• Previous Article
    Novel correlation coefficients under the intuitionistic multiplicative environment and their applications to decision-making process
  • JIMO Home
  • This Issue
  • Next Article
    Parameter identification and numerical simulation for the exchange coefficient of dissolved oxygen concentration under ice in a boreal lake
October  2018, 14(4): 1479-1500. doi: 10.3934/jimo.2018017

Scheduling family jobs on an unbounded parallel-batch machine to minimize makespan and maximum flow time

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, China

* Corresponding author: Jinjiang Yuan

Received  February 2017 Revised  August 2017 Published  January 2018

Fund Project: The authors are supported by NSFC (11671368), NSFC (11571321), and NSFC (11771406).

This paper investigates the scheduling of family jobs with release dates on an unbounded parallel-batch machine. The involved objective functions are makespan and maximum flow time. It was reported in the literature that the single-criterion problem for minimizing makespan is strongly NP-hard when the number of families is arbitrary, and is polynomially solvable when the number of families is fixed. We first show in this paper that the single-criterion problem for minimizing maximum flow time is also strongly NP-hard when the number of families is arbitrary. We further show that the Pareto optimization problem (also called bicriteria problem) for minimizing makespan and maximum flow time is polynomially solvable when the number of families is fixed, by enumerating all Pareto optimal points in polynomial time. This implies that the single-criterion problem for minimizing maximum flow time is also polynomially solvable when the number of families is fixed.

Citation: Zhichao Geng, Jinjiang Yuan. Scheduling family jobs on an unbounded parallel-batch machine to minimize makespan and maximum flow time. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1479-1500. doi: 10.3934/jimo.2018017
References:
[1]

A. AllahverdiC. T. NgT. C. E. Cheng and M. Y. Kovalyov, A survey of scheduling problems with setup times or costs, European Journal of Operational Research, 187 (2008), 985-1032.  doi: 10.1016/j.ejor.2006.06.060.  Google Scholar

[2]

J. BaiZ. R. Li and X. Huang, Single-machine group scheduling with general deterioration and learning effects, Applied Mathematical Modelling, 36 (2012), 1267-1274.  doi: 10.1016/j.apm.2011.07.068.  Google Scholar

[3]

P. BruckerA. GladkyH. HoogeveenM. Y. KovalyovC. N. Potts and T. Tautenhahn, Scheduling a batching machine, Journal of Scheduling, 1 (1998), 31-54.  doi: 10.1002/(SICI)1099-1425(199806)1:1<31::AID-JOS4>3.0.CO;2-R.  Google Scholar

[4]

K. ChakhlevitchC. A. Glass and H. Kellerer, Batch machine production with perishability time windows and limited batch size, European Journal of Operational Research, 210 (2011), 39-47.  doi: 10.1016/j.ejor.2010.10.033.  Google Scholar

[5]

T. C. E. ChengZ. H. Liu and W. C. Yu, Scheduling jobs with release dates and deadlines on a batch processing machine, IIE Transactions, 33 (2001), 685-690.   Google Scholar

[6]

Y. Gao and J. J. Yuan, A note on Pareto minimizing total completion time and maximum cost, Operational Research Letter, 43 (2015), 80-82.  doi: 10.1016/j.orl.2014.12.001.  Google Scholar

[7]

Z. C. Geng and J. J. Yuan, Pareto optimization scheduling of family jobs on a p-batch machine to minimize makespan and maximum lateness, Theoretical Computer Science, 570 (2015), 22-29.  doi: 10.1016/j.tcs.2014.12.020.  Google Scholar

[8]

Z. C. Geng and J. J. Yuan, A note on unbounded parallel-batch scheduling, Information Processing Letters, 115 (2015), 969-974.  doi: 10.1016/j.ipl.2015.07.002.  Google Scholar

[9]

C. HeY. X. Lin and J. J. Yuan, Bicriteria scheduling on a batching machine to minimize maximum lateness and makespan, Theoretical Computer Science, 381 (2007), 234-240.  doi: 10.1016/j.tcs.2007.04.034.  Google Scholar

[10]

H. Hoogeveen, Multicriteria scheduling, European Journal of Operational Research, 167 (2005), 592-623.  doi: 10.1016/j.ejor.2004.07.011.  Google Scholar

[11]

J. A. Hoogeveen, Single-machine scheduling to minimize a function of two or three maximum cost criteria, Journal of Algorithms, 21 (1996), 415-433.  doi: 10.1006/jagm.1996.0051.  Google Scholar

[12]

J. A. Hoogeveen and S. L. van de Velde, Minimizing total completion time and maximum cost simultaneously is solvable in polynomial time, Operations Research Letters, 17 (1995), 205-208.  doi: 10.1016/0167-6377(95)00023-D.  Google Scholar

[13]

J. A. Hoogeveen and S. L. van de Velde, Scheduling with target start times, European Journal of Operational Research, 129 (2001), 87-94.  doi: 10.1016/S0377-2217(99)00426-9.  Google Scholar

[14]

Y. Ikura and M. Gimple, Efficient scheduling algorithms for a single batch processing machine, Operations Research Letters, 5 (1986), 61-65.  doi: 10.1016/0167-6377(86)90104-5.  Google Scholar

[15]

F. Jolai, Minimizing number of tardy jobs on a batch processing machine with incompatible job families, European Journal of Operational Research, 162 (2005), 184-190.  doi: 10.1016/j.ejor.2003.10.011.  Google Scholar

[16]

C. Y. LeeR. Uzsoy and L. A. Martin-Vega, Efficient algorithms for scheduling semi-conductor burn-in operations, Operations Research, 40 (1992), 764-775.  doi: 10.1287/opre.40.4.764.  Google Scholar

[17]

S. S. Li and R. X. Chen, Single-machine parallel-batching scheduling with family jobs to minimize weighted number of tardy jobs, Computers and Industrial Engineering, 73 (2014), 5-10.  doi: 10.1016/j.cie.2014.04.007.  Google Scholar

[18]

S. S. LiC. T. NgT. C. E. Cheng and J. J. Yuan, Parallel-batch scheduling of deteriorating jobs with release dates to minimize the makespan, European Journal of Operational Research, 210 (2011), 482-488.  doi: 10.1016/j.ejor.2010.11.021.  Google Scholar

[19]

S. S. Li and J. J. Yuan, Parallel-machine parallel-batching scheduling with family jobs and release dates to minimize makespan, Journal of Combinatorial Optimization, 19 (2010), 84-93.  doi: 10.1007/s10878-008-9163-z.  Google Scholar

[20]

Z. H. Liu and W. C. Yu, Scheduling one batch processor subject to job release dates, Discrete Applied Mathematics, 105 (2000), 129-136.  doi: 10.1016/S0166-218X(00)00181-5.  Google Scholar

[21]

L. L. Liu and F. Zhang, Minimizing the number of tardy jobs on a batch processing machine with incompatible job families, ISECS International Colloquium on Computing, Communication, Control, and Management, 3 (2008), 277-280.  doi: 10.1109/CCCM.2008.107.  Google Scholar

[22]

Z. H. LiuJ. J. Yuan and T. C. E. Cheng, On scheduling an unbounded batch machine, Operations Research Letters, 31 (2003), 42-48.  doi: 10.1016/S0167-6377(02)00186-4.  Google Scholar

[23]

S. Malve and R. Uzsoy, A genetic algorithm for minimizing maximum lateness on parallel identical batch processing machines with dynamic job arrivals and incompatible job families, Computers and Operations Research, 34 (2007), 3016-3028.  doi: 10.1016/j.cor.2005.11.011.  Google Scholar

[24]

C. X. MiaoY. Z. Zhang and Z. G. Cao, Bounded parallel-batch scheduling on single and multi machines for deteriorating jobs, Information Processing Letters, 111 (2011), 798-803.  doi: 10.1016/j.ipl.2011.05.018.  Google Scholar

[25]

Q. Q. NongC. T. Ng and T. C. E. Cheng, The bounded single-machine parallel-batching scheduling problem with family jobs and release dates to minimize makespan, Operations Research Letters, 36 (2008), 61-66.  doi: 10.1016/j.orl.2007.01.007.  Google Scholar

[26]

E. S. PanG. N. WangL. F. XiL. Chen and X. L. Han, Single-machine group scheduling problem considering learning, forgetting effects and preventive maintenance, International Journal of Production Research, 52 (2014), 5690-5704.  doi: 10.1080/00207543.2014.904967.  Google Scholar

[27]

J. PeiX. B. LiuP. M. PardalosA. Migdalas and S. L. Yang, Serial-batching scheduling with time-dependent Setup time and effects of deterioration and learning on a single-machine, Journal of Global Optimization, 67 (2017), 251-262.  doi: 10.1007/s10898-015-0320-5.  Google Scholar

[28]

J. PeiB. Y. ChengX. B. LiuP. M. Pardalos and M. Kong, Single-machine and parallel-machine serial-batching scheduling problems with position-based learning effect and linear setup time, Annals of Operations Research, (2017), 1-25.  doi: 10.1007/s10479-017-2481-8.  Google Scholar

[29]

J. PeiX. B. LiuP. M. PardalosW. J. Fan and S. L. Yang, Single machine serial-batching scheduling with independent setup time and deteriorating job processing times, Optimization Letters, 9 (2015), 91-104.  doi: 10.1007/s11590-014-0740-z.  Google Scholar

[30]

J. PeiX. B. LiuP. M. PardalosW. J. Fan and S. L. Yang, Scheduling deteriorating jobs on a single serial-batching machine with multiple job types and sequence-dependent setup times, Annals of Operations Research, 249 (2017), 175-195.  doi: 10.1007/s10479-015-1824-6.  Google Scholar

[31]

J. PeiP. M. PardalosX. B. LiuW. J. Fan and S. L. Yang, Serial batching scheduling of deteriorating jobs in a two-stage supply chain to minimize the makespan, European Journal of Operational Research, 244 (2015), 13-25.  doi: 10.1016/j.ejor.2014.11.034.  Google Scholar

[32]

C. N. Potts and M. Y. Kovalyov, Scheduling with batching: a review, European Journal of Operational Research, 120 (2000), 228-249.  doi: 10.1016/S0377-2217(99)00153-8.  Google Scholar

[33]

X. L. QiS. G. Zhou and J. J. Yuan, Single machine parallel-batch scheduling with dete-riorating jobs, Theoretical Computer Science, 410 (2009), 830-836.  doi: 10.1016/j.tcs.2008.11.009.  Google Scholar

[34] V. T'kindt and J. C. Billaut, Multicriteria Scheduling: Theory, Models and Algorithms, 2$^{nd}$ edition, Springer, Berlin, 2006.  doi: 10.1007/978-3-662-04986-0.  Google Scholar
[35]

H. Xuan and L. X. Tang, Scheduling a hybrid flowshop with batch production at the last stage, Computers and Operations Research, 34 (2007), 2718-2733.  doi: 10.1016/j.cor.2005.10.014.  Google Scholar

[36]

J. J. YuanZ. H. LiuC. T. Ng and T. C. E. Cheng, The unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan, Theoretical Computer Science, 320 (2004), 199-212.  doi: 10.1016/j.tcs.2004.01.038.  Google Scholar

show all references

References:
[1]

A. AllahverdiC. T. NgT. C. E. Cheng and M. Y. Kovalyov, A survey of scheduling problems with setup times or costs, European Journal of Operational Research, 187 (2008), 985-1032.  doi: 10.1016/j.ejor.2006.06.060.  Google Scholar

[2]

J. BaiZ. R. Li and X. Huang, Single-machine group scheduling with general deterioration and learning effects, Applied Mathematical Modelling, 36 (2012), 1267-1274.  doi: 10.1016/j.apm.2011.07.068.  Google Scholar

[3]

P. BruckerA. GladkyH. HoogeveenM. Y. KovalyovC. N. Potts and T. Tautenhahn, Scheduling a batching machine, Journal of Scheduling, 1 (1998), 31-54.  doi: 10.1002/(SICI)1099-1425(199806)1:1<31::AID-JOS4>3.0.CO;2-R.  Google Scholar

[4]

K. ChakhlevitchC. A. Glass and H. Kellerer, Batch machine production with perishability time windows and limited batch size, European Journal of Operational Research, 210 (2011), 39-47.  doi: 10.1016/j.ejor.2010.10.033.  Google Scholar

[5]

T. C. E. ChengZ. H. Liu and W. C. Yu, Scheduling jobs with release dates and deadlines on a batch processing machine, IIE Transactions, 33 (2001), 685-690.   Google Scholar

[6]

Y. Gao and J. J. Yuan, A note on Pareto minimizing total completion time and maximum cost, Operational Research Letter, 43 (2015), 80-82.  doi: 10.1016/j.orl.2014.12.001.  Google Scholar

[7]

Z. C. Geng and J. J. Yuan, Pareto optimization scheduling of family jobs on a p-batch machine to minimize makespan and maximum lateness, Theoretical Computer Science, 570 (2015), 22-29.  doi: 10.1016/j.tcs.2014.12.020.  Google Scholar

[8]

Z. C. Geng and J. J. Yuan, A note on unbounded parallel-batch scheduling, Information Processing Letters, 115 (2015), 969-974.  doi: 10.1016/j.ipl.2015.07.002.  Google Scholar

[9]

C. HeY. X. Lin and J. J. Yuan, Bicriteria scheduling on a batching machine to minimize maximum lateness and makespan, Theoretical Computer Science, 381 (2007), 234-240.  doi: 10.1016/j.tcs.2007.04.034.  Google Scholar

[10]

H. Hoogeveen, Multicriteria scheduling, European Journal of Operational Research, 167 (2005), 592-623.  doi: 10.1016/j.ejor.2004.07.011.  Google Scholar

[11]

J. A. Hoogeveen, Single-machine scheduling to minimize a function of two or three maximum cost criteria, Journal of Algorithms, 21 (1996), 415-433.  doi: 10.1006/jagm.1996.0051.  Google Scholar

[12]

J. A. Hoogeveen and S. L. van de Velde, Minimizing total completion time and maximum cost simultaneously is solvable in polynomial time, Operations Research Letters, 17 (1995), 205-208.  doi: 10.1016/0167-6377(95)00023-D.  Google Scholar

[13]

J. A. Hoogeveen and S. L. van de Velde, Scheduling with target start times, European Journal of Operational Research, 129 (2001), 87-94.  doi: 10.1016/S0377-2217(99)00426-9.  Google Scholar

[14]

Y. Ikura and M. Gimple, Efficient scheduling algorithms for a single batch processing machine, Operations Research Letters, 5 (1986), 61-65.  doi: 10.1016/0167-6377(86)90104-5.  Google Scholar

[15]

F. Jolai, Minimizing number of tardy jobs on a batch processing machine with incompatible job families, European Journal of Operational Research, 162 (2005), 184-190.  doi: 10.1016/j.ejor.2003.10.011.  Google Scholar

[16]

C. Y. LeeR. Uzsoy and L. A. Martin-Vega, Efficient algorithms for scheduling semi-conductor burn-in operations, Operations Research, 40 (1992), 764-775.  doi: 10.1287/opre.40.4.764.  Google Scholar

[17]

S. S. Li and R. X. Chen, Single-machine parallel-batching scheduling with family jobs to minimize weighted number of tardy jobs, Computers and Industrial Engineering, 73 (2014), 5-10.  doi: 10.1016/j.cie.2014.04.007.  Google Scholar

[18]

S. S. LiC. T. NgT. C. E. Cheng and J. J. Yuan, Parallel-batch scheduling of deteriorating jobs with release dates to minimize the makespan, European Journal of Operational Research, 210 (2011), 482-488.  doi: 10.1016/j.ejor.2010.11.021.  Google Scholar

[19]

S. S. Li and J. J. Yuan, Parallel-machine parallel-batching scheduling with family jobs and release dates to minimize makespan, Journal of Combinatorial Optimization, 19 (2010), 84-93.  doi: 10.1007/s10878-008-9163-z.  Google Scholar

[20]

Z. H. Liu and W. C. Yu, Scheduling one batch processor subject to job release dates, Discrete Applied Mathematics, 105 (2000), 129-136.  doi: 10.1016/S0166-218X(00)00181-5.  Google Scholar

[21]

L. L. Liu and F. Zhang, Minimizing the number of tardy jobs on a batch processing machine with incompatible job families, ISECS International Colloquium on Computing, Communication, Control, and Management, 3 (2008), 277-280.  doi: 10.1109/CCCM.2008.107.  Google Scholar

[22]

Z. H. LiuJ. J. Yuan and T. C. E. Cheng, On scheduling an unbounded batch machine, Operations Research Letters, 31 (2003), 42-48.  doi: 10.1016/S0167-6377(02)00186-4.  Google Scholar

[23]

S. Malve and R. Uzsoy, A genetic algorithm for minimizing maximum lateness on parallel identical batch processing machines with dynamic job arrivals and incompatible job families, Computers and Operations Research, 34 (2007), 3016-3028.  doi: 10.1016/j.cor.2005.11.011.  Google Scholar

[24]

C. X. MiaoY. Z. Zhang and Z. G. Cao, Bounded parallel-batch scheduling on single and multi machines for deteriorating jobs, Information Processing Letters, 111 (2011), 798-803.  doi: 10.1016/j.ipl.2011.05.018.  Google Scholar

[25]

Q. Q. NongC. T. Ng and T. C. E. Cheng, The bounded single-machine parallel-batching scheduling problem with family jobs and release dates to minimize makespan, Operations Research Letters, 36 (2008), 61-66.  doi: 10.1016/j.orl.2007.01.007.  Google Scholar

[26]

E. S. PanG. N. WangL. F. XiL. Chen and X. L. Han, Single-machine group scheduling problem considering learning, forgetting effects and preventive maintenance, International Journal of Production Research, 52 (2014), 5690-5704.  doi: 10.1080/00207543.2014.904967.  Google Scholar

[27]

J. PeiX. B. LiuP. M. PardalosA. Migdalas and S. L. Yang, Serial-batching scheduling with time-dependent Setup time and effects of deterioration and learning on a single-machine, Journal of Global Optimization, 67 (2017), 251-262.  doi: 10.1007/s10898-015-0320-5.  Google Scholar

[28]

J. PeiB. Y. ChengX. B. LiuP. M. Pardalos and M. Kong, Single-machine and parallel-machine serial-batching scheduling problems with position-based learning effect and linear setup time, Annals of Operations Research, (2017), 1-25.  doi: 10.1007/s10479-017-2481-8.  Google Scholar

[29]

J. PeiX. B. LiuP. M. PardalosW. J. Fan and S. L. Yang, Single machine serial-batching scheduling with independent setup time and deteriorating job processing times, Optimization Letters, 9 (2015), 91-104.  doi: 10.1007/s11590-014-0740-z.  Google Scholar

[30]

J. PeiX. B. LiuP. M. PardalosW. J. Fan and S. L. Yang, Scheduling deteriorating jobs on a single serial-batching machine with multiple job types and sequence-dependent setup times, Annals of Operations Research, 249 (2017), 175-195.  doi: 10.1007/s10479-015-1824-6.  Google Scholar

[31]

J. PeiP. M. PardalosX. B. LiuW. J. Fan and S. L. Yang, Serial batching scheduling of deteriorating jobs in a two-stage supply chain to minimize the makespan, European Journal of Operational Research, 244 (2015), 13-25.  doi: 10.1016/j.ejor.2014.11.034.  Google Scholar

[32]

C. N. Potts and M. Y. Kovalyov, Scheduling with batching: a review, European Journal of Operational Research, 120 (2000), 228-249.  doi: 10.1016/S0377-2217(99)00153-8.  Google Scholar

[33]

X. L. QiS. G. Zhou and J. J. Yuan, Single machine parallel-batch scheduling with dete-riorating jobs, Theoretical Computer Science, 410 (2009), 830-836.  doi: 10.1016/j.tcs.2008.11.009.  Google Scholar

[34] V. T'kindt and J. C. Billaut, Multicriteria Scheduling: Theory, Models and Algorithms, 2$^{nd}$ edition, Springer, Berlin, 2006.  doi: 10.1007/978-3-662-04986-0.  Google Scholar
[35]

H. Xuan and L. X. Tang, Scheduling a hybrid flowshop with batch production at the last stage, Computers and Operations Research, 34 (2007), 2718-2733.  doi: 10.1016/j.cor.2005.10.014.  Google Scholar

[36]

J. J. YuanZ. H. LiuC. T. Ng and T. C. E. Cheng, The unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan, Theoretical Computer Science, 320 (2004), 199-212.  doi: 10.1016/j.tcs.2004.01.038.  Google Scholar

Figure 1.  The structure of the scheduling problem
Figure 2.  An intuitive interpretation of the proposed algorithm
Table 1.  The definitions of the abbreviations/notations
Abbreviation/Notation Definition
ERD earliest release date first (rule)
PoP Pareto optimal point
PoS Pareto optimal schedule
ERD-family scheule a schedule of family jobs defined in the first paragraph in section 3.1
${\mathcal F}_f$ the $f$-th job family
$f$-job a job which belongs to family ${\mathcal F}_f$
$f$-batch a p-bath which only includes $f$-jobs
$J_{f, j}$ the $j$-th job in family ${\mathcal F}_f$
$J_{j}$ a general job
$p_{j}~(p_{f, j})$/$r_{j}~(r_{f, j})$ the processing time/the release dates of job $J_{j}~(J_{f, j})$
$n_f$ the number of jobs in family ${\mathcal F}_f$
$(b<n)/(b\geq n)$ the bounded/unbounded capacity of a p-batch
$p(B)$ the processing time of batch $B$
$\pi/\sigma$ a feasible schedule
$C_{j}(\pi)(C_{f, j}(\pi))$ the completion time of job $J_{j}$ ($J_{f, j}$) in $\pi$
$S_{j}(\pi)(S_{f, j}(\pi))$ the starting time of job $J_{j}$ ($J_{f, j}$) in $\pi$
$C_{B}(\pi)$ the completion time of batch $B$ in $\pi$
$S_{B}(\pi)$ the starting time of batch $B$ in $\pi$
$F_{j}(\pi)(F_{f, j}(\pi))$ the flow time of job $J_{j}$ ($J_{f, j}$) in $\pi$ which is $F_{j}(\pi)=C_{j}(\pi)-r_{j}(\pi)$
$C_{\max}(\pi)$ the maximum completion time of all jobs in $\pi$
$F_{\max}(\pi)$ the maximum flow time of all jobs in $\pi$
${\mathcal F}_f^{(i)}$ the set composed by the first $i$ jobs of family ${\mathcal F}_f$, i.e., $\{J_{f, 0}, J_{f, 1}, \cdots, J_{f, i}\}$
$(i_1, \cdots, i_K)$ the instance composed by the job set ${\mathcal F}_1^{(i_1)} \cup \cdots \cup {\mathcal F}_K^{(i_K)}$
$P(i_1, \cdots, i_K)$ the problem $1|\beta |^\#(C_{\max}, F_{\max})$ restricted on the instance $(i_1, \cdots, i_K)$
$P_{Y}(i_1, \cdots, i_K)$ the problem $1|\beta |C_{\max}: F_{\max}\leq Y$ restricted on the instance $(i_1, \cdots, i_K)$
$P_{Y}^{(f)}(i_1, \cdots, i_K)$ a restricted version of $P_{Y}(i_1, \cdots, i_K)$ with $i_f \geq 1$ for which it is required that feasible schedules end with an $f$-batch
$C_Y^{(f)}(i_1, \cdots, i_K)$ the optimal makespan of problem $P_{Y}^{(f)}(i_1, \cdots, i_K)$
$C_Y^{(f, k_f)}(i_1, \cdots, i_K)$ defined in equation (3)
$D_Y^{(f)}(i_1, \cdots, i_K)$ defined in equation (4)
${\mathcal X}_{Y}(i_1, \cdots, i_K)$ defined in equation (5), the set of family indices attaining the minimum in equation (1)
$\Psi^{(f)}_{Y}(i_1, \cdots, i_K)$ defined in equation (6), the set of the $f$-job indices attaining the minimum in equation (2)
$C_Y(i_1, \cdots, i_K)$ the optimal makespan of problem $P_{Y}(i_1, \cdots, i_K)$
Algorithm DP($Y$) the proposed dynamic programming algorithm for problem $1|\beta |C_{\max}: F_{\max}\leq Y$
Algorithm Family-CF the proposed algorithm for problem $1|\beta |^\#(C_{\max}, F_{\max})$
Abbreviation/Notation Definition
ERD earliest release date first (rule)
PoP Pareto optimal point
PoS Pareto optimal schedule
ERD-family scheule a schedule of family jobs defined in the first paragraph in section 3.1
${\mathcal F}_f$ the $f$-th job family
$f$-job a job which belongs to family ${\mathcal F}_f$
$f$-batch a p-bath which only includes $f$-jobs
$J_{f, j}$ the $j$-th job in family ${\mathcal F}_f$
$J_{j}$ a general job
$p_{j}~(p_{f, j})$/$r_{j}~(r_{f, j})$ the processing time/the release dates of job $J_{j}~(J_{f, j})$
$n_f$ the number of jobs in family ${\mathcal F}_f$
$(b<n)/(b\geq n)$ the bounded/unbounded capacity of a p-batch
$p(B)$ the processing time of batch $B$
$\pi/\sigma$ a feasible schedule
$C_{j}(\pi)(C_{f, j}(\pi))$ the completion time of job $J_{j}$ ($J_{f, j}$) in $\pi$
$S_{j}(\pi)(S_{f, j}(\pi))$ the starting time of job $J_{j}$ ($J_{f, j}$) in $\pi$
$C_{B}(\pi)$ the completion time of batch $B$ in $\pi$
$S_{B}(\pi)$ the starting time of batch $B$ in $\pi$
$F_{j}(\pi)(F_{f, j}(\pi))$ the flow time of job $J_{j}$ ($J_{f, j}$) in $\pi$ which is $F_{j}(\pi)=C_{j}(\pi)-r_{j}(\pi)$
$C_{\max}(\pi)$ the maximum completion time of all jobs in $\pi$
$F_{\max}(\pi)$ the maximum flow time of all jobs in $\pi$
${\mathcal F}_f^{(i)}$ the set composed by the first $i$ jobs of family ${\mathcal F}_f$, i.e., $\{J_{f, 0}, J_{f, 1}, \cdots, J_{f, i}\}$
$(i_1, \cdots, i_K)$ the instance composed by the job set ${\mathcal F}_1^{(i_1)} \cup \cdots \cup {\mathcal F}_K^{(i_K)}$
$P(i_1, \cdots, i_K)$ the problem $1|\beta |^\#(C_{\max}, F_{\max})$ restricted on the instance $(i_1, \cdots, i_K)$
$P_{Y}(i_1, \cdots, i_K)$ the problem $1|\beta |C_{\max}: F_{\max}\leq Y$ restricted on the instance $(i_1, \cdots, i_K)$
$P_{Y}^{(f)}(i_1, \cdots, i_K)$ a restricted version of $P_{Y}(i_1, \cdots, i_K)$ with $i_f \geq 1$ for which it is required that feasible schedules end with an $f$-batch
$C_Y^{(f)}(i_1, \cdots, i_K)$ the optimal makespan of problem $P_{Y}^{(f)}(i_1, \cdots, i_K)$
$C_Y^{(f, k_f)}(i_1, \cdots, i_K)$ defined in equation (3)
$D_Y^{(f)}(i_1, \cdots, i_K)$ defined in equation (4)
${\mathcal X}_{Y}(i_1, \cdots, i_K)$ defined in equation (5), the set of family indices attaining the minimum in equation (1)
$\Psi^{(f)}_{Y}(i_1, \cdots, i_K)$ defined in equation (6), the set of the $f$-job indices attaining the minimum in equation (2)
$C_Y(i_1, \cdots, i_K)$ the optimal makespan of problem $P_{Y}(i_1, \cdots, i_K)$
Algorithm DP($Y$) the proposed dynamic programming algorithm for problem $1|\beta |C_{\max}: F_{\max}\leq Y$
Algorithm Family-CF the proposed algorithm for problem $1|\beta |^\#(C_{\max}, F_{\max})$
Table 2.  The jobs in family ${\mathcal F}_1$
${\mathcal F}_1$ $J_{1, 1}$ $J_{1, 2}$ $J_{1, 3}$ $J_{1, 4}$ $J_{1, 5}$ $J_{1, 6}$ $J_{1, 7}$ $J_{1, 8}$ $J_{1, 9}$ $J_{1, 10}$
$r_{1, i}$ 0 2 3 4 6 7 9 11 14 17
$p_{1, i}$ 2 2 4 5 7 12 3 6 1 3
${\mathcal F}_1$ $J_{1, 1}$ $J_{1, 2}$ $J_{1, 3}$ $J_{1, 4}$ $J_{1, 5}$ $J_{1, 6}$ $J_{1, 7}$ $J_{1, 8}$ $J_{1, 9}$ $J_{1, 10}$
$r_{1, i}$ 0 2 3 4 6 7 9 11 14 17
$p_{1, i}$ 2 2 4 5 7 12 3 6 1 3
Table 3.  The jobs in family ${\mathcal F}_2$
${\mathcal F}_1$ $J_{1, 1}$ $J_{1, 2}$ $J_{1, 3}$ $J_{1, 4}$ $J_{1, 5}$ $J_{1, 6}$ $J_{1, 7}$ $J_{1, 8}$ $J_{1, 9}$ $J_{1, 10}$
$r_{1, i}$ 1 2 4 6 8 10 11 14 16 19
$p_{1, i}$ 2 1 1 4 3 8 10 2 11 9
${\mathcal F}_1$ $J_{1, 1}$ $J_{1, 2}$ $J_{1, 3}$ $J_{1, 4}$ $J_{1, 5}$ $J_{1, 6}$ $J_{1, 7}$ $J_{1, 8}$ $J_{1, 9}$ $J_{1, 10}$
$r_{1, i}$ 1 2 4 6 8 10 11 14 16 19
$p_{1, i}$ 2 1 1 4 3 8 10 2 11 9
Table 4.  The PoPs of the instances
jobs PoPs jobs PoPs jobs PoPs jobs PoPs jobs PoPs
${\mathcal J}(1, 1)$ $(4, 3)$ ${\mathcal J}(1, 2)$ $(4, 3)$ ${\mathcal J}(2, 1)$ $(6, 4)$ ${\mathcal J}(2, 2)$ $(6, 4)$ ${\mathcal J}(1, 3)$ $(5, 3)$
$(5, 5)$
${\mathcal J}(2, 3)$ $(5, 4)$ ${\mathcal J}(3, 1)$ $(8, 6)$ ${\mathcal J}(3, 2)$ $(8, 5)$ ${\mathcal J}(3, 3)$ $(9, 5)$ ${\mathcal J}(1, 4)$ $(2, 2)$
$(7, 7)$ $(7, 7)$ $(8, 7)$
${\mathcal J}(2, 4)$ $(4, 2)$ ${\mathcal J}(3, 4)$ $(7, 5)$ ${\mathcal J}(4, 1)$ $(3, 2)$ ${\mathcal J}(4, 2)$ $(4, 2)$ ${\mathcal J}(4, 3)$ $(5, 2)$
${\mathcal J}(4, 4)$ $(10, 4)$ ${\mathcal J}(1, 5)$ $(2, 2)$ ${\mathcal J}(2, 5)$ $(4, 2)$ ${\mathcal J}(3, 5)$ $(7, 5)$ ${\mathcal J}(4, 5)$ $(13, 5)$
$(9, 6)$ $(9, 6)$
${\mathcal J}(5, 1)$ $(3, 2)$ ${\mathcal J}(5, 2)$ $(4, 2)$ ${\mathcal J}(5, 3)$ $(5, 2)$ ${\mathcal J}(5, 4)$ $(10, 4)$ ${\mathcal J}(5, 5)$ $(13, 5)$
$(12, 6)$
${\mathcal J}(1, 6)$ $(2, 2)$ ${\mathcal J}(2, 6)$ $(4, 2)$ ${\mathcal J}(3, 6)$ $(7, 5)$ ${\mathcal J}(4, 6)$ $(9, 6)$ ${\mathcal J}(5, 6)$ $(13, 10)$
${\mathcal J}(6, 1)$ $(3, 2)$ ${\mathcal J}(6, 2)$ $(4, 2)$ ${\mathcal J}(6, 3)$ $(5, 2)$ ${\mathcal J}(6, 4)$ $(10, 4)$ ${\mathcal J}(6, 5)$ $(13, 5)$
$(12, 6)$
${\mathcal J}(6, 6)$ $(18, 10)$ ${\mathcal J}(1, 7)$ $(2, 2)$ ${\mathcal J}(2, 7)$ $(4, 2)$ ${\mathcal J}(3, 7)$ $(7, 5)$ ${\mathcal J}(4, 7)$ $(9, 6)$
${\mathcal J}(5, 7)$ $(13, 10)$ ${\mathcal J}(6, 7)$ $(22, 12)$ ${\mathcal J}(7, 1)$ $(3, 2)$ ${\mathcal J}(7, 2)$ $(4, 2)$ ${\mathcal J}(7, 3)$ $(5, 2)$
$(21, 13)$
$(19, 15)$
${\mathcal J}(7, 4)$ $(10, 4)$ ${\mathcal J}(7, 5)$ $(13, 5)$ ${\mathcal J}(7, 6)$ $(18, 10)$ ${\mathcal J}(7, 7)$ $(22, 12)$ ${\mathcal J}(1, 8)$ $(2, 2)$
$(12, 6)$ $(21, 13)$
${\mathcal J}(2, 8)$ $(4, 2)$ ${\mathcal J}(3, 8)$ $(7, 5)$ ${\mathcal J}(4, 8)$ $(9, 6)$ ${\mathcal J}(5, 8)$ $(13, 10)$ ${\mathcal J}(6, 8)$ $(24, 12)$
$(23, 13)$
$(19, 15)$
${\mathcal J}(7, 8)$ $(24, 12)$ ${\mathcal J}(8, 1)$ $(3, 2)$ ${\mathcal J}(8, 2)$ $(4, 2)$ ${\mathcal J}(8, 3)$ $(5, 2)$ ${\mathcal J}(8, 4)$ $(10, 4)$
$(23, 13)$
$(21, 15)$
${\mathcal J}(8, 5)$ $(13, 5)$ ${\mathcal J}(8, 6)$ $(18, 10)$ ${\mathcal J}(8, 7)$ $(22, 12)$ ${\mathcal J}(8, 8)$ $(24, 12)$ ${\mathcal J}(1, 9)$ $(2, 2)$
$(12, 6)$ $(21, 13)$ $(23, 13)$
${\mathcal J}(2, 9)$ $(4, 2)$ ${\mathcal J}(3, 9)$ $(7, 5)$ ${\mathcal J}(4, 9)$ $(9, 6)$ ${\mathcal J}(5, 9)$ $(13, 10)$ ${\mathcal J}(6, 9)$ $(19, 15)$
${\mathcal J}(7, 9)$ $(21, 15)$ ${\mathcal J}(8, 9)$ $(25, 16)$ ${\mathcal J}(9, 1)$ $(3, 2)$ ${\mathcal J}(9, 2)$ $(4, 2)$ ${\mathcal J}(9, 3)$ $(5, 2)$
$(23, 17)$
${\mathcal J}(9, 4)$ $(10, 4)$ ${\mathcal J}(9, 5)$ $(13, 5)$ ${\mathcal J}(9, 6)$ $(18, 10)$ ${\mathcal J}(9, 7)$ $(22, 12)$ ${\mathcal J}(9, 8)$ $(24, 12)$
$(12, 6)$ $(21, 13)$ $(23, 13)$
${\mathcal J}(9, 9)$ $(25, 16)$ ${\mathcal J}(1, 10)$ $(2, 2)$ ${\mathcal J}(2, 10)$ $(4, 2)$ ${\mathcal J}(3, 10)$ $(7, 5)$ ${\mathcal J}(4, 10)$ $(9, 6)$
$(24, 17)$
${\mathcal J}(5, 10)$ $(13, 10)$ ${\mathcal J}(6, 10)$ $(19, 15)$ ${\mathcal J}(7, 10)$ $(21, 15)$ ${\mathcal J}(8, 10)$ $(25, 16)$ ${\mathcal J}(9, 10)$ $(25, 16)$
$(23, 17)$ $(24, 17)$
${\mathcal J}(10, 1)$ $(3, 2)$ ${\mathcal J}(10, 2)$ $(4, 2)$ ${\mathcal J}(10, 3)$ $(5, 2)$ ${\mathcal J}(10, 4)$ $(13, 5)$ ${\mathcal J}(10, 5)$ $(18, 10)$
$(12, 6)$
${\mathcal J}(10, 1)$ $(3, 2)$ ${\mathcal J}(10, 2)$ $(4, 2)$ ${\mathcal J}(10, 3)$ $(5, 2)$ ${\mathcal J}(10, 4)$ $(10, 4)$ ${\mathcal J}(10, 5)$ $(13, 5)$
$(12, 6)$
${\mathcal J}(10, 6)$ $(18, 10)$ ${\mathcal J}(10, 7)$ $(22, 12)$ ${\mathcal J}(10, 8)$ $(24, 12)$ ${\mathcal J}(10, 9)$ $(25, 16)$ ${\mathcal J}(10, 10)$ $(25, 16)$
$(21, 13)$ $(23, 13)$
jobs PoPs jobs PoPs jobs PoPs jobs PoPs jobs PoPs
${\mathcal J}(1, 1)$ $(4, 3)$ ${\mathcal J}(1, 2)$ $(4, 3)$ ${\mathcal J}(2, 1)$ $(6, 4)$ ${\mathcal J}(2, 2)$ $(6, 4)$ ${\mathcal J}(1, 3)$ $(5, 3)$
$(5, 5)$
${\mathcal J}(2, 3)$ $(5, 4)$ ${\mathcal J}(3, 1)$ $(8, 6)$ ${\mathcal J}(3, 2)$ $(8, 5)$ ${\mathcal J}(3, 3)$ $(9, 5)$ ${\mathcal J}(1, 4)$ $(2, 2)$
$(7, 7)$ $(7, 7)$ $(8, 7)$
${\mathcal J}(2, 4)$ $(4, 2)$ ${\mathcal J}(3, 4)$ $(7, 5)$ ${\mathcal J}(4, 1)$ $(3, 2)$ ${\mathcal J}(4, 2)$ $(4, 2)$ ${\mathcal J}(4, 3)$ $(5, 2)$
${\mathcal J}(4, 4)$ $(10, 4)$ ${\mathcal J}(1, 5)$ $(2, 2)$ ${\mathcal J}(2, 5)$ $(4, 2)$ ${\mathcal J}(3, 5)$ $(7, 5)$ ${\mathcal J}(4, 5)$ $(13, 5)$
$(9, 6)$ $(9, 6)$
${\mathcal J}(5, 1)$ $(3, 2)$ ${\mathcal J}(5, 2)$ $(4, 2)$ ${\mathcal J}(5, 3)$ $(5, 2)$ ${\mathcal J}(5, 4)$ $(10, 4)$ ${\mathcal J}(5, 5)$ $(13, 5)$
$(12, 6)$
${\mathcal J}(1, 6)$ $(2, 2)$ ${\mathcal J}(2, 6)$ $(4, 2)$ ${\mathcal J}(3, 6)$ $(7, 5)$ ${\mathcal J}(4, 6)$ $(9, 6)$ ${\mathcal J}(5, 6)$ $(13, 10)$
${\mathcal J}(6, 1)$ $(3, 2)$ ${\mathcal J}(6, 2)$ $(4, 2)$ ${\mathcal J}(6, 3)$ $(5, 2)$ ${\mathcal J}(6, 4)$ $(10, 4)$ ${\mathcal J}(6, 5)$ $(13, 5)$
$(12, 6)$
${\mathcal J}(6, 6)$ $(18, 10)$ ${\mathcal J}(1, 7)$ $(2, 2)$ ${\mathcal J}(2, 7)$ $(4, 2)$ ${\mathcal J}(3, 7)$ $(7, 5)$ ${\mathcal J}(4, 7)$ $(9, 6)$
${\mathcal J}(5, 7)$ $(13, 10)$ ${\mathcal J}(6, 7)$ $(22, 12)$ ${\mathcal J}(7, 1)$ $(3, 2)$ ${\mathcal J}(7, 2)$ $(4, 2)$ ${\mathcal J}(7, 3)$ $(5, 2)$
$(21, 13)$
$(19, 15)$
${\mathcal J}(7, 4)$ $(10, 4)$ ${\mathcal J}(7, 5)$ $(13, 5)$ ${\mathcal J}(7, 6)$ $(18, 10)$ ${\mathcal J}(7, 7)$ $(22, 12)$ ${\mathcal J}(1, 8)$ $(2, 2)$
$(12, 6)$ $(21, 13)$
${\mathcal J}(2, 8)$ $(4, 2)$ ${\mathcal J}(3, 8)$ $(7, 5)$ ${\mathcal J}(4, 8)$ $(9, 6)$ ${\mathcal J}(5, 8)$ $(13, 10)$ ${\mathcal J}(6, 8)$ $(24, 12)$
$(23, 13)$
$(19, 15)$
${\mathcal J}(7, 8)$ $(24, 12)$ ${\mathcal J}(8, 1)$ $(3, 2)$ ${\mathcal J}(8, 2)$ $(4, 2)$ ${\mathcal J}(8, 3)$ $(5, 2)$ ${\mathcal J}(8, 4)$ $(10, 4)$
$(23, 13)$
$(21, 15)$
${\mathcal J}(8, 5)$ $(13, 5)$ ${\mathcal J}(8, 6)$ $(18, 10)$ ${\mathcal J}(8, 7)$ $(22, 12)$ ${\mathcal J}(8, 8)$ $(24, 12)$ ${\mathcal J}(1, 9)$ $(2, 2)$
$(12, 6)$ $(21, 13)$ $(23, 13)$
${\mathcal J}(2, 9)$ $(4, 2)$ ${\mathcal J}(3, 9)$ $(7, 5)$ ${\mathcal J}(4, 9)$ $(9, 6)$ ${\mathcal J}(5, 9)$ $(13, 10)$ ${\mathcal J}(6, 9)$ $(19, 15)$
${\mathcal J}(7, 9)$ $(21, 15)$ ${\mathcal J}(8, 9)$ $(25, 16)$ ${\mathcal J}(9, 1)$ $(3, 2)$ ${\mathcal J}(9, 2)$ $(4, 2)$ ${\mathcal J}(9, 3)$ $(5, 2)$
$(23, 17)$
${\mathcal J}(9, 4)$ $(10, 4)$ ${\mathcal J}(9, 5)$ $(13, 5)$ ${\mathcal J}(9, 6)$ $(18, 10)$ ${\mathcal J}(9, 7)$ $(22, 12)$ ${\mathcal J}(9, 8)$ $(24, 12)$
$(12, 6)$ $(21, 13)$ $(23, 13)$
${\mathcal J}(9, 9)$ $(25, 16)$ ${\mathcal J}(1, 10)$ $(2, 2)$ ${\mathcal J}(2, 10)$ $(4, 2)$ ${\mathcal J}(3, 10)$ $(7, 5)$ ${\mathcal J}(4, 10)$ $(9, 6)$
$(24, 17)$
${\mathcal J}(5, 10)$ $(13, 10)$ ${\mathcal J}(6, 10)$ $(19, 15)$ ${\mathcal J}(7, 10)$ $(21, 15)$ ${\mathcal J}(8, 10)$ $(25, 16)$ ${\mathcal J}(9, 10)$ $(25, 16)$
$(23, 17)$ $(24, 17)$
${\mathcal J}(10, 1)$ $(3, 2)$ ${\mathcal J}(10, 2)$ $(4, 2)$ ${\mathcal J}(10, 3)$ $(5, 2)$ ${\mathcal J}(10, 4)$ $(13, 5)$ ${\mathcal J}(10, 5)$ $(18, 10)$
$(12, 6)$
${\mathcal J}(10, 1)$ $(3, 2)$ ${\mathcal J}(10, 2)$ $(4, 2)$ ${\mathcal J}(10, 3)$ $(5, 2)$ ${\mathcal J}(10, 4)$ $(10, 4)$ ${\mathcal J}(10, 5)$ $(13, 5)$
$(12, 6)$
${\mathcal J}(10, 6)$ $(18, 10)$ ${\mathcal J}(10, 7)$ $(22, 12)$ ${\mathcal J}(10, 8)$ $(24, 12)$ ${\mathcal J}(10, 9)$ $(25, 16)$ ${\mathcal J}(10, 10)$ $(25, 16)$
$(21, 13)$ $(23, 13)$
[1]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069

[2]

Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281

[3]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393

[4]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005

[5]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

[6]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

[7]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[8]

Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20.

[9]

Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021004

[10]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

[11]

Cécile Carrère, Grégoire Nadin. Influence of mutations in phenotypically-structured populations in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3609-3630. doi: 10.3934/dcdsb.2020075

[12]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

[13]

Guillermo Reyes, Juan-Luis Vázquez. Long time behavior for the inhomogeneous PME in a medium with slowly decaying density. Communications on Pure & Applied Analysis, 2009, 8 (2) : 493-508. doi: 10.3934/cpaa.2009.8.493

[14]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[15]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[16]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1

[17]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (189)
  • HTML views (1444)
  • Cited by (0)

Other articles
by authors

[Back to Top]