A robust bus evacuation model with delayed scenario information

Abstract

Due to natural or man-made disasters, the evacuation of a whole region or city may become necessary. Apart from private traffic, the emergency services also need to consider transit-dependent evacuees which have to be transported from collection points to secure shelters outside the endangered region with the help of a bus fleet. We consider a simplified version of the arising bus evacuation problem (BEP), which is a vehicle scheduling problem that aims at minimizing the network clearance time, i.e., the time needed until the last person is brought to safety. In this paper, we consider an adjustable robust formulation without recourse for the BEP, the robust bus evacuation problem (RBEP), in which the exact numbers of evacuees are not known in advance. Instead, a set of likely scenarios is known. After some reckoning time, this uncertainty is eliminated and planners are given exact figures. The problem is to decide for each bus, if it is better to send it right away—using uncertain information on the evacuees—or to wait until the the scenario becomes known. We present a mixed-integer linear programming formulation for the RBEP and discuss solution approaches; in particular, we present a tabu search framework for finding heuristic solutions of acceptable quality within short computation time. In computational experiments using both randomly generated instances and the real-world scenario of evacuating the city of Kaiserslautern, Germany, we compare our solution approaches.

Appendices

Appendix A: A MIP formulation for the BEP

We present a MIP model for the BEP, that is similar to the one presented in Bish (2011). Table 12 summarizes the variables we use. We choose the concept of rounds to model subsequent bus tours; i.e., one round consists of traveling from one source to one sink. We estimate the maximum number $R$ of rounds a bus needs to do in advance. A trivial way to do so is to set $R = \sum _{i\in \mathcal {S}} \ell _i$.

Table 12 Variables of the BEP MIP formulation

$$\begin{aligned} \min T^\mathrm{evac}&\end{aligned}$$

(32)

$$\begin{aligned} \text {s.t.} T^\mathrm{evac}&\ge \sum _{r\in \mathcal {R}}\left( t^{br}_\mathrm{to} + t^{br}_\mathrm{back} \right) + \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} d^\mathrm{start}_i x^{b1}_{ij} \qquad \forall b\in \mathcal {B} \end{aligned}$$

(33)

$$\begin{aligned} t^{br}_\mathrm{to}&= \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} d_{ij} x^{br}_{ij} \qquad \forall b\in \mathcal {B}, r\in \mathcal {R}\end{aligned}$$

(34)

$$\begin{aligned} t^{br}_\mathrm{back}&\ge d_{ij} \left( \sum _{k\in \mathcal {S}} x^{br}_{kj} + \sum _{l\in \mathcal {T}} x^{b,r+1}_{il} - 1 \right) \nonumber \\&\forall b\in \mathcal {B}, r\in \{1,\ldots ,R-1\}, i\in \mathcal {S}, j\in \mathcal {T}\end{aligned}$$

(35)

$$\begin{aligned} \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} x^{br}_{ij}&\le 1 \qquad \forall b\in \mathcal {B}, r\in \mathcal {R}\end{aligned}$$

(36)

$$\begin{aligned} \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} x^{br}_{ij}&\ge \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} x^{b,r+1}_{ij} \qquad \forall b\in \mathcal {B}, r\in \{1,\ldots ,R-1\} \end{aligned}$$

(37)

$$\begin{aligned} \sum _{j\in \mathcal {T}} \sum _{b\in \mathcal {B}} \sum _{r\in \mathcal {R}} x^{br}_{ij}&\ge \ell _i \qquad \forall i\in \mathcal {S} \end{aligned}$$

(38)

$$\begin{aligned} \sum _{i\in \mathcal {S}} \sum _{b\in \mathcal {B}} \sum _{r\in \mathcal {R}} x^{br}_{ij}&\le u_j \qquad \forall j\in \mathcal {T}\end{aligned}$$

(39)

$$\begin{aligned}&x^{br}_{ij} \in \mathbb {B} \qquad \forall i\in \mathcal {S},j\in \mathcal {T},b\in \mathcal {B},r\in \mathcal {R}\end{aligned}$$

(40)

$$\begin{aligned}&t^{br}_\mathrm{to}, t^{br}_\mathrm{back} \in \mathbb {R_+} \qquad \forall b\in \mathcal {B},r\in \mathcal {R}\end{aligned}$$

(41)

$$\begin{aligned}&T^\mathrm{evac} \in \mathbb {R_+} \end{aligned}$$

(42)

Constraint (33) models the evacuation time as the maximum over the travel times of all buses, for which the auxiliary variables $t^{br}_\mathrm{to}$ and $t^{br}_\mathrm{back}$ are used. These are determined with the help of the following two constraints: Constraint (34) defines the travel time of bus $b$ in round $r$ from its source to its sink, i.e., $t^{br}_\mathrm{to}$. In constraint (35), the value $\sum _{k\in \mathcal {S}} x^{br}_{kj} + \sum _{l\in \mathcal {T}} x^{b,r+1}_{il} - 1$ is one if and only if bus $b$ ends round $r$ at sink $j$, and start round $r+1$ at source $i$. This way, $t^{br}_\mathrm{back}$ is equal to $d_{ij}$ in an optimal solution.

From (36), it follows that each bus can make at most one tour per round. Constraint (37) ensures that a bus can only drive in round $r+1$, if it was also underway in round $r$; this way, “empty” rounds that are not at the end are forbidden.

Finally, constraints (38) and (39) determine that all persons are evacuated, and brought to shelters of sufficient capacity. In this formulation, we assume symmetric travel times between the sources and the sinks; note however, that non-symmetric travel times could be easily included.

Appendix B: A MIP formulation for the linear search approach

We assume a fixed number $B^{hn}$ of here-and-now buses, and a fixed number $B^{ws}$ of wait-and-see buses. Set $\mathcal {B}^{hn} = \{1,\ldots ,B^{hn}\}$, and $\mathcal {B}^{ws} = \{1,\ldots ,B^{ws}\}$. We modify the MIP formulation presented in Sect. 2.2.2 in the following way:

$$\begin{aligned} \min T^\mathrm{evac}&\\ \text {s.t.} \qquad T^\mathrm{evac}&\ge \sum _{r\in \mathcal {R}}\left( t^{br}_\mathrm{to} + t^{br}_\mathrm{back} \right) + \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} d^\mathrm{start}_i x^{b1}_{ij} \qquad \forall b\in \mathcal {B}^{hn}\\ \quad T^\mathrm{evac}&\ge p_\mathrm{wait} + \sum _{r\in \mathcal {R}} \left( t^{crz}_\mathrm{to} + t^{crz}_\mathrm{back} \right) + \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} d^\mathrm{start}_i x^{c1z}_{ij} \quad \forall c\in \mathcal {B}^{ws}, z\in \mathcal {Z}\\ \quad t^{br}_\mathrm{to}&= \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} d_{ij} x^{br}_{ij} \quad \forall b\in \mathcal {B}^{hn}, r\in \mathcal {R}\\ \quad t^{crz}_\mathrm{to}&= \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} d_{ij} x^{crz}_{ij} \quad \forall c\in \mathcal {B}^{ws}, r\in \mathcal {R}, z\in \mathcal {Z}\\ t^{br}_\mathrm{back}&\ge d_{ij} \left( \sum _{k\in \mathcal {S}} x^{br}_{kj} + \sum _{l\in \mathcal {T}} x^{b,r+1}_{il} - 1 \right) \\&\forall b\in \mathcal {B}^{hn}, r\in \{1,\ldots ,R-1\}, i\in \mathcal {S}, j\in \mathcal {T}\\ t^{crz}_\mathrm{back}&\ge d_{ij} \left( \sum _{k\in \mathcal {S}} x^{crz}_{kj} + \sum _{l\in \mathcal {T}} x^{c,r+1,z}_{il} - 1 \right) \\&\forall c\in \mathcal {B}^{ws}, r\in \{1,\ldots ,R-1\}, i\in \mathcal {S}, j\in \mathcal {T}, z\in \mathcal {Z}\\ \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} x^{br}_{ij}&\le 1 \qquad \forall b\in \mathcal {B}^{hn}, r\in \mathcal {R}\\ \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} x^{crz}_{ij}&\le 1 \qquad \forall c\in \mathcal {B}^{ws}, r\in \mathcal {R}, z\in \mathcal {Z}\\ \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} x^{br}_{ij}&\ge \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} x^{b,r+1}_{ij} \quad \forall b\in \mathcal {B}^{hn}, r\in \{1,\ldots ,R-1\}\\ \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} x^{crz}_{ij}&\ge \sum _{i\in \mathcal {S}} \sum _{j\in \mathcal {T}} x^{c,r+1,z}_{ij} \\&\forall c\in \mathcal {B}^{ws}, r\in \{1,\ldots ,R-1\}, z\in \mathcal {Z}\\&\Delta ^{brz}_{ij} \le x^{br}_{ij} \qquad \forall i\in \mathcal {S}, j\in \mathcal {T}, b\in \mathcal {B}^{hn}, r\in \mathcal {R}\\&\quad \sum _{j\in \mathcal {T}} \sum _{r\in \mathcal {R}} \left( \sum _{b\in \mathcal {B}^{hn}} \Delta ^{brz}_{ij} + \sum _{c\in \mathcal {B}^{ws}} x^{crz}_{ij}\right) \ge \ell ^z_i \qquad \forall i\in \mathcal {S}, z\in \mathcal {Z}\\&\quad \sum _{i\in \mathcal {S}} \sum _{r\in \mathcal {R}} \left( \sum _{b\in \mathcal {B}^{hn}} \Delta ^{brz}_{ij} + \sum _{c\in \mathcal {B}^{ws}} x^{crz}_{ij}\right) \le u_j \qquad \forall j\in \mathcal {T}, z\in \mathcal {Z}\\&\quad x^{br}_{ij} \in \mathbb {B} \qquad \forall i\in \mathcal {S},j\in \mathcal {T},b\in \mathcal {B}^{hn},r\in \mathcal {R}\\&\quad x^{crz}_{ij} \in \mathbb {B} \qquad \forall i\in \mathcal {S},j\in \mathcal {T},c\in \mathcal {B}^{ws},r\in \mathcal {R},z\in \mathcal {Z}\\&\quad \Delta _{ij}^z \in \mathbb {Z}_+ \qquad \forall i \in \mathcal {S}, j\in \mathcal {T}, z\in \mathcal {Z}\\&\quad t^{br}_\mathrm{to}, t^{br}_\mathrm{back} \in \mathbb {R_+} \qquad \forall b\in \mathcal {B}^{hn}, r\in \mathcal {R}\\&\quad t^{crz}_\mathrm{to}, t^{crz}_\mathrm{back} \in \mathbb {R_+} \qquad \forall c\in \mathcal {B}^{ws}, r\in \mathcal {R}, z\in \mathcal {Z}\\&\quad T^\mathrm{evac} \in \mathbb {R_+} \end{aligned}$$

Appendix C: Further results for Kaiserslautern

We present the development of upper and lower bounds for the Kaiserslautern instance in Fig. 5. The lower bound LB2 with value $74$ was found in less than one second. The lower bound LBC is not included in the plot, and stalls at a value of $44$ over the considered time horizon. Concerning the upper bounds, Fig. 5 shows that the tabu search approach is able to find a solution after less than one second which is better than the best solution CPLEX produces after the full 180 s computation time.

Fig. 5

Solution quality (in min) over time (in s)