Volume 12, no. 4Pages 52 - 66
Mathematical Model of Multichannel Advertising Management with Continuously Distributed LagsI.V. Lutoshkin, N.R. Yamaltdinova
The dynamic model of optimal distribution of advertising costs on the planning period is analyzed. It is assumed that companies can use several media channels of different quality and influence on demand. We consider the delayed reaction of consumers to advertising and non-advertising factors. Unlike such classical dynamic optimization models as Nerlove-Arrow model, Vidal-Wolf model and their extensions, the proposed model takes into account accumulated effects of advertising of several media channels and previous sales. In the framework of the proposed model, we consider the problem of optimal control of advertising costs with a nonlinear Volterra integral equation, which is generated by the natural constraints. The theorem on existence of solution to this equation is proved. Also, we formulate the theorem on existence of solution to the total profit maximization problem for the planning period under restriction. The maximum principle is applied to the problem and the necessary conditions for constructing an optimal strategy are found.Full text
- mathematical model; advertising strategy; Volterra equation; optimal control; maximum principle; existence of solution.
- 1. Granger C.W.J. Investigating Causal Relations by Econometric Model and Cross-Spectral Methods. Econometrica, 1969, vol. 37, no. 3, pp. 424-438.
2. Bensoussan A., Bultezn A., Naert P. A Generalization of the Nerlove-Arrow Optimality Condition. Brussels, European Institute for Advanced Studies in Management, 1973.
3. Pauwels W. Optimal Dynamic Advertising Policies in the Presence Of Continuously Distributed Time Lags. Journal of Optimization Theory and Applications, 1977, vol. 22, no. 1, pp. 79-89.
4. Lutoshkin I.V., Martynenko E.V. Comparison of Sales of Products of Different Types Depending on Advertising Influences [Sravnenie prodazh produkcii razlichnyh vidov v zavisimosti ot reklamnyh vozdeystviy]. News Of Higher Educational Institutions. A Series: Economy, Finance And Production Management, 2015, vol. 3, no. 25, pp. 113-121. (in Russian)
5. Nerlove M.K., Arrow J. Optimal Advertising Policy Under Dynamic. Economica, 1962, vol. 29, no. 114, pp. 129-142.
6. Erickson G.M. A Differential Game Model of the Marketing-Operations Interface. European Journal of Operational Research, 2011, vol. 211, no. 2, pp. 394-402.
7. Fruchter G.E. Signaling Quality: Dynamic Price-Advertising Model. Journal of Optimization Theory and Applications, 2009, vol. 143, no. 3, pp. 479-496.
8. Lambertini L. Advertising in a Dynamic Spatial Monopoly. European Journal of Operational Research, 2005, vol. 166, no. 2, pp. 547-556.
9. Giovanni P.D. Quality Improvement vs. Advertising Support: Which Strategy Works Better for a Manufacturer? European Journal of Operational Research, 2011, vol. 208, no. 2, pp. 119-130.
10. Buratto A., Zaccour G. Coordination of Advertising Strategies in a Fashion Licensing Contract. Journal of Optimization Theory and Applications, 2009, vol. 142, no. 1, pp. 31-53.
11. Vidale M.L., Wolfe H.B. An Operations-Research Study of Sales Response to Advertising. Operations Research, 1957, vol. 5, no. 3, pp. 370-381.
12. Deal K.R. Optimizing Advertising Expenditures in a Dynamic Duopoly. Operations Research, 1979, vol. 27, no. 4, pp. 682-692.
13. Mukundan R.W., Elsner B. Linear Feedback Strategies in Non-Zero-Sum Differential International. Journal of System Science, 1975, vol. 6, no. 6, pp. 513-532.
14. Wang Q., Wu A. A Duopolistic Model of Dynamic Competitive Advertising. European Journal of Operational Research, 2001, vol. 128, no. 1, pp. 213-226.
15. Fershtman C. Goodwill and Market Shares in Oligopoly. Economica, 1984, vol. 51, no. 203, pp. 271-281.
16. He X.L., Krishnamoorthy A., Prasad A., Sethi S.P. Retail Competition and Cooperative Advertising. Operations Research Letters, 2011, vol. 39, no. 1, pp. 11-16.
17. Fershtman C., Vijay M., Eitan M. Market Share Pioneering Advantage: A Theoretical Approach. Management Science, 1990, vol. 36, no. 8, pp. 900-918.
18. Kimball G.E. Some Industrial Applications of Military Operations Research Methods. Operations Research, 1957, vol. 5, no. 2, pp. 201-204.
19. Buratto A., Grosset L., Viscolani B. Advertising Channel Selection in a Segmented Market. Automatica, 2006, vol. 42, no. 8, pp. 1343-1347.
20. Lutoshkin I.V., Yamaltdinova N.R. The Dynamic Model of Advertising Costs with Continuously Distributed Lags. CEUR Workshop Proceedings, vol. 2018, pp. 103-112.
21. Krasnov M.L. Integral Equation [Integral'nye uravneniya]. Moscow, Nauka, 1975. (in Russian)
22. Dmitruk A.V., Osmolovskiy N.P. Neobhodimye usloviya slabogo minimuma v zadachah s integral'nymi uravneniyami [The Necessary Conditions for a Weak Minimum in Problems with Integral Equation]. Proceedings of the XII Russian Meeting on Management Problems, 2014, pp. 709-713. (in Russian)
23. Lutoshkin I.V., Yamaltdinova N.R. The Dynamic Model of Advertising Costs. ECECSR Journal, 2018, vol. 52, no. 1, pp. 201-214.