A medical resource allocation model for serving emergency victims with deteriorating health conditions

Abstract

Large-scale disasters typically result in a shortage of essential medical resources, and thus it is critical to optimize resource allocation to improve the quality of the relief operations. One important factor that has been largely neglected when optimizing the available medical resources is the deterioration of victims’ health condition in the aftermath of a disaster; e.g., a victim’s health condition could deteriorate from mild to severe if not treated promptly. In this paper, we first present a novel queueing network to model this deterioration in health conditions. Second, we provide both analytical solutions and numerical illustrations for this queueing network. Finally, we formulate two resource allocation models in order to minimize the total expected death rate and total waiting time, respectively. Numerical examples are provided to illustrate the properties of optimal policies.

Appendices

Appendix 1

The solution method used for solving the 2-D Markov process in this paper involves decomposing the 2-D process into a set of 1-D Markov processes. Instead, we can obtain the numerical results by solving the steady-state equations for the 2-D Markov process directly. Table 1 provides the results comparing the decomposition versus the numerical methods in three examples where $C_1=C_2=4, \lambda _2=1, q_{21}=0.2, q_{10}=0.1$: Example 1 ($\lambda _1=0.8, \mu _1=1.0, \mu _2=1.5$), Example 2 ($\lambda _1=0.5, \mu _1=0.5, \mu _2=1.5$), and Example 3 ($\lambda _1=0.8, \mu _1=1.0, \mu _2=1.5$). From Table 1 we observe that the comparison results are pretty good: the absolute errors are very small.

Table 1 Comparing analytical with approximating probabilities for each states and each queue

Further more, we study how such absolute errors change when the number of states $C$ increases. For each of the three examples in Table 1, we extend to study $C=5, 10, 15, 50$ as shown in Table 2. The results show that both the average and the standard deviation of the absolute errors (across states; it is not meaningful to report probabilities for each of the states $i=1,C$ for each queue) decrease when $C$ increases. This confirms that our approximation method is stable.

Table 2 Average and standard deviations for absolute errors between analytical and approximating probabilities when $C$ changes

Appendix 2

To test the optimality of the Local Unimodal Sampling algorithm in Sect. 3.1, we randomly generate twelve instances of the medical resource allocation problem of our interest, and compare the results obtained from unimodal sampling algorithm against the direct grid-search method (exhaustive search). The grid-search method involves setting up a suitable grid in the design space, evaluating the objective function at all grid points, and finding the grid point corresponding to the lowest function value (Rao 2009). The reason for choosing the simple grid-search approach for comparison purpose is that it may not be safe to use approximation or heuristic methods that avoid doing an exhaustive parameter search. For the twelve problem instances we tested, the results (expected total death rate) from unimodal sampling algorithm are 0.18 % higher than those from the grid-search on average, and a more detailed comparison is shown in Fig. 9. The comparison results provide evidence that the Local Unimodal Sampling Algorithm is acceptable for the optimization problems studied in this paper.

Fig. 9

Unimodal sampling algorithm versus direct grid-search