Mathematical analysis of the optimizing acquisition and retention over time problem
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 1, p. 119-137

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.

DOI : https://doi.org/10.1051/m2an:2008043
Classification:  34B15,  34B60,  34B93,  34C11,  34E05,  49N05,  65L10
Keywords: ODE nonlinear boundary value problems; ODE applications; ODE growth, boundedness, comparison of solutions; ODE asymptotic expansions; optimal control; numerical methods ODE boundary value problems
@article{M2AN_2009__43_1_119_0,
     author = {Ditkowski, Adi},
     title = {Mathematical analysis of the optimizing acquisition and retention over time problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {1},
     year = {2009},
     pages = {119-137},
     doi = {10.1051/m2an:2008043},
     zbl = {1156.91432},
     mrnumber = {2494796},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2009__43_1_119_0}
}
Ditkowski, Adi. Mathematical analysis of the optimizing acquisition and retention over time problem. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 1, pp. 119-137. doi : 10.1051/m2an:2008043. http://www.numdam.org/item/M2AN_2009__43_1_119_0/

[1] T. Ambler, Abandon lifetime value theories and take care of customers now. Marketing 12 (2001) 18.

[2] C.M. Bender and S.A. Orszag, Advanced mathematical methods for scientists and engineers, McGraw-Hill International Series in Pure and Applied Mathematics. McGraw-Hill, New York (1978). | MR 538168 | Zbl 0417.34001

[3] R.C. Blattberg and J. Deighton, Managing marketing by the customer equity test. Harvard Business Rev. 74 (1996) 136-144.

[4] R.C. Blattberg, G. Getz and J.S. Thomas, Customer Equity: Building and Managing Relationships as Valuable Assets. Harvard Business School Press, Boston, MA, USA (2001).

[5] H.-C. Chang, D. Gottlieb, M. Marion and B.W. Sheldon, Mathematical analysis and optimization of infiltration processes. J. Scientific Computing 13 (1998) 303-321. | MR 1656916 | Zbl 0933.76089

[6] A. Ditkowski, Numerical method for solving non-contiguous initial/final-value problems. J. Scientific Computing (to appear).

[7] A. Ditkowski, D. Gottlieb and B.W. Sheldon, On the mathematical analysis and optimization of chemical vapor infiltration in materials science. ESAIM: M2AN 34 (2000) 337-351. | Numdam | MR 1765663 | Zbl 0962.76083

[8] A. Ditkowski, B. Libai and E. Muller, Optimizing acquisition and retention over time. Marketing Lett. (submitted).

[9] C. Fornell and B. Wernerfelt, Defensive marketing strategy by customer complaint management: A theoretical analysis. J. Marketing Research 24 (1987) 337-346.

[10] S. Gupta and D.R. Lehmann, Customers as assets. J. Interactive Marketing 17 (2003) 9-24.

[11] S. Gupta, D.R. Lehmann and J.A. Stuart, Valuing customers. J. Marketing Research 41 (2004) 7-18.

[12] D. Hanssens, Allocating marketing communication expenditures: A long run view, in Measuring and Allocating Marcom Budgets: Seven Expert Points of View, Marketing Science Institute, Cambridge, MA, USA (2003).

[13] J.E. Hogan, K.N. Lemon and R.T. Rust, Customer equity management: Charting new directions for the future of marketing. J. Service Research 5 (2002) 4-12.

[14] D. Jain and S.S. Singh, Customer lifetime value research in marketing: A review and future directions. J. Interactive Marketing 16 (2002) 34-46.

[15] M. Kamien and N.L. Schwartz, Dynamic Optimization. Second edition, New York: North Holland (1991). | MR 1159711 | Zbl 0727.90002

[16] L.Y. Lester, CRM meets Wall Street. Target Marketing 26 (2003) 50-54.

[17] R.J. Leveque, Numerical Methods for Conservation Laws, Lectures in Mathematics ETH Zürich. Second edition, Birkhäuser Verlag (1992). | MR 1153252 | Zbl 0723.65067

[18] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in C, The Art of Scientific Computing. Second edition, Cambridge University Press (1992). | MR 1201159 | Zbl 0845.65001

[19] W.J. Reinartz and V. Kumar, On the profitability of long-life customers in a non-contractual setting: An empirical investigation and implications for marketing. J. Marketing 64 (2000) 17-35.

[20] W.J. Reinartz, J. Thomas and V. Kumar, Balancing acquisition and retention resources to maximize customer profitability. J. Marketing 69 (2005) 63-79.

[21] R.T. Rust, K.N. Lemon and V.A. Zeithaml, Return on marketing: Using customer equity to focus marketing strategy. J. Marketing 68 (2004) 109-127.

[22] L. Selden and G. Colvin, How to measure the profitability of your customers. Harvard Business Rev. (2003) 74-81.

[23] J.S. Thomas, A methodology for linking customer acquisition to customer retention. J. Marketing Research 38 (2001) 262-268.

[24] B. Vatanasombut, A.C. Stylianou and M. Igbaria, How to retain online customers. Communication of the ACM 47 (2004) 65-78.

[25] R. Venkatesan and V. Kumar, A customer lifetime value framework for customer selection and optimal resource allocation strategy. J. Marketing 68 (2004) 106-125.

[26] R. Weinstock, Calculus of Variations. Dover Punlications, Inc., NY (1974). | MR 443487 | Zbl 0296.49001

[27] V.A. Zeithaml, Service quality, profitability, and the economic worth of customers: What we know and what we need to learn. J. Acad. Mark. Sci. 28 (2000) 67-85.