# American Institute of Mathematical Sciences

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

Received  May 2015 Published  March 2016

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.

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 and 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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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.
Production process flow
Example of bill of materials from one type of first-stage product
A disconnected subtour and a main sequence
A subtour connected to a main sequence at the beginning of period
Relax and fix heuristic on multi-stage and over the periods
Comparison of total setup time between the company planning and our model
Comparison of total inventory level between the company planning and our model
Comparison of total overtime between the company planning and our model
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
Numerical results of small problems
 Problem size($N \times M \times T$) $<$1000 1000—4000 4000—6000 MIP Heu. MIP Heu. MIP Heu. Avg. Time (sec) 8716 958 35226 1090 81646 1774 Avg. Gap (%) 3.94 5.67 5.44 8.63 6.71 9.22 StDev. Gap 1.81 4.06 1.88 6.19 2.73 3.36
 Problem size($N \times M \times T$) $<$1000 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
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
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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