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Staff Scheduling for Demand-Responsive Services

Debsankha Manik, Rico Raber

TL;DR

The paper tackles staff scheduling for demand-responsive services by directly integrating demand modelling into the shift optimization. It formulates a mixed-integer convex program that maximizes the total reward $\sum_t f_t(y_t)$ with a concave $f_t$, given a fixed workforce and time-varying demand $d_t$. A shift-agnostic optimum provides a theoretical benchmark, yielding a closed-form $y_t^* = \frac{sN\delta}{\sum_t d_t} d_t$ under the chosen $f_t$, against which solutions are evaluated via a relative gap. Empirical results in an on-demand mobility setting show that the integrated approach achieves higher total rewards than methods that separate demand modelling from scheduling, and it exhibits favorable robustness to internal parameter choices. The work lays groundwork for extensions to multiple shift types and full shift assignment, with practical implications for MaaS operators seeking revenue-maximizing, demand-aware staffing.

Abstract

Staff scheduling is a well-known problem in operations research and finds its application at hospitals, airports, supermarkets, and many others. Its goal is to assign shifts to staff members such that a certain objective function, e.g. revenue, is maximized. Meanwhile, various constraints of the staff members and the organization need to be satisfied. Typically in staff scheduling problems, there are hard constraints on the minimum number of employees that should be available at specific points of time. Often multiple hard constraints guaranteeing the availability of specific number of employees with different roles need to be considered. Staff scheduling for demand-responsive services, such as, e.g., ride-pooling and ride-hailing services, differs in a key way from this: There are often no hard constraints on the minimum number of employees needed at fixed points in time. Rather, the number of employees working at different points in time should vary according to the demand at those points in time. Having too few employees at a point in time results in lost revenue, while having too many employees at a point in time results in not having enough employees at other points in time, since the total personnel-hours are limited. The objective is to maximize the total reward generated over a planning horizon, given a monotonic relationship between the number of shifts active at a point in time and the instantaneous reward generated at that point in time. This key difference makes it difficult to use existing staff scheduling algorithms for planning shifts in demand-responsive services. In this article, we present a novel approach for modelling and solving staff scheduling problems for demand-responsive services that optimizes for the relevant reward function.

Staff Scheduling for Demand-Responsive Services

TL;DR

The paper tackles staff scheduling for demand-responsive services by directly integrating demand modelling into the shift optimization. It formulates a mixed-integer convex program that maximizes the total reward with a concave , given a fixed workforce and time-varying demand . A shift-agnostic optimum provides a theoretical benchmark, yielding a closed-form under the chosen , against which solutions are evaluated via a relative gap. Empirical results in an on-demand mobility setting show that the integrated approach achieves higher total rewards than methods that separate demand modelling from scheduling, and it exhibits favorable robustness to internal parameter choices. The work lays groundwork for extensions to multiple shift types and full shift assignment, with practical implications for MaaS operators seeking revenue-maximizing, demand-aware staffing.

Abstract

Staff scheduling is a well-known problem in operations research and finds its application at hospitals, airports, supermarkets, and many others. Its goal is to assign shifts to staff members such that a certain objective function, e.g. revenue, is maximized. Meanwhile, various constraints of the staff members and the organization need to be satisfied. Typically in staff scheduling problems, there are hard constraints on the minimum number of employees that should be available at specific points of time. Often multiple hard constraints guaranteeing the availability of specific number of employees with different roles need to be considered. Staff scheduling for demand-responsive services, such as, e.g., ride-pooling and ride-hailing services, differs in a key way from this: There are often no hard constraints on the minimum number of employees needed at fixed points in time. Rather, the number of employees working at different points in time should vary according to the demand at those points in time. Having too few employees at a point in time results in lost revenue, while having too many employees at a point in time results in not having enough employees at other points in time, since the total personnel-hours are limited. The objective is to maximize the total reward generated over a planning horizon, given a monotonic relationship between the number of shifts active at a point in time and the instantaneous reward generated at that point in time. This key difference makes it difficult to use existing staff scheduling algorithms for planning shifts in demand-responsive services. In this article, we present a novel approach for modelling and solving staff scheduling problems for demand-responsive services that optimizes for the relevant reward function.
Paper Structure (28 sections, 2 theorems, 32 equations, 9 figures)

This paper contains 28 sections, 2 theorems, 32 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Demand Modelling for Staff Scheduling
  3. Our Contribution: Integrating Demand Modelling and Shift Optimization into a Single Step
  4. Problem setting
  5. Application to shift planning for on-demand mobility
  6. The Variables and Constraints
  7. The shift types
  8. The reward function
  9. Demanded rides
  10. The constraints
  11. Shift plan must match the number of employed drivers
  12. Available vehicles constraint
  13. The Full Program
  14. Benchmarking against shift agnostic optimum
  15. Benchmarking against approaches using a separate demand modelling step
  16. ...and 13 more sections

Key Result

Lemma 1

If ( ) and ( ) hold, then there exists an assignment of shifts to drivers such that each driver works exactly $s$ shifts of $\delta$ hours each, and each driver has a break of at least $\beta$ time steps between any two consecutive shifts.

Figures (9)

  • Figure 1: The reward $r_t$ as a function of supply $y_t$ as defined in \ref{['eq:chosen-reward-function']} for $d_t = 1$ and different values of the parameter $a$.
  • Figure 2: The demanded rides $d_t$ as a function of time $t$.
  • Figure 3: Proof of Part 2 of \ref{['lemma:shift-assignment']}. The dashed rectangle shows a connected component of the bipartite graph $G$. $P_1$ can $P_2$ can be swapped, keeping the graph bipartite, because there is no edge between the connected component and the rest of the graph.
  • Figure 4: A solution to the MIP formulation \ref{['mip']} that is close to the shift agnostic optimum. Solved for $N=10$, $s=5$, $\delta=8$, $\beta=8$, $d_{\max}=10$, $a=2$.
  • Figure 5: As the number of drivers $N$ increases, the difference between the shift agnostic optimum and the optimal shift plan decreases. (a) The supply curve, for different values of total drivers (coloured dashed lines), as well as the shift agnostic optimum (solid black line). Each supply curve is normalized, i.e., divided by the total working hours $sN\delta$. (b) The relative gap from optimum supply compared to the shift agnostic optimum, which decreases as the number of drivers increases.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1: Relative gap from optimum supply