4.3 Model solution

The optimal portfolio model is a binary integer program, and its exact solution can be found using exhaustive search. If tailored testing and modification policies for each donor group are allowed, there will be \(\sum_{i=0}^I \binom{I}{i} 2^{i(L+J)}\) feasible policies. If only universal testing and modification policies are considered, the feasible state space will be \(1+\sum_{i=1}^I \binom{I}{i} 2^{(L+J)}\) which is smaller by a factor of approximately \(2^I\). Because the state space increases exponentially in the number of available interventions, exhaustive search is not feasible for larger problems. Here we describe structural properties that allow for more efficient identification of the optimal policy in certain cases.

4.3.1 Tailored policies

When policies can be tailored to individual donor groups, the objective function is linearly separable, and we can solve a single, smaller optimization problem for each donor group. In this case we can identify the optimal portfolio by evaluating \(I \times (1 + 2^{L+J})\) policies, rather than the \(\sum_{i=0}^I \binom{I}{i} 2^{i(L+J)}\) evaluated by naive exhaustive search.

4.3.2 Elimination of infeasible tests or modifications

Eliminating some tests or modifications from consideration in advance can considerably reduce the time needed to find a solution. We can do so by leveraging the following theorem.

Theorem 1: If use of a single test or modification is not preferred over a ‘no interventions’ scenario in any donor group then it cannot be part of an optimal portfolio.

To see why this theorem holds, consider the following. Tests and modifications reduce the cost function by reducing the multipliers on the expected cost of releasing an infectious donation (i.e., making elements of \((1- \textbf{v}_2) \circ [(\textbf{B}_1 \circ \textbf{B}_2 \circ \textbf{P} ) \textbf{c}]\) smaller by decreasing one or more entries in \((1 - \textbf{v}_2)\) and \(\textbf{B}_1\) [tests] or \(\textbf{B}_2\) [modifications]). Addition of a test or modification will generate the greatest reduction in expected cost when the term \((1- \textbf{v}_2) \circ [(\textbf{B}_1 \circ \textbf{B}_2 \circ \textbf{P} ) \textbf{c}]\) is largest, i.e., when no interventions are in use. Therefore, any test or modification that is part of an optimal portfolio must be preferred (i.e., yield a lower objective function value) compared to use of no intervention in at least one donor group.

To apply this theorem, we use a simple equation to calculate the per-donor incremental objective function cost of each test or modification as compared to using no intervention. We let \(\gamma_{ji}^{test}\) represent the cost of using only test \(j\) (no other tests or modifications) compared to using no interventions in donor group \(i\). Similarly, we let \(\gamma_{li}^{mod}\) denote the cost of using only modification \(l\) compared to using no interventions in donor group \(i\). These quantities can be computed for test \(j^*\) and modification \(l^*\) in donor group \(i^*\) as:

\[ \begin{align} \gamma^\text{test}_{j, i} (j^*,i^*) &= \textbf{w} + \phi_{j^*} + (1-s^{i^*j^*}) ((\textbf{1} - \textbf{r}_{*j^*}) \circ \textbf{p}_{*i^*} ) \textbf{c} + gs^{i^*j^*} - (\textbf{w} + \textbf{p}_{*i^*} \textbf{c})\\ &= \phi_{j^*} + gs^{i^*j^*} + (1-s^{i^*j^*}) ((\textbf{1} - \textbf{r}_{*j^*}) \circ \textbf{p}_{*i^*} ) \textbf{c} - \textbf{p}_{*i^*} \textbf{c}\\ &=\phi_{j^*} + gs^{i^*j^*} - (1-s^{i^*j^*}) (\textbf{r}_{*j^*} \circ \textbf{p}_{*i^*} ) \textbf{c}\\ \gamma^\text{mod}_{li} (l^*,i^*) &= w + \phi_{l^*} + ( \textbf{h}_{{*l^*}} \circ \textbf{p}_{*i^*})\textbf{c} - (\textbf{w} + \textbf{p}_{*i^*} \textbf{c})\\ &= \phi_{l^*} - ((\textbf{1} - \textbf{h}_{{*l^*}}) \circ \textbf{p}_{*i^*})\textbf{c} \end{align} \]

where \(\textbf{p}_{*i^*}\), \(\textbf{r}_{*j^*}\), and \(\textbf{h}_{*l^*}\) indicate the \(i^*\)th, \(j^*\)th, and \(l^*\)th rows of \(\textbf{P}\), \(\textbf{R}\), and \(\textbf{H}\), respectively. The term \(s^{i^*j^*}\) is the percent of donations removed by testing when only test \(j^*\) is used in donor group \(i^*\) and is computed as \(s^{i^*j^*} = \det \text{diag} [(\textbf{q}_{*j^*}(1 - \textbf{p}_{*i^*}) + (1-\textbf{r}_{*j^*}) \textbf{p}_{*i^*} )]\). The derivation of this equation along with a weaker simplification that can be useful for evaluating the potential benefit of developing a new test or modification are shown in the supplement.

In the case where tests and modifications are tailored by donor group, a test or modification can be removed from consideration for any donor groups for which the associated \(\gamma_{ji}^{test}\) or \(\gamma_{li}^{mod}\) is positive. Additionally, tests or modifications can be removed from consideration across all donor groups if \(\gamma_{ji}^{test}\) or \(\gamma_{li}^{mod}\) is positive across all donor groups, even if the state space has additional constraints (e.g., when only universal testing and modification policies are under consideration). Calculating this incremental cost for all tests and modifications across all donor groups before running the binary integer program requires \(I(L+J)\) calculations. Each test or modification removed from consideration reduces the state space for the binary integer program by approximately a factor of 2. The supplement includes a tutorial on applying this theorem to solve both the tailored and universal versions of the portfolio problem with example R code.

4.3.3 Mapping prevalence to the optimal policy

When donor groups are identical except for the number of donors (\(n_i\)) and the prevalence of each TTI (\(p_{ik}\)), as is often the case, then the optimal policy \(\pi = (z, \textbf{m}, \textbf{a})\) will depend only on prevalence. In that case, a \(k\)-dimensional function \(f(\textbf{p})\) mapping the prevalence by TTI to the optimal policy (\(\pi^*\)) can be defined as follows:

\[ \pi^* = f(\textbf{p}), \quad \text{where} \text{ } f(\textbf{p}) = {\arg\min}_{\textbf{m}, \textbf{a}, z} \mathbb{E}[\mathcal{C}(z, \textbf{m}, \textbf{a}, \textbf{p}) ]. \]

When the number of donor groups is large relative to the number of tests and modifications, it can be more efficient to solve or approximate this function than to explicitly evaluate each donor group separately. Exact and approximate methods for learning this function are provided in the supplement.

4.3.4 Linear approximation by excluding some test costs

For a given policy, the objective function cost is a nonlinear function of prevalence due to the term \(\mathbf{v}_2\), the proportion of donations removed by testing in each group, which can include high-order polynomial expressions. In most real-world settings, only a small fraction of donations will be removed by testing. Because of this, many cost-utility analyses only include the up-front test cost and savings associated with reducing infections in transfusion recipients, assuming that costs such as those for confirmatory testing and replacing donations that test positive are negligible [7–9]. In our framework, this is equivalent to setting \(\textbf{v}_2 = 0\), which yields an approximation of the cost function that is linear in TTI prevalence:

\[ \tilde{\mathbb{E}} [\mathcal{C}(\textbf{P}) \mid \pi] = \big((\textbf{1}-\textbf{z} )\circ \textbf{n}\big)^\top \textbf{d} + (\textbf{z} \circ \textbf{n})^\top \Big(\textbf{w} + \textbf{A}^\top\boldsymbol{\phi} + \textbf{M}^\top \boldsymbol{\psi} + (\textbf{B}_1 \circ \textbf{B}_2 \circ \textbf{P} ) \textbf{c} \Big). \] Because this approximation is linear in \(\textbf{P}\), the resulting problem can be solved for each policy as a linear integer program, a class of problems for which many efficient solution techniques are available [124]. Differences between \(\tilde{\mathbb{E}} [\mathcal{C}(\textbf{p}) \mid \pi]\) and \(\mathbb{E}[\mathcal{C}(\textbf{p}) \mid \pi]\) will be largest when elements of \(\textbf{v}_2\) are large, which can occur when testing policies are used and prevalence and test sensitivity are high, test specificity is low, or many tests are used.