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A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations
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 
References:
[1] 
K. J. Arrow, T. E. Harris and J. Marschak, Optimal inventory policy, Econometrica, 19 (1951), 250272. doi: 10.2307/1906813. 
[2] 
D. Bertsekas, "Dynamic Programming and Optimal Control," Vol. 2, 1^{st} edition, Athena Scientific, Belmont, MA, 1995. 
[3] 
D. Blackwell, Discrete dynamic programming, Annals of Mathematics Statistics, 33 (1962), 719726. doi: 10.1214/aoms/1177704593. 
[4] 
S. Bollapragada and T. E. Morton, A simple heuristic for computing nonstationary $(s, S)$ policies, Operations Research, 47 (1999), 576584. doi: 10.1287/opre.47.4.576. 
[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), 123147. 
[6] 
F. Chen, Auctioning supply contracts, Management Science, 53 (2007), 15621576. doi: 10.1287/mnsc.1070.0716. 
[7] 
X. Chen and D. SimchiLevi, Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: The finite horizon case, Operations Research, 52 (2004), 887896. doi: 10.1287/opre.1040.0127. 
[8] 
X. Chen and D. SimchiLevi, Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: The infinite horizon case, Mathematics of Operations Research, 29 (2004), 698723. doi: 10.1287/moor.1040.0093. 
[9] 
Y. Chen, S. Ray and Y. Song, Optimal pricing and inventory control policy in periodicreview sysytems with fixed ordering cost and lost sales, Naval Research Logistics, 53 (2006), 117136. doi: 10.1002/nav.20127. 
[10] 
H. A. David, "Order Statistics," 2^{nd} edition, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 1981. 
[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), 617631. 
[12] 
A. Federgruen and A. Heching, Combined pricing and inventory control under uncertainty, Operations Research, 47 (1999), 454475. doi: 10.1287/opre.47.3.454. 
[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. 
[14] 
Q. Feng, S. P. Sethi, H. Yan and H. Zhang, Optimality and nonoptimality of the basestock policy in inventory problems with multiple delivery modes, Journal of Industrial and Management Optimization, 2 (2006), 1942. doi: 10.3934/jimo.2006.2.19. 
[15] 
Q. Hu and W. Yue, "Markov Decision Processes with Their Applications," Advances in Mechanics and Mathematics, 14, Springer, New York, 2008. 
[16] 
W. T. Huh and G. Janakiraman, $(s, S)$ optimality in joint inventorypricing control: An alternate approach, Operations Research, 56 (2008), 783790. doi: 10.1287/opre.1070.0462. 
[17] 
W. T. Huh and G. Janakiraman, Inventory management with auctions and other sales channels: Optimality of $(s, S)$ policies, Management Science, 54 (2008), 139150. doi: 10.1287/mnsc.1070.0767. 
[18] 
D. Iglehart, Optimality of $(s, S)$ policies in the infinitehorizon dynamic inventory problem, Management Science, 9 (1963), 259267. doi: 10.1287/mnsc.9.2.259. 
[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. 
[20] 
E. Maskin and J. Riley, Optimal multiunit auctions, in "The Economic Theory of Auctions" (ed. P. Klemperer), Vol. 2, E. Elgar Publishing, U. K., (2000), 312336. 
[21] 
R. Myerson, Optimal auction design, Mathematics of Operations Research, 6 (1981), 5873. doi: 10.1287/moor.6.1.58. 
[22] 
S. Nahmias, "Production and Operation Analysis," 4^{th} edition, McGrawHill/Irwin, Boston, 2001. 
[23] 
E. L. Porteus, "Foundations of Stochastic Inventory Theory," Stanford University Press, Stanford, California, Stanford University Press, Stanford, California, 2002. 
[24] 
E. Pinker, A. Seidmann and Y. Vakrat, Using transaction data for the design of sequential, multiunit, online auctions, Working paper CIS0003, William E. Simon Graduate School of Business Administration, University of Rochaster, Rochaster, NY, 2001. 
[25] 
E. Pinker, A. Seidmann and Y. Vakrat, Managing online auctions: Current business and research issues, Management Science, 49 (2003), 14571484. doi: 10.1287/mnsc.49.11.1457.20584. 
[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), 196202. 
[27] 
A. Segev, C. Beam and J. Shanthikumar, Optimal design of internetbased auctions, Information Technology and Mangement, 2 (2001), 121163. doi: 10.1023/A:1011411801246. 
[28] 
S. P. Sethi and F. Cheng, Optimality of $(s, S)$ policies in inventory models with Markovian demand, Operations Research, 45 (1997), 931939. doi: 10.1287/opre.45.6.931. 
[29] 
Y. Song, S. Ray and T. Boyaci, Optimal dynamic joint inventorypricing control for multiplicative demand with fixed order costs and lost sales, Operations Research, 57 (2009), 245250. doi: 10.1287/opre.1080.0530. 
[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), 10671083. doi: 10.1137/0114086. 
[31] 
G. Vulcano, G. J. van Ryzin and C. Maglaras, Optimal dynamic auctions for revenue management, Management Science, 48 (2002), 13881407. doi: 10.1287/mnsc.48.11.1388.269. 
[32] 
R. J. Weber, Multipleobject auctions, in "Auctions, Bidding, and Contracting: Uses and Theory" (eds. Richard EngelbrechtpWiggans, Martin Shubik and Robert M. Stark), Chapter 3, New York University Press, New York, 165191; 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), 240266. 
[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. 
[34] 
Y. S. Zheng, A simple proof for optimality of $(s, S)$ policies in the infinitehorizon inventory systems, Journal of Applied Probability, 28 (1991), 802810. doi: 10.2307/3214683. 
[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), 654665. doi: 10.1287/opre.39.4.654. 
[36] 
P. H. Zipkin, "Foundations of Inventory Management," McGraw Hill, Boston, 2000. 
show all references
References:
[1] 
K. J. Arrow, T. E. Harris and J. Marschak, Optimal inventory policy, Econometrica, 19 (1951), 250272. doi: 10.2307/1906813. 
[2] 
D. Bertsekas, "Dynamic Programming and Optimal Control," Vol. 2, 1^{st} edition, Athena Scientific, Belmont, MA, 1995. 
[3] 
D. Blackwell, Discrete dynamic programming, Annals of Mathematics Statistics, 33 (1962), 719726. doi: 10.1214/aoms/1177704593. 
[4] 
S. Bollapragada and T. E. Morton, A simple heuristic for computing nonstationary $(s, S)$ policies, Operations Research, 47 (1999), 576584. doi: 10.1287/opre.47.4.576. 
[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), 123147. 
[6] 
F. Chen, Auctioning supply contracts, Management Science, 53 (2007), 15621576. doi: 10.1287/mnsc.1070.0716. 
[7] 
X. Chen and D. SimchiLevi, Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: The finite horizon case, Operations Research, 52 (2004), 887896. doi: 10.1287/opre.1040.0127. 
[8] 
X. Chen and D. SimchiLevi, Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: The infinite horizon case, Mathematics of Operations Research, 29 (2004), 698723. doi: 10.1287/moor.1040.0093. 
[9] 
Y. Chen, S. Ray and Y. Song, Optimal pricing and inventory control policy in periodicreview sysytems with fixed ordering cost and lost sales, Naval Research Logistics, 53 (2006), 117136. doi: 10.1002/nav.20127. 
[10] 
H. A. David, "Order Statistics," 2^{nd} edition, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 1981. 
[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), 617631. 
[12] 
A. Federgruen and A. Heching, Combined pricing and inventory control under uncertainty, Operations Research, 47 (1999), 454475. doi: 10.1287/opre.47.3.454. 
[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. 
[14] 
Q. Feng, S. P. Sethi, H. Yan and H. Zhang, Optimality and nonoptimality of the basestock policy in inventory problems with multiple delivery modes, Journal of Industrial and Management Optimization, 2 (2006), 1942. doi: 10.3934/jimo.2006.2.19. 
[15] 
Q. Hu and W. Yue, "Markov Decision Processes with Their Applications," Advances in Mechanics and Mathematics, 14, Springer, New York, 2008. 
[16] 
W. T. Huh and G. Janakiraman, $(s, S)$ optimality in joint inventorypricing control: An alternate approach, Operations Research, 56 (2008), 783790. doi: 10.1287/opre.1070.0462. 
[17] 
W. T. Huh and G. Janakiraman, Inventory management with auctions and other sales channels: Optimality of $(s, S)$ policies, Management Science, 54 (2008), 139150. doi: 10.1287/mnsc.1070.0767. 
[18] 
D. Iglehart, Optimality of $(s, S)$ policies in the infinitehorizon dynamic inventory problem, Management Science, 9 (1963), 259267. doi: 10.1287/mnsc.9.2.259. 
[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. 
[20] 
E. Maskin and J. Riley, Optimal multiunit auctions, in "The Economic Theory of Auctions" (ed. P. Klemperer), Vol. 2, E. Elgar Publishing, U. K., (2000), 312336. 
[21] 
R. Myerson, Optimal auction design, Mathematics of Operations Research, 6 (1981), 5873. doi: 10.1287/moor.6.1.58. 
[22] 
S. Nahmias, "Production and Operation Analysis," 4^{th} edition, McGrawHill/Irwin, Boston, 2001. 
[23] 
E. L. Porteus, "Foundations of Stochastic Inventory Theory," Stanford University Press, Stanford, California, Stanford University Press, Stanford, California, 2002. 
[24] 
E. Pinker, A. Seidmann and Y. Vakrat, Using transaction data for the design of sequential, multiunit, online auctions, Working paper CIS0003, William E. Simon Graduate School of Business Administration, University of Rochaster, Rochaster, NY, 2001. 
[25] 
E. Pinker, A. Seidmann and Y. Vakrat, Managing online auctions: Current business and research issues, Management Science, 49 (2003), 14571484. doi: 10.1287/mnsc.49.11.1457.20584. 
[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), 196202. 
[27] 
A. Segev, C. Beam and J. Shanthikumar, Optimal design of internetbased auctions, Information Technology and Mangement, 2 (2001), 121163. doi: 10.1023/A:1011411801246. 
[28] 
S. P. Sethi and F. Cheng, Optimality of $(s, S)$ policies in inventory models with Markovian demand, Operations Research, 45 (1997), 931939. doi: 10.1287/opre.45.6.931. 
[29] 
Y. Song, S. Ray and T. Boyaci, Optimal dynamic joint inventorypricing control for multiplicative demand with fixed order costs and lost sales, Operations Research, 57 (2009), 245250. doi: 10.1287/opre.1080.0530. 
[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), 10671083. doi: 10.1137/0114086. 
[31] 
G. Vulcano, G. J. van Ryzin and C. Maglaras, Optimal dynamic auctions for revenue management, Management Science, 48 (2002), 13881407. doi: 10.1287/mnsc.48.11.1388.269. 
[32] 
R. J. Weber, Multipleobject auctions, in "Auctions, Bidding, and Contracting: Uses and Theory" (eds. Richard EngelbrechtpWiggans, Martin Shubik and Robert M. Stark), Chapter 3, New York University Press, New York, 165191; 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), 240266. 
[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. 
[34] 
Y. S. Zheng, A simple proof for optimality of $(s, S)$ policies in the infinitehorizon inventory systems, Journal of Applied Probability, 28 (1991), 802810. doi: 10.2307/3214683. 
[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), 654665. doi: 10.1287/opre.39.4.654. 
[36] 
P. H. Zipkin, "Foundations of Inventory Management," McGraw Hill, Boston, 2000. 
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