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Optimal double control problem for a PDE model of goodwill dynamics

Author

Listed:
  • Mariusz Górajski

    (University of Łódź)

  • Dominika Machowska

    (University of Łódź)

Abstract

We propose a new optimal control model of product goodwill in a segmented market where the state variable behaviour is described by a partial differential equation of the Lotka–Sharp–McKendrick type. In order to maximize the sum of discounted profits over a finite time horizon, we control the marketing communication activities which influence the state equation and the boundary condition. Moreover, we introduce the mathematical representation of heterogeneous electronic word of mouth. Based on the semigroup approach, we prove the existence and uniqueness of optimal controls. Using a maximum principle, we describe a numerical algorithm to find the optimal solution. Finally, we examine several examples on the optimal goodwill model and discover two types of marketing strategies.

Suggested Citation

  • Mariusz Górajski & Dominika Machowska, 2017. "Optimal double control problem for a PDE model of goodwill dynamics," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(3), pages 425-452, June.
    • Handle: RePEc:spr:mathme:v:85:y:2017:i:3:d:10.1007_s00186-017-0577-1
    DOI: 10.1007/s00186-017-0577-1
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    References listed on IDEAS

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    Cited by:

    1. Mariusz Górajski & Dominika Machowska, 2019. "How do loyalty programs affect goodwill? An optimal control approach," 4OR, Springer, vol. 17(3), pages 297-316, September.

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    More about this item

    Keywords

    Optimal boundary control; Nonlinear Lotka–Sharp–McKendrick equation; Product goodwill; eWOM;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D42 - Microeconomics - - Market Structure, Pricing, and Design - - - Monopoly
    • M37 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Marketing and Advertising - - - Advertising

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