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.

[2]

K. J. ArrowT. Harris and J. Marshak, Optimal inventory policy, Econometrica, 19 (1951), 250-272. doi: 10.2307/1906813.

[3]

J. A. Bather, A continuous time inventory model, Journal of Applied Probability, 3 (1966), 538-549. doi: 10.1017/S0021900200114317.

[4]

R. Bellman, Dynamic Programming, Dover Books on Computer Science, 2003.

[5]

A. Bensoussan, Dynamic Programming and Inventory Control, IOS Press, Studies in Probability, Optimization and Statistics, 2011.

[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.

[7]

A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities, Dunod, 1982.

[8]

A. BensoussanR. 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.

[9]

A. Bensoussan and Y. Houmin, Inventory Control with Pricing Optimization, 2013.

[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.

[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.

[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.

[13]

T. DohiN. 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.

[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.

[15]

Q. FengG. GallegoS. SethiH. 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.

[16]

Q. FengS. 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.

[17]

F. S. Gökhan, Effect of the Guess Function & Continuation Method on the Run Time of MATLAB BVP Solvers, 2011.

[18]

F. W. Harris, How many parts to make of one, Factory, The Magazine of Management, 10 (1913), 135-136.

[19]

J. Harrison and A. Taylor, Optimal control of a brownian storage system, Stochastic Process and Their Applications, 6 (1978), 179-194.

[20]

L. Gimpl-HeersinkC. RudloffM. 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.

[21]

S. C. Graves, A Base Stock Inventory Model for Remanufacturable Product, MIT, http://hdl.handle.net/1721.1/3735.

[22]

R. Güllü, Base Stock policies for production/invenotry problems with uncertain capacity levels, European Journal of Operational Research, 105 (1998), 43-51.

[23]

http://se.mathworks.com/help/matlab/ref/bvp5c.html?requestedDomain=www.mathworks.com#

[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.

[25]

Y. LuY. ChenM. 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.

[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.

[27]

M. L. Puterman, A diffusion process model for a storage system, TIMS Studies in Management Sciences, 1 (1975), 143-159.

[28]

Y. QinR. WangJ. V. AsooY. 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.

[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.

[30] L. F. ShampineI. Gladwell and S. Thompson, Solving ODEs with MATLAB, Cambridge University Press, 2013. doi: 10.1017/CBO9780511615542.
[31]

T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-68. doi: 10.1287/mnsc.2.1.61.

[32]

R. Zhang, An Introduction to Joint Pricing and Inventory Management under Stochastic Demand, 2013.

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.

[2]

K. J. ArrowT. Harris and J. Marshak, Optimal inventory policy, Econometrica, 19 (1951), 250-272. doi: 10.2307/1906813.

[3]

J. A. Bather, A continuous time inventory model, Journal of Applied Probability, 3 (1966), 538-549. doi: 10.1017/S0021900200114317.

[4]

R. Bellman, Dynamic Programming, Dover Books on Computer Science, 2003.

[5]

A. Bensoussan, Dynamic Programming and Inventory Control, IOS Press, Studies in Probability, Optimization and Statistics, 2011.

[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.

[7]

A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities, Dunod, 1982.

[8]

A. BensoussanR. 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.

[9]

A. Bensoussan and Y. Houmin, Inventory Control with Pricing Optimization, 2013.

[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.

[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.

[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.

[13]

T. DohiN. 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.

[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.

[15]

Q. FengG. GallegoS. SethiH. 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.

[16]

Q. FengS. 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.

[17]

F. S. Gökhan, Effect of the Guess Function & Continuation Method on the Run Time of MATLAB BVP Solvers, 2011.

[18]

F. W. Harris, How many parts to make of one, Factory, The Magazine of Management, 10 (1913), 135-136.

[19]

J. Harrison and A. Taylor, Optimal control of a brownian storage system, Stochastic Process and Their Applications, 6 (1978), 179-194.

[20]

L. Gimpl-HeersinkC. RudloffM. 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.

[21]

S. C. Graves, A Base Stock Inventory Model for Remanufacturable Product, MIT, http://hdl.handle.net/1721.1/3735.

[22]

R. Güllü, Base Stock policies for production/invenotry problems with uncertain capacity levels, European Journal of Operational Research, 105 (1998), 43-51.

[23]

http://se.mathworks.com/help/matlab/ref/bvp5c.html?requestedDomain=www.mathworks.com#

[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.

[25]

Y. LuY. ChenM. 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.

[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.

[27]

M. L. Puterman, A diffusion process model for a storage system, TIMS Studies in Management Sciences, 1 (1975), 143-159.

[28]

Y. QinR. WangJ. V. AsooY. 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.

[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.

[30] L. F. ShampineI. Gladwell and S. Thompson, Solving ODEs with MATLAB, Cambridge University Press, 2013. doi: 10.1017/CBO9780511615542.
[31]

T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-68. doi: 10.1287/mnsc.2.1.61.

[32]

R. Zhang, An Introduction to Joint Pricing and Inventory Management under Stochastic Demand, 2013.

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
[1]

Yusuke Murase, Atsushi Kadoya, Nobuyuki Kenmochi. Optimal control problems for quasi-variational inequalities and its numerical approximation. Conference Publications, 2011, 2011 (Special) : 1101-1110. doi: 10.3934/proc.2011.2011.1101

[2]

Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457

[3]

Yusuke Murase, Risei Kano, Nobuyuki Kenmochi. Elliptic Quasi-variational inequalities and applications. Conference Publications, 2009, 2009 (Special) : 583-591. doi: 10.3934/proc.2009.2009.583

[4]

Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383

[5]

Laura Scrimali. Mixed behavior network equilibria and quasi-variational inequalities. Journal of Industrial & Management Optimization, 2009, 5 (2) : 363-379. doi: 10.3934/jimo.2009.5.363

[6]

Haisen Zhang. Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities. Mathematical Control & Related Fields, 2014, 4 (3) : 365-379. doi: 10.3934/mcrf.2014.4.365

[7]

Yiju Wang, Wei Xing, Hengxia Gao. Optimal ordering policy for inventory mechanism with a stochastic short-term price discount. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018199

[8]

Qi Feng, Suresh P. Sethi, Houmin Yan, Hanqin Zhang. Optimality and nonoptimality of the base-stock policy in inventory problems with multiple delivery modes. Journal of Industrial & Management Optimization, 2006, 2 (1) : 19-42. doi: 10.3934/jimo.2006.2.19

[9]

Xiaojun Chen, Guihua Lin. CVaR-based formulation and approximation method for stochastic variational inequalities. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 35-48. doi: 10.3934/naco.2011.1.35

[10]

Xin Zhou, Liangping Shi, Bingzhi Huang. Integrated inventory model with stochastic lead time and controllable variability for milk runs. Journal of Industrial & Management Optimization, 2012, 8 (3) : 657-672. doi: 10.3934/jimo.2012.8.657

[11]

Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165

[12]

Nobuyuki Kenmochi. Parabolic quasi-variational diffusion problems with gradient constraints. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 423-438. doi: 10.3934/dcdss.2013.6.423

[13]

Xiao-Qian Jiang, Lun-Chuan Zhang. Stock price fluctuation prediction method based on time series analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 915-927. doi: 10.3934/dcdss.2019061

[14]

Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. Deteriorating inventory with preservation technology under price- and stock-sensitive demand. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-28. doi: 10.3934/jimo.2019019

[15]

Zhengyan Wang, Guanghua Xu, Peibiao Zhao, Zudi Lu. The optimal cash holding models for stochastic cash management of continuous time. Journal of Industrial & Management Optimization, 2018, 14 (1) : 1-17. doi: 10.3934/jimo.2017034

[16]

Andrzej Nowakowski, Jan Sokolowski. On dual dynamic programming in shape control. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2473-2485. doi: 10.3934/cpaa.2012.11.2473

[17]

Paolo Rinaldi, Heinz Schättler. Minimization of the base transit time in semiconductor devices using optimal control. Conference Publications, 2003, 2003 (Special) : 742-751. doi: 10.3934/proc.2003.2003.742

[18]

Junfeng Yang. Dynamic power price problem: An inverse variational inequality approach. Journal of Industrial & Management Optimization, 2008, 4 (4) : 673-684. doi: 10.3934/jimo.2008.4.673

[19]

Takeshi Fukao, Nobuyuki Kenmochi. Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2523-2538. doi: 10.3934/dcds.2015.35.2523

[20]

Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1345-1373. doi: 10.3934/jimo.2018098

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (13)
  • HTML views (72)
  • Cited by (0)

Other articles
by authors

[Back to Top]