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An $(s,S)$ inventory model with level-dependent $G/M/1$-Type structure
Optimal investment strategy on advertisement in duopoly
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 |
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. |
[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. |
[4] |
G. M. Erickson, Advertising competition in a dynamic oligopoly with multiple brands,, Operations Research, 57 (2009), 1106.
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). 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
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.
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.
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.
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.
doi: 10.3934/jimo.2010.6.895. |
[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. |
[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. |
[4] |
G. M. Erickson, Advertising competition in a dynamic oligopoly with multiple brands,, Operations Research, 57 (2009), 1106.
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). 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
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.
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.
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.
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.
doi: 10.3934/jimo.2010.6.895. |
[31] |
J. K. Zhang, Advertising Economics Practical Tutorial,, Shanghai Far East Publishers, (1998). Google Scholar |
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