Comptes Rendus
no. 7
Optimal Control
On the optimal control of partially observed inventory systems
[Contrôle optimal des stocks avec information partielle]

Présenté par : Alain Bensoussan

Alain Bensoussan1 ; Metin Çakanyıldırım1 ; Suresh P. Sethi1
1 International Center for Decision and Risk Analysis, School of Management, P.O. Box 830688, SM 30, University of Texas at Dallas, Richardson, TX 75083-0688, USA
Comptes Rendus. Mathématique, Volume 341 (2005) no. 7, pp. 419-426.

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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Accepté le :
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DOI : 10.1016/j.crma.2005.08.003
Alain Bensoussan 1 ; Metin Çakanyıldırım 1 ; Suresh P. Sethi 1

1 International Center for Decision and Risk Analysis, School of Management, P.O. Box 830688, SM 30, University of Texas at Dallas, Richardson, TX 75083-0688, USA 
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Alain Bensoussan; Metin Çakanyıldırım; Suresh P. Sethi. On the optimal control of partially observed inventory systems. Comptes Rendus. Mathématique, Volume 341 (2005) no. 7, pp. 419-426. doi : 10.1016/j.crma.2005.08.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.08.003/

[1] K.J. Arrow; S. Karlin; H.E. Scarf Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, California, 1958

[2] A. Bensoussan Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, UK, 1992

[3] A. Bensoussan, M. Çakanyıldırım, S.P. Sethi, Partially observed inventory systems: The case of zero balance walk, Working paper SOM 200548, School of Management, University of Texas at Dallas, TX, 2004

[4] A. Bensoussan, M. Çakanyıldırım, S.P. Sethi, Optimality of base stock and (s,S) policies for inventory problems with information delays, Working paper SOM 200547, School of Management, University of Texas at Dallas, TX, 2005

[5] A. Bensoussan, M. Çakanyıldırım, S.P. Sethi, A multiperiod newsvendor problem with partially observed demand, Working paper SOM 200550, School of Management, University of Texas at Dallas, TX, 2005

[6] A. Bensoussan, M. Çakanyıldırım, S.P. Sethi, Optimal ordering policies for inventory problems with dynamic information delays, Working paper 200560, School of Management, University of Texas at Dallas, TX, 2005

[7] X. Ding; M.L. Puterman; A. Bisi The censored newsvendor and the optimal acquisition of information, Oper. Res., Volume 50 (2002) no. 3, pp. 517-527

[8] M.L. Fisher; A. Raman; A.S. McClelland Rocket science retailing is almost here: Are you ready?, Harvard Business Review (July–August 2000), pp. 115-124

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[10] H.J. Kushner Dynamic equations for nonlinear filtering, J. Differential Equations, Volume 3 (1967), pp. 179-190

[11] M. Zakai On the optimal filtering of diffusion processes, Z. Wahrsch. Verw. Gebiete, Volume 11 (1969), pp. 230-243

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