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January
2017, 13(1): 413-428.
doi: 10.3934/jimo.2016024
Multiple-stage multiple-machine capacitated lot-sizing and scheduling with sequence-dependent setup: A case study in the wheel industry
Graduate School of Decision Science and Technology, Tokyo Institute of Technology, Tokyo 152-8552, Japan |
This paper studies a real-world problem of simultaneous lot-sizing and scheduling in a capacitated flow shop. The problem combines two significant characteristics in production which are multiple-stage production with heterogeneous multiple machines and sequence-dependent setup time. Setup time does not hold the triangle inequality, thus there may be a setup for a product without actual production. Consequently, a novel mixed integer programming (MIP) formulation is proposed and tested on real data sets of wheel production. Exact approaches cannot find a feasible solution for the model in a reasonable time, so MIP-based heuristics are developed to solve the model more quickly. Test results show that the formulation is able to contain the problem requirements and the heuristics are computationally effective. Moreover, the obtained solution can improve on a real practice at the plant.
Keywords:
Lot-sizing,
scheduling,
multiple-stage production,
sequence-dependent setup time,
mixed integer programming,
relax-and-fix heuristic solution.
Mathematics Subject Classification:
Primary: 58F15, 58F17; Secondary: 53C35.
Citation:
Lalida Deeratanasrikul, Shinji Mizuno. Multiple-stage multiple-machine capacitated lot-sizing and scheduling with sequence-dependent setup: A case study in the wheel industry. Journal of Industrial & Management Optimization,
2017, 13
(1)
: 413-428.
doi: 10.3934/jimo.2016024
![]()
References:
| [1] |
A. Allahverdi, C. Ng, T. 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. |
| [2] |
B. Almada-lobo, D. Klabjan, M. Antnia carravilla and J. F. Oliveira,
Single machine multi-product capacitated lot sizing with sequence-dependent setups, International Journal of Production Research, 45 (2007), 4873-4894.
doi: 10.1080/00207540601094465. |
| [3] |
A. Drexl and A. Kimms,
Lot sizing and scheduling survey and extensions, European Journal of Operational Research, 99 (1997), 221-235.
doi: 10.1016/S0377-2217(97)00030-1. |
| [4] |
M. Gnoni, R. Iavagnilio, G. Mossa, G. Mummolo and A. D. Leva, Production planning of a multisite, manufacturing system by hybrid modelling: A case study from the automotive industry, International Journal of Production Economics, 85 (2003), 251-262. Google Scholar |
| [5] |
K. Haase,
Capacitated lot-sizing with sequence dependent setup costs, Operations-Research-Spektrum, 18 (1996), 51-59.
doi: 10.1007/BF01539882. |
| [6] |
R. J. James and B. Almada-Lobo,
Single and parallel machine capacitated lotsizing and scheduling: New iterative mip-based neighborhood search heuristics, Computers & Operations Research, 38 (2011), 1816-1825.
doi: 10.1016/j.cor.2011.02.005. |
| [7] |
R. Jans and Z. Degraeve,
Meta-heuristics for dynamic lot sizing: A review and comparison of solution approaches, European Journal of Operational Research, 177 (2007), 1855-1875.
doi: 10.1016/j.ejor.2005.12.008. |
| [8] |
M. Gnoni, R. Iavagnilio, G. Mossa, G. Mummolo and A. D. Leva, Fix-and-Optimize heuristics for capacitated lot-sizing with sequence-dependent setups and substitutions, European Journal of Operational Research, 214 (2011), 595-605. Google Scholar |
| [9] |
A. Menezes, A. Clark and B. Almada-Lobo,
Capacitated lot-sizing and scheduling with sequencedependent, period-overlapping and non-triangular setups, Journal of Scheduling, 14 (2011), 209-219.
doi: 10.1007/s10951-010-0197-6. |
| [10] |
C. E. Miller, A. W. Tucker and R. A. Zemlin,
Integer programming formulation of traveling salesman problems, Journal of the ACM, 7 (1960), 326-329.
doi: 10.1145/321043.321046. |
| [11] |
OICA Production statistics, Report of International Organization of Motor Vehicle Manufacturers, 2014. Available from: http://www.oica.net/category/production-statistics.Google Scholar |
| [12] |
D. Quadt and H. Kuhn,
Capacitated lot-sizing with extensions: A review, 4OR, 6 (2008), 61-83.
doi: 10.1007/s10288-007-0057-1. |
| [13] |
F. Seeanner, B. Almada-Lobo and H. Meyr,
Combining the principles of variable neighborhood decomposition search and the fix & optimize heuristic to solve multi-level lot-sizing and scheduling problems, Computers & Operations Research, 40 (2003), 303-317.
doi: 10.1016/j.cor.2012.07.002. |
| [14] |
F. Seeanner and H. Meyr,
Multi-stage simultaneous lot-sizing and scheduling for flow line production, OR Spectrum, 35 (2013), 33-73.
doi: 10.1007/s00291-012-0296-1. |
| [15] |
F. Seeanner,
Multi-Stage Simultaneous Lot-Sizing and Scheduling: Planning of Flow Lines with Shifting Bottlenecks, Damstadt: Springer Fachmedien Wiesbaden, 2013.
doi: 10.1007/978-3-658-02089-7. |
| [16] |
H. Stadtler and F. Sahling,
A lot-sizing and scheduling model for multi-stage flow lines with zero lead times, European Journal of Operational Research, 225 (2013), 404-419.
doi: 10.1016/j.ejor.2012.10.011. |
| [17] |
J. Xiao, C. Zhang, L. Zheng and J. N. D. Gupta, Mip-based Fix-and-Optimize algorithms for the parallel machine capacitated lot-sizing and scheduling problem, International Journal of Production Research, 51 (2013), 5011-5028. Google Scholar |
| [18] |
X. Zhu and W. E. Wilhelm,
Scheduling and lot sizing with sequence-dependent setup: A literature review, IIE Transactions, 38 (2006), 987-1007.
doi: 10.1080/07408170600559706. |
show all references
References:
| [1] |
A. Allahverdi, C. Ng, T. 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. |
| [2] |
B. Almada-lobo, D. Klabjan, M. Antnia carravilla and J. F. Oliveira,
Single machine multi-product capacitated lot sizing with sequence-dependent setups, International Journal of Production Research, 45 (2007), 4873-4894.
doi: 10.1080/00207540601094465. |
| [3] |
A. Drexl and A. Kimms,
Lot sizing and scheduling survey and extensions, European Journal of Operational Research, 99 (1997), 221-235.
doi: 10.1016/S0377-2217(97)00030-1. |
| [4] |
M. Gnoni, R. Iavagnilio, G. Mossa, G. Mummolo and A. D. Leva, Production planning of a multisite, manufacturing system by hybrid modelling: A case study from the automotive industry, International Journal of Production Economics, 85 (2003), 251-262. Google Scholar |
| [5] |
K. Haase,
Capacitated lot-sizing with sequence dependent setup costs, Operations-Research-Spektrum, 18 (1996), 51-59.
doi: 10.1007/BF01539882. |
| [6] |
R. J. James and B. Almada-Lobo,
Single and parallel machine capacitated lotsizing and scheduling: New iterative mip-based neighborhood search heuristics, Computers & Operations Research, 38 (2011), 1816-1825.
doi: 10.1016/j.cor.2011.02.005. |
| [7] |
R. Jans and Z. Degraeve,
Meta-heuristics for dynamic lot sizing: A review and comparison of solution approaches, European Journal of Operational Research, 177 (2007), 1855-1875.
doi: 10.1016/j.ejor.2005.12.008. |
| [8] |
M. Gnoni, R. Iavagnilio, G. Mossa, G. Mummolo and A. D. Leva, Fix-and-Optimize heuristics for capacitated lot-sizing with sequence-dependent setups and substitutions, European Journal of Operational Research, 214 (2011), 595-605. Google Scholar |
| [9] |
A. Menezes, A. Clark and B. Almada-Lobo,
Capacitated lot-sizing and scheduling with sequencedependent, period-overlapping and non-triangular setups, Journal of Scheduling, 14 (2011), 209-219.
doi: 10.1007/s10951-010-0197-6. |
| [10] |
C. E. Miller, A. W. Tucker and R. A. Zemlin,
Integer programming formulation of traveling salesman problems, Journal of the ACM, 7 (1960), 326-329.
doi: 10.1145/321043.321046. |
| [11] |
OICA Production statistics, Report of International Organization of Motor Vehicle Manufacturers, 2014. Available from: http://www.oica.net/category/production-statistics.Google Scholar |
| [12] |
D. Quadt and H. Kuhn,
Capacitated lot-sizing with extensions: A review, 4OR, 6 (2008), 61-83.
doi: 10.1007/s10288-007-0057-1. |
| [13] |
F. Seeanner, B. Almada-Lobo and H. Meyr,
Combining the principles of variable neighborhood decomposition search and the fix & optimize heuristic to solve multi-level lot-sizing and scheduling problems, Computers & Operations Research, 40 (2003), 303-317.
doi: 10.1016/j.cor.2012.07.002. |
| [14] |
F. Seeanner and H. Meyr,
Multi-stage simultaneous lot-sizing and scheduling for flow line production, OR Spectrum, 35 (2013), 33-73.
doi: 10.1007/s00291-012-0296-1. |
| [15] |
F. Seeanner,
Multi-Stage Simultaneous Lot-Sizing and Scheduling: Planning of Flow Lines with Shifting Bottlenecks, Damstadt: Springer Fachmedien Wiesbaden, 2013.
doi: 10.1007/978-3-658-02089-7. |
| [16] |
H. Stadtler and F. Sahling,
A lot-sizing and scheduling model for multi-stage flow lines with zero lead times, European Journal of Operational Research, 225 (2013), 404-419.
doi: 10.1016/j.ejor.2012.10.011. |
| [17] |
J. Xiao, C. Zhang, L. Zheng and J. N. D. Gupta, Mip-based Fix-and-Optimize algorithms for the parallel machine capacitated lot-sizing and scheduling problem, International Journal of Production Research, 51 (2013), 5011-5028. Google Scholar |
| [18] |
X. Zhu and W. E. Wilhelm,
Scheduling and lot sizing with sequence-dependent setup: A literature review, IIE Transactions, 38 (2006), 987-1007.
doi: 10.1080/07408170600559706. |
Figure 2.
Example of bill of materials from one type of first-stage product
Figure 3.
A disconnected subtour and a main sequence
Figure 4.
A subtour connected to a main sequence at the beginning of period
Figure 5.
Relax and fix heuristic on multi-stage and over the periods
Figure 6.
Comparison of total setup time between the company planning and our model
Figure 7.
Comparison of total inventory level between the company planning and our model
Figure 8.
Comparison of total overtime between the company planning and our model
Table 1.
Average objective values in detailed
| q | Setup time (sec) | Inventory level (pieces) | Overtime (sec) | ||||||||
| W=1000 | W=100 | W=10 | W=1000 | W=100 | W=10 | W=1000 | W=100 | W=10 | |||
| 20 | 573,750 | 511,500 | 407,850 | 3,843 | 11,437 | 13,906 | 2,247,857 | 8,029 | 7,712 | ||
| 100 | 529,500 | 521,100 | 404,400 | 10,356 | 11,557 | 14,133 | 17,100 | 7,713 | 7,712 | ||
| 200 | 539,100 | 521,250 | 395,280 | 11,535 | 11,409 | 13,680 | 7,868 | 7,713 | 7,712 | ||
| 300 | 545,250 | 506,850 | 398,450 | 11,443 | 11,257 | 13,380 | 8,245 | 7,712 | 7,712 | ||
| 400 | 559,350 | 519,300 | 404,850 | 11,757 | 11,579 | 13,737 | 7,725 | 7,712 | 7,712 | ||
| q | Setup time (sec) | Inventory level (pieces) | Overtime (sec) | ||||||||
| W=1000 | W=100 | W=10 | W=1000 | W=100 | W=10 | W=1000 | W=100 | W=10 | |||
| 20 | 573,750 | 511,500 | 407,850 | 3,843 | 11,437 | 13,906 | 2,247,857 | 8,029 | 7,712 | ||
| 100 | 529,500 | 521,100 | 404,400 | 10,356 | 11,557 | 14,133 | 17,100 | 7,713 | 7,712 | ||
| 200 | 539,100 | 521,250 | 395,280 | 11,535 | 11,409 | 13,680 | 7,868 | 7,713 | 7,712 | ||
| 300 | 545,250 | 506,850 | 398,450 | 11,443 | 11,257 | 13,380 | 8,245 | 7,712 | 7,712 | ||
| 400 | 559,350 | 519,300 | 404,850 | 11,757 | 11,579 | 13,737 | 7,725 | 7,712 | 7,712 | ||
Table 2.
Numerical results of small problems
| Problem size( |
1000—4000 | 4000—6000 | ||||
| MIP | Heu. | MIP | Heu. | MIP | Heu. | |
| Avg. Time (sec) | 8716 | 958 | 35226 | 1090 | 81646 | 1774 |
| Avg. Gap (%) | 3.94 | 5.67 | 5.44 | 8.63 | 6.71 | 9.22 |
| StDev. Gap | 1.81 | 4.06 | 1.88 | 6.19 | 2.73 | 3.36 |
| Problem size( |
1000—4000 | 4000—6000 | ||||
| MIP | Heu. | MIP | Heu. | MIP | Heu. | |
| Avg. Time (sec) | 8716 | 958 | 35226 | 1090 | 81646 | 1774 |
| Avg. Gap (%) | 3.94 | 5.67 | 5.44 | 8.63 | 6.71 | 9.22 |
| StDev. Gap | 1.81 | 4.06 | 1.88 | 6.19 | 2.73 | 3.36 |
Table 3.
Numerical results of real problems by our heuristics
| Avg. Time(sec) | Avg. LBDev(%) | |
| High variant of products family | 8330 | 18.54 |
| Low variant of products family | 2756 | 1.47 |
| Avg. Time(sec) | Avg. LBDev(%) | |
| High variant of products family | 8330 | 18.54 |
| Low variant of products family | 2756 | 1.47 |
Table 4.
Total objective value between the company solutions and our model solutions
| Week | 1 | 2 | 3 | 4 |
| Company | 1,473,400 | 1,973,405 | 2,008,300 | 15,855,500 |
| Model | 1,209,100 | 1,294,400 | 1,885,500 | 11,345,500 |
| Week | 1 | 2 | 3 | 4 |
| Company | 1,473,400 | 1,973,405 | 2,008,300 | 15,855,500 |
| Model | 1,209,100 | 1,294,400 | 1,885,500 | 11,345,500 |
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