Positionally dependent processing time
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
focused on constant processing times (see  for single
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Computers & Operations Research 39 (2012) 12181224rithms for various models with positional and time deterioration/(V. Strusevich).machine problems). Gordon and Shafransky , Tanaev et al. and Wang et al.  consider time deterioration singlemachine problems under the so-called seriesparallel precedenceconstraints. Gordon et al.  propose polynomial time algo-
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Corresponding author at: School of Computing and Mathematical Sciences,University of Greenwich, Old Royal Naval College, Park Row, Greenwich, SE10 9LS
London, UK. Tel.: +442083318662; fax: +442083318665.
E-mail addresses: firstname.lastname@example.org (A. Dolgui), V.Strusevich@greenwich.ac.ukpositional deterioration/learning model, while if the processing imposing precedence constraints on the set of jobs.Most of research on scheduling with precedence constraints isthe 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 a
decreases (a learning effect) as the jobs position in a scheadvances.
It is often found in practice that some products are mfactured in a certain order implied, for example, by technolomarketing or assembly requirements. This can be modeleaffecting 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 on
and Biskup . In these survey papers, one can also nd 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) or1. Introduction
For majority of deterministicprocessing conditions, including jobconsidered as given constants. Hsystems, the processing conditionsbounded width (this class of precedence constraints includes, in particular, seriesparallel 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.
duling problems, thessing times, are usuallyr, in various real-lifeary over time, thereby
time of a job depends on its start time, we refer to the timedeterioration/learning model.
For a survey of scheduling with controllable processing timeswe refer to the paper by Shabtay and Steiner , while for state-of-the-art reviews on deterioration/learning scheduling modelswe refer to the surveys of Alidaee and Womer , Cheng et al. Single machine scheduling with preceddependent processing times
Alexandre Dolgui a, Valery Gordonb , Vitaly Strusea Centre for Industrial Engineering and Computer Science, Ecole Nationale Superieure db United Institute of Informatics Problems, National Academy of Sciences of Belarus, Suc School of Computing and Mathematical Sciences, University of Greenwich, Old Royal
a r t i c l e i n f o
Available online 3 July 2010
a b s t r a c t
In many real-life situation
constants since they vary
machine scheduling prob
depends on its position (wce constraints and positionally
ines de Saint-Etienne, 158, Cours Fauriel, 42023 Saint-Etienne, France
ova 6, 220012 Minsk, Belarus
al College, Park Row, Greenwich, SE10 9LS London, UK
e processing conditions in scheduling models cannot be viewed as given
er time thereby affecting actual durations of jobs. We consider single
s of minimizing the makespan in which the processing time of a job
either cumulative deterioration or exponential learning). It is often found
area. The denitions related to seriesparallel graphs and to
A. Dolgui et al. / Computers & Operations Research 39 (2012) 12181224 1219priority-generating functions are presented formally in Sections 2and 3, respectively.
For sequencing problems with precedence constraints, Sidneyand Steiner  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 theseriesparallel precedence constraints. The problems of this classpossess the recursion property of DP and the so-called job moduleproperty (considered in details in Section 3); this class ofsequencing problems includes the total weighted completion timeproblem , the total discounted cost problem , the least-costfault detection problem  and the jump number problem .Gordon et al.  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 denitions 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 setN 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
prj pjgr, 1
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 effectlearning under precedence constraints. Janiak and Kovalyov 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 , scheduling problems with seriesparallel precedenceconstraints have been studied extensively. A class of objectivefunctions (the so-called priority-generating functions) has beenidentied that allows these functions to be minimized inpolynomial time over seriesparallel precedence constraints. Werefer to Gordon and Shafransky , Monma and Sidney , andTanaev et al.  for the description of the main results in thisfor r 1,2, . . . ,n1. For each individual deterioration model, therelearning models.Formally, the precedence constraints are dened by a binary
relation-. 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 dened 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 |, if X X1 [ X2 and U U1 [ 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 +, if X X1 [ X2 and U U1 [ 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 seriesparallel (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 seriesparallel. ASP-graph can be dened 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 dened by/i,jSAR if and only if i-j. So, in our case, the notions ofprecedence 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 thanseriesparallel, that admit decomposition into the so-called jobmodules of a special structure. In the description of this class ofconstraints, we follow Buer and Mohring , Mohring andRademacher [18,19], Muller and Spinrad . 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) ik for all iANu, where we write ik 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 ~GN,E. We may assume that ~GN,E 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 adjacenttheposito amakespan under given precedence constraints fortional cumulative deterioration and for positional exponentialby Cmax. In this paper, we consider the problems of minimizing
theote the value of the objective function by F(p). In particular,makespan (the maximum completion time) is denotedis a particular rule that explains how exactly the value of pjchanges, 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 specied by a permutation p of jobs, well vertices of N0, or is adjacent to no vertex in N0.
predecessors of i are also in S. For a set A, dene the set
A. Dolgui et al. / Computers & Operations Research 39 (2012) 121812241220IA f jj jAA, j has no successors in Ag.For all initial sets S, the maximum size of any set I(S) is called
the 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 xed width inpolynomial time.
Given a poset, or, equivalently, a directed graph, the decom-position process of partitioning the graph into modules is calledmodular decomposition . 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 describemodular decomposition formally, we use the followingnotation of Sidney and Steiner . For N0 fi1,i2, . . . ,img, let P0(N0,R0) be a poset of m elements. Let Ph(Nh, Rh) for h 1,2, . . . ,m bedisjoint posets. The composition poset P(N, R) is dened byN Smh 1 Nh and R
Smh 1 Rh
Sf/i, jSjiANh, jANk,/ih, ikSAR0g.For this composition, we write P P0P1, . . . ,Pm, where P0 is
referred 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 Pis implemented as an iterative process until all factors have beendecomposed into one-element sets. In each iteration, we decidewhich type of composition applies and nd 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, Muller and Spinrad , Sidney and Steiner , Dahlhauset al. .
3. Priority-generating functions and modular decomposition
For problems in which an optimal schedule is dened 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,  or...