4.2 Model specification
In this section we derive our model by first developing the components for donor deferral, risk-reducing modification, and disease marker testing. All notation is summarized in Table 4.1.
| Notation | Description |
|---|---|
| \(p, m, a\) | single variable |
| \(\textbf{p}, \textbf{m}, \textbf{a}\) with elements \(p_k, m_n, a_j\) | vector |
| \(\textbf{P}, \textbf{M}, \textbf{A}\) with elements \(p_{ik}, m_{ni}, a_{ji}\) | matrix |
| \(\mathbb{1}\) | indicator variable |
| \(\textbf{1}\) | vector for which every entry is 1 |
| \(\mathbb{P}(x = y)\) | probability \(x\) equals \(y\) |
| \(A \circ B\) | Hadamard (element-wise) product of same-dimensioned vectors or matrices |
| Indices | |
| \(k=1,..., K\) | transfusion-transmissible infections (TTIs) |
| \(i=1,..., I\) | segments of the donor population |
| \(j=1,..., J\) | available disease marker tests |
| \(l=1,..., L\) | available risk-reducing modifications |
| Decision variables | |
| \(\textbf{z}\) where \(z_i \in \{0, 1\}\) | 1 if donations from donor group \(i\) are accepted |
| \(\textbf{M}\) where \(m_{li} \in \{0, 1\}\) | 1 if modification \(l\) is used in donations from donors in group \(i\) |
| \(\textbf{A}\) where \(a_{ji} \in \{0, 1\}\) | 1 if disease marker test \(j\) is used for donations from donors in group \(i\) |
| Parameters related to transfusion-transmissible infections (TTIs) | |
| \(\textbf{y}\) where \(y_k \in \{0, 1\}\) | 1 if a donor is infectious with TTI \(k\) |
| \(\textbf{p}\) where \(p_k \in [0,1]\) | probability that a donor is infectious with TTI \(k\) |
| \(\textbf{c}\) where \(c_k > 0\) | net health cost of releasing a donation infectious for TTI \(k\) |
| Parameters related to donor groups | |
| \(\textbf{P}\) where \(p_{ik} \in [0,1]\) | probability a donation from donor group \(i\) will be infectious for TTI \(k\) |
| \(\textbf{d}\) where \(d_i \geq 0\) | cost of replacing a deferred donation from a deferred donor from group \(i\) |
| \(\textbf{w}\) where \(w_i \geq 0\) | cost of processing a donation for a donor from group \(i\) |
| \(\textbf{n}\) where \(n_i \in \mathbb{N}\) | number of donors in subgroup \(i\) |
| Parameters related to disease marker tests | |
| \(\boldsymbol{\phi}\) where \(\phi_j > 0\) | per-donation cost of disease marker test \(j\) |
| \(\textbf{R}\) where \(r_{jk} \in [0,1]\) | sensitivity of test \(j\) for TTI \(k\) |
| \(\textbf{Q}\) where \(q_{jk} \in [0,1]\) | specificity of test \(j\) for TTI \(k\) |
| \(g \geq 0\) | disposal cost for collected donations that test positive |
| Parameters related to modification interventions | |
| \(\boldsymbol{\psi}\) where \(\psi_l > 0\) | per-donation cost of modification intervention \(l\) |
| \(\textbf{H}\) where \(h_{lk} \in [0,1]\) | percent reduction in risk of TTI \(k\) from modification intervention \(l\) |
4.2.1 Donor deferral model
We begin with a simplified model for deciding whether to accept a donation based on risk of a single TTI. A donation may be infectious for the TTI (\(y=1\)) or not (\(y=0\)), and the decision is to accept (\(z=1\)) or reject (\(z=0\)) the donation. If the donation is rejected (also called deferring the donor), a replacement cost \(d\) is incurred because another donor must be recruited to meet demand. If the donation is accepted, a processing cost \(w\) is incurred. We assume \(w < d\); otherwise, the optimal decision would be to always reject donations regardless of blood safety concerns.
If the donation is accepted and is infectious for the TTI (\(z=1, y=1\)) then, in the absence of testing or modification interventions, an infectious donation is released for transfusion. Because donations are typically processed into multiple components, one infectious donation can expose multiple recipients to infection. We use the variable \(c\) to represent the expected net monetary cost of releasing an infectious donation. We estimate this cost as \(c = l + \gamma q\), where \(l\) is the net present expected cost of a breakthrough infectious donation, \(\gamma\) is the decision maker’s willingness to pay to avert the loss of one quality-adjusted life year (QALY), and \(q\) is the net present expected QALYs lost. Estimating \(c\) is a nontrivial exercise; its value depends on the TTI, the donor’s stage of infection, transfusion recipient characteristics, and how recipient exposures are treated in a particular health system.
Using the above notation, we express the cost function \(\mathcal{C}(y,z)\) as:
\[ \mathcal{C}(y,z) = \mathbb{1}_{\{z=0\}}d + \mathbb{1}_{\{z=1\}}w + \mathbb{1}_{\{z=1, y=1\}}c. \] For TTIs of concern, we assume \(c > d\) (the net monetary cost of releasing an infectious donation exceeds the replacement cost of rejecting a donation), which ensures that the optimal decision is to reject a donation if it is known to be infectious. In practice, a blood center does not know whether a donation will be infectious, but several methods are available for estimating the risk, which we denote by \(p = \mathbb{P}(y=1)\).
Given \(p\), the optimal policy is to choose \(z\) (accept or reject) such that the expected cost \(\mathbb{E}[\mathcal{C}(z)]\) is minimized:
\[ \min_z \quad \mathbb{E}[\mathcal{C}(z \mid p)] = (1-z)d + z\big(pc + w\big). \]
From this equation, one can see that a decision maker should be indifferent between rejecting and accepting the donation when \(p = \frac{d-w}{c}\). When \(p < \frac{d-w}{c}\), the optimal decision is to accept the donation (\(z=1\)), and when \(p > \frac{d-w}{c}\), the optimal decision is to reject the donation (\(z=0\)).
Policymakers are typically concerned about multiple TTIs. We now consider multiple TTIs indexed by \(k=1, ..., K\). We define a vector \(\textbf{y}\) where entry \(y_k \in \{0, 1\}\) indicates whether the potential donation is infectious for TTI \(k\), and a vector \(\textbf{c}\) where entry \(c_k = l_k + \gamma q_k\) (\(c_k \geq 0\)) represents the expected cost of releasing a donation that is infectious for TTI \(k\). Our new cost function is
\[ \mathcal{C}(\textbf{y},z) = \mathbb{1}_{\{z=0\}}d + \mathbb{1}_{\{z=1\}}[w + \textbf{y}^\top \textbf{c}]. \] Taking the expectation, we obtain \[ \mathbb{E}[\mathcal{C}(z \mid \textbf{p})] = (1-z)d + z\big(w + \textbf{p}^\top \textbf{c}\big), \]
where \(\textbf{p}\) is a vector for which \(p_k = \mathbb{P}(y_k=1)\). The decision maker should reject the donation when \(\textbf{p}^\top \textbf{c} > d - w\), accept when \(\textbf{p}^\top \textbf{c} < d - w\), and be indifferent when \(\textbf{p}^\top \textbf{c} = d - w\).
Finally, we consider the case of deferral with multiple TTIs and donor groups. Rather than deciding whether to accept the entire potential donor population, policymakers often consider various donor groups, defined based on factors such as geographic location or the donor’s response to a pre-donation questionnaire. Donor groups may be defined in ways that facilitate temporary or lifetime deferrals (e.g., “travel to Mexico within the past 60 days” or “ever tested positive for HIV”). While less common, donor groups could also be based on factors related to the replacement cost of rejecting a donor (e.g., a rare blood type).
We assume that the potential donor population has been segmented into \(I\) mutually exclusive and collectively exhaustive groups indexed by \(i\), and the decision to accept or reject donations from a specific group is represented by a vector \(\textbf{z}\) with elements \(z_i \in \{0, 1\}\). We introduce a prevalence matrix \(\textbf{P}\) with rows that correspond to donor groups and with columns that correspond to TTIs. Entry \(p_{ik}\) represents the risk of infectiousness for TTI \(k\) in donations from donor group \(i\) (i.e., \(\mathbb{P}(y_k = 1 \mid i=i)\)). It is possible that donor groups have different replacement and processing costs, so we define \(\textbf{d}\) where \(d_i\) represents the replacement cost of a donation from group \(i\) and \(\textbf{w}\) where \(w_i\) represents the cost of processing a donation from group \(i\). Finally, we define \(\textbf{n}\) where \(n_i\) is the estimated number of donors from each group to present for donation in the period of analysis. Using this notation, the total expected cost of a given deferral policy is
\[ \mathbb{E}[C(\textbf{z} \mid \textbf{P})] = \big((\textbf{1}-\textbf{z} )\circ \textbf{n}\big)^\top \textbf{d} + (\textbf{z} \circ \textbf{n})^\top(\textbf{w} + \textbf{P} \textbf{c}), \] where the \(\circ\) operator indicates element-wise multiplication and \(\textbf{1}\) is a vector of all 1’s, in this case with length \(I\).
4.2.2 Disease marker testing model
We now consider disease marker testing starting with the case of one test for one TTI. We introduce a binary decision variable \(a\), where \(a=1\) if the test is used and 0 otherwise. The test has an associated cost \(\phi\), sensitivity \(r\), and specificity \(q\). The probability of a positive test result is \(r p+(1-q)(1-p)\) and the probability of a false negative is \((1-r)p\). We assume the blood center will always dispose of donations that test positive for a TTI, incurring a per donation cost of \(g\). The constant \(g\) should reflect the costs of any confirmatory testing, donor notification and counseling, and the cost of replacing the donation. Expected cost is
\[ \mathbb{E}[\mathcal{C}(z, a \mid p)] = (1-z)d + z\bigg(w + a\phi + ag\big(r p+(1-q)(1-p)\big) + a(1-r)pc + (1-a)pc \bigg). \] Note that not using a test is equivalent to using a test with sensitivity of 0 and specificity of 1; the expression \(ra\) will equal \(r\) when \(a=1\) and \(0\) when \(a=0\) and the expression \(1+a(q-1)\) will equal \(q\) when \(a=1\) and will equal \(1\) when \(a=0\). Using these, we can simplify the cost function as
\[ \mathbb{E}[\mathcal{C}(z, a \mid p)] = (1-z)d + z\bigg(w + a\phi + ag(rp + (1-q)(1-p)) + (1-ra)pc \bigg). \]
For the case of one test that can detect multiple TTIs (such as multiplex tests), we define \(\textbf{r}\) where \(r_k\) is the sensitivity for detecting TTI \(k\), \(\textbf{q}\) where \(q_k\) is the specificity for detecting TTI \(k\), and \(\textbf{p}\) is as defined above. The probability that a single test returns a negative result is \((1-p)q+p(1-r)\). The probability that any TTI tests positive is one minus the probability that all TTIs test negative and is computed as \(s_1 = a\big (1 - \prod_{k=1}^K \big[(1-p_k) q_k+ p_k (1-r_k)\big]\big)\). Using \(s_1\), the expected cost for one test and multiple TTIs is
\[ \mathbb{E}[\mathcal{C}(z, a \mid \textbf{p})] = (1-z)d + z\Big(w + a\phi + (1-s_1) ((\textbf{1}-a \textbf{r}) \circ \textbf{p})^\top \textbf{c} + g s_1 \Big). \]
If a test does not detect disease \(k\), this can be modeled by setting the sensitivity to 0 and specificity to 1 (\(r_k=0\), \(q_k=1\)).
We now consider multiple tests and multiple TTIs. We assume that tests are independent conditional on the true disease state of the donation (i.e., errors are uncorrelated). This means that given that a donation is infectious (not infectious) for a disease, the probability that a test will yield a false negative (false positive) is independent of other tests. This assumption is made in most, if not all, decision analytic models for blood safety, including cost-effectiveness analyses [7–9,15,37] and studies of optimal blood safety test selection [119–121]. We define a matrix \(\textbf{Q}\) where \(q_{jk}\) is the sensitivity of test \(j\) for TTI \(k\) and a matrix \(\textbf{R}\) where \(r_{jk}\) is the specificity of test \(j\) for TTI \(k\), and a vector \(\boldsymbol{\phi}\) where \(\phi_j\) is the cost of test \(j\). We use the decision variable \(\textbf{a}\) where \(a_j=1\) when disease marker test \(j\) is used.
Assuming that every available test is used, the probability of any positive result is \(1 - \prod_{k=1}^{K} \prod_{j=1}^{J} [(1 - p_k) q_{jk} + p_k (1 - r_{jk})]\). Replacing \(\textbf{R}_{jk}\) and \(\textbf{Q}_{jk}\) with expressions that evaluate correctly when \(a_j = 0\), we obtain \(s_2 = 1 - \prod_{k=1}^{K} \prod_{j=1}^{J} [ (1 - p_k) (1 + a_j(q_{jk} - 1)) + p_k (1 - a_j r_{jk})]\).
To calculate the probability of a false negative test result for each TTI, we take the element-wise product of \(\textbf{p}\) and the following vector:
\[ \textbf{v}_1 = \begin{bmatrix} \prod_{j=1}^J 1-r_{j1} a_j \\ \vdots \\ \prod_{j=1}^J 1-r_{jk} a_j \\ \vdots \\ \prod_{j=1}^J 1-r_{jK} a_j \end{bmatrix} . \]
Using \(\textbf{v}_1\) and \(s_2\), the expected cost function for multiple tests and multiple TTIs is
\[ \mathbb{E}[\mathcal{C}(z, \textbf{a} \mid \textbf{p})] = (1-z)d + z\bigg(w + \textbf{a}^\top \boldsymbol{\phi}+ (1- s_2) (\textbf{v}_1 \circ \textbf{p})^\top \textbf{c}\Big) + g s_2 \bigg). \]
Finally, we develop an expected cost function for the case of multiple tests, TTIs, and donor groups. We define the decision variable \(\textbf{A}\) where \(a_{ji} = 1\) if test \(j\) is used on donor group \(i\). To calculate the risk of each TTI in each group after tests are applied, we take the element-wise product of \(\textbf{P}\) and the following matrix:
\[ \textbf{B}_1 = \begin{bmatrix} \prod_{j=1}^J 1-r_{j1} a_{j1} & & \dots \\ \vdots & \prod_{j=1}^J 1-r_{jk} a_{ji} &\ddots \\ \prod_{j=1}^J 1-r_{j1} a_{jI} & & \prod_{j=1}^J 1-r_{jK} a_{jI} \end{bmatrix}. \]
Additionally, we define a vector that represents the probability that a unit is disposed of in each donor group:
\[ \textbf{v}_2 = \begin{bmatrix} 1 - \prod_{k=1}^K \prod_{j=1}^J (1- p_{1k})(1+a_{j1} (q_{jk} - 1)) + p_{1k}(1- a_{j1} r_{jk}) \\ \vdots \\ 1 - \prod_{k=1}^K \prod_{j=1}^J (1- p_{ik})(1+a_{ji} (q_{jk} - 1)) + p_{ik}(1- a_{ji} r_{jk}) \\ \vdots \\ 1 - \prod_{k=1}^K \prod_{j=1}^J (1- p_{Ik})(1+a_{jI} (q_{jk} - 1)) + p_{Ik}(1- a_{jI} r_{jk}) \end{bmatrix}. \]
Using \(\textbf{B}_1\) and \(\textbf{v}_2\), the expected cost function for multiple donor groups, TTIs, and tests is
\[ \mathbb{E}[\mathcal{C}(\textbf{z}, \textbf{A} \mid \textbf{P})] = \big((\textbf{1}-\textbf{z} )\circ \textbf{n}\big)^\top \textbf{d} + (\textbf{z} \circ \textbf{n})^\top \Big(\textbf{w} + \textbf{A}^\top\boldsymbol{\phi} + (1- \textbf{v}_2) \circ [(\textbf{B}_1 \circ \textbf{P} ) \textbf{c}] + g\textbf{v}_2 \Big). \]
4.2.3 Risk-reducing modification model
Risk-reducing modifications (e.g., pathogen inactivation or leukoreduction) can decrease the risk of TTI in components derived from blood donations. We first consider one available modification and one TTI. We define \(h \in [0,1]\) as the risk-reduction multiplier for the modification and \(\psi\) as the per-donation cost. Often, modifications are applied to only some of the components derived from a donation rather than the whole donation. For example, pathogen inactivation is currently FDA approved for platelet and plasma components but not red blood cells [123]. In this case \(h\) can be scaled proportionally to the fraction of components modified, or the same TTI in different components can be modeled as different TTIs (e.g. HIV in platelets vs. HIV in red blood cells). Because not applying a modification is equivalent to applying a modification with a risk multiplier of 1, we use the expression \(1 + m(h-1)\), which equals 1 when \(m=0\) and \(h\) when \(m=1\). Expected cost is
\[ \mathbb{E}[\mathcal{C}(z, m \mid p)] = (1-z)d + z\bigg(w + m\psi + (1 + m(h-1))pC \bigg). \]
A modification can sometimes reduce the risk of multiple TTIs. We model this by introducing \(\textbf{h}\) where \(h_k\) is the risk-reduction multiplier for TTI \(k\). The expected cost for a single modification with multiple TTIs is
\[ \mathbb{E}[\mathcal{C}(z, m \mid \textbf{p})] = (1-z)d + z\bigg(w + m\psi + \big((\textbf{1}+m(\textbf{h}-\textbf{1})) \circ \textbf{p}\big)^\top \textbf{c} \bigg). \]
If a modification does not reduce risk for disease \(k\), this can be modeled by setting the risk-reduction multiplier to 1 (\(h_k=1\)).
Often multiple modifications are available, each of which might reduce the risk for multiple TTIs. To model this, we define the vector \(\boldsymbol\psi\) where \(\psi_l\) is the cost for modification \(l\), and \(\textbf{H}\) where \(h_{lk}\) is the risk-reduction multiplier for modification \(l\) and TTI \(k\). We replace the single decision variable \(m\) with the vector \(\textbf{m}\) where \(m_l=1\) if modification \(l\) is added to the portfolio. We assumed that combinations of modifications would have multiplicative rather than additive effects because we assume that the mechanistic targets of each modification (e.g., pathogen inactivation, leukoreduction) are different. The product of risk-reducing multipliers for each modification in use can be calculated as follows:
\[ \textbf{v}_3 = \begin{bmatrix} \prod_{l=1}^L 1 + m_{l} (h_{l1} - 1) \\ \vdots \\ \prod_{l=1}^L 1 + m_{l} (h_{lk} - 1) \\ \vdots \\ \prod_{l=1}^L 1 + m_{l} ( h_{lK} - 1) \end{bmatrix} . \]
Using this, the new expected cost is
\[ \mathbb{E}[\mathcal{C}(z, \textbf{m} \mid \textbf{p})] = (1-z)d + z\bigg(w + \textbf{m}^\top \boldsymbol\psi + \big(\textbf{v}_3 \circ \textbf{p}\big)^\top \textbf{c} \bigg). \]
Lastly, we integrate the model for multiple modifications with the model for multiple donor groups. We define a new decision variable \(\textbf{M}\) where \(m_{li}=1\) if modification \(l\) is used on donor group \(i\). Because each element in \(\textbf{P}\) must be multiplied by the product of any risk-reduction modifiers that are used in that sub-population, we define the following matrix:
\[ \textbf{B}_2 = \begin{bmatrix} \prod_{l=1}^L 1 + m_{l1} (h_{l1} - 1) & & \dots \\ \vdots & \prod_{l=1}^L 1 + m_{li} (h_{lk} - 1) &\ddots \\ \prod_{l=1}^L 1 + m_{lI} ( h_{l1} - 1) & & \prod_{l=1}^L 1 + m_{lI} ( h_{lK} - 1) \end{bmatrix}. \]
Using this, the expected cost with multiple donor groups and modifications is
\[ \mathbb{E}[\mathcal{C}(\textbf{z}, \textbf{M} \mid \textbf{P})] = \big((\textbf{1}-\textbf{z} )\circ \textbf{n}\big)^\top \textbf{d} + (\textbf{z} \circ \textbf{n})^\top \Big(\textbf{w} + \textbf{M}^\top \boldsymbol{\psi} + (\textbf{B}_2 \circ \textbf{P} ) \textbf{c} \Big). \]
4.2.4 Optimal portfolio model
We can now write the expected cost function for a portfolio containing any combination of donor deferral policies, disease marker tests, and risk-reducing modifications:
\[ \mathbb{E}[\mathcal{C}(\textbf{z}, \textbf{M}, \textbf{A} \mid \textbf{P})] = \big((\textbf{1}-\textbf{z} )\circ \textbf{n}\big)^\top \textbf{d} + (\textbf{z} \circ \textbf{n})^\top \Big(\textbf{w} + \textbf{A}^\top\boldsymbol{\phi} + \textbf{M}^\top \boldsymbol{\psi} + (1- \textbf{v}_2) \circ [(\textbf{B}_1 \circ \textbf{B}_2 \circ \textbf{P} ) \textbf{c}] + g\textbf{v}_2 \Big). \]
This cost function expresses the net present net monetary cost of a policy (all future costs are discounted to the present when calculating \(\textbf{c}\)). The first term in the above expression is the cost of deferral if a donation is deferred. The second term is the cost incurred if the donation is not deferred and includes (in the bracketed term) the costs of processing, testing, and modifying a donation plus the expected cost incurred from release of any infectious donations.
The optimal combination of interventions solves the following optimization problem:
\[ \begin{aligned} \min_{\textbf{z}, \textbf{M}, \textbf{A}} \quad & \mathbb{E}[\mathcal{C}(\textbf{z}, \textbf{M}, \textbf{A} \mid \textbf{P})]\\ \textrm{s.t.} \quad & m_{li} = a_{ji} = 0 \quad \forall l,j \quad \text{ when } \quad z_i=0 \end{aligned} \] The constraint ensures that no tests or modifications are applied to deferred donor groups.
The above formulation allows each non-deferred donor group to receive a tailored portfolio of tests and modifications, but many health systems use the same set of tests and modifications for all accepted donations regardless of donor group. Such policies may produce less benefit at a fixed willingness-to-pay as compared to tailored policies, but they are easier to implement and might be perceived as fairer. To consider only universal testing and modification policies, two additional constraints can be introduced:
\[ \begin{aligned} \min_{\textbf{z}, \textbf{M}, \textbf{A}} \quad & \mathbb{E}[\mathcal{C}(\textbf{z}, \textbf{M}, \textbf{A} \mid \textbf{P})]\\ \textrm{s.t.} \quad & m_{li} = a_{ji} = 0 \quad \forall l,j \quad \text{ when } \quad z_i=0\\ & m_{li_1} = m_{li_2} \quad \forall i_1, i_2 \quad \text{when} \quad z_{i_1} = z_{i_2} = 1\\ & a_{ji_1} = a_{ji_2} \quad \forall i_1, i_2 \quad \text{when} \quad z_{i_1} = z_{i_2} = 1 \end{aligned} \]
The cost function can be used to derive many other performance measures, summarized in Table 4.2, that may be more interpretable to policymakers. These include, for example, the number of donors deferred, the donation yield, and the number of infectious donations released over a given time horizon. Performance measures can also be used to impose additional constraints on the optimization problem. For instance, one could limit the number of donors deferred (\(\big((\textbf{1}-\textbf{z} )\circ \textbf{n}\big)^\top \textbf{1} \leq \pi\), where \(\pi\) is an upper bound) or the total budget for tests and modifications (\((\textbf{z} \circ \textbf{n})^\top (\textbf{A}^\top \boldsymbol{\phi} + \textbf{M}^\top \boldsymbol{\psi}) \leq \rho\), where \(\rho\) is an upper bound).
| Measure | Formula |
|---|---|
| Number of donors deferred | \((\textbf{1}-\textbf{z} )\circ \textbf{n}\) |
| Risk reduction for TTI \(k\) in group \(i\) by testing (\(I \times K\)) | \((1-\textbf{v}_2)\textbf{B}_1\) |
| Risk reduction for TTI \(k\) in group \(i\) by modifications (\(I \times K\)) | \(\textbf{B}_2\) |
| Residual risk for TTI \(k\) in group \(i\) (returns \(I \times K\)) | \((1-\textbf{v}_2)\textbf{B}_1 \circ \textbf{B}_2 \circ \textbf{P}\) |
| Donation yield | \((\textbf{z} \circ (1 - \textbf{v}_2) )^\top \textbf{n}\) |
| Residual risk of infection for TTI \(k\) (\(K \times 1\)) | \((\textbf{B}_1 \circ \textbf{B}_2 \circ \textbf{P})^\top (\textbf{z} \circ (1- \textbf{v}_2) \circ \textbf{n}) / (\textbf{z} \circ (1 - \textbf{v}_2) )^\top \textbf{n}\) |
| Number of infectious donations released for TTI \(k\) (\(K \times 1\)) | \((\textbf{B}_1 \circ \textbf{B}_2 \circ \textbf{P})^\top (\textbf{z} \circ (1- \textbf{v}_2) \circ \textbf{n})\) |
| Total modification cost | \((\textbf{z} \circ \textbf{n})^\top (\textbf{A}^\top \boldsymbol{\phi})\) |
| Total cost of initial tests | \((\textbf{z} \circ \textbf{n})^\top (\textbf{M}^\top \boldsymbol{\psi})\) |
| Total donor replacement cost | \(\big((\textbf{1}-\textbf{z} )\circ \textbf{n}\big)^\top \textbf{d}\) |
| Total processing cost | \((\textbf{z} \circ \textbf{n})^\top \textbf{w}\) |
| Total cost due to released infectious donations | \((\textbf{z} \circ \textbf{n})^\top \Big( (1- \textbf{v}_2) \circ [(\textbf{B}_1 \circ \textbf{P} ) \textbf{c}] \Big)\) |
| Total cost due to removed donations testing positive | \(g (\textbf{z} \circ \textbf{v}_2)^\top \textbf{n}\) |
| Number of donations testing positive | \((\textbf{z} \circ \textbf{v}_2)^\top \textbf{n}\) |