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Theoretical Analysis of a Dynamic Pricing Problem with Linear and Isoelastic Demand Functions

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Optimization and Learning (OLA 2021)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1443))

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Abstract

Dynamic pricing strategies are usually adopted to dynamically adjust the products’ prices taking into account demand function characteristics to maximize the revenue. This paper addresses the problem in which a firm has to make decisions about its selling prices in each period to maximize the total profit over the whole horizon. We propose a theoretical analysis of this problem from which we show that: first, when the demand function is linear, the problem can be formulated as a quadratic programming problem. We also present the Karush-Kuhn-Tucker system, which can be used to find the optimal pricing policy when the objective function is concave. Then, when the demand is isoelastic, we also show that the problem can be reduced to the maximization of N independent functions in bounded intervals. Some numerical examples are provided to illustrate the results obtained for both the linear and isoelastic cases.

Supported by the European Regional Development Fund (FEDER) and the Industrial Chair Connected-Innovation (https://chaire-connected-innovation.fr/).

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Correspondence to Mourad Terzi .

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Appendix A KKT system for Example 1

Appendix A KKT system for Example 1

From the parameters values presented in the Table 1, the problem optimization problem is formulated as:

$$\begin{aligned}&\min _{P_{1},P_{2}} \varPi ' = \frac{1}{2}(P_{1}, P_{2}) \begin{pmatrix} 2 &{} -3 \\ -3 &{} 6 \end{pmatrix} \begin{pmatrix} P_{1} \\ P_{2} \end{pmatrix} + (-2,-13) \begin{pmatrix} P_{1} \\ P_{2} \end{pmatrix} \\&s.t: \begin{pmatrix} -1 &{} 2 \\ 1 &{} -2 \\ 1 &{} 0 \\ -1 &{} 0 \\ 1 &{} -3 \\ -1 &{} 3 \\ 0 &{} 1 \\ 0 &{} -1 \end{pmatrix} \begin{pmatrix} P_{1} \\ P_{2} \end{pmatrix} \le E_{8,1} = \begin{pmatrix} 1 \\ 0 \\ 8 \\ -1 \\ 1 \\ 6 \\ 8 \\ -1 \end{pmatrix} \\ \end{aligned}$$

The KKT system is defined as:

$$\begin{aligned}&\begin{pmatrix} 2 &{} -3 \\ -3 &{} 6 \\ \end{pmatrix} \begin{pmatrix} P_{1} \\ P_{2} \end{pmatrix} + \begin{pmatrix} -2 \\ -13 \end{pmatrix} + \begin{pmatrix} -1 &{} 1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 0 &{} 0 \\ 2 &{} -2 &{} 0 &{} 0 &{} -3 &{} 3 &{} 1 &{} -1 \end{pmatrix} \begin{pmatrix} \lambda _{1} \\ \lambda _{2} \\ \lambda _{3} \\ \lambda _{4} \\ \lambda _{5} \\ \lambda _{6} \\ \lambda _{7} \\ \lambda _{8} \\ \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \\&\begin{pmatrix} -1 &{} 2 \\ 1 &{} -2 \\ 1 &{} 0 \\ -1 &{} 0 \\ 1 &{} -3 \\ -1 &{} 3 \\ 0 &{} 1 \\ 0 &{} -1 \end{pmatrix} \begin{pmatrix} P_{1} \\ P_{2} \end{pmatrix} - \begin{pmatrix} 1 \\ 0 \\ 8 \\ -1 \\ 1 \\ 6 \\ 8 \\ -1 \end{pmatrix} + \begin{pmatrix} s_{1} \\ s_{2} \\ s_{3} \\ s_{4} \\ s_{5} \\ s_{6} \\ s_{7} \\ s_{8} \end{pmatrix} =\begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} \\&(\lambda _{1}, \lambda _{2}, \lambda _{3}, \lambda _{4}, \lambda _{5}, \lambda _{6}, \lambda _{7}, \lambda _{8}) \begin{pmatrix} s_{1} \\ s_{2} \\ s_{3} \\ s_{4} \\ s_{5} \\ s_{6} \\ s_{7} \\ s_{8} \end{pmatrix} = 0 \\&\lambda _{j}, s_{j} \ge 0, \qquad j = 1,2,...,8 \end{aligned}$$

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Terzi, M., Ouazene, Y., Yalaoui, A., Yalaoui, F. (2021). Theoretical Analysis of a Dynamic Pricing Problem with Linear and Isoelastic Demand Functions. In: Dorronsoro, B., Amodeo, L., Pavone, M., Ruiz, P. (eds) Optimization and Learning. OLA 2021. Communications in Computer and Information Science, vol 1443. Springer, Cham. https://doi.org/10.1007/978-3-030-85672-4_23

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  • DOI: https://doi.org/10.1007/978-3-030-85672-4_23

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