# American Institute of Mathematical Sciences

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

 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.
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), 250-272. doi: 10.2307/1906813.  Google Scholar [2] D. Bertsekas, "Dynamic Programming and Optimal Control," Vol. 2, 1st edition, Athena Scientific, Belmont, MA, 1995. Google Scholar [3] D. Blackwell, Discrete dynamic programming, Annals of Mathematics Statistics, 33 (1962), 719-726. 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), 576-584. 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), 123-147.  Google Scholar [6] F. Chen, Auctioning supply contracts, Management Science, 53 (2007), 1562-1576. 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), 887-896. 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), 698-723. 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), 117-136. doi: 10.1002/nav.20127.  Google Scholar [10] H. A. David, "Order Statistics," 2nd edition, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 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, Discrete and Impulsive System, Series B: Application and Algorithms, 12 (2005), 617-631.  Google Scholar [12] A. Federgruen and A. Heching, Combined pricing and inventory control under uncertainty, Operations Research, 47 (1999), 454-475. 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, Technical Report TR1442, Cornell University, 2006. Available from: http://legacy.orie.cornell.edu/techreports/TR1442.pdf. 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), 19-42. 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, Springer, New York, 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), 783-790. 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), 139-150. 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), 259-267. doi: 10.1287/mnsc.9.2.259.  Google Scholar [19] D. Iglehart, Dynamic programming and the analysis of inventory problems, in "Multistage Inventory Models and Technique" (eds. H. Scarf, D. Gilford and M. Shelly), Chap. 1, Stanford, 1963. Google Scholar [20] E. Maskin and J. Riley, Optimal multi-unit auctions, in "The Economic Theory of Auctions" (ed. P. Klemperer), Vol. 2, E. Elgar Publishing, U. K., (2000), 312-336. Google Scholar [21] R. Myerson, Optimal auction design, Mathematics of Operations Research, 6 (1981), 58-73. doi: 10.1287/moor.6.1.58.  Google Scholar [22] S. Nahmias, "Production and Operation Analysis," 4th edition, McGraw-Hill/Irwin, Boston, 2001. Google Scholar [23] E. L. Porteus, "Foundations of Stochastic Inventory Theory," Stanford University Press, Stanford, California, Stanford University Press, Stanford, California, 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, William E. Simon Graduate School of Business Administration, University of Rochaster, Rochaster, NY, 2001. Google Scholar [25] E. Pinker, A. Seidmann and Y. Vakrat, Managing online auctions: Current business and research issues, Management Science, 49 (2003), 1457-1484. 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 "Mathematical Methods in the Social Sciences," 1959, Stanford University Press, Stanford, California, (1960), 196-202.  Google Scholar [27] A. Segev, C. Beam and J. Shanthikumar, Optimal design of internet-based auctions, Information Technology and Mangement, 2 (2001), 121-163. 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), 931-939. 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), 245-250. 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), 1067-1083. 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), 1388-1407. doi: 10.1287/mnsc.48.11.1388.269.  Google Scholar [32] R. J. Weber, Multiple-object auctions, in "Auctions, Bidding, and Contracting: Uses and Theory" (eds. Richard Engelbrechtp-Wiggans, Martin Shubik and Robert M. Stark), Chapter 3, New York University Press, New York, 165-191; also in "The Economics Theory of Auctions" (ed. Paul Klemperer), Vol. II, Edward Elgar Publishing Limited, Cheltenham, UK / Edward Elgar Publishing, Inc., Northampton, USA, (1983), 240-266. Google Scholar [33] C. A. Yano and S. M. Gilbert, Coordinated pricing and producion/procurement decisions, A review, in "Managing Business Interfaces: Marketing, Engineering, and Manufacturing Perspectives" (eds. A. K. Chakravarty and J. Eliashberg), Kluwer Academic Publshers, Boston, Massachusetts, 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), 802-810. 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), 654-665. doi: 10.1287/opre.39.4.654.  Google Scholar [36] P. H. Zipkin, "Foundations of Inventory Management," McGraw Hill, Boston, 2000. Google Scholar

show all references

##### References:
 [1] K. J. Arrow, T. E. Harris and J. Marschak, Optimal inventory policy, Econometrica, 19 (1951), 250-272. doi: 10.2307/1906813.  Google Scholar [2] D. Bertsekas, "Dynamic Programming and Optimal Control," Vol. 2, 1st edition, Athena Scientific, Belmont, MA, 1995. Google Scholar [3] D. Blackwell, Discrete dynamic programming, Annals of Mathematics Statistics, 33 (1962), 719-726. 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), 576-584. 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), 123-147.  Google Scholar [6] F. Chen, Auctioning supply contracts, Management Science, 53 (2007), 1562-1576. 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), 887-896. 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), 698-723. 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), 117-136. doi: 10.1002/nav.20127.  Google Scholar [10] H. A. David, "Order Statistics," 2nd edition, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 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, Discrete and Impulsive System, Series B: Application and Algorithms, 12 (2005), 617-631.  Google Scholar [12] A. Federgruen and A. Heching, Combined pricing and inventory control under uncertainty, Operations Research, 47 (1999), 454-475. 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, Technical Report TR1442, Cornell University, 2006. Available from: http://legacy.orie.cornell.edu/techreports/TR1442.pdf. 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), 19-42. 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, Springer, New York, 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), 783-790. 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), 139-150. 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), 259-267. doi: 10.1287/mnsc.9.2.259.  Google Scholar [19] D. Iglehart, Dynamic programming and the analysis of inventory problems, in "Multistage Inventory Models and Technique" (eds. H. Scarf, D. Gilford and M. Shelly), Chap. 1, Stanford, 1963. Google Scholar [20] E. Maskin and J. Riley, Optimal multi-unit auctions, in "The Economic Theory of Auctions" (ed. P. Klemperer), Vol. 2, E. Elgar Publishing, U. K., (2000), 312-336. Google Scholar [21] R. Myerson, Optimal auction design, Mathematics of Operations Research, 6 (1981), 58-73. doi: 10.1287/moor.6.1.58.  Google Scholar [22] S. Nahmias, "Production and Operation Analysis," 4th edition, McGraw-Hill/Irwin, Boston, 2001. Google Scholar [23] E. L. Porteus, "Foundations of Stochastic Inventory Theory," Stanford University Press, Stanford, California, Stanford University Press, Stanford, California, 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, William E. Simon Graduate School of Business Administration, University of Rochaster, Rochaster, NY, 2001. Google Scholar [25] E. Pinker, A. Seidmann and Y. Vakrat, Managing online auctions: Current business and research issues, Management Science, 49 (2003), 1457-1484. 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 "Mathematical Methods in the Social Sciences," 1959, Stanford University Press, Stanford, California, (1960), 196-202.  Google Scholar [27] A. Segev, C. Beam and J. Shanthikumar, Optimal design of internet-based auctions, Information Technology and Mangement, 2 (2001), 121-163. 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), 931-939. 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), 245-250. 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), 1067-1083. 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), 1388-1407. doi: 10.1287/mnsc.48.11.1388.269.  Google Scholar [32] R. J. Weber, Multiple-object auctions, in "Auctions, Bidding, and Contracting: Uses and Theory" (eds. Richard Engelbrechtp-Wiggans, Martin Shubik and Robert M. Stark), Chapter 3, New York University Press, New York, 165-191; also in "The Economics Theory of Auctions" (ed. Paul Klemperer), Vol. II, Edward Elgar Publishing Limited, Cheltenham, UK / Edward Elgar Publishing, Inc., Northampton, USA, (1983), 240-266. Google Scholar [33] C. A. Yano and S. M. Gilbert, Coordinated pricing and producion/procurement decisions, A review, in "Managing Business Interfaces: Marketing, Engineering, and Manufacturing Perspectives" (eds. A. K. Chakravarty and J. Eliashberg), Kluwer Academic Publshers, Boston, Massachusetts, 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), 802-810. 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), 654-665. doi: 10.1287/opre.39.4.654.  Google Scholar [36] P. H. Zipkin, "Foundations of Inventory Management," McGraw Hill, Boston, 2000. Google Scholar
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