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January  2013, 9(1): 13-30. doi: 10.3934/jimo.2013.9.13

Inventory policies for a partially observed supply capacity model

Qing Yang 1, , Shiji Song 1, and  Cheng Wu 1,

1. 

Department of Automation, Tsinghua University, Beijing 100084, China, China

Received  April 2011 Revised  April 2012 Published  December 2012

This paper considers a multi-period inventory problem with partially observed supply capacity in the lost sales case. Partially observed supply means that exact available supply in a period is observed only when the order quantity is not less than the supply capacity. Then, these observations are used to update the supply capacity distribution from one period to the next. For this inventory problem with partially observed supply information and random demand, we establish the inventory model according to a known Markov decision process(MDP) space. The existence of an optimal policy for this inventory problem is proved. Finally, some numerical examples considering Poisson distributed demand are given to verify the ability to find an optimal order quantity.
Keywords: Markov decision process., Partially observed supply-capacity, lost sales demand.
Mathematics Subject Classification: Primary: 90B05; Secondary: 90C4.
Citation: Qing Yang, Shiji Song, Cheng Wu. Inventory policies for a partially observed supply capacity model. Journal of Industrial and Management Optimization, 2013, 9 (1) : 13-30. doi: 10.3934/jimo.2013.9.13
References:
[1]

A. Bensoussan, M. Cakanyildirim, J. Minjárez-Sosa, A. Royal and S. Sethi, Inventory problems with partially observed demands and lost sales, Journal of Optimization Theory and Applications, 136 (2008), 321-340. doi: 10.1007/s10957-007-9311-0.

[2]

A. Bensoussan, M. Cakanyildirim, J. A. Minjárez-Sosa, S. P. Sethi and R. X. Shi, Partially observed inventory systems: The case of rain checks, SIAM Journal on Control and Optimization, 47 (2008), 2490-2519. doi: 10.1137/070688663.

[3]

A. Bensoussan, M. Cakanyildirim and S. P. Sethi, On the optimal control of partially observed inventory systems, Comptes Rendus Mathematique, 341 (2005), 419-426. doi: 10.1016/j.crma.2005.08.003.

[4]

A. Bensoussan, M. Cakanyildirim and S. P. Sethi, A multiperiod newsvendor problem with partially observed demand, Mathematics of Operations Research, 32 (2007), 322-344. doi: 10.1287/moor.1060.0236.

[5]

A. Bensoussan, M. Cakanyildirim and S. P. Sethi, Partially observed inventory systems: The case of zero-balance walk, SIAM Journal on Control and Optimization, 46 (2007), 176-209. doi: 10.1137/040620321.

[6]

A. Bensoussan, M. Cakanyildirim and S. P. Sethi, Censored newsvendor model revisited with unnormalized probabilities, Journal of Industrial and Management Optimization, 5 (2009), 391-402. doi: 10.3934/jimo.2009.5.391.

[7]

A. Bensoussan, M. Cakanyildirim and S. P. Sethi, A note on "The Censored Newsvendor and the Optimal Acquisition of Information", Operations Research, 57 (2009), 791-794. doi: 10.1287/opre.1080.0609.

[8]

E. Bayraktar and M. Ludkovski, Inventory management with partially observed nonstationary demand, Annals of Operations Research, 176 (2010), 37-39. doi: 10.1007/s10479-009-0513-8.

[9]

F. Cheng and S. P. Sethi, Optimality of state-dependent (s,S) policies in inventory models with Markov-modulated demand and lost sales, Production and Operations Management, 8 (1999), 183-192. doi: 10.1111/j.1937-5956.1999.tb00369.x.

[10]

G. Gallego and L. B. Toktay, All-or-nothing ordering under a capacity constraint, Operations Research, 52 (2004), 1001-1002. doi: 10.1287/opre.1040.0153.

[11]

H. Wang, B. Chen and H. Yan, Optimal inventory decisions in a multiperiod newsvendor problem with partially observed Markovian supply capacities, European Journal of Operational Research, 51 (2010), 502-517. doi: 10.1016/j.ejor.2009.05.042.

[12]

J. Gallien and L. M. Wein, A smart market for industrial procurement with capacity constraints, Management Science, 51 (2005), 76-91. doi: 10.1287/mnsc.1040.0230.

[13]

J. T. Treharne and C. R. Sox, Adaptive inventory control for nonstationary demand and partial information, Management Science, 48 (2002), 607-624. doi: 10.1287/mnsc.48.5.607.7807.

[14]

K. R. Kamath and T. P. M. Pakkala, A bayesian approach to a dynamic inventory model under an unknown demand distribution, Computers and Operations Research, 29 (2002), 403-422.

[15]

K. S. Azoury, Bayes solution to dynamic inventory models under unknown demand distribution, Management Science, 31 (1985), 1150-1160. doi: 10.1287/mnsc.31.9.1150.

[16]

M. Parlar, Y. Wang and Y. Gerchak, A periodic review inventory model with Markovian supply availability, International Journal of Production Economics, 42 (1995), 131-136. doi: 10.1016/0925-5273(95)00115-8.

[17]

R. Yin and K. Rajaram, Joint pricing and inventory control with a Markovian demand model, European Journal of Operational Research, 182 (2007), 113-126. doi: 10.1016/j.ejor.2006.06.054.

[18]

S. A. Conrad, Sales data and the estimation of demand, Operational Research Quarterly (1970-1977), 27 (1976), 123-127.

[19]

S. P. Sethi and F. Cheng, Optimality of (s, S) policies in inventory models with Markovian demand, Operations Research, 45 (1997), 931-939. doi: 10.1287/opre.45.6.931.

[20]

X. Ding, M. L. Puterman and B. Arnab, The censored newsvendor and the optimal acquisition of information, Operations Research, 50 (2002), 517-527. doi: 10.1287/opre.50.3.517.7752.

[21]

Y. Aviv and A. Pazgal, A partially observed markov decision process for dynamic pricing, Management Science, 5 (2005), 1400-1416. doi: 10.1287/mnsc.1050.0393.

show all references

References:
[1]

A. Bensoussan, M. Cakanyildirim, J. Minjárez-Sosa, A. Royal and S. Sethi, Inventory problems with partially observed demands and lost sales, Journal of Optimization Theory and Applications, 136 (2008), 321-340. doi: 10.1007/s10957-007-9311-0.

[2]

A. Bensoussan, M. Cakanyildirim, J. A. Minjárez-Sosa, S. P. Sethi and R. X. Shi, Partially observed inventory systems: The case of rain checks, SIAM Journal on Control and Optimization, 47 (2008), 2490-2519. doi: 10.1137/070688663.

[3]

A. Bensoussan, M. Cakanyildirim and S. P. Sethi, On the optimal control of partially observed inventory systems, Comptes Rendus Mathematique, 341 (2005), 419-426. doi: 10.1016/j.crma.2005.08.003.

[4]

A. Bensoussan, M. Cakanyildirim and S. P. Sethi, A multiperiod newsvendor problem with partially observed demand, Mathematics of Operations Research, 32 (2007), 322-344. doi: 10.1287/moor.1060.0236.

[5]

A. Bensoussan, M. Cakanyildirim and S. P. Sethi, Partially observed inventory systems: The case of zero-balance walk, SIAM Journal on Control and Optimization, 46 (2007), 176-209. doi: 10.1137/040620321.

[6]

A. Bensoussan, M. Cakanyildirim and S. P. Sethi, Censored newsvendor model revisited with unnormalized probabilities, Journal of Industrial and Management Optimization, 5 (2009), 391-402. doi: 10.3934/jimo.2009.5.391.

[7]

A. Bensoussan, M. Cakanyildirim and S. P. Sethi, A note on "The Censored Newsvendor and the Optimal Acquisition of Information", Operations Research, 57 (2009), 791-794. doi: 10.1287/opre.1080.0609.

[8]

E. Bayraktar and M. Ludkovski, Inventory management with partially observed nonstationary demand, Annals of Operations Research, 176 (2010), 37-39. doi: 10.1007/s10479-009-0513-8.

[9]

F. Cheng and S. P. Sethi, Optimality of state-dependent (s,S) policies in inventory models with Markov-modulated demand and lost sales, Production and Operations Management, 8 (1999), 183-192. doi: 10.1111/j.1937-5956.1999.tb00369.x.

[10]

G. Gallego and L. B. Toktay, All-or-nothing ordering under a capacity constraint, Operations Research, 52 (2004), 1001-1002. doi: 10.1287/opre.1040.0153.

[11]

H. Wang, B. Chen and H. Yan, Optimal inventory decisions in a multiperiod newsvendor problem with partially observed Markovian supply capacities, European Journal of Operational Research, 51 (2010), 502-517. doi: 10.1016/j.ejor.2009.05.042.

[12]

J. Gallien and L. M. Wein, A smart market for industrial procurement with capacity constraints, Management Science, 51 (2005), 76-91. doi: 10.1287/mnsc.1040.0230.

[13]

J. T. Treharne and C. R. Sox, Adaptive inventory control for nonstationary demand and partial information, Management Science, 48 (2002), 607-624. doi: 10.1287/mnsc.48.5.607.7807.

[14]

K. R. Kamath and T. P. M. Pakkala, A bayesian approach to a dynamic inventory model under an unknown demand distribution, Computers and Operations Research, 29 (2002), 403-422.

[15]

K. S. Azoury, Bayes solution to dynamic inventory models under unknown demand distribution, Management Science, 31 (1985), 1150-1160. doi: 10.1287/mnsc.31.9.1150.

[16]

M. Parlar, Y. Wang and Y. Gerchak, A periodic review inventory model with Markovian supply availability, International Journal of Production Economics, 42 (1995), 131-136. doi: 10.1016/0925-5273(95)00115-8.

[17]

R. Yin and K. Rajaram, Joint pricing and inventory control with a Markovian demand model, European Journal of Operational Research, 182 (2007), 113-126. doi: 10.1016/j.ejor.2006.06.054.

[18]

S. A. Conrad, Sales data and the estimation of demand, Operational Research Quarterly (1970-1977), 27 (1976), 123-127.

[19]

S. P. Sethi and F. Cheng, Optimality of (s, S) policies in inventory models with Markovian demand, Operations Research, 45 (1997), 931-939. doi: 10.1287/opre.45.6.931.

[20]

X. Ding, M. L. Puterman and B. Arnab, The censored newsvendor and the optimal acquisition of information, Operations Research, 50 (2002), 517-527. doi: 10.1287/opre.50.3.517.7752.

[21]

Y. Aviv and A. Pazgal, A partially observed markov decision process for dynamic pricing, Management Science, 5 (2005), 1400-1416. doi: 10.1287/mnsc.1050.0393.

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