Hub Location Problems with Price Sensitive Demands

Abstract

This work addresses the hub location problem with price-sensitive demands. The article analyzes sensitivity of demand to quality of service in such systems, and enables the solution of large-scale instances. Two distinct working formulations are provided, and an improved Benders decomposition algorithm is deployed. Simulation of consumer choice between competing services is addressed in the computational experiments. Further, a specialized sub-problem solution procedure which is able to deliver good Benders cuts in reasonable time is developed. The results are illustrated with standard test data sets. The research contributes to a better understanding of hub traffic with varying service levels, as well as price equilibrium in competitive markets.

Appendix: A: Economic equilibrium conditions

Considering formulation (1–13), for a given (fixed) state of the γ vector (γ=γ ^h), the resulting problem is searching for a maximization of a concave cost function subject to linear constraints. As the matrix W is assumed to be symmetric, the line integral in Eq. 1 is analytically tractable and the Karush-Kuhn-Tucker Conditions are are necessary and sufficient for a global optimum. Therefore, associating the dual variables u _{i j k} ≥0, $\ddot {\rho }_{ij} \geq 0$, $\bar {\rho }_{ij} \geq 0$ and $\tilde {\rho }_{ij} \geq 0$ respectively to constraints Eqs. 2, 3, 4 and 5, the following relations are verified at an optimal solution for each ij pair, provided that constraints Eqs. 2–13 are also satisfied

$$\begin{array}{@{}rcl@{}} \boldsymbol {\rho} = \boldsymbol {\phi}(\mathbf{w}) && \end{array} $$

(43)

$$\begin{array}{@{}rcl@{}} \ddot{c}_{ijkm} - \ddot{\rho}_{ij} + u_{ijk} + u_{ijm} \geq 0 && \forall (k,m) \end{array} $$

(44)

$$\begin{array}{@{}rcl@{}} \bar{c}_{ijl} - \bar{\rho}_{ij} + u_{ijl} \geq 0 && \forall l \end{array} $$

(45)

$$\begin{array}{@{}rcl@{}} \tilde{c}_{ij} - \tilde{\rho}_{ij} \geq 0 && \end{array} $$

(46)

$$\begin{array}{@{}rcl@{}} [ \ddot{c}_{ijkm} - \ddot{\rho}_{ij} + u_{ijk} + u_{ijm} ] \ x_{ijkm} = 0 && \forall (k,m) \end{array} $$

(47)

$$\begin{array}{@{}rcl@{}} [ \bar{c}_{ijl} - \bar{\rho}_{ij} + u_{ijl} ] \ y_{ijl} = 0 && \forall l \end{array} $$

(48)

$$\begin{array}{@{}rcl@{}} [ \tilde{c}_{ij} - \tilde{\rho}_{ij} ] \ z_{ij} = 0 && \end{array} $$

(49)

$$\begin{array}{@{}rcl@{}} [ y_{ijk} + {\sum}_{m , m \neq k} x_{ijkm} + {\sum}_{m, m \neq k} x_{ijmk} - b_{ij} {\gamma_{k}^{h}} ] \ u_{ijk} = 0 && \forall k \end{array} $$

(50)

$$\begin{array}{@{}rcl@{}} [ \ddot{w}_{ij} - {\sum}_{(k,m)} x_{ijkm} ] \ \ddot{\rho}_{ij} = 0 && \end{array} $$

(51)

$$\begin{array}{@{}rcl@{}} [ \bar{w}_{ij} - {\sum}_{l} y_{ijl} ] \ \bar{\rho}_{ij} = 0 && \end{array} $$

(52)

$$\begin{array}{@{}rcl@{}} [ \tilde{w}_{ij} - z_{ij} ] \ \tilde{\rho}_{ij} = 0 && \end{array} $$

(53)

Note that at the optimal solution, for each l,k,m,k≠m such that ${\gamma _{l}^{h}} = {\gamma _{k}^{h}} = {\gamma _{m}^{h}} = 1$, complementary slackness conditions (50) will enforce u _{i j l} =u _{i j k} =u _{i j m} =0, as required for an uncapacitated approach (b _{i j} very large, not disturbing the economic equilibrium). Observing conditions (47–49), is straightforward to conclude that $\tilde {\rho }_{ij} - \tilde {c}_{ij} = \bar {\rho }_{ij} - \bar {c}_{ijl} = \ddot {\rho }_{ij} - \ddot {c}_{ijkm} = 0$.