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Optimal Control of the Vidale-Wolfe Advertising Model

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  • Suresh P. Sethi

    (University of Toronto, Toronto, Ontario, Canada)

Abstract

This paper considers an optimal-control problem for the dynamics of the Vidale-Wolfe advertising model, the optimal control being the rate of advertising expenditure to achieve a terminal market share within specified limits in a way that maximizes the present value of net profit streams over a finite horizon. First, the special polar cases of fixed and free end points are solved with and without an upper limit on advertising rate. The complete solution to the general problem is then constructed from these polar cases. The fixed-end-point case with no upper limit on the advertising rate is solved by using Green's theorem, while the other cases require additional use of switching-point analysis based on the maximum principle. The optimal control is characterized by a combination of bang-bang, impulse, and singular control, with the singular arc forming a turnpike.

Suggested Citation

  • Suresh P. Sethi, 1973. "Optimal Control of the Vidale-Wolfe Advertising Model," Operations Research, INFORMS, vol. 21(4), pages 998-1013, August.
    • Handle: RePEc:inm:oropre:v:21:y:1973:i:4:p:998-1013
    DOI: 10.1287/opre.21.4.998
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    Cited by:

    1. Chahim, M. & Hartl, R.F. & Kort, P.M., 2011. "The Deterministic Impulse Control Maximum Principle in Operations Research : Necessary and Sufficient Optimality Conditions (replaces CentER DP 2011-052)," Discussion Paper 2011-133, Tilburg University, Center for Economic Research.
    2. Fred M. Feinberg, 2001. "On Continuous-Time Optimal Advertising Under S-Shaped Response," Management Science, INFORMS, vol. 47(11), pages 1476-1487, November.
    3. Schlosser, Rainer & Chenavaz, Régis Y. & Dimitrov, Stanko, 2021. "Circular economy: Joint dynamic pricing and recycling investments," International Journal of Production Economics, Elsevier, vol. 236(C).
    4. Yanwu Yang & Baozhu Feng & Joni Salminen & Bernard J. Jansen, 2022. "Optimal advertising for a generalized Vidale–Wolfe response model," Electronic Commerce Research, Springer, vol. 22(4), pages 1275-1305, December.
    5. Reza Mousavi & Monica Johar & Vijay S. Mookerjee, 2020. "The Voice of the Customer: Managing Customer Care in Twitter," Information Systems Research, INFORMS, vol. 31(2), pages 340-360, June.
    6. A. Prasad & S. P. Sethi, 2004. "Competitive Advertising Under Uncertainty: A Stochastic Differential Game Approach," Journal of Optimization Theory and Applications, Springer, vol. 123(1), pages 163-185, October.
    7. Yanwu Yang & Daniel Zeng & Yinghui Yang & Jie Zhang, 2015. "Optimal Budget Allocation Across Search Advertising Markets," INFORMS Journal on Computing, INFORMS, vol. 27(2), pages 285-300, May.
    8. Dengpan Liu & Subodha Kumar & Vijay S. Mookerjee, 2012. "Advertising Strategies in Electronic Retailing: A Differential Games Approach," Information Systems Research, INFORMS, vol. 23(3-part-2), pages 903-917, September.
    9. Danaher, Peter J. & Rust, Roland T., 1996. "Determining the optimal return on investment for an advertising campaign," European Journal of Operational Research, Elsevier, vol. 95(3), pages 511-521, December.
    10. Frank M. Bass & Anand Krishnamoorthy & Ashutosh Prasad & Suresh P. Sethi, 2005. "Generic and Brand Advertising Strategies in a Dynamic Duopoly," Marketing Science, INFORMS, vol. 24(4), pages 556-568, February.
    11. Alessandra Buratto & Stefan Wrzaczek, 2018. "Advertising a product to face a competitor entry: a differential game approach," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 41(2), pages 463-487, November.
    12. El Ouardighi, Fouad & Feichtinger, Gustav & Grass, Dieter & Hartl, Richard & Kort, Peter M., 2016. "Autonomous and advertising-dependent ‘word of mouth’ under costly dynamic pricing," European Journal of Operational Research, Elsevier, vol. 251(3), pages 860-872.
    13. Gsottbauer, Elisabeth & van den Bergh, Jeroen C.J.M., 2014. "Environmental policy when pollutive consumption is sensitive to advertising: Norms versus status," Ecological Economics, Elsevier, vol. 107(C), pages 39-50.
    14. S.D. Deshmukh & S.D.Chikte, 1975. "Optimal Stochastic Development Strategy," Discussion Papers 159, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    15. Suresh Chand & Vernon Ning Hsu & Suresh Sethi, 2002. "Forecast, Solution, and Rolling Horizons in Operations Management Problems: A Classified Bibliography," Manufacturing & Service Operations Management, INFORMS, vol. 4(1), pages 25-43, September.
    16. Mesak, Hani I. & Calloway, James A., 1995. "A pulsing model of advertising competition: A game theoretic approach, part A -- Theoretical foundation," European Journal of Operational Research, Elsevier, vol. 86(2), pages 231-248, October.
    17. Schlosser, Rainer, 2017. "Stochastic dynamic pricing and advertising in isoelastic oligopoly models," European Journal of Operational Research, Elsevier, vol. 259(3), pages 1144-1155.

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