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  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 & Management Optimization, 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. ArenaC. 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 AM. DawandeS. GavirneniY. 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. BylkaS. 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. ChandV. N. HsuS. 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. ChandV. 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 SSe 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. ChandS. 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. CheevaprawatdomrongI. E. SchochetmanR. 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. DawandeS. GavirneniY. P. MuS. Sethi and C. Sriskandarajah, On the interaction between demand substitution and production changeovers, Manufacturing and Service Operations Management, 12 (2010), 682-691.

[16]

M. DawandeS. GavirneniS. 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. DawandeS. GavirneniS. 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. DawandeS. GavirneniS. 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. EppenF. 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. HellionF. 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. HellionF. 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. TeyarachakulS. 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. ArenaC. 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 AM. DawandeS. GavirneniY. 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. BylkaS. 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. ChandV. N. HsuS. 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. ChandV. 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 SSe 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. ChandS. 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. CheevaprawatdomrongI. E. SchochetmanR. 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. DawandeS. GavirneniY. P. MuS. Sethi and C. Sriskandarajah, On the interaction between demand substitution and production changeovers, Manufacturing and Service Operations Management, 12 (2010), 682-691.

[16]

M. DawandeS. GavirneniS. 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. DawandeS. GavirneniS. 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. DawandeS. GavirneniS. 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. EppenF. 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. HellionF. 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. HellionF. 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. TeyarachakulS. 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.

Figure 1.  Median forecast horizon as a function of minimum order quantities
Figure 2.  Median forecast horizon as a function of lifetime
Figure 3.  Median forecast horizon as a function of backlogging cost
Figure 4.  Median forecast horizon as a function of inventory holding cost
Table 1.  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 $
[1]

M. M. Ali, L. Masinga. A nonlinear optimization model for optimal order quantities with stochastic demand rate and price change. Journal of Industrial & Management Optimization, 2007, 3 (1) : 139-154. doi: 10.3934/jimo.2007.3.139

[2]

Daniele Bartoli, Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco. The minimum order of complete caps in $PG(4,4)$. Advances in Mathematics of Communications, 2011, 5 (1) : 37-40. doi: 10.3934/amc.2011.5.37

[3]

Carlos Munuera, Fernando Torres. A note on the order bound on the minimum distance of AG codes and acute semigroups. Advances in Mathematics of Communications, 2008, 2 (2) : 175-181. doi: 10.3934/amc.2008.2.175

[4]

Giovanni Colombo, Thuy T. T. Le. Higher order discrete controllability and the approximation of the minimum time function. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4293-4322. doi: 10.3934/dcds.2015.35.4293

[5]

Andrei V. Dmitruk, Alexander M. Kaganovich. Quadratic order conditions for an extended weak minimum in optimal control problems with intermediate and mixed constraints. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 523-545. doi: 10.3934/dcds.2011.29.523

[6]

Zhenwei Luo, Jinting Wang. The optimal price discount, order quantity and minimum quantity in newsvendor model with group purchase. Journal of Industrial & Management Optimization, 2015, 11 (1) : 1-11. doi: 10.3934/jimo.2015.11.1

[7]

Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control & Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012

[8]

Bin Dan, Huali Gao, Yang Zhang, Ru Liu, Songxuan Ma. Integrated order acceptance and scheduling decision making in product service supply chain with hard time windows constraints. Journal of Industrial & Management Optimization, 2018, 14 (1) : 165-182. doi: 10.3934/jimo.2017041

[9]

T. Diogo, N. B. Franco, P. Lima. High order product integration methods for a Volterra integral equation with logarithmic singular kernel. Communications on Pure & Applied Analysis, 2004, 3 (2) : 217-235. doi: 10.3934/cpaa.2004.3.217

[10]

José M. Amigó, Karsten Keller, Valentina A. Unakafova. On entropy, entropy-like quantities, and applications. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3301-3343. doi: 10.3934/dcdsb.2015.20.3301

[11]

Sonomi Kakizaki, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura. Conserved quantities of the integrable discrete hungry systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 889-899. doi: 10.3934/dcdss.2015.8.889

[12]

Rafael G. L. D'Oliveira, Marcelo Firer. Minimum dimensional Hamming embeddings. Advances in Mathematics of Communications, 2017, 11 (2) : 359-366. doi: 10.3934/amc.2017029

[13]

Jake Bouvrie, Boumediene Hamzi. Kernel methods for the approximation of some key quantities of nonlinear systems. Journal of Computational Dynamics, 2017, 4 (1&2) : 1-19. doi: 10.3934/jcd.2017001

[14]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072

[15]

Guodong Yi, Xiaohong Chen, Chunqiao Tan. Optimal pricing of perishable products with replenishment policy in the presence of strategic consumers. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2018112

[16]

Stefano Luzzatto, Marks Ruziboev. Young towers for product systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1465-1491. doi: 10.3934/dcds.2016.36.1465

[17]

Nir Avni, Benjamin Weiss. Generating product systems. Journal of Modern Dynamics, 2010, 4 (2) : 257-270. doi: 10.3934/jmd.2010.4.257

[18]

Petr Lisoněk, Layla Trummer. Algorithms for the minimum weight of linear codes. Advances in Mathematics of Communications, 2016, 10 (1) : 195-207. doi: 10.3934/amc.2016.10.195

[19]

Victor Berdichevsky. Distribution of minimum values of stochastic functionals. Networks & Heterogeneous Media, 2008, 3 (3) : 437-460. doi: 10.3934/nhm.2008.3.437

[20]

Giovanni Colombo, Khai T. Nguyen. On the minimum time function around the origin. Mathematical Control & Related Fields, 2013, 3 (1) : 51-82. doi: 10.3934/mcrf.2013.3.51

2017 Impact Factor: 0.994

Metrics

  • PDF downloads (8)
  • HTML views (155)
  • Cited by (0)

Other articles
by authors

[Back to Top]