We consider two heuristics for our model. The first heuristic, which we call FCFS, reflects the allocation policy commonly used by global public health managers. The second heuristic, which we refer to as PNS (standing for probability of no shortfall), computes the probability that all state 2 patients will treated in the next period and uses that information to make the allocation decision for state 1 patients in the current period. In what follows, we discuss the two heuristics in greater detail.
3.4.1 Heuristic FCFS
The FCFS heuristic is very simple to understand and easy to implement. In every period,
a2= min{rt, n2t}anda1 = min{(rt−n2t)+, n1t}, i.e., after treating state 2 patients, treat as many state 1 patients as possible with the funding available on–hand. Notice that the FCFS heuristic is naive since it does not take into account future funding availability for state 2 patients when making the allocation decision for state 1 patients in the current period. Nevertheless, it remains a popular approach in the humanitarian health sector and we are interested in evaluating its performance relative to the PNS heuristic and the optimal policy.
3.4.2 Heuristic PNS
The PNS heuristic is based on the calculation of the probability that all state 2 patients would be treated in period t-1, given that the allocation for state 1 patients in period t is a1. Of course, 0≤a1 ≤(rt−n2t)+. Whenrt≥n2t, for anya1, this probability can be easily computed
as shown below.
P r{all state 2 patients will be treated in t−1} |a1 =P r{rt−1 ≥n2t−1} |a1
=P r{rt−n2t −a1+ (OFt−OFt−1)≥α12(n1t −a1) +β2nNt−1} =P r{OFt−1 ≤rt−n2t −a1+OFt−α12(n1t−a1)−β2nNt−1}
Then, the PNS heuristic chooses the allocation level a1 as follows: max{a1 : P r{r t−1 ≥
n2
t−1} |a1 ≥ K,0 ≤ a1 ≤ (rt−n2t)+} where K is a threshold value between 0 and 1. If no such a1 exists, then a1=0. In our numerical experiments, we optimize over the range [0,1] to determine the optimal value of K.
The PNS heuristic is appealing to us for two reasons. First, notice that the PNS heuristic is the same as FCFS if we set K=0. Thus, by optimizing over the set of possible values forK, the PNS heuristic offers a natural and intuitive way to improve upon the performance of the FCFS heuristic.
To understand the other reason why we are interested in the PNS heuristic, consider expres- sions (3.9) and (3.10), which represent the minimum expected disease–adjusted life months lost corresponding to allocationsa1 and a1−δ in periodt. We are only interested in small values of
δ since our aim is to capture the impact of making incremental changes to the allocation level
a1. In writing these equations, we assume thatrt≥n2t since it is only under this situation that the question of how much to allocate to state 1 patients arises.
b1n2t+ ˆb1(n1t −a1) +EnN t−1EOFt−1|OFtVt−1(n 1 t−1, n2t−1, rt−1, OFt−1) (3.9) b1n2t + ˆb1(n1t −a1+δ) +EnN t−1EOFt−1|OFtVt−1(n 1 t−1+α11δ, n2t−1+α12δ, rt−1+δ, OFt−1) (3.10)
In the above expressions,n1t−1 =n2t+α11(n1t−a1)+β1ntN−1,n2t−1 =α12(n1t−a1)+β2nNt−1 and
rt−1=rt−n2t−a1+OFt−OFt−1. Notice that when we look at the difference between (3.9) and (3.10), we are essentially considering the expected value of−ˆb1δ+Vt−1(nt1−1, n2t−1, rt−1, OFt−1)−
Vt−1(n1t−1 +α11δ, n2t−1 +α12δ, rt−1+δ, OFt−1), with the possibility of either rt−1 ≥ n2t−1 or
rt−1 < n2t−1, i.e., funding available at the beginning of period t-1 may or may not be sufficient to treat all state 2 patients in that period. Now, ifVt−1(n1t−1, n2t−1, rt−1, OFt−1)−Vt−1(n1t−1+
α11δ, n2t−1 +α12δ, rt−1+δ, OFt−1) ≤0 when rt−1 ≥ n2t−1, then we can reasonably expect the difference between (3.9) and (3.10) to be negative if the probability ofrt−1 ≥n2t−1is greater than some threshold value. Let us analyze the differenceVt−1(n1t−1, n2t−1, rt−1, OFt−1)−Vt−1(n1t−1+
α11δ, n2t−1+α12δ, rt−1+δ, OFt−1) when rt−1 ≥n2t−1. Vt−1(n1t−1, n2t−1, rt−1, OFt−1) =b1n2t−1 + min 0≤a1 ≤n1t−1 a1 ≤rt−n2t−1 ˆ b1(n1t−1−a1) +EVt−2(n1t−2, n2t−2, rt−2, OFt−2) (3.11) and Vt−1(n1t−1+α11δ, n2t−1+α12δ, rt−1+δ, OFt−1) =b1(n2t−1+α12δ) + min 0≤a1≤n1t−1+α11δ a1≤rt−n2t−1+α11δ ˆb1(n1 t−1+α11δ−a1) +EVt−2(n1t−2+ ∆1, nt2−2+ ∆2, rt−2+ ∆3, OFt−2) (3.12)
In equations (3.11) and (3.12), the expectation is taken with respect tonNt−2andOFt−2|OFt−1, and in (3.12), ∆1 = (α12+α211)δ, ∆2 = α12α11δ, ∆3 = α11δ. If α11=0, then, clearly, (3.11)- (3.12)≤0. Whenα11>0, let ˆa1be the optimal solution for expression (3.12). Now, if ˆa1≥α11δ, then using ˆa1−α
11δas a solution for expression (3.11) again yieldsVt−1(n1t−1, n2t−1, rt−1, OFt−1)≤
Vt−1(n1t−1 +α11δ, nt2−1 +α12δ, rt−1 +δ, OFt−1). While it is not possible to guarantee that ˆ
a1 ≥α11δ will always hold, it is clear that the possibility of the condition holding increases as the value ofα11goes down. This suggests that the PNS heuristic may perform well for low val- ues ofα11 but the performance may be sensitive toα11. In Section 3.6, we test this hypothesis and more broadly, evaluate the performance of the two heuristics relative to the optimal policy.
So far, we have focused on answering the first of our two main research questions: how to optimally allocate funding between the two health states in every period, taking into account the current funding availability and future financial inflows. In the next section, we turn our attention to the second research question: what is the impact of system parameters and funding changes on the number of disease–adjusted life months lost? Specifically, we focus on the impact of funding changes.