Optimal investment strategy on advertisement in duopoly

Previous Article
Tail asymptotics of fluid queues in a distributed server system fed by a heavy-tailed ON-OFF flow
JIMO Home
This Issue
Next Article
An $(s,S)$ inventory model with level-dependent $G/M/1$-Type structure

April 2016, 12(2): 625-636. doi: 10.3934/jimo.2016.12.625

Optimal investment strategy on advertisement in duopoly

Fengjun Wang ^1, , Qingling Zhang ^2, , Bin Li ^3, and Wanquan Liu ^4,

1.	Institute of Systems Science, Northeastern University, Shenyang, Liaoning Province, 110819, China
2.	Institute of Systems Science, Northeastern University, Shenyang, Liaoning, 110819
3.	Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, WA 6845
4.	Department of Computing, Curtin University of Technology, Perth, WA 6102

Received September 2014 Revised March 2015 Published June 2015

Abstract
Related Papers

In this paper, we will investigate a duopoly competition issue in a commencing period of horizontal expansion. This is an important problem in marketing investment for new products in free market. First, we propose a new market model characterized by nonlinear differential-algebraic equations with continuous inequality constraints, which aims to maximize an enterprise's product market share rather than its profit in the commencing period in an environment of the duopoly market. In order to solve the investment problem numerically based on proposed model, the control parameterization technique together with the constraint transcription method is used by transforming the proposed problem into a sequence of optimal parameter selection problems. Finally, a practical example on beer sales is used to show the effectiveness of proposed model and we present the optimal advertising strategies corresponding to different competition situations.

Keywords: nonlinear differential-algebraic model, Optimal control, inequality continuous constraints, control parameterization, investment on advertising..

Mathematics Subject Classification: Primary: 90-08; Secondary: 49M3.

Citation: Fengjun Wang, Qingling Zhang, Bin Li, Wanquan Liu. Optimal investment strategy on advertisement in duopoly. Journal of Industrial and Management Optimization, 2016, 12 (2) : 625-636. doi: 10.3934/jimo.2016.12.625

References:

[1]	B. L. Bai and R. X. Bai, The Modern Western Economic Theory, Economic Science Press, 2011.
[2]	F. M. Bass, A. Krishnamoorthy, A. Prasad and S. P. Sethi, Generic and brand advertising strategies in a dynamic duopoly, Marketing Science, 24 (2005), 556-568. doi: 10.1287/mksc.1050.0119.
[3]	G. M. Erickson, An oligopoly model of dynamic advertising competition, European Journal of Operational Research, 19 (2009), 374-388. doi: 10.1016/j.ejor.2008.06.023.
[4]	G. M. Erickson, Advertising competition in a dynamic oligopoly with multiple brands, Operations Research, 57 (2009), 1106-1113. doi: 10.1287/opre.1080.0663.
[5]	G. Fasano and J. Pintér, Modeling and Optimization in Space Engineering, Springer, 2013. doi: 10.1007/978-1-4614-4469-5.
[6]	H. Gao, Western Economics: Macro Part, China Renmin University Press, 2011.
[7]	L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER 3: Optimal Control Software, Version 2.0, Theory and user manual, 2002.
[8]	C. H. Jiang, Q. Lin, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints, Journal of Optimization Theory and Applications, 154 (2012), 30-53. doi: 10.1007/s10957-012-0006-9.
[9]	A. Krishnamoorthy, A. Prasad and S. P. Sethi, Optimal pricing and advertising in a durable-good duopoly, European Journal of Operational Research, 200 (2010), 486-497. doi: 10.1016/j.ejor.2009.01.003.
[10]	B. Li, K. L. Teo and G. R. Duan, Optimal control computation for discrete time time-delayed optimal control problem with all-time-step inequality constraints, International Journal of Innovative Computing, Information and Control, 6 (2010), 3157-3175.
[11]	B. Li, K. L. Teo, C. C. Lim and G. R. Duan, An optimal PID controller design for nonlinear constrained optimal control problems, Discrete and Continuous Dynamical Systems-Series B, 16 (2011), 1101-1117. doi: 10.3934/dcdsb.2011.16.1101.
[12]	B. Li, K. L. Teo, G. H. Zhao and G. R. Duan, An efficient computational approach to a class of minmax optimal control problems with applications, The ANZIAM Journal, 51 (2009), 162-177. doi: 10.1017/S1446181110000040.
[13]	B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866-875. doi: 10.1016/j.amc.2013.08.092.
[14]	B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260-291. doi: 10.1007/s10957-011-9904-5.
[15]	R. C. Loxtonnd, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control, Automatica, 45 (2009), 2250-2257. doi: 10.1016/j.automatica.2009.05.029.
[16]	T. Matsumura and T. Sunada, Advertising competition in a mixed oligopoly, Economics Letters, 119 (2013), 183-185. doi: 10.1016/j.econlet.2013.02.021.
[17]	J. P. Nelson, Beer advertising and marketing update: structure, conduct, and social costs, Review of Industrial Organization, 26 (2005), 269-306.
[18]	A. Prasad and S. P. Sethi, Competitive advertising under uncertainty: A stochastic differential game approach, Journal of Optimization Theory and Applications, 123 (2004), 163-185. doi: 10.1023/B:JOTA.0000043996.62867.20.
[19]	A. Prasad, S. P. Sethi and P. A. Naik, Understanding the impact of churn in dynamic oligopoly markets, Automatica, 48 (2012), 2882-2887. doi: 10.1016/j.automatica.2012.08.031.
[20]	J. Qi and D. W. Wang, On analysis of chaotic synchronization in an advertising competition model, Journal of Management Sciences in China, 7 (2004), 27-31.
[21]	J. Qi and D. W. Wang, Optimal control strategies for an advertising competing model, Systems Engineering-Theory & Practice, 27 (2007), 39-44.
[22]	S. P. Sethi, Optimal control of the Vidale-Wolfe advertising model, Operations Research, 21 (1973), 998-1013. doi: 10.1287/opre.21.4.998.
[23]	S. P. Sethi, A. Prasad and X. L. He, Optimal advertising and pricing in a new-product adoption model, Journal of Optimization Theory and Applications, 139 (2008), 351-360. doi: 10.1007/s10957-008-9472-5.
[24]	J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer Heidelberg, 2002. doi: 10.1007/978-0-387-21738-3.
[25]	K. L. Teo, C. J. Goh and K. H. Wong, A unified computational approach to optimal control problems, Longman Scientific and Technical, 1991.
[26]	K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, The control parameterization enhancing transform for constrained optimal control problems, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 40 (1999), 314-335. doi: 10.1017/S0334270000010936.
[27]	M. L. Vidale and H. B. Wolfe, An operations-research study of sales response to advertising, Operations Research, 5 (1957), 370-381. doi: 10.1287/opre.5.3.370.
[28]	Q. Wang and Z. Wu, A duopolistic model of dynamic competitive advertising, European Journal of Operational Research, 128 (2001), 213-226. doi: 10.1016/S0377-2217(99)00346-X.
[29]	C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, Journal of Industrial Management and Optimization, 8 (2012), 485-491. doi: 10.3934/jimo.2012.8.485.
[30]	C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895-910. doi: 10.3934/jimo.2010.6.895.
[31]	J. K. Zhang, Advertising Economics Practical Tutorial, Shanghai Far East Publishers, 1998.

show all references

References:

[1]	B. L. Bai and R. X. Bai, The Modern Western Economic Theory, Economic Science Press, 2011.
[2]	F. M. Bass, A. Krishnamoorthy, A. Prasad and S. P. Sethi, Generic and brand advertising strategies in a dynamic duopoly, Marketing Science, 24 (2005), 556-568. doi: 10.1287/mksc.1050.0119.
[3]	G. M. Erickson, An oligopoly model of dynamic advertising competition, European Journal of Operational Research, 19 (2009), 374-388. doi: 10.1016/j.ejor.2008.06.023.
[4]	G. M. Erickson, Advertising competition in a dynamic oligopoly with multiple brands, Operations Research, 57 (2009), 1106-1113. doi: 10.1287/opre.1080.0663.
[5]	G. Fasano and J. Pintér, Modeling and Optimization in Space Engineering, Springer, 2013. doi: 10.1007/978-1-4614-4469-5.
[6]	H. Gao, Western Economics: Macro Part, China Renmin University Press, 2011.
[7]	L. S. Jennings, M. E. Fisher, K. L. Teo and C. J. Goh, MISER 3: Optimal Control Software, Version 2.0, Theory and user manual, 2002.
[8]	C. H. Jiang, Q. Lin, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints, Journal of Optimization Theory and Applications, 154 (2012), 30-53. doi: 10.1007/s10957-012-0006-9.
[9]	A. Krishnamoorthy, A. Prasad and S. P. Sethi, Optimal pricing and advertising in a durable-good duopoly, European Journal of Operational Research, 200 (2010), 486-497. doi: 10.1016/j.ejor.2009.01.003.
[10]	B. Li, K. L. Teo and G. R. Duan, Optimal control computation for discrete time time-delayed optimal control problem with all-time-step inequality constraints, International Journal of Innovative Computing, Information and Control, 6 (2010), 3157-3175.
[11]	B. Li, K. L. Teo, C. C. Lim and G. R. Duan, An optimal PID controller design for nonlinear constrained optimal control problems, Discrete and Continuous Dynamical Systems-Series B, 16 (2011), 1101-1117. doi: 10.3934/dcdsb.2011.16.1101.
[12]	B. Li, K. L. Teo, G. H. Zhao and G. R. Duan, An efficient computational approach to a class of minmax optimal control problems with applications, The ANZIAM Journal, 51 (2009), 162-177. doi: 10.1017/S1446181110000040.
[13]	B. Li, C. Xu, K. L. Teo and J. Chu, Time optimal Zermelo's navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866-875. doi: 10.1016/j.amc.2013.08.092.
[14]	B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, 151 (2011), 260-291. doi: 10.1007/s10957-011-9904-5.
[15]	R. C. Loxtonnd, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control, Automatica, 45 (2009), 2250-2257. doi: 10.1016/j.automatica.2009.05.029.
[16]	T. Matsumura and T. Sunada, Advertising competition in a mixed oligopoly, Economics Letters, 119 (2013), 183-185. doi: 10.1016/j.econlet.2013.02.021.
[17]	J. P. Nelson, Beer advertising and marketing update: structure, conduct, and social costs, Review of Industrial Organization, 26 (2005), 269-306.
[18]	A. Prasad and S. P. Sethi, Competitive advertising under uncertainty: A stochastic differential game approach, Journal of Optimization Theory and Applications, 123 (2004), 163-185. doi: 10.1023/B:JOTA.0000043996.62867.20.
[19]	A. Prasad, S. P. Sethi and P. A. Naik, Understanding the impact of churn in dynamic oligopoly markets, Automatica, 48 (2012), 2882-2887. doi: 10.1016/j.automatica.2012.08.031.
[20]	J. Qi and D. W. Wang, On analysis of chaotic synchronization in an advertising competition model, Journal of Management Sciences in China, 7 (2004), 27-31.
[21]	J. Qi and D. W. Wang, Optimal control strategies for an advertising competing model, Systems Engineering-Theory & Practice, 27 (2007), 39-44.
[22]	S. P. Sethi, Optimal control of the Vidale-Wolfe advertising model, Operations Research, 21 (1973), 998-1013. doi: 10.1287/opre.21.4.998.
[23]	S. P. Sethi, A. Prasad and X. L. He, Optimal advertising and pricing in a new-product adoption model, Journal of Optimization Theory and Applications, 139 (2008), 351-360. doi: 10.1007/s10957-008-9472-5.
[24]	J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer Heidelberg, 2002. doi: 10.1007/978-0-387-21738-3.
[25]	K. L. Teo, C. J. Goh and K. H. Wong, A unified computational approach to optimal control problems, Longman Scientific and Technical, 1991.
[26]	K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, The control parameterization enhancing transform for constrained optimal control problems, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 40 (1999), 314-335. doi: 10.1017/S0334270000010936.
[27]	M. L. Vidale and H. B. Wolfe, An operations-research study of sales response to advertising, Operations Research, 5 (1957), 370-381. doi: 10.1287/opre.5.3.370.
[28]	Q. Wang and Z. Wu, A duopolistic model of dynamic competitive advertising, European Journal of Operational Research, 128 (2001), 213-226. doi: 10.1016/S0377-2217(99)00346-X.
[29]	C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, Journal of Industrial Management and Optimization, 8 (2012), 485-491. doi: 10.3934/jimo.2012.8.485.
[30]	C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895-910. doi: 10.3934/jimo.2010.6.895.
[31]	J. K. Zhang, Advertising Economics Practical Tutorial, Shanghai Far East Publishers, 1998.

[1]	Ryan Loxton, Qun Lin, Volker Rehbock, Kok Lay Teo. Control parameterization for optimal control problems with continuous inequality constraints: New convergence results. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 571-599. doi: 10.3934/naco.2012.2.571
[2]	Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial and Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275
[3]	Canghua Jiang, Zhiqiang Guo, Xin Li, Hai Wang, Ming Yu. An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1845-1865. doi: 10.3934/dcdss.2020109
[4]	IvÁn Area, FaÏÇal NdaÏrou, Juan J. Nieto, Cristiana J. Silva, Delfim F. M. Torres. Ebola model and optimal control with vaccination constraints. Journal of Industrial and Management Optimization, 2018, 14 (2) : 427-446. doi: 10.3934/jimo.2017054
[5]	Xin Jiang, Kam Chuen Yuen, Mi Chen. Optimal investment and reinsurance with premium control. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2781-2797. doi: 10.3934/jimo.2019080
[6]	Yingjie Bi, Siyu Liu, Yong Li. Periodic solutions of differential-algebraic equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1383-1395. doi: 10.3934/dcdsb.2019232
[7]	Vu Hoang Linh, Volker Mehrmann. Spectral analysis for linear differential-algebraic equations. Conference Publications, 2011, 2011 (Special) : 991-1000. doi: 10.3934/proc.2011.2011.991
[8]	Marcelo J. Villena, Mauricio Contreras. Global and local advertising strategies: A dynamic multi-market optimal control model. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1017-1048. doi: 10.3934/jimo.2018084
[9]	Roderick V.N. Melnik, Ningning Song, Per Sandholdt. Dynamics of torque-speed profiles for electric vehicles and nonlinear models based on differential-algebraic equations. Conference Publications, 2003, 2003 (Special) : 610-617. doi: 10.3934/proc.2003.2003.610
[10]	Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial and Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082
[11]	Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control and Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006
[12]	Changjun Yu, Shuxuan Su, Yanqin Bai. On the optimal control problems with characteristic time control constraints. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1305-1320. doi: 10.3934/jimo.2021021
[13]	Piermarco Cannarsa, Hélène Frankowska, Elsa M. Marchini. On Bolza optimal control problems with constraints. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 629-653. doi: 10.3934/dcdsb.2009.11.629
[14]	Ellina Grigorieva, Evgenii Khailov. Optimal control of a nonlinear model of economic growth. Conference Publications, 2007, 2007 (Special) : 456-466. doi: 10.3934/proc.2007.2007.456
[15]	Jason R. Scott, Stephen Campbell. Auxiliary signal design for failure detection in differential-algebraic equations. Numerical Algebra, Control and Optimization, 2014, 4 (2) : 151-179. doi: 10.3934/naco.2014.4.151
[16]	Nidhal Gammoudi, Hasnaa Zidani. A differential game control problem with state constraints. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022008
[17]	Elimhan N. Mahmudov. Optimal control of second order delay-discrete and delay-differential inclusions with state constraints. Evolution Equations and Control Theory, 2018, 7 (3) : 501-529. doi: 10.3934/eect.2018024
[18]	Elimhan N. Mahmudov. Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2503-2520. doi: 10.3934/jimo.2019066
[19]	Jan-Hendrik Webert, Philip E. Gill, Sven-Joachim Kimmerle, Matthias Gerdts. A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1259-1282. doi: 10.3934/dcdss.2018071
[20]	Eduardo Casas, Fredi Tröltzsch. Sparse optimal control for the heat equation with mixed control-state constraints. Mathematical Control and Related Fields, 2020, 10 (3) : 471-491. doi: 10.3934/mcrf.2020007

2020 Impact Factor: 1.801

Tools

Metrics

Other articles
by authors

on AIMS
Fengjun Wang Qingling Zhang Bin Li Wanquan Liu
on Google Scholar
Fengjun Wang Qingling Zhang Bin Li Wanquan Liu

[Back to Top]