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September  2020, 16(5): 2389-2406. doi: 10.3934/jimo.2019059

## A dynamic lot sizing model with production-or-outsourcing decision under minimum production quantities

 1 School of Management and Economics, University of Electronic Science and Technology of China, Chengdu 611731, China 2 College of Communication Science and Art, Chengdu University of Technology, Chengdu 610059, China

* Corresponding author: chaoxr@uestc.edu.cn

Received  September 2018 Revised  February 2019 Published  September 2020 Early access  May 2019

In the real-world production process, the firms need to determine the optimal production planning under minimum production quantity constraint in order to achieve economies of scale. However, the inventory cost will hugely increase when there is a very large amount of production in a period and also a large amount of total demands for the next few periods. This paper considers a single-item dynamic lot sizing problem with production-or-outsourcing decisions. In each period, the production level cannot be lower than a given quantity in order to make full use of resources, but the outsourcing is unrestricted. The demands in a period can be backlogged. The production and outsourcing costs are fixed-plus-linear, and the inventory and backlogging costs are linear. We establish a mathematical programming model according to the real problem in the firm. We explore some structural properties of the optimal solution and use them to develop a dynamic programming algorithm to solve the proposed problem. We further present a special case with stationary production and outsourcing costs which can be solved with reduced computational complexities. In the end, we use three numerical instances to show how to obtain the optimal solutions by using the dynamic programming algorithm. Furthermore, we show that the policy of backlogging or outsourcing can reduce the total cost.

Citation: Min Tang, Fuying Jing, Xiangrui Chao. A dynamic lot sizing model with production-or-outsourcing decision under minimum production quantities. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2389-2406. doi: 10.3934/jimo.2019059
##### References:
 [1] N. Absi, S. K. Sidhoum and S. D. Peres, Uncapacitated lot-sizing problem with production time window, early production, backlog and lost sale, International Journal of Production Research, 49 (2011), 2551-2566.  doi: 10.1080/00207543.2010.532920. [2] N. Absi and S. K. Sidhoum, The multi-item capacitated lot-sizing problem with setup times and shortage costs, European Journal of Operational Research, 185 (2008), 1351-1374.  doi: 10.1016/j.ejor.2006.01.053. [3] D. Aksen, K. Alt$\imath$nkemer and S. Chand, The single-item lot-sizing problem with immediate lost sales, European Journal of Operational Research, 147 (2003), 558-566.  doi: 10.1016/S0377-2217(02)00331-4. [4] D. Aksen, Loss of customer goodwill in the uncapacitated lot-sizing problem, Computer and Operations Research, 34 (2007), 2805-2823.  doi: 10.1016/j.cor.2005.10.012. [5] F. Altiparmak, M. Gen, L. Lin and T. Paksoy, A genetic algorithm approach for multi-ojective optimization of supply chain networks, Computer and Industrial Engineering, 51 (2006), 196-215. [6] E. J. Anderson and B. S. Cheah, Capacitated lot-sizing with minimum batch sizes and setup times, International Journal of Production Economics, 30/31 (1993), 137-152.  doi: 10.1016/0925-5273(93)90087-2. [7] E. Berk, A. O. Toy and O. Hazir, Single item lot-sizing problem for a worm/cold process with immediate lost sales, European Journal of Operational Research, 187 (2008), 1251-1267.  doi: 10.1016/j.ejor.2006.06.070. [8] N. Brahimi, N. Absi, S. D. Peres and A. Nordli, Single-item dynamic lot-sizing problems: An updated survey, European Journal of Operational Research, 263 (2017), 838-863.  doi: 10.1016/j.ejor.2017.05.008. [9] M. B. Cheng, S. X. Xiao and G. S. Liu, Single-machine rescheduling problems with learning effect under disruptions, Journal of Industrial & Management Optimization, 14 (2018), 967-980.  doi: 10.3934/jimo.2017085. [10] C. C. Chu, F. Chu, J. H. Zhong and S. L. Yang, A polynomial algorithm for lot-sizing problem with backlogging, outsourcing and limited inventory, Computer and Industrial Engineering, 64 (2013), 200-210.  doi: 10.1016/j.cie.2012.08.007. [11] F. Chu and C. C. Chu, Polynomial algorithms for single item lot-sizing models with bounded inventory and backlogging, or outsourcing, IEEE Transactions on Automation Science and Engineering, 4 (2007), 233-251.  doi: 10.1109/TASE.2006.880855. [12] F. Chu and C. C. Chu, Single-Item dynamic lot-sizing models with bounded inventory and outsourcing, IEEE Transactions on Systems, Man, and Cybernetics, 38 (2008), 70-77. [13] M. Constantino, Lower Bounds in Lot-Sizing Models: A Polyhedral Study, Mathematics of Operations Research, 23 (1998), 101-118.  doi: 10.1287/moor.23.1.101. [14] A. Federgruen and M. Tzur, The dynamic lot sizing model with backlogging: A simple O(nlogn) algorithm and minimum forecast horizon procedure, Naval Research Logistic, 40 (1993), 459-478.  doi: 10.1002/1520-6750(199306)40:4<459::AID-NAV3220400404>3.0.CO;2-8. [15] J. P. Gayon, G. Massonnet, C. Rapine and G. Satuffer, Constant approximation algorithms for the one warehouse multiple retailers problem with backlog or lost-sales, European Journal of Operational Research, 250 (2016), 155-163.  doi: 10.1016/j.ejor.2015.10.054. [16] M. Ghaniabadi and A. Mazinani, Dynamic lot sizing with multiple suppliers, backlogging and quantity discounts, Computer and Industrial Engineering, 110 (2017), 67-74.  doi: 10.1016/j.cie.2017.05.031. [17] A. Goerler and S. Voß, Dynamic lot-sizing with rework of defective items and minimum lot-size constraints, International Journal of Production Research, 54 (2016), 2284-2297.  doi: 10.1080/00207543.2015.1070970. [18] B. Hellion, F. Mangione and B. Penz, A polynomial time algorithm to solve the single-item capacitated lot sizing problem with minimum order quantities and concave costs, European Journal of Operational Research, 222 (2012), 10–16. doi: 10.1016/j.ejor.2012.04.024. [19] B. Hellion, F. Mangione and B. Penz, Corrigendum to "A polynomial time algorithm to solve the single-item capacitated lot sizing problem with minimum order quantities and concave costs", European Journal of Operational Research, 229 (2013), 279. doi: 10.1016/j.ejor.2013.01.051. [20] B. Hellion, F. Mangione and B. Penz, A polynomial time algorithm for the single-item lot sizing problem with capacities, minimum order quantities and dynamic time windows, Operations Research Letters, 42 (2014), 500-504.  doi: 10.1016/j.orl.2014.08.010. [21] C. Y. Lee, Inventory replenishment model: Lot sizing versus just-in-time delivery, Operations Research Letters, 32 (2004), 581-590.  doi: 10.1016/j.orl.2003.12.008. [22] C. Y. Lee, S. Cetinkaya and A. P. M.Wagelmans, A dynamic lot sizing model with demand time windows, Management Science, 47 (2001), 1311-1440.  doi: 10.1287/mnsc.47.10.1384. [23] S. B. Lee and P. H. Zipkin, A dynamic lot-size model with make-or-buy decisions, Management Science, 35 (1989), 447-458.  doi: 10.1287/mnsc.35.4.447. [24] C. L. Li, V. N. Hsu and W. Q. Xiao, Dynamic lot sizing with batch ordering and truckload discounts, Operations Research, 52 (2004), 639–654. doi: 10.1287/opre.1040.0121. [25] T. E. Morton, An improved algorithm for the stationary cost dynamic lot size model with backlogging, Management Science, 24 (1978), 869-873.  doi: 10.1287/mnsc.24.8.869. [26] E. P. Musalem and R. Dekker, Controlling inventories in a supply chain: a case study, International Journal of Production Economics, 93/94 (2005), 179-188.  doi: 10.1016/j.ijpe.2004.06.016. [27] M. Narenji, S. M. T. F. Ghomi and S. V. R. Nooraie, Grouping in decomposition method for multi-item capacitated lot-szing problem with immediate lost sales and joint and item-dependent setup cost, International Journal of System Science, 42 (2011), 489-498. [28] D. Neslihan, E. Özceylan, T. Paksoy and H. Gökçen, A genetic algorithm approach for optimising a closed-loop supply chain network with crisp and fuzzy objectives, International Journal of Production Research, 52 (2014), 3637-3664. [29] I. Okhrin and K. Richter, An $O(T^{3})$ algorithm for the capacitated dynamic lot sizing problem with minimum order quantities, European Journal of Operational Research, 211 (2011), 507-514.  doi: 10.1016/j.ejor.2011.01.007. [30] I. Okhrin and K. Richter, The linear dynamic lot size problem with minimum order quantity, International Journal of Production Economics, 133 (2011), 688-693.  doi: 10.1016/j.ijpe.2011.05.017. [31] E. Özceylan and T. Paksoy, Fuzzy multi-objective linear programming approach for optimising a closed loop supply chain network, International Journal of Production Research, 51 (2013), 2443-2461. [32] E. Özceylan and T. Paksoy, A mixed integer programming model for a closed loop supply-chain network, International Journal of Production Research, 51 (2013), 718-734. [33] E. Özceylan and T. Paksoy, Interactive fuzzy programming approaches to the strategic and tactical planning of a closed loop supply chain under uncertainty, International Journal of Production Research, 52 (2014), 2363-2387. [34] E. Özceylan, T. Paksoy and T. Bektas, Modelling and optimizing the integrated problem of closed-loop supply chain network design and disassembly line balancing, Transportation Research Part E, 61 (2014), 142-164. [35] T. Paksoy, T. Bektas and E. Özceylan, Operational and environmental performance measures in a multi-product closed-loop supply chain, Transportation Research Part E, 47 (2011), 532-546.  doi: 10.1016/j.tre.2010.12.001. [36] T. Paksoy and C. T. Chang, Revised multi-choice goal programming for multi-period, multi-stage inventory controlled supply chain model with popup stores in Guerrilla marketing, Applied Mathematical Modelling, 34 (2010), 3586-3598.  doi: 10.1016/j.apm.2010.03.008. [37] Y. W. Park and D. Klabjan, Lot sizing with minimum order quantity, Discrete Applied Mathematics, 181 (2015), 235-254.  doi: 10.1016/j.dam.2014.09.015. [38] M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price and stock-dependent demand: A trade-credit policy, Journal of Industrial & Management Optimization, 2018. [39] E. Porras and R. Dekker, An efficient optimal solution method for the joint replenishment problem with minimum order quantities, European Journal of Operational Research, 174 (2006), 1595-1615.  doi: 10.1016/j.ejor.2005.02.056. [40] S. K. Roy, M. Gurupada and G. W. Weber, Multi-objective two-stage grey transportation problem using utility function with goals, Central European Journal of Operations Research, 25 (2017), 417-439.  doi: 10.1007/s10100-016-0464-5. [41] S. K. Roy and P. Mula, Solving matrix game with rough payoffs using genetic algorithm, Operational Research-An International Journal, 16 (2016), 117-130.  doi: 10.1007/s12351-015-0189-6. [42] R. A. Sandbothe and G. L. Thompson, Decision horizons for the capacitated lot size model with inventory bounds and stockouts, Computers & Operations Research, 20 (1993), 455-465. [43] R. A. Sandbothe and G. L. Thompson, A forward algorithm for the capacitated lot size model with stockouts, Operations Research, 38 (1990), 474-486.  doi: 10.1287/opre.38.3.474. [44] C. Suerie, Campaign planning in time-indexed model formulations, International Journal of Production Research, 43 (2005), 49-66.  doi: 10.1080/00207540412331285823. [45] S. Voß and D. L. Woodruff, Introduction to Computational Optimization Models for Production Planning in a Supply Chain, 2nd ed. Berlin, Heidelberg: Springer, 2006. [46] H. M. Wagner and T. M. Whitin, Dynamic version of the economic lot size model, Management Science, 5 (1958), 89-96.  doi: 10.1287/mnsc.5.1.89. [47] N. Wang, Z. W. He, J. C. Sun, H. Y. Xie and W. Shi, A single-item uncapacitated lot-sizing problem with remanufacturing and outsourcing, Procedia Engineering, 15 (2011), 5170-5178.  doi: 10.1016/j.proeng.2011.08.959. [48] G. D. Yi, X. H. Chen and C. Q. Tan, Optimal pricing of perishable products with replenishment policy in the presence of strategic consumers, Journal of Industrial & Management Optimization, 2018. doi: 10.3934/jimo.2018112. [49] W. I. Zangwill, A backlogging model and a multi-echelon model of a dynamic economic lot size production system–-a network approach, Management Science, 15 (1969), 506-527.  doi: 10.1287/mnsc.15.9.506. [50] J. H. Zhong, F. Chu, H. B. Chu and S. L. Yang, Polynomial dynamic programming algorithms for lot sizing models with bounded inventory and stockout and or backlogging, Journal of Systems Science and Systems Engineering, 25 (2016), 370-397.  doi: 10.1007/s11518-015-5277-x.

show all references

##### References:
 [1] N. Absi, S. K. Sidhoum and S. D. Peres, Uncapacitated lot-sizing problem with production time window, early production, backlog and lost sale, International Journal of Production Research, 49 (2011), 2551-2566.  doi: 10.1080/00207543.2010.532920. [2] N. Absi and S. K. Sidhoum, The multi-item capacitated lot-sizing problem with setup times and shortage costs, European Journal of Operational Research, 185 (2008), 1351-1374.  doi: 10.1016/j.ejor.2006.01.053. [3] D. Aksen, K. Alt$\imath$nkemer and S. Chand, The single-item lot-sizing problem with immediate lost sales, European Journal of Operational Research, 147 (2003), 558-566.  doi: 10.1016/S0377-2217(02)00331-4. [4] D. Aksen, Loss of customer goodwill in the uncapacitated lot-sizing problem, Computer and Operations Research, 34 (2007), 2805-2823.  doi: 10.1016/j.cor.2005.10.012. [5] F. Altiparmak, M. Gen, L. Lin and T. Paksoy, A genetic algorithm approach for multi-ojective optimization of supply chain networks, Computer and Industrial Engineering, 51 (2006), 196-215. [6] E. J. Anderson and B. S. Cheah, Capacitated lot-sizing with minimum batch sizes and setup times, International Journal of Production Economics, 30/31 (1993), 137-152.  doi: 10.1016/0925-5273(93)90087-2. [7] E. Berk, A. O. Toy and O. Hazir, Single item lot-sizing problem for a worm/cold process with immediate lost sales, European Journal of Operational Research, 187 (2008), 1251-1267.  doi: 10.1016/j.ejor.2006.06.070. [8] N. Brahimi, N. Absi, S. D. Peres and A. Nordli, Single-item dynamic lot-sizing problems: An updated survey, European Journal of Operational Research, 263 (2017), 838-863.  doi: 10.1016/j.ejor.2017.05.008. [9] M. B. Cheng, S. X. Xiao and G. S. Liu, Single-machine rescheduling problems with learning effect under disruptions, Journal of Industrial & Management Optimization, 14 (2018), 967-980.  doi: 10.3934/jimo.2017085. [10] C. C. Chu, F. Chu, J. H. Zhong and S. L. Yang, A polynomial algorithm for lot-sizing problem with backlogging, outsourcing and limited inventory, Computer and Industrial Engineering, 64 (2013), 200-210.  doi: 10.1016/j.cie.2012.08.007. [11] F. Chu and C. C. Chu, Polynomial algorithms for single item lot-sizing models with bounded inventory and backlogging, or outsourcing, IEEE Transactions on Automation Science and Engineering, 4 (2007), 233-251.  doi: 10.1109/TASE.2006.880855. [12] F. Chu and C. C. Chu, Single-Item dynamic lot-sizing models with bounded inventory and outsourcing, IEEE Transactions on Systems, Man, and Cybernetics, 38 (2008), 70-77. [13] M. Constantino, Lower Bounds in Lot-Sizing Models: A Polyhedral Study, Mathematics of Operations Research, 23 (1998), 101-118.  doi: 10.1287/moor.23.1.101. [14] A. Federgruen and M. Tzur, The dynamic lot sizing model with backlogging: A simple O(nlogn) algorithm and minimum forecast horizon procedure, Naval Research Logistic, 40 (1993), 459-478.  doi: 10.1002/1520-6750(199306)40:4<459::AID-NAV3220400404>3.0.CO;2-8. [15] J. P. Gayon, G. Massonnet, C. Rapine and G. Satuffer, Constant approximation algorithms for the one warehouse multiple retailers problem with backlog or lost-sales, European Journal of Operational Research, 250 (2016), 155-163.  doi: 10.1016/j.ejor.2015.10.054. [16] M. Ghaniabadi and A. Mazinani, Dynamic lot sizing with multiple suppliers, backlogging and quantity discounts, Computer and Industrial Engineering, 110 (2017), 67-74.  doi: 10.1016/j.cie.2017.05.031. [17] A. Goerler and S. Voß, Dynamic lot-sizing with rework of defective items and minimum lot-size constraints, International Journal of Production Research, 54 (2016), 2284-2297.  doi: 10.1080/00207543.2015.1070970. [18] B. Hellion, F. Mangione and B. Penz, A polynomial time algorithm to solve the single-item capacitated lot sizing problem with minimum order quantities and concave costs, European Journal of Operational Research, 222 (2012), 10–16. doi: 10.1016/j.ejor.2012.04.024. [19] B. Hellion, F. Mangione and B. Penz, Corrigendum to "A polynomial time algorithm to solve the single-item capacitated lot sizing problem with minimum order quantities and concave costs", European Journal of Operational Research, 229 (2013), 279. doi: 10.1016/j.ejor.2013.01.051. [20] B. Hellion, F. Mangione and B. Penz, A polynomial time algorithm for the single-item lot sizing problem with capacities, minimum order quantities and dynamic time windows, Operations Research Letters, 42 (2014), 500-504.  doi: 10.1016/j.orl.2014.08.010. [21] C. Y. Lee, Inventory replenishment model: Lot sizing versus just-in-time delivery, Operations Research Letters, 32 (2004), 581-590.  doi: 10.1016/j.orl.2003.12.008. [22] C. Y. Lee, S. Cetinkaya and A. P. M.Wagelmans, A dynamic lot sizing model with demand time windows, Management Science, 47 (2001), 1311-1440.  doi: 10.1287/mnsc.47.10.1384. [23] S. B. Lee and P. H. Zipkin, A dynamic lot-size model with make-or-buy decisions, Management Science, 35 (1989), 447-458.  doi: 10.1287/mnsc.35.4.447. [24] C. L. Li, V. N. Hsu and W. Q. Xiao, Dynamic lot sizing with batch ordering and truckload discounts, Operations Research, 52 (2004), 639–654. doi: 10.1287/opre.1040.0121. [25] T. E. Morton, An improved algorithm for the stationary cost dynamic lot size model with backlogging, Management Science, 24 (1978), 869-873.  doi: 10.1287/mnsc.24.8.869. [26] E. P. Musalem and R. Dekker, Controlling inventories in a supply chain: a case study, International Journal of Production Economics, 93/94 (2005), 179-188.  doi: 10.1016/j.ijpe.2004.06.016. [27] M. Narenji, S. M. T. F. Ghomi and S. V. R. Nooraie, Grouping in decomposition method for multi-item capacitated lot-szing problem with immediate lost sales and joint and item-dependent setup cost, International Journal of System Science, 42 (2011), 489-498. [28] D. Neslihan, E. Özceylan, T. Paksoy and H. Gökçen, A genetic algorithm approach for optimising a closed-loop supply chain network with crisp and fuzzy objectives, International Journal of Production Research, 52 (2014), 3637-3664. [29] I. Okhrin and K. Richter, An $O(T^{3})$ algorithm for the capacitated dynamic lot sizing problem with minimum order quantities, European Journal of Operational Research, 211 (2011), 507-514.  doi: 10.1016/j.ejor.2011.01.007. [30] I. Okhrin and K. Richter, The linear dynamic lot size problem with minimum order quantity, International Journal of Production Economics, 133 (2011), 688-693.  doi: 10.1016/j.ijpe.2011.05.017. [31] E. Özceylan and T. Paksoy, Fuzzy multi-objective linear programming approach for optimising a closed loop supply chain network, International Journal of Production Research, 51 (2013), 2443-2461. [32] E. Özceylan and T. Paksoy, A mixed integer programming model for a closed loop supply-chain network, International Journal of Production Research, 51 (2013), 718-734. [33] E. Özceylan and T. Paksoy, Interactive fuzzy programming approaches to the strategic and tactical planning of a closed loop supply chain under uncertainty, International Journal of Production Research, 52 (2014), 2363-2387. [34] E. Özceylan, T. Paksoy and T. Bektas, Modelling and optimizing the integrated problem of closed-loop supply chain network design and disassembly line balancing, Transportation Research Part E, 61 (2014), 142-164. [35] T. Paksoy, T. Bektas and E. Özceylan, Operational and environmental performance measures in a multi-product closed-loop supply chain, Transportation Research Part E, 47 (2011), 532-546.  doi: 10.1016/j.tre.2010.12.001. [36] T. Paksoy and C. T. Chang, Revised multi-choice goal programming for multi-period, multi-stage inventory controlled supply chain model with popup stores in Guerrilla marketing, Applied Mathematical Modelling, 34 (2010), 3586-3598.  doi: 10.1016/j.apm.2010.03.008. [37] Y. W. Park and D. Klabjan, Lot sizing with minimum order quantity, Discrete Applied Mathematics, 181 (2015), 235-254.  doi: 10.1016/j.dam.2014.09.015. [38] M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price and stock-dependent demand: A trade-credit policy, Journal of Industrial & Management Optimization, 2018. [39] E. Porras and R. Dekker, An efficient optimal solution method for the joint replenishment problem with minimum order quantities, European Journal of Operational Research, 174 (2006), 1595-1615.  doi: 10.1016/j.ejor.2005.02.056. [40] S. K. Roy, M. Gurupada and G. W. Weber, Multi-objective two-stage grey transportation problem using utility function with goals, Central European Journal of Operations Research, 25 (2017), 417-439.  doi: 10.1007/s10100-016-0464-5. [41] S. K. Roy and P. Mula, Solving matrix game with rough payoffs using genetic algorithm, Operational Research-An International Journal, 16 (2016), 117-130.  doi: 10.1007/s12351-015-0189-6. [42] R. A. Sandbothe and G. L. Thompson, Decision horizons for the capacitated lot size model with inventory bounds and stockouts, Computers & Operations Research, 20 (1993), 455-465. [43] R. A. Sandbothe and G. L. Thompson, A forward algorithm for the capacitated lot size model with stockouts, Operations Research, 38 (1990), 474-486.  doi: 10.1287/opre.38.3.474. [44] C. Suerie, Campaign planning in time-indexed model formulations, International Journal of Production Research, 43 (2005), 49-66.  doi: 10.1080/00207540412331285823. [45] S. Voß and D. L. Woodruff, Introduction to Computational Optimization Models for Production Planning in a Supply Chain, 2nd ed. Berlin, Heidelberg: Springer, 2006. [46] H. M. Wagner and T. M. Whitin, Dynamic version of the economic lot size model, Management Science, 5 (1958), 89-96.  doi: 10.1287/mnsc.5.1.89. [47] N. Wang, Z. W. He, J. C. Sun, H. Y. Xie and W. Shi, A single-item uncapacitated lot-sizing problem with remanufacturing and outsourcing, Procedia Engineering, 15 (2011), 5170-5178.  doi: 10.1016/j.proeng.2011.08.959. [48] G. D. Yi, X. H. Chen and C. Q. Tan, Optimal pricing of perishable products with replenishment policy in the presence of strategic consumers, Journal of Industrial & Management Optimization, 2018. doi: 10.3934/jimo.2018112. [49] W. I. Zangwill, A backlogging model and a multi-echelon model of a dynamic economic lot size production system–-a network approach, Management Science, 15 (1969), 506-527.  doi: 10.1287/mnsc.15.9.506. [50] J. H. Zhong, F. Chu, H. B. Chu and S. L. Yang, Polynomial dynamic programming algorithms for lot sizing models with bounded inventory and stockout and or backlogging, Journal of Systems Science and Systems Engineering, 25 (2016), 370-397.  doi: 10.1007/s11518-015-5277-x.
The sketch for production and outsourcing decisions
The sketch for $F(i,j,t)$
The sketch for $F(i,j,j_1 ,j_1 ',g_2 ,j_2 ,...,g_n ,j_n ,t)$
The sketch for $F(i,j,g_1 ,j_1 ,...,g_n ,j_n ,t)$
The sketch for $F(e,i,t)$
The sketch for $F(e,i,j_1 ,j_1 ',g_2 ,j_2 ,...,g_n ,j_n ,t)$
The sketch for $F(e,i,g_1 ,j_1 ,...,g_n ,j_n ,t)$
The sketch for $F(g_n ,j_n ,t)$
The basic notation
 $K_t :$ fixed (setup) costs for production in period $t$, $t=1,2,..,T$; $c_t :$ per unit production cost in period $t$, $t=1,2,..,T$; $k_t :$ fixed (setup) costs for outsourcing in period $t$, $t=1,2,..,T$; $o_t :$ per unit outsourcing cost in period $t$, $t=1,2,..,T$; $h_t :$ per unit holding cost in period $t$, $t=1,2,..,T$; $b_t :$ per unit backlogging cost in period $t$, $t=1,2,..,T$
 $K_t :$ fixed (setup) costs for production in period $t$, $t=1,2,..,T$; $c_t :$ per unit production cost in period $t$, $t=1,2,..,T$; $k_t :$ fixed (setup) costs for outsourcing in period $t$, $t=1,2,..,T$; $o_t :$ per unit outsourcing cost in period $t$, $t=1,2,..,T$; $h_t :$ per unit holding cost in period $t$, $t=1,2,..,T$; $b_t :$ per unit backlogging cost in period $t$, $t=1,2,..,T$
Summary of Computations of Example 1
 $t$ 1 2 3 4 5 6 7 8 9 10 $d_t$ 8 6 9 29 5 6 5 15 5 6 $X_t^\ast$ 30 $X_t^\ast$ 30 0 $X_t^\ast$ 30 0 0 $X_t^\ast$ 30 0 0 30 $X_t^\ast$ 30 0 0 30 0 $X_t^\ast$ 30 0 0 33 0 0 $X_t^\ast$ 30 0 0 38 0 0 0 $X_t^\ast$ 30 0 0 53 0 0 0 0 $X_t^\ast$ 30 0 0 58 0 0 0 0 0 $X_t^\ast$ 30 0 0 38 0 0 0 30 0 0 $F(t)$ 254 286 300 526 532 556 606 786 856 874
 $t$ 1 2 3 4 5 6 7 8 9 10 $d_t$ 8 6 9 29 5 6 5 15 5 6 $X_t^\ast$ 30 $X_t^\ast$ 30 0 $X_t^\ast$ 30 0 0 $X_t^\ast$ 30 0 0 30 $X_t^\ast$ 30 0 0 30 0 $X_t^\ast$ 30 0 0 33 0 0 $X_t^\ast$ 30 0 0 38 0 0 0 $X_t^\ast$ 30 0 0 53 0 0 0 0 $X_t^\ast$ 30 0 0 58 0 0 0 0 0 $X_t^\ast$ 30 0 0 38 0 0 0 30 0 0 $F(t)$ 254 286 300 526 532 556 606 786 856 874
Summary of Computations of Example 2
 $t$ 1 2 3 4 5 6 7 8 9 10 $d_t$ 8 6 9 29 5 6 5 15 5 6 $X_t^\ast$ 30 $X_t^\ast$ 0 30 $X_t^\ast$ 0 0 30 $X_t^\ast$ 0 0 0 52 $X_t^\ast$ 0 0 0 57 0 $X_t^\ast$ 0 0 0 63 0 0 $X_t^\ast$ 0 0 0 68 0 0 0 $X_t^\ast$ 0 0 0 57 0 0 0 30 $X_t^\ast$ 0 0 0 57 0 0 0 31 0 $X_t^\ast$ 0 0 0 57 0 0 0 37 0 0 $F(t)$ 254 258 268 388 418 466 546 682 688 748
 $t$ 1 2 3 4 5 6 7 8 9 10 $d_t$ 8 6 9 29 5 6 5 15 5 6 $X_t^\ast$ 30 $X_t^\ast$ 0 30 $X_t^\ast$ 0 0 30 $X_t^\ast$ 0 0 0 52 $X_t^\ast$ 0 0 0 57 0 $X_t^\ast$ 0 0 0 63 0 0 $X_t^\ast$ 0 0 0 68 0 0 0 $X_t^\ast$ 0 0 0 57 0 0 0 30 $X_t^\ast$ 0 0 0 57 0 0 0 31 0 $X_t^\ast$ 0 0 0 57 0 0 0 37 0 0 $F(t)$ 254 258 268 388 418 466 546 682 688 748
Summary of Computations of Example 3
 $t$ 1 2 3 4 5 6 7 8 9 10 $d_t$ 8 6 9 29 5 6 5 15 5 6 $X_t^\ast ,O_t^\ast$ 0, 8 $X_t^\ast ,O_t^\ast$ 0, 14 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 30, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 34, 0 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 40, 0 0, 0 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 45, 0 0, 0 0, 0 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 45, 0 0, 0 0, 0 0, 0 0, 15 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 45, 0 0, 0 0, 0 0, 0 0, 20 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 45, 0 0, 0 0, 0 0, 0 0, 26 0, 0 0, 0 $F(t)$ 98 146 236 448 472 532 582 722 762 822
 $t$ 1 2 3 4 5 6 7 8 9 10 $d_t$ 8 6 9 29 5 6 5 15 5 6 $X_t^\ast ,O_t^\ast$ 0, 8 $X_t^\ast ,O_t^\ast$ 0, 14 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 30, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 34, 0 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 40, 0 0, 0 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 45, 0 0, 0 0, 0 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 45, 0 0, 0 0, 0 0, 0 0, 15 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 45, 0 0, 0 0, 0 0, 0 0, 20 0, 0 $X_t^\ast ,O_t^\ast$ 0, 23 0, 0 0, 0 45, 0 0, 0 0, 0 0, 0 0, 26 0, 0 0, 0 $F(t)$ 98 146 236 448 472 532 582 722 762 822
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