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Mathematical analysis of the optimizing acquisition and retention over time problem. (English) Zbl 1156.91432
Summary: While making informed decisions regarding investments in customer retention and acquisition becomes a pressing managerial issue, formal models and analysis, which may provide insight into this topic, are still scarce. In this study we examine two dynamic models for optimal acquisition and retention models of a monopoly, the total cost and the cost per customer models. These models are analytically analyzed using classical, direct, methods and asymptotic expansions (for the total cost model). In order to numerically simulated the models, an innovative numerical method was developed for solving ODE systems with initial/final value problems.

MSC:
91B62 Economic growth models
90B50 Management decision making, including multiple objectives
34B15 Nonlinear boundary value problems for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B60 Applications of boundary value problems involving ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
49N05 Linear optimal control problems
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