei ¼ 3A for v¼ 1,…,4t,
where i¼ 1,…,t and y¼∑ti ¼ 1½3ði−1ÞBþðt−iþ2ÞB�þ3tAþð3tþ1ÞB
and n¼4t. Observe that we analyze stepwise functions of jobprocessing times.To prove the strong NP-completeness of DAE2 the transforma-
tion has to be pseudopolynomial. Note that it is fulfilled sinceDAE2 belongs to NP class, the problem size of DAE2 does notdecrease exponentially in reference to 3PP (n¼4t), the maximalvalue of DAE2 does not increase exponentially in reference to 3PP(i.e., O(tB)), each value of DAE2 can be calculated in pseudopoly-nomial time (depends polynomially on t and B) and each para-meter of DAE2 has an integer value.Observe also that the high job processing times of the con-
structed instance of DAE2 (i.e., pð1Þei ðvÞ ¼ 2y if v > 4ði−1Þþ2)describe cases when, for example, the current durability and thequality of the tool (due to the tool wear caused by previouslyprocessed jobs) is not efficient to process the given job. In suchcase the tool has to be exchanged to prevent its breakdown,however, it usually increases significantly the processing time.Therefore, the objective is to determine such a schedule thatallows one to process the given batch of jobs but withoutadditional and time consuming operations such as tool replace-ment, i.e., in other words it allows one to efficiently exploitthe tool.Let J ¼ J1∪J2∪⋯∪Jt denote the set containing indices of all
partition jobs, where Ji is a set of indices of jobs processed betweenthe enforcer jobs ei and eiþ1 for i¼ 1,…,t−1 and Jt contains jobsscheduled after et. Moreover, ⟨j⟩ denotes a position of job j in asequence. At first, note that in the optimal schedule the enforcerjobs are sequenced according to the increasing order of theirindexes ei and position vi of each ei cannot be greater than4ði−1Þþ1, otherwise Cmax > pð1Þei ðviÞ > y for i¼ 1,…,t. Next, wewill show that each enforcer job ei occupies exactly positionvi ¼ 4ði−1Þþ1, i.e., jJij ¼ 3 for i¼ 1,…,t.Consider an arbitrary schedule, where jJt j < 3. It results that the
position of et is at least 4ðt−1Þþ2, thus, Cmax > y. On the otherhand, if jJt j ¼ 3þu and u > 0, then the completion time of et on M2
A. Rudek, R. Rudek / Computers & Operations Research 40 (2013) 2071–20822074
is equal to
Cð2Þet ¼ ∑
t
i ¼ 1pð1Þei
ðiÞþ ∑j∈J⧹Jt
að1Þj það2Þet ¼ y−uA−ð3tþ1ÞB,
and the criterion value Cmax is
Cmax≥Cð2Þet þ∑
j∈Jtpð2Þj ð⟨j⟩Þ > y−uA−ð3tþ1ÞBþ4tB > y:
Consider now an arbitrary schedule, where jJt j ¼ 3 (otherwiseCmax > y) and jJt−1j ¼ 3þu, where u > 0. The criterion value can beestimated similarly as in the previous case:
Cmax≥Cð2Þet−1 þ ∑
j∈Jt−1pð2Þj ð⟨j⟩Þþað2Þet þ∑
j∈Jtpð2Þj ð⟨j⟩Þ > yþðt−1ÞB−uA > y:
Following this way and using inductive principle, we can easilyprove that jJij ¼ 3 for i¼1,…,t in the optimal schedule of theconsidered instances of DAE2. Thus, Ji denotes a set of indices ofjobs, but from now on, constructed on the basis of elements fromXi for i¼ 1,…,t. Hence, for convenience, instead of Ji, we will use Xi.The schedule π for DAE2 can be given as follows ðe1,X1,e2,X2,
e3,…,ei,Xi,eiþ1,Xiþ1,…,et ,XtÞ, where the sequence of jobs withineach set Xi is arbitrary, since it does not affect the objectivefunction. Thus, in the further part, we only consider schedulesconsistent with π.Based on (1), the completion times of the enforcer jobs onM1 for
the schedule π are as follows:
Cð1Þei
¼ ∑i
l ¼ 1½3ðl−1ÞBþðt−lþ2ÞB�þ3ði−1ÞA
¼ Cð1Þei−1
þ3Aþ3ði−1ÞBþðt−iþ2ÞB, ð3Þ
for i¼ 1,…,t−1 and
Cð1Þet ¼ y−3A−ð3tþ1ÞB: ð4Þ
Let CðzÞXi
denote the completion time of the last job in Xi on Mz forthe schedule π (i¼1,…,t) that is equal to
Cð1ÞXi
¼ Cð1Þei
þ ∑j∈Xi
að1Þj ¼ Cð1Þei
það2Þei
for i−1,…,t−1. Note that the first job j in Xi (for i¼ 1,…,t) starts onM2 at maxfCð1Þ
j ,Cð2Þei g, since Cð1Þ
j ≤Cð1ÞXi
¼ Cð1Þei þpð2Þei ≤C
ð2Þei , thereby it
starts at Cð2Þei . Thus, the completion times on M2 for enforcer and
partition jobs are
Cð2Þe1 ¼ ðtþ1ÞBþ3A¼ Cð1Þ
e1 þ3A,
Cð2ÞX1
¼ Cð2Þe1 þ3Bþt ∑
j∈X1
xj,
Cð2Þe2 ¼maxfCð1Þ
e2 ,Cð2ÞX1gþ3A,
Cð2ÞX2
¼ Cð2Þe2 þ6Bþðt−1Þ ∑
j∈X2
xj,
Cð2Þei ¼maxfCð1Þ
ei ,Cð2ÞXi−1
gþ3A, ð5Þ
Cð2ÞXi
¼ Cð2Þei
þ3iBþðt−iþ1Þ∑j∈Xi
xj, ð6Þ
for i¼ 1,…,t, where Cð2ÞX0
¼ 0.Now we will show that the answer for the constructed instances
of DAE2 is yes (i.e., Cmax≤y) if and only if it is yes for 3PP (i.e.,∑q∈Xi
xq ¼ B for i¼1,…,t).
“Only if.” Assume that the answer for 3PP is yes. Thus, for eachsubset Xi (i¼1,…,t) ∑q∈Xi
xq ¼ B holds. Based on (3), (5) and (6), thecompletion times of jobs for the schedule π on M2 are as follows:
Cð2Þe1 ¼ ðtþ1ÞBþ3A¼ Cð1Þ
e1 þ3A,
Cð2ÞX1
¼ Cð2Þe1 þ3BþtB¼ Cð1Þ
e1 þ3Aþ3BþtB¼ Cð1Þe2 ,
Cð2Þe2 ¼maxfCð1Þ
e2 ,Cð2ÞX1gþ3A¼ Cð1Þ
e2 þ3A,
Cð2ÞX2
¼ Cð2Þe2 þ6Bþðt−1ÞB¼ Cð1Þ
e2 þ3Aþ6Bþðt−1ÞB¼ Cð1Þe3 :
Concluding in this way, we can show that
Cð2Þei
¼maxfCð1Þei,Cð2Þ
Xi−1gþ3A¼ Cð1Þ
eiþ3A,
Cð2ÞXi
¼ Cð2Þei þ3iBþðt−iþ1ÞB¼ Cð1Þ
ei þ3Aþ3iBþðt−iþ1ÞB¼ Cð1Þeiþ 1
,
for i¼ 1,…,t. Since Cð2Þei ¼ Cð1Þ
ei þ3A and Cð2ÞXi
¼ Cð1Þeiþ 1
, thenCmax ¼ Cð1Þ
et þ3Aþð3tþ1ÞB¼ y.“If.” Assume now that the answer for 3PP is no. Thus, there is no
partition of the set X such that ∑q∈Xixq ¼ B holds for all i¼ 1,…,t;
note that jXij ¼ 3 for i¼1,…,t (follows from the property of theoptimal schedule) regardless of the partition of 3PP.Let ∑q∈Xi
xq ¼ Bþλi for i¼ 1,…,t and from the assumption of 3PPB=4 < xq < B=2 (for q¼ 1,…,3t) follows that ð3=4ÞB < ∑q∈Xi
xq <ð3=2ÞB, thereby λi∈ð−B=4,B=2Þ.Since the answer for 3PP is no, thus, for any partition of the set
f1,…,3tg into disjoint subsets X1,…,Xt , there must exist at leasttwo subsets Xu and Xw (u≠w) such that ∑q∈Xuxq≠∑q∈Xwxq foru,w∈f1,…,tg and u < w.At first, let us present the idea of the proof, which is similar to
[29]. It is sufficient (from the perspective of the criterion value andthe relation Cmax > y, taking into consideration the given transfor-mation) to consider only two types of cases, since any distributionof λi (following the partition of jobs) can be reduced to them(without increasing the criterion value). Note that such cases donot have to be sufficient for different types of problems. Theparameters of jobs for the transformation from 3PP to DPAE wereobtained using similar approach as in [29], based on the followingcases (in this paper, we have simplified their description):
(a)
λu < 0, λw > 0 and ∑il ¼ uλl < 0 for i¼ u,…,w−1,(b)
λu > 0, λw < 0 and ∑il ¼ uλl > 0 for i¼ u,…,w−1,where u,w∈f1,…,mg and u < w and λi ¼ 0 for i∈f1,…,u−1g∪fwþ1,…,mg, thereby ∑w
l ¼ uλl ¼ 0.Note that in [29], we analyzed the following. Namely, consider
case (a) and assume that Xw is the first one such that ∑wl ¼ uλl < 0
does not hold (i.e., ∑wl ¼ uλl ¼ 0) and if ∑w
i ¼ uλiþλwþ1 > 0 (i.e.,λwþ1 > 0), then there must exist such v > wþ1, for which λv < 0and ∑v
i ¼ wþ1λi≤0, but this is represented by case (b). By this part,we wanted to say nothing more than that (a) and (b) can beanalyzed separately for searching the transformation and duringthe proving process (although it can be perceived as a redundantinformation).Although (a) and (b) are sufficient to prove the strong NP-
hardness (due to the contribution to the criterion value), i.e., toshow that Cmax > y, they may be less formal (see [29]) andsomehow confusing. Therefore, we will provide calculations formore general (formal) cases, which are given as follows:
ða′Þ
λw > 0,
A. Rudek, R. Rudek / Computers & Operations Research 40 (2013) 2071–2082 2075
ðb′Þ
λu > 0 and λw < 0 and ∑il ¼ uλl > 0 for i¼ u,…,w−1 (and∑wl ¼ uλl≥0),
where for both λi ¼ 0 for i¼wþ1,…,t and 1≤u < w≤t. Note thatthe calculations for the proof in [29] are also relevant for theseabove cases (a0) and (b0).Observe that ða′Þ covers cases, where λw > 0 is the first (starting
from t towards 1) non-zero element; the values of λi for i < w arearbitrary. On the other hand, ðb′Þ covers other cases. Namely, λw <
0 is the first (starting from t towards 1) non-zero element and λu isthe first element (starting from t towards 1) such that ∑w
l ¼ uλl≥0.Thus, λu > 0 and ∑i
l ¼ uλi > 0 for i¼ u,…,w−1; the values of λi fori < u are arbitrary. Therefore, ða′Þ and ðb′Þ cover all possibledistributions of λi for i¼ 1,…,t.Nevertheless, after the calculations, we will show that for
holding Cmax > y the analyzed cases ða′Þ and ðb′Þ may be reducedto cases (a) and (b). Namely, if Cmax > y for cases described by(a) and (b), it still holds for all possible distributions of λi (i.e.,general (a0) and (b0)). Furthermore, for the clarity of presentation,we will also provide the transition steps of calculating the criterionvalue for cases ða′Þ and ðb′Þ.Consider case ða′Þ. Based on (5), note that Cð2Þ
ew≥Cð1Þew þ3A regard-
less of λi for i < w. Therefore, the completion time of the last job inXw on M2 can be estimated
Cð2ÞXw
¼ Cð2Þew þ3wBþðt−wþ1ÞðBþλwÞ
≥Cð1Þew þ3Aþ3wBþðt−wþ1ÞðBþλwÞ
¼ Cð1Þewþ 1
þðt−wþ1Þλw > Cð1Þewþ 1
:
Since λw > 0 and λi ¼ 0 for i¼wþ1,…,t, then Cð2ÞXi≥Cð1Þ
eiþ 1þ
ðt−wþ1Þλw > Cð1Þeiþ 1
(for i¼w,…,t−1) and Cð2Þet ≥C
ð1Þet þ3Aþ
ðt−wþ1Þλw. Based on (4) and (6), we have
Cmax ¼ Cð2Þet þð3tþ1ÞB≥Cð1Þ
et þ3Aþðt−wþ1Þλwþð3tþ1ÞB¼ yþðt−wþ1Þλw > y:
Consider now case ðb′Þ. Since Cð2Þeu ≥C
ð1Þeu þ3A, the completion time
of the last job in Xu on M2 can be estimated
Cð2ÞXu≥Cð1Þ
eu þ3Aþ3uBþðt−uþ1ÞðBþλuÞ¼ Cð1Þ
euþ 1þðt−uþ1Þλu > Cð1Þ
euþ 1:
Since λu > 0, thereby Cð2ÞXu
> Cð1Þeuþ 1
, then the completion time of euþ1
on M2 is
Cð2Þeuþ 1
≥Cð1Þeuþ 1
þðt−uþ1Þλuþ3A > Cð1Þeuþ 1
þ3A:
The completion times of the last job in Xuþ1 and of euþ2 on M2 are
Cð2ÞXuþ 1
≥Cð1Þeuþ 1
þ3Aþ3ðuþ1ÞBþðt−uÞðBþλuþ1Þþðt−uþ1Þλu¼ Cð1Þ
euþ 2þðt−uþ1Þλuþðt−uÞλuþ1 > Cð1Þ
euþ 2,
Cð2Þeuþ 2
≥Cð1Þeuþ 2
þ3Aþðt−uþ1Þλuþðt−uÞλuþ1 > Cð1Þeuþ 2
þ3A:
Following this way and knowing that ∑il ¼ uðt−lþ1Þλl ¼
ðt−iþ1Þ∑il ¼ uλlþ∑i−1
l ¼ u∑lv ¼ uλv, we have
Cð2ÞXi≥Cð1Þ
eiþ 1þ ∑
i
l ¼ uðt−lþ1Þλl ¼ Cð1Þ
eiþ 1þðt−iþ1Þ ∑
i
l ¼ uλlþ ∑
i−1
l ¼ u∑l
v ¼ uλv: ð7Þ
Since ∑iv ¼ uλl > 0 for i¼ u,…,w−1, hence Cð2Þ
Xi> Cð1Þ
eiþ 1. On this basis,
the completion time of ei on M2 for i¼ u,…,w can be estimated
Cð2Þei≥Cð1Þ
eiþ3Aþðt−iþ1Þ ∑
i
l ¼ uλl
þ ∑i−1
l ¼ u∑l
v ¼ uλv > Cð1Þ
eiþ3A: ð8Þ
Based on (7) and (8), we have
Cð2ÞXw≥Cð1Þ
ewþ 1þðt−wþ1Þ ∑
w
l ¼ uλlþ ∑
w−1
l ¼ u∑l
v ¼ uλv
≥Cð1Þewþ 1
þ ∑w−1
l ¼ u∑l
v ¼ uλv > Cð1Þ
ewþ 1,
Cð2Þewþ 1
≥Cð1Þewþ 1
þ3Aþ ∑w−1
l ¼ u∑l
v ¼ uλv:
Since λi ¼ 0 for i¼wþ1,…,t, then Cð2ÞXi≥Cð1Þ
eiþ 1þ∑w−1
l ¼ u∑lv ¼ uλv (for
i¼ 1,…,t) and
Cð2Þet ≥C
ð1Þet þ ∑
w−1
l ¼ u∑l
v ¼ uλvþ3A:
On this basis, and taking into consideration (4) and (6), we have
Cmax≥Cð1Þet þ3Aþ ∑
w−1
l ¼ u∑l
v ¼ uλvþð3tþ1ÞB¼ yþ ∑
w−1
l ¼ u∑l
v ¼ uλv > y:
Therefore, for all the cases the criterion value Cmax is greaterthan y.We hereby showed that DAE2 has an answer yes if and only if
the answer for 3PP problem is yes, which means that F2jAEjCmax isstrongly NP-hard, since 3PP is NP-complete in the strong sense.Based on the calculations, it can be easily observed for case ða′Þ
that Cmax > y regardless of the values of λi for i < w. However, Xw
onM2 starts later for λw−1 > 0 than for λw−1≤0 (and can start later ifthe same relation is hold for i < w−1). Thereby, for this proof thereis no need to analyze the distribution of λl for l < w. Thus, withoutincreasing the criterion value, we may assume that λl ¼ 0 forl¼ 1,…,u−1 and ∑i
l ¼ uλl < 0 for i¼ u,…,w−1 (which have theminimal contribution to Cð2Þ
Xw) and ∑w
l ¼ uλl ¼ 0. Therefore, theanalyzed case ða′Þ may be reduced to (a) without increasing thecriterion value, i.e., case (a) is sufficient.For case ðb′Þ note that Cmax > y regardless of the distribution of λi
for i < u, since Xu can start on M2 not earlier than at Cð1ÞXu
(accordingto the transformation) regardless of the distribution of λi for i < u.Additionally the criterion value is minimized if ∑w
l ¼ uλl≥0 isminimized, thereby for ∑w
l ¼ uλl ¼ 0. Observe also that Xu on M2
starts later for λu−1 > 0 than for λu−1≤0 (and can start later if thesame relation is hold for i < u−1). Therefore, λi ¼ 0 for i¼ 1,…,u−1have the smallest contribution to Cð2Þ
Xu(actually none contribution).
Thus, without increasing the criterion value, for this proof we mayassume ∑w
l ¼ uλl ¼ 0 and ∑il ¼ uλl > 0 for i¼ u,…,w−1 and λi ¼ 0 for
i∈f1,…,u−1g∪fwþ1,…,mg. Thereby, without increasing the criter-ion value the analyzed case ðb′Þ may be reduced to (b) (in the sameway as in [29,30]), i.e., case (b) is sufficient. However, it has to bekept in mind that (a) and (b) are less formal than cases ða′Þ and ðb′Þ.It is worth mentioning that cases (a) and (b) were very helpful
during the process of searching the transformation that guaran-tees Cmax≤y if and only if ∑q∈Xi
xq ¼ B for i¼1,…,t. In other words,the presented approach (method considering (a) and (b)) allows usto construct the transformation such that to include into thecompletion times of relevant jobs the terms ∑i
l ¼ ulλl (see[29,30]), which support the proving process. After the transforma-tion is obtained the calculations are (in fact) done for ða′Þ and ðb′Þ(see [29,30]).Finally recall that the given proof is similar to the proof
presented in [29]. Although the calculations in [29] were relevantfor ða′Þ and ðb′Þ, but these cases were omitted there. Thus, the
A. Rudek, R. Rudek / Computers & Operations Research 40 (2013) 2071–20822076
proof in [29] may be less formal and perceived as a sketch, therebymay be onfusing. Therefore, for the better comprehension of theproof in [29] and its logic, cases ða′Þ and ðb′Þ (as well as additionalhelpful comments as presented here) may be taken into consid-eration. Furthermore, in [29], for case (a), only the calculations forCð2ÞXw
and Cmax are crucial (as it is done in this paper). Therefore, Cð2ÞXu
and Cð2Þeuþ 1
may be omitted. Obviously for job completion times in[29], the estimation “≥” should be used (instead of “¼”), e.g.,Cð2ÞXu≥Sð1Þeuþ 1
þV ð2ÞðXuÞ. The similar situation is for the proof presentedin [30]. □
4. Algorithms
In this section, we provide heuristic and metaheuristic algo-rithms for the general version (that covers cases with learning andaging effects) of the problem FmjpðzÞj ðvÞjCmax, namely, Fast NEH(following the algorithms presented in [27,33]), tabu search(following [11]) and simulated annealing (based on [20]) withcomputational complexity Oðmn2Þ, OðIterations �mn2Þ andOðIterations �mnÞ, respectively.
4.1. Fast NEH
The NEH algorithm of Nawaz et al. [27] is considered as one ofthe most efficient heuristics for solving the classical permutationflowshop problem to minimize the makespan with constant jobprocessing times (see [18]). The computational complexity of itsprimary version is Oðmn3Þ, however, Taillard [33] decreased it toOðmn2Þ (by decreasing the computational complexity of a neigh-borhood searching process). A comprehensive survey on NEH andits application for solving this classical flowshop problem can befound in [18]. It is pointed out that undisputed superiority of otheralgorithms (including metaheuristics) over NEH can be demon-strated if they improve a random initial solution (sequence of jobs)within a time limit similar to the time needed by NEH such thatthe provided solution is better than obtained by NEH. Althoughsome attempts to design such algorithms were presented in [19], itwas also claimed that it is highly unlikely to develop a non-NEH-type Oðmn2Þ heuristic that would be competitive with NEH.
NEH was observed to be efficient as a stand-alone algorithm,but it is also often applied to determine the initial solution inmany other algorithms (usually metaheuristics) for the classicalflowshop scheduling problem (see [19,28]). Furthermore, thestandard version of NEH proposed in [27] (with the computationalcomplexity Oðmn3Þ) can be easily (almost straightforwardly)applied for combinatorial optimization problems, where a solutioncan be represented by a permutation. Therefore, it was strongmotivation to use NEH for solving other (especially scheduling)problems (e.g., [34]). Nevertheless, design and development of fastversions of NEH (based on Oðmn2Þ) for such problems is not knowto be possible (e.g., for maximum lateness minimization schedul-ing problems as analyzed in [5]) or require more effort andapplication of different approaches than the standard version(e.g., [34]).
Now let us focus on the analyzed permutation flowshopproblem with position dependent job processing times. Takinginto consideration the successes of NEH for solving the classicalflowshop problem, its standard version with the computationalcomplexity Oðmn3Þ was applied for the makespan minimizationproblem with the learning effect in [6]. However, we propose itsimproved version—the fast NEH algorithm (denoted as FNEH) withcomplexity Oðmn2Þ. The low complexity of FNEH is achieved bycalculating the criterion value using the idea presented in [33].However, note that the algorithm presented in [33] is dedicated
for the classical scheduling problem with constat job processingtimes, whereas we extend it to be used for problems with variableprocessing times. Since the designed FNEH does not followstraightforward from [33], thus, let us briefly describe the pro-posed approach (Algorithm 1).
Algorithm 1. Fast NEH.
1:
Initialize:Determine initial solution πinit , π� ¼∅,
qðzÞv ¼ 0 for v¼ 2,…,nþ1 and z¼ 1,…,mþ1,
f ðzÞv ¼ 0 for v¼ 1,…,n−1 and z¼ 0,…,m,
CðzÞv ¼ 0 for v¼ 0,…,n and z¼ 0,…,m
2:
For i¼1 To n 3: Insert job πinitðiÞ in the last (i.e., ith)position in π�
4:
Assign π¼ π�% calculate criterion value for π� in O(m) ⟶
5: For z¼1 To mCðzÞi ¼maxfCðzÞ
i−1,Cðz−1Þi gþpðzÞπ�ðiÞðiÞ
%⟵
6: Assign v� ¼ i and C� ¼ CðmÞi
7:
For v¼ i−1 To 1 % insert i into v by swapping in Oð1Þ⟶8:
Swap jobs in positions v and vþ1 in thepermutation π
% ⟵ % calculate auxiliary f and q in O(m)⟶9:
For z¼1 To mf ðzÞv ¼maxff ðz−1Þv ,CðzÞv−1gþpðzÞπðvÞðvÞ
10:
For z¼m To 1qðzÞvþ1 ¼maxfqðzÞvþ2,qðzþ1Þvþ1 gþpðzÞπðvþ1Þðvþ1Þ
% ⟵
% calculate criterion for π in O(m)⟶11:
Cmax ¼max1≤z≤mff ðzÞv þqðzÞvþ1g %⟵12:
If Cmax < C� ThenAssign v� ¼ v and C� ¼ Cmax
% insert job πinitðiÞ in position v* in π� for which thecriterion is minimal,
% but only if v* is different than i (in O(n)); vector CðzÞv is
calculated in O(mn)
% only for jobs that completion times have changed ⟶13:
If v�≠i Then 14: Insert job πinitðiÞ in position v* in thepermutation π�
15:
For v¼ v� To i For z¼1 To mCðzÞv ¼maxfCðzÞ
v−1,Cðz−1Þv gþpðzÞπ�ðvÞðvÞ
%⟵
16: π� is the given solutionThe algorithm FNEH starts with an initial solution πinit thatdetermines the order of jobs that are inserted subsequently intothe resulting solution π�. For each job πinitðiÞ (i¼1,…,n) its positionv* in π� has to be determined that minimizes the (partial) criterionvalue CðmÞ
π�ðiÞ. To achieve the complexity Oðmn2Þ of FNEH, position v*
for each πinitðiÞ has to be found in O(mn), thereby the criterionvalue for each insertion has to be calculated in O(m).
Let us analyze the determining of position v* in π� for the ithjob from πinit (denoted by j). At first, we express the insertion of j asan interchanging (swapping) with an adjacent job in π�. At thebeginning j occupies the last position in π�, i.e., π�ðiÞ ¼ j, then it isinserted (by swapping) in positions v¼ i−1,…,1.
A. Rudek, R. Rudek / Computers & Operations Research 40 (2013) 2071–2082 2077
Consider the temporary permutation πvþ1 that is obtained fromπ� by the insertion of job j into position vþ1 (v¼ 1,…,i−1) andgiven as follows:
πvþ1 ¼ ⟨π�ð1Þ,…,π�ðv−1Þ,π�ðvÞ,π�ðiÞ,π�ðvþ1Þ,…,π�ði−1Þ⟩:Let us now consider the insertion of job j into position v by theinterchanging adjacent jobs πvþ1ðvÞ and πvþ1ðvþ1Þ, i.e., π�ðvÞ andπ�ðiÞ, respectively (Step 7). Denote this new permutation by πv andit is given as follows:
πv ¼ ⟨π�ð1Þ,…,π�ðv−1Þ,π�ðiÞ,π�ðvÞ,π�ðvþ1Þ,…,π�ði−1Þ⟩:After each such swapping of adjacent jobs the auxiliary vectors ofparameters f and q are calculated. To simplify the notation a temporarypermutation is denoted by π (in particular the permutation πv).
The parameter f ðzÞπðvÞ is the earliest relative completion time of ajob scheduled in position v (i.e., job j) on machine Mz calculated asfollows (Step 9):
f ðzÞπðvÞ ¼maxff ðz−1ÞπðvÞ ,CðzÞπðv−1ÞgþpðzÞπðvÞðvÞ, ð9Þ
where f ð0ÞπðlÞ ¼ 0 (l¼1,…,n). Note that this parameter refers to thecurrent job j and it is calculated on the basis of its predecessorπðv−1Þ. Observe also that πðlÞ ¼ π�ðlÞ for l¼ 1,…,v−1, therefore, valuesCðzÞπðlÞ ¼ CðzÞ
π�ðlÞ are already known since they were calculated (Step 5 orStep 15) when jobs from πinit were added into π�. The computationalcomplexity to calculate f ðzÞπðvÞ for z¼1,…,m is O(m) (Step 9).
The parameter qðzÞπðvþ1Þ is called tail that is the period of timebetween the completion time of the last job in π (i.e., πðiÞ) onmachine Mm and the starting time of job πðvÞ on machine Mz and itis calculated as follows (Step 10):
qðzÞπðvþ1Þ ¼maxfqðzÞπðvþ2Þ,qðzþ1Þπðvþ1ÞgþpðzÞπðvþ1Þðvþ1Þ, ð10Þ
where qðmþ1ÞπðlÞ ¼ 0 for l¼ 1,…,i and qðzÞπðiþ1Þ ¼ 0 for z¼1,…,m. Note that
qðzÞπðvþ1Þ is calculated for job πðvþ1Þ (that is after j) on the basis of jobπðvþ2Þ. Since qðzÞπðlÞ for z¼1,…,m and l > vþ1 is not affected by theinterchanging of adjacent jobs v and vþ1, therefore, qðzÞπðvþ1Þ can becalculated on the basis of values qðzÞπðvþ2Þ obtained in the previousinterchanging of job jwith adjacent jobs. The computational complex-ity to calculate qðzÞπðvþ1Þ for z¼1,…,m is O(m) (Step 10).
On this basis, we can calculate the criterion value for π afterinsertion j into v (by interchanging of adjacent jobs) as follows(Step 11):
Cmax ¼ max1≤z≤m
ff ðzÞπðvÞ þqðzÞπðvþ1Þg: ð11Þ
It is easy to observe that Cmax is calculated in O(m).Therefore, after each insertion (swapping) of job j in position v
the criterion value is calculated and if it is better than alreadyfound criterion value for the considered j, then position v of thecorresponding insertion is stored as v* (Step 12). After all inser-tions of j are examined, then job j is inserted into position v* in π�
(Step 14) that takes O(n). Finally CðzÞπ� is updated in O(mn) (Step 15)
and a new job πinitðiþ1Þ is considered (Step 3).Let us analyze the complexity of FNEH. There are O(n) jobs to be
examined (Step 2) for each of them an auxiliary permutation π isassigned in O(n) (Step 4) and each job is inserted in O(n) positions(Step 7). The criterion value of an insertion of a job in position v iscalculated in O(m) (Steps 8–11) and position v* with the minimalcriterion value is stored in Oð1Þ (Step 12). After all possiblepositions for a given job are examined and v* is found, the new(best) found solution is constructed in O(mn) (Step 14) and therelated parameters are updated in O(mn) (Step 15). Therefore, thebest insertion of a job is obtained in O(mn) steps. Thereby, the totalcomplexity of the proposed FNEH algorithm is Oðmn2Þ.
The presented FNEH algorithm can be used as a stand-aloneheuristic or it can determine the initial solution for other algo-rithms (e.g., presented in the further part of this paper). Moreover,
the developed fast neighborhood search can be applied in meta-heuristics (tabu search based) or in branch and bound algorithms(e.g., elimination procedures).
4.2. Tabu search
Tabu search (TS) algorithm [11] uses local search with a shortterm memory, called tabu list, which stores forbidden moves,hence allows TS to escape from local minima. In the proposedalgorithm, the move for the given permutation π is defined as theinsertion of a job being in position i in a sequence π into position v.Since the insertion can be expressed as the interchanging ofadjacent jobs, then we design a fast neighborhood search of asolution π with complexity Oðmn2Þ that is based on the same ideaas FNEH (Algorithm 1). The formal definition of the implementedTS is given below (Algorithm 2).
Algorithm 2. Tabu search.
1:
Initialize:Determine πinit and π� ¼ πbest ¼ πinit , C�max ¼ Cbest ¼ CðmÞ
π�ðnÞ,
TabuList ¼∅
qðzÞv ¼ 0 for v¼ 2,…,nþ1 and z¼ 1,…,mþ1,f ðzÞv ¼ 0 for v¼ 1,…,n−1 and z¼ 0,…,m,
CðzÞv ¼ 0 for v¼ 0,…,n and z¼ 0,…,m
2:
For iter¼1 To Iterations 3: first¼1 4: For i¼n To 1 5: Assign π¼ πbest 6: Insert job πbestðiÞ in position n in thepermutation π
7: For v¼1 To nFor z¼1 To m
Calculate CðzÞv for π
8:
For v¼ n−1 To i For z¼m To 1Calculate qðzÞvþ1 for π
9: If (CðmÞn < Cbest or first¼¼1) and (i,v) is not in
TabuList Then
ðibest ,vbestÞ ¼ ði,nÞ, Cbest ¼ CðmÞn , first¼0
10:
For v¼ i−1 To 1 11: Swap jobs in positions v and vþ1 in thepermutation π
12: For z¼1 To mCalculate f ðzÞv for π
13: For z¼m To 1Calculate qðzÞvþ1 for π
14: Cmax ¼max1≤z≤mff ðzÞv þqðzÞvþ1g 15: If i≠v and (Cmax < Cbest or first¼¼1) and(i,v) is not in TabuList Then
ðibest ,vbestÞ ¼ ði,vÞ, Cbest ¼ Cmax, first¼016:
Insert job from position ibest into position vbestin πbest17:
Add ðibest ,vbestÞ to TabuList 18: If Cbest < C� Then19:
π� ¼ πbest and C� ¼ Cbest20:
π� is the given solutionThe applied tabu list stores pairs (i,v) of forbidden moves. If (i,v) is in tabu list, then for any sequence the insertion of a job fromposition i into position v is forbidden. The tabu list is organized asFIFO (First In First Out), thereby, if the list is full, then the new pair(i,v) is added at its beginning overriding the previous pair
A. Rudek, R. Rudek / Computers & Operations Research 40 (2013) 2071–20822078
occupying this position. The computational complexity of theimplemented TS is OðIterations �mn2Þ. In this paper, the initialsolution of TS is provided by FNEH.
Note that the proposed TS offers fast neighborhood searchOðmn2Þ, thus, it constitutes a solid background for its furtherdevelopment, e.g., by using more advanced and effective intensi-fication and diversification strategies.
4.3. Simulated annealing
We also provide an algorithm SA that is based on the simulatedannealing approach (see [20]). This algorithm starts with an initialsolution πinit and generates in each iteration a new permutation π′based on the current solution π by interchanging of random two jobsin π (for the given problem it provides better results than an insertionof a random job into a random position in a permutation). This newsolution π′ replaces π (i.e., π¼ π′) with the following probabilityPðT ,π′,πÞ ¼minf1,expð−ðCπ′ðnÞ−CπðnÞÞ=Tg, where T is the temperaturethat decreases in a logarithmical manner T ¼ T=ð1þλTÞ, and thevalues of the initial temperature T0 and of the parameter λ are chosenempirically. The solution with the minimal found criterion valueCπ�ðnÞ is also stored. The stopping condition of this algorithm is thenumber of Iterations, thus, its overall computational complexity isOðIterations �mnÞ. In this paper, the initial solution of SA is providedby FNEH.
Algorithm 3. SA.
1 TIntelA
1:
Initialize:Determine initial solution πinit and π¼ π� ¼ πinit ,T ¼ T0
2:
For i¼1 to Iterations 3: Choose π′ by a random interchange of two jobs inπ
4: Assign π¼ π′ with probability� �n oPðT ,π′,πÞ ¼min 1,exp − Cπ′ðnÞ−CπðnÞT
5:
If CπðnÞ < Cπ�ðnÞ Then π� ¼ π 6: T ¼ T1þ λT
7: π� is the given solution5. Numerical experiments
The algorithms are designed for the general problemFmjpðzÞj ðvÞjCmax, i.e., processing times are arbitrary functions depen-dent on a job position in a sequence. Thus, these algorithms arevalid for the problem with the learning effect as well as with theaging effect.1
Note that there are no benchmark instances for schedulingproblems with position dependent processing times. Therefore,the values of parameters for problems with aging and learning toanalyze the algorithms have to be generated.
Furthermore, exact solution algorithms that are able to find anoptimal criterion value of the considered problem with n≥50 inreasonable time do not exist, since the algorithms dedicated forthe classical problemwith constant job processing times cannot beapplied. Namely, there are no exact solution algorithms for theproblem with the aging effect. On the other hand, some exactbranch and bound algorithms for the problem with the learningeffect were provided in [40,6], but they can find optimal solutionsin reasonable time only for instances up to 12 (m¼2) and 18(m¼5) jobs, respectively. Therefore, to find an optimal solution for
he algorithms were coded in Cþþ and simulations were run on PC, CPUCoreTMi7-2600K 3.40 GHz and 8 GB RAM.
the considered problem with learning and aging we use theexhaustive search that is sufficient for n¼10 and m¼10, whereasfor n≥50 the proposed algorithms are compared with each other.
5.1. The aging effect
In this part, we analyze the efficiency of the algorithms for theproblem with general non-decreasing position dependent func-tions, FmjAEjCmax, that is strongly NP-hard for m≥2.
The proposed algorithms are evaluated for the followingproblem sizes n∈f10,50,100g and m∈f2,5,10g. For each pair of nand m, 100 different random instances (replications) were gener-ated from the uniform distribution over the integers in thefollowing ranges of parameters: pðzÞj ðvÞ∈½1,10� and ½1,30� (i.e.,f1,2,…,10g and f1,2,…,30g), where z∈f1,…,mg and v¼1,…,n suchthat processing times are non-decreasing. The chosen ranges ofparameters are typical for the pickling process, which is under-stood as a factory process during which steel is immersed in hotacid, removing mill scale and rust [31]. Namely, the time requiredto chemically clean a single item usually increases with thenumber of cleaned items (due to decreasing activity of an activesubstance) and it can take 1/30 min (see [12,31]).
Values of the parameters of the algorithms were chosenempirically as follows: TS1: Iterations¼500 and jTabuListj ¼ 250(where jTabuListj is the size of the tabu list); TS2: Iterations¼2000and jTabuListj ¼ 500; SA: Iterations¼ 100,000, T0 ¼ 1,000,000 andλ¼ 0:01. The initial solution of SA and TS is provided by FNEHalgorithm and its initial sequence of jobs is random.
The considered algorithms are evaluated, for each instance I,according to the relative error δAðIÞ that is calculated in thefollowing way: δAðIÞ ¼ ðCmaxðπAI Þ=Cmaxðπ�I Þ−1Þ � 100%, whereCmaxðπAI Þ denotes the criterion value provided by algorithmA∈fFNEH,TS1,TS2,SAg for instance I and Cmaxðπ�I Þ is the optimalsolution of instance I (if n¼ 10) or the best found solution ofinstance I (if n¼ 50 or 100) provided by the considered approx-imation algorithms. Recall that the optimal solution (for n¼10) isprovided by the exhaustive search.
The results concerning mean and maximum relative errorsprovided by the analyzed algorithms and their mean runningtimes are presented in Tables 1 and 2, respectively. The runningtimes of SA and TS include FNEH. However, to better illustrate thecomputational complexity of the algorithms (especially NEH andFNEH), the running times are shown for the following problemsizes n∈f10,50,100,250,500,750,1000g and m∈f2,5,10g.
At first observe (Table 1) that among two settings of theimplemented tabu search, TS2 provides solutions with smallerrelative errors than TS1 for n≥50. It follows from the fact that TS2visits more potential solutions due to the greater number ofIterations, whereas for the smaller n it makes no difference.
During numerical analysis, we also observed that TS providesbetter results for greater sizes of TabuList. However, if the sizeexceeds a certain value (for the given instance and the number ofIterations), then the accuracy of TS decreases. Among differentsettings of Iterations and of jTabuListj, we chose TS1 and TS2, sincefor the analyzed instances and Iterations¼500 the best resultswere provided for jTabuListj ¼ 250 (TS1) and if Iterations¼2000the best results were provided for jTabuListj ¼ 500 (TS2).
It can be seen that the analyzed metaheuristic algorithmsprovide very low mean relative errors, where in reference to theoptimum solution (n¼10) for TS and SA they do not exceed 1.24%and 0.07%, respectively, whereas the maximum errors do notexceed 6.25% and 2.04%, respectively. It means that if the optimalpickling of 10 elements takes 45 min, then TS and SA provideschedules for which the pickling process is longer than theoptimum about 3 min and 1 min, respectively. However, if nincreases, then the difference between results provided by TS and
Table 1Mean and maximum (in square brackets) relative percentage errors of thealgorithms; symbol < means that the value is lower than the given.
n m pðzÞj ðvÞ FNEH TS1 TS2 SA
10 2 [1, 10] 13.15 [31.91] 0.90 [6.25] 0.90 [6.25] 0.07 [2.04][1, 30] 12.93 [36.52] 1.24 [5.48] 1.24 [5.48] 0.02 [0.65]
5 [1, 10] 8.59 [24.05] 0.82 [4.05] 0.82 [4.05] 0.03 [1.23][1, 30] 9.27 [23.14] 0.84 [3.68] 0.84 [3.68] 0.02 [1.33]
10 [1, 10] 7.18 [15.12] 0.40 [3.25] 0.40 [3.25] <0.01 [0.80][1, 30] 6.77 [14.16] 0.56 [3.09] 0.56 [3.09] 0.01 [0.89]
50 2 [1, 10] 14.49 [19.60] 4.92 [8.83] 1.15 [3.60] 0.02 [1.22][1, 30] 15.23 [22.67] 4.62 [8.50] 1.48 [3.27] <0.01 [0.45]
5 [1, 10] 10.26 [15.27] 2.91 [6.94] 0.55 [2.08] 0.12 [1.39][1, 30] 10.90 [15.93] 2.79 [5.13] 0.73 [2.50] 0.03 [0.61]
10 [1, 10] 8.55 [11.81] 2.05 [4.04] 0.41 [2.02] 0.19 [1.15][1, 30] 8.81 [14.32] 1.98 [3.83] 0.49 [1.83] 0.10 [0.81]
100 2 [1, 10] 12.14 [15.94] 7.97 [12.00] 3.85 [5.97] 0.00 [0.00][1, 30] 12.31 [15.95] 6.28 [9.79] 3.66 [6.75] 0.00 [0.00]
5 [1, 10] 7.97 [10.21] 4.38 [6.97] 1.55 [3.93] 0.01 [0.70]
[1, 30] 8.35 [11.01] 3.65 [5.87] 1.93 [3.43] <0.01 [0.06]10 [1, 10] 6.60 [8.94] 3.48 [5.43] 1.04 [2.40] 0.01 [0.47]
[1, 30] 6.76 [9.45] 2.70 [4.27] 1.31 [2.65] <0.01 [0.50]
Table 2
Mean running times (in seconds) of the algorithms for pðzÞj ðvÞ∈½1,10�.
n m FNEH NEH TS1 TS2 SA
10 2 <0.01 <0.01 0.01 0.04 0.075 <0.01 <0.01 0.02 0.10 0.1510 <0.01 <0.01 0.05 0.20 0.29
50 2 <0.01 <0.01 0.25 1.04 0.305 <0.01 <0.01 0.64 2.62 0.7310 0.01 0.01 1.35 5.45 1.48
100 2 <0.01 0.01 1.01 4.08 0.585 <0.01 0.04 2.62 10.56 1.4410 <0.01 0.09 5.40 21.65 2.93
250 2 <0.01 0.29 6.30 25.28 1.445 0.01 0.73 16.05 64.38 3.5510 0.02 1.50 33.25 133.01 7.19
500 2 0.01 2.34 25.25 101.19 2.875 0.04 5.87 63.51 254.74 7.1210 0.08 11.96 131.08 525.11 14.41
750 2 0.03 7.89 – – –
5 0.09 19.75 – – –
10 0.18 40.19 – – –
1000 2 0.06 18.67 – – –
5 0.15 46.76 – – –
10 0.32 94.97 – – –
Table 3Mean and maximum (in square brackets) relative percentage errors of thealgorithms; symbol < means that the value is lower than the given (the stoppingcondition of TS and SA is 0.5 s for n¼10 and 60 s for n∈f50,100g).
n m pðzÞj ðvÞ FNEH TS1 TS2 SA
10 2 [1, 10] 13.15 [31.91] 0.86 [6.25] 0.90 [6.25] 0.00 [0.00][1, 30] 12.93 [36.52] 1.17 [4.02] 1.23 [5.48] 0.00 [0.00]
5 [1, 10] 8.59 [24.05] 0.78 [4.05] 0.82 [4.05] 0.00 [0.00][1, 30] 9.27 [23.14] 0.81 [3.68] 0.83 [3.68] 0.00 [0.00]
10 [1, 10] 7.18 [15.12] 0.39 [3.25] 0.40 [3.25] 0.00 [0.00][1, 30] 6.77 [14.16] 0.56 [3.08] 0.56 [3.08] 0.00 [0.00]
50 2 [1, 10] 16.67 [22.17] 1.58 [4.87] 0.43 [2.01] 0.05 [0.41][1, 30] 15.94 [24.28] 0.90 [3.05] 0.26 [1.28] 0.03 [0.44]
5 [1, 10] 12.48 [18.46] 0.59 [2.41] 0.09 [0.69] 0.48 [1.07][1, 30] 11.93 [17.37] 0.28 [1.21] 0.08 [0.50] 0.38 [1.10]
10 [1, 10] 9.96 [15.52] 0.47 [2.61] 0.07 [0.88] 0.42 [1.19][1, 30] 10.25 [14.52] 0.19 [1.34] 0.06 [0.80] 0.38 [0.94]
100 2 [1, 10] 13.24 [17.39] 5.13 [10.06] 3.85 [6.42] 0.00 [0.00][1, 30] 13.54 [17.16] 4.82 [6.93] 3.18 [6.52] 0.00 [0.00]
5 [1, 10] 9.16 [11.92] 2.26 [4.29] 1.63 [3.08] 0.00 [0.00][1, 30] 9.18 [11.59] 2.20 [4.06] 1.43 [2.75] 0.00 [0.00]
10 [1, 10] 7.34 [9.42] 1.28 [2.57] 0.94 [1.91] <0.01 [0.15][1, 30] 7.45 [9.82] 1.60 [3.06] 1.01 [2.38] <0.01 [0.16]
A. Rudek, R. Rudek / Computers & Operations Research 40 (2013) 2071–2082 2079
SA also increases, e.g., for TS2 and n¼100 the mean and themaximum difference is less than 4% and 7%, respectively. It can beobserved that for n∈f50,100g the relative error of SA is greater thanzero, thus, TS provides better results than SA for some instances.
The numerical analysis revealed very good performances of theproposed algorithms for possible practical applications that are evenmore relevant taking into consideration that the running times of SAfor 2 machines and 10 jobs is about 0.07 s and it is less than 0.6 s for100 jobs, whereas the running times of TS1 for the same instances areabout 0.01 s and 1 s and of TS2 they are about 0.05 s and 4 s.
Note that the proposed TS is equipped with the fast neighb-orhood search, which constitutes a solid background for the furtherdevelopment of TS by using more advanced and effective intensifica-tion and diversification strategies. Although the designed TS includes
only basic strategies, it is comparable with SA, which is simple, but hasproved its high efficiency for many combinatorial optimization pro-blems, where a solution can be expressed by a permutation (e.g., [35]).
On the other hand, the results provided by FNEH are not soimpressive comparing to the criterion values of TS and SA, sincethe mean relative error is about 15% and the maximum relativeerror is about 36%. However, FNEH is characterized by a lowcomputational complexity and that is a significant advantage.Furthermore, any of the presented algorithms do not provide asolution that is even comparable with FNEH within a time limitdetermined by its running time. Thus, FNEH is strong as a stand-alone algorithm, especially if a calculation time is a crucialcriterion. It also provides a relatively good initial solution for TSand SA. Finally note that the running times of FNEH are signifi-cantly improved in reference to the standard NEH, i.e., it is about0.28n times faster than NEH, e.g., 30 and 290 times for n¼100 andn¼1000, respectively. It can also be observed that the impact ofvalues of parameters (job processing times) on the algorithms isnegligible that reveals their stability for industrial applications.
The real impact of the problem size (n and m) on the runningtimes of TS and SA is shown in Table 2 (for fixed numbers ofiterations). However, to conduct a fair comparison between TS andSA, it is also worth analyzing their accuracy in finding efficientschedules when their running times are the same. Thus, weprovide an additional experiment, where the stopping conditionof TS and SA is the running time that is set to 0.5 s for n¼10 and60 s for n∈f50,100g; the running times of the exhaustive search forn¼10 and m¼ 2,5,10 are about 0.6 s, 1.5 s, 2.8 s, respectively. Theresults of this experiment are shown in Table 3.
It can be seen that SA finds solutions with smaller criterionvalues more often than TS (especially if m¼2), but it is notsuperior over TS (see n¼50 and m¼5, 10).
5.2. The learning effect
Since the proposed algorithms solve the problem with thelearning effect, therefore, we analyze it for the most popularlearning model (see [1]):
pðzÞj ðvÞ ¼ aðzÞj vα, ð12Þ
A. Rudek, R. Rudek / Computers & Operations Research 40 (2013) 2071–20822080
where α is a learning index (slope of the learning curve) that dependson the learning rate LR, i.e., α¼ log2 LR. The learning rate is defined asrate of each redoubling the output, i.e., LR¼ pjð2vÞ=pjðvÞ. For thepopular 80% hypothesis the learning index is calculated as followsα¼ log2 0:8¼ −0:322. Note that the computational complexity ofF2jpjðvÞ ¼ ajvαjCmax is not determined, however, some exact andheuristic algorithms have been already proposed (see [6,40]).
Similar to the previous case the algorithms are evaluated forthe following problem sizes n∈f10,50,100g and m∈f2,5,10g. Foreach pair of n and m, 100 different random instances (replications)
Table 4Mean and maximum (in square brackets) relative percentage errors of the algorithms.
n m α aðzÞjFNEH
10 2 −0.152 [1, 10] 0.78 [2.83][1, 30] 0.74 [3.86]
−0.322 [1, 10] 1.55 [5.47][1, 30] 1.55 [6.19]
−0.515 [1, 10] 1.97 [7.45][1, 30] 2.11 [7.33]
5 −0.152 [1, 10] 2.68 [10.27][1, 30] 2.62 [10.54]
−0.322 [1, 10] 2.38 [12.19][1, 30] 2.68 [13.00]
−0.515 [1, 10] 2.58 [12.50][1, 30] 2.73 [10.37]
10 −0.152 [1, 10] 2.53 [8.59][1, 30] 2.88 [8.64]
−0.322 [1, 10] 2.11 [6.64][1, 30] 2.49 [9.85]
−0.515 [1, 10] 1.84 [7.89][1, 30] 2.90 [11.01]
50 2 −0.152 [1, 10] 1.36 [3.88][1, 30] 1.46 [3.93]
−0.322 [1, 10] 2.49 [5.27][1, 30] 3.01 [5.64]
−0.515 [1, 10] 3.64 [6.87][1, 30] 4.06 [7.29]
5 −0.152 [1, 10] 2.09 [5.37][1, 30] 2.16 [5.91]
−0.322 [1, 10] 3.10 [5.57][1, 30] 3.48 [7.62]
−0.515 [1, 10] 4.24 [7.08][1, 30] 4.44 [9.71]
10 −0.152 [1, 10] 2.54 [5.45][1, 30] 2.80 [5.52]
−0.322 [1, 10] 2.92 [8.09][1, 30] 3.06 [7.69]
−0.515 [1, 10] 3.13 [6.43][1, 30] 3.60 [9.23]
100 2 −0.152 [1, 10] 1.32 [3.23][1, 30] 1.57 [3.26]
−0.322 [1, 10] 0.52 [5.53][1, 30] 2.87 [5.18]
−0.515 [1, 10] 3.80 [5.59][1, 30] 4.24 [6.25]
5 −0.152 [1, 10] 1.69 [3.35][1, 30] 1.95 [3.50]
−0.322 [1, 10] 1.09 [5.70][1, 30] 1.52 [6.13]
−0.515 [1, 10] 3.70 [8.85][1, 30] 4.19 [8.53]
10 −0.152 [1, 10] 1.80 [3.71][1, 30] 1.84 [4.30]
−0.322 [1, 10] 2.17 [3.73][1, 30] 2.35 [4.34]
−0.515 [1, 10] 3.06 [6.67][1, 30] 3.14 [6.56]
were generated from the uniform distribution over the integers inthe following ranges of parameters (following [6,40]): aðzÞj ∈½1,10�and ½1,30� (i.e., f1,2,…,10g and f1,2,…,30g), where z∈f1,…,mg andthe learning curves are 90%, 80% and 70%, thereby α∈f−0:152,−0:322,−0:515g.
The results concerning mean and maximum relative errorsprovided by the analyzed algorithms are presented in Table 4.Since the running times of the algorithms do not depend on theproblem parameters, thus, mean running times are the same as inTable 2.
TS1 TS2 SA
0.01 [0.34] 0.01 [0.34] 0.00 [0.00]0.07 [1.86] 0.07 [1.86] <0.01 [0.05]0.12 [1.79] 0.12 [1.79] 0.00 [0.00]0.12 [2.02] 0.12 [2.02] <0.01 [0.24]0.10 [1.26] 0.10 [1.26] 0.00 [0.00]0.14 [1.96] 0.14 [1.96] <0.01 [0.21]
0.25 [3.36] 0.25 [3.36] 0.03 [1.39]0.22 [2.54] 0.21 [2.54] 0.04 [1.15]0.27 [6.39] 0.27 [6.39] 0.03 [1.24]0.25 [1.78] 0.25 [1.78] 0.04 [1.10]0.22 [4.90] 0.22 [4.90] <0.01 [0.27]0.34 [3.00] 0.34 [3.00] 0.08 [3.04]
0.19 [1.57] 0.19 [1.57] 0.08 [1.83]0.37 [5.95] 0.37 [5.95] 0.08 [1.61]0.32 [3.00] 0.32 [3.00] 0.05 [0.94]0.42 [2.84] 0.42 [2.84] 0.05 [0.84]0.18 [1.68] 0.18 [1.68] <0.01 [0.45]0.34 [7.69] 0.34 [7.69] 0.03 [1.11]
0.15 [1.23] 0.07 [0.91] 0.01 [0.17]0.15 [1.30] 0.05 [0.63] 0.02 [0.24]0.34 [1.25] 0.13 [0.95] 0.03 [0.30]0.43 [2.10] 0.15 [1.27] 0.05 [0.68]0.59 [2.81] 0.24 [2.66] 0.05 [0.53]0.82 [3.81] 0.30 [3.74] 0.10 [0.72]
0.34 [2.00] 0.09 [1.26] 0.19 [1.62]0.27 [1.26] 0.07 [0.98] 0.21 [1.44]0.57 [2.42] 0.11 [1.37] 0.36 [1.78]0.59 [2.50] 0.18 [1.92] 0.29 [1.72]1.04 [4.68] 0.53 [3.53] 0.25 [1.42]1.14 [5.90] 0.55 [5.32] 0.25 [2.21]
0.38 [2.46] 0.08 [1.16] 0.56 [2.72]0.49 [3.12] 0.13 [3.05] 0.69 [3.69]0.56 [3.29] 0.19 [2.71] 0.60 [3.90]0.61 [2.93] 0.16 [2.60] 0.77 [4.52]0.77 [3.15] 0.30 [2.26] 0.31 [2.73]0.89 [4.41] 0.39 [4.30] 0.50 [3.91]
0.18 [0.76] 0.10 [0.60] <0.01 [0.12]0.24 [0.85] 0.15 [0.72] 0.01 [0.17]0.55 [2.47] 0.37 [1.50] <0.01 [0.07]0.60 [1.88] 0.40 [1.46] <0.01 [0.10]0.96 [2.42] 0.69 [2.04] <0.01 [0.35]1.13 [2.88] 0.80 [2.21] <0.01 [0.51]
0.21 [0.89] 0.09 [0.85] 0.14 [1.25]0.27 [1.12] 0.13 [0.84] 0.15 [0.85]0.74 [2.55] 0.44 [2.31] 0.14 [1.84]0.72 [2.80] 0.36 [2.22] 0.19 [2.12]1.04 [3.01] 0.69 [2.68] 0.16 [1.91]1.27 [4.92] 0.87 [4.72] 0.20 [2.47]
0.17 [0.86] 0.02 [0.69] 0.78 [3.23]0.20 [1.21] 0.05 [0.85] 0.69 [2.58]0.33 [1.60] 0.12 [1.60] 0.62 [2.93]0.38 [2.17] 0.13 [2.03] 0.78 [3.92]0.77 [4.79] 0.52 [4.49] 0.47 [3.65]0.83 [4.06] 0.52 [3.98] 0.55 [4.32]
A. Rudek, R. Rudek / Computers & Operations Research 40 (2013) 2071–2082 2081
The experiments confirm the previous observations for theproblem with the aging effect. Among the analyzed algorithms SAprovides solutions with the smallest criterion values, however, TShas similar efficiency and for some cases it outperforms SA. It canbe observed that the parameters of the learning curves havenoticeable impact on the algorithms accuracy. It usually slightlyincreases when α decreases or values of aj increase.
Moreover, the algorithms provide solutions with smaller rela-tive errors for the problem with a common learning curve than forthe instances with arbitrary non-decreasing processing times
Table 5Mean and maximum (in square brackets) relative percentage errors of the algorithms (
n m α aðzÞjFNEH
10 2 −0.152 [1, 10] 0.78 [2.83][1, 30] 0.74 [3.86]
−0.322 [1, 10] 1.55 [5.47][1, 30] 1.55 [6.19]
−0.515 [1, 10] 1.97 [7.45][1, 30] 2.11 [7.33]
5 −0.152 [1, 10] 2.68 [10.27][1, 30] 2.62 [10.54]
−0.322 [1, 10] 2.38 [12.19][1, 30] 2.68 [13.00]
−0.515 [1, 10] 2.58 [12.50][1, 30] 2.73 [10.37]
10 −0.152 [1, 10] 2.53 [8.59][1, 30] 2.88 [8.64]
−0.322 [1, 10] 2.11 [6.64][1, 30] 2.49 [9.85]
−0.515 [1, 10] 1.84 [7.89][1, 30] 2.90 [11.01]
50 2 −0.152 [1, 10] 1.25 [3.25][1, 30] 1.56 [3.60]
−0.322 [1, 10] 2.57 [6.03][1, 30] 2.71 [5.19]
−0.515 [1, 10] 3.83 [7.97][1, 30] 4.06 [7.77]
5 −0.152 [1, 10] 2.13 [4.36][1, 30] 2.50 [6.69]
−0.322 [1, 10] 3.54 [6.84][1, 30] 4.01 [6.33]
−0.515 [1, 10] 4.66 [8.38][1, 30] 5.01 [10.02]
10 −0.152 [1, 10] 2.87 [6.43][1, 30] 3.24 [6.66]
−0.322 [1, 10] 3.32 [7.19][1, 30] 3.33 [9.79]
−0.515 [1, 10] 3.97 [9.45][1, 30] 4.23 [8.02]
100 2 −0.152 [1, 10] 1.45 [2.76][1, 30] 1.59 [3.89]
−0.322 [1, 10] 3.01 [4.97][1, 30] 3.16 [6.14]
−0.515 [1, 10] 4.05 [8.39][1, 30] 4.30 [7.81]
5 −0.152 [1, 10] 1.98 [4.32][1, 30] 2.10 [3.89]
−0.322 [1, 10] 3.22 [5.13][1, 30] 3.59 [6.19]
−0.515 [1, 10] 4.30 [7.17][1, 30] 4.90 [7.05]
10 −0.152 [1, 10] 1.81 [3.75][1, 30] 2.12 [4.69]
−0.322 [1, 10] 2.51 [4.39][1, 30] 2.56 [4.37]
−0.515 [1, 10] 3.53 [7.83][1, 30] 3.79 [7.47]
(see Table 1). The results provided by the algorithms are alsosignificantly better and closer to the optimum than the heuristicsproposed in [40], where two machine cases were considered.
Finally, as for the problem with the aging effect, we alsocompare the efficiency of TS and SA if they have the same runningtimes; the stopping condition of TS and SA is set to 0.5 s for n¼10and 60 s for n∈f50,100g. The results are shown in Table 5.
Again SA finds solutions with smaller criterion values moreoften than TS, but for some cases TS is better (see n¼50 andm¼10).
the stopping condition of TS and SA is 0.5 s for n¼10 and 60 s for n∈f50,100g).
TS1 TS2 SA
0.01 [0.34] 0.01 [0.34] 0.00 [0.00]0.06 [0.94] 0.07 [1.86] <0.01 [0.01]0.12 [1.70] 0.12 [1.79] 0.00 [0.00]0.12 [2.02] 0.12 [2.02] 0.00 [0.00]0.10 [1.26] 0.09 [1.26] 0.00 [0.00]0.11 [1.41] 0.14 [1.96] 0.00 [0.00]
0.23 [3.36] 0.25 [3.36] 0.02 [1.33]0.21 [2.45] 0.21 [2.54] 0.02 [0.90]0.26 [6.39] 0.27 [6.39] 0.02 [1.08]0.25 [1.78] 0.25 [1.78] 0.03 [0.63]0.19 [4.02] 0.22 [4.90] <0.01 [0.09]0.33 [3.00] 0.33 [3.00] 0.01 [0.72]
0.19 [1.57] 0.19 [1.57] 0.07 [0.88]0.37 [5.81] 0.37 [5.95] 0.06 [1.38]0.32 [3.00] 0.32 [3.00] 0.01 [0.57]0.40 [2.84] 0.41 [2.84] 0.03 [0.74]0.18 [1.68] 0.18 [1.68] <0.01 [0.29]0.30 [4.77] 0.27 [3.25] 0.02 [0.49]
0.08 [0.58] 0.03 [0.45] <0.01 [0.09]0.14 [0.89] 0.06 [0.56] 0.01 [0.11]0.25 [1.14] 0.10 [0.79] 0.02 [0.22]0.25 [1.06] 0.10 [0.76] 0.03 [0.26]0.53 [1.81] 0.22 [1.05] 0.02 [0.21]0.59 [2.75] 0.22 [2.46] 0.03 [0.25]
0.24 [1.27] 0.08 [1.23] 0.09 [0.56]0.30 [1.50] 0.11 [1.25] 0.13 [0.79]0.58 [1.95] 0.25 [1.60] 0.13 [0.83]0.60 [2.14] 0.32 [1.54] 0.14 [0.95]1.14 [6.48] 0.58 [6.02] 0.07 [0.81]1.07 [3.89] 0.61 [3.30] 0.10 [1.02]
0.30 [2.35] 0.12 [2.16] 0.46 [1.97]0.46 [3.60] 0.28 [3.47] 0.41 [2.06]0.72 [4.73] 0.33 [3.01] 0.30 [1.43]0.49 [2.79] 0.24 [2.58] 0.30 [1.38]1.19 [6.11] 0.87 [5.43] 0.05 [0.90]1.09 [4.84] 0.66 [4.63] 0.15 [1.18]
0.22 [0.74] 0.15 [0.72] <0.01 [0.09]0.21 [0.91] 0.14 [0.87] <0.01 [0.07]0.67 [1.96] 0.48 [1.44] <0.01 [0.06]0.69 [2.21] 0.48 [1.67] <0.01 [0.07]0.98 [2.44] 0.73 [2.07] <0.01 [0.05]1.20 [3.04] 0.92 [2.23] <0.01 [0.02]
0.31 [1.09] 0.23 [0.98] 0.02 [0.37]0.33 [1.38] 0.24 [1.35] 0.03 [0.50]0.83 [2.39] 0.59 [2.34] 0.01 [0.56]0.85 [2.98] 0.60 [2.87] 0.05 [1.13]1.70 [4.24] 1.42 [3.92] <0.01 [0.17]1.73 [4.26] 1.41 [3.83] <0.01 [0.27]
0.16 [0.98] 0.09 [1.00] 0.24 [0.98]0.20 [2.19] 0.12 [2.07] 0.33 [1.50]0.55 [2.22] 0.40 [2.08] 0.15 [1.81]0.48 [2.14] 0.35 [1.88] 0.20 [1.45]1.34 [5.40] 1.15 [5.27] 0.02 [0.50]1.42 [5.42] 1.25 [5.20] 0.06 [1.02]
A. Rudek, R. Rudek / Computers & Operations Research 40 (2013) 2071–20822082
6. Conclusions
We proved that the makespan minimization problem in two-machine flowshop environments, where job processing times aredescribed by non-decreasing position dependent functions (agingeffect) is at least NP-hard if only one machine deteriorates andstrongly NP-hard if both machines deteriorate. Its computationalcomplexity had not been determined.
Moreover, we constructed the fast NEH with complexity Oðmn2Þ(that is faster than NEH with Oðmn3Þ in [27,6]), tabu search withfast neighborhood search Oðmn2Þ and simulated annealing algo-rithms that solve the problem with job processing times describedby arbitrary position dependent functions modelling both learningand aging effects. The proposed fast neighborhood can be appliedfor other metaheuristics or branch and bound algorithms. Theefficiency of the proposed methods was numerically analyzed forthe problems with aging and learning. The extensive numericalanalysis revealed that the algorithms are characterized by lowrelative errors and short running times and, moreover, are stablein both accuracy and running times.
The proposed TS offers fast neighborhood search, thus, itconstitutes a solid background for its further development, e.g.,by using more advanced and effective intensification and diversi-fication strategies, which will be our future work.
Acknowledgments
We are grateful to the Editor and the Referees for their valuablecomments on an earlier version of our paper. The researchpresented in this paper has been partially supported by the PolishMinistry of Science and Higher Education in years 2012–2013under Grant no. IP2011 046771.
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