Robust Facility Location Problem for Hazardous Waste Transportation

Abstract

We consider a robust facility location problem for hazardous materials (hazmat) transportation considering routing decisions of hazmat carriers. Given a network and a known set of nodes from which hazmat originate, we compute the locations of hazmat processing sites (e.g. incinerators) which will minimize total cost, in terms of fixed facility cost, transportation cost, and exposure risk. We assume that hazmat will be taken to the closest existing processing site. We present an exact full enumeration method, which is useful for small or medium-size problems. For larger problems, the use of a genetic algorithm is explored. Through numerical experiments, we discuss the impact of uncertainty and robust optimization in the hazmat combined location-routing problem.

Appendix

1.1 Karush Kuhn-Tucker Conditions for the Shortest Path Problem

In order to convert the two-level problem into a single level optimization problem, we want to be able to replace the inner minimization (shortest path) problem with an equivalent set of constraints. As we recall, the inner shortest path problem for shipment s was

$$\min_{x^{s}} \sum\limits_{(i,j) \in \mathcal{A}} L_{ij} x^{s}_{ij} $$

subject to

$$\begin{array}{@{}rcl@{}} &&\sum\limits_{(i,k)\in\mathcal{A}} x^{s}_{ik} - \sum\limits_{(k,i)\in\mathcal{A}} x^{s}_{ki}\left\{ \begin{aligned} = +1 & i=o(s) \\ \geq -y_{i} & i \in \mathcal{M} \\ = 0 & \,\, \mathrm{otherwise} \end{aligned}\right. \quad \forall i\in\mathcal{N} \\ && - x^{s}_{ij} \leq 0 \quad \forall (i,j)\in\mathcal{A} \end{array} $$

which we put into the standard g(x) ≤ 0, h(x) = 0 form:

$$\begin{array}{@{}rcl@{}} - \sum\limits_{(i,k)\in\mathcal{A}} x^{s}_{ik} + \sum\limits_{(k,i)\in\mathcal{A}} x^{s}_{ki} - y_{i} & \leq & 0 \quad \forall i \in \mathcal{M} \\ - x^{s}_{ij} & \leq & 0 \quad \forall (i,j)\in\mathcal{A} \\ - \sum\limits_{(o(s),k)\in\mathcal{A}} x^{s}_{o(s),k} + \sum\limits_{(k,o(s))\in\mathcal{A}} x^{s}_{k,o(s)} + 1 & = & 0 \\ - \sum\limits_{(i,k)\in\mathcal{A}} x^{s}_{ik} + \sum\limits_{(k,i)\in\mathcal{A}} x^{s}_{ki} & = & 0 \quad \forall i \in \mathcal{N} , i \neq o(s), i \notin \mathcal{M} \end{array} $$

The corresponding KKT conditions are, first, stationarity

$$\begin{array}{@{}rcl@{}} L_{ij} - \zeta_{i}^{s} + \zeta_{j}^{s} - \phi_{ij}^{s} & = & 0 \qquad \forall (i,j) \in \mathcal{A} \end{array} $$

then complementary slackness

$$\begin{array}{@{}rcl@{}} \phi^{s}_{ij} x^{s}_{ij} & = & 0 \quad \forall (i,j) \in \mathcal{A} \\ \left( - \sum\limits_{(i,k)\in\mathcal{A}} x^{s}_{ik} + \sum\limits_{(k,i)\in\mathcal{A}} x^{s}_{ki} - y_{i} \right) \zeta_{i} & = & 0 \quad \forall i=\mathcal{M} \end{array} $$

and finally dual feasibility

$$\begin{array}{@{}rcl@{}} \phi^{s}_{ij} & \geq & 0 \qquad \forall (i,j) \in \mathcal{A} \\ \zeta^{s}_{i} & \geq & 0 \qquad \forall i \in \mathcal{M} \end{array} $$