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<ul><li><p>Information Sciences xxx (2014) xxxxxxContents lists available at ScienceDirect</p><p>Information Sciences</p><p>journal homepage: www.elsevier .com/locate / insSingle machine group scheduling with time dependentprocessing times and ready timeshttp://dx.doi.org/10.1016/j.ins.2014.02.0340020-0255/ 2014 Elsevier Inc. All rights reserved.</p><p> Corresponding author. Tel.: +86 411 84707834.E-mail addresses: wangjibo75@163.com (J.-B. Wang), wangjianjun_2008@163.com, wmeagle717@163.com (J.-J. Wang).</p><p>Please cite this article in press as: J.-B. Wang, J.-J. Wang, Single machine group scheduling with time dependent processing times antimes, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.034Ji-Bo Wang a, Jian-Jun Wang b,a School of Science, Shenyang Aerospace University, Shenyang 110136, Chinab Faculty of Management and Economics, Dalian University of Technology, Dalian 116024, China</p><p>a r t i c l e i n f o a b s t r a c tArticle history:Received 13 March 2009Accepted 6 February 2014Available online xxxx</p><p>Keywords:SchedulingSingle machineStart time dependent processing timeGroup technologyReady timeIn this paper we investigate a single machine scheduling problem with time dependentprocessing times and group technology (GT) assumption. By time dependent processingtimes and group technology assumption, we mean that the group setup times and jobprocessing times are both increasing functions of their starting times, i.e., group setuptimes and job processing times are both described by function which is proportional to alinear function of time. We attempt to minimize the makespan with ready times ofthe jobs. We show that the problem can be solved in polynomial time when start timedependent processing times and group technology are considered simultaneously.</p><p> 2014 Elsevier Inc. All rights reserved.1. Introduction</p><p>Traditional scheduling problems usually involve jobs with constant, independent processing times (Janiak et al. [12],Wang and Wei [30]). In practice, however, we often encounter settings in which the job processing times vary with learningeffect (Cheng et al. [5,6], Lee et al. [18,19], Rudek [21], Sun et al. [22,23], Wang et al. [31], Wu et al. [35], Yeh et al. [40], Yinet al. [41]) and deterioration effect (Alidaee and Womer [1], Cheng et al. [2] and Gawiejnowicz [7]). Scheduling problemsinvolving deterioration effect (time dependent processing times) appears, e.g., in scheduling maintenance jobs, steelproduction, national defense, emergency medicine or cleaning assignments, where any delay in processing a job is penalizedby incurring additional time for accomplishing the job. Extensive surveys of different scheduling models and problemsinvolving jobs with start time dependent processing times can be found in Alidaee and Womer [1], Cheng et al. [2] andGawiejnowicz [7]. More recent papers which have considered scheduling jobs with deterioration effects include Chenget al. [3,4], Gawiejnowicz [8], Gawiejnowicz et al. [9], Hsu et al. [11], Janiak and Kovalyov [13], Kang and Ng [15], Leeet al. [17], Wang [24], Wang et al. [2528], Wang and Sun [29], Wei and Wang [33], Wu and Lee [34], Wu et al. [3638], Yangand Wang [39], Yin et al. [42], and Xu et al. [43].</p><p>On the other hand, many manufacturers have implemented the concept of group technology (GT) in order to reduce setupcosts, lead times, work-in-process inventory costs, and material handling costs. In GT scheduling, it is conventional toschedule continuously all jobs from the same group. Group technology that groups similar products into families helpsincrease the efficiency of operations and decrease the requirement of facilities (Janiak et al. [14], Liaee and Emmons [16],Potts and Van Wassenhove [20], Webster and Baker [32]).</p><p>It is natural to study the situations where scheduling with group technology and start time dependent processing times(time-dependent scheduling) are combined. For the case of the setup time of each group is a fixed constant: Wang et al. [27]d ready</p><p>http://dx.doi.org/10.1016/j.ins.2014.02.034mailto:wangjibo75@163.commailto:wangjianjun_2008@163.commailto:wmeagle717@163.comhttp://dx.doi.org/10.1016/j.ins.2014.02.034http://www.sciencedirect.com/science/journal/00200255http://www.elsevier.com/locate/inshttp://dx.doi.org/10.1016/j.ins.2014.02.034</p></li><li><p>2 J.-B. Wang, J.-J. Wang / Information Sciences xxx (2014) xxxxxxconsidered single machine group scheduling in which the actual processing time of a job is a general linear decreasing func-tion of its starting time. For the makespan minimization problem and total completion time minimization problem theyshowed that some problems can be solved in polynomial time. Wang et al. [25] and Xu et al. [43] considered the single ma-chine scheduling problems with group technology and release times. Wang et al. [25] considered a simple linear deteriora-tion function and Xu et al. [43] considered a proportional linear deterioration function.</p><p>For the case of the setup time of each group is a linear deterioration function of its starting time: Since longer setup orpreparation might be necessary as food quality deteriorates or a patients condition worsens, Wu et al. [37] considered a sit-uation where the setup time groups and jobs in each group deteriorate as they wait for processing, i.e., group setup times andjob processing times are both described by a simple linear deterioration function. They proved that the single machine make-span minimization problem and the total completion time minimization problem can be solved in polynomial time. Wanget al. [28] considered single machine scheduling where group setup times and processing times of jobs are both described bya proportional linear deterioration function. For the makespan and the total completion time minimization problems, theyproposed a polynomial time solution, respectively. Wu and Lee [34] considered a situation where group setup times and jobprocessing times are both described by a linear deterioration function. They showed that the makespan minimization prob-lem remain polynomially solvable. For the sum of completion times problem, they showed that the problem remains poly-nomially solvable for a special case. Wang et al. [26] considered a situation where the group setup times and processingtimes of jobs are both described by a general linear deterioration function. They proved that the single machine makespanminimization problem can be solved in polynomial time.</p><p>In reality, longer setup or preparation might be necessary as food quality deteriorates or a patients condition worsens. Inthis paper, we consider the single machine scheduling with ready times of the jobs under the group technology assumptionand starting time-dependent setup times. The remaining part of the paper is organized as follows. In the next section, a pre-cise formulation of the problem is given. The problem of minimization of the makespan is given in the Section 3. The lastsection contains some conclusions.</p><p>2. Problem formulation</p><p>The single machine group scheduling problem with group setup times can be stated as follows: We consider nnon-preemptive jobs to be grouped into m groups and to be processed on a single machine. The jobs in the same group areconsecutively processed and a setup time is required if the machine switches from one group to another and all setup timesof groups for processing at time t0 > 0. Let Jij denote the jth job in group Gi; i 1;2; . . . ;m; j 1;2; . . . ;ni, rij > 0 denote theready (arrival) time of job Jij, where ni denotes the number of jobs belonging to group Gi (i.e., n1 n2 . . . nm n). As inWang et al. [28], we consider the following proportional deterioration model, i.e., the actual processing time of job Jij isPleasetimes,pij aija bt;where aij is the deterioration rate of job Jij and t is its starting time. As in the above proportional model, we also assume thatthe setup time of group Gi is a proportional deterioration model, i.e., the actual setup time of group Gi issi dia bt;where di is the deterioration rate of the group Gi. Our objective is to find the optimal group sequence and the optimal jobsequence within each group to minimize the maximum completion time of all jobs, i.e., the makespan.</p><p>For a given schedule p, let Cij Cijp denote the completion time of job Jij, and Cmax maxfCijji 1;2; . . . ;m; j 1;2; . . . ;nig represent the makespan of a given schedule. Using the three-field notation schema in scheduling problems(Graham et al. [10]) the makespan minimization problem is denoted as 1jrj; pij aija bt; si dia bt; GTjCmax.</p><p>3. The makespan minimization problem</p><p>In the following section, we will prove that the single machine makespan minimization scheduling problem withdeteriorating jobs and ready times of the jobs is polynomially solvable.</p><p>Lemma 1. For the problem 1jrj; pj aja btjCmax, the optimal job sequence can be obtained by sequencing the jobs in thenondecreasing order of rj.Proof. Suppose that p S1; Ji; Jj; S2 and p0 S1; Jj; Ji; S2 are two job sequences, where S1 and S2 denote a partialsequence (note that S1 and S2 may be empty) and the difference between p and p0 is a pairwise interchange of two adjacentjobs Ji and Jj. In addition, denote by A as the completion time of the last job of S1 in sequence p (p0). Then the completiontimes of jobs Ji and Jj under p areCip maxfA; rig aia bmaxfA; rig max Aab</p><p> 1 bai; ri </p><p>ab</p><p> 1 bai</p><p>n o ab</p><p>1cite this article in press as: J.-B. Wang, J.-J. Wang, Single machine group scheduling with time dependent processing times and readyInform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.034</p><p>http://dx.doi.org/10.1016/j.ins.2014.02.034</p></li><li><p>J.-B. Wang, J.-J. Wang / Information Sciences xxx (2014) xxxxxx 3andPleasetimes,Cjp maxfCi; rjg aja bmaxfCi; rjg</p><p> max A ab</p><p> 1 bai1 baj; ri </p><p>ab</p><p> 1 bai1 baj; rj </p><p>ab</p><p> 1 baj</p><p>n o ab: 2Similarly, the completion times of jobs Jj and Ji under p0 areCjp0 max Aab</p><p> 1 baj; rj </p><p>ab</p><p> 1 baj</p><p>n o ab</p><p>3andCip0 max Aab</p><p> 1 baj1 bai; rj </p><p>ab</p><p> 1 baj1 bai; ri </p><p>ab</p><p> 1 bai</p><p>n o ab: 4Suppose ri 6 rj, based on Eqs. (2) and (4), we haveCip0 Cjp max Aab</p><p> 1 baj1 bai; rj </p><p>ab</p><p> 1 baj1 bai; ri </p><p>ab</p><p> 1 bai</p><p>n omax A a</p><p>b</p><p> 1 bai1 baj; ri </p><p>ab</p><p> 1 bai1 baj; rj </p><p>ab</p><p> 1 baj</p><p>n oP 0:Hence, we have Cip0 P Cjp. Repeating this interchange argument, we conclude that an optimal schedule can beobtained by sequencing the jobs in nondecreasing order of rj. h</p><p>Now we consider the problem 1jrj; pij aija bt; si dia bt; GTjCmax. We assume that A denotes the completiontime of the i 1th group and ri1 6 ri2 6 . . . 6 rini is satisfied in the ith group Gi. Then, for the ith group Gi, we haveCiniGimax Aab</p><p> 1bdi</p><p>Ynil1</p><p>1bail; ri1 ab</p><p> Ynil1</p><p>1bail; ri2 ab</p><p> Ynil2</p><p>1bail; . . . ; rini ab</p><p> 1baini</p><p>( )ab</p><p>max Aab;</p><p>rBi ab1bdi</p><p>QBi1l1 1bail</p><p>( )1bdi</p><p>Ynil1</p><p>1bailab</p><p>5where rBi abQni</p><p>lBi1 bail max ri1 ab Qni</p><p>l11 bail; ri2 ab Qni</p><p>l21 bail; . . . ; rini ab </p><p>1 baini </p><p>, Bi 2f1;2; . . . ; nig.</p><p>Theorem 1. For the problem 1jrj; pij aija bt; si dia bt; GTjCmax, the optimal schedule satisfies the following:</p><p>(1) The job sequence in each group is in nondecreasing order of rj, i.e.,ri1 6 ri2 6 . . . 6 rini; i 1;2; . . . ;m:(2) The groups are arranged in nondecreasing order ofrBi ab1 bdi</p><p>QBi1l1 1 bail</p><p>;</p><p>where rBi ab Qni</p><p>lBi1 bail max ri1 abQni</p><p>l11 bail; ri2 ab Qni</p><p>l21 bail; . . . ; rini ab </p><p>1 baini </p><p>, Bi 2f1;2; . . . ; nig.Proof. In the same group, the result of (1) can be easily obtained by Lemma 1.Next, we consider the case in item (2). Let p and p0 be two schedules where the difference between p and p0 is a pairwise</p><p>interchange of two adjacent groups Gi and Gj, that is, p S1;Gi;Gj; S2;p0 S1;Gj;Gi; S2, where S1 and S2 are partialsequences. Furthermore, we assume that A denote the completion time of the last job in S1. To show p dominates p0, itsuffices to show that Cjnjp 6 Cinip0. Under p, using Eq. (5), we obtain that the completion time of the group Gi is( )Cinip max Aab;</p><p>rBi ab1 bdi</p><p>QBi1l1 1 bail</p><p>1 bdiYni</p><p>l11 bail </p><p>ab</p><p>and the completion time of the group Gj isCjnjp max Cinip ab;</p><p>rBj ab1 bdj</p><p>QBj1l1 1 bajl</p><p>( )1 bdj</p><p>Ynjl1</p><p>1 bail ab</p><p> max A ab</p><p> 1 bdi</p><p>Ynil1</p><p>1 bail;rBi ab</p><p>1 bdiQBi1</p><p>l1 1 bail1 bdi</p><p>Ynil1</p><p>1 bail;(</p><p>rBj ab1 bdj</p><p>QBj1l1 1 bajl</p><p>)1 bdj</p><p>Ynjl1</p><p>1 bajl ab</p><p>6cite this article in press as: J.-B. Wang, J.-J. Wang, Single machine group scheduling with time dependent processing times and readyInform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.034</p><p>http://dx.doi.org/10.1016/j.ins.2014.02.034</p></li><li><p>4 J.-B. Wang, J.-J. Wang / Information Sciences xxx (2014) xxxxxxUnder p0, the completion times of the groups Gj and Gi arePleasetimes,Cjnjp0 max A a</p><p>b;</p><p>rBj ab1 bdj</p><p>QBj1l1 1 bajl</p><p>( )1 bdj</p><p>Ynjl1</p><p>1 bajl ab</p><p>and the completion time of the group Gi isCinip0 max Cjnjp</p><p>0 ab;</p><p>rBi ab1 bdi</p><p>QBi1l1 1 bail</p><p>( )1 bdi</p><p>Ynil1</p><p>1 bail ab</p><p> max A ab</p><p> 1 bdj</p><p>Ynjl1</p><p>1 bajl;rBj ab</p><p>1 bdjQBj1</p><p>l1 1 bajl1 bdj</p><p>Ynjl1</p><p>1 bajl;(</p><p>rBi ab1 bdi</p><p>QBi1l1 1 bail</p><p>)1 bdi</p><p>Ynil1</p><p>1 bail ab</p><p>7SupposerBi ab1 bdi</p><p>QBi1l1 1 bail</p><p>6rBj ab</p><p>1 bdjQBj1</p><p>l1 1 bajl;</p><p>based on Eqs. (6) and (7), we haveCjnjp Cinip0</p><p> max A ab</p><p> 1 bdi</p><p>Ynil1</p><p>1 bail;(</p><p>rBi ab1 bdi</p><p>QBi1l1 1 bail</p><p>1 bdiYnil1</p><p>1 bail;</p><p>rBj ab1 bdj</p><p>QBj1l1 1 bajl</p><p>)1 bdj</p><p>Ynjl1</p><p>1 bajl max Aab</p><p> 1 bdj</p><p>Ynjl1</p><p>1 bajl;n</p><p>rBj ab1 bdj</p><p>QBj1l1 1 bajl</p><p>1 bdjYnj</p><p>l11 bajl;</p><p>rBi ab1 bdi</p><p>QBi1l1 1 bail</p><p>)1 bdi</p><p>Ynil1</p><p>1 bail</p><p>6 max A ab</p><p> 1 bdi1 bdj</p><p>Ynil1</p><p>1 bailYnj</p><p>l11 bajl;</p><p>(</p><p>rBj abQBj1l1 1 bajl</p><p>1 bdiYnj</p><p>l11 bajl</p><p>Ynil1</p><p>1 bail;rBj abQBj1</p><p>l1 1 bajl</p><p>Ynjl1</p><p>1 bajl)</p><p>max A ab</p><p> 1 bdj1 bdi</p><p>Ynjl1</p><p>1 bajlYnil1</p><p>1 bail;(</p><p>rBj abQBj1l1 1 bajl</p><p>1 bdiYnjl1</p><p>1 bajlYnil1</p><p>1 bail;rBi abQBi1</p><p>l1 1 bail</p><p>Ynil1</p><p>1 bail)</p><p> 0:Therefore,Cjnjp 6 Cinip0:This completes the proof. h</p><p>Based on the result of Theorem 1, for the problem 1jrj; pij aija bt; si dia bt;GTjCmax, we provide a simple algo-rithm presented in the following:</p><p>Algorithm 1.</p><p>Step 1. Jobs in each group scheduled in nondecreasing order of rj, i.e.,ri1 6 ri2 6 . . . 6 rini; i 1;2; . . . ;m:</p><p>Step 2. Let rBi ab Qni</p><p>lBi1 bail max ri1 ab Qni</p><p>l11 bail; ri2 ab Qni</p><p>l21 bail; . . . ; rini ab </p><p>1 baini </p><p>,</p><p>Bi 2 f1;2; . . . ;nig, calculate Bi andrBiab</p><p>1bdiQBi1</p><p>l1 1bail; i 1;2; . . . ;m.</p><p>Step 3. Groups scheduled in nondecreasing order ofcite this article in press as: J.-B. Wang, J.-J. Wang, Single machine group scheduling with time dependent processing times and readyInform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.034</p><p>http://dx.doi.org/10.1016/j.ins.2014.02.034</p></li><li><p>J.-B. Wang, J.-J. Wang / Information Sciences xxx (2014) xxxxxx 5</p><p>Pleasetimes,qGi rBi ab</p><p>1 bdiQBi1</p><p>l1 1 bail:</p><p>Obviously, the complexity of obtaining the optimal job sequence within a certain group Gi is Oni logni and that ofobtaining the optimal group sequence is Om logm. It is easy to show that</p><p>Pmi1Oni logni 6 On logn. Hence, the complex-</p><p>ity of Algorithm 1 is at most On logn. In addition, we demonstrate Algorithm 1 by the following example.</p><p>Example 1. There are eight jobs (n 8) divided into three groups (m 3) to be processed on a single machine. Leta 1; b 0:1; t0 1 and G1 : f J11; J12g; d1 1; a11 1; a12 3; r11 16; r12 6; G2 : fJ21; J22; J23g; d2 2; a21 1;a22 2; a23 3;...</p></li></ul>