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.

翻译：人员调度是运筹学中的一个经典问题，广泛应用于医院、机场、超市等诸多场景。其目标是为员工分配班次，以最大化特定目标函数（如收入），同时满足员工与组织的各类约束条件。在传统的人员调度问题中，通常存在关于特定时间点最低在岗员工数量的硬性约束，且往往需同时考虑多个硬性约束以确保不同岗位的员工数量达标。然而，针对需求响应服务（如拼车与网约车服务）的人员调度存在一个关键差异：这类服务通常没有固定时间点所需最低员工数量的硬性约束。相反，不同时间点的在岗员工数量应根据该时间点的需求动态调整。若某时间点员工过少会导致收入损失，而员工过多则因总工时有限，将导致其他时间点人手不足。该问题的目标是在规划周期内最大化总收益，其中任意时间点的活跃班次数量与该时间点的瞬时收益呈单调关系。这一关键差异使得现有人员调度算法难以直接应用于需求响应服务的班次规划。本文提出一种新颖的建模与求解方法，专门针对需求响应服务的人员调度问题，以优化相关收益函数。