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A difference of convex functions algorithm for optimal scheduling and real-time assignment of preventive maintenance jobs on parallel processors
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January  2014, 10(1): 259-273. doi: 10.3934/jimo.2014.10.259

## Heuristics for parallel machine scheduling with batch delivery consideration

 1 Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China, China

Received  February 2012 Revised  March 2013 Published  October 2013

We consider the parallel machine scheduling problem in which the finished jobs need to be delivered to a customer in batches by a single vehicle. The goal is to minimize the makespan, i.e., the time by which the vehicle has delivered all jobs and returned to its initial location. We distinguish two types of batching strategies. The strategy of Type 1 permits the jobs processed on different machines to compose a delivery batch, and the strategy of Type 2 assumes that only the jobs processed on the same machine can compose a batch. For both types of the $m$-machine case, we propose $(2-\frac{1}{m})$-approximation algorithms respectively. For both types of the two-machine case, we obtain two improved $\frac{4}{3}$-approximation algorithms.
Citation: Leiyang Wang, Zhaohui Liu. Heuristics for parallel machine scheduling with batch delivery consideration. Journal of Industrial & Management Optimization, 2014, 10 (1) : 259-273. doi: 10.3934/jimo.2014.10.259
##### References:
 [1] J. H. Ahmadi, R. H. Ahmadi, S. Dasu and C. S. Tang, Batching and scheduling jobs on batch and discrete processors,, Operations Research, 40 (1992), 750. doi: 10.1287/opre.40.4.750. Google Scholar [2] Y.-C. Chang and C.-Y. Lee, Machine scheduling with job delivery coordination,, European Journal of Operational Research, 158 (2004), 470. doi: 10.1016/S0377-2217(03)00364-3. Google Scholar [3] Z.-L. Chen, Integrated production and outbound distribution scheduling: Review and extensions,, Operations Research, 58 (2010), 130. doi: 10.1287/opre.1080.0688. Google Scholar [4] Z.-L. Chen and G. Pundoor, Order assignment and scheduling in a supply chain,, Operations Research, 54 (2006), 555. doi: 10.1287/opre.1060.0280. Google Scholar [5] Z.-L. Chen and G. Vairaktarakis, Integrated scheduling of production and distribution operations,, Management Science, 51 (2005), 614. doi: 10.1287/mnsc.1040.0325. Google Scholar [6] H. N. Geismar, G. Laporte, L. Lei and C. Sriskandarajah, The integrated production and transportation scheduling problem for a product with a short lifespan,, INFORMS Journal on Computing, 20 (2008), 21. doi: 10.1287/ijoc.1060.0208. Google Scholar [7] H. Gong and L. X. Tang, Scheduling production on parallel machines and batch delivery with limited waiting time constraint,, Control and Decision, 26 (2011), 921. Google Scholar [8] R. L. Graham, Bounds for certain multiprocessing anomalies,, The Bell System Technical Journal, 45 (1966), 1563. doi: 10.1002/j.1538-7305.1966.tb01709.x. Google Scholar [9] R. L. Graham, Bounds on multiprocessing timing anomalies,, SIAM Journal on Applied Mathematics, 17 (1969), 416. doi: 10.1137/0117039. Google Scholar [10] N. G. Hall and C. N. Potts, The coordination of scheduling and batch deliveries,, Annals of Operations Research, 135 (2005), 41. doi: 10.1007/s10479-005-6234-8. Google Scholar [11] C.-Y. Lee and Z.-L. Chen, Machine scheduling with transportation considerations,, Journal of Scheduling, 4 (2001), 3. doi: 10.1002/1099-1425(200101/02)4:1<3::AID-JOS57>3.0.CO;2-D. Google Scholar [12] C.-L. Li, G. Vairaktarakis and C.-Y. Lee, Machine scheduling with deliveries to multiple customer locations,, European Journal of Operational Research, 164 (2005), 39. doi: 10.1016/j.ejor.2003.11.022. Google Scholar [13] C.-S. Su, J. C.-H. Pan and T.-S. Hsu, A new heuristic algorithm for the machine scheduling problem with job delivery coordination,, Theoretical Computer Science, 410 (2009), 2581. doi: 10.1016/j.tcs.2009.02.019. Google Scholar [14] G. Wang and T. C. E. Cheng, Parallel machine scheduling with batch delivery costs,, International Journal of Production Economics, 68 (2000), 177. doi: 10.1016/S0925-5273(99)00105-X. Google Scholar [15] X. Wang and T. C. E. Cheng, Machine scheduling with an availability constraint and job delivery coordination,, Naval Research Logistics, 54 (2007), 11. doi: 10.1002/nav.20175. Google Scholar [16] W. Y. Zhong, G. Dosa and Z. Y. Tan, On the machine scheduling problem with job delivery coordination,, European Journal of Operational Research, 182 (2007), 1057. doi: 10.1016/j.ejor.2006.09.059. Google Scholar

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##### References:
 [1] J. H. Ahmadi, R. H. Ahmadi, S. Dasu and C. S. Tang, Batching and scheduling jobs on batch and discrete processors,, Operations Research, 40 (1992), 750. doi: 10.1287/opre.40.4.750. Google Scholar [2] Y.-C. Chang and C.-Y. Lee, Machine scheduling with job delivery coordination,, European Journal of Operational Research, 158 (2004), 470. doi: 10.1016/S0377-2217(03)00364-3. Google Scholar [3] Z.-L. Chen, Integrated production and outbound distribution scheduling: Review and extensions,, Operations Research, 58 (2010), 130. doi: 10.1287/opre.1080.0688. Google Scholar [4] Z.-L. Chen and G. Pundoor, Order assignment and scheduling in a supply chain,, Operations Research, 54 (2006), 555. doi: 10.1287/opre.1060.0280. Google Scholar [5] Z.-L. Chen and G. Vairaktarakis, Integrated scheduling of production and distribution operations,, Management Science, 51 (2005), 614. doi: 10.1287/mnsc.1040.0325. Google Scholar [6] H. N. Geismar, G. Laporte, L. Lei and C. Sriskandarajah, The integrated production and transportation scheduling problem for a product with a short lifespan,, INFORMS Journal on Computing, 20 (2008), 21. doi: 10.1287/ijoc.1060.0208. Google Scholar [7] H. Gong and L. X. Tang, Scheduling production on parallel machines and batch delivery with limited waiting time constraint,, Control and Decision, 26 (2011), 921. Google Scholar [8] R. L. Graham, Bounds for certain multiprocessing anomalies,, The Bell System Technical Journal, 45 (1966), 1563. doi: 10.1002/j.1538-7305.1966.tb01709.x. Google Scholar [9] R. L. Graham, Bounds on multiprocessing timing anomalies,, SIAM Journal on Applied Mathematics, 17 (1969), 416. doi: 10.1137/0117039. Google Scholar [10] N. G. Hall and C. N. Potts, The coordination of scheduling and batch deliveries,, Annals of Operations Research, 135 (2005), 41. doi: 10.1007/s10479-005-6234-8. Google Scholar [11] C.-Y. Lee and Z.-L. Chen, Machine scheduling with transportation considerations,, Journal of Scheduling, 4 (2001), 3. doi: 10.1002/1099-1425(200101/02)4:1<3::AID-JOS57>3.0.CO;2-D. Google Scholar [12] C.-L. Li, G. Vairaktarakis and C.-Y. Lee, Machine scheduling with deliveries to multiple customer locations,, European Journal of Operational Research, 164 (2005), 39. doi: 10.1016/j.ejor.2003.11.022. Google Scholar [13] C.-S. Su, J. C.-H. Pan and T.-S. Hsu, A new heuristic algorithm for the machine scheduling problem with job delivery coordination,, Theoretical Computer Science, 410 (2009), 2581. doi: 10.1016/j.tcs.2009.02.019. Google Scholar [14] G. Wang and T. C. E. Cheng, Parallel machine scheduling with batch delivery costs,, International Journal of Production Economics, 68 (2000), 177. doi: 10.1016/S0925-5273(99)00105-X. Google Scholar [15] X. Wang and T. C. E. Cheng, Machine scheduling with an availability constraint and job delivery coordination,, Naval Research Logistics, 54 (2007), 11. doi: 10.1002/nav.20175. Google Scholar [16] W. Y. Zhong, G. Dosa and Z. Y. Tan, On the machine scheduling problem with job delivery coordination,, European Journal of Operational Research, 182 (2007), 1057. doi: 10.1016/j.ejor.2006.09.059. Google Scholar
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