[Contrôle optimal des stocks avec information partielle]
On présente un certain nombre de modèles de systèmes de stocks avec information partielle. Ils sont formalisés comme des problèmes de contrôle où l'état est une probabilité conditionnelle, dans un espace de dimension infinie. On introduit des probabilités non normalisées, permettant de transformer des équations non linéaires en équations linéaires. On peut alors montrer l'existence de feedbacks optimaux pour deux modèles où la demande et le stock sont partiellement observables. Dans un troisième modèle, le stock n'est pas observé, mais un stock antérieur est observé. Une statistique exhaustive est obtenue, et l'état est de dimension finie. On établit l'optimalité des politiques « stock de base », généralisant les modèles classiques avec information complète.
This Note introduces recent developments in the analysis of inventory systems with partial observations. The states of these systems are typically conditional distributions, which evolve in infinite dimensional spaces over time. Our analysis involves introducing unnormalized probabilities to transform nonlinear state transition equations to linear ones. With the linear equations, the existence of the optimal feedback policies are proved for two models where demand and inventory are partially observed. In a third model where the current inventory is not observed but a past inventory level is fully observed, a sufficient statistic is provided to serve as a state. The last model serves as an example where a partially observed model has a finite dimensional state. In that model, we also establish the optimality of the basestock policies, hence generalizing the corresponding classical models with full information.
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@article{CRMATH_2005__341_7_419_0,
author = {Bensoussan, Alain and \c{C}akany{\i}ld{\i}r{\i}m, Metin and Sethi, Suresh P.},
title = {On the optimal control of partially observed inventory systems},
journal = {Comptes Rendus. Math\'ematique},
pages = {419--426},
publisher = {Elsevier},
volume = {341},
number = {7},
year = {2005},
doi = {10.1016/j.crma.2005.08.003},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2005.08.003/}
}
TY - JOUR AU - Bensoussan, Alain AU - Çakanyıldırım, Metin AU - Sethi, Suresh P. TI - On the optimal control of partially observed inventory systems JO - Comptes Rendus. Mathématique PY - 2005 SP - 419 EP - 426 VL - 341 IS - 7 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2005.08.003/ DO - 10.1016/j.crma.2005.08.003 LA - en ID - CRMATH_2005__341_7_419_0 ER -
%0 Journal Article %A Bensoussan, Alain %A Çakanyıldırım, Metin %A Sethi, Suresh P. %T On the optimal control of partially observed inventory systems %J Comptes Rendus. Mathématique %D 2005 %P 419-426 %V 341 %N 7 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2005.08.003/ %R 10.1016/j.crma.2005.08.003 %G en %F CRMATH_2005__341_7_419_0
Bensoussan, Alain; Çakanyıldırım, Metin; Sethi, Suresh P. On the optimal control of partially observed inventory systems. Comptes Rendus. Mathématique, Tome 341 (2005) no. 7, pp. 419-426. doi : 10.1016/j.crma.2005.08.003. http://www.numdam.org/articles/10.1016/j.crma.2005.08.003/
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