Single machine group scheduling with decreasing time-dependent processing times subject to release dates

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<ul><li><p> 2014 Published by Elsevier Inc.</p><p>d prottinglantse res</p><p>o the problems inating job p-increasinging problem</p><p>decreasing linear deterioration, i.e., the (actual) processing time of job Jj on machine Mh is phj ahj1h 1;2; . . . ;m; j 1;2; . . . ;n, where ahj is the normal (basic) processing time of job Jj on machine Mh; t is its startinb &gt; 0 is a decreasing rate such that 1 bt &gt; 0. When some dominance relations between m 1 machines can be sathey showed that the makespan minimization problem can be solved in polynomial time. Wang et al. [27] considered single</p><p>http://dx.doi.org/10.1016/j.amc.2014.01.1680096-3003/ 2014 Published by Elsevier Inc.</p><p> Corresponding author.E-mail addresses: luyuanyuan_jilin@163.com (Y.-Y. Lu), wangjibo75@163.com (J.-B. Wang).</p><p>Applied Mathematics and Computation 234 (2014) 286292</p><p>Contents lists available at ScienceDirect</p><p>Applied Mathematics and ComputationGenerally, there are two types of models describing this kind of scheduling. The rst type is devoted twhich the processing time of a job is an increasing (non-decreasing) function of its starting time (deterioring times). The second type concerns problems in which the processing time of a job is a decreasing (nontion of its starting time (shortening processing times). Wang et al. [24] considered ow shop schedulrocess-) func-with</p><p> bt;g time,tised,times. Extensive surveys of different scheduling models and problems involving jobs with start time dependent processingtimes can be found in Alidaee and Womer [1], Cheng et al. [2] and Gawiejnowicz [3]. More recent papers which have con-sidered scheduling problems with job time-dependent processing times include Lee et al. [4], Wu et al. [5], Wu and Lee [6],Wang et al. [7], Wang et al. [8], Lee et al. [9], He et al. [10], Chung et al. [11], Li et al. [12], Yang and Kuo [13], Wang and Sun[14], Wei and Wang [15], Yang and Yang [16], Yang et al. [17], Zhang and Yan [18], Zhao and Tang [19], Lee et al. [20], Zhuet al. [21], Huang and Wang [22], Zhao and Tang [23], Wang et al. [24], Yang and Wang [25], Sun et al. [26], Wang et al. [27],Wang et al. [28], Lee and Lu [29], Zhang and Luo [30], Liu et al. [31], Wu et al. [32], Wang and Wang [3337], Xu et al. [38].1. Introduction</p><p>Traditional scheduling models antice, however, we often encounter seling of the forging process in steel pwhy in recent years more and morblems usually involve jobs with constant, independent processing times. In prac-s in which job processing times increase or decrease over time, e.g., in the model-, nance management and scheduling maintenance or learning activities. This isearchers are considered scheduling problems with time-dependent processingSingle machine group scheduling with decreasingtime-dependent processing times subject to release dates</p><p>Yuan-Yuan Lu a,, Jian-Jun Wang b, Ji-Bo Wang caCollege of Mathematics, Jilin Normal University, Siping, Jilin 136000, Chinab Faculty of Management and Economics, Dalian University of Technology, Dalian 116024, Chinac School of Science, Shenyang Aerospace University, Shenyang 110136, China</p><p>a r t i c l e i n f o</p><p>Keywords:SchedulingSingle machineTime-dependent processing timesGroup technologyReady times</p><p>a b s t r a c t</p><p>In this paper we investigate a single machine scheduling problem with decreasing time-dependent processing times and group technology assumption. By the decreasing time-dependent processing times and group technology assumption, we mean that the groupsetup times and job processing times are both decreasing linear functions of their startingtimes. We want to minimize the makespan subject to release dates. We show that theproblem can be solved in polynomial time.</p><p>journal homepage: www.elsevier .com/ locate /amc</p></li><li><p>Y.-Y. Lu et al. / Applied Mathematics and Computation 234 (2014) 286292 287machine scheduling problem with decreasing linear deterioration, i.e., the (actual) processing time of job Jj ispj aja bt; j 1;2; . . . ;n, where a &gt; 0; b &gt; 0 and a bt &gt; 0. For the total absolute differences in waiting times mini-mization problem, they proved several properties of an optimal schedule, and proposed two heuristic algorithms.</p><p>On the other hand, scheduling models and problems in a group technology (GT) environment have attracted numerousresearchers due to their frequent real-life occurrence (Potts and Van Wassenhove [39], Webster and Baker [40], Liaee andEmmons [41], Janiak et al. [42], Bozorgirad and Logendran [43], Ji et al. [44]). Group technology that groups similar productsinto families helps increase the efciency of operations and decrease the requirement of facilities.</p><p>It is natural to study the situations where group scheduling and time dependent processing times are combined. To thebest of our knowledge, only a few results concerning scheduling problems with time dependent processing times and grouptechnology simultaneously are known.Wu et al. [5] considered a situation where group setup times and job processing timesare both described by a simple linear deterioration function, i.e., the (actual) processing time of job Jj in group Gi ispij bijt; i 1;2; . . . ;m; j 1;2; . . . ; ni, where bij &gt; 0 is the deterioration rate of job Jj in group Gi; the (actual) setup timeof group Gi is si git, where gi &gt; 0 is the deterioration rate of the setup time for group Gi. Using the extended three-eldnotation scheme (Graham et al. [45]), they proved that the makespan minimization problem (1jpij bijt; si git;GTjCmax)and the total completion time minimization problem (1jpij bijt; si git;GTj</p><p>PPCij) can be solved in polynomial time,</p><p>where Cij represents the completion time of job Jj in group Gi, Cmax maxfCijji 1;2; . . . ;m; j 1;2; . . . ;nig represent make-span of a given schedule. Wu and Lee [6] considered a situation where group setup times and job processing times are bothdescribed by a linear deterioration function, i.e., pij aij bt; i 1;2; . . . ;m; j 1;2; . . . ;ni, and si di gt; i 1;2; . . . ;m,where aij P 0 is the normal (basic) processing time of job Jj in group Gi; b &gt; 0 is the deterioration rate of jobs, di P 0 isthe normal (basic) setup time for group Gi; g is a deterioration rate of setup times. They showed that the makespan minimi-zation problem (1jpij aij bt; si di gt;GTjCmax) remain polynomially solvable. For the sum of completion times problem,they showed that the problem remains polynomially solvable under the assumption that the numbers of jobs in each groupare equal. Wang et al. [7] considered the following model: pij aija bt; i 1;2; . . . ;m; j 1;2; . . . ;ni, andsi dia bt; i 1;2; . . . ;m. They proved that the problems 1jpij aija bt; si dia bt;GTjCmax and1jpij aija bt; si dia bt;GTj</p><p>PPwijCij can be solved in polynomial time, where wij denote the weight of job Jj in</p><p>group Gi. Wang et al. [8] considered a situation where group setup times and job processing times are both described bya general linear deterioration function, i.e., pij aij bijt; i 1;2; . . . ;m; j 1;2; . . . ;ni, and si di git; i 1;2; . . . ;m. Theyproved that the makespan minimization problem (1jpij aij bijt; si di git;GTjCmax) can be solved in polynomial time.Wei and Wang [15] proved that the problems 1jpij bijt; si git;GTj</p><p>PwijC</p><p>2ij and 1jpij bijt; si git;GTj</p><p>PwijW</p><p>2ij can be</p><p>solved in polynomial time, whereWij Cij pij is the waiting time of job Jj in group Gi. Yang and Yang [16] considered sched-uling problems under the effects of deterioration and learning under group technology, i.e.,pij aijrai ; pij aij1</p><p>Pr1q1aiq</p><p>ai; si dit, where ai 0 denote the learning factor of group Gi, and r denote the job position.</p><p>They showed that the makespan minimization problems can be solved in polynomial time. They also showed that the totalcompletion time minimization problem have a polynomial optimal solution under agreeable restrictions. Zhang and Yan [18]considered group scheduling with deterioration and learning effect on a single machine, i.e., pij aij btra1ka2 ; si dira1 ,where a1 0 and a2 0 denote the learning effect, and r and k denote the group position and the job position. They showedthat the makespan and the total completion time minimization problems can be solved in polynomial time. They alsoshowed that the maximum lateness minimization problem have a polynomial optimal solution under agreeable conditions.Lee and Lu [29] considered the problem 1jpij bijt; si git;GTj</p><p>PwjUj, where Ujp 1 if Cjp &gt; dj and 0 otherwise, where</p><p>dj denote the due date of job Jj, they proposed a branch-and-bound algorithm to solve this problem.Wang and Sun [14] considered the group scheduling with linearly decreasing time-dependent setup times and job pro-</p><p>cessing times on a single machine, i.e., pij aij bijt; si di git, where bij is the decreasing rate of job Jj in group Gi. Theyproved that the problem 1jpij aij bijt; si di git;GTjCmax can be solved in polynomial time. For a special casebij baij; gi bdi, they proved that the problem 1jpij aij1 bt; si di1 bt;GTj</p><p>PwijCij can be solved in polynomial</p><p>time.Wang et al. [28] considered scheduling with independent setup times, ready times, and deteriorating job processing times</p><p>under group technology assumption on a single machine. They proved that the problem 1jrij; pij bijt; si;GTjCmax have a poly-nomial optimal solution under an agreeable condition, where rij denote the ready times (release dates) of job Jj in group Gi.Xu et al. [38] considered the problem 1jrij; pij bija bt; si;GTjCmax. For some special cases, they proved that the problemcan be solved in polynomial time.</p><p>In this paper we consider single machine group scheduling with release dates, decreasing time-dependent setup timesand job processing times (to our knowledge for the rst time) at the same time. We show that the makespan minimizationproblem can be solved in polynomial time. The remaining part of the paper is organized as follows. In the next section, aprecise formulation of the problem is given. The problem of minimizing the makespan is given in the Section 3. The last sec-tion 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: There are n jobs groupedinto m groups, and these n jobs are to be processed on a single machine. A setup time is required if the machine switches</p></li><li><p>a job mjth jobing tim</p><p>i</p><p>mode</p><p>i i</p><p>startin</p><p>wherejob prexplan</p><p>minim1jrij; p</p><p>3. Res</p><p>Lemma 1. For the problem 1jrj; p aj1 btjCmax, an optimal schedule can be obtained by sequencing the jobs in nondecreasing</p><p>Sim</p><p>0</p><p>288 Y.-Y. Lu et al. / Applied Mathematics and Computation 234 (2014) 286292b b b</p><p>max B 1b</p><p> 1 bai1 baj; ri 1b</p><p> 1 bai1 baj; rj 1b</p><p> 1 baj</p><p> P 0:</p><p>if and only if ri 6 rj.Cip0 Cjp max B 1 </p><p>1 baj1 bai; rj 1 </p><p>1 baj1 bai; ri 1 </p><p>1 bai Cip0 max B b 1 baj1 bai; rj b 1 baj1 bai; ri b 1 bai b : 4</p><p>Based on Eqs. (2) and (4), we have1 </p><p>1 </p><p>1 </p><p>1Cjp max B b 1 baj; rj b 1 baj b 3</p><p>andilarly, the completion times of jobs Jj and Ji in p0 are respectively</p><p>1 </p><p>1 </p><p>1j</p><p>order of rj.</p><p>Proof. Suppose that p and p0 are two job schedules. The difference between p and p0 is a pairwise interchange of two adja-cent jobs Ji and Jj, i.e., p S1; Ji; Jj; S2 and p0 S1; Jj; Ji; S2, where S1 and S2 denote a partial sequence. In addition, let Bdenote the completion time of the last job in S1. Under p, the completion times of jobs Ji and Jj are</p><p>Cip maxfB; rig ai1 bmaxfB; rig</p><p> max B 1b</p><p> 1 bai; ri 1b</p><p> 1 bai</p><p> 1b</p><p>1</p><p>and</p><p>Cjp maxfCi; rjg aj1 bmaxfCi; rjg</p><p> max B 1b</p><p> 1 bai1 baj; ri 1b</p><p> 1 bai1 baj; rj 1b</p><p> 1 baj</p><p> 1b:</p><p>2First, we consider a single machine scheduling problem with decreasing time-dependent processing times and readytimes of the jobs. The objective function is to minimize the makespan of all jobs.ize the makespan (the maximum completion time of all jobs), i.e., the problem</p><p>ij aij1 bt; si di1 bt;GTjCmax.</p><p>ultsFor a given schedule p;Cijp represents the completion time of job Jij in group Gi under schedule p. The objective is tormax maxfrjjj 1;2; . . . ;ng and hmin minfaij; diji 1;2; . . . ;m; j 1;2; . . . ;nig. The condition ensures that all theocessing times are positive in any feasible schedule (see also Wang and Sun [14], and Ho et al. [46] for detailedations).b t0 rmax i1 j1</p><p>aij i1</p><p>di hmin &lt; 1;g at time 0. It is assumed that normal processing times and setup times satisfy the following condition:</p><p>Xm Xni Xm !</p><p>where di &gt; 0 is the normal setup time of the group Gi, i.e., di is the initial setup requirement to adjust the group Gi if it isl, we also assume that the setup time of group Gi is</p><p>s d 1 bt;where aij &gt; 0 is the normal processing time of the jth job in the group Gi, i.e., aij is the initial processing requirement to com-plete the jth job in the group G if it is starting at time 0, t is its start time, and 0 &lt; b &lt; 1 is the decreasing rate. As in the aboveay not be interrupted. Let ni be the number of jobs belonging to group Gi, thus, n1 n2 . . . nm n. Let Jij denote thein group Gi, i 1;2; . . . ;m; j 1;2; . . . ;ni; rij &gt; 0 denote the ready (arrival) time of job Jij. Let pij be the actual process-e of the jth job in the group Gi; si be the setup time of group Gi. In this paper, we consider the following model:</p><p>pij aij1 bt;from one group to another and all setup times of groups for processing at time t0 P 0. We also assume that the processing of</p></li><li><p>nonde</p><p>Noprobleand ri</p><p>wheremaxf</p><p>follow</p><p>1. The</p><p>2. The</p><p>f1;2; .</p><p>Proof. The form of 1 follows from Lemma 1.</p><p>1 r Y 1</p><p>l1</p><p>and th</p><p>Y.-Y. Lu et al. / Applied Mathematics and Computation 234 (2014) 286292 289max Ab</p><p>1bdjl1</p><p>1bajl; jBj b1bdjQBj1</p><p>l1 1bajl1bdj</p><p>l11bajl; iBi b1bdi</p><p>QBi1l1 1bail</p><p>1bdiYnil1</p><p>1bail1b 7</p><p>Based on Eqs. (6) and (7), we havee completion time of the group Gi is</p><p>Cinip0max Cjnjp01b;</p><p>riBi 1b1bdi</p><p>QBi1l1 1bail</p><p>( )1bdi</p><p>Ynil1</p><p>1bail1b1</p><p> Ynj r 1 Ynj r 1( )bl1 1bdi l1 1bail l1 1bdj l1 1bajl</p><p>1bdjYnjl1</p><p>1bajl1b 6</p><p>Under p0, the completion times of the groups Gj and Gi are</p><p>Cjnjp0 max A1b;</p><p>rjBj 1b1 bdj</p><p>QBj11 bajl( )</p><p>1 bdjYnjl1</p><p>1 bajl 1bmax A1 1bdiYni</p><p>1bail;riBi 1bQBi1 1bdiY</p><p>ni</p><p>1bail;rjBj 1bQBj1jnj ini b 1bdjQBj1</p><p>l1 1bajlj</p><p>l1il b ( )e completion time of the group Gj is</p><p>C pmax C p1; rjBj 1b</p><p>( )1bd </p><p>Ynj1ba 1Cinip max A b ;iBi b</p><p>1 bdiQBi1</p><p>l1 1 bail1 bdi</p><p>l11 bail b</p><p>and thNext, we consider the case in item 2. Let p and p0 be two schedules where the difference between p and p0 is a pairwiseinterchange 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, itsufces to show that Cjnjp 6 Cinip0. Under p, using Eq. (5), we obtain that the completion time of the group Gi is</p><p>1( )</p><p>niriBi b ilBi1 bail maxfri1 b il11 bail; ri2 b il21 bail; . . . ; rini b1 bainig, Bi 2. . ; nig.1 di l1 1 bailwhere 1</p><p>Qn 1 Qn 1 Qn 1ing way:</p><p>job sequence in each group is in nondecreasing order of ri, i.e.,</p><p>ri1 6 ri2 6 . . . 6 rini; i 1;2; . . . ;m:groups are arranged in nondecreasing order of</p><p>riBi 1bQBi1 ;Theorem 1. For t...</p></li></ul>

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