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<ul><li><p>Ruling</p><p>. D</p><p>lytech</p><p>l Chi</p><p>; acce</p><p>Abstract</p><p>1. Introduction namic behaviors characterized by a set of dynamic</p><p>namic parameters considered by Gupta et al.</p><p>(1987) or some earlier Russian papers (e.g. Tanaev</p><p>et al., 1994), Gupta and Gupta (1988) introduced</p><p>an interesting scheduling model in which the pro-*</p><p>European Journal of Operational ReCorresponding author. Tel.: +852-2766-5215; fax: +852-</p><p>2364-5245.Machine scheduling problems with time-de-</p><p>pendent processing times have received increas-</p><p>ing attention in recent years. For many years,</p><p>most scheduling research has focused on problems</p><p>with deterministic parameters. As mentioned in</p><p>Rinnooy Kan (1976), the traditional restrictiveassumptions may correspond to a somewhat over-</p><p>simplied picture of reality, though they can take</p><p>great advantage of computational convenience. In</p><p>real-life applications, many systems exhibit dy-</p><p>parameters. This fact is commonly recognized in</p><p>control theory, systems engineering and many</p><p>other areas. But scheduling problems with dy-</p><p>namic parameters have been studied only in a few</p><p>papers. It should, however, be noted that a con-</p><p>siderable body of literature has existed for sto-</p><p>chastic scheduling that deals with schedulingproblems in an environment of uncertainty, see,</p><p>Mohring and Rademacher (1985), Righter (1994)</p><p>and Pinedo (1995).</p><p>Based on some scheduling problems with dy-We consider a class of machine scheduling problems in which the processing time of a task is dependent on its</p><p>starting time in a schedule. On reviewing the literature on this topic, we provide a framework to illustrate how models</p><p>for this class of problems have been generalized from the classical scheduling theory. A complexity boundary is pre-</p><p>sented for each model and related existing results are consolidated. We also introduce some enumerative solution al-</p><p>gorithms and heuristics and analyze their performance. Finally, we suggest a few interesting areas for future research.</p><p> 2003 Elsevier B.V. All rights reserved.</p><p>Keywords: Survey; Scheduling; Sequencing; Time dependence; Computational complexityInvited</p><p>A concise survey of schedprocess</p><p>T.C.E. Cheng a,*, Qa Department of Management, The Hong Kong Pob Department of Information Management, Nationa</p><p>Received 22 April 2002E-mail addresses: mscheng@polyu.edu.hk (T.C.E. Cheng),</p><p>mtlin@ncnu.edu.tw (B.M.T. Lin).</p><p>0377-2217/$ - see front matter 2003 Elsevier B.V. All rights reservdoi:10.1016/S0377-2217(02)00909-8eview</p><p>ing with time-dependenttimes</p><p>ing a, B.M.T. Lin b</p><p>nic University, Hung Hom, Kowloon, Hong Kong</p><p>Nan University, Pu-Li, Nan-Tou County, Taiwan</p><p>pted 4 December 2002</p><p>search 152 (2004) 113</p><p>www.elsevier.com/locate/dswcessing time of a task is a polynomial function of</p><p>its starting time. From a modeling perspective,</p><p>ed.</p></li><li><p>l of Ohowever, the makespan scheduling problem withquadratic time-dependent processing times is al-</p><p>ready very intricate. For this reason, most subse-</p><p>quent research along this line has concentrated on</p><p>problems with linear or piecewise linear time-</p><p>dependent processing times.</p><p>This model reects some real-life situations in</p><p>which the expected processing time of a task in-</p><p>creases/decreases linearly or piecewise linearly withits starting time. Examples can be found in nan-</p><p>cial management, steel production, resource allo-</p><p>cation and national defense, where any delay in</p><p>tackling a task may result in an increasing/de-</p><p>creasing eort (time, cost, etc.) to accomplish the</p><p>task. The reader is referred to Kunnathur and</p><p>Gupta (1990), Mosheiov (1994, 1996a) and Sun-</p><p>dararaghavan and Kunnathur (1994) for a list ofapplications. Moreover, it seems that in other</p><p>cases, for example, re ghting, learning eect and</p><p>maintenance scheduling, a linear or piecewise lin-</p><p>ear function is a close approximation of the actual</p><p>nonlinear phenomenon.</p><p>Research on time-dependent problems has</p><p>spawned a new area in the scheduling eld. It has</p><p>uncovered many new properties neglected in theclassical scheduling theory and led to ecient</p><p>methodological approaches to algorithm design</p><p>and NP-complete reduction. For example, tech-</p><p>niques based on reductions from a multiplicative</p><p>type NP-complete problem, such as SUBSETSUBSET</p><p>PRODUCTPRODUCT, are crucial to the NP-completeness</p><p>proofs for many time-dependent scheduling prob-</p><p>lems. Regarding the development of polynomialtime algorithms, a very interesting phenomenon is</p><p>related to the existence of algorithms with time</p><p>complexities of On5 or On6 log n, which is notso common in the deterministic scheduling litera-</p><p>ture. Thus, research on these problems is signi-</p><p>cant in both practical and theoretical senses.</p><p>Alidaee and Womer (1999) presented a review</p><p>on scheduling problems with time-dependent pro-cessing times. Our study aims not only at survey-</p><p>ing recent developments in this line of research but</p><p>also at investigating several unsolved problems.</p><p>Based upon state-of-the-art status of research on</p><p>scheduling problems with time-dependent pro-</p><p>cessing times, we discuss the relationships of dif-</p><p>2 T.C.E. Cheng et al. / European Journaferent models and explicate how they aregenerated from a basic linear model. A complexityboundary is presented for each model and existing</p><p>and new results are consolidated. For the intrac-</p><p>table problems, we also introduce some enumera-</p><p>tive solution algorithms and heuristics and analyze</p><p>their performance. Finally, we give some insights</p><p>into scheduling problems of this type, which reveal</p><p>several potential future directions of research for</p><p>this exciting eld of study.In Section 2, we introduce a notation-and-</p><p>model system for the scheduling problem with</p><p>time-dependent processing times. In Section 3, we</p><p>present a set of complexity results for each model.</p><p>In Section 4, we illustrate a series of polynomial</p><p>and pseudo-polynomial algorithms. In Section 5,</p><p>we discuss some enumerative and heuristic solu-</p><p>tion algorithms in the literature. Some concludingremarks and suggestions for future research are</p><p>given in Section 6.</p><p>2. Notation and models</p><p>Since most of the time-dependent scheduling</p><p>problems are a natural generalization of their</p><p>classical counterpart, we adopt the notation, de-</p><p>nitions and assumptions prevalent in classical</p><p>scheduling theory, see the survey of Graham et al.</p><p>(1976).Research on time-dependent problems has</p><p>mainly dealt with the single machine model, with</p><p>only a few exceptions dealing with the parallel</p><p>machine and ow shop situations. The objective</p><p>has been conned to the minimization of a handful</p><p>of traditional regular performance measures, such</p><p>as makespan, ow time, maximum lateness and</p><p>number of tardy tasks. The time-dependent func-tion used to model the processing time of a task is</p><p>usually a linear or piecewise linear function of the</p><p>starting time of the task in a schedule. We give</p><p>below a formal statement for the basic linear</p><p>model.</p><p>A task system consisting of n independent tasksis denoted by TS fTig; faig; fbig; frig; fdig.Each task Ti is associated with a release time ri anda due date di, and is characterized by a normalprocessing time ai P 0 and a processing rate bi.The actual processing time of Ti depends on its</p><p>perational Research 152 (2004) 113start time si and is given by pi ai bisi.</p></li><li><p>pi bisi, if si 0, then every actual processingtime is 0, a trivial case. Thus, we assume that si P e</p><p>l of OGupta et al. (1987), Gupta and Gupta (1988),Browne and Yechiali (1990), Gawiejnowicz and</p><p>Pankowska (1995) and some earlier Russian pa-</p><p>pers (e.g. Tanaev et al., 1994) proposed the model</p><p>pi ai bisi from dierent perspectives. Themodel reects real-life situations, such as searching</p><p>for an object under worsening weather or perfor-</p><p>mance of medical treatments under deteriorating</p><p>health conditions, where any delay may incur extraeorts to accomplish the task (Mosheiov, 1994).</p><p>Motivated by a military application concerning</p><p>aerial treats, Ho et al. (1993) proposed the model</p><p>pi ai bisi with deadline di. Some special casesof the model pi ai bisi have also been investi-gated in the literature.</p><p>Mosheiov (1994) rst considered the special</p><p>case with ai 0, i.e., the model pi bisi. For thecases with bi b or ai a, the problems with adeadline or release time restriction have been</p><p>studied by Cheng and Ding (1998a,b, 1999, 2000,</p><p>2003). Next we introduce some piecewise linear</p><p>models.</p><p>In some situations, if a task starts after a time</p><p>di, its processing time deteriorates with its startingtime. Note that the parameter di is not the classicaldeadline or due date, but a kind of deteriorating</p><p>(decreasing) break point. Practical examples arise</p><p>from jobbing production where jobs are produced</p><p>at a normal or overtime cost depending on whe-</p><p>ther the job is started before a specied time point,</p><p>i.e., the break point. Kunnathur and Gupta (1990)</p><p>proposed a model with piecewise increasing pro-</p><p>cessing times, denoted by pi maxfai; aibisi dig. Sundararaghavan and Kunnathur(1994) and Mosheiov (1995) independently intro-</p><p>duced another model with step deteriorating pro-</p><p>cessing times, denoted by pi ai or pi ai bi,where pi ai, if si6 di; pi ai bi, otherwise.When the parameter di is considered, only themakespan and ow time problems have been</p><p>considered in the literature.To approximate the learning eect, Cheng et al.</p><p>(2003) studied a model with piecewise linear de-</p><p>creasing processing times, denoted by pi ai biminfsi; dig. In this model, if a task startsbefore or at di, its processing time decreases withits starting time linearly; otherwise, the decrease of</p><p>T.C.E. Cheng et al. / European Journaits processing time is a constant bidi. Bachman et al.for the model pi bisi, where e is a given smallpositive number.</p><p>For the model pi ai or pi ai bi, all pa-rameters are assumed to be integers. For the other</p><p>models, the normal processing times ai are as-sumed to be integers. Since the deteriorating rates</p><p>bi are rational numbers in most practical cases andthe actual processing times are always greatly af-</p><p>fected by some exponential terms of the corre-</p><p>sponding deteriorating rates, the other parameters</p><p>are allowed to be positive rational numbers.Discussions and justications for these as-</p><p>sumptions are provided in the remainder of this</p><p>paper when necessary. Table 1 presents a list of the</p><p>dierent models that have appeared in the litera-</p><p>ture, along with the corresponding references. Fig.</p><p>1 illustrates the time-dependent function of each</p><p>model.</p><p>3. NP-complete problems</p><p>The NP-complete results for the class of clas-</p><p>sical scheduling problems have been well re-</p><p>searched and surveyed in several papers and</p><p>books, for example, Rinnooy Kan (1976), Graham</p><p>et al. (1976) and Lawler et al. (1993). However, thetime-dependent scheduling problems present quite</p><p>dierent boundaries for the computational com-</p><p>plexities of the problems. In this section, we in-</p><p>troduce the new scheduling results, starting from</p><p>the simple models and gradually progressing to(2002b) investigated the model pi ai bisi, illus-trating with an example where a worker gains</p><p>knowledge and skills when assembling a large</p><p>quantity of similar products.</p><p>For the models pi ai bisi and pi ai biminfsi; dig, if bi > 1, then some performancemeasures become nonregular. If ai < bidi, then theactual processing time may reduce to 0. To avoid</p><p>these unrealistic and uninteresting cases, we as-sume that 0 < bi < 1 and ai > bidi for these twomodels. For the case without the deadline restric-</p><p>tion, we assume ai > biP</p><p>aj. For the model</p><p>perational Research 152 (2004) 113 3their generalizations.</p></li><li><p>7), Gu</p><p>, 1996</p><p>d Pan</p><p>; Hsie</p><p>wiejn</p><p>upta</p><p>nd va</p><p>and</p><p>l of OTable 1</p><p>Time-dependent scheduling models and references</p><p>Model References</p><p>pi ai bisi Gupta et al. (198(1991, 1994, 1995</p><p>Gawiejnowicz an</p><p>1999, 2000, 2003)</p><p>Kononov and Ga</p><p>pi maxfai; ai bisi dig Kunnathur and G(1998), Kubiak a</p><p>pi ai or pi ai bI Sundararaghavan</p><p>4 T.C.E. Cheng et al. / European JournaIn the literature, studies involving multiple</p><p>machines or shops center around the two models</p><p>pi bisi and pi ai bisi. For the model pi bisi,Mosheiov (1994, 1998), Chen (1996) and Mos-</p><p>heiov (1996a, 2002) reduced the SUBSET PRODUCTSUBSET PRODUCT</p><p>problem (Garey and Johnson, 1979) to P2jpi bisijCmax, P2jpi bisij</p><p>PCi and F 3jpi bisijCmax</p><p>respectively, where the respective parameters were</p><p>explained at the beginning of Section 2 and the</p><p>three-eld notation is adopted from Graham et al.</p><p>(1976). Kononov (1997) independently established</p><p>(2001), Jeng and Lin (2</p><p>pi ai biminfsi; dig Cheng et al. (2003)</p><p>Fig. 1. Time-dependent functions.the ordinary NP-completeness of P2jpi bisijCmaxand P2jpi bisij</p><p>PCi. As mentioned in Chen</p><p>(1997), SUBSET PRODUCTSUBSET PRODUCT is NP-complete in the</p><p>ordinary sense, not strongly NP-complete as used</p><p>in the above three papers. See also Johnson (1981)for a correction of the complexity results of SUB-SUB-</p><p>SET PRODUCTSET PRODUCT. Thus, these three problems are</p><p>only NP-complete in the ordinary sense and the</p><p>respective original reductions need some modi-</p><p>cation for their correctness. For the open shop and</p><p>job shop problem, Mosheiov (2002) showed that</p><p>O3jpi bisijCmax and J2jpi bisijCmax are NP-complete. The above results demonstrate a com-plexity structure similar to that of classical shop</p><p>oor scheduling problems where pi ai.For the model pi ai bisi on multiple</p><p>machines, Kononov and Gawiejnowicz (2001) pre-</p><p>sented a strong NP-completeness proof for F 3jpi ai bisijCmax by a reduction from 3-PARTITIONPARTITION,</p><p>pta and Gupta (1988), Browne and Yechiali (1990), Mosheiov</p><p>a, 2002), Ho et al. (1993), Woeginger (1995), Chen (1995, 1996),</p><p>kowska (1995), Kononov (1997); Cheng and Ding (1998a,b,</p><p>h and Bricker (1997); Bachman and Janiak (1997, 2000),</p><p>owicz (2001); Ng et al. (2002), Bachman et al. (2002a)</p><p>(1990); Sundararaghavan and Kunnathur (1990), Cai et al.</p><p>n de Velde (1998)</p><p>Kunnathur (1994), Mosheiov (1995, 1998), Cheng and Ding</p><p>002)</p><p>perational Research 152 (2004) 113which is known to be strongly NP-complete</p><p>(Garey and Johnson, 1979). They also presented areduction from PARTITIONPARTITION for the model pi ai bisi with two machines. Moreover, by a re-duction from SUBSET PRODUCTSUBSET PRODUCT, they showed that</p><p>makespan minimization on a three-machine open</p><p>shop with the model pi ai bisi remains NP-complete even if all jobs have the same deteriora-</p><p>tion rate on the third machine.</p><p>For the model pi ai bisi with bi b and dion a single machine, Cheng and Ding (1998a) re-</p><p>duced PARTITIONPARTITION to 1jpi ai bsi; di 2 fD1;D2gjCmax and Cheng and Ding (1999) reduced 3-PAR-PAR-TITIONTITION to 1jpi ai bsi; dijCmax. Naturally, thecorresponding maximum lateness problems are</p></li><li><p>l of Oalso NP-complete and strongly NP-complete,respectively.</p><p>Given a schedule, one convenient approach to</p><p>analyze it is to interchange the positions of one</p><p>pair of adjacent tasks and compare the properties</p><p>(such as the makespan and ow time) of the</p><p>original and the new schedules. Such an approach</p><p>of analysis is called the adjacent jobs interchange</p><p>argument. The use of this approach to facilitate theNP-completeness analysis of a problem is outlined</p><p>in the following exam...</p></li></ul>