Embed Size (px)
Citation preview
Computers & Operations Research 39 (2012) 1218–1224
Contents lists available at ScienceDirect
Computers & Operations Research
0305-05
doi:10.1
� Corr
Univers
London
E-m
(V. Stru
journal homepage: www.elsevier.com/locate/caor
Single machine scheduling with precedence constraints and positionallydependent processing times
Alexandre Dolgui a, Valery Gordonb , Vitaly Strusevich c,�
a Centre for Industrial Engineering and Computer Science, Ecole Nationale Superieure des Mines de Saint-Etienne, 158, Cours Fauriel, 42023 Saint-Etienne, Franceb United Institute of Informatics Problems, National Academy of Sciences of Belarus, Surganova 6, 220012 Minsk, Belarusc School of Computing and Mathematical Sciences, University of Greenwich, Old Royal Naval College, Park Row, Greenwich, SE10 9LS London, UK
a r t i c l e i n f o
Available online 3 July 2010
Keywords:
Scheduling
Precedence constraints
Deterioration
Learning
Positionally dependent processing time
48/$ - see front matter & 2010 Elsevier Ltd. A
016/j.cor.2010.06.004
esponding author at: School of Computing
ity of Greenwich, Old Royal Naval College, Pa
, UK. Tel.: +44 20 8331 8662; fax: +44 20 833
ail addresses: [email protected] (A. Dolgui), V.St
sevich).
a b s t r a c t
In many real-life situations the processing conditions in scheduling models cannot be viewed as given
constants since they vary over time thereby affecting actual durations of jobs. We consider single
machine scheduling problems of minimizing the makespan in which the processing time of a job
depends on its position (with either cumulative deterioration or exponential learning). It is often found
in practice that some products are manufactured in a certain order implied, for example, by
technological, marketing or assembly requirements. This can be modeled by imposing precedence
constraints on the set of jobs. We consider scheduling models with positional deterioration or learning
under precedence constraints that are built up iteratively from the prime partially ordered sets of a
bounded width (this class of precedence constraints includes, in particular, series–parallel precedence
constraints). We show that objective functions of the considered problems satisfy the job module
property and possess the recursion property. As a result, the problems under consideration are solvable
in polynomial time.
& 2010 Elsevier Ltd. All rights reserved.
1. Introduction
For majority of deterministic scheduling problems, theprocessing conditions, including job processing times, are usuallyconsidered as given constants. However, in various real-lifesystems, the processing conditions may vary over time, therebyaffecting actual durations of jobs. This leads to the study ofscheduling models in which the processing times are controllableby allocating resources, or in which the actual processing time of ajob depends on its place in the schedule (either with adeterioration effect or a learning effect). Under the deteriorationmodel, the later a job starts, the longer it takes to process, and onthe contrary, under the model with a learning effect, the actualprocessing time of a job gets shorter, provided that the job isscheduled later. There are two categories of models that addressscheduling problems with deterioration and learning: if theprocessing time of a job depends on its position, we refer to apositional deterioration/learning model, while if the processing
ll rights reserved.
and Mathematical Sciences,
rk Row, Greenwich, SE10 9LS
1 8665.
time of a job depends on its start time, we refer to the time
deterioration/learning model.For a survey of scheduling with controllable processing times
we refer to the paper by Shabtay and Steiner [1], while for state-of-the-art reviews on deterioration/learning scheduling modelswe refer to the surveys of Alidaee and Womer [2], Cheng et al. [3]and Biskup [4]. In these survey papers, one can also find thereferences to practical applications of the models mentionedabove.
In this paper, we consider single machine scheduling modelswith positional deterioration or learning, i.e., in which theprocessing time of a job grows (a deterioration effect) ordecreases (a learning effect) as the job’s position in a scheduleadvances.
It is often found in practice that some products are manu-factured in a certain order implied, for example, by technological,marketing or assembly requirements. This can be modeled byimposing precedence constraints on the set of jobs.
Most of research on scheduling with precedence constraints isfocused on constant processing times (see [5–10] for singlemachine problems). Gordon and Shafransky [11], Tanaev et al.[12] and Wang et al. [13] consider time deterioration singlemachine problems under the so-called series–parallel precedenceconstraints. Gordon et al. [14] propose polynomial time algo-rithms for various models with positional and time deterioration/
A. Dolgui et al. / Computers & Operations Research 39 (2012) 1218–1224 1219
learning under precedence constraints. Janiak and Kovalyov [15]consider scheduling problems in which the jobs are partiallyordered and the processing machine must have rest periodswhose durations are start time dependent.
The factors that may affect the complexity of a schedulingproblem include the structure of precedence constraints and theobjective function involved. Starting from the seminal paper ofLawler [6], scheduling problems with series–parallel precedenceconstraints have been studied extensively. A class of objectivefunctions (the so-called priority-generating functions) has beenidentified that allows these functions to be minimized inpolynomial time over series–parallel precedence constraints. Werefer to Gordon and Shafransky [11], Monma and Sidney [16], andTanaev et al. [12] for the description of the main results in thisarea. The definitions related to series–parallel graphs and topriority-generating functions are presented formally in Sections 2and 3, respectively.
For sequencing problems with precedence constraints, Sidneyand Steiner [17] propose the combined use of dynamic program-ming (DP) and substitution decomposition introduced by Mohringand Rademacher [18,19]. They show that the proposed combina-tion enables one to solve in polynomial time a class of sequencingproblems under more general precedence constraints than theseries–parallel precedence constraints. The problems of this classpossess the recursion property of DP and the so-called job module
property (considered in details in Section 3); this class ofsequencing problems includes the total weighted completion time
problem [20], the total discounted cost problem [21], the least-cost
fault detection problem [22] and the jump number problem [23].Gordon et al. [24] show that the problem of minimizing earlinesspenalties on a single machine under SLK due date assignment alsobelongs to this class and can be tackled by the combination of DPand substitution decomposition. In this paper, we show that thisapproach can be used for the single machine problem ofminimizing the makespan with one particular positional dete-rioration effect and one particular positional learning effect.
The remainder of the paper is organized as follows: Section 2presents the problem formulation and some definitions related topartially ordered sets. In Section 3, the priority-generatingfunctions, job module and recursion properties are considered.Single machine problems of minimizing the makespan underprecedence constraints and positional deterioration/learning ofjob processing times are studied in Sections 4 and 5 for thecumulative type of deterioration and the exponential learningeffect, respectively. Both problems are shown to have the jobmodule property and the recursion property, and therefore arepolynomially solvable under the precedence constraints that arebuilt up iteratively from the prime partially ordered sets of abounded width. Finally, conclusions are reported in Section 6.
2. Problem formulation
In the models that we consider in this paper, the jobs of setN¼ f1,2, . . . ,ng have to be processed without preemption on asingle machine. The jobs are simultaneously available at timezero. The machine can handle only one job at a time and ispermanently available from time zero. For each job j, where jAN,the value of its ‘standard’ or ‘normal’ processing time pj is known.In the most general case of positional deterioration/learning, theprocessing time of a job j scheduled in position r is equal to
p½r�j ¼ pjgðrÞ, ð1Þ
where pj is the ‘normal’ processing time of job j and g(r)rg(r+1)in case of deterioration and g(r)Zg(r+1) in case of learning effectfor r¼ 1,2, . . . ,n�1. For each individual deterioration model, there
is a particular rule that explains how exactly the value of pj
changes, depending on the position of the job in the processingsequence.
The purpose of scheduling is to minimize a certain non-decreasing objective function F of the job completion times. For acertain schedule, let Cj denote the completion time of job j, forjAN. Given a schedule specified by a permutation p of jobs, wedenote the value of the objective function by F(p). In particular,the makespan (the maximum completion time) is denotedby Cmax. In this paper, we consider the problems of minimizingthe makespan under given precedence constraints forpositional cumulative deterioration and for positional exponentiallearning models.
Formally, the precedence constraints are defined by a binaryrelation -. We write i-j and say that job i precedes job j if in anyfeasible schedule job i must be completed before job j starts. Theset of constraints is usually given by a directed acyclic graph G, inwhich the set of vertices is defined by the set of jobs N and there isan arc from vertex i to vertex j if and only if i-j. It is convenient torepresent the constraints in the form of a reduction graph,obtained from G by removing all transitive arcs. A sequence (or apermutation) of jobs is feasible if no pair of jobs violates theprecedence constraints.
Let G(X, U) be a directed acyclic graph (dag), where X is the setof vertices and U is the set of arcs. A dag G(X, U) is said to be aparallel composition of two dags G1(X1, U1) and G2(X2, U2), whereX1 \ X2 ¼ |, if X ¼ X1 [ X2 and U ¼U1 [ U2. A dag G(X, U) is said tobe a series composition of two dags G1(X1, U1) and G2(X2, U2),where X1 \ X2 ¼+, if X ¼ X1 [ X2 and U ¼U1 [ U2 [
~U , where ~U isthe set of arcs from each vertex of the graph G1 with zerooutdegree to each vertex of the graph G2 with zero indegree. Agraph is called series–parallel (or a SP-graph) if either it consists ofonly one vertex, or it can be obtained from a set of single-vertexgraphs by the successive application of series and/or parallelcomposition operations. The graph obtained from a SP-graph byremoving all of its transitive arcs is also called series–parallel. ASP-graph can be defined by its so-called decomposition tree whichcan be found in O(|X|2) time (see for the details [6,12,24,25]).
The elements of a partially ordered set (poset) P(N, R) are givenby the job set N, and the order relation R that is defined by/i,jSAR if and only if i-j. So, in our case, the notions of‘‘precedence constraints’’, ‘‘dag’’ and ‘‘poset’’ are interchangeable.For a dag, let (i, j) denote an arc that goes from vertex i to vertex j.The transitive closure of a dag G is a dag GT such that GT contains anarc (i, j) if and only if ia j and there is a path from i to j in G.
There is a type of precedence constraints, more general thanseries–parallel, that admit decomposition into the so-called jobmodules of a special structure. In the description of this class ofconstraints, we follow Buer and Mohring [26], Mohring andRademacher [18,19], Muller and Spinrad [27]. Given a poset P(N,R), a subset NuAN is a (job) module of P if for every job kAN\Nu oneof the following holds:
(a)
k-i for all iANu, or (b) i-k for all iANu, or (c) i�k for all iANu, where we write i�k if jobs are independent,i.e., neither i-k nor k-i.
Let G be a dag corresponding to poset P(N, R). Replacing all arcsof the transitive closure GT by undirected edges, we obtain the(undirected) graph ~GðN,EÞ. We may assume that ~GðN,EÞ is given tous in the form of the adjacency matrix. A module N0 is a set ofvertices that is indistinguishable in graph ~G by the verticesoutside N0; that is, in graph ~G any vertex in N\Nu is either adjacentto all vertices of N0, or is adjacent to no vertex in N0.
A. Dolgui et al. / Computers & Operations Research 39 (2012) 1218–12241220
There are three distinct types of modules: parallel, series, andneighborhood. To introduce these types of modules, we firstintroduce the notions of a complement graph and a complement-connected graph. The complement graph of ~GðN,EÞ is the graphGuðN,EuÞ, where ðu,vÞAEu if and only if ðu,vÞ=2E. A graph iscomplement-connected if its complement graph is connected.Parallel modules are characterized by the property that thesubgraph induced by the vertices of the module is not connected.A module is a series module if the subgraph induced by thevertices of the module is not complement-connected. In aneighborhood module, the subgraph induced by the vertices ofthe module is both connected and complement-connected.
For a poset P(N, R), a subset SDN is initial if for each iAS allpredecessors of i are also in S. For a set A, define the setIðAÞ ¼ f jj jAA, j has no successors in Ag.
For all initial sets S, the maximum size of any set I(S) is calledthe width of P. If the poset is decomposed into modules, thewidth of any module does not exceed the width of the originalposet. Provided that the objective function possesses certainproperties, it can be minimized over a poset of a fixed width inpolynomial time.
Given a poset, or, equivalently, a directed graph, the decom-position process of partitioning the graph into modules is calledmodular decomposition [27]. At any stage of the process, thecurrent subgraph to be decomposed is a module of the originalgraph. Each of these subgraphs is decomposed recursively. Thisprocess continues until each obtained subgraph contains only asingle vertex. The decomposition procedure partitions a givengraph in the unique way, provided that in each iteration wedecompose into maximal nontrivial modules.
To describe modular decomposition formally, we use the followingnotation of Sidney and Steiner [17]. For N0 ¼ fi1,i2, . . . ,img, let P0(N0,R0) be a poset of m elements. Let Ph(Nh, Rh) for h¼ 1,2, . . . ,m bedisjoint posets. The composition poset P(N, R) is defined byN¼
Smh ¼ 1 Nh and R¼
Smh ¼ 1 Rh
Sf/i, jSjiANh, jANk,/ih, ikSAR0g.
For this composition, we write P¼ P0½P1, . . . ,Pm�, where P0 isreferred to as the outer factor, and each of P1, . . . ,Pm is called aninner factor. Then P is the series composition S of the inner factors ifP0 is a chain, and the parallel composition P if ih� ik for all ih,ikAN0.In any other case, P is called a neighborhood composition N. In theobtained composition each inner factor is a module of P.
The modular decomposition procedure of the original poset P
is implemented as an iterative process until all factors have beendecomposed into one-element sets. In each iteration, we decidewhich type of composition applies and find the correspondinginner factors. The composition tree T(P) is a data structurethat represents this process. For details on the decompositionprocess and on the constructing of T(P) see Buer and Mohring[26], Muller and Spinrad [27], Sidney and Steiner [17], Dahlhauset al. [28].
3. Priority-generating functions and modular decomposition
For problems in which an optimal schedule is defined by apermutation of jobs, an optimal permutation can often be foundby a priority rule (index policy), i.e., by assigning certain prioritiesto jobs, which are then sorted in the order of these priorities (see,for example, [29] or Tanaev et al. [12]). A generalization of apriority rule for subsequences of jobs (rather than individual jobs)leads to priority functions and priority-generating objective func-tions (see the definition and the theorem below by Tanaevet al. [12]).
Definition 1. Let pab¼(p0abp00) and pba¼(p0bap00) be twopermutations of n jobs that differ only in the order of the
subsequences a and b (each of the sequences p0 and p00 may beempty). For a function F(p) that depends on a permutation,suppose that there exists a function o(p) such that for any twopermutations pab and pba the inequality o(a)4o(b) implies thatF(pab)F(pba), while the equality o(a)¼o(b) implies thatF(pab)¼F(pba). In this case, function F is called a priority-
generating function, while function o is called its priority function.For a (partial) permutation p, the value of o(p) is called thepriority of p.
Note that a notion of a priority-generating function isequivalent to the strong adjacent sequence interchange propertydiscussed below.
Theorem 1. If for a priority-generating function F(p) its priority
o(p) can be computed in linear time, then F(p) can be minimized in
Oðn log nÞ time, provided that the reduction graph is series–parallel
and its decomposition tree is given.
Algorithms for minimizing priority-generating functions underseries–parallel precedence constraints are proposed in[6,11,16,30].
The objective functions for the problems of minimizing themakespan with positional deterioration which we consider in thispaper are shown to be priority-generating by Gordon et al. [14].In Sections 4 and 5, we show that these problems are polynomiallysolvable for a wider class of precedence constraints thanseries–parallel.
Consider the problem of minimizing some function F over aposet P(N, R). For a poset P(N, R) and Nu �N, the subset PuðNu,RuÞ isan induced poset if Ru ¼ f/i,jSARji,jANug. Let PuðNu,RuÞ be the subsetof P induced by Nu �N. For an arbitrary feasible sequence s of theelements of N, let sjNu denote the restriction of s to N0, i.e., sjNu isa sequence of the elements of N0 in which they appear exactly inthe same order as in s.
The following definition [17,31] determines sequencing func-tions, for which any optimal sequence on a job module can beextended into an optimal sequence for the whole sequencingproblem.
Definition 2. An objective function F possesses the job module
property if it satisfies the following condition: If N0 is a job moduleof a poset P(N, R) and s0 is an optimal sequence for the problem ofminimizing F over PuðNu,RuÞ, where PuðNu,RuÞ is a subposet of P
induced by Nu �N, then there exists an optimal permutation s forP such that su ¼ sjNu.
Informally, the job module property states that any optimalsolution to a problem defined by a job module is consistent withat least one optimal solution to the entire problem. The jobmodule property is required for the development of efficientsequencing algorithms [17,31]. These algorithms obtain optimalsequences by finding optimal subsequences for progressivelylarger modules and use efficient procedures [18,27] for locatingmodules in the precedence network.
Monma and Sidney [31] have shown that the following threeconditions are sufficient for the job module property to hold:
1.
Strong adjacent sequence interchange property which is valid foran objective function F if there exists a (transitive) ‘‘pre-ference’’ relation ! defined on all pairs of sequences asfollows: for all sequences a, b, p0 and p00, a!b if and only ifF(p0,a, b, p00)rF(p0, b, a, p00).2.
Strong series network decomposition (strong SND) propertywhich is valid for an objective function F if the followingcondition holds for all permutations a and b of the same set:for all sequences p0 and p00, the inequality F(a)rF(b) is valid ifand only if F(p0, a, p00)rF(p0, b, p00).
A. Dolgui et al. / Computers & Operations Research 39 (2012) 1218–1224 1221
3.
Consistency property which is valid for an objective function Fwith the preference relation ! defined on all pairs ofsequences satisfying the following condition: for all permuta-tions a and b of the same set, if F(a)rF(b) then a!b.
The strong adjacent sequence interchange property is equiva-lent to the fact that the objective function is priority-generating,i.e., the preference relation ! used in this property and in theconsistency property (a!b) is equivalent to o(a)Zo(b), whereo is a priority function of the objective function F. Thus, thisproperty holds for all problems under consideration in this papersince their objective functions are priority-generating.
The consistency property can be formulated in terms ofpriority functions as follows. A priority-generating function F(p)with a priority function o(p) possesses the consistency property ifthe following condition holds: for all permutations a and b of thesame set, the inequality F(a)rF(b) implies o(a)Zo(b).
Sidney and Steiner [17] propose a combination of DP withmodular decomposition which enlarges the polynomially solvableclasses of sequencing problems. The DP approach uses thefollowing recursion property of the objective function F(N) overthe initial sets S:
FðSÞ ¼minfgðFðS\fjgÞ,S,jÞjjA IðSÞg,
where Fð+Þ ¼ 0 and g is a problem-dependent recursion formula.If the recursion g can be computed in constant time for any
given set of arguments, then the computation of F(N) requiresO(Kw) time, where K denotes the number of initial subsets of agiven poset P and w is the width of P. After F(N) has beencalculated, an optimal permutation can be found in O(nw) time bybacktracking: we start with S¼N and identify a job jAS for whichthe minimum is obtained in the recursion property; this job j isplaced last among the jobs in S, and S is replaced by S\{j} until weobtain S\fjg ¼+. Note that the running time reduces to O(n),provided that the last element j for each created optimalsubsequence is stored.
Sidney and Steiner [17] define the class Dw of posets such thatfor any positive w a poset P belongs to Dw if and only if P can bebuilt by a finite number of successive compositions of posets inwhich every outer factor has a width at most w. The members ofDw can be recognized in polynomial time using canonicaldecomposition of posets and their composition trees [17,26,27].It is important that D2 contains all series–parallel posets.
The following statement is due to Sidney and Steiner [17].
Theorem 2. [17]. Consider a function F that satisfies the recursion
property and the job module property, and a poset P that belongs to
Dw for wZ2. Then the sequencing problem of minimizing F over P is
solvable in O(nw + 1) time. Moreover, if in P the cardinality of each
N-type outer factor is at most m for a fixed constant m, then the
problem is solvable in time O(nm�1).
Thus, the algorithm that combines DP and the poset modulardecomposition solves in polynomial time those sequencingproblems in which the objective functions have both the recursionproperty and the job module property, provided that the width ofeach outer factor of the corresponding poset is bounded.
In Sections 4 and 5, we consider the cumulative model ofpositional deterioration and the exponential model of positionallearning for a single machine scheduling problem of minimizingthe makespan under precedence constraints. We show that theseproblems have both the recursion and job module properties, andtherefore are polynomially solvable if precedence constraintsbelong to the class in which the width of each outer factor isbounded by some fixed value.
4. Cumulative positional deterioration
In this section, we consider the deterioration model in whichthe processing time of a job depends polynomially on the total‘normal’ processing time of the jobs that have been sequencedbefore it, i.e., the actual processing time pj
[r] of a job j scheduled inposition r of a permutation p is given by
p½r�j ¼ pj
�1þ
Xr�1
k ¼ 1
ppðkÞ
�A, ð2Þ
where pj is the ‘normal’ processing time of job j, while A is a givenpositive constant which is common for all jobs.
This model is introduced (not for deteriorating, but for alearning effect with Ao0), by Kuo and Yang [32,33]. Gordon et al.[14] show that, for the cumulative positional deterioration model,the problem of minimizing the makespan is solvable in O(n) timeunder general precedence constraints if A¼1, and in Oðn log nÞ
time under series–parallel precedence constraints if A¼2.The following statements are valid for the problem of minimizing
the makespan under cumulative positional deterioration.
Lemma 1. The strong SND property holds for the objective function
Cmax if the processing time of each job is defined by (2) and A¼2.
Proof. Let p be a (partial) permutation of jobs contained as asubsequence in some schedule. Assume that (i) the first job in p startsat time t; and (ii) the sum of the ‘standard’ processing times of thejobs that precede the first job in p, i.e., those completed by time t, isequal to t. Under these assumptions, let Cmaxðp,t,tÞ denote themaximum completion time of the jobs in p. By definition,
Cmaxðp,t,tÞ ¼ tþCmaxðp,0,tÞ
¼ tþXjpj
j ¼ 1
ppðjÞ
�1þtþ
Xj�1
i ¼ 1
ppðiÞ
�A: ð3Þ
Let a and b be permutations of the same set, and m be the
number of elements in a and b, i.e., m¼ |a|¼ |b|. If the objective
function F is Cmax, then
FðaÞ ¼ Cmaxða,0,0Þ ¼Xm
j ¼ 1
paðjÞ
�1þ
Xj�1
i ¼ 1
paðiÞ
�A,
FðbÞ ¼ Cmaxðb,0,0Þ ¼Xm
j ¼ 1
pbðjÞ
�1þ
Xj�1
i ¼ 1
pbðiÞ
�A
and the inequality F(a)rF(b) can be rewritten as
Xm
j ¼ 1
paðjÞ
�1þ
Xj�1
i ¼ 1
paðiÞ
�ArXm
j ¼ 1
pbðjÞ
�1þ
Xj�1
i ¼ 1
pbðiÞ
�A: ð4Þ
Let pa ¼ ðpu,a,p00 Þ and pb ¼ ðpu,b,p00 Þ be two permutations of all
jobs that only differ in the subsequences a and b of the same set of
jobs. To prove the lemma we demonstrate that (4) is valid for A¼2
if and only if FðpaÞrFðpbÞ, i.e., if and only if
Cmaxðpa,0,0ÞrCmaxðpb,0,0Þ: ð5Þ
Let tu denote the total sum of the ‘standard’ processing timesu u
00
of the jobs in p . It follows from (3) that Cmaxðp ap ,0,0Þ ¼
Cmaxðpu,0,0ÞþCmaxðap00
,0,tuÞ and Cmaxðpubp00 ,0,0Þ ¼ Cmaxðpu,0,0Þ
þCmaxðbp00
,0,tuÞ, so that (5) reduces to
Cmaxðap00
,0,tuÞrCmaxðbp00
,0,tuÞ: ð6Þ
Furthermore,
Cmaxðap00
,0,tuÞ ¼ Cmaxða,0,tuÞ
þXjp00 j
k ¼ 1
pp00 ðkÞ
�1þtuþ
Xm
i ¼ 1
paðiÞ þXk�1
i ¼ 1
pp00 ðiÞ
�A
A. Dolgui et al. / Computers & Operations Research 39 (2012) 1218–12241222
and
Cmaxðbp00
,0,tuÞ ¼ Cmaxðb,0,tuÞ
þXjp00 j
k ¼ 1
pp00 ðkÞ
�1þtuþ
Xm
i ¼ 1
pbðiÞ þXk�1
i ¼ 1
pp00 ðiÞ
�A,
so that (taking into account thatPm
i ¼ 1 paðiÞ ¼Pm
i ¼ 1 pbðiÞ) (6)
reduces to Cmaxða,0,tuÞrCmaxðb,0,tuÞ. Further, we rewrite this
inequality as
Xm
j ¼ 1
paðjÞ�
1þtuþXj�1
i ¼ 1
paðiÞ�A
rXm
j ¼ 1
pbðjÞ
�1þtuþ
Xj�1
i ¼ 1
pbðiÞ
�A: ð7Þ
In the case that A¼2, inequality (7) is transformed into
Xm
j ¼ 1
paðjÞ�
1þtuþXj�1
i ¼ 1
paðiÞ�2
rXm
j ¼ 1
pbðjÞ
�1þtuþ
Xj�1
i ¼ 1
pbðiÞ
�2,
and reduces to
Xm
j ¼ 1
paðjÞðtuÞ2þXm
j ¼ 1
paðjÞ2tu�
1þXj�1
i ¼ 1
paðiÞ
�
þXm
j ¼ 1
paðjÞ
�1þ
Xj�1
i ¼ 1
paðiÞ
�2rXm
j ¼ 1
pbðjÞðtuÞ2
þXm
j ¼ 1
pbðjÞ2tu�
1þXj�1
i ¼ 1
pbðiÞ
�þXm
j ¼ 1
pbðjÞ
�1þ
Xj�1
i ¼ 1
pbðiÞ
�2,
which due toPm
i ¼ 1 paðiÞ ¼Pm
i ¼ 1 pbðiÞ is equivalent to
2tuXm
j ¼ 1
paðjÞ
�1þ
Xj�1
i ¼ 1
paðiÞ
�þXm
j ¼ 1
paðjÞ
�1þ
Xj�1
i ¼ 1
paðiÞ
�2
r2tuXm
j ¼ 1
pbðjÞ
�1þ
Xj�1
i ¼ 1
pbðiÞ
�þXm
j ¼ 1
pbðjÞ
�1þ
Xj�1
i ¼ 1
pbðiÞ
�2:
Taking into account that bothPm
j ¼ 1 paðjÞPj�1
i ¼ 1 paðiÞ andPm Pj�1
j ¼ 1 pbðjÞ i ¼ 1 pbðiÞ are the sums of all possible productspaðkÞpaðlÞ, ka l, and pbðkÞpbðlÞ, ka l, of the elements of the
same set, and thereforePm
j ¼ 1 paðjÞPj�1
i ¼ 1 paðiÞ ¼Pm
j ¼ 1 pbðjÞPj�1i ¼ 1 pbðiÞ, the last displayed inequality reduces toPmj ¼ 1 paðjÞð1þ
Pj�1i ¼ 1 paðiÞÞ
2rPm
j ¼ 1 pbðjÞð1þPj�1
i ¼ 1 pbðiÞÞ2, which ex-
actly corresponds to (4) if A¼2. This completes the proof of the
lemma. &
Lemma 2. The consistency property holds for the objective
function Cmax if the processing time of each job is defined by (2)and A¼2.
Proof. In the case that the processing times are defined by (2) andA¼2 we have from Gordon et al. [14] that for the objectivefunction FðpÞ ¼ Cmax the priority function is given by
oðpÞ ¼Pjpj
j ¼ 1 ppðjÞPjpjj ¼ 1 p2
pðjÞ:
Thus
oðaÞ ¼Pjaj
j ¼ 1 paðjÞPjajj ¼ 1 p2
aðjÞ¼
Pjbjj ¼ 1 pbðjÞPjbjj ¼ 1 p2
bðjÞ
¼oðbÞ
and the consistency property holds immediately since theequality o(a)¼o(b) takes place independently of the inequalityF(a)rF(b). &
Lemma 3. The objective function Cmax possesses the recursion
property if the processing time of each job is defined by (2)and A¼2.
Proof. To verify that the recursion property holds for Cmax, it issufficient to use the following recursion formula:
gðCmaxðS\fjgÞ,S,jÞ ¼ CmaxðS\fjgÞþpjð1þXjSj
k ¼ 1
ppðkÞÞ2,
where p is derived by taking an optimal sequence p0 for S\{ j} andappending j to it in the last position. This formula is computed inO(|S|) time, i.e., no more than in O(n) time. &
Theorem 3. The problem of minimizing the makespan is polynomi-
ally solvable, provided that the processing time of each job is defined
by (2) with A¼2 and the precedence constraints are defined by a
poset with a fixed width of each outer factor.
Proof. The objective function Cmax is priority-generating if A¼2[14] and therefore possesses the strong adjacent sequenceinterchange property. According to Lemmas 1 and 2, Cmax alsopossesses the strong SND property and the consistency property,and therefore possesses the job module property. Thus, fromLemma 3 and Theorem 2 the proof is completed. &
5. Exponential positional learning
In this section we consider the exponential positional learningeffect. Under this model, the actual processing time of a job j thatis sequenced in position r is given by
p½r�j ¼ pjgr�1, ð8Þ
where 0ogo1 is a given constant representing a learning ratewhich is common for all jobs.
Let p be a (partial) permutation of jobs contained as asubsequence in some schedule. Assume that (i) the first job inthis permutation is sequenced in position r of an overall schedule;and (ii) this first job starts at time tZ0. Under these assumptions,let Cmaxðp,t,rÞ denote the maximum completion time of thejobs in p.
It is clear that
Cmaxðp,0,1Þ ¼Xjpj
j ¼ 1
ppðjÞgj�1
and
Cmaxðp,t,rÞ ¼ tþgr�1Cmaxðp,0,1Þ:
Gordon et al. [14] show that the objective function Cmax is
priority-generating with the priority function given byoðpÞ ¼�Cmaxðp,0,1Þ
1�gjpj ¼ �
Pjpjj ¼ 1 ppðjÞgj�1
1�gjpj ð9Þ
and therefore the problem of minimizing the makespan underseries–parallel precedence constraints is solvable in Oðn log nÞ
time.The following statements are valid for the problem of
minimizing the makespan with an exponential positional learningeffect.
Lemma 4. The strong SND property holds for the objective
function Cmax if the processing time of each job is defined by (8)and 0ogo1.
Proof. Let a and b be permutations of the same set, |a|¼ |b|¼m.
Then CmaxðaÞ ¼ Cmaxða,0,1Þ ¼Pm
j ¼ 1 paðjÞgj�1 and CmaxðbÞ ¼ Cmax
ðb,0,1Þ ¼Pm
j ¼ 1 pbðjÞgj�1.
A. Dolgui et al. / Computers & Operations Research 39 (2012) 1218–1224 1223
Let pa ¼ ðpu,a,p00 Þ and pb ¼ ðpu,b,p00 Þ. Consider
DC ¼ CmaxðpaÞ�CmaxðpbÞ ¼ Cmaxðpu,a,p00 ,0,1Þ�Cmaxðpu,b,p00 ,0,1Þ:To prove the lemma we have to show that CmaxðaÞrCmaxðbÞ if
and only if DCr0.
Assume that jpuj ¼ r�1. It follows that Cmaxðpu,a,p00 ,0,1Þ
¼ Cmaxðpu,0,1ÞþCmaxðap00
,0,rÞ and Cmaxðpu,b,p00 ,0,1Þ ¼ Cmaxðpu,0,1Þ
þCmaxðbp00
,0,rÞ, so that
DC ¼ Cmaxðap00
,0,rÞ�Cmaxðbp00
,0,rÞ:
Using this equation, we further deduce that
DC ¼ ðCmaxða,0,rÞþCmaxðp00
,0,rþmÞÞ
�ðCmaxðb,0,rÞþCmaxðp00
,0,rþmÞÞ
¼ Cmaxða,0,rÞ�Cmaxðb,0,rÞ
¼ gr�1ðCmaxða,0,1Þ�Cmaxðb,0,1ÞÞ
¼ gr�1ðCmaxðaÞ�CmaxðbÞÞ
as required. &
Lemma 5. The consistency property holds for the objective function
Cmax if the processing time of each job is defined by (8) and 0ogo1.
Proof. For the problem of minimizing the makespan under theexponential learning scenario with 0ogo1, the objective func-tion is priority-generating and (9) holds, i.e.,
oðpÞ ¼�Cmaxðp,0,1Þ
1�gjpj¼ �
Pjpjj ¼ 1 ppðjÞgj�1
1�gjpj:
If Cmax(a)rCmax(b), then �CmaxðaÞZ�CmaxðbÞ and for 0ogo1
we have o(a)Zo(b), as required. &
Lemma 6. The objective function Cmax possesses the recursion
property if the processing time of each job is defined by (8) and
0ogo1.
Proof. To verify that the recursion property holds for Cmax, it issufficient to use the following recursion formula:
gðCmaxðS\fjgÞ,S,jÞ ¼ CmaxðS\fjgÞþgjSj�1pj,
which is computed in constant time given the number of jobs in S,and in any case no more than in O(n) time. &
Theorem 4. The makespan minimization problem is polynomially
solvable if the processing time of each job is defined by (8) with
0ogo1 and the precedence constraints are defined by a poset with
the fixed width of each outer factor.
Proof. The objective function Cmax is priority-generating if0ogo1 [14] and therefore possesses the strong adjacentsequence interchange property. According to Lemmas 4 and 5,Cmax possesses also the strong SND property and the consistencyproperty and therefore possesses the job module property. Thus,from Lemma 6 and Theorem 2 the proof is completed. &
6. Conclusions
In this paper, we present an approach to solving singlemachine scheduling problems with positionally dependent pro-cessing times under precedence constraints. We consider pro-blems of minimizing the makespan with positional cumulativedeterioration or positional exponential learning. We show thatthese problems have both the recursion and the job moduleproperties and therefore they can be solved in polynomial time ifthe precedence constraints belong to the class in which the widthof each outer factor is bounded by a fixed value.
Our further research will be directed to finding some otherclasses of scheduling problems with positionally dependent
processing times which can be polynomially solvable under givenprecedence constraints.
References
[1] Shabtay D, Steiner G. A survey of scheduling with controllable processingtimes. Discrete Applied Mathematics 2007;155:1643–66.
[2] Alidaee B, Womer NK. Scheduling with time dependent processing times:review and extensions. Journal of the Operational Research Society 1999;50:711–20.
[3] Cheng TCE, Ding Q, Lin BMT. A concise survey of scheduling with timedependent processing times. European Journal of Operational Research2004;152:1–13.
[4] Biskup D. A state-of-the-art review on scheduling with learning effects.European Journal of Operational Research 2008;188:315–29.
[5] Lawler EL. Optimal sequencing of a single machine subject to precedenceconstraints. Management Science 1973;19:544–6.
[6] Lawler EL. Sequencing jobs to minimize total weighted completion timesubject to precedence constraints. Annals of Discrete Mathematics 1978;2:75–90.
[7] Baker KR, Lawler EL, Lenstra JK, Rinnooy Kan AHG. Preemptive scheduling of asingle machine to minimize maximum cost subject to release dates andprecedence constraints. Operations Research 1983;31:381–6.
[8] Gordon VS, Tanaev VS. On minimax problems of scheduling theory for asingle machine. Vetsi Akademii Navuk BSSR, Seria Fizika-MatematychnyhNavuk 1983;3:3–9. (in Russian).
[9] Gordon VS. A note on optimal assignment of slack due-dates insingle-machine scheduling. European Journal of Operational Research 1993;70:311–5.
[10] Gordon VS, Strusevich VA. Earliness penalties on a single machine subject toprecedence constraints: SLK due date assignment. Computers and OperationsResearch 1999;26:157–77.
[11] Gordon VS, Shafransky YM. Optimal ordering with series–parallel precedenceconstraints. Doklady Akademii Nauk BSSR 1978;22:244–7. (in Russian).
[12] Tanaev VS, Gordon VS, Shafransky YM. 1984. Scheduling theory. Single-stagesystems. Nauka: Moscow (in Russian) (translated into English by KluwerAcademic, Dordrecht, 1994).
[13] Wang J-B, Ng CT, Cheng TCE. Single-machine scheduling with deterioratingjobs under a series–parallel graph constraint. Computers and OperationsResearch 2008;35:2684–93.
[14] Gordon VS, Potts CN, Strusevich VA, Whitehead JD. Single machinescheduling models with deterioration and learning: handling precedenceconstraints via priority generation. Journal of Scheduling 2008;11:357–70.
[15] Janiak A, Kovalyov MY. Scheduling in a contaminated area: a model and poly-nomial algorithms. European Journal of Operational Research 2006;173:125–32.
[16] Monma CL, Sidney JB. Sequencing with series–parallel precedence con-straints. Mathematics of Operations Research 1979;4:215–34.
[17] Sidney JB, Steiner G. Optimal sequencing by modular decomposition:polynomial algorithms. Operations Research 1986;34:606–12.
[18] Mohring RH, Rademacher FJ. Substitution decomposition for discretestructures and connections with combinatorial optimization. Annals ofDiscrete Mathematics 1984;19:257–356.
[19] Mohring RH, Rademacher FJ. Generalized results on the polynomiality ofcertain weighted sum scheduling problems. Methods of Operations Research1985;49:405–17.
[20] Sidney JB. Decomposition algorithms for single-machine sequencing withprecedence relations and deferral costs. Operations Research 1975;23:238–98.
[21] Glazebrook KD, Gittins JC. On single machine scheduling with precedencerelations and linear or discounted costs. Operations Research 1981;29:161–74.
[22] Kadane JB, Simon HA. Optimal strategies for a class of constrained sequentialproblems. Annals of Statistics 1977;5:237–55.
[23] Steiner G. On finding the jump number of a partial order by substitutiondecomposition. Order 1985;2:9–23.
[24] Gordon VS, Proth J-M, Strusevich VA. Single machine scheduling and due dateassignment under series–parallel precedence constraints. Central EuropeanJournal of Operations Research 2005;13:15–35.
[25] Valdes JR, Tarjan E, Lawler EL. The recognition of series–parallel digraphs.SIAM Journal on Computing 1982;11:361–70.
[26] Buer H, Mohring RH. A fast algorithm for the decomposition of graphs andposets. Mathematics of Operations Research 1983;8:170–84.
[27] Muller JH, Spinrad J. Incremental modular decomposition: polynomial algo-rithms. Journal of the Association for Computing Machinery 1989;36:1–19.
[28] Dahlhaus E, Gustedt J, McConnell RM. Efficient and practical algorithms forsequential modular decomposition. Journal of Algorithms 2001;41:360–87.
[29] Lawler EL, Lenstra JK, Rinnooy Kan AHG, Shmoys DB. Sequencing and scheduling:algorithms and complexity. In: Graves SC, Rinnooy Kan AHG, Zipkin PH, editors.Handbooks in operations research and management science, vol. 4, logistics ofproduction and inventory. Amsterdam: North-Holland; 1993. p. 445–522.
A. Dolgui et al. / Computers & Operations Research 39 (2012) 1218–12241224
[30] Shafransky YM. On a problem of minimizing functions over a set ofpermutations of partially ordered elements, Part I. Vestsi Akademii NavukBSSR, Seria Fizika-Matematychnykh Navuk 1980;5:132. (in Russian).
[31] Monma CL, Sidney JB. Optimal sequencing via modular decomposition:characterization of sequencing functions. Mathematics of Operations Re-search 1987;12:22–31.
[32] Kuo W-H, Yang D-L. Minimizing the makespan in a single machinescheduling problem with a with a time-based learning effect. InformationProcessing Letters 2006;97:64–7.
[33] Kuo W-H, Yang D-L. Minimizing the total completion time in a single-machine scheduling problem with a time-dependent learning effect.European Journal of Operational Research 2006;174:1184–90.
