Single machine scheduling problems with position-dependent processing times

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<ul><li><p>JAMCJ Appl Math Comput (2009) 30: 293304DOI 10.1007/s12190-008-0173-x</p><p>Single machine scheduling problemswith position-dependent processing times</p><p>Ji-Bo Wang Li-Yan Wang Dan Wang Xiao-Yuan Wang Wen-Jun Gao Na Yin</p><p>Received: 14 August 2007 / Revised: 2 November 2007 / Published online: 7 October 2008 Korean Society for Computational and Applied Mathematics 2008</p><p>Abstract We consider single machine scheduling problems in which the processingtime of a job depends on its position in a processing sequence. The objectives are tominimize the weighted sum of completion times, to minimize the maximum latenessand to minimize the number of tardy jobs. For some special cases, we prove that theweighted shortest processing time (WSPT) rule, the earliest due date (EDD) rule andMoores algorithm can construct an optimal sequence for these objective functions,respectively. For the general cases, we also give the worst-case bound of these threerules.</p><p>Keywords Scheduling Single machine Learning effect</p><p>Mathematics Subject Classification (2000) 90B35</p><p>1 Introduction</p><p>Machine scheduling problems have received considerable attentions for many years.In traditional scheduling theory, the processing time of a job is independent of itsposition in the production sequence. However, in many realistic scheduling settings,both machines and workers can improve their performance by repeating the produc-tion operations. Therefore, the actual processing time of a job is shorter if it is sched-uled later in a sequence. This phenomenon is known as the learning effect in theliterature (Badiru [2]).</p><p>Biskup [3] is the first one who investigated the learning effect in the framework ofscheduling. He proved that single machine scheduling with a learning effect remains</p><p>J.-B. Wang () L.-Y. Wang D. Wang X.-Y. Wang W.-J. Gao N. YinDepartment of Science, Shenyang Institute of Aeronautical Engineering, Shenyang 110136,Peoples Republic of Chinae-mail: wangjibo75@yahoo.com.cn</p></li><li><p>294 J.-B. Wang et al.</p><p>polynomial solvable if its objective is to minimize the deviation of job completiontimes from a common due date or to minimize the sum of job flow times. Cheng andWang [5] considered a single machine scheduling problem in which the job process-ing times decrease as a result of learning. A piecewise linear processing time functionwas used to model the learning effect. They showed that the maximum lateness mini-mization problem is NP-hard in the strong sense and then identified two special casesthat are polynomially solvable. They also proposed two heuristics and analysed theirworst-case performance. Mosheiov [10, 11] investigated several other single machineproblems and the minimum total flow time problem on identical parallel machines.Liu et al. [8] proved that the weighted shortest processing time (WSPT) rule, theearliest due date (EDD) rule and the modified Moore-Hodgson algorithm can con-struct optimal sequences under certain conditions for the following three objectives:the total weighted completion time, the maximum lateness and the number of tardyjobs, respectively. They also gave an error estimation for each of these rules for thegeneral cases. Mosheiov and Sidney [12] considered a job-dependent learning curve,where the learning rate of some jobs is faster than that of the others. Lee et al. [7]studied a bi-criterion single machine scheduling problem, with the objective of mini-mizing a linear combination of the total completion time and the maximum tardiness.They presented a branch-and-bound and a heuristic algorithm to search for optimaland near optimal solutions. Biskup and Simons [4] considered a scheduling prob-lem where the processing times decrease according to a learning rate, which can beinfluenced by an initial cost-incurring investment. They presented a formulation ofthe common due date scheduling problem with autonomous and induced learningeffects. They further proved some structural properties, which enable the develop-ment of a polynomial bound solution procedure. Mosheiov and Sidney [13] intro-duced a polynomial time solution for the single machine scheduling problem to min-imize the number of tardy jobs with general non-increasing job-dependent learningcurves and a common due-date. Wang and Xia [19] considered flow shop schedulingproblems with a learning effect. The objective is to minimize one of the two regu-lar performance criteria, namely, makespan and total flowtime. They gave a heuristicalgorithm with worst-case bound m for each criteria, where m was the number ofmachines. They also found polynomial time solutions to two special cases of theproblems, i.e., identical processing time on each machine and an increasing seriesof dominating machines. Wang [14] considered flow shop scheduling problems witha learning effect. He suggested the use of Johnsons rule as a heuristic algorithmfor two-machine flow shop scheduling to minimize the makespan. He also developedpolynomial time solution algorithms for some special cases of the following objectivefunctions: the weighted sum of completion times and the maximum lateness. Wanget al. [18] considered a single machine scheduling problems with a general learningeffect. They introduced polynomial solutions for the single machine makespan mini-mization problem, total flow time minimization problem and two class multi-criteriasingle machine problems. Wang [15, 16] and Wang et al. [17] considered schedul-ing problems with the effects of learning and deterioration. A survey of this line ofscheduling research can be found in Bachman and Janiak [1].</p><p>In this paper we consider the same model as that of Wang et al. [18] and Wangand Xia [19], but with different objective functions. The remaining part of this paper</p></li><li><p>Single machine scheduling problems 295</p><p>is organized as follows. In Sect. 2 we formulate the model. In Sects. 3, 4 and 5, weconsider the weighted sum of completion times minimization problem, the maximumlateness minimization problem and the number of tardy jobs minimization problem,respectively. The last section presents the conclusions.</p><p>2 Assumptions</p><p>There are n independent and non-preemptive jobs that are immediately available.They have to be processed on a continuously available machine. The machine canhandle one job at a time and preemption is not allowed. Let pj be the normal process-ing time of job j , the normal processing time of a job is incurred if the job is sched-uled first in a sequence, and p[k] be the normal processing time of a job if it is sched-uled in the kth position in a sequence. Associated with each job j (j = 1,2, . . . , n)is a weight wj and a due-date dj . Let pjr be the processing time of job j if it isscheduled in position r in a sequence. As in Wang et al. [18] and Wang and Xia [19],we consider two special models of job processing time characterized by position-dependent function, namely:</p><p>pjr = pj [M + (1 M)ra], j, r = 1,2, . . . , n, (1)pjr = pj [M + (1 M)br1], j, r = 1,2, . . . , n, (2)</p><p>where a 0 (0 &lt; b 1) denotes a constant learning index of model (1) (model (2)),and 0 M 1.</p><p>For a given schedule = (1,2, . . . , n), Cj = Cj () represents the completiontime of job j . Let Cmax = max{Cj |j = 1,2, . . . , n}, wjCj , Lmax = max{Cj dj |j = 1,2, . . . , n} and Uj , where Uj = 1 if Cj &gt; dj (i.e., the job is late) andUj = 0 otherwise, j = 1,2, . . . , n, represent the makespan, the weighted sum ofcompletion times, the maximum lateness and the number of tardy jobs of a givenschedule, respectively. In the remaining part of the paper, all the problems consideredwill be denoted using the three-field notation scheme introduced by Graham et al. [6].</p><p>3 The weighted sum of completion times minimization problem</p><p>First, we give two lemmas, which are useful for the following theorems.</p><p>Lemma 3.1 [18] For the problem 1|pjr = pj [M + (1 M)ra]|Cmax, an optimalschedule can be obtained by sequencing the jobs in non-decreasing order of pj (i.e.,the smallest processing time (SPT) rule).</p><p>Lemma 3.2 [19] For the problem 1|pjr = pj [M + (1 M)br1]|Cmax, an optimalschedule can be obtained by sequencing the jobs in non-decreasing order of pj (i.e.,the smallest processing time (SPT) rule).</p></li><li><p>296 J.-B. Wang et al.</p><p>Similar to Mosheiov [10], for the objective function of minimizing the weightedsum of completion times, we know that the optimal schedule of the WSPT rule isnot optimal for the problems 1|pjr = pj [M + (1 M)ra]|wjCj , and 1|pjr =pj [M + (1 M)br1]|wjCj , respectively.</p><p>In order to solve 1|pjr = pj [M + (1 M)ra]|wjCj and 1|pjr = pj [M +(1 M)br1]|wjCj approximately, we will use the WSPT rule as a heuristic forthe problems. The performance of the heuristic will be evaluated by its worst-casebound.</p><p>Theorem 3.1 Let S be an optimal schedule and S be a WSPT schedule for theproblem 1|pjr = pj [M + (1 M)ra]|wjCj . Then</p><p>wjCj (S)</p><p>wjCj (S)</p><p> 1M + (1 M)na ,</p><p>and the bound is tight.</p><p>Proof We assume without loss of generality that the jobs are renumbered in the orderof WSPT, i.e., p1/w1 p2/w2 pn/wn. Then</p><p>wjCj (S) = w1p1 + w2[p1 + p2(M + (1 M)2a)] + </p><p>+ wn[p1 + p2(M + (1 M)2a) + + pn(M + (1 M)na)]</p><p>n</p><p>j=1wj</p><p>(j</p><p>k=1pk</p><p>)</p><p>,</p><p>wjCj (S</p><p>) = w[1]p[1] + w[2][p[1] + p[2](M + (1 M)2a)] + + w[n][p[1] + p[2](M + (1 M)2a) + + p[n](M + (1 M)na)]</p><p>n</p><p>j=1w[j ]</p><p>(j</p><p>k=1p[k]</p><p>)</p><p>(M + (1 M)na)</p><p>n</p><p>j=1wj</p><p>(j</p><p>k=1pk</p><p>)</p><p>(M + (1 M)na),</p><p>hence</p><p>wjCj (S)</p><p>wjCj (S) 1</p><p>M + (1 M)na .</p><p>It is not difficult to see that the bound is tight, since if a = 0 or M = 1, we havewjCj (S)wjCj (S</p><p>) = 1. This result is intuitive as when a = 0 or M = 1, the WSPT scheduleis optimal. </p></li><li><p>Single machine scheduling problems 297</p><p>Theorem 3.2 Let S be an optimal schedule and S be a WSPT schedule for theproblem 1|pjr = pj [M + (1 M)br1]|wjCj . Then</p><p>wjCj (S)</p><p>wjCj (S)</p><p> 1M + (1 M)bn1 ,</p><p>and the bound is tight.</p><p>Proof Similar to the proof of Theorem 3.1. </p><p>Obviously, the result obtained</p><p>wjCj (S)/</p><p>wjCj (S) depends greatly on</p><p>the parameter values. We now show that the problems 1|pjr = pj [M + (1 M)ra]|wjCj and 1|pjr = pj [M + (1M)br1]|wjCj can be solved in poly-nomial time under some special conditions.</p><p>Theorem 3.3 For the problem 1|pjr = pj [M + (1 M)ra]|wjCj , if the jobshave agreeable weights, i.e., pj pk implies wj wk for all the jobs j and k, anoptimal schedule can be obtained by the WSPT rule.</p><p>Proof (By contradiction). Consider an optimal schedule that does not follow theWSPT rule. In this schedule there must be at least two adjacent jobs, say job j fol-lowed by job k, such that pj/wj &gt; pk/wk , which implies pj pk . Assume that job jis scheduled in position r . Perform an adjacent pair-wise interchange of jobs j and k.Whereas under the original schedule job j is scheduled in position r and job k isscheduled in position r + 1, under the new schedule job k is scheduled in position rand job j is scheduled in position r + 1. All other jobs remain in their original posi-tions. Call the new schedule . The completion times of the jobs processed beforejobs j and k are not affected by the interchange. Furthermore, the completion timesof the jobs processed after jobs j and k cannot be increased by the interchange sincepj pk . Under ,</p><p>Cj() = p[1] + p[2](M + (1 M)2a) + + pj (M + (1 M)ra),Ck() = p[1] + p[2](M + (1 M)2a) + + pj (M + (1 M)ra)</p><p>+ pk(M + (1 M)(r + 1)a),whereas under , they are</p><p>Ck() = p[1] + p[2](M + (1 M)2a) + + pk(M + (1 M)ra),</p><p>Cj () = p[1] + p[2](M + (1 M)2a) + + pk(M + (1 M)ra)</p><p>+ pj (M + (1 M)(r + 1)a).So we have</p><p>wjCj () </p><p>wjCj (</p><p>)</p><p> (wj + wk)(pj pk)(1 M)(ra (r + 1)a)+ (wkpj wjpk)(M + (1 M)(r + 1)a).</p></li><li><p>298 J.-B. Wang et al.</p><p>Since pj pk , wkpj &gt; wjpk (because pj/wj &gt; pk/wk), and ra (r + 1)a thenwjCj () wjCj ( ) &gt; 0. It follows that the weighted sum of completion</p><p>times under is strictly less than that under . This contradicts the optimality of and proves the theorem. </p><p>Theorem 3.4 For the problem 1|pjr = pj [M + (1 M)br1]|wjCj , if the jobshave agreeable weights, i.e., pj pk implies wj wk for all the jobs j and k, anoptimal schedule can be obtained by the WSPT rule.</p><p>Proof Similar to the proof of Theorem 3.3. </p><p>The following theorems are corollaries of Theorems 3.3 and 3.4.</p><p>Theorem 3.5 For the problem 1|pjr = pj [M + (1 M)ra],pj = p|wjCj , anoptimal schedule can be obtained by sequencing the jobs in non-increasing orderof wj .</p><p>Theorem 3.6 For the problem 1|pjr = pj [M + (1 M)ra],wj = kpj |wjCj , anoptimal schedule can be obtained by the SPT rule.</p><p>Theorem 3.7 For the problem 1|pjr = pj [M + (1 M)br1],pj = p|wjCj , anoptimal schedule can be obtained by sequencing the jobs in non-increasing orderof wj .</p><p>Theorem 3.8 For the problem 1|pjr = pj [M + (1 M)br1],wj = kpj |wjCj ,an optimal schedule can be obtained by the SPT rule.</p><p>4 The maximum lateness minimization problem</p><p>Similar to Mosheiov [10], for the objective function of minimizing the maximumlateness, we know that the optimal schedule of the EDD rule is not optimal for theproblems 1|pjr = pj [M+(1M)ra]|Lmax and 1|pjr = pj [M+(1M)br1|Lmax,respectively.</p><p>In order to solve 1|pjr = pj [M + (1 M)ra]|Lmax and 1|pjr = pj [M +(1 M)br1]|Lmax approximately, we will use the EDD rule as a heuristic for theproblems. To develop a worst-case performance ratio for a heuristic, we have to avoidcases involving nonpositive Lmax. Similar to Cheng and Wang [5], the worst-casebound is defined as follows:</p><p>Lmax(S) + dmaxLmax(S) + dmax ,</p><p>where S and Lmax(S) denote the heuristic schedule and the corresponding maxi-mum lateness, respectively, while S and Lmax(S) denote the optimal scheduleand the minimum maximum lateness value, respectively, and dmax = max{dj |j =1,2, . . . , n}.</p></li><li><p>Single machine scheduling problems 299</p><p>Theorem 4.1 Let S be an optimal schedule and S be an EDD schedule for theproblem 1|pjr = pj [M + (1 M)ra]|Lmax. Then</p><p>Lmax(S) + dmaxLmax(S) + dmax </p><p>ni=1 piCmax</p><p>,</p><p>and the bound is tight, where Cmax is the optimal makespan of the problem 1|pjr =pj [M + (1 M)ra]|Cmax.</p><p>Proof We assume without loss of generality that the jobs are renumbered in the orderof EDD, i.e., d1 d2 dn, then</p><p>Lmax(S) = max{p1 + p2(M + (1 M)2a) + + pj (M + (1 M)ja) dj |j = 1,2, . . . , n}</p><p> max{p1 + p2 + + pj dj |j = 1,2, . . . , n}= Lmax(S),</p><p>where Lmax(S) is the optimal value of the classical version of the problem, i.e.,pj,r = pj</p><p>Lmax(S) = max{p[1] + p[2](M + (1 M)2a) + </p><p>+ p[j ](M + (1 M)ja) d[j ]|j = 1,2, . . . , n}</p><p>= max{</p><p>j</p><p>i=1p[i] d[j ] </p><p>j</p><p>i=1p[i]</p><p>+j</p><p>i=1p[i](M + (1 M)ia)|j = 1,2, . . . , n</p><p>}</p><p> max{</p><p>j</p><p>i=1p[i] d[j ]|j = 1,2, . . . , n</p><p>}</p><p>n</p><p>i=1p[i]</p><p>+n</p><p>i=1p[i](M + (1 M)ia)</p><p> Lmax(S) n</p><p>i=1pi + Cmax,</p><p>hence,</p><p>Lmax(S) Lmax(S) n</p><p>i=1pi Cmax,</p></li><li><p>300 J.-B. Wang et al.</p><p>and so</p><p>Lmax(S) + dmaxLmax(S) + dmax 1 +</p><p>ni=1 pi Cmax</p><p>Lmax(S) + dmax 1 +n</p><p>i=1 pi CmaxCmax</p><p>n</p><p>i=1 piCmax</p><p>,</p><p>where Cmax can be obtained by the SPT rule (see Lemma 3.1).It is not difficult to see that the bound is tight, since if a = 0, we have Cmax =ni=1 pi and</p><p>Lmax(S)+dmaxLmax(S)+dmax = 1. This result is intuitive as when a = 0 or M = 1, the</p><p>EDD schedule is optimal. </p><p>Theorem 4.2 Let S be an optimal schedule and S be an EDD schedule for thep...</p></li></ul>

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