Exact computational approaches to a stochastic uncapacitated single allocation p-hub center problem

Abstract

The stochastic uncapacitated single allocation p-hub center problem is an extension of the deterministic version which aims to minimize the longest origin-destination path in a hub and spoke network. Considering the stochastic nature of travel times on links is important when designing a network to guarantee the quality of service measured by a maximum delivery time for a proportion of all deliveries. We propose an efficient reformulation for a stochastic p-hub center problem and develop exact solution approaches based on variable reduction and a separation algorithm. We report numerical results to show effectiveness of our new reformulations and approaches by finding global solutions of small-medium sized problems. The combination of model reformulation and a separation algorithm is particularly noteworthy in terms of computational speed.

Appendix

Proof of Lemma 1

(a) Suppose that X _jm =1. Let j=i and m=k. Then constraint (7) reduces to $\beta\geq t_{jjmm} + z_{\gamma} \delta_{jjmm} = 2 d_{jm} + z_{\gamma} \sqrt{2} \sigma_{jm}$, which implies that a feasible solution with X _jm =1 gives a worse objective function value than the current available upper bound β ^U. Hence, X _jm =0 is a valid cut for SpHCP^L.

(b) Suppose that X _jm =1. Then constraint (7) reduces to

$$\sum_{k = 1}^N (t_{ijkm} + z_{\gamma} \delta_{ijkm}) X_{ik} \geq \min _{k=1}^N (t_{ijkm} + z_{\gamma} \delta_{ijkm}) \sum_{k =1}^N X_{ik} = \min_{k=1}^N (t_{ijkm} + z_{\gamma} \delta_{ijkm}), $$

which is greater than β ^U according to the assumption. This shows that any feasible solution with X _jm =1 gives a worse objective function value than β ^U. Therefore, X _jm =0 is a valid cut for SpHCP^L.

(c) Suppose that X _jm =1. Then constraint (7) reduces to

$$\begin{aligned} \beta \geq&\sum_{k =1}^N (t_{ijkm} + z_{\gamma} \delta_{ijkm}) X_{ik} \leq \max _{k=1}^N (t_{ijkm} + z_{\gamma} \delta_{ijkm}) \sum_{k =1}^N X_{ik} \\ =& \max_{k=1}^N (t_{ijkm} + z_{\gamma} \delta_{ijkm}) < \beta^L, \end{aligned}$$

which is redundant for the given i,j,m.

(d) Suppose that X _jm =1 and assume $d_{mi} = \max_{\ell=1}^{N} d_{m\ell}$ and X _in =1. It follows from constraint (7) and the triangle inequality property that

$$\begin{aligned} \beta \geq& (t_{ijnm} + z_{\gamma} \delta_{ijnm}) \\ = & d_{in} + \alpha d_{nm} + d_{mj} + z_{\gamma} \sqrt{(\sigma_{in})^2 + \alpha^2 (\sigma_{nm})^2 + (\sigma _{mj})^2} \\ \geq& \alpha d_{im} + d_{mj} + z_{\gamma} \sigma_{mj} \\ = & d_{mj} + \alpha\max_{\ell=1}^N d_{m\ell} + z_{\gamma} \sigma _{mj} \\ > & \beta^U. \end{aligned}$$

This shows that any feasible solution with X _jm =1 gives a worse objective function value than β ^U. Hence X _jm =0 is a valid cut.

(e) When X _jm =0, the right-hand side of constraint (7) for any i and the corresponding j,m is non-positive. Clearly, this constraint is redundant for SpHCP^L. □