Creating contiguous service areas around points of dispensing for resource distribution during bio-emergencies

Abstract

Response plans in preparation for public health emergencies often involve the setup of facilities like shelters, ad-hoc clinics, etc. to serve the affected population. While public health authorities frequently have prospective facility locations, balancing the demand or population at these facilities can be challenging. Assigning populations to their closest facilities may lead to uneven distribution of demand. This research proposes a novel greedy heuristic algorithm to create service areas around given facilities such that the population to be served by each facility is uniform or proportional to available resources. This algorithm has been implemented in the context of response plans for bio-emergencies in Denton County, Texas, USA. Given the location of Points of Dispensing (PODs), the objective is to create contiguous catchment areas, each served by one POD such that demand distribution constraints are satisfied. While the demand distribution constraints are hard constraints, it is also preferred that populations are mapped to PODs as close to them as possible. A response plan defines a mapping of populations to facilities and presents a combinatorial optimization problem in which the average distance between population locations and PODs is the cost function value, and demand equity and contiguity of catchment areas are hard constraints. We present a decision support system for planners to select solutions based on the compactness of catchment areas, the average distance between populations and PODs, and execution time, given that all solutions have contiguous catchment areas and balanced demand.

Appendices

Appendix : A

A.1 Forcing contiguity

A demand point is transferred from facility f_j to facility f_k only if the service area corresponding to f_j is adjacent to the service area for f_k. While this condition is a necessary condition for contiguity of service areas, it does not guarantee contiguity of the resulting service areas for both f_j and f_k. As can be seen in Fig. 12 the donating facility service area will lose contiguity if the shaded demand point is selected for transfer to another facility. Figure 13 shows an example where a facility service area (f₂) becomes disconnected after a demand point is transferred to it (from f₁). Hence, if the contiguity of service areas is a hard constraint, then while selecting a demand point for transfer this condition needs to be taken into account. If the selection of the demand point from f_j that minimizes the cost of transfer results in non-contiguous service areas then the next closest demand point is selected until this condition is met. If no such demand point can be found, the path is discarded.

Each service area can either be of polygon type or multi-polygon type [44]. If the service area is a polygon, it signifies that it is a contiguous region, however, if it is a multi-polygon it may be a collection of connected or disconnected polygons. Figure 14 shows examples of both a multi-polygon with connected polygons (I) and a multi-polygon with disconnected polygons(II). The contiguity condition is checked by traversing the graph formed by the polygons in a multi-polygon service area if any polygon is unreachable, it shows that the service area is non-contiguous.

Fig. 12

Example of loss of contiguity for the donating facility service area. If this service area loses the shaded demand point, the service area becomes disconnected

Fig. 13

Example of loss of contiguity for the accepting facility service area. The service area corresponding to facility f₂ becomes disconnected after gaining its closest demand point from f₁

Fig. 14

Types of multi-polygons

A.2 Computational complexity

Theorem A.1

The number of iterations required to balance the demand at facilities is less than the number of demand points.

Proof

To simplify the proof we assume that each demand point has one unit of demand, hence the total demand is n and the average demand is n/p. Only one unit of demand is transferred between any two facilities in each iteration, hence the source can only lose a unit and a sink can gain only one unit of demand in each iteration. To maximize the number of iterations, we examine the scenario when the demand is most unbalanced. Each facility will have to have at least one demand point assigned to it, as the demand point where the facility is located at, is assigned to the facility. To create the most unbalanced scenario, we assume that one of p facilities, $f^{\prime }$ has been assigned the remaining demand points. Hence $demand(f^{\prime }) = (n-p+1)$ while, the demand at all remaining facilities is equal to demand(f_i) = 1, for all f_i ∈ F, $f_i \neq f^{\prime }$. Since only one demand point is transferred from a source ($f^{\prime }$) to a sink f_j in each iteration, to achieve the balance at $f^{\prime }$, number of required iterations is equal to (n − p + 1 − n/p) = n + 1 − (p + n/p). The transfer may be indirect. b = (n + 1 − (p + n/p)). The required number of iterations is zero if the demands are already balanced. Therefore, 0 ≤ b < n. The condition of achieving balance in the more realistic scenario where demands are not uniform across all demand points is less restrictive because the algorithm terminates with a non-zero balance as discussed in the previous section, hence requires even less number of iterations. Hence the bound holds for the general case.