# American Institute of Mathematical Sciences

January  2017, 22(1): 1-28. doi: 10.3934/dcdsb.2017001

## Base stock list price policy in continuous time

 International Center for Decision and Risk Analysis, Jindal School of Management, University of Texas -Dallas, Richardson, TX 75080-5298, USA

* Corresponding author: Alain Bensoussan

Received  January 2016 Revised  June 2016 Published  November 2016

Fund Project: 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.

Citation: Alain Bensoussan, Sonny Skaaning. Base stock list price policy in continuous time. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 1-28. doi: 10.3934/dcdsb.2017001
##### References:
 [1] 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.  Google Scholar [2] K. J. Arrow, T. Harris and J. Marshak, Optimal inventory policy, Econometrica, 19 (1951), 250-272.  doi: 10.2307/1906813.  Google Scholar [3] J. A. Bather, A continuous time inventory model, Journal of Applied Probability, 3 (1966), 538-549.  doi: 10.1017/S0021900200114317.  Google Scholar [4] R. Bellman, Dynamic Programming, Dover Books on Computer Science, 2003.  Google Scholar [5] A. Bensoussan, Dynamic Programming and Inventory Control, IOS Press, Studies in Probability, Optimization and Statistics, 2011.  Google Scholar [6] 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.  Google Scholar [7] A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities, Dunod, 1982. Google Scholar [8] 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.  Google Scholar [9] A. Bensoussan and Y. Houmin, Inventory Control with Pricing Optimization, 2013. Google Scholar [10] 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.  Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] 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.  Google Scholar [14] 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 [15] 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.  Google Scholar [16] 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.  Google Scholar [17] F. S. Gökhan, Effect of the Guess Function & Continuation Method on the Run Time of MATLAB BVP Solvers, 2011. Google Scholar [18] F. W. Harris, How many parts to make of one, Factory, The Magazine of Management, 10 (1913), 135-136.   Google Scholar [19] J. Harrison and A. Taylor, Optimal control of a brownian storage system, Stochastic Process and Their Applications, 6 (1978), 179-194.   Google Scholar [20] 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.  Google Scholar [21] S. C. Graves, A Base Stock Inventory Model for Remanufacturable Product, MIT, http://hdl.handle.net/1721.1/3735. Google Scholar [22] R. Güllü, Base Stock policies for production/invenotry problems with uncertain capacity levels, European Journal of Operational Research, 105 (1998), 43-51.   Google Scholar [23] [24] 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.  Google Scholar [25] 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.  Google Scholar [26] 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.  Google Scholar [27] M. L. Puterman, A diffusion process model for a storage system, TIMS Studies in Management Sciences, 1 (1975), 143-159.   Google Scholar [28] 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.  Google Scholar [29] 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.   Google Scholar [30] L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs with MATLAB, Cambridge University Press, 2013.  doi: 10.1017/CBO9780511615542.  Google Scholar [31] T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-68.  doi: 10.1287/mnsc.2.1.61.  Google Scholar [32] R. Zhang, An Introduction to Joint Pricing and Inventory Management under Stochastic Demand, 2013. Google Scholar

show all references

##### References:
 [1] 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.  Google Scholar [2] K. J. Arrow, T. Harris and J. Marshak, Optimal inventory policy, Econometrica, 19 (1951), 250-272.  doi: 10.2307/1906813.  Google Scholar [3] J. A. Bather, A continuous time inventory model, Journal of Applied Probability, 3 (1966), 538-549.  doi: 10.1017/S0021900200114317.  Google Scholar [4] R. Bellman, Dynamic Programming, Dover Books on Computer Science, 2003.  Google Scholar [5] A. Bensoussan, Dynamic Programming and Inventory Control, IOS Press, Studies in Probability, Optimization and Statistics, 2011.  Google Scholar [6] 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.  Google Scholar [7] A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities, Dunod, 1982. Google Scholar [8] 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.  Google Scholar [9] A. Bensoussan and Y. Houmin, Inventory Control with Pricing Optimization, 2013. Google Scholar [10] 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.  Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] 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.  Google Scholar [14] 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 [15] 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.  Google Scholar [16] 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.  Google Scholar [17] F. S. Gökhan, Effect of the Guess Function & Continuation Method on the Run Time of MATLAB BVP Solvers, 2011. Google Scholar [18] F. W. Harris, How many parts to make of one, Factory, The Magazine of Management, 10 (1913), 135-136.   Google Scholar [19] J. Harrison and A. Taylor, Optimal control of a brownian storage system, Stochastic Process and Their Applications, 6 (1978), 179-194.   Google Scholar [20] 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.  Google Scholar [21] S. C. Graves, A Base Stock Inventory Model for Remanufacturable Product, MIT, http://hdl.handle.net/1721.1/3735. Google Scholar [22] R. Güllü, Base Stock policies for production/invenotry problems with uncertain capacity levels, European Journal of Operational Research, 105 (1998), 43-51.   Google Scholar [23] [24] 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.  Google Scholar [25] 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.  Google Scholar [26] 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.  Google Scholar [27] M. L. Puterman, A diffusion process model for a storage system, TIMS Studies in Management Sciences, 1 (1975), 143-159.   Google Scholar [28] 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.  Google Scholar [29] 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.   Google Scholar [30] L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs with MATLAB, Cambridge University Press, 2013.  doi: 10.1017/CBO9780511615542.  Google Scholar [31] T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-68.  doi: 10.1287/mnsc.2.1.61.  Google Scholar [32] R. Zhang, An Introduction to Joint Pricing and Inventory Management under Stochastic Demand, 2013. Google Scholar
Finding the value of $S$
$H_\epsilon$ with value of $S_\epsilon$ for $b=20$
$H_\epsilon$ with value of $S_\epsilon$ for $b=30$
$H_\epsilon$ with value of $S_\epsilon$ for $b=100$
$H_\epsilon$ with value of $S_\epsilon$ for $b=250$
$H_\epsilon(x)$ for decreasing epsilon
$H_S(x)$ with value $S$ for $b=30$
Price depending on Inventory Level

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