Real-time management of intra-hospital patient transport requests

Abstract

This paper addresses the management of patients’ transportation requests within a hospital, a very challenging problem where requests must be scheduled among the available porters so that patients arrive at their destination timely and the resources invested in patient transport are kept as low as possible. Transportation requests arrive during the day in an unpredictable manner, so they need to be scheduled in real-time. To ensure that the requests are scheduled in the best possible manner, one should also reconsider the decisions made on pending requests that have not yet been completed, a process that will be referred to as rescheduling. This paper proposes several policies to trigger and execute the rescheduling of pending requests and three approaches (a mathematical formulation, a constructive heuristic, and a local search heuristic) to solve each rescheduling problem. A simulation tool is proposed to assess the performance of the rescheduling strategies and the proposed scheduling methods to tackle instances inspired by a real mid-size hospital. Compared to a heuristic that mimics the way requests are currently handled in our partner hospital, the best combination of scheduling method and rescheduling strategy produces an average 5.7 minutes reduction in response time and a 13% reduction in the percentage of late requests. Furthermore, since the total distance walked by porters is substantially reduced, our experiments demonstrate that it is possible to reduce the number of porters – and therefore the operating costs – without reducing the current level of service.

Appendix A Assessing the performance of methods based on the mathematical formulation

Table 6 Evaluation of MP computational performance in terms of average, minimum, maximum and of confidence intervals at 95% of the percentage of optimal solutions and the reported optimality gap for reschedules that failed to give proof of optimality

In this subsection, we report additional results allowing us to investigate the performance of the methods based on the mathematical formulation presented in Subsection 4.1 (i.e., methods XXMP) with the enhancements described in Section 6.2.

We are first interested in the number of times each approach triggered a rescheduling, and for each rescheduling, the number of requests considered, as well as the time they consumed. The results are reported in Table 4 for each policy ($\varPhi 1$, $\varPhi 2$, $\varPhi 3$, and $\varPhi 4$) over all 36 instances, considering the three profiles. The half-width of confidence intervals computed at 95% is also provided under headers CI. In addition, Table 4 also provides the average computational time per reschedule (in seconds) and total execution (in hours).

To give the reader an overview of the comparison of the computational time between the scheduling approaches, the average of total computational time, in seconds, is presented in Table 5, also, for each policy over all 36 instances. The haft width of confidence intervals computed at 95% and the maximum value are also provided under headers CI and MAX, respectively. The results clearly demonstrate that both the Constructive Heuristic (CH) and Local Search (LS) approaches are well-suited for real-time applications in terms of computational efficiency.

We are now interested in the formulation’s effectiveness, always with the described enhancements, which include two key components: 1) providing a warm initialization to the solver using the solution produced by the LS approach, and 2) improving the MP formulation with a lower bound. To incorporate the LB into the mathematical formulation, we add the following constraints:

$$\begin{aligned}&LB = \sum _{i \in R}\alpha _i \left( min_{p \in P} \{C_{ip} - t^{d}_{i} \}\right) \end{aligned}$$

(13)

$$\begin{aligned}&LB \le \sum _{i \in R}\alpha _i L_{i} \end{aligned}$$

(14)

Constraint Eq. 13 defines the LB, while constraint Eq. 14 imposes a limit on the objective function. Table 6 presents the percentage of times the solver gave proof of optimality and the half-width of the confidence interval at 95% around it (columns C.I.), as well as the reported gap for those reschedules for which such proof could not be made. It is worth mentioning that the gap has been computed for every reschedule within each instance. Consequently, the gap results presented herein represent the average of the aggregation within the instances of the average, minimum, maximum, and standard deviation values. The best average value is highlighted in bold and underlined for ease of identification.

Policy $\varPhi 1$ leads to the most favorable results, as the MP achieves nearly 100% optimal solutions. Additionally, the average gaps presented for this policy are the smallest ones. However, it is noteworthy that for all four policies, there were instances in which all the reschedules were solved to optimality. On the other hand, Policies $\varPhi 2$ and $\varPhi 3$ produced less optimal reschedules and larger optimal gaps.