- Email: [email protected]
Parallel-machine scheduling with non-simultaneous machine available time
Applied Mathematical Modelling 37 (2013) 5227–5232
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Parallel-machine scheduling with non-simultaneous machine available time Lixin Shen a, Dan Wang b, Xiao-Yuan Wang c,⇑ a
College of Transportation Management, Dalian Maritime University, Dalian 116026, China School of Computer, Shenyang Aerospace University, Shenyang 110136, China c School of Science, Shenyang Aerospace University, Shenyang 110136, China b
a r t i c l e
i n f o
Article history: Received 11 August 2011 Received in revised form 20 August 2012 Accepted 25 September 2012 Available online 17 October 2012 Keywords: Scheduling Parallel machines Machine availability constraint
a b s t r a c t We consider a problem of scheduling n independent jobs on m parallel identical machines. The jobs are available at time zero, but the machines may not be available simultaneously at time zero. We concentrate on two goals separately, namely, minimizing a cost function containing total completion time and total absolute differences in completion times; minimizing a cost function containing total waiting time and total absolute differences in waiting times. In this paper, we present polynomial time algorithm to solve this problem. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction In many branches of industry and logistics, there arise problems of ordering jobs on machines (Lee and Wu [1], Wang et al. [2], Lee et al. [3], Wang and Guo [4], Wang and Li [5], Liu et al. [6], Wang et al. [7], Wei et al. [8], Bai et al. [9], and Wang and Wang [10]). Lee [11] first considered the problem of parallel machines scheduling with non-simultaneous machine available. His goal is to minimize the makespan. The goal to maximize the minimum machine completion time of the classical parallel machines scheduling was proposed by Deuermeyer et al. [12]. Lin et al. [13] combined the models of Deuermeyer et al. [12] and Lee [11]. They studied parallel machines scheduling with non-simultaneous machine available time to maximize the minimum machine completion time. Zhao et al. [14] considered the two-parallel machine scheduling problem with rate-modifying activities. For the total completion time minimization problem, they provided a polynomial algorithm to solve the problem optimally. Wang et al. [15] considered parallel identical machines scheduling problems with a deteriorating maintenance activity. They proved that the total completion time minimization problem can be solved in polynomial time. This paper addresses two scheduling problems of parallel identical machines in which the machines may not be available simultaneously at time zero, and the objectives are to minimize an objective function which includes costs for total completion time and total absolute differences in completion times and to minimize an objective function which includes costs for total waiting time and total absolute differences in waiting times, respectively. The remaining part of this paper is organized as follows. In Section 3.2 we formulate the model. In Section 3, the identical parallel machines problem with minimizing the linear combination of total completion time and total absolute differences in completion times is considered. In Section 4, the identical parallel machines problem with minimizing the linear
⇑ Corresponding author at: School of Science, Shenyang Aerospace University, Shenyang 110136, China. E-mail address: [email protected] (X.-Y. Wang). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.09.053
5228
L. Shen et al. / Applied Mathematical Modelling 37 (2013) 5227–5232
combination of total waiting time and total absolute differences in waiting times is considered. In Section 5, an example is given. The last section presents the conclusions. 2. Problem description The following notations will be used throughout the paper: N = {J1, J2, . . ., Jn}: The set of jobs to be processed. M = {M1, M2, . . ., Mm}: The set of parallel identical machines. s: The sequence of jobs to be processed by the machines. [i]: job in the ith position. pj: The processing time of job Jj. ai: The ready (release) time of machine Mi. Cj: The completion time of job Jj. Wj: The waiting time of job Jj, that is Wj = Cj pj. P TC: The total completion times, that is TC = i=1nCi. P n TW: The total waiting times, that is TW = i=1 Wi. TADC: The total absolute differences in completion times, that is
TADC ¼
n X n X jC i C j j: i¼1 j¼i
TADW: The total absolute differences in waiting times, that is
TADW ¼
n X n X jW i W j j: i¼1 j¼i
The problem under investigation can be described as follows: There are n independent jobs N ¼ fJ 1 ; J 2 ; . . . ; J n g to be processed on m parallel identical machines M ¼ fM 1 ; M 2 ; . . . ; M m g. As in Lee [11], let pj denote the processing time required by job J j ; ai denote the earliest time that machine M i can start to process the jobs. Without loss of generality, we assume that the machines are indexed by nondecreasing ready times, i.e., 0 6 a1 6 a2 6 6 am . The machine can handle one job at a time and preemption is not allowed. We also assume that all jobs are ready at time zero. Any job cannot be processed on two or more machines at the same time. Let ni be the number of jobs processed on machine M i , thus, n1 þ n2 þ þ nm ¼ n; Ai be the job set processed on machine Mi ; i ¼ 1; 2; . . . ; m. Let J i½j denote the jth job on machine M i ; pi½j and C i½j denote the processing and completion time of job J i½j . The objective is to determine the optimal sequence of jobs in the machine so that the corresponding value of the following cost functions be optimal:
Z ¼ d1 TC þ d2 TADC;
ð1Þ
Z ¼ d1 TW þ d2 TADW;
ð2Þ
where weights d1 P 0 and d2 P 0 are given constants (the decision-maker selects the weights d1 and d2 ). 3. The problem Pm; ai jjd1 TC þ d2 TADC 3.1. Main results for m ¼ 2 In this subsection we consider problem P2; ai jjd1 TC þ d2 TADC. When the sets A1 and A2 , and a sequence of job on each machine are given, we have
C 1½j ¼ a1 þ
j X
p1½k ;
j ¼ 1; 2; . . . ; n1 ;
p2½k ;
j ¼ 1; 2; . . . ; n2 :
k¼1
C 2½j ¼ a2 þ
j X k¼1
Hence, for the total completion time and the total absolute differences in completion times (Kanet [16]), we have
TCðn1 ; n2 Þ ¼ n1 a1 þ n2 a2 þ
n1 n2 X X ðn1 k þ 1Þp1½k þ ðn2 k þ 1Þp2½k ; k¼1
TADCðn1 ; n2 Þ ¼
n1 X k¼1
ðk 1Þðn1 k þ 1Þp1½k þ
k¼1 n2 X ðk 1Þðn2 k þ 1Þp2½k : k¼1
5229
L. Shen et al. / Applied Mathematical Modelling 37 (2013) 5227–5232
Hence
d1 TCðn1 ; n2 Þ þ d2 TADCðn1 ; n2 Þ ¼ d1 ðn1 a1 þ n2 a2 Þ þ
n1 n2 X X ½d1 ðn1 k þ 1Þ þ d2 ðk 1Þðn1 k þ 1Þp1½k þ ½d1 ðn2 k þ 1Þ k¼1
k¼1
þ d2 ðk 1Þðn2 k þ 1Þp2½k : We need to determine the sets A1 and A2 , and a sequence of job on each machine to minimize d1 TC þ d2 TADC. In the following, we will show that when a ðn1 ; n2 Þ vector is given, the problem can be formulated as a assignment problem. Let
Bðn1 ; n2 Þ ¼
n1 n2 X X ½d1 ðn1 k þ 1Þ þ d2 ðk 1Þðn1 k þ 1Þp1½k þ ½d1 ðn2 k þ 1Þ þ d2 ðk 1Þðn2 k þ 1Þp2½k : k¼1
k¼1
Since d1 TCðn1 ; n2 Þ þ d2 TADCðn1 ; n2 Þ ¼ d1 ðn1 a1 þ n2 a2 Þ þ Bðn1 ; n2 Þ, for a given a ðn1 ; n2 Þ vector, minimize d1 TCðn1 ; n2 Þþ d2 TADCðn1 ; n2 Þ is equivalent to minimizing Bðn1 ; n2 Þ. Suppose job J j ðj ¼ 1; 2; . . . ; nÞ is scheduled in position r ðr ¼ 1; 2; . . . ; ni Þ on machine Mi ði ¼ 1; 2; . . . ; mÞ, then its contribution for objective is d1 ðni r þ 1Þ þ d2 ðr 1Þðni r þ 1Þpj . In order to formulate this problem as an assignment problem, we define the following standard binary variables: Let xjir be a 0/1 variable such that xjir ¼ 1 if job J j (j ¼ 1; 2; . . . ; n) is assigned to machine M i (i ¼ 1; 2) at position r (r ¼ 1; 2; . . . ; ni ), and xjir ¼ 0, otherwise. Let wjir denote the cost associated with assigning job J j (j ¼ 1; 2; . . . ; n) is assigned to machine M i (i ¼ 1; 2) at position r (r ¼ 1; 2; . . . ; ni ):
wjir ¼ d1 ðni r þ 1Þ þ d2 ðr 1Þðni r þ 1Þpj :
ð3Þ
For a given ðn1 ; n2 Þ vector, the problem P2; ai jjd1 TC þ d2 TADC can be solved by the following assignment problem: Minimize
Z¼
ni X 2 X n X wjir xjir þ d1 ðn1 a1 þ n2 a2 Þ;
ð4Þ
i¼1 r¼1 j¼1
subject to ni 2 X X
xjir ¼ 1;
i¼1 r¼1 n X
xjir ¼ 1;
j ¼ 1; 2; . . . ; n;
i ¼ 1; 2; r ¼ 1; 2; . . . ; ni ;
j¼1
xjir ¼ 0 or 1;
j ¼ 1; 2; . . . ; n; i ¼ 1; 2; r ¼ 1; 2; . . . ; ni ;
where wjir are calculated by (3). We have to solve such an assignment problem for each ðn1 ; n2 Þ vector. Hence the vectors ðn1 ; n2 Þ are ð0; nÞ; ð1; n 1Þ; . . . ; ðn; 0Þ, and the number of vectors ðn1 ; n2 Þ is n þ 1. We have to solve a weighted-bipartite matching problems for each ðn1 ; n2 Þ vector. The number of all assignment problems is n þ 1. We now to give a polynomial time algorithm for the problem P2; ai jjd1 TC þ d2 TADC. Algorithm 1. Step 0. Set n1 ¼ 0. Step 1. Set n2 ¼ n n1 . Step 2. For a given pair of ðn1 ; n2 Þ create an assignment problem with weight wjir defined above. Solve the assignment problem (4) to obtain a local optimal schedule and the total cost. Step 3. Set n1 ¼ n1 þ 1, If n1 > n, then go to Step 4. Otherwise go to Step 1. Step 4. The global optimal schedule is the one with the minimum total cost.
Theorem 1. Algorithm 1 will find an optimal solution for the problem P2; ai jjd1 TC þ d2 TADC with time complexity Oðn4 Þ. Proof. As discussed above, we can convert the problem P2; ai jjd1 TC þ d2 TADC to an assignment problem and find an optimal solution by Algorithm 1. Step 2 can be solved in Oðn3 Þ for each assignment problem. Step 1 is executed n þ 1 times. Consequently, the overall time requirement of Algorithm 1 is Oðn4 Þ. h
5230
L. Shen et al. / Applied Mathematical Modelling 37 (2013) 5227–5232
3.2. Extensions to m-parallel identical machines Now we discuss how the results for two-parallel machines can be extended to m-parallel machines (m > 2Þ. Similarly to the case of two-parallel machines, when ðn1 ; n2 ; . . . ; nm Þ ðn1 þ n2 þ . . . nm ¼ nÞ and a sequence of job on each machine are given, we have
d1 TCðn1 ; n2 ; . . . ; nm Þ þ d2 TADCðn1 ; n2 ; . . . ; nm Þ ¼
ni m X m X X ½d1 ðni j þ 1Þ þ d2 ðj 1Þðni j þ 1Þpi½j þ d1 ni ai : i¼1 j¼1
i¼1
Hence, for a given ðn1 ; n2 ; . . . ; nm Þ vector, the problem Pm; ai jjd1 TC þ d2 TADC can be solved by an assignment problem. The remaining question is how many ðn1 ; n2 ; . . . ; nm Þ vectors exist. This problem is equal to that of allocation n identical balls (=jobs) into m different urns (=machines). The number of ball allocations in the problem is known to be (see Stirzaker nþm1 ð2nÞm ð2nÞm [17, p. 64]) : An upper bound on this number m is given in Stirzaker [17, p. 64]: ðm1Þ! . Obviously, ðm1Þ! is a polym1 nomial upper bound in n (similar to Mosheiov [18]). Hence, we have that the problem Pm; ai jjd1 TC þ d2 TADC can be solved in polynomial time, i.e., Theorem 2. When m > 2 is given, the problem Pm; ai jjd1 TC þ d2 TADC can be solved in Oðnmþ3 Þ times. Remark 1. The solution procedure for minimizing the problem Pm; ai jjd1 TC þ d2 TADC can be easily extended to the case of unrelated machines. Using a similar model to that of (4), let xjir be a 0/1 variable such that xjir ¼ 1 if job J j (j ¼ 1; 2; . . . ; n) is assigned to machine M i (i ¼ 1; 2; . . . ; m) at position r (r ¼ 1; 2; . . . ; ni ), and xjir ¼ 0, otherwise. For a given ðn1 ; n2 ; . . . ; nm Þ vector, the problem Rm; ai jjd1 TC þ d2 TADC can be solved by the following assignment problem:
min
ni X m X n m X X wjir xjir =sij þ d1 ni ai i¼1 r¼1 j¼1
i¼1
st ni m X X xjir ¼ 1; i¼1 r¼1 n X
xjir ¼ 1;
j ¼ 1; 2; . . . ; n;
i ¼ 1; 2; . . . ; m; r ¼ 1; 2; . . . ; ni ;
j¼1
xjir ¼ 0 or 1;
j ¼ 1; 2; . . . ; n; i ¼ 1; 2; . . . ; m; r ¼ 1; 2; . . . ; ni ;
where sij denote the process speed of job J j (j ¼ 1; 2; . . . ; n) on machine M i (i ¼ 1; 2; . . . ; m), wjir are calculated by (3). 4. The problem Pm; ai jjd1 TW þ d2 TADW 4.1. Main results for m ¼ 2 In this subsection we consider problems P2; ai jjd1 TW þ d2 TADW. When the sets A1 and A2 , and a sequence of job on each machine are given, we have
W 1½j ¼ a1 þ
j1 X p1½k ;
j ¼ 1; 2; . . . ; n1 ;
k¼1
W 2½j ¼ a2 þ
j1 X p2½k ;
j ¼ 1; 2; . . . ; n2 :
k¼1
Hence, for the total waiting time and the total absolute differences in waiting times (Bagchi [19]), we have
TWðn1 ; n2 Þ ¼ n1 a1 þ n2 a2 þ
n1 n2 X X ðn1 kÞp1½k þ ðn2 kÞp2½k ; k¼1
TADWðn1 ; n2 Þ ¼
k¼1
n1 n2 X X kðn1 kÞp1½k þ kðn2 kÞp2½k : k¼1
k¼1
Hence
d1 TWðn1 ; n2 Þ þ d2 TADWðn1 ; n2 Þ ¼ d1 ðn1 a1 þ n2 a2 Þ þ
n1 n2 X X ½d1 ðn1 kÞ þ d2 kðn1 kÞp1½k þ ½d1 ðn2 kÞ þ d2 kðn2 kÞp2½k : k¼1
k¼1
5231
L. Shen et al. / Applied Mathematical Modelling 37 (2013) 5227–5232
Similar the Section 3, for a given ðn1 ; n2 Þ vector, the problem P2; ai jjd1 TW þ d2 TADW can be solved by the following assignment problem: Minimize
Z¼
ni X 2 X n X
v jir xjir þ d1 ðn1 a1 þ n2 a2 Þ
ð5Þ
i¼1 r¼1 j¼1
subject to ni 2 X X
xjir ¼ 1;
i¼1 r¼1 n X
xjir ¼ 1;
j ¼ 1; 2; . . . ; n;
i ¼ 1; 2; r ¼ 1; 2; . . . ; ni ;
j¼1
xjir ¼ 0 or 1;
j ¼ 1; 2; . . . ; n; i ¼ 1; 2; r ¼ 1; 2; . . . ; ni ;
where v jir ¼ d1 ðni rÞ þ d2 rðni rÞpj . Similar the Section 3, we now to give a polynomial time algorithm for the problem P2; ai jjd1 TW þ d2 TADW. Algorithm 2. Step 0. Set n1 ¼ 0. Step 1. Set n2 ¼ n n1 . Step 2. For a given pair of ðn1 ; n2 Þ, solve the assignment problem (5) to obtain a local optimal schedule and the total cost. Step 3. Set n1 ¼ n1 þ 1, If n1 > n, then go to Step 4. Otherwise go to Step 1. Step 4. The global optimal schedule is the one with the minimum total cost.
Theorem 3. Algorithm 2 will find an optimal solution for the problem P2; ai jjd1 TW þ d2 TADW with time complexity Oðn4 Þ. Proof. Similar to the proof of Theorem 1. h 4.2. Extensions to m-parallel identical machines Similar to the Section 3.2, we have. Theorem 4. When m > 2 is given, the problem Pm; ai jjd1 TW þ d2 TADW can be solved in Oðnmþ3 Þ times. Remark 2. The solution procedure for the problem Pm; ai jjd1 TW þ d2 TADW can be easily extended to the case of unrelated machines. Using a similar model to that of (4), let xjir be a 0/1 variable such that xjir ¼ 1 if job J j (j ¼ 1; 2; . . . ; n) is assigned to machine M i (i ¼ 1; 2; . . . ; m) at position r (r ¼ 1; 2; . . . ; ni ), and xjir ¼ 0, otherwise. For a given ðn1 ; n2 ; . . . ; nm Þ vector, the problem Rm; ai jjd1 TW þ d2 TADW can be solved by the following assignment problem:
min
ni X m X n X
m X ni ai
i¼1 r¼1 j¼1
i¼1
v jir xjir =sij þ d1
st ni m X X
xjir ¼ 1;
i¼1 r¼1 n X
xjir ¼ 1;
j ¼ 1; 2; . . . ; n;
i ¼ 1; 2; . . . ; m; r ¼ 1; 2; . . . ; ni ;
j¼1
xjir ¼ 0 or 1;
j ¼ 1; 2; . . . ; n; i ¼ 1; 2; . . . ; m; r ¼ 1; 2; . . . ; ni ;
where sij denote the process speed of job J j (j ¼ 1; 2; . . . ; n) on machine M i (i ¼ 1; 2; . . . ; m), 5. An example The following example illustrates the working of Algorithms 1 and 2.
v jir ¼ d1 ðni rÞ þ d2 rðni rÞpj .
5232
L. Shen et al. / Applied Mathematical Modelling 37 (2013) 5227–5232
Example 1 In the modern manufacturing and service industries (like the beverage industry, the capital-intensive printing industry, the pharmaceutical industry), many production environments consist of several workcenters, each with a number of parallel identical machines with non-simultaneous machine available time, i.e., each machine M i ð1 6 i 6 mÞ has an availability time ai on which it becomes continuously ready for working. Consider 4 jobs to be processed on machines M 1 and M 2 with: p1 ¼ 6; p2 ¼ 3; p3 ¼ 8; p4 ¼ 5; a1 ¼ 1; a2 ¼ 4; d1 ¼ 2; d2 ¼ 1. According Algorithm 1, when ðn1 ; n2 Þ ¼ ð0; 4Þ, the cost is 187; when ðn1 ; n2 Þ ¼ ð1; 3Þ, the cost is 114; when ðn1 ; n2 Þ ¼ ð2; 2Þ, the cost is 94; when ðn1 ; n2 Þ ¼ ð3; 1Þ, the cost is 102; when ðn1 ; n2 Þ ¼ ð4; 0Þ, the cost is 163. Hence for the objective function d1 TC þ d2 TADC, an optimal schedule is ½J 2 ; J 6 on machine M 1 and ½J 4 ; J 8 on machine M 2 , and the optimal cost is 94. According Algorithm 2, when ðn1 ; n2 Þ ¼ ð0; 4Þ, the cost is 129; when ðn1 ; n2 Þ ¼ ð1; 3Þ, the cost is 64; when ðn1 ; n2 Þ ¼ ð2; 2Þ, the cost is 44; when ðn1 ; n2 Þ ¼ ð3; 1Þ, the cost is 60; when ðn1 ; n2 Þ ¼ ð4; 0Þ, the cost is 105. Hence for the objective function d1 TW þ d2 TADW, an optimal schedule is ½J 2 ; J 6 on machine M 1 and ½J 4 ; J 8 on machine M 2 , and the optimal cost is 44. 6. Conclusions In this paper we studied parallel identical machines scheduling with non-simultaneous machine available time. The objectives are to minimize the linear combination of total completion (waiting) time and total absolute differences in completion (waiting) times. We proved that the problem can be solved in polynomial time. Further research includes the investigation of other function objectives. Acknowledgments We are grateful to the editor and two anonymous referees for their helpful comments on an earlier version of this paper. This research was fully supported by the National Natural Science Foundation of China (Grant No. 11001181). References [1] W.-C. Lee, C.-C. Wu, A note on single-machine group scheduling problems with position-based learning effect, Appl. Math. Model. 33 (2009) 2159– 2163. [2] J.-B. Wang, L.-H. Sun, L.-Y. Sun, Single machine scheduling with exponential sum-of-logarithm-processing-times based learning effect, Appl. Math. Model. 34 (2010) 2813–2819. [3] W.-C. Lee, W.-J. Wang, Y.-R. Shiau, C.-C. Wu, A single-machine scheduling problem with two-agent and deteriorating jobs, Appl. Math. Model. 34 (2010) 3098–3107. [4] J.-B. Wang, Q. Guo, A due-date assignment problem with learning effect and deteriorating jobs, Appl. Math. Model. 34 (2010) 309–313. [5] J.-B. Wang, J.-X. Li, Single machine past-sequence-dependent setup times scheduling with general position-dependent and time-dependent learning effects, Appl. Math. Model. 35 (2011) 1388–1395. [6] P. Liu, N. Yi, X. Zhou, Two-agent single-machine scheduling problems under increasing linear deterioration, Appl. Math. Model. 35 (2011) 2290–2296. [7] X.-R. Wang, X. Huang, J.-B. Wang, Single-machine scheduling with linear decreasing deterioration to minimize earliness penalties, Appl. Math. Model. 35 (2011) 3509–3515. [8] C.-M. Wei, J.-B. Wang, P. Ji, Single-machine scheduling with time-and-resource-dependent processing times, Appl. Math. Model. 62 (2012) 792–798. [9] J. Bai, Z.-R. Li, X. Huang, Single-machine group scheduling with general deterioration and learning effects, Appl. Math. Model. 36 (2012) 1267–1274. [10] X.-R. Wang, J.-J. Wang, Single-machine scheduling with convex resource dependent processing times and deteriorating jobs, Appl. Math. Model., http:// dx.doi.org/10.1016/j.apm.2012.05.025. [11] C.-Y. Lee, Parallel machines scheduling with nonsimultaneous machine available time, Disc. Appl. Math. 30 (1991) 53–61. [12] B.L. Deuermeyer, D.K. Friesen, M.A. Langston, Scheduling to maximize the minimum processor finish time in a multi-processor system, SIAM J. Algebraic Disc. Methods 3 (1982) 190–196. [13] G.-H. Lin, E.-Y. Yao, Y. He, Parallel machine scheduling to maximize the minimum load with nonsimultaneous machine available times, Oper. Res. Lett. 22 (1998) 75–81. [14] C.-L. Zhao, H.-Y. Tang, C.-D. Cheng, Two-parallel machines scheduling with rate-modifying activities to minimize total completion time, Eur. J. Oper. Res. 198 (2009) 354–357. [15] J.-J. Wang, J.-B. Wang, F. Liu, Parallel machines scheduling with a deteriorating maintenance activity, J. Oper. Res. Soc. 62 (2011) 1898–1902. [16] J.J. Kanet, Minimizing variation of flow time in single machine systems, Manage. Sci. 27 (1981) 1453–1459. [17] D. Stirzaker, Elementary Probability, Cambridge University Press, Cambridge, 1995. [18] G. Mosheiov, Parallel machine scheduling with a learning effect, J. Oper. Res. Soc. 52 (2001) 1165–1169. [19] U.B. Bagchi, Simultaneous minimization of mean and variation of flow-time and waiting time in single machine systems, Oper. Res. 37 (1989) 118– 125.
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Parallel-machine scheduling with non-simultaneous machine available time Lixin Shen a, Dan Wang b, Xiao-Yuan Wang c,⇑ a
College of Transportation Management, Dalian Maritime University, Dalian 116026, China School of Computer, Shenyang Aerospace University, Shenyang 110136, China c School of Science, Shenyang Aerospace University, Shenyang 110136, China b
a r t i c l e
i n f o
Article history: Received 11 August 2011 Received in revised form 20 August 2012 Accepted 25 September 2012 Available online 17 October 2012 Keywords: Scheduling Parallel machines Machine availability constraint
a b s t r a c t We consider a problem of scheduling n independent jobs on m parallel identical machines. The jobs are available at time zero, but the machines may not be available simultaneously at time zero. We concentrate on two goals separately, namely, minimizing a cost function containing total completion time and total absolute differences in completion times; minimizing a cost function containing total waiting time and total absolute differences in waiting times. In this paper, we present polynomial time algorithm to solve this problem. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction In many branches of industry and logistics, there arise problems of ordering jobs on machines (Lee and Wu [1], Wang et al. [2], Lee et al. [3], Wang and Guo [4], Wang and Li [5], Liu et al. [6], Wang et al. [7], Wei et al. [8], Bai et al. [9], and Wang and Wang [10]). Lee [11] first considered the problem of parallel machines scheduling with non-simultaneous machine available. His goal is to minimize the makespan. The goal to maximize the minimum machine completion time of the classical parallel machines scheduling was proposed by Deuermeyer et al. [12]. Lin et al. [13] combined the models of Deuermeyer et al. [12] and Lee [11]. They studied parallel machines scheduling with non-simultaneous machine available time to maximize the minimum machine completion time. Zhao et al. [14] considered the two-parallel machine scheduling problem with rate-modifying activities. For the total completion time minimization problem, they provided a polynomial algorithm to solve the problem optimally. Wang et al. [15] considered parallel identical machines scheduling problems with a deteriorating maintenance activity. They proved that the total completion time minimization problem can be solved in polynomial time. This paper addresses two scheduling problems of parallel identical machines in which the machines may not be available simultaneously at time zero, and the objectives are to minimize an objective function which includes costs for total completion time and total absolute differences in completion times and to minimize an objective function which includes costs for total waiting time and total absolute differences in waiting times, respectively. The remaining part of this paper is organized as follows. In Section 3.2 we formulate the model. In Section 3, the identical parallel machines problem with minimizing the linear combination of total completion time and total absolute differences in completion times is considered. In Section 4, the identical parallel machines problem with minimizing the linear
⇑ Corresponding author at: School of Science, Shenyang Aerospace University, Shenyang 110136, China. E-mail address: [email protected] (X.-Y. Wang). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.09.053
5228
L. Shen et al. / Applied Mathematical Modelling 37 (2013) 5227–5232
combination of total waiting time and total absolute differences in waiting times is considered. In Section 5, an example is given. The last section presents the conclusions. 2. Problem description The following notations will be used throughout the paper: N = {J1, J2, . . ., Jn}: The set of jobs to be processed. M = {M1, M2, . . ., Mm}: The set of parallel identical machines. s: The sequence of jobs to be processed by the machines. [i]: job in the ith position. pj: The processing time of job Jj. ai: The ready (release) time of machine Mi. Cj: The completion time of job Jj. Wj: The waiting time of job Jj, that is Wj = Cj pj. P TC: The total completion times, that is TC = i=1nCi. P n TW: The total waiting times, that is TW = i=1 Wi. TADC: The total absolute differences in completion times, that is
TADC ¼
n X n X jC i C j j: i¼1 j¼i
TADW: The total absolute differences in waiting times, that is
TADW ¼
n X n X jW i W j j: i¼1 j¼i
The problem under investigation can be described as follows: There are n independent jobs N ¼ fJ 1 ; J 2 ; . . . ; J n g to be processed on m parallel identical machines M ¼ fM 1 ; M 2 ; . . . ; M m g. As in Lee [11], let pj denote the processing time required by job J j ; ai denote the earliest time that machine M i can start to process the jobs. Without loss of generality, we assume that the machines are indexed by nondecreasing ready times, i.e., 0 6 a1 6 a2 6 6 am . The machine can handle one job at a time and preemption is not allowed. We also assume that all jobs are ready at time zero. Any job cannot be processed on two or more machines at the same time. Let ni be the number of jobs processed on machine M i , thus, n1 þ n2 þ þ nm ¼ n; Ai be the job set processed on machine Mi ; i ¼ 1; 2; . . . ; m. Let J i½j denote the jth job on machine M i ; pi½j and C i½j denote the processing and completion time of job J i½j . The objective is to determine the optimal sequence of jobs in the machine so that the corresponding value of the following cost functions be optimal:
Z ¼ d1 TC þ d2 TADC;
ð1Þ
Z ¼ d1 TW þ d2 TADW;
ð2Þ
where weights d1 P 0 and d2 P 0 are given constants (the decision-maker selects the weights d1 and d2 ). 3. The problem Pm; ai jjd1 TC þ d2 TADC 3.1. Main results for m ¼ 2 In this subsection we consider problem P2; ai jjd1 TC þ d2 TADC. When the sets A1 and A2 , and a sequence of job on each machine are given, we have
C 1½j ¼ a1 þ
j X
p1½k ;
j ¼ 1; 2; . . . ; n1 ;
p2½k ;
j ¼ 1; 2; . . . ; n2 :
k¼1
C 2½j ¼ a2 þ
j X k¼1
Hence, for the total completion time and the total absolute differences in completion times (Kanet [16]), we have
TCðn1 ; n2 Þ ¼ n1 a1 þ n2 a2 þ
n1 n2 X X ðn1 k þ 1Þp1½k þ ðn2 k þ 1Þp2½k ; k¼1
TADCðn1 ; n2 Þ ¼
n1 X k¼1
ðk 1Þðn1 k þ 1Þp1½k þ
k¼1 n2 X ðk 1Þðn2 k þ 1Þp2½k : k¼1
5229
L. Shen et al. / Applied Mathematical Modelling 37 (2013) 5227–5232
Hence
d1 TCðn1 ; n2 Þ þ d2 TADCðn1 ; n2 Þ ¼ d1 ðn1 a1 þ n2 a2 Þ þ
n1 n2 X X ½d1 ðn1 k þ 1Þ þ d2 ðk 1Þðn1 k þ 1Þp1½k þ ½d1 ðn2 k þ 1Þ k¼1
k¼1
þ d2 ðk 1Þðn2 k þ 1Þp2½k : We need to determine the sets A1 and A2 , and a sequence of job on each machine to minimize d1 TC þ d2 TADC. In the following, we will show that when a ðn1 ; n2 Þ vector is given, the problem can be formulated as a assignment problem. Let
Bðn1 ; n2 Þ ¼
n1 n2 X X ½d1 ðn1 k þ 1Þ þ d2 ðk 1Þðn1 k þ 1Þp1½k þ ½d1 ðn2 k þ 1Þ þ d2 ðk 1Þðn2 k þ 1Þp2½k : k¼1
k¼1
Since d1 TCðn1 ; n2 Þ þ d2 TADCðn1 ; n2 Þ ¼ d1 ðn1 a1 þ n2 a2 Þ þ Bðn1 ; n2 Þ, for a given a ðn1 ; n2 Þ vector, minimize d1 TCðn1 ; n2 Þþ d2 TADCðn1 ; n2 Þ is equivalent to minimizing Bðn1 ; n2 Þ. Suppose job J j ðj ¼ 1; 2; . . . ; nÞ is scheduled in position r ðr ¼ 1; 2; . . . ; ni Þ on machine Mi ði ¼ 1; 2; . . . ; mÞ, then its contribution for objective is d1 ðni r þ 1Þ þ d2 ðr 1Þðni r þ 1Þpj . In order to formulate this problem as an assignment problem, we define the following standard binary variables: Let xjir be a 0/1 variable such that xjir ¼ 1 if job J j (j ¼ 1; 2; . . . ; n) is assigned to machine M i (i ¼ 1; 2) at position r (r ¼ 1; 2; . . . ; ni ), and xjir ¼ 0, otherwise. Let wjir denote the cost associated with assigning job J j (j ¼ 1; 2; . . . ; n) is assigned to machine M i (i ¼ 1; 2) at position r (r ¼ 1; 2; . . . ; ni ):
wjir ¼ d1 ðni r þ 1Þ þ d2 ðr 1Þðni r þ 1Þpj :
ð3Þ
For a given ðn1 ; n2 Þ vector, the problem P2; ai jjd1 TC þ d2 TADC can be solved by the following assignment problem: Minimize
Z¼
ni X 2 X n X wjir xjir þ d1 ðn1 a1 þ n2 a2 Þ;
ð4Þ
i¼1 r¼1 j¼1
subject to ni 2 X X
xjir ¼ 1;
i¼1 r¼1 n X
xjir ¼ 1;
j ¼ 1; 2; . . . ; n;
i ¼ 1; 2; r ¼ 1; 2; . . . ; ni ;
j¼1
xjir ¼ 0 or 1;
j ¼ 1; 2; . . . ; n; i ¼ 1; 2; r ¼ 1; 2; . . . ; ni ;
where wjir are calculated by (3). We have to solve such an assignment problem for each ðn1 ; n2 Þ vector. Hence the vectors ðn1 ; n2 Þ are ð0; nÞ; ð1; n 1Þ; . . . ; ðn; 0Þ, and the number of vectors ðn1 ; n2 Þ is n þ 1. We have to solve a weighted-bipartite matching problems for each ðn1 ; n2 Þ vector. The number of all assignment problems is n þ 1. We now to give a polynomial time algorithm for the problem P2; ai jjd1 TC þ d2 TADC. Algorithm 1. Step 0. Set n1 ¼ 0. Step 1. Set n2 ¼ n n1 . Step 2. For a given pair of ðn1 ; n2 Þ create an assignment problem with weight wjir defined above. Solve the assignment problem (4) to obtain a local optimal schedule and the total cost. Step 3. Set n1 ¼ n1 þ 1, If n1 > n, then go to Step 4. Otherwise go to Step 1. Step 4. The global optimal schedule is the one with the minimum total cost.
Theorem 1. Algorithm 1 will find an optimal solution for the problem P2; ai jjd1 TC þ d2 TADC with time complexity Oðn4 Þ. Proof. As discussed above, we can convert the problem P2; ai jjd1 TC þ d2 TADC to an assignment problem and find an optimal solution by Algorithm 1. Step 2 can be solved in Oðn3 Þ for each assignment problem. Step 1 is executed n þ 1 times. Consequently, the overall time requirement of Algorithm 1 is Oðn4 Þ. h
5230
L. Shen et al. / Applied Mathematical Modelling 37 (2013) 5227–5232
3.2. Extensions to m-parallel identical machines Now we discuss how the results for two-parallel machines can be extended to m-parallel machines (m > 2Þ. Similarly to the case of two-parallel machines, when ðn1 ; n2 ; . . . ; nm Þ ðn1 þ n2 þ . . . nm ¼ nÞ and a sequence of job on each machine are given, we have
d1 TCðn1 ; n2 ; . . . ; nm Þ þ d2 TADCðn1 ; n2 ; . . . ; nm Þ ¼
ni m X m X X ½d1 ðni j þ 1Þ þ d2 ðj 1Þðni j þ 1Þpi½j þ d1 ni ai : i¼1 j¼1
i¼1
Hence, for a given ðn1 ; n2 ; . . . ; nm Þ vector, the problem Pm; ai jjd1 TC þ d2 TADC can be solved by an assignment problem. The remaining question is how many ðn1 ; n2 ; . . . ; nm Þ vectors exist. This problem is equal to that of allocation n identical balls (=jobs) into m different urns (=machines). The number of ball allocations in the problem is known to be (see Stirzaker nþm1 ð2nÞm ð2nÞm [17, p. 64]) : An upper bound on this number m is given in Stirzaker [17, p. 64]: ðm1Þ! . Obviously, ðm1Þ! is a polym1 nomial upper bound in n (similar to Mosheiov [18]). Hence, we have that the problem Pm; ai jjd1 TC þ d2 TADC can be solved in polynomial time, i.e., Theorem 2. When m > 2 is given, the problem Pm; ai jjd1 TC þ d2 TADC can be solved in Oðnmþ3 Þ times. Remark 1. The solution procedure for minimizing the problem Pm; ai jjd1 TC þ d2 TADC can be easily extended to the case of unrelated machines. Using a similar model to that of (4), let xjir be a 0/1 variable such that xjir ¼ 1 if job J j (j ¼ 1; 2; . . . ; n) is assigned to machine M i (i ¼ 1; 2; . . . ; m) at position r (r ¼ 1; 2; . . . ; ni ), and xjir ¼ 0, otherwise. For a given ðn1 ; n2 ; . . . ; nm Þ vector, the problem Rm; ai jjd1 TC þ d2 TADC can be solved by the following assignment problem:
min
ni X m X n m X X wjir xjir =sij þ d1 ni ai i¼1 r¼1 j¼1
i¼1
st ni m X X xjir ¼ 1; i¼1 r¼1 n X
xjir ¼ 1;
j ¼ 1; 2; . . . ; n;
i ¼ 1; 2; . . . ; m; r ¼ 1; 2; . . . ; ni ;
j¼1
xjir ¼ 0 or 1;
j ¼ 1; 2; . . . ; n; i ¼ 1; 2; . . . ; m; r ¼ 1; 2; . . . ; ni ;
where sij denote the process speed of job J j (j ¼ 1; 2; . . . ; n) on machine M i (i ¼ 1; 2; . . . ; m), wjir are calculated by (3). 4. The problem Pm; ai jjd1 TW þ d2 TADW 4.1. Main results for m ¼ 2 In this subsection we consider problems P2; ai jjd1 TW þ d2 TADW. When the sets A1 and A2 , and a sequence of job on each machine are given, we have
W 1½j ¼ a1 þ
j1 X p1½k ;
j ¼ 1; 2; . . . ; n1 ;
k¼1
W 2½j ¼ a2 þ
j1 X p2½k ;
j ¼ 1; 2; . . . ; n2 :
k¼1
Hence, for the total waiting time and the total absolute differences in waiting times (Bagchi [19]), we have
TWðn1 ; n2 Þ ¼ n1 a1 þ n2 a2 þ
n1 n2 X X ðn1 kÞp1½k þ ðn2 kÞp2½k ; k¼1
TADWðn1 ; n2 Þ ¼
k¼1
n1 n2 X X kðn1 kÞp1½k þ kðn2 kÞp2½k : k¼1
k¼1
Hence
d1 TWðn1 ; n2 Þ þ d2 TADWðn1 ; n2 Þ ¼ d1 ðn1 a1 þ n2 a2 Þ þ
n1 n2 X X ½d1 ðn1 kÞ þ d2 kðn1 kÞp1½k þ ½d1 ðn2 kÞ þ d2 kðn2 kÞp2½k : k¼1
k¼1
5231
L. Shen et al. / Applied Mathematical Modelling 37 (2013) 5227–5232
Similar the Section 3, for a given ðn1 ; n2 Þ vector, the problem P2; ai jjd1 TW þ d2 TADW can be solved by the following assignment problem: Minimize
Z¼
ni X 2 X n X
v jir xjir þ d1 ðn1 a1 þ n2 a2 Þ
ð5Þ
i¼1 r¼1 j¼1
subject to ni 2 X X
xjir ¼ 1;
i¼1 r¼1 n X
xjir ¼ 1;
j ¼ 1; 2; . . . ; n;
i ¼ 1; 2; r ¼ 1; 2; . . . ; ni ;
j¼1
xjir ¼ 0 or 1;
j ¼ 1; 2; . . . ; n; i ¼ 1; 2; r ¼ 1; 2; . . . ; ni ;
where v jir ¼ d1 ðni rÞ þ d2 rðni rÞpj . Similar the Section 3, we now to give a polynomial time algorithm for the problem P2; ai jjd1 TW þ d2 TADW. Algorithm 2. Step 0. Set n1 ¼ 0. Step 1. Set n2 ¼ n n1 . Step 2. For a given pair of ðn1 ; n2 Þ, solve the assignment problem (5) to obtain a local optimal schedule and the total cost. Step 3. Set n1 ¼ n1 þ 1, If n1 > n, then go to Step 4. Otherwise go to Step 1. Step 4. The global optimal schedule is the one with the minimum total cost.
Theorem 3. Algorithm 2 will find an optimal solution for the problem P2; ai jjd1 TW þ d2 TADW with time complexity Oðn4 Þ. Proof. Similar to the proof of Theorem 1. h 4.2. Extensions to m-parallel identical machines Similar to the Section 3.2, we have. Theorem 4. When m > 2 is given, the problem Pm; ai jjd1 TW þ d2 TADW can be solved in Oðnmþ3 Þ times. Remark 2. The solution procedure for the problem Pm; ai jjd1 TW þ d2 TADW can be easily extended to the case of unrelated machines. Using a similar model to that of (4), let xjir be a 0/1 variable such that xjir ¼ 1 if job J j (j ¼ 1; 2; . . . ; n) is assigned to machine M i (i ¼ 1; 2; . . . ; m) at position r (r ¼ 1; 2; . . . ; ni ), and xjir ¼ 0, otherwise. For a given ðn1 ; n2 ; . . . ; nm Þ vector, the problem Rm; ai jjd1 TW þ d2 TADW can be solved by the following assignment problem:
min
ni X m X n X
m X ni ai
i¼1 r¼1 j¼1
i¼1
v jir xjir =sij þ d1
st ni m X X
xjir ¼ 1;
i¼1 r¼1 n X
xjir ¼ 1;
j ¼ 1; 2; . . . ; n;
i ¼ 1; 2; . . . ; m; r ¼ 1; 2; . . . ; ni ;
j¼1
xjir ¼ 0 or 1;
j ¼ 1; 2; . . . ; n; i ¼ 1; 2; . . . ; m; r ¼ 1; 2; . . . ; ni ;
where sij denote the process speed of job J j (j ¼ 1; 2; . . . ; n) on machine M i (i ¼ 1; 2; . . . ; m), 5. An example The following example illustrates the working of Algorithms 1 and 2.
v jir ¼ d1 ðni rÞ þ d2 rðni rÞpj .
5232
L. Shen et al. / Applied Mathematical Modelling 37 (2013) 5227–5232
Example 1 In the modern manufacturing and service industries (like the beverage industry, the capital-intensive printing industry, the pharmaceutical industry), many production environments consist of several workcenters, each with a number of parallel identical machines with non-simultaneous machine available time, i.e., each machine M i ð1 6 i 6 mÞ has an availability time ai on which it becomes continuously ready for working. Consider 4 jobs to be processed on machines M 1 and M 2 with: p1 ¼ 6; p2 ¼ 3; p3 ¼ 8; p4 ¼ 5; a1 ¼ 1; a2 ¼ 4; d1 ¼ 2; d2 ¼ 1. According Algorithm 1, when ðn1 ; n2 Þ ¼ ð0; 4Þ, the cost is 187; when ðn1 ; n2 Þ ¼ ð1; 3Þ, the cost is 114; when ðn1 ; n2 Þ ¼ ð2; 2Þ, the cost is 94; when ðn1 ; n2 Þ ¼ ð3; 1Þ, the cost is 102; when ðn1 ; n2 Þ ¼ ð4; 0Þ, the cost is 163. Hence for the objective function d1 TC þ d2 TADC, an optimal schedule is ½J 2 ; J 6 on machine M 1 and ½J 4 ; J 8 on machine M 2 , and the optimal cost is 94. According Algorithm 2, when ðn1 ; n2 Þ ¼ ð0; 4Þ, the cost is 129; when ðn1 ; n2 Þ ¼ ð1; 3Þ, the cost is 64; when ðn1 ; n2 Þ ¼ ð2; 2Þ, the cost is 44; when ðn1 ; n2 Þ ¼ ð3; 1Þ, the cost is 60; when ðn1 ; n2 Þ ¼ ð4; 0Þ, the cost is 105. Hence for the objective function d1 TW þ d2 TADW, an optimal schedule is ½J 2 ; J 6 on machine M 1 and ½J 4 ; J 8 on machine M 2 , and the optimal cost is 44. 6. Conclusions In this paper we studied parallel identical machines scheduling with non-simultaneous machine available time. The objectives are to minimize the linear combination of total completion (waiting) time and total absolute differences in completion (waiting) times. We proved that the problem can be solved in polynomial time. Further research includes the investigation of other function objectives. Acknowledgments We are grateful to the editor and two anonymous referees for their helpful comments on an earlier version of this paper. This research was fully supported by the National Natural Science Foundation of China (Grant No. 11001181). References [1] W.-C. Lee, C.-C. Wu, A note on single-machine group scheduling problems with position-based learning effect, Appl. Math. Model. 33 (2009) 2159– 2163. [2] J.-B. Wang, L.-H. Sun, L.-Y. Sun, Single machine scheduling with exponential sum-of-logarithm-processing-times based learning effect, Appl. Math. Model. 34 (2010) 2813–2819. [3] W.-C. Lee, W.-J. Wang, Y.-R. Shiau, C.-C. Wu, A single-machine scheduling problem with two-agent and deteriorating jobs, Appl. Math. Model. 34 (2010) 3098–3107. [4] J.-B. Wang, Q. Guo, A due-date assignment problem with learning effect and deteriorating jobs, Appl. Math. Model. 34 (2010) 309–313. [5] J.-B. Wang, J.-X. Li, Single machine past-sequence-dependent setup times scheduling with general position-dependent and time-dependent learning effects, Appl. Math. Model. 35 (2011) 1388–1395. [6] P. Liu, N. Yi, X. Zhou, Two-agent single-machine scheduling problems under increasing linear deterioration, Appl. Math. Model. 35 (2011) 2290–2296. [7] X.-R. Wang, X. Huang, J.-B. Wang, Single-machine scheduling with linear decreasing deterioration to minimize earliness penalties, Appl. Math. Model. 35 (2011) 3509–3515. [8] C.-M. Wei, J.-B. Wang, P. Ji, Single-machine scheduling with time-and-resource-dependent processing times, Appl. Math. Model. 62 (2012) 792–798. [9] J. Bai, Z.-R. Li, X. Huang, Single-machine group scheduling with general deterioration and learning effects, Appl. Math. Model. 36 (2012) 1267–1274. [10] X.-R. Wang, J.-J. Wang, Single-machine scheduling with convex resource dependent processing times and deteriorating jobs, Appl. Math. Model., http:// dx.doi.org/10.1016/j.apm.2012.05.025. [11] C.-Y. Lee, Parallel machines scheduling with nonsimultaneous machine available time, Disc. Appl. Math. 30 (1991) 53–61. [12] B.L. Deuermeyer, D.K. Friesen, M.A. Langston, Scheduling to maximize the minimum processor finish time in a multi-processor system, SIAM J. Algebraic Disc. Methods 3 (1982) 190–196. [13] G.-H. Lin, E.-Y. Yao, Y. He, Parallel machine scheduling to maximize the minimum load with nonsimultaneous machine available times, Oper. Res. Lett. 22 (1998) 75–81. [14] C.-L. Zhao, H.-Y. Tang, C.-D. Cheng, Two-parallel machines scheduling with rate-modifying activities to minimize total completion time, Eur. J. Oper. Res. 198 (2009) 354–357. [15] J.-J. Wang, J.-B. Wang, F. Liu, Parallel machines scheduling with a deteriorating maintenance activity, J. Oper. Res. Soc. 62 (2011) 1898–1902. [16] J.J. Kanet, Minimizing variation of flow time in single machine systems, Manage. Sci. 27 (1981) 1453–1459. [17] D. Stirzaker, Elementary Probability, Cambridge University Press, Cambridge, 1995. [18] G. Mosheiov, Parallel machine scheduling with a learning effect, J. Oper. Res. Soc. 52 (2001) 1165–1169. [19] U.B. Bagchi, Simultaneous minimization of mean and variation of flow-time and waiting time in single machine systems, Oper. Res. 37 (1989) 118– 125.