American Institute of Mathematical Sciences

Advanced
January  2012, 8(1): 19-40. doi: 10.3934/jimo.2012.8.19

Optimal inventory control with fixed ordering cost for selling by internet auctions

Shuren Liu 1, , Qiying Hu 2, and  Yifan Xu 2,

1. 

School of Mathematics and Computational Science, Xiangtan University, Hunan, 411105, China
2. 

School of Management, Fudan University, Shanghai 200433

Received  July 2009 Revised  June 2011 Published  November 2011

We study an optimal inventory control problem for a seller to sell a replenishment product via sequential Internet auctions. At the beginning of each auction, the seller may purchase his good from an outside supplier with a fixed ordering cost. There is a holding cost for inventory and backordering is not allowed. We address the total expected discounted criteria in both finite and infinite horizons and the average criterion in an infinite horizon. We show that the classic $(j, J)$ policy is optimal for each criterion. Moreover, we obtain integer programming with bounded decision variables $j$ and $J$ for computing the optimal $(j, J)$ policies for both the discounted and average criteria in an infinite horizon. Finally, numerical results show that it is meaningful for the seller to reduce randomness in the number of buyers with certainly remaining the average number of arriving buyers, but to enhance randomness in the buyers' valuation.
Keywords: Markov decision process, Internet auction, J) policy., inventory control, (j.
Mathematics Subject Classification: Primary: 90B05; Secondary: 90C4.
Citation: Shuren Liu, Qiying Hu, Yifan Xu. Optimal inventory control with fixed ordering cost for selling by internet auctions. Journal of Industrial & Management Optimization, 2012, 8 (1) : 19-40. doi: 10.3934/jimo.2012.8.19
References:
[1]

K. J. Arrow, T. E. Harris and J. Marschak, Optimal inventory policy,, Econometrica, 19 (1951), 250.  doi: 10.2307/1906813.  Google Scholar

[2]

D. Bertsekas, "Dynamic Programming and Optimal Control,", Vol. 2, (1995).   Google Scholar

[3]

D. Blackwell, Discrete dynamic programming,, Annals of Mathematics Statistics, 33 (1962), 719.  doi: 10.1214/aoms/1177704593.  Google Scholar

[4]

S. Bollapragada and T. E. Morton, A simple heuristic for computing nonstationary $(s, S)$ policies,, Operations Research, 47 (1999), 576.  doi: 10.1287/opre.47.4.576.  Google Scholar

[5]

L. Caccetta and E. Mardaneh, Joint pricing and production planning for fixed priced multiple products with backorders,, Journal of Industrial and Management Optimization, 6 (2010), 123.   Google Scholar

[6]

F. Chen, Auctioning supply contracts,, Management Science, 53 (2007), 1562.  doi: 10.1287/mnsc.1070.0716.  Google Scholar

[7]

X. Chen and D. Simchi-Levi, Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: The finite horizon case,, Operations Research, 52 (2004), 887.  doi: 10.1287/opre.1040.0127.  Google Scholar

[8]

X. Chen and D. Simchi-Levi, Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: The infinite horizon case,, Mathematics of Operations Research, 29 (2004), 698.  doi: 10.1287/moor.1040.0093.  Google Scholar

[9]

Y. Chen, S. Ray and Y. Song, Optimal pricing and inventory control policy in periodic-review sysytems with fixed ordering cost and lost sales,, Naval Research Logistics, 53 (2006), 117.  doi: 10.1002/nav.20127.  Google Scholar

[10]

H. A. David, "Order Statistics,", 2nd edition, (1981).   Google Scholar

[11]

L. Du, Q. Hu and W. Yue, Analysis and evaluation for optimal allocation in sequential internet auction systems with reserve price,, Dynamics of Continuous, 12 (2005), 617.   Google Scholar

[12]

A. Federgruen and A. Heching, Combined pricing and inventory control under uncertainty,, Operations Research, 47 (1999), 454.  doi: 10.1287/opre.47.3.454.  Google Scholar

[13]

E. L. Feiberg and M. E. Lewis, Optimality inequalities for average cost Markov decision processes and the optimality of $(s, S)$ policies,, Working paper, (2006).   Google Scholar

[14]

Q. Feng, S. P. Sethi, H. Yan and H. Zhang, Optimality and nonoptimality of the base-stock policy in inventory problems with multiple delivery modes,, Journal of Industrial and Management Optimization, 2 (2006), 19.  doi: 10.3934/jimo.2006.2.19.  Google Scholar

[15]

Q. Hu and W. Yue, "Markov Decision Processes with Their Applications,", Advances in Mechanics and Mathematics, 14 (2008).   Google Scholar

[16]

W. T. Huh and G. Janakiraman, $(s, S)$ optimality in joint inventory-pricing control: An alternate approach,, Operations Research, 56 (2008), 783.  doi: 10.1287/opre.1070.0462.  Google Scholar

[17]

W. T. Huh and G. Janakiraman, Inventory management with auctions and other sales channels: Optimality of $(s, S)$ policies,, Management Science, 54 (2008), 139.  doi: 10.1287/mnsc.1070.0767.  Google Scholar

[18]

D. Iglehart, Optimality of $(s, S)$ policies in the infinite-horizon dynamic inventory problem,, Management Science, 9 (1963), 259.  doi: 10.1287/mnsc.9.2.259.  Google Scholar

[19]

D. Iglehart, Dynamic programming and the analysis of inventory problems,, in, (1963).   Google Scholar

[20]

E. Maskin and J. Riley, Optimal multi-unit auctions,, in, (2000), 312.   Google Scholar

[21]

R. Myerson, Optimal auction design,, Mathematics of Operations Research, 6 (1981), 58.  doi: 10.1287/moor.6.1.58.  Google Scholar

[22]

S. Nahmias, "Production and Operation Analysis,", 4th edition, (2001).   Google Scholar

[23]

E. L. Porteus, "Foundations of Stochastic Inventory Theory,", Stanford University Press, (2002).   Google Scholar

[24]

E. Pinker, A. Seidmann and Y. Vakrat, Using transaction data for the design of sequential, multi-unit, online auctions,, Working paper CIS-00-03, (2001), 00.   Google Scholar

[25]

E. Pinker, A. Seidmann and Y. Vakrat, Managing online auctions: Current business and research issues,, Management Science, 49 (2003), 1457.  doi: 10.1287/mnsc.49.11.1457.20584.  Google Scholar

[26]

H. Scarf, The optimality of $(s, S)$ policies in the dynamic inventory problem,, in, (1960), 196.   Google Scholar

[27]

A. Segev, C. Beam and J. Shanthikumar, Optimal design of internet-based auctions,, Information Technology and Mangement, 2 (2001), 121.  doi: 10.1023/A:1011411801246.  Google Scholar

[28]

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

[29]

Y. Song, S. Ray and T. Boyaci, Optimal dynamic joint inventory-pricing control for multiplicative demand with fixed order costs and lost sales,, Operations Research, 57 (2009), 245.  doi: 10.1287/opre.1080.0530.  Google Scholar

[30]

A. F. Veinott, On the optimality of $(s, S)$ inventory policies: New conditions and a new proof,, SIAM Journal on Applied Mathematics, 14 (1966), 1067.  doi: 10.1137/0114086.  Google Scholar

[31]

G. Vulcano, G. J. van Ryzin and C. Maglaras, Optimal dynamic auctions for revenue management,, Management Science, 48 (2002), 1388.  doi: 10.1287/mnsc.48.11.1388.269.  Google Scholar

[32]

R. J. Weber, Multiple-object auctions,, in, (1983), 165.   Google Scholar

[33]

C. A. Yano and S. M. Gilbert, Coordinated pricing and producion/procurement decisions, A review,, in, (2004).   Google Scholar

[34]

Y. S. Zheng, A simple proof for optimality of $(s, S)$ policies in the infinite-horizon inventory systems,, Journal of Applied Probability, 28 (1991), 802.  doi: 10.2307/3214683.  Google Scholar

[35]

Y. S. Zheng and A. Federgruen, Finding optimal $(s, S)$ policies is about as simple as evaluating a single policy,, Operations Research, 39 (1991), 654.  doi: 10.1287/opre.39.4.654.  Google Scholar

[36]

P. H. Zipkin, "Foundations of Inventory Management,", McGraw Hill, (2000).   Google Scholar

show all references

References:
[1]

K. J. Arrow, T. E. Harris and J. Marschak, Optimal inventory policy,, Econometrica, 19 (1951), 250.  doi: 10.2307/1906813.  Google Scholar

[2]

D. Bertsekas, "Dynamic Programming and Optimal Control,", Vol. 2, (1995).   Google Scholar

[3]

D. Blackwell, Discrete dynamic programming,, Annals of Mathematics Statistics, 33 (1962), 719.  doi: 10.1214/aoms/1177704593.  Google Scholar

[4]

S. Bollapragada and T. E. Morton, A simple heuristic for computing nonstationary $(s, S)$ policies,, Operations Research, 47 (1999), 576.  doi: 10.1287/opre.47.4.576.  Google Scholar

[5]

L. Caccetta and E. Mardaneh, Joint pricing and production planning for fixed priced multiple products with backorders,, Journal of Industrial and Management Optimization, 6 (2010), 123.   Google Scholar

[6]

F. Chen, Auctioning supply contracts,, Management Science, 53 (2007), 1562.  doi: 10.1287/mnsc.1070.0716.  Google Scholar

[7]

X. Chen and D. Simchi-Levi, Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: The finite horizon case,, Operations Research, 52 (2004), 887.  doi: 10.1287/opre.1040.0127.  Google Scholar

[8]

X. Chen and D. Simchi-Levi, Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: The infinite horizon case,, Mathematics of Operations Research, 29 (2004), 698.  doi: 10.1287/moor.1040.0093.  Google Scholar

[9]

Y. Chen, S. Ray and Y. Song, Optimal pricing and inventory control policy in periodic-review sysytems with fixed ordering cost and lost sales,, Naval Research Logistics, 53 (2006), 117.  doi: 10.1002/nav.20127.  Google Scholar

[10]

H. A. David, "Order Statistics,", 2nd edition, (1981).   Google Scholar

[11]

L. Du, Q. Hu and W. Yue, Analysis and evaluation for optimal allocation in sequential internet auction systems with reserve price,, Dynamics of Continuous, 12 (2005), 617.   Google Scholar

[12]

A. Federgruen and A. Heching, Combined pricing and inventory control under uncertainty,, Operations Research, 47 (1999), 454.  doi: 10.1287/opre.47.3.454.  Google Scholar

[13]

E. L. Feiberg and M. E. Lewis, Optimality inequalities for average cost Markov decision processes and the optimality of $(s, S)$ policies,, Working paper, (2006).   Google Scholar

[14]

Q. Feng, S. P. Sethi, H. Yan and H. Zhang, Optimality and nonoptimality of the base-stock policy in inventory problems with multiple delivery modes,, Journal of Industrial and Management Optimization, 2 (2006), 19.  doi: 10.3934/jimo.2006.2.19.  Google Scholar

[15]

Q. Hu and W. Yue, "Markov Decision Processes with Their Applications,", Advances in Mechanics and Mathematics, 14 (2008).   Google Scholar

[16]

W. T. Huh and G. Janakiraman, $(s, S)$ optimality in joint inventory-pricing control: An alternate approach,, Operations Research, 56 (2008), 783.  doi: 10.1287/opre.1070.0462.  Google Scholar

[17]

W. T. Huh and G. Janakiraman, Inventory management with auctions and other sales channels: Optimality of $(s, S)$ policies,, Management Science, 54 (2008), 139.  doi: 10.1287/mnsc.1070.0767.  Google Scholar

[18]

D. Iglehart, Optimality of $(s, S)$ policies in the infinite-horizon dynamic inventory problem,, Management Science, 9 (1963), 259.  doi: 10.1287/mnsc.9.2.259.  Google Scholar

[19]

D. Iglehart, Dynamic programming and the analysis of inventory problems,, in, (1963).   Google Scholar

[20]

E. Maskin and J. Riley, Optimal multi-unit auctions,, in, (2000), 312.   Google Scholar

[21]

R. Myerson, Optimal auction design,, Mathematics of Operations Research, 6 (1981), 58.  doi: 10.1287/moor.6.1.58.  Google Scholar

[22]

S. Nahmias, "Production and Operation Analysis,", 4th edition, (2001).   Google Scholar

[23]

E. L. Porteus, "Foundations of Stochastic Inventory Theory,", Stanford University Press, (2002).   Google Scholar

[24]

E. Pinker, A. Seidmann and Y. Vakrat, Using transaction data for the design of sequential, multi-unit, online auctions,, Working paper CIS-00-03, (2001), 00.   Google Scholar

[25]

E. Pinker, A. Seidmann and Y. Vakrat, Managing online auctions: Current business and research issues,, Management Science, 49 (2003), 1457.  doi: 10.1287/mnsc.49.11.1457.20584.  Google Scholar

[26]

H. Scarf, The optimality of $(s, S)$ policies in the dynamic inventory problem,, in, (1960), 196.   Google Scholar

[27]

A. Segev, C. Beam and J. Shanthikumar, Optimal design of internet-based auctions,, Information Technology and Mangement, 2 (2001), 121.  doi: 10.1023/A:1011411801246.  Google Scholar

[28]

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

[29]

Y. Song, S. Ray and T. Boyaci, Optimal dynamic joint inventory-pricing control for multiplicative demand with fixed order costs and lost sales,, Operations Research, 57 (2009), 245.  doi: 10.1287/opre.1080.0530.  Google Scholar

[30]

A. F. Veinott, On the optimality of $(s, S)$ inventory policies: New conditions and a new proof,, SIAM Journal on Applied Mathematics, 14 (1966), 1067.  doi: 10.1137/0114086.  Google Scholar

[31]

G. Vulcano, G. J. van Ryzin and C. Maglaras, Optimal dynamic auctions for revenue management,, Management Science, 48 (2002), 1388.  doi: 10.1287/mnsc.48.11.1388.269.  Google Scholar

[32]

R. J. Weber, Multiple-object auctions,, in, (1983), 165.   Google Scholar

[33]

C. A. Yano and S. M. Gilbert, Coordinated pricing and producion/procurement decisions, A review,, in, (2004).   Google Scholar

[34]

Y. S. Zheng, A simple proof for optimality of $(s, S)$ policies in the infinite-horizon inventory systems,, Journal of Applied Probability, 28 (1991), 802.  doi: 10.2307/3214683.  Google Scholar

[35]

Y. S. Zheng and A. Federgruen, Finding optimal $(s, S)$ policies is about as simple as evaluating a single policy,, Operations Research, 39 (1991), 654.  doi: 10.1287/opre.39.4.654.  Google Scholar

[36]

P. H. Zipkin, "Foundations of Inventory Management,", McGraw Hill, (2000).   Google Scholar

[1]

Tassilo Küpper. Bifurcation revisited along footprints of Jürgen Scheurle. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1031-1041. doi: 10.3934/dcdss.2020061
[2]

Yury Arlinskiĭ, Eduard Tsekanovskiĭ. Constant J-unitary factor and operator-valued transfer functions. Conference Publications, 2003, 2003 (Special) : 48-56. doi: 10.3934/proc.2003.2003.48
[3]

Thomas Hagen, Andreas Johann, Hans-Peter Kruse, Florian Rupp, Sebastian Walcher. Dynamical systems and geometric mechanics: A special issue in Honor of Jürgen Scheurle. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : ⅰ-ⅲ. doi: 10.3934/dcdss.20204i
[4]

Sung-Seok Ko. A nonhomogeneous quasi-birth-death process approach for an $ (s, S) $ policy for a perishable inventory system with retrial demands. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019009
[5]

Mathias Staudigl. A limit theorem for Markov decision processes. Journal of Dynamics & Games, 2014, 1 (4) : 639-659. doi: 10.3934/jdg.2014.1.639
[6]

Alberto Bressan, Yilun Jiang. The vanishing viscosity limit for a system of H-J equations related to a debt management problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 793-824. doi: 10.3934/dcdss.2018050
[7]

Pierluigi Colli, Gianni Gilardi, Dietmar Hömberg, Pavel Krejčí, Elisabetta Rocca. Preface: Special issue dedicated to Jürgen Sprekels on the occasion of his 65th birthday. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : i-ii. doi: 10.3934/dcds.2015.35.6i
[8]

Vladimir Ejov, Anatoli Torokhti. How to transform matrices $U_1, \ldots, U_p$ to matrices $V_1, \ldots, V_p$ so that $V_i V_j^T= {\mathbb O} $ if $ i \neq j $?. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 293-299. doi: 10.3934/naco.2012.2.293
[9]

Shan Gao, Jinting Wang. On a discrete-time GI$^X$/Geo/1/N-G queue with randomized working vacations and at most $J$ vacations. Journal of Industrial & Management Optimization, 2015, 11 (3) : 779-806. doi: 10.3934/jimo.2015.11.779
[10]

Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653
[11]

A. Mittal, N. Hemachandra. Learning algorithms for finite horizon constrained Markov decision processes. Journal of Industrial & Management Optimization, 2007, 3 (3) : 429-444. doi: 10.3934/jimo.2007.3.429
[12]

Alexander O. Brown, Christopher S. Tang. The impact of alternative performance measures on single-period inventory policy. Journal of Industrial & Management Optimization, 2006, 2 (3) : 297-318. doi: 10.3934/jimo.2006.2.297
[13]

Shu Zhang, Yuan Yuan. The Filippov equilibrium and sliding motion in an internet congestion control model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1189-1206. doi: 10.3934/dcdsb.2017058
[14]

Volker Rehbock, Iztok Livk. Optimal control of a batch crystallization process. Journal of Industrial & Management Optimization, 2007, 3 (3) : 585-596. doi: 10.3934/jimo.2007.3.585
[15]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005
[16]

Feyza Gürbüz, Panos M. Pardalos. A decision making process application for the slurry production in ceramics via fuzzy cluster and data mining. Journal of Industrial & Management Optimization, 2012, 8 (2) : 285-297. doi: 10.3934/jimo.2012.8.285
[17]

Harish Garg. Novel correlation coefficients under the intuitionistic multiplicative environment and their applications to decision-making process. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1501-1519. doi: 10.3934/jimo.2018018
[18]

Qiuli Liu, Xiaolong Zou. A risk minimization problem for finite horizon semi-Markov decision processes with loss rates. Journal of Dynamics & Games, 2018, 5 (2) : 143-163. doi: 10.3934/jdg.2018009
[19]

Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. Ergodic control for a mean reverting inventory model. Journal of Industrial & Management Optimization, 2018, 14 (3) : 857-876. doi: 10.3934/jimo.2017079
[20]

Jui-Jung Liao, Wei-Chun Lee, Kuo-Nan Huang, Yung-Fu Huang. Optimal ordering policy for a two-warehouse inventory model use of two-level trade credit. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1661-1683. doi: 10.3934/jimo.2017012

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences