Two-agent integrated scheduling of production and distribution operations with fixed departure times

Yunqing Zou; Zhengkui Lin; Dongya Han; T. C. Edwin Cheng; Chin-Chia Wu

doi:10.3934/jimo.2021005

Previous Article
Portfolio optimization for jump-diffusion risky assets with regime switching: A time-consistent approach
JIMO Home
This Issue
Next Article
Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs

doi: 10.3934/jimo.2021005

Two-agent integrated scheduling of production and distribution operations with fixed departure times

Yunqing Zou ^1, , Zhengkui Lin ^1, , Dongya Han ^2,, , T. C. Edwin Cheng ^3, and Chin-Chia Wu ^4,

1.	School of Maritime Economics and Management, Dalian Maritime University, Dalian, 116023, China
2.	International Institute of Financial, University of Science and Technology of China, Hefei, 230601, China
3.	Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
4.	Department of Statistics, Feng Chia University, Taichung, Taiwan

^* Corresponding author: Dongya Han

Received July 2020 Revised September 2020 Published January 2021

Fund Project: Cheng was supported in part by The Hong Kong Polytechnic University under the Fung Yiu King-Wing Hang Bank Endowed Professorship in Business Administration. Wu was supported by the Ministry of Science and Technology of Taiwan under grant number MOST 109-2410-H-035-019

We consider integrated scheduling of production and distribution operations associated with two customers (agents). Each customer has a set of orders to be processed on the single production line at a supplier on a competitive basis. The finished orders of the same customer are then packed and delivered to the customer by a third-party logistics (3PL) provider with a limited number of delivery transporters. The number of orders carried in a delivery transporter cannot exceed its delivery capacity. Each transporter incurs a fixed delivery cost regardless of the number of orders it carries, and departs from the 3PL provider to a customer at fixed times. Each customer desires to minimise a certain optimality criterion involving simultaneously the customer service level and the total delivery cost for its orders only. The customer service level for a customer is related to the times when its orders are delivered to it. The problem is to determine a joint schedule of production and distribution to minimise the objective of one customer, while keeping the objective of the other customer at or below a predefined level. Using several optimality criteria to measure the customer service level, we obtain different scenarios that depend on optimality criterion of each customer. For each scenario, we either devise an efficient solution procedure to solve it or demonstrate that such a solution procedure is impossible to exist.

Keywords: Two agents, dynamic programming, integrated production and delivery scheduling, departure times.

Mathematics Subject Classification: Primary: 90B35; Secondary: 68M20.

Citation: Yunqing Zou, Zhengkui Lin, Dongya Han, T. C. Edwin Cheng, Chin-Chia Wu. Two-agent integrated scheduling of production and distribution operations with fixed departure times. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021005

References:

[1]	A. Agnetis, M. A. Aloulou and M. Y. Kovalyov, Integrated production scheduling and batch delivery with fixed departure times and inventory holding costs, International Journal of Production Research, 55 (2017), 6193-6206. doi: 10.1080/00207543.2017.1346323. Google Scholar
[2]	A. Agnetis, P. B. Mirchandani, D. Pacciarelli and A. Pacifici, Scheduling problems with two competing agents, Operations Research, 52 (2004), 229-242. doi: 10.1287/opre.1030.0092. Google Scholar
[3]	K. R. Baker and J. C. Smith, A multiple-criterion model for machine scheduling, Journal of Scheduling, 6 (2003), 7-16. doi: 10.1023/A:1022231419049. Google Scholar
[4]	Z.-L. Chen, Integrated production and outbound distribution scheduling: Review and extensions, Operations Research, 58 (2010), 130-148. doi: 10.1287/opre.1080.0688. Google Scholar
[5]	E. Gerstl and G. Mosheiov, Single machine just-in-time scheduling problems with two competing agents, Naval Research Logistics, 61 (2014), 1-16. doi: 10.1002/nav.21562. Google Scholar
[6]	A. M. Fathollahi-Fard, M. Hajiaghaei-Keshteli, G. Tian and Z. Li, An adaptive Lagrangian relaxation-based algorithm for a coordinated water supply and wastewater collection network design problem, Information Sciences, 512 (2020), 1335-1359. doi: 10.1016/j.ins.2019.10.062. Google Scholar
[7]	N. G. Hall, M. Lesaoana and C. N. Potts, Scheduling with fixed delivery dates, Operations Research, 49 (2001), 134-144. doi: 10.1287/opre.49.1.134.11192. Google Scholar
[8]	N. G. Hall and C. N. Potts, Supply chain scheduling: Batching and delivery, Operations Research, 51 (2003), 566-584. doi: 10.1287/opre.51.4.566.16106. Google Scholar
[9]	D. Han, Y. Yang, D. Wang, T. C. E. Cheng and Y. Yin, Integrated production, inventory, and outbound distribution operations with fixed departure times in a three-stage supply chain, Transportation Research Part E: Logistics and Transportation Review, 125 (2019), 334-347. doi: 10.1016/j.tre.2019.03.014. Google Scholar
[10]	D. Hermelina, J.-M. Kubitza, D. Shabtay, N. Talmon and G. J. Woeginger, Scheduling two agents on a single machine: A parameterized analysis of $NP$-hard problems, Omega, 83 (2011), 275-286. doi: 10.1016/j.omega.2018.08.001. Google Scholar
[11]	M. E. Johnson, Learning from toys: Lessons in managing supply chain risk from the toy industry, California Management Review, 43 (2001), 106-124. doi: 10.2307/41166091. Google Scholar
[12]	M. Y. Kovalyov, A. Oulamara and A. Soukhal, Two-agent scheduling with agent specific batches on an unbounded serial batching machine, Journal of Scheduling, 18 (2015), 423-434. doi: 10.1007/s10951-014-0410-0. Google Scholar
[13]	J. Y.-T. Leung and Z.-L. Chen, Integrated production and distribution with fixed delivery departure dates, Operations Research Letters, 41 (2013), 290-293. doi: 10.1016/j.orl.2013.02.006. Google Scholar
[14]	J. Y.-T. Leung, M. Pinedo and G. Wan, Competitive two agents scheduling and its applications, Operations Research, 58 (2010), 458-469. doi: 10.1287/opre.1090.0744. Google Scholar
[15]	F. Li, Z.-L. Chen and L. Tang, Integrated production, inventory and delivery problems: Complexity and algorithms, INFORMS Journal on Computing, 29 (2017), 232-250. doi: 10.1287/ijoc.2016.0726. Google Scholar
[16]	S. Li and J. Yuan, Unbounded parallel-batching scheduling with two competitive agents, Journal of Scheduling, 15 (2012), 629-640. doi: 10.1007/s10951-011-0253-x. Google Scholar
[17]	H. Matsuo, The weighted total tardiness problem with fixed shipping times and overtime utilization, Operations Research, 36 (1988), 293-307. doi: 10.1287/opre.36.2.293. Google Scholar
[18]	R. A. Melo and L. A. Wolsey, Optimizing production and transportation in a commit-to-delivery business mode, European Journal of Operational Research, 203 (2010), 614-618. doi: 10.1016/j.ejor.2009.09.011. Google Scholar
[19]	B. Mor and G. Mosheiov, Single machine batch scheduling with two competing agents to minimize total flowtime, European Journal of Operational Research, 215 (2011), 524-531. doi: 10.1016/j.ejor.2011.06.037. Google Scholar
[20]	P. Perez-Gonzalez and J. M. Framinan, A common framework and taxonomy for multicriteria scheduling problem with interfering and competing jobs: Multi-agent scheduling problems, European Journal of Operational Research, 235 (2014), 1-16. doi: 10.1016/j.ejor.2013.09.017. Google Scholar
[21]	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
[22]	M. Safaeian, A. M. Fathollahi-Fard, G. Tian, Z. Li and H. Ke, A multi-objective supplier selection and order allocation through incremental discount in a fuzzy environment, Journal of Intelligent & Fuzzy Systems, 37 (2019), 1435-1455. doi: 10.3233/JIFS-182843. Google Scholar
[23]	Y. Seddik, C. Gonzales and S. Kedad-Sidhoum, Single machine scheduling with delivery dates and cumulative payoffs, Journal of Scheduling, 16 (2013), 313-329. doi: 10.1007/s10951-012-0302-0. Google Scholar
[24]	K. E. Stecke and X. Zhao, Production and transportation integrationfor a make-to-order manufacturing company with a commit-to-delivery business mode, Manufacturing & Service Operations Management, 9 (2007), 206-224. doi: 10.1287/msom.1060.0138. Google Scholar
[25]	G. Tian, X. Liu, M. Zhang, Y. Yang, H. Zhang, Y. Lin, F. Ma, X. Wang, T. Qu and Z. Li, Selection of take-back pattern of vehicle reverse logistics in China via Grey-DEMATEL and Fuzzy-VIKOR combined method, Journal of Cleaner Production, 220 (2019), 1088-1100. doi: 10.1016/j.jclepro.2019.01.086. Google Scholar
[26]	G. Tian, H. Zhang, Y. Feng, H. Jia, C. Zhang, Z. Jiang, Z. Li and P. Li, Operation patterns analysis of automotive components remanufacturing industry development in China, Journal of Cleaner Production, 64 (2017), 1363-1375. doi: 10.1016/j.jclepro.2017.07.028. Google Scholar
[27]	G. Wan, S. R. Vakati, J. Y.-T. Leung and M. Pinedo, Scheduling two agents with controllable processing times, European Journal of Operational Research, 205 (2010), 528-539. doi: 10.1016/j.ejor.2010.01.005. Google Scholar
[28]	D.-J. Wang, Y. Yin, J. Xu, W. H. Wu, S.-R. Cheng and C.-C. Wu, Some due date determination scheduling problems with two agents on a single machine, International Journal of Production Economics, 168 (2015), 81-90. doi: 10.1016/j.ijpe.2015.06.018. Google Scholar
[29]	D. Wang, Y. Yu, H. Qiu, Y. Yin and T. C. E. Cheng, Two-agent scheduling with linear resource-dependent processing times, Naval Research Logistics, 67 (2020), 573-591. doi: 10.1002/nav.21936. Google Scholar
[30]	D.-Y. Wang, O. Grunderand and A. E. Moudni, Integrated scheduling of production and distribution operations: A review, International Journal of Industrial and Systems Engineering, 19 (2015), 94-122. doi: 10.1504/IJISE.2015.065949. Google Scholar
[31]	W. Wang, G. Tian, M. Chen, F. Tao, C. Zhang, A. Al-Ahmari, Z. Li and Z. Jiang, Dual-objective program and improved artificial bee colony for the optimization of energy-conscious milling parameters subject to multiple constraints, Journal of Cleaner Production, 245 (2020), 118714. doi: 10.1016/j.jclepro.2019.118714. Google Scholar
[32]	Y. Yin, S.-R. Cheng, T. C. E. Cheng, D.-J. Wang and C.-C. Wu, Just-in-time scheduling with two competing agents on unrelated parallel machines, Omega, 63 (2016), 41-47. doi: 10.1016/j.omega.2015.09.010. Google Scholar
[33]	Y. Yin, Y. Chen, K. Qin and D. Wang, Two-agent scheduling on unrelated parallel machines with total completion time and weighted number of tardy jobs criteria, Journal of Scheduling, 22 (2019), 315-333. doi: 10.1007/s10951-018-0583-z. Google Scholar
[34]	Y. Yin, D. Li, D. Wang and T. C. E. Cheng, Single-machine serial-batch delivery scheduling with two competing agents and due date assignment, Annals of Operations Research, (2018). doi: 10.1007/s10479-018-2839-6. Google Scholar
[35]	Y. Yin, Y. Wang, T. C. E. Cheng, D. Wang and C. C. Wu, Two-agent single-machine scheduling to minimize the batch delivery cost, Computers & Industrial Engineering, 92 (2016), 16-30. Google Scholar
[36]	Y. Yin, W. Wang, D. Wang and T. C. E. Cheng, Multi-agent single-machine scheduling and unrestricted due date assignment with a fixed machine unavailability interval, Computers & Industrial Engineering, 111 (2017), 202-215. doi: 10.1016/j.cie.2017.07.013. Google Scholar
[37]	Y. Yin, Y. Yang, D. Wang, T. C. E. Cheng and C.-C. Wu, Integrated production, inventory, and batch delivery scheduling with due date assignment and two competing agents, Naval Research Logistics, 65 (2018), 393-409. doi: 10.1002/nav.21813. Google Scholar

show all references

References:

[1]	A. Agnetis, M. A. Aloulou and M. Y. Kovalyov, Integrated production scheduling and batch delivery with fixed departure times and inventory holding costs, International Journal of Production Research, 55 (2017), 6193-6206. doi: 10.1080/00207543.2017.1346323. Google Scholar
[2]	A. Agnetis, P. B. Mirchandani, D. Pacciarelli and A. Pacifici, Scheduling problems with two competing agents, Operations Research, 52 (2004), 229-242. doi: 10.1287/opre.1030.0092. Google Scholar
[3]	K. R. Baker and J. C. Smith, A multiple-criterion model for machine scheduling, Journal of Scheduling, 6 (2003), 7-16. doi: 10.1023/A:1022231419049. Google Scholar
[4]	Z.-L. Chen, Integrated production and outbound distribution scheduling: Review and extensions, Operations Research, 58 (2010), 130-148. doi: 10.1287/opre.1080.0688. Google Scholar
[5]	E. Gerstl and G. Mosheiov, Single machine just-in-time scheduling problems with two competing agents, Naval Research Logistics, 61 (2014), 1-16. doi: 10.1002/nav.21562. Google Scholar
[6]	A. M. Fathollahi-Fard, M. Hajiaghaei-Keshteli, G. Tian and Z. Li, An adaptive Lagrangian relaxation-based algorithm for a coordinated water supply and wastewater collection network design problem, Information Sciences, 512 (2020), 1335-1359. doi: 10.1016/j.ins.2019.10.062. Google Scholar
[7]	N. G. Hall, M. Lesaoana and C. N. Potts, Scheduling with fixed delivery dates, Operations Research, 49 (2001), 134-144. doi: 10.1287/opre.49.1.134.11192. Google Scholar
[8]	N. G. Hall and C. N. Potts, Supply chain scheduling: Batching and delivery, Operations Research, 51 (2003), 566-584. doi: 10.1287/opre.51.4.566.16106. Google Scholar
[9]	D. Han, Y. Yang, D. Wang, T. C. E. Cheng and Y. Yin, Integrated production, inventory, and outbound distribution operations with fixed departure times in a three-stage supply chain, Transportation Research Part E: Logistics and Transportation Review, 125 (2019), 334-347. doi: 10.1016/j.tre.2019.03.014. Google Scholar
[10]	D. Hermelina, J.-M. Kubitza, D. Shabtay, N. Talmon and G. J. Woeginger, Scheduling two agents on a single machine: A parameterized analysis of $NP$-hard problems, Omega, 83 (2011), 275-286. doi: 10.1016/j.omega.2018.08.001. Google Scholar
[11]	M. E. Johnson, Learning from toys: Lessons in managing supply chain risk from the toy industry, California Management Review, 43 (2001), 106-124. doi: 10.2307/41166091. Google Scholar
[12]	M. Y. Kovalyov, A. Oulamara and A. Soukhal, Two-agent scheduling with agent specific batches on an unbounded serial batching machine, Journal of Scheduling, 18 (2015), 423-434. doi: 10.1007/s10951-014-0410-0. Google Scholar
[13]	J. Y.-T. Leung and Z.-L. Chen, Integrated production and distribution with fixed delivery departure dates, Operations Research Letters, 41 (2013), 290-293. doi: 10.1016/j.orl.2013.02.006. Google Scholar
[14]	J. Y.-T. Leung, M. Pinedo and G. Wan, Competitive two agents scheduling and its applications, Operations Research, 58 (2010), 458-469. doi: 10.1287/opre.1090.0744. Google Scholar
[15]	F. Li, Z.-L. Chen and L. Tang, Integrated production, inventory and delivery problems: Complexity and algorithms, INFORMS Journal on Computing, 29 (2017), 232-250. doi: 10.1287/ijoc.2016.0726. Google Scholar
[16]	S. Li and J. Yuan, Unbounded parallel-batching scheduling with two competitive agents, Journal of Scheduling, 15 (2012), 629-640. doi: 10.1007/s10951-011-0253-x. Google Scholar
[17]	H. Matsuo, The weighted total tardiness problem with fixed shipping times and overtime utilization, Operations Research, 36 (1988), 293-307. doi: 10.1287/opre.36.2.293. Google Scholar
[18]	R. A. Melo and L. A. Wolsey, Optimizing production and transportation in a commit-to-delivery business mode, European Journal of Operational Research, 203 (2010), 614-618. doi: 10.1016/j.ejor.2009.09.011. Google Scholar
[19]	B. Mor and G. Mosheiov, Single machine batch scheduling with two competing agents to minimize total flowtime, European Journal of Operational Research, 215 (2011), 524-531. doi: 10.1016/j.ejor.2011.06.037. Google Scholar
[20]	P. Perez-Gonzalez and J. M. Framinan, A common framework and taxonomy for multicriteria scheduling problem with interfering and competing jobs: Multi-agent scheduling problems, European Journal of Operational Research, 235 (2014), 1-16. doi: 10.1016/j.ejor.2013.09.017. Google Scholar
[21]	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
[22]	M. Safaeian, A. M. Fathollahi-Fard, G. Tian, Z. Li and H. Ke, A multi-objective supplier selection and order allocation through incremental discount in a fuzzy environment, Journal of Intelligent & Fuzzy Systems, 37 (2019), 1435-1455. doi: 10.3233/JIFS-182843. Google Scholar
[23]	Y. Seddik, C. Gonzales and S. Kedad-Sidhoum, Single machine scheduling with delivery dates and cumulative payoffs, Journal of Scheduling, 16 (2013), 313-329. doi: 10.1007/s10951-012-0302-0. Google Scholar
[24]	K. E. Stecke and X. Zhao, Production and transportation integrationfor a make-to-order manufacturing company with a commit-to-delivery business mode, Manufacturing & Service Operations Management, 9 (2007), 206-224. doi: 10.1287/msom.1060.0138. Google Scholar
[25]	G. Tian, X. Liu, M. Zhang, Y. Yang, H. Zhang, Y. Lin, F. Ma, X. Wang, T. Qu and Z. Li, Selection of take-back pattern of vehicle reverse logistics in China via Grey-DEMATEL and Fuzzy-VIKOR combined method, Journal of Cleaner Production, 220 (2019), 1088-1100. doi: 10.1016/j.jclepro.2019.01.086. Google Scholar
[26]	G. Tian, H. Zhang, Y. Feng, H. Jia, C. Zhang, Z. Jiang, Z. Li and P. Li, Operation patterns analysis of automotive components remanufacturing industry development in China, Journal of Cleaner Production, 64 (2017), 1363-1375. doi: 10.1016/j.jclepro.2017.07.028. Google Scholar
[27]	G. Wan, S. R. Vakati, J. Y.-T. Leung and M. Pinedo, Scheduling two agents with controllable processing times, European Journal of Operational Research, 205 (2010), 528-539. doi: 10.1016/j.ejor.2010.01.005. Google Scholar
[28]	D.-J. Wang, Y. Yin, J. Xu, W. H. Wu, S.-R. Cheng and C.-C. Wu, Some due date determination scheduling problems with two agents on a single machine, International Journal of Production Economics, 168 (2015), 81-90. doi: 10.1016/j.ijpe.2015.06.018. Google Scholar
[29]	D. Wang, Y. Yu, H. Qiu, Y. Yin and T. C. E. Cheng, Two-agent scheduling with linear resource-dependent processing times, Naval Research Logistics, 67 (2020), 573-591. doi: 10.1002/nav.21936. Google Scholar
[30]	D.-Y. Wang, O. Grunderand and A. E. Moudni, Integrated scheduling of production and distribution operations: A review, International Journal of Industrial and Systems Engineering, 19 (2015), 94-122. doi: 10.1504/IJISE.2015.065949. Google Scholar
[31]	W. Wang, G. Tian, M. Chen, F. Tao, C. Zhang, A. Al-Ahmari, Z. Li and Z. Jiang, Dual-objective program and improved artificial bee colony for the optimization of energy-conscious milling parameters subject to multiple constraints, Journal of Cleaner Production, 245 (2020), 118714. doi: 10.1016/j.jclepro.2019.118714. Google Scholar
[32]	Y. Yin, S.-R. Cheng, T. C. E. Cheng, D.-J. Wang and C.-C. Wu, Just-in-time scheduling with two competing agents on unrelated parallel machines, Omega, 63 (2016), 41-47. doi: 10.1016/j.omega.2015.09.010. Google Scholar
[33]	Y. Yin, Y. Chen, K. Qin and D. Wang, Two-agent scheduling on unrelated parallel machines with total completion time and weighted number of tardy jobs criteria, Journal of Scheduling, 22 (2019), 315-333. doi: 10.1007/s10951-018-0583-z. Google Scholar
[34]	Y. Yin, D. Li, D. Wang and T. C. E. Cheng, Single-machine serial-batch delivery scheduling with two competing agents and due date assignment, Annals of Operations Research, (2018). doi: 10.1007/s10479-018-2839-6. Google Scholar
[35]	Y. Yin, Y. Wang, T. C. E. Cheng, D. Wang and C. C. Wu, Two-agent single-machine scheduling to minimize the batch delivery cost, Computers & Industrial Engineering, 92 (2016), 16-30. Google Scholar
[36]	Y. Yin, W. Wang, D. Wang and T. C. E. Cheng, Multi-agent single-machine scheduling and unrestricted due date assignment with a fixed machine unavailability interval, Computers & Industrial Engineering, 111 (2017), 202-215. doi: 10.1016/j.cie.2017.07.013. Google Scholar
[37]	Y. Yin, Y. Yang, D. Wang, T. C. E. Cheng and C.-C. Wu, Integrated production, inventory, and batch delivery scheduling with due date assignment and two competing agents, Naval Research Logistics, 65 (2018), 393-409. doi: 10.1002/nav.21813. Google Scholar

Table 1. Computational complexity results

Problem	Complexity
$ (\sum w_j^AD_j^A+TC^A, \gamma^B\leq V^B) $	SNP, even if there is no capacity constraint on the delivery transporters, Theorems 5.1 and 7.2
$ (\sum D_j^A+TC^A, f_{\max}^B+TC^B\leq V^B) $	PS, $ O({{s}^{2}}{{n}_{A}}n_{B}^{2}n_{\max d}^{A}n_{\max d}^{B}\min \{{{v}^{B}},{{n}_{B}}\}) $, Theorem 4.4
$ (f_{\max}^B+TC^B, \sum D_j^A+TC^A\leq V^A) $	PS, $ O(s^2n_An_B^2n^A_{\max d}n^B_{\max d} $ $ \min \{{{v}^{B}},{{n}_{B}}\}\log (Q_{u}^{B}-Q_{l}^{B})) $, Theorem 4.5
$ (\sum w_j^AD_j^A+TC^A, f_{\max}^B+TC^B\leq V^B)_{s=\overline{s}} $	ONP, $ O({{n}_{A}}n_{B}^{2}{{(n_{\max d}^{A})}^{\bar{s}-1}}n_{\max d}^{B}{{P}^{\bar{s}-1}}\min \{{{v}^{B}},{{n}_{B}}\}) $, Theorem 7.2
$ (f_{\max}^B+TC^B, \sum w_j^AD_j^A+TC^A\leq V^A)_{s=\overline{s}} $	ONP, $ O({{n}_{A}}n_{B}^{2}{{(n_{\max d}^{A})}^{\bar{s}-1}}n_{\max d}^{B}{{P}^{\bar{s}-1}}\min \{{{v}^{B}},{{n}_{B}}\}\log (Q_{u}^{B}-Q_{l}^{B})) $, Theorem 7.2
$ (\sum w_j^AD_j^A+TC^A, \sum D_k^B+TC^B\leq V^B)_{s=\overline{s}} $	ONP, $ O(n_An_B(n^A_{\max d})^{\overline{s}-1} $ $ n^B_{\max d}P^{\overline{s}-1}V^B) $, Theorem 5.6
$ (\sum w_j^AD_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B)_{s=\overline{s}} $	ONP, $ O({n_A}{n_B}{(n_{\max d}^A)^{\bar s - 1}} $ $ n_{\max d}^B{P^{\bar s - 1}}{P^B}{V^B}) $, Theorem 7.2
$ (\sum D_j^A+TC^A, \sum D_k^B+TC^B\leq V^B) $	Open, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}V^B) $, Theorem 7.2
$ (\sum D_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B) $	ONP, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}P^BV^B) $, Theorem 6.5
$ (\sum w_j^AU_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B) $	ONP, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}PV^B) $, Theorem 7.1

Table Options

Download as excel

Problem	Complexity
$ (\sum w_j^AD_j^A+TC^A, \gamma^B\leq V^B) $	SNP, even if there is no capacity constraint on the delivery transporters, Theorems 5.1 and 7.2
$ (\sum D_j^A+TC^A, f_{\max}^B+TC^B\leq V^B) $	PS, $ O({{s}^{2}}{{n}_{A}}n_{B}^{2}n_{\max d}^{A}n_{\max d}^{B}\min \{{{v}^{B}},{{n}_{B}}\}) $, Theorem 4.4
$ (f_{\max}^B+TC^B, \sum D_j^A+TC^A\leq V^A) $	PS, $ O(s^2n_An_B^2n^A_{\max d}n^B_{\max d} $ $ \min \{{{v}^{B}},{{n}_{B}}\}\log (Q_{u}^{B}-Q_{l}^{B})) $, Theorem 4.5
$ (\sum w_j^AD_j^A+TC^A, f_{\max}^B+TC^B\leq V^B)_{s=\overline{s}} $	ONP, $ O({{n}_{A}}n_{B}^{2}{{(n_{\max d}^{A})}^{\bar{s}-1}}n_{\max d}^{B}{{P}^{\bar{s}-1}}\min \{{{v}^{B}},{{n}_{B}}\}) $, Theorem 7.2
$ (f_{\max}^B+TC^B, \sum w_j^AD_j^A+TC^A\leq V^A)_{s=\overline{s}} $	ONP, $ O({{n}_{A}}n_{B}^{2}{{(n_{\max d}^{A})}^{\bar{s}-1}}n_{\max d}^{B}{{P}^{\bar{s}-1}}\min \{{{v}^{B}},{{n}_{B}}\}\log (Q_{u}^{B}-Q_{l}^{B})) $, Theorem 7.2
$ (\sum w_j^AD_j^A+TC^A, \sum D_k^B+TC^B\leq V^B)_{s=\overline{s}} $	ONP, $ O(n_An_B(n^A_{\max d})^{\overline{s}-1} $ $ n^B_{\max d}P^{\overline{s}-1}V^B) $, Theorem 5.6
$ (\sum w_j^AD_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B)_{s=\overline{s}} $	ONP, $ O({n_A}{n_B}{(n_{\max d}^A)^{\bar s - 1}} $ $ n_{\max d}^B{P^{\bar s - 1}}{P^B}{V^B}) $, Theorem 7.2
$ (\sum D_j^A+TC^A, \sum D_k^B+TC^B\leq V^B) $	Open, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}V^B) $, Theorem 7.2
$ (\sum D_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B) $	ONP, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}P^BV^B) $, Theorem 6.5
$ (\sum w_j^AU_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B) $	ONP, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}PV^B) $, Theorem 7.1

Table 2. Overview of the problem characteristics in recent recent studies on integrate production and distribution

Article	Number of agents	Delivery capacity	Delivery cost	Delivery mode	Departure times
Agnetis et al. [1]	One	Bounded	Yes	Non-splittable	Fixed
Hall et al. [7]	One	Unbounded	No	Non-splittable	Fixed
Han et al. [9]	One	Bounded	Yes	Non-splittable	Fixed
Kovalyov et al. [12]	Two	Unbounded	No	Non-splittable	Fixed
Leung and Chen [14]	One	Bounded	No	Non-splittable	Fixed
Li et al. [15]	One	Bounded	No	Splittable or Non-splittable	Fixed
Melo and Wolsey [18]	One	Bounded	Yes	Non-splittable	Fixed
Mor and Mosheiov [19]	Two	Unbounded	No	Non-splittable	Fixed
Seddik et al. [23]	One	Not involve	No	Not involve	Fixed
Stecke and Zhao [24]	One	Bounded	Yes	Splittable or Non-splittable	Fixed
Yin et al. [35]	Two	Unbounded	Yes	Non-splittable	Fixed
Yin et al. [37,34]	Two	Unbounded	Yes	Non-splittable	No
Our paper	Multiple	Bounded	Yes	Non-splittable	Yes

Table Options

Download as excel

Article	Number of agents	Delivery capacity	Delivery cost	Delivery mode	Departure times
Agnetis et al. [1]	One	Bounded	Yes	Non-splittable	Fixed
Hall et al. [7]	One	Unbounded	No	Non-splittable	Fixed
Han et al. [9]	One	Bounded	Yes	Non-splittable	Fixed
Kovalyov et al. [12]	Two	Unbounded	No	Non-splittable	Fixed
Leung and Chen [14]	One	Bounded	No	Non-splittable	Fixed
Li et al. [15]	One	Bounded	No	Splittable or Non-splittable	Fixed
Melo and Wolsey [18]	One	Bounded	Yes	Non-splittable	Fixed
Mor and Mosheiov [19]	Two	Unbounded	No	Non-splittable	Fixed
Seddik et al. [23]	One	Not involve	No	Not involve	Fixed
Stecke and Zhao [24]	One	Bounded	Yes	Splittable or Non-splittable	Fixed
Yin et al. [35]	Two	Unbounded	Yes	Non-splittable	Fixed
Yin et al. [37,34]	Two	Unbounded	Yes	Non-splittable	No
Our paper	Multiple	Bounded	Yes	Non-splittable	Yes

Table 3. Computational complexity results

Problem	Complexity
$ (\sum w_j^AD_j^A+TC^A, \gamma^B\leq V^B) $	SNP, even if there is no capacity constraint on the delivery transporters, Theorems 5.1 and 7.2
$ (\sum D_j^A+TC^A, f_{\max}^B+TC^B\leq V^B) $	PS, $ O(s^2n_An_B^2n^A_{\max d}n^B_{\max d}\min v^B, n_B ) $, Theorem 4.4
$ (f_{\max}^B+TC^B, \sum D_j^A+TC^A\leq V^A) $	PS, $ O(s^2n_An_B^2n^A_{\max d}n^B_{\max d} $ $ \min v^B, n_B \log(Q_u^B-Q_l^B)) $, Theorem 4.5
$ (\sum w_j^AD_j^A+TC^A, f_{\max}^B+TC^B\leq V^B)_{s=\overline{s}} $	ONP, $ O(n_An_B^2(n^A_{\max d})^{\overline{s}-1}n^B_{\max d}P^{\overline{s}-1}\min v^B, n_B ) $, Theorem 7.2
$ (f_{\max}^B+TC^B, \sum w_j^AD_j^A+TC^A\leq V^A)_{s=\overline{s}} $	ONP, $ O(n_An_B^2(n^A_{\max d})^{\overline{s}-1}n^B_{\max d}P^{\overline{s}-1}\min v^B, n_B \log(Q_u^B-Q_l^B)) $, Theorem 7.2
$ (\sum w_j^AD_j^A+TC^A, \sum D_k^B+TC^B\leq V^B)_{s=\overline{s}} $	ONP, $ O(n_An_B(n^A_{\max d})^{\overline{s}-1} $ $ n^B_{\max d}P^{\overline{s}-1}V^B) $, Theorem 5.6
$ (\sum w_j^AD_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B)_{s=\overline{s}} $	ONP, $ O(n_An_B(n^A_{\max d})^{\overline{s}-1} $ $ n^B_{\max d}P^{\overline{s}-1}P^BV^B) $, Theorem 7.2
$ (\sum D_j^A+TC^A, \sum D_k^B+TC^B\leq V^B) $	Open, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}V^B) $, Theorem 7.2
$ (\sum D_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B) $	ONP, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}P^BV^B) $, Theorem 6.5
$ (\sum w_j^AU_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B) $	ONP, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}PV^B) $, Theorem 7.1

Table Options

Download as excel

Problem	Complexity
$ (\sum w_j^AD_j^A+TC^A, \gamma^B\leq V^B) $	SNP, even if there is no capacity constraint on the delivery transporters, Theorems 5.1 and 7.2
$ (\sum D_j^A+TC^A, f_{\max}^B+TC^B\leq V^B) $	PS, $ O(s^2n_An_B^2n^A_{\max d}n^B_{\max d}\min v^B, n_B ) $, Theorem 4.4
$ (f_{\max}^B+TC^B, \sum D_j^A+TC^A\leq V^A) $	PS, $ O(s^2n_An_B^2n^A_{\max d}n^B_{\max d} $ $ \min v^B, n_B \log(Q_u^B-Q_l^B)) $, Theorem 4.5
$ (\sum w_j^AD_j^A+TC^A, f_{\max}^B+TC^B\leq V^B)_{s=\overline{s}} $	ONP, $ O(n_An_B^2(n^A_{\max d})^{\overline{s}-1}n^B_{\max d}P^{\overline{s}-1}\min v^B, n_B ) $, Theorem 7.2
$ (f_{\max}^B+TC^B, \sum w_j^AD_j^A+TC^A\leq V^A)_{s=\overline{s}} $	ONP, $ O(n_An_B^2(n^A_{\max d})^{\overline{s}-1}n^B_{\max d}P^{\overline{s}-1}\min v^B, n_B \log(Q_u^B-Q_l^B)) $, Theorem 7.2
$ (\sum w_j^AD_j^A+TC^A, \sum D_k^B+TC^B\leq V^B)_{s=\overline{s}} $	ONP, $ O(n_An_B(n^A_{\max d})^{\overline{s}-1} $ $ n^B_{\max d}P^{\overline{s}-1}V^B) $, Theorem 5.6
$ (\sum w_j^AD_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B)_{s=\overline{s}} $	ONP, $ O(n_An_B(n^A_{\max d})^{\overline{s}-1} $ $ n^B_{\max d}P^{\overline{s}-1}P^BV^B) $, Theorem 7.2
$ (\sum D_j^A+TC^A, \sum D_k^B+TC^B\leq V^B) $	Open, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}V^B) $, Theorem 7.2
$ (\sum D_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B) $	ONP, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}P^BV^B) $, Theorem 6.5
$ (\sum w_j^AU_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B) $	ONP, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}PV^B) $, Theorem 7.1

[1]	Karl-Peter Hadeler, Frithjof Lutscher. Quiescent phases with distributed exit times. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 849-869. doi: 10.3934/dcdsb.2012.17.849
[2]	Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021
[3]	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
[4]	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
[5]	Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[6]	M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072
[7]	Xianchao Xiu, Ying Yang, Wanquan Liu, Lingchen Kong, Meijuan Shang. An improved total variation regularized RPCA for moving object detection with dynamic background. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1685-1698. doi: 10.3934/jimo.2019024
[8]	Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021035
[9]	Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881
[10]	Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009
[11]	Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933

2019 Impact Factor: 1.366

Tools

Metrics

Tools

Article outline

Figures and Tables

2019 Impact Factor: 1.366

Tools

Figures and Tables

2019 Impact Factor: 1.366

American Institute of Mathematical Sciences

Two-agent integrated scheduling of production and distribution operations with fixed departure times

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Two-agent integrated scheduling of production and distribution operations with fixed departure times

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors