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A nonhomogeneous quasi-birth-death process approach for an $(s, S)$ policy for a perishable inventory system with retrial demands
May  2020, 16(3): 1435-1456. doi: 10.3934/jimo.2019010

## Forecast horizon of dynamic lot size model for perishable inventory with minimum order quantities

 1 University of Electronic Science and Technology of China, Chengdu 611731, China 2 School of Economic and Management, Tianjin University of Science and Technology, Tianjin 300222, China

* Corresponding author: Zirui Lan

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

We consider the dynamic lot size problem for perishable inventory under minimum order quantities. The stock deterioration rates and inventory costs depend on both the age of the stocks and their periods of order. Based on two structural properties of the optimal solution, we develop a dynamic programming algorithm to solve the problem without backlogging. We also extend the model by considering backlogging. By establishing the regeneration set, we give a sufficient condition for obtaining forecast horizon under without and with backlogging. Finally, based on a detailed test bed of instance, we obtain useful managerial insights on the impact of minimum order quantities and perishability of product and the costs on the length of forecast horizon.

Citation: Fuying Jing, Zirui Lan, Yang Pan. Forecast horizon of dynamic lot size model for perishable inventory with minimum order quantities. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1435-1456. doi: 10.3934/jimo.2019010
##### References:
 [1] 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. [2] C. Arena, C. Cannarozzo and M. R. Mazzola, Exploring the potential and the boundaries of the rolling horizon technique for the management of reservoir systems with over-year behaviour, Water Resources Management, 31 (2017), 867-884.  doi: 10.1007/s11269-016-1550-0. [3] A. Bardhan A, M. Dawande, S. Gavirneni, Y. P. Mu and S. Sethi, Forecast and rolling horizons under demand substitution and production changeovers: analysis and insights, IIE Transactions, 45 (2013), 323-340. [4] J. D. Blackburn and H. Kunreuther, Planning horizons for the dynamic lot size model with backlogging, Management Science, 21 (1974), 251-255.  doi: 10.1287/mnsc.21.3.251. [5] S. Bylka, S. P. Sethi and G. Sorger, Minimal forecast horizons in equipment replacement models with multiple technologies and general switching costs, Naval Research Logistics, 39 (1992), 487-507.  doi: 10.1002/1520-6750(199206)39:4<487::AID-NAV3220390405>3.0.CO;2-6. [6] S. Chand, V. N. Hsu, S. Sethi and V. Deshpande, A dynamic lot sizing problem with multiple customers: Customer-specific shipping and backlogging costs, IIE Transactions, 39 (2007), 1059-1069. [7] S. Chand, V. N. Hsu and S. Sethi, Forecast, solution, and rolling horizons in operations management problems: A classified bibliography, Manufacturing and Service Operations Management, 4 (2002), 25-43. [8] S. Chand and T. E. Morton, Minimal forecast horizon procedures for dynamic lot size models, Naval Research Logistics Quarterly, 33 (1986), 111-122.  doi: 10.1002/nav.3800330110. [9] S. Chand S, Se thi S P and J. M. Proth, Existence of forecast horizons in undisconted discrete time lot-size model, Operations Research, 38 (1990), 884-892.  doi: 10.1287/opre.38.5.884. [10] S. Chand, S. P. Sethi and G. Sorger, Forecast horizons in the discounted dynamic lot size model, Management Science, 38 (1992), 1034-1048. [11] S. Chand and S. P. Sethi, A dynamic lot sizing model with learning in setups, Operations Research, 38 (1990), 644-655.  doi: 10.1287/opre.38.4.644. [12] T. Cheevaprawatdomrong, I. E. Schochetman, R. L. Smith and A. Garcia, Solution and forecast horizons for infinite-horizon nonhomogeneous Markov Decision Process, Mathematics of Operations Research, 32 (2007), 51-72.  doi: 10.1287/moor.1060.0224. [13] T. Cheevaprawatdomrong and R. L. Smith, Infinite horizon production scheduling in time-varying systems under stochastic demand, Operations Research, 52 (2004), 105-115.  doi: 10.1287/opre.1030.0080. [14] 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. [15] M. Dawande, S. Gavirneni, Y. P. Mu, S. Sethi and C. Sriskandarajah, On the interaction between demand substitution and production changeovers, Manufacturing and Service Operations Management, 12 (2010), 682-691. [16] M. Dawande, S. Gavirneni, S. Naranpanawe and S. Sethi, Computing minimal forecast horizons: An integer programming approach, Journal of Mathematical Modelling and Algorithm, 5 (2006), 239-258.  doi: 10.1007/s10852-005-9012-3. [17] M. Dawande, S. Gavirneni, S. Naranpanawe and S. Sethi, Forecast horizons for a class of dynamic lot-size problems under discrete future demand, Operations Research, 55 (2007), 688-702.  doi: 10.1287/opre.1060.0378. [18] M. Dawande, S. Gavirneni, S. Naranpanawe and S. Sethi, Discrete forecast horizons for two-product variants of the dynamic lot-size problem, International Journal of Production Economics, 120 (2009), 430-436. [19] G. D. Eppen, F. J. Gould and B. P. Pashigian, Extensions of the planning horizon theorem in the dynamic lot size model, Management Science, 15 (1969), 268-277.  doi: 10.1287/mnsc.15.5.268. [20] A. Federgruen and M. Tzur, The dynamic lot-sizing model with backlogging: A simple O(nlogn) algorithm and minimal forecast horizon procedure, Naval Research Logistics, 40 (1993), 459-478.  doi: 10.1002/1520-6750(199306)40:4<459::AID-NAV3220400404>3.0.CO;2-8. [21] A. Federgruen and M. Tzur, Minimal forecast horizons and a new planning procedure for the general dynamic lot sizing model: Nervousness revisited, Operations Research, 42 (1994), 390-575.  doi: 10.1287/opre.42.3.456. [22] A. Federgruen and M. Tzur, Fast solution and detection of minimal forecast horizons in dynamic programs with a single indicator of the future: Applications to dynamic lot-sizing models, Management Science, 41 (1995), 749-936.  doi: 10.1287/mnsc.41.5.874. [23] A. Federgruen and M. Tzur, Detection of minimal forecast horizons in dynamic programs with multiple indicators of the future, Naval Research Logistics, 43 (1996), 169-189.  doi: 10.1002/(SICI)1520-6750(199603)43:2<169::AID-NAV2>3.0.CO;2-8. [24] A. Garcia, Forecast horizons for a class of dynamic Games, Journal of Optimization Theory and Applications, 122 (2004), 471-486.  doi: 10.1023/B:JOTA.0000042591.71156.89. [25] A. Ghate and R. L. Smith, Optimal backlogging over an infinite horizon under time-varying convex production and inventory costs, Manufacturing and Service Operations Management, 11 (2009), 191-372.  doi: 10.1287/msom.1080.0218. [26] 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. [27] 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. [28] 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" [Eur. J. Oper. Res., 222 (2012), 10–16], European Journal of Operational Research, 229 (2013), 279. [29] 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. [30] V. N. Hsu, Dynamic economic lot size model with perishable inventory, Management Science, 46 (2000), 1013-1169.  doi: 10.1287/mnsc.46.8.1159.12021. [31] V. N. Hsu, An economic lot size model for perishable products with age-dependent inventory and backorder costs, IIE Transactions, 35 (2003), 775-780.  doi: 10.1080/07408170304352. [32] F. Y. Jing and Z. R. Lan, Forecast horizon of multi-item dynamic lot size model with perishable inventory, PLOS ONE, 12 (2017), e0187725. doi: 10.1371/journal.pone.0187725. [33] R. A. Lundin and T. E. Morton, Planning horizons for the dynamic lot size model: Zabel vs. protective procedures and computational results, Operations Research, 23 (1975), 711-734.  doi: 10.1287/opre.23.4.711. [34] 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. [35] 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. [36] 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. [37] P. Pineyro and O. Viera, Inventory policies for the economic lot-sizing problem with remanufacturing and final disposal options, Journal of Industrial & Management Optimization, 5 (2009), 217-238.  doi: 10.3934/jimo.2009.5.217. [38] 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. [39] 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. [40] F. Z. Sargut and G. Içık, Dynamic economic lot size model with perishable inventory and capacity constraints, Applied Mathematical Modelling, 48 (2017), 806-820.  doi: 10.1016/j.apm.2017.02.024. [41] F. Z. Sargut and H. E. Romeijn, Capacitated requirements planning with pricing flexibility and general cost and revenue functions, Journal of Industrial & Management Optimization, 3 (2007), 87-98.  doi: 10.3934/jimo.2007.3.87. [42] S. Sethi and S. Chand, Multiple finite production rate dynamic lot size inventory models, Operations Research, 29 (1981), 931-944.  doi: 10.1287/opre.29.5.931. [43] R. L. Smith and R. Q. Zhang, Infinite horizon production planning in time-varying systems with convex production and inventory costs, Management Science, 44 (1998), 1167-1320.  doi: 10.1287/mnsc.44.9.1313. [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. Teyarachakul, S. Chand and M. Tzur, Lot sizing with learning and forgetting in setups: Analytical results and insights, Naval Research Logistics, 63 (2016), 93-108.  doi: 10.1002/nav.21681. [46] M. Tzur, Learning in setups: Analysis, minimal forecast horizons, and algorithms, Management Science, 42 (1996), 1627-1752.  doi: 10.1287/mnsc.42.12.1732. [47] S. Voß and D. L. Woodruff, Introduction to Computational Optimization Models for Production Planning in a Supply Chain, Second edition. Springer-Verlag, Berlin, 2006. [48] 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. [49] J. Y. You and X. M. Cai, Determining forecast and decision horizons for reservoir operations under hedging policies, Water Resources Research, 44 (2008), 2276-2283.  doi: 10.1061/40927(243)553. [50] J. Y. You and C. W. Yu, Theoretical error convergence of limited forecast horizon in optimal reservoir operating decisions, Water Resources Research, 49 (2013), 1728-1734.  doi: 10.1002/wrcr.20114. [51] E. Zabel, Some generalizations of an inventory planning horizon theorem, Management Science, 10 (1964), 397-600.  doi: 10.1287/mnsc.10.3.465. [52] 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), 459-472.  doi: 10.1287/mnsc.15.9.506. [53] T. T. G. Zhao, D. W. Yang, X. M. Cai, J. S. Zhao and H. Wang, Identifying effective forecast horizon for real-time reservoir operation under a limited inflow forecast, Water Resources Research, 48 (2012), 1540. doi: 10.1029/2011WR010623.

show all references

##### References:
 [1] 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. [2] C. Arena, C. Cannarozzo and M. R. Mazzola, Exploring the potential and the boundaries of the rolling horizon technique for the management of reservoir systems with over-year behaviour, Water Resources Management, 31 (2017), 867-884.  doi: 10.1007/s11269-016-1550-0. [3] A. Bardhan A, M. Dawande, S. Gavirneni, Y. P. Mu and S. Sethi, Forecast and rolling horizons under demand substitution and production changeovers: analysis and insights, IIE Transactions, 45 (2013), 323-340. [4] J. D. Blackburn and H. Kunreuther, Planning horizons for the dynamic lot size model with backlogging, Management Science, 21 (1974), 251-255.  doi: 10.1287/mnsc.21.3.251. [5] S. Bylka, S. P. Sethi and G. Sorger, Minimal forecast horizons in equipment replacement models with multiple technologies and general switching costs, Naval Research Logistics, 39 (1992), 487-507.  doi: 10.1002/1520-6750(199206)39:4<487::AID-NAV3220390405>3.0.CO;2-6. [6] S. Chand, V. N. Hsu, S. Sethi and V. Deshpande, A dynamic lot sizing problem with multiple customers: Customer-specific shipping and backlogging costs, IIE Transactions, 39 (2007), 1059-1069. [7] S. Chand, V. N. Hsu and S. Sethi, Forecast, solution, and rolling horizons in operations management problems: A classified bibliography, Manufacturing and Service Operations Management, 4 (2002), 25-43. [8] S. Chand and T. E. Morton, Minimal forecast horizon procedures for dynamic lot size models, Naval Research Logistics Quarterly, 33 (1986), 111-122.  doi: 10.1002/nav.3800330110. [9] S. Chand S, Se thi S P and J. M. Proth, Existence of forecast horizons in undisconted discrete time lot-size model, Operations Research, 38 (1990), 884-892.  doi: 10.1287/opre.38.5.884. [10] S. Chand, S. P. Sethi and G. Sorger, Forecast horizons in the discounted dynamic lot size model, Management Science, 38 (1992), 1034-1048. [11] S. Chand and S. P. Sethi, A dynamic lot sizing model with learning in setups, Operations Research, 38 (1990), 644-655.  doi: 10.1287/opre.38.4.644. [12] T. Cheevaprawatdomrong, I. E. Schochetman, R. L. Smith and A. Garcia, Solution and forecast horizons for infinite-horizon nonhomogeneous Markov Decision Process, Mathematics of Operations Research, 32 (2007), 51-72.  doi: 10.1287/moor.1060.0224. [13] T. Cheevaprawatdomrong and R. L. Smith, Infinite horizon production scheduling in time-varying systems under stochastic demand, Operations Research, 52 (2004), 105-115.  doi: 10.1287/opre.1030.0080. [14] 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. [15] M. Dawande, S. Gavirneni, Y. P. Mu, S. Sethi and C. Sriskandarajah, On the interaction between demand substitution and production changeovers, Manufacturing and Service Operations Management, 12 (2010), 682-691. [16] M. Dawande, S. Gavirneni, S. Naranpanawe and S. Sethi, Computing minimal forecast horizons: An integer programming approach, Journal of Mathematical Modelling and Algorithm, 5 (2006), 239-258.  doi: 10.1007/s10852-005-9012-3. [17] M. Dawande, S. Gavirneni, S. Naranpanawe and S. Sethi, Forecast horizons for a class of dynamic lot-size problems under discrete future demand, Operations Research, 55 (2007), 688-702.  doi: 10.1287/opre.1060.0378. [18] M. Dawande, S. Gavirneni, S. Naranpanawe and S. Sethi, Discrete forecast horizons for two-product variants of the dynamic lot-size problem, International Journal of Production Economics, 120 (2009), 430-436. [19] G. D. Eppen, F. J. Gould and B. P. Pashigian, Extensions of the planning horizon theorem in the dynamic lot size model, Management Science, 15 (1969), 268-277.  doi: 10.1287/mnsc.15.5.268. [20] A. Federgruen and M. Tzur, The dynamic lot-sizing model with backlogging: A simple O(nlogn) algorithm and minimal forecast horizon procedure, Naval Research Logistics, 40 (1993), 459-478.  doi: 10.1002/1520-6750(199306)40:4<459::AID-NAV3220400404>3.0.CO;2-8. [21] A. Federgruen and M. Tzur, Minimal forecast horizons and a new planning procedure for the general dynamic lot sizing model: Nervousness revisited, Operations Research, 42 (1994), 390-575.  doi: 10.1287/opre.42.3.456. [22] A. Federgruen and M. Tzur, Fast solution and detection of minimal forecast horizons in dynamic programs with a single indicator of the future: Applications to dynamic lot-sizing models, Management Science, 41 (1995), 749-936.  doi: 10.1287/mnsc.41.5.874. [23] A. Federgruen and M. Tzur, Detection of minimal forecast horizons in dynamic programs with multiple indicators of the future, Naval Research Logistics, 43 (1996), 169-189.  doi: 10.1002/(SICI)1520-6750(199603)43:2<169::AID-NAV2>3.0.CO;2-8. [24] A. Garcia, Forecast horizons for a class of dynamic Games, Journal of Optimization Theory and Applications, 122 (2004), 471-486.  doi: 10.1023/B:JOTA.0000042591.71156.89. [25] A. Ghate and R. L. Smith, Optimal backlogging over an infinite horizon under time-varying convex production and inventory costs, Manufacturing and Service Operations Management, 11 (2009), 191-372.  doi: 10.1287/msom.1080.0218. [26] 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. [27] 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. [28] 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" [Eur. J. Oper. Res., 222 (2012), 10–16], European Journal of Operational Research, 229 (2013), 279. [29] 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. [30] V. N. Hsu, Dynamic economic lot size model with perishable inventory, Management Science, 46 (2000), 1013-1169.  doi: 10.1287/mnsc.46.8.1159.12021. [31] V. N. Hsu, An economic lot size model for perishable products with age-dependent inventory and backorder costs, IIE Transactions, 35 (2003), 775-780.  doi: 10.1080/07408170304352. [32] F. Y. Jing and Z. R. Lan, Forecast horizon of multi-item dynamic lot size model with perishable inventory, PLOS ONE, 12 (2017), e0187725. doi: 10.1371/journal.pone.0187725. [33] R. A. Lundin and T. E. Morton, Planning horizons for the dynamic lot size model: Zabel vs. protective procedures and computational results, Operations Research, 23 (1975), 711-734.  doi: 10.1287/opre.23.4.711. [34] 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. [35] 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. [36] 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. [37] P. Pineyro and O. Viera, Inventory policies for the economic lot-sizing problem with remanufacturing and final disposal options, Journal of Industrial & Management Optimization, 5 (2009), 217-238.  doi: 10.3934/jimo.2009.5.217. [38] 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. [39] 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. [40] F. Z. Sargut and G. Içık, Dynamic economic lot size model with perishable inventory and capacity constraints, Applied Mathematical Modelling, 48 (2017), 806-820.  doi: 10.1016/j.apm.2017.02.024. [41] F. Z. Sargut and H. E. Romeijn, Capacitated requirements planning with pricing flexibility and general cost and revenue functions, Journal of Industrial & Management Optimization, 3 (2007), 87-98.  doi: 10.3934/jimo.2007.3.87. [42] S. Sethi and S. Chand, Multiple finite production rate dynamic lot size inventory models, Operations Research, 29 (1981), 931-944.  doi: 10.1287/opre.29.5.931. [43] R. L. Smith and R. Q. Zhang, Infinite horizon production planning in time-varying systems with convex production and inventory costs, Management Science, 44 (1998), 1167-1320.  doi: 10.1287/mnsc.44.9.1313. [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. Teyarachakul, S. Chand and M. Tzur, Lot sizing with learning and forgetting in setups: Analytical results and insights, Naval Research Logistics, 63 (2016), 93-108.  doi: 10.1002/nav.21681. [46] M. Tzur, Learning in setups: Analysis, minimal forecast horizons, and algorithms, Management Science, 42 (1996), 1627-1752.  doi: 10.1287/mnsc.42.12.1732. [47] S. Voß and D. L. Woodruff, Introduction to Computational Optimization Models for Production Planning in a Supply Chain, Second edition. Springer-Verlag, Berlin, 2006. [48] 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. [49] J. Y. You and X. M. Cai, Determining forecast and decision horizons for reservoir operations under hedging policies, Water Resources Research, 44 (2008), 2276-2283.  doi: 10.1061/40927(243)553. [50] J. Y. You and C. W. Yu, Theoretical error convergence of limited forecast horizon in optimal reservoir operating decisions, Water Resources Research, 49 (2013), 1728-1734.  doi: 10.1002/wrcr.20114. [51] E. Zabel, Some generalizations of an inventory planning horizon theorem, Management Science, 10 (1964), 397-600.  doi: 10.1287/mnsc.10.3.465. [52] 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), 459-472.  doi: 10.1287/mnsc.15.9.506. [53] T. T. G. Zhao, D. W. Yang, X. M. Cai, J. S. Zhao and H. Wang, Identifying effective forecast horizon for real-time reservoir operation under a limited inflow forecast, Water Resources Research, 48 (2012), 1540. doi: 10.1029/2011WR010623.
Median forecast horizon as a function of minimum order quantities
Median forecast horizon as a function of lifetime
Median forecast horizon as a function of backlogging cost
Median forecast horizon as a function of inventory holding cost
Summary of Computations of Example 1
 $t$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $d_t$ $6$ $8$ $9$ $12$ $11$ $7$ $26$ $x_t^\ast$ $25$ $x_t^\ast$ $25$ $0$ $x_t^\ast$ $25$ $0$ $0$ $x_t^\ast$ $35$ $0$ $0$ $0$ $x_t^\ast$ $25$ $0$ $0$ $25$ $0$ $x_t^\ast$ $25$ $0$ $0$ $32$ $0$ $0$ $x_t^\ast$ $25$ $0$ $0$ $32$ $0$ $0$ $26$ $C(t)$ $263$ $285$ $289$ $399$ $544$ $607$ $837$
 $t$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $d_t$ $6$ $8$ $9$ $12$ $11$ $7$ $26$ $x_t^\ast$ $25$ $x_t^\ast$ $25$ $0$ $x_t^\ast$ $25$ $0$ $0$ $x_t^\ast$ $35$ $0$ $0$ $0$ $x_t^\ast$ $25$ $0$ $0$ $25$ $0$ $x_t^\ast$ $25$ $0$ $0$ $32$ $0$ $0$ $x_t^\ast$ $25$ $0$ $0$ $32$ $0$ $0$ $26$ $C(t)$ $263$ $285$ $289$ $399$ $544$ $607$ $837$
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