Developing childhood vaccine administration and inventory replenishment policies that minimize open vial wastage

Abstract

In the last century, many infectious diseases have been completely eradicated or significantly reduced because of childhood vaccinations. Ample evidence suggests that low vaccination coverage in developing countries is caused by vaccine stockout and high rates of vaccine wastage. Wastage occurs when a vaccine vial is physically damaged or exposed to extreme temperatures, or when doses from an open vial are discarded after their safe-use time expires. The latter is referred to as open vial wastage (OVW). Clinics can use single-dose vials to reduce OVW; however, such an approach is more expensive than using multi-dose vials. The focus of this research is to develop new policies that support vaccine administration and inventory replenishment. These policies are expected to reduce OVW, reduce the cost of vaccinations, and improve vaccination coverage levels in developing countries. This paper proposes a two-stage stochastic programming model that identifies an optimal combination of differently sized vaccine vials and the corresponding decisions that clinics make about opening vials in the face of uncertain patient arrivals. This work develops a case study with data gathered from Bangladesh. Experimental results indicate that using a combination of vials of different sizes reduces OVW, as opposed to the current practice of using single-sized multi-dose vials. Experimental results also point to simple and economic vaccine administration policies that health care administrators can use to minimize OVW. The model is solved using an extension of the stochastic Benders decomposition algorithm, the L-shaped method (LS). This algorithm uses Gomory mixed integer and mixed-integer rounding cuts to address the problem’s non-convexity. Computational results reveal that the solution approach presented here outperforms the standard LS method.

Appendices

Appendix A: Two-stage stochastic programming model

See Table 6.

$$\begin{aligned} \min ~&\sum _{t \in \mathcal {T}} \left( f_t z_t + \sum _{\nu \in \mathcal {V}} c_{\nu }r_{\nu t} \right) +\sum _{\nu \in \mathcal {V}} \sum _{n \in \mathcal {N}} d_{\nu }s_{\nu n} + \mathbb {E} \{ H(x,\omega ) \}, \end{aligned}$$

(15a)

$$\begin{aligned} s.t.~&\nonumber \\ s_{\nu Nt}&= s_{\nu (Nt-1)} + r_{\nu t} - u_{\nu Nt} \quad \quad \forall t \in \mathcal {T}, \end{aligned}$$

(15b)

$$\begin{aligned} s_{\nu n}&= s_{\nu n-1} - u_{\nu n} \quad \quad \forall n \in \mathcal {N} {\setminus } \{N,2N,\ldots ,TN\}, \end{aligned}$$

(15c)

$$\begin{aligned} \sum _{\nu \in \mathcal {V}} r_{\nu t}&\le M_{t} z_{t} \quad \quad \forall t \in \mathcal {T},\end{aligned}$$

(15d)

$$\begin{aligned}&z_{t} \in \{0,1\}; s_{\nu n},u_{\nu n},r_{\nu t} \in \mathbb {Z}^+ \quad \quad \forall t \in \mathcal {T}, \nu \in \mathcal {V}, n \in \mathcal {N}, \end{aligned}$$

(15e)

where,

$$\begin{aligned}&H(x,\omega ) = \min ~\sum _{n \in \mathcal {N}}(gy_{n(n+\tau )}+p\ell _{n})\nonumber \\&s.t.~ \end{aligned}$$

(16a)

$$\begin{aligned}&\sum _{m=n}^{n+\tau -1} y_{nm} + y_{n(n+\tau )} = \sum _{\nu \in \mathcal {V}}q_{\nu }u_{\nu n} \quad \quad \forall n \in \mathcal {N}, \end{aligned}$$

(16b)

$$\begin{aligned}&\sum _{m=n-\tau +1}^{n}y_{mn}+\ell _{n} = \omega \quad \quad \forall n \in \mathcal {N}, \end{aligned}$$

(16c)

$$\begin{aligned}&y_{mn}, \ell {n} \in \mathbb {Z}^+ \quad \quad \forall m,n \in \mathcal {N}. \end{aligned}$$

(16d)

Table 6 Decision variables and parameters

Appendix B: Performance evaluation of stochastic solutions

This section presents the performance evaluation of stochastic solutions obtained from solving the 2-SIP with first-stage problem (6) and second-stage problem (7). To achieve this goal, metrics, such as expected value of perfect information (EVPI) and the value of stochastic solution (VSS), are computed. EVPI and VSS are obtained for a problem instance created with data from the Barisal region. The number of scenarios generated for these experiments is $S=1000$.

EVPI measures the price one is willing to pay to gain access to perfect information, and EVPI uses the difference between the objective function value of the wait-and-see and here-and-now problems. To calculate the objective function value of the wait-and-see problem, the SAA problem in formulation (9) is solved separately for each single scenario. Next, the corresponding expected value is calculated over the scenarios generated. The here-and-now problem is indeed the SAA problem in formulation (9). Then, $EVPI = \$ 3300 - \$ 2870 = \$ 428$. This means, if the number of patients arriving in a session is known, the total costs would only be $2870. Thus, the cost of not knowing the future is $EVPI = \$428$.

Fig. 11

The value of the stochastic solution

VSS measures the impact of random patient arrivals has on the performance of the system (Birge and Francois 2011). VSS is the difference between the objective function value of SAA problem in formulation (9) and the deterministic mean value problem. To compare the objective function values of SAA and deterministic mean value problem, both problems are initially solved for a given set of scenarios. Next, the values of the first-stage solutions are fixed, and second-stage problem (7) is simulated over a different set of scenarios. The corresponding objective function values are reported in Fig. 11. Fewer variations in the objective function values obtained from the stochastic solution are observed. Also, the average objective function value of the stochastic solution is lower than the solutions to the mean value problem. These results indicate that solving the proposed stochastic model formulation, rather than the corresponding mean value problem, is valuable.