Online scheduling for outpatient services with heterogeneous patients and physicians

Abstract

In outpatient services, it is critical to schedule patients for physicians to reduce both patients waiting and physicians overtime working. In this paper, we regard the problem as an online scheduling problem and based on analysis of a real data set from a big hospital in China, we develop a dynamic programming model to solve the problem. We propose a Policy Iteration Algorithm to find the optimal solution in the steady state, and obtain the structural properties of the policy. We conduct numerical experiments to compare the performance of the policy with that of the two policies used in practice by simulating various scenarios. The numerical results show that the policy has the best performance across all scenarios, especially when the system is heavily loaded. We also discuss the managerial implications of the study for practitioners. The model and solution method can be easily extended to multi-server case and can be applied to the general service scheduling problems with heterogeneous customers and service providers.

Appendix: Proofs

1.1 A.1 Proof of Proposition 1

If $h(\cdot )$ is a linear function, then in the minimization problem of each iteration, the objective function in is also a linear function of $a({\varvec{s}})$, then the optimal solution must be achieved among extreme points, which is either 1 or 0.

1.2 A.2 Proof of Proposition 2

If $f({\varvec{s}}|{\varvec{a}})$ only depends on $a({\varvec{s}})$ and is a linear function of $a({\varvec{s}})$, and under policy ${\varvec{a}}$, $a(n_G-1,n_S)=1$, $a(n_G+1,n_S)=1$, $a(n_G,n_S-1)=1$ and $a(n_G,n_S+1)=1$, then we have, $g((n_G,n_S)|{\varvec{a}})\ge g((n_G-1,n_S+1)|{\varvec{a}})$, $g((n_G+2,n_S)|{\varvec{a}})\ge g((n_G+1,n_S+1)|{\varvec{a}})$, $g((n_G+1,n_S-1)|{\varvec{a}})\ge g((n_G,n_S)|{\varvec{a}})$ and $g((n_G+1,n_S+1)|{\varvec{a}})\ge g((n_G,n_S+2)|{\varvec{a}})$. Since $g((n_G+1,n_S)|{\varvec{a}})=\frac{1-p_1-p_2}{2}(g((n_G,n_S)|{\varvec{a}})+g((n_G+1,n_S-1)|{\varvec{a}}))+p_1\min (g((n_G+2,n_S)|{\varvec{a}}),g((n_G+1,n_S+1)|{\varvec{a}}))+p_2g((n_G+1,n_S+1)|{\varvec{a}})$ and $g((n_G,n_S+1)|{\varvec{a}})=\frac{1-p_1-p_2}{2}(g((n_G,n_S)|{\varvec{a}})+g((n_G-1,n_S+1)|{\varvec{a}}))+p_1\min (g((n_G,n_S+2)|{\varvec{a}}),g((n_G+1,n_S+1)|{\varvec{a}}))+p_2g((n_G,n_S+2)|{\varvec{a}})$, where $p_1=\frac{\lambda _G}{\lambda _G+\lambda _C+2\mu }$ and $p_2=\frac{\lambda _C}{\lambda _G+\lambda _C+2\mu }$, thus we have $g((n_G+1,n_S)|{\varvec{a}})\ge g((n_G,n_S+1)|{\varvec{a}})$, which means that $a(n_G,n_S)=1$.

If $f({\varvec{s}}|{\varvec{a}})$ only depends on $a({\varvec{s}})$ and is a linear function of $a({\varvec{s}})$, and under policy ${\varvec{a}}$, $a(n_G-1,n_S)=0$, $a(n_G+1,n_S)=0$, $a(n_G,n_S-1)=0$ and $a(n_G,n_S+1)=0$, then we have, $g((n_G,n_S)|{\varvec{a}})\le g((n_G-1,n_S+1)|{\varvec{a}})$, $g((n_G+2,n_S)|{\varvec{a}})\le g((n_G+1,n_S+1)|{\varvec{a}})$, $g((n_G+1,n_S-1)|{\varvec{a}})\le g((n_G,n_S)|{\varvec{a}})$ and $g((n_G+1,n_S+1)|{\varvec{a}})\le g((n_G,n_S+2)|{\varvec{a}})$. Since $g((n_G+1,n_S)|{\varvec{a}})=\frac{1-p_1-p_2}{2}(g((n_G,n_S)|{\varvec{a}})+g((n_G+1,n_S-1)|{\varvec{a}}))+p_1\min (g((n_G+2,n_S)|{\varvec{a}}),g((n_G+1,n_S+1)|{\varvec{a}}))+p_2g((n_G+1,n_S+1)|{\varvec{a}})$ and $g((n_G,n_S+1)|{\varvec{a}})=\frac{1-p_1-p_2}{2}(g((n_G,n_S)|{\varvec{a}})+g((n_G-1,n_S+1)|{\varvec{a}}))+p_1\min (g((n_G,n_S+2)|{\varvec{a}}),g((n_G+1,n_S+1)|{\varvec{a}}))+p_2g((n_G,n_S+2)|{\varvec{a}})$, where $p_1=\frac{\lambda _G}{\lambda _G+\lambda _C+2\mu }$ and $p_2=\frac{\lambda _C}{\lambda _G+\lambda _C+2\mu }$, thus we have $g((n_G+1,n_S)|{\varvec{a}})\le g((n_G,n_S+1)|{\varvec{a}})$, which means that $a(n_G,n_S)=0$.