American Institute of Mathematical Sciences

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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

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 & Management Optimization, 2016, 12 (2) : 625-636. doi: 10.3934/jimo.2016.12.625
References:
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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.  doi: 10.1287/mksc.1050.0119.  Google Scholar

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G. M. Erickson, Advertising competition in a dynamic oligopoly with multiple brands,, Operations Research, 57 (2009), 1106.  doi: 10.1287/opre.1080.0663.  Google Scholar

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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.  doi: 10.1007/s10957-012-0006-9.  Google Scholar

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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.  doi: 10.1016/j.ejor.2009.01.003.  Google Scholar

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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, 6 (2010), 3157.   Google Scholar

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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.  doi: 10.3934/dcdsb.2011.16.1101.  Google Scholar

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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.  doi: 10.1017/S1446181110000040.  Google Scholar

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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.  doi: 10.1016/j.amc.2013.08.092.  Google Scholar

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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.  doi: 10.1007/s10957-011-9904-5.  Google Scholar

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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.  doi: 10.1016/j.automatica.2009.05.029.  Google Scholar

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T. Matsumura and T. Sunada, Advertising competition in a mixed oligopoly,, Economics Letters, 119 (2013), 183.  doi: 10.1016/j.econlet.2013.02.021.  Google Scholar

[17]

J. P. Nelson, Beer advertising and marketing update: structure, conduct, and social costs,, Review of Industrial Organization, 26 (2005), 269.   Google Scholar

[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.  doi: 10.1023/B:JOTA.0000043996.62867.20.  Google Scholar

[19]

A. Prasad, S. P. Sethi and P. A. Naik, Understanding the impact of churn in dynamic oligopoly markets,, Automatica, 48 (2012), 2882.  doi: 10.1016/j.automatica.2012.08.031.  Google Scholar

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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.   Google Scholar

[21]

J. Qi and D. W. Wang, Optimal control strategies for an advertising competing model,, Systems Engineering-Theory & Practice, 27 (2007), 39.   Google Scholar

[22]

S. P. Sethi, Optimal control of the Vidale-Wolfe advertising model,, Operations Research, 21 (1973), 998.  doi: 10.1287/opre.21.4.998.  Google Scholar

[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.  doi: 10.1007/s10957-008-9472-5.  Google Scholar

[24]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis,, Springer Heidelberg, (2002).  doi: 10.1007/978-0-387-21738-3.  Google Scholar

[25]

K. L. Teo, C. J. Goh and K. H. Wong, A unified computational approach to optimal control problems,, Longman Scientific and Technical, (1991).   Google Scholar

[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.  doi: 10.1017/S0334270000010936.  Google Scholar

[27]

M. L. Vidale and H. B. Wolfe, An operations-research study of sales response to advertising,, Operations Research, 5 (1957), 370.  doi: 10.1287/opre.5.3.370.  Google Scholar

[28]

Q. Wang and Z. Wu, A duopolistic model of dynamic competitive advertising,, European Journal of Operational Research, 128 (2001), 213.  doi: 10.1016/S0377-2217(99)00346-X.  Google Scholar

[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.  doi: 10.3934/jimo.2012.8.485.  Google Scholar

[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.  doi: 10.3934/jimo.2010.6.895.  Google Scholar

[31]

J. K. Zhang, Advertising Economics Practical Tutorial,, Shanghai Far East Publishers, (1998).   Google Scholar

show all references

References:
[1]

B. L. Bai and R. X. Bai, The Modern Western Economic Theory,, Economic Science Press, (2011).   Google Scholar

[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.  doi: 10.1287/mksc.1050.0119.  Google Scholar

[3]

G. M. Erickson, An oligopoly model of dynamic advertising competition,, European Journal of Operational Research, 19 (2009), 374.  doi: 10.1016/j.ejor.2008.06.023.  Google Scholar

[4]

G. M. Erickson, Advertising competition in a dynamic oligopoly with multiple brands,, Operations Research, 57 (2009), 1106.  doi: 10.1287/opre.1080.0663.  Google Scholar

[5]

G. Fasano and J. Pintér, Modeling and Optimization in Space Engineering,, Springer, (2013).  doi: 10.1007/978-1-4614-4469-5.  Google Scholar

[6]

H. Gao, Western Economics: Macro Part,, China Renmin University Press, (2011).   Google Scholar

[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).   Google Scholar

[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.  doi: 10.1007/s10957-012-0006-9.  Google Scholar

[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.  doi: 10.1016/j.ejor.2009.01.003.  Google Scholar

[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, 6 (2010), 3157.   Google Scholar

[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.  doi: 10.3934/dcdsb.2011.16.1101.  Google Scholar

[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.  doi: 10.1017/S1446181110000040.  Google Scholar

[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.  doi: 10.1016/j.amc.2013.08.092.  Google Scholar

[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.  doi: 10.1007/s10957-011-9904-5.  Google Scholar

[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.  doi: 10.1016/j.automatica.2009.05.029.  Google Scholar

[16]

T. Matsumura and T. Sunada, Advertising competition in a mixed oligopoly,, Economics Letters, 119 (2013), 183.  doi: 10.1016/j.econlet.2013.02.021.  Google Scholar

[17]

J. P. Nelson, Beer advertising and marketing update: structure, conduct, and social costs,, Review of Industrial Organization, 26 (2005), 269.   Google Scholar

[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.  doi: 10.1023/B:JOTA.0000043996.62867.20.  Google Scholar

[19]

A. Prasad, S. P. Sethi and P. A. Naik, Understanding the impact of churn in dynamic oligopoly markets,, Automatica, 48 (2012), 2882.  doi: 10.1016/j.automatica.2012.08.031.  Google Scholar

[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.   Google Scholar

[21]

J. Qi and D. W. Wang, Optimal control strategies for an advertising competing model,, Systems Engineering-Theory & Practice, 27 (2007), 39.   Google Scholar

[22]

S. P. Sethi, Optimal control of the Vidale-Wolfe advertising model,, Operations Research, 21 (1973), 998.  doi: 10.1287/opre.21.4.998.  Google Scholar

[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.  doi: 10.1007/s10957-008-9472-5.  Google Scholar

[24]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis,, Springer Heidelberg, (2002).  doi: 10.1007/978-0-387-21738-3.  Google Scholar

[25]

K. L. Teo, C. J. Goh and K. H. Wong, A unified computational approach to optimal control problems,, Longman Scientific and Technical, (1991).   Google Scholar

[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.  doi: 10.1017/S0334270000010936.  Google Scholar

[27]

M. L. Vidale and H. B. Wolfe, An operations-research study of sales response to advertising,, Operations Research, 5 (1957), 370.  doi: 10.1287/opre.5.3.370.  Google Scholar

[28]

Q. Wang and Z. Wu, A duopolistic model of dynamic competitive advertising,, European Journal of Operational Research, 128 (2001), 213.  doi: 10.1016/S0377-2217(99)00346-X.  Google Scholar

[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.  doi: 10.3934/jimo.2012.8.485.  Google Scholar

[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.  doi: 10.3934/jimo.2010.6.895.  Google Scholar

[31]

J. K. Zhang, Advertising Economics Practical Tutorial,, Shanghai Far East Publishers, (1998).   Google Scholar

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