Modelling beneficiaries’ choice in disaster relief logistics

Abstract

This paper introduces the element of beneficiaries’ choice into a location-routing problem for disaster relief logistics suited for decision support systems. Decision makers in humanitarian logistics face the challenge where to establish distribution centers (DCs) for relief goods. For this purpose, two objectives are considered: the impact of the relief operations on the beneficiaries and the efficient use of monetary resources. The proposed multi-objective location-routing model minimizes unserved demand as well as cost for opening DCs and for routing relief goods. It anticipates the choice of beneficiaries to which DC to go (if at all), based on a model adopted from the literature on competitive location analysis. A mathematical programming formulation is presented. For small instances, the Pareto front can be determined exactly using an epsilon constraint method. For solving also realistic instances, an evolutionary algorithm has been implemented and evaluated. The algorithms are tested on real-world instances from Mozambique. The results show that when designing a distribution network, improvements can be achieved by taking the predicted behavior of beneficiaries into account.

Appendix: Capacity calculation

The DC capacity calculation is explained here in more detail:

A first idea (see Equation (50)) for calculating appropriate capacities of DCs in a beneficiaries choice model was to sum up the number of inhabitants of the PCs within the sphere of influence of the DC and dividing it by the average number of DCs available to the PCs considered this way. Here , $M_{i_k}$ is the set of PCs k within the sphere of influence of DC i, $m_i$ is the number of PCs within sphere of influence from DC i and $n_k$ is the number of DCs within sphere of influence of PC k. This resulted in very low levels of capacity, forcing the opening of the majority of DCs to supply the population.

$$\begin{aligned} \gamma _i=\left( \sum \limits _{k \in M_{i_k}}^{}p_k\right) \left( \frac{\sum \nolimits _{k \in M_{i_k}}^{}n_k}{m_i}\right) ^{-1} \end{aligned}$$

(50)

Therefore, a concept following the logic of the Economic Order Quantity (EOQ) of Harris (1990) is employed. For this purpose, the cost to deliver to a DC has been approximated by considering full truckloads in $c_{FTL_i}=2 \cdot d_{0i}$. If the fixed cost for the DC is now divided by the approximated cost to deliver goods, one can see the ratio of those cost types $r_{EOQ}$ (see Equation (51)). A high ratio would signal high opening cost, and favoring less opened DC with higher capacities from a cost perspective, while lower ratios describe a situation in which the fixed opening cost is not so dominant, allowing for more DCs with lower capacities.

$$\begin{aligned} r_{EOQ} = \frac{c_i}{c_{FTL_i}} \end{aligned}$$

(51)

The $r_{EOQ}$ is used to adjust the capacities for this cost ratios to increase the capacities if it is advisable from a cost perspective while the focus on the unserved remains intact due to considering only everything within the sphere of influence.

$$\begin{aligned} \gamma _i = \left( \sum \limits _{k \in M_{i_k}}^{}p_k\right) \cdot \left( \frac{\sum \nolimits _{k \in M_{i_k}}^{}n_k}{m_i}\right) ^{-1} \cdot \frac{c_i}{2 \cdot d_{0i}} \end{aligned}$$

(52)

Constraint (53) makes sure that at least all demand can be fulfilled if every DC is opened, and that the capacities of the DCs are not higher than necessary to supply all PCs within the sphere of influence.

$$\begin{aligned} 1\le \left( \frac{\sum \nolimits _{k \in M_{i_k}}^{}n_k}{m_i}\right) ^{-1} \cdot \frac{c_i}{2 \cdot d_{0i}} \le \left( \frac{\sum \nolimits _{k \in M_{i_k}}^{}n_k}{m_i}\right) \end{aligned}$$

(53)