Multi-start Local Search Procedure for the Maximum Fire Risk Insured Capital Problem

Abstract

A recently European Commission regulation requires insurance companies to determine the maximum value of insured fire risk policies of all buildings that are partly or fully located within circle of a radius of 200 m. In this work, we present the multi-start local search meta-heuristics that has been developed to solve the real case of an insurance company having more than 400 thousand insured buildings in mainland Portugal. A random sample of the data set was used and the solutions of the meta-heuristic were compared with the optimal solution of a MILP model based on the Maximal Covering Location Problem. The results show the proposed approach to be very efficient and effective in solving the problem.

Appendix

Let $(a_i,b_i)$ be the longitude and latitude coordinates of building i, $i=1,...,n$, $R_i$ be the risk associated with building i, and $D_{ij}$ the Euclidean distance between buildings i and j, and consider the set

$$\varDelta _{ij}=\{(i,j): D_{ij} \le 2k+\epsilon \},$$

with $\epsilon >0$ and k the circle radius.

Let two binary variables be defined as: $x_{ij}=1$ if building i is covered by the circle centred at j, 0 otherwise; and $y_j=1$ if j is the centre of the circle.

$$\begin{aligned} \begin{aligned}&\text {max} \quad \sum _{ij\in \varDelta _{ij}}R_{ij}x_{ij} \\&\text {s.t.} \quad D_{ij}x_{ij} \le k y_i \, \quad (i,j)\in \varDelta _{ij}\\&\quad \quad \sum _{i:=1}^n y_i =1 \\&\quad \quad x_{ii} \le y_i \, \quad i\in \{1,...,n\}\\&\quad \quad x_{ij}, y_i \in \{ 0, 1\} \\ \end{aligned} \end{aligned}$$

The objective function is defined by the sum of the fire risks insured of the buildings covered by the circle. The first constraint assures that only the buildings distancing less that k from the circle centre $y_j$ will be considered. The second constraint assures that only one circle is determined. The last constraint is needed since one has a maximization model. Notice that $D_{ii}=0, \, i=1,...,n$. Therefore, all $x_{ii}=1$ will verify the first constraint, whatever the value of $y_i$.