Article Contents
Article Contents

Base stock list price policy in continuous time

• * Corresponding author: Alain Bensoussan

Alain Bensoussan is also with the department of Systems Engineeringand Engineering Management, the City University of Hong Kong :Research supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region (City U 500111)

• We study the problem of inventory control, with simultaneous pricing optimization in continuous time. For the classical inventory control problem in continuous time, see [5], as a recent reference. We incorporate pricing decisions together with inventory decisions. We consider the situation without fixed cost for an infinite horizon. Without pricing, under very natural assumptions, the optimal ordering policy is given by a Base stock, which we review briefly. With pricing, the natural generalization is the so called "Base Stock list price" (BSLP) term coined by E. Porteus, see [36], and was shown in discrete time by A. Federgruen and A. Herching to be the optimal strategy, see [14]. We extend the concept to continuous time which not only complicates the dynamics of the problem, which has never been considered before.

Mathematics Subject Classification: Primary:58F15, 58F17;Secondary:53C35.

 Citation:

• Figure 1.  Finding the value of $S$

Figure 2.  $H_\epsilon$ with value of $S_\epsilon$ for $b=20$

Figure 3.  $H_\epsilon$ with value of $S_\epsilon$ for $b=30$

Figure 4.  $H_\epsilon$ with value of $S_\epsilon$ for $b=100$

Figure 5.  $H_\epsilon$ with value of $S_\epsilon$ for $b=250$

Figure 6.  $H_\epsilon(x)$ for decreasing epsilon

Figure 7.  $H_S(x)$ with value $S$ for $b=30$

Figure 8.  Price depending on Inventory Level

•  G. Allon  and  A. Zeevi , A Note on the Relationship Among Capacity, Pricing and Inventory in a Make-to-Stock System, Production and Operations Management, 20 (2011) , 143-151.  doi: 10.1111/j.1937-5956.2010.01193.x. K. J. Arrow , T. Harris  and  J. Marshak , Optimal inventory policy, Econometrica, 19 (1951) , 250-272.  doi: 10.2307/1906813. J. A. Bather , A continuous time inventory model, Journal of Applied Probability, 3 (1966) , 538-549.  doi: 10.1017/S0021900200114317. R. Bellman, Dynamic Programming, Dover Books on Computer Science, 2003. A. Bensoussan, Dynamic Programming and Inventory Control, IOS Press, Studies in Probability, Optimization and Statistics, 2011. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers Ⅰ, Springer Science & Business Media, 1999. doi: 10.1007/978-1-4757-3069-2. A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities, Dunod, 1982. A. Bensoussan , R. H. Liu  and  S. Sethi , Optimality of and $(s,S)$ policy with compound poisson and diffusion demands: A QVI approach, SIAM J. Control Optim., 44 (2005) , 1650-1676.  doi: 10.1137/S0363012904443737. A. Bensoussan and Y. Houmin, Inventory Control with Pricing Optimization, 2013. S. Browne  and  P. Zipkin , Inventory models with continuous, stochastic demands, The Annals of Applied Probability, 1 (1991) , 419-435.  doi: 10.1214/aoap/1177005875. X. Chen  and  D. Simchi-Levi , Pricing and inventory management, The Oxford Handbook of Pricing Management eds. R. Phillips and O. Ozalp, Oxford University Press, (2012) , 784-822.  doi: 10.1093/oxfordhb/9780199543175.013.0030. X. Chen  and  J. Zhang , Production control and supplier selection under demand, Journal of Industrial Engineering and Management, 3 (2010) , 421-446.  doi: 10.3926/jiem.2010.v3n3.p421-446. T. Dohi , N. Kaio  and  S. Osaki , A continuous time inventory control for wiener process demand, Computers Math. Applic., 26 (1993) , 11-22.  doi: 10.1016/0898-1221(93)90002-D. 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. Q. Feng , G. Gallego , S. Sethi , H. Yan  and  H. Zhang , Are base-stock policies optimal in inventory problems with multiple delivery modes?, Operations Research, 54 (2006) , 801-807.  doi: 10.1287/opre.1050.0271. Q. Feng , S. Luo  and  D. Zhang , Dynamic inventory-pricing control under backorder: Demand estimation and policy optimization, Manufactoring and Service Operations Management, 16 (2013) , 149-160.  doi: 10.1287/msom.2013.0459. F. S. Gökhan, Effect of the Guess Function & Continuation Method on the Run Time of MATLAB BVP Solvers, 2011. F. W. Harris , How many parts to make of one, Factory, The Magazine of Management, 10 (1913) , 135-136. J. Harrison  and  A. Taylor , Optimal control of a brownian storage system, Stochastic Process and Their Applications, 6 (1978) , 179-194. L. Gimpl-Heersink , C. Rudloff , M. Fleischmann  and  A. Taudes , Integrating pricing and inventory control: Is it worth the efffort?, Business Reasearch Official Open Access Journal of VHB, 1 (2008) , 106-123.  doi: 10.1007/BF03342705. S. C. Graves, A Base Stock Inventory Model for Remanufacturable Product, MIT, http://hdl.handle.net/1721.1/3735. R. Güllü , Base Stock policies for production/invenotry problems with uncertain capacity levels, European Journal of Operational Research, 105 (1998) , 43-51. http://se.mathworks.com/help/matlab/ref/bvp5c.html?requestedDomain=www.mathworks.com# G. van Ryzin  and  G. Vulcano , Optimal auctioning and ordering in an infinite horizon inventory-pricing system, Operations Research, 52 (2004) , 346-367.  doi: 10.1287/opre.1040.0105. Y. Lu , Y. Chen , M. Song  and  X. Yan , Optimal pricing and inventory control policy with quantity-based price differentiation, Operations Research, 62 (2014) , 512-523.  doi: 10.1287/opre.2013.1240. E. L. Porteus, Stochastic Inventory Theory, in Handbooks in O. R. and M. S. , (eds. D. Heyman, M. J. Sobel), Elsevier, 2 (1990), 605-652. doi: 10.1016/S0927-0507(05)80176-8. M. L. Puterman , A diffusion process model for a storage system, TIMS Studies in Management Sciences, 1 (1975) , 143-159. Y. Qin , R. Wang , J. V. Asoo , Y. Chen  and  M. M. H. Seref , The newsvendor problem: Review and directions for future research, European Journal of Operational Research, 213 (2011) , 361-374.  doi: 10.1016/j.ejor.2010.11.024. K. Sato  and  K. Sawaki , A continuous-time inventory model with procurement from spot market, Journal of the Operations Research Society of Japan, 53 (2010) , 136-148. L. F. Shampine,  I. Gladwell and  S. Thompson,  Solving ODEs with MATLAB, Cambridge University Press, 2013.  doi: 10.1017/CBO9780511615542. T. M. Whitin , Inventory control and price theory, Management Science, 2 (1955) , 61-68.  doi: 10.1287/mnsc.2.1.61. R. Zhang, An Introduction to Joint Pricing and Inventory Management under Stochastic Demand, 2013.

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