The subject is a deterministic continuous-time continuous-state inventory control model. Stock is replenished by ordering from one of a number of suppliers incurring a different cost per item and a different set-up cost. Taking the cost of procurement into account, the objective is to minimize the total discounted cost over an infinite planning horizon. The size of the order that is to be placed and the supplier with which it is to be placed are to be decided. Earlier studies of the problem have relied substantially on the assumption that the set-up cost of every supplier is strictly positive. Removing this restriction calls for a significant modification of the adopted approach. This is realized in the present study. It is shown that there is a stable unique optimal policy of a type that encompasses (s, S) and generalized (s, S) policies. Conditions that are necessary and sufficient for it to reduce to each of these types are established. The case of two suppliers is studied in detail, properties of the solution are investigated, numerical examples illustrating various aspects are included, and the connection with antecedent results is assessed.
Keywords: Optimal inventory policy, quasi-variational inequality, ($$, $$) policy, generalized ($$, $$) policy, hyper-generalized ($$, $$) policy
@article{RO_2022__56_3_1453_0,
author = {Gilding, Brian H.},
title = {Optimal policies for a deterministic continuous-time inventory model with several suppliers: when a supplier incurs no set-up cost},
journal = {RAIRO. Operations Research},
pages = {1453--1490},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {3},
doi = {10.1051/ro/2022059},
mrnumber = {4437999},
zbl = {1493.90008},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ro/2022059/}
}
TY - JOUR AU - Gilding, Brian H. TI - Optimal policies for a deterministic continuous-time inventory model with several suppliers: when a supplier incurs no set-up cost JO - RAIRO. Operations Research PY - 2022 SP - 1453 EP - 1490 VL - 56 IS - 3 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ro/2022059/ DO - 10.1051/ro/2022059 LA - en ID - RO_2022__56_3_1453_0 ER -
%0 Journal Article %A Gilding, Brian H. %T Optimal policies for a deterministic continuous-time inventory model with several suppliers: when a supplier incurs no set-up cost %J RAIRO. Operations Research %D 2022 %P 1453-1490 %V 56 %N 3 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ro/2022059/ %R 10.1051/ro/2022059 %G en %F RO_2022__56_3_1453_0
Gilding, Brian H. Optimal policies for a deterministic continuous-time inventory model with several suppliers: when a supplier incurs no set-up cost. RAIRO. Operations Research, Tome 56 (2022) no. 3, pp. 1453-1490. doi: 10.1051/ro/2022059
[1] , and , Optimal inventory policy. Econometrica 19 (1951) 250–272. | MR | Zbl | DOI
[2] , and (editors), Studies in the Mathematical Theory of Inventory and Production. Stanford University Press, Stanford, CA (1958). | MR | Zbl
[3] , A continuous time inventory model. J. Appl. Probab. 3 (1966) 538–549. | MR | Zbl | DOI
[4] , and , Optimal policies for inventory systems with concave ordering costs. Nav. Res. Logistics 65 (2018) 291–302. | MR | Zbl | DOI
[5] and , Optimal policies for a deterministic continuous-time inventory model with several suppliers. RAIRO: Oper. Res. 55 (2021) S947–S966. | MR | Zbl | Numdam | DOI
[6] and , Optimal policies for a deterministic continuous-time inventory model with several suppliers: a hyper-generalized (s, S) policy. RAIRO: Oper. Res. 55 (2021) 1841–1863. | MR | Zbl | Numdam | DOI
[7] and , Optimality of policies for jump inventory models. Math. Methods Oper. Res. 76 (2012) 377–393. | MR | Zbl | DOI
[8] , Dynamic Programming and Inventory Control. IOS Press, Amsterdam (2011). | MR | Zbl
[9] , and , Introduction to Operations Research. John Wiley & Sons, New York (1957). | MR | Zbl
[10] , An analysis of vendor selection systems and decisions. J. Purchasing 2 (1966) 5–17. | DOI
[11] , and , Optimal inventory policies with two suppliers. Oper. Res. 54 (2006) 389–393. | Zbl | DOI
[12] and , Recent trends in modeling of deteriorating inventory. Eur. J. Oper. Res. 134 (2001) 1–16. | MR | Zbl | DOI
[13] and , Analysis of Inventory Systems. Prentice-Hall, Englewood Cliffs, NJ (1963). | Zbl
[14] , and , Optimal ordering policy for inventory systems with quantity-dependent setup costs. Math. Oper. Res. 42 (2017) 979–1006. | MR | Zbl | DOI
[15] , , and , A mathematical model for optimal inventory policies with backlog sales. Int. J. Traffic Transp. Eng. 11 (2021) 323–340. | DOI
[16] , and , A measure approach for continuous inventory models: discounted cost criterion. SIAM J. Control Optim. 53 (2015) 2100–2140. | MR | Zbl | DOI
[17] , and , Continuous inventory models of diffusion type: long-term average cost criterion. Ann. Appl. Probab. 27 (2017) 1831–1885. | MR | Zbl | DOI
[18] , and , A weak convergence approach to inventory control using a long-term average criterion. Adv. Appl. Probab. 50 (2018) 1032–1074. | MR | Zbl | DOI
[19] and , Introduction to Operations Research. Holden-Day, San Francisco (1967). | Zbl
[20] , Optimality of policies in the infinite horizon dynamic inventory problem. Manage. Sci. 9 (1963) 259–267. | DOI
[21] , and , Optimal inventory control with jump diffusion and nonlinear dynamics in the demand. SIAM J. Control Optim. 56 (2018) 53–74. | MR | Zbl | DOI
[22] , Multiple-supplier inventory models in supply chain management: a review. Int. J. Prod. Econ. 81–82 (2003) 265–279. | DOI
[23] , Inventory Systems. John Wiley & Sons, New York (1966).
[24] , and , Optimality of policies in EOQ models with general cost structures. Int. J. Prod. Econ. 187 (2017) 216–228. | DOI
[25] , and , Optimality of inventory policies under renewal demand and general cost structures. Prod. Oper. Manage. 27 (2018) 368–383. | DOI
[26] , On the optimality of generalized policies. Manage. Sci. 17 (1971) 411–426. | Zbl | DOI
[27] , The optimality of generalized policies under uniform demand densities. Manage. Sci. 18 (1972) 644–646. | MR | Zbl | DOI
[28] , Foundations of Stochastic Inventory Theory. Stanford University Press, Stanford, CA (2002). | DOI
[29] , A note on measurement of utility. Rev. Econ. Stud. 4 (1937) 155–161. | DOI
[30] , and , Operations Research Methods and Problems. John Wiley & Sons, New York, (1959). | MR | Zbl
[31] , The optimality of policies in the dynamic inventory problem. In: Mathematical Methods in the Social Sciences 1959, edited by , and . Stanford University Press, Stanford, CA (1960) 196–202. | MR | Zbl
[32] , A solvable one-dimensional model of a diffusion inventory system. Math. Oper. Res. 11 (1986) 125–133. | MR | Zbl | DOI
[33] , and , Typology and literature review on multiple supplier inventory control models. Eur. J. Oper. Res. 293 (2021) 1–23. | MR | Zbl | DOI
[34] , Inventory models with inventory-level-dependent demand: a comprehensive review and unifying theory. Eur. J. Oper. Res. 162 (2005) 792–804. | Zbl | DOI
[35] , On the optimality of inventory policies: new conditions and a new proof. SIAM J. Appl. Math. 14 (1966) 1067–1083. | MR | Zbl | DOI
[36] , and , Vendor selection criteria and methods. Eur. J. Oper. Res. 50 (1991) 2–18. | Zbl | DOI
[37] , and , Impulse control with discontinuous setup costs: discounted cost criterion. SIAM J. Control Optim. 59 (2021) 267–295. | MR | Zbl | DOI
[38] , and , Optimal control policy for a Brownian inventory system with concave ordering cost. J. Appl. Probab. 52 (2015) 909–925. | MR | Zbl | DOI
[39] , and , Optimal policies for Brownian inventory systems with a piecewise linear ordering cost. IEEE Trans. Autom. Control 62 (2017) 3235–3248. | MR | Zbl | DOI
[40] , A note on the optimality of policies in inventory theory. Manage. Sci. 9 (1962) 123–125. | DOI
[41] , Foundations of Inventory Management. McGraw–Hill, Boston, MA, (2000). | Zbl
Cité par Sources :
