An Integrated Approach to Pricing, Inventory, and Market Segmentation Decisions with Demand Leakage

Abstract

Differentiated pricing is among the widely practised Revenue Management (RM) tactics in which a firm offers its products/services at differentiated prices to distinct markets. Earlier researches have shown that the benefits from differentiated pricing are evident when the market segmentation is assumed perfect which are regarded as distinct markets with deterministic demands. In perfect market segmentation customers associated with a market segment do not cannibalize (move) between market segments. However, it is not uncommon to notice that the market segmentation a firm exercises is seldom perfect, and due to imperfect segmentation customers cannibalize between market segments which is also referred as demand leakage. In addition to this, the demand is often uncertain, and thus a firm also experiences short sales and leftovers due to uncertain demand. This research addresses the issue of establishing an integrated framework to optimize price differentiation strategy, pricing, and order quantity for a firm that experiences demand leakage. The models to determine the optimal market segmentation strategy, pricing, and order quantities for a firm are developed facing price dependent deterministic demand, stochastic demand, and when the demand is stochastic, yet the distribution is unknown. The models are analyzed to identify the optimal pricing, order quantities, and price differentiation strategy. Numerical experimentation show that optimizing the price differentiation strategy (market segmentation) along with optimizing the joint pricing and order quantity decisions price significantly improve the revenue to a firm although it experiences customer cannibalization. This paper, however, only highlights the deterministic model and its analysis.

Appendix

Proof of Proposition 1

Using the Karush Kuhn Tucker Optimality condition we get the following Lagrange function for the problem

$$\begin{aligned} L(p_1,p_2,\upsilon ,\lambda _1, \lambda _2)= (p_1 -c) z_1 + (p_2- c) z_2 + \lambda _1(p_1 -\upsilon ) + \lambda _2(\upsilon - p_2) \end{aligned}$$

(4.6)

Where in Eq. 4.6, $z_1= (1 -\theta ) (\alpha _1 - \beta p_1) + \mu _1$, and $z_2= (\beta \upsilon - \beta p_2) + \theta (\alpha _1 -~\beta \upsilon )+\mu _2$. Also $\lambda _1$, and $\lambda _2$ are non-negative. The First order Optimality Conditions (FOCs) are:

$$\begin{aligned} \displaystyle \frac{\partial L}{\partial p_1}&= (p_1 -c) \frac{\partial z_1}{\partial p_1} + z_1 + \lambda _1 \le 0, p_1 \ge 0, \left( (p_1 -c) \frac{\partial z_1}{\partial p_1} + z_1 + \lambda _1 \right) p_1 =0 \end{aligned}$$

(4.7)

$$\begin{aligned} \displaystyle \frac{\partial L}{\partial p_2}&= (p_2 -c) \frac{\partial z_2}{\partial p_2} + z_2 - \lambda _2 \le 0, p_2 \ge 0, \left( (p_2 -c) \frac{\partial z_2}{\partial p_2} + z_2 - \lambda _2\right) p_2=0 \end{aligned}$$

(4.8)

$$\begin{aligned} \displaystyle \frac{\partial L}{\partial \upsilon }&= (1 - \theta ) \beta - \lambda _1 + \lambda _2 \le 0, \left( (1 - \theta ) \beta - \lambda _1 + \lambda _2\right) \upsilon =0 \end{aligned}$$

(4.9)

$$\begin{aligned} \displaystyle \frac{\partial L}{\partial \lambda _1}&= p_1 - \upsilon \ge 0, \lambda _1 \ge 0, \left( p_1 - \upsilon \right) \lambda _1 =0 \end{aligned}$$

(4.10)

$$\begin{aligned} \displaystyle \frac{\partial L}{\partial \lambda _2}&= \upsilon - p_2 \ge 0,\lambda _2 \ge 0, \left( \upsilon - p_2 \right) \lambda _2 =0 \end{aligned}$$

(4.11)

Consider Eq. 4.10, since $p_1 \ge \upsilon$, and $\left( \upsilon - p_1\right) \lambda _1 =0$, and $\lambda _1 \ge 0$, thus the only feasible solution is, $\upsilon = p_1$, and $\lambda _1 > 0$. Now, since $\upsilon = p_1$, from Eq. 4.11, we can only have $p_2 < \upsilon$, which is possible only when $\lambda _2=0$. Using Eq. 4.9, as $\upsilon >0$, therefore,$\lambda _1= (1-\theta ) \beta$, and $\lambda _2= 0$. To find optimal prices, $p_1$, and $p_2$, we need to solve Eqs. 4.7 and 4.8. Notice here, $\displaystyle \frac{\partial z_1}{\partial p_1} =-(1- \theta ) \beta$, and $\displaystyle \frac{\partial z_2}{\partial p_2}= - \beta$. The unique optimal prices would be: $p_1 = \displaystyle \frac{\mu _1}{2 \beta (1-\theta )}+\frac{\alpha _1+\beta +\beta c}{2 \beta }$; and $p_2 = \displaystyle \frac{\alpha _1+\beta +\theta \left( \alpha _1-\beta (c+1)\right) +3 \beta c+\mu _1+2 \mu _2}{4 \beta }$.