Enhancing Fairness in High-Speed Railway Crew Scheduling: A Two-Stage Heuristic Optimization Framework Under Daily-Adjusted Timetables

Featured Application

The proposed flexible high-speed railway crew scheduling optimization framework and two-stage heuristic algorithm can be directly applied to decision-support systems. The method enables automated and fair crew task allocation while ensuring compliance with operational safety and regulatory constraints, significantly reducing manual scheduling efforts and enhancing resource utilization efficiency.

Abstract

The existing crew base assignment system in high-speed railway operations struggles to cope with the frequent deployment of additional and coupled trains under the “One-Day-One-Operation Plan” dynamic scheduling paradigm. This often results in unequal overtime distribution among crews, low scheduling efficiency, and limited operational adaptability. To address the above-mentioned application challenges, this study proposes a shift from the fixed crew-based system towards a fully flexible pool-based system. Specifically, we develop a novel integer programming model designed to optimize monthly crew schedules with the primary objective of balancing total working hours across all crew teams. In this model, crew teams are treated as unified entities but are no longer permanently tied to specific train services. Instead, they are dynamically allocated to all available train tasks within the network. Numerical results, based on a real-world case study from Shanghai, China, demonstrate that the proposed model effectively automates the scheduling process. It significantly enhances fairness in working hour distribution while fully complying with all operational rules. Furthermore, by enabling crews to undertake a diverse range of services, the model substantially improves the flexibility of human resource allocation and the overall robustness of the crew management system. This research provides an efficient and scientific decision-support tool for tackling crew scheduling difficulties in dynamic railway operations.

1. Introduction

As China’s high-speed railway system rapidly evolves and passenger service demand escalates, railway operations are shifting from a conventional static periodic timetable to a daily-adjusted timetable [1,2,3], a short-horizon service plan regenerated each operating day from a baseline schedule using updated demand forecasts and operational resource constraints, such that departure times, stopping patterns, and train frequencies may vary across days while remaining operationally feasible [4,5]. Figure 1a,b illustrates a multi-day comparison of the periodic and daily-adjusted paradigms. This high-frequency, short-term adjustment technique facilitates swift adaptation to varying passenger demand, but it also leads to numerous unscheduled, temporary crew assignments. The existing high-speed railway crew management system typically utilizes a fixed assignment model, wherein crews are consistently assigned to designated passenger routes. The absence of flexibility results in temporary tasks being managed through significant manual modifications, leading to considerable disparities in the monthly distribution of work hours among crews, which adversely impacts both the equity of labor resource allocation and the efficiency of scheduling [6,7]. Consequently, exploring flexible optimization methods for high-speed railway crew scheduling has emerged as a pivotal scientific concern for enhancing the operational management of high-speed rail services.

Existing railway crew scheduling studies [8,9] typically decompose planning into the Crew Scheduling Problem (CSP) and Crew Rostering Problem (CRP) under a fixed periodic timetable [10,11], which is not well suited to routine, network-wide variability under daily-adjusted operations. Meanwhile, disruption-management studies mainly aim to restore feasibility and minimize deviation/cost after localized incidents, rather than systematically optimizing monthly-scale crew assignment under recurring daily adjustments. Finally, although fairness has attracted growing attention, most fairness-oriented models are still validated under static or low-frequency timetable changes [12,13]. Therefore, a gap remains for a fairness-first crew (re)assignment framework that treats daily adjustment as the operational norm and scales to monthly (and multi-month) planning [14].

To fill this gap, we propose a fairness-first, pooled-resource crew scheduling framework for daily-adjusted high-speed railway operations. The framework (i) unifies baseline and day-specific ad hoc duties over a monthly horizon, (ii) treats crews as a flexible pool rather than route-bound units, and (iii) explicitly optimizes workload equity by minimizing the range of total monthly working hours under operational and fatigue-regulation constraints. To enable large-scale deployment, we further develop a two-stage “construct-then-improve” heuristic (greedy construction + local search) and validate its effectiveness using real-world data.

The contribution of this study lies in both theoretical and practical dimensions. Theoretically, we formulate an integrated optimization framework tailored to daily-adjusted timetables, which unifies baseline and day-specific ad hoc duties, treats crews as a pooled resource instead of route-bound, and explicitly models range-based fairness of monthly working hours under operational and regulatory constraints, thereby adapting the classical crew scheduling and crew rostering pipeline to a fairness-oriented setting driven by daily-adjusted timetables. Practically, we customize and implement a simple two-stage heuristic, greedy assignment followed by local search, within this framework and demonstrate, using real high-speed railway data, that this problem-tailored implementation achieves substantial improvements in workload equity while maintaining acceptable computational times at monthly and multi-month scales.

The structure of this study is organized as follows. Section 2 reviews the current research on railway crew scheduling; Section 3 introduces the problem definition, model formulation, and algorithm design for the flexible crew scheduling optimization model; Section 4 validates the efficiency and solution quality of the proposed method; Section 5 discusses how to design, deploy, and promote a flexible crew scheduling system for high-speed rail based on the proposed model; Section 6 summarize the conclusions, limitations, and future research directions.

2. Literature Review

The railway crew scheduling problem (RCSP) is generally formulated as a set-partitioning/covering model over a duty pool within a space-time network and resolved by column generation along with branch-and-price techniques. These studies collectively assume a fixed timetable and optimize primarily for cost or crew size [15,16,17]. China’s high-speed train has implemented daily-adjusted timetables to align with short-term demand fluctuations. This operational reality generates persistent, extensive re-assignment demands on crew members that differ qualitatively from the occasional disruptions anticipated in traditional planning. As a result, plans that are best for a singular, periodic diagram may become mismatched with daily service variations, leading to persistent workload inequities if crew allocation is route-specific [18].

To solve RCSPs under periodic timetables, exact approaches have centered on decomposition, especially column generation (CG), which iterates between a master set-partitioning problem and pricing subproblems to avoid enumeration. For example, Tapkan et al. [19] cast the master as a set-partitioning model and generate new, constraint-feasible duties via pricing to iteratively reduce cost. Van develops a rolling horizon approach with column generation to achieve individual-level fairness in railway crew scheduling, satisfying personalized work distribution rules for over 95% of employees in real-world instances. Dong et al. [20] pioneered modeling and solution frameworks (sequential and integrated) to address the crew scheduling problem with fairness considerations under flexible on-train supervision in metro automated operations, validated effectively with real-world case data. Nevertheless, as the quantity of feasible duties increases exponentially with the crew size and duty sets, even decomposition-based exact methods may encounter difficulties in producing an acceptable solution within an acceptable period for large instances. Consequently, constructive heuristics, graph search, and population-based metaheuristics are widely used to trade off solution quality and runtime, often serving either as stand-alone solvers or as warm starts for CG.

Most dynamic work treats crew planning as rescheduling after localized disruptions (e.g., delays, blockages) [21,22]. In this disruption-management line, the timetable, rolling stock, and crew plans are typically recovered sequentially, with models and heuristics designed to restore feasibility quickly and minimize deviations from the published plan. Representative studies include crew rescheduling with limited train retiming and real-time insertion heuristics for small disruptions: effective for one-off incidents, but not designed for the routine, network-wide re-planning that a daily-adjusted regime creates day after day. For instance, Wang et al. [23] propose a co-adjustment of timetable and crew plans on mixed passenger-freight lines, and other studies construct rapid re-planning at the individual-crew level; yet these designs target emergency response and may not scale to the recurring, large-scale re-assignment induced by daily timetable updates.

Beyond responsiveness, objective design is another limitation: many formulations retain cost minimization as the dominant goal while under-weighting fairness, which affects the acceptability and executability of schedules in practice. If duty allocation, working hours, or workload intensity are imbalanced, crews’ willingness to execute the plan may erode, potentially degrading service quality. Recent studies begin to incorporate equity, e.g., Breugem et al. [24] co-optimize duty attractiveness and fairness; Bansal et al. [25] balance workload and align roster cycles; and Neufeld, Scheffler, Tamke, Hoffmann and Buscher [16] examine balanced task distributions under cost controls. However, most of these contributions are evaluated under static or low-frequency adjustments and thus fall short when timetables change frequently.

Beyond railway applications, fairness-oriented optimization methods have been extensively studied in related fields of crew scheduling and resource allocation, particularly in aviation operations. For example, Zhou et al. [26] proposed a multi-objective ant colony algorithm for airline crew scheduling that simultaneously optimizes fairness and preference satisfaction, demonstrating that the multi-objective metaheuristic algorithm can effectively approximate the Pareto front on large-scale monthly instances. Breugem et al. [27] developed a fairness-oriented crew scheduling framework that explicitly quantifies the trade-off between fairness and scheduling attractiveness, introduces an approximate resource allocation model to analyze fairness schemes, and uses real-world data from Dutch Railways to show that fairness considerations can be integrated into precise branch pricing and reduction methods without sacrificing operational relevance. More broadly, the survey of airline crew scheduling by Kasirzadeh et al. [28] demonstrates how modern models must adapt to complex protocols and individual considerations, providing methodological reference that are also relevant in designing more equitable crew scheduling models in the high-speed railway operation.

While recent studies have begun to incorporate fairness into crew scheduling and rostering, and others have addressed dynamic or disruption-aware timetabling, their focus are different. Fairness-oriented works typically assume a static or periodic timetable and evaluate equity on fixed rosters or within rolling-horizon schemes, often treating the timetable as exogenous and crew duties as essentially stable across days. Conversely, dynamic and disruption-management studies emphasize rapid feasibility restoration and cost or deviation minimization, and are not designed to handle fairness-first, network-wide assignment driven by a systematic daily adjusted timetable. In contrast, this paper takes daily-adjusted timetables as the norm rather than an exception, and formulates a fairness-first integrated framework that unifies baseline and day-specific ad hoc duties over a monthly horizon, treats crews as a pooled resource instead of route-bound, and explicitly models range-based fairness of monthly working hours under operational and regulatory constraints.

In sum, the literature offers (i) mature periodic RCSP formulations and solvers, (ii) fast disruption-recovery tools for localized shocks, and (iii) emerging fairness objectives mostly validated on static rosters. The comparison of the representative studies is denoted in

Table 1. Overview of railway crew scheduling optimization representative studies.

Author/Year	Model	Objective	Key Constraints	Main Takeaways
Neufeld et al. (2021) [16]	Overlapping multi-period RCSP	Cost, attendance rates	Coverage, depot, attendance	Efficient CG for practical multi-period variants.
Veelenturf et al. (2012) [13]	Crew rescheduling with retiming	Minimize change/cancelations	Legal rules; limited retiming	Feasibility restoration after disruptions
Breugem et al. (2022) [27]	Fairness-oriented crew rostering	Equity and attractiveness	Labor rules	Formal equity objectives
Feng et al.(2024) [29] Feng et al. (2023) [30]	Integrated crew scheduling + rostering	Cost and duration variance	Operation constraints	Integration across layers improves quality
Wang et al. (2024) [31]	Individual crew re-planning model	Minimize assignment cost, reduce deviation from the original plan	Task coverage, duty, and roster rules	The proposed decomposition strategy is effective for long-period scheduling.
Fuentes et al. (2019) [7]	Hybrid network-flow task assignment model	Minimize total operational cost	Task coverage, continuous working time	The fix-and-relax heuristic method can obtain near-optimal solutions in a short computational time.
Wang et al. (2022) [32]	Time-space-state network for railway crew scheduling	Minimize total crew pairing cost	Standard railway crew rules	The Lagrangian relaxation approach is highly efficient, fast, and achieves near-optimality
Hanczar et al. (2021) [33]	Duty generation and assignment model	Minimize total working time	Total working time limits, Rest/break time rules, Task coverage	Demonstrates considerable cost savings over manual plans
Wang et al. (2024) [23]	Coordinated Rescheduling MILP	Minimize passenger delay and crew deviation	Crew connection rules, train operation constraints	The rolling horizon algorithm enhances efficiency and real-time response
This study	Integrated crew scheduling optimization framework	Fairness (Minimize the range of total working hours among all crews)	Task assignment constraints, fatigue control regulation constraints, and fairness constraints	Unify baseline and day-specific duties, treat crews as a pooled resource rather than route-fixed.

. While Table 1 provides a broad overview,

Table 2. Comparison among closest related studies.

Closest Related Study	Timetable Setting	Planning Scope	Primary Goal	Key Difference vs. This Paper
Fairness-oriented crew planning (static/periodic)	periodic/low-frequency	roster or rolling horizon	fairness + cost	does not model routine daily-adjusted duty generation at network scale
Disruption-management crew rescheduling	incident-driven changes	short-term recovery	feasibility + deviation/cost	focuses on localized shocks, not systematic monthly re-planning
This study	daily-adjusted as norm	monthly/multi-month	fairness-first (range of hours)	unifies baseline + ad hoc duties; pooled crews; scalable two-stage heuristic

make the novelty explicit by contrasting our framework with the most closely related fairness-oriented railway crew planning studies and disruption-management rescheduling studies. This targeted comparison highlights that our work uniquely (i) treats daily-adjusted timetables as the norm, (ii) co-allocates baseline and ad hoc duties at the monthly horizon, and (iii) prioritizes fairness as the primary objective within a pooled-crew assignment setting.

3. Methodology

3.1. Problem Formulation and Notation

This study addresses the high-speed railway crew scheduling optimization problem under daily-adjusted timetables. The daily timetable consists of a baseline set of tasks and a set of temporary tasks added based on short-term passenger flow forecasts, experiencing high-frequency changes. Traditional crew management models, which rely on fixed routes, cannot accommodate this operational flexibility and often lead to significant imbalances in the total work hours distribution among crews over the month. To address this issue, the study proposes an integrated high-speed railway crew scheduling optimization model. The core concept of this model is to treat all crews as a pooled resource that can be dynamically allocated, breaking the traditional binding relationship between crews and fixed routes. This allows for coordinated distribution of both baseline and temporary tasks.

The crew scheduling problem is formalized as a combinatorial optimization problem, with the primary objective of improving the fairness of the workload distribution among crews. The challenge lies in assigning dynamically changing tasks to various crews while satisfying a series of complex operational constraints, including maximum working hours, limits on consecutive working days, minimum rest time requirements, and the feasibility of task continuity. The key is to assign each task to a unique crew while ensuring that the schedule satisfies both operational feasibility and fairness in resource allocation. The notation utilized in this model is summarized in Appendix A 

Table A1. Notation of the high-speed railway crew scheduling optimization model.

Symbol	Type	Explanation
Indices
$g$	Index	Element of crew set
$j$	Index	Element of the task set
$d$	Index	Element of date set
$w$	Index	Element of the time window set
Sets
$\mathcal{G}$	Set	Set of all train crews
$\mathcal{D}$	Set	Set of operating dates in the planning horizon
$J_d^B$	Set	Set of baseline duties operated on the date $d$
$J_d^{\Delta }$	Set	Set of adjustment duties on the date $d$
$J_d$	Set	Set of all duties on the date $d$
$\mathcal{J}$	Set	Unified monthly duty set
$J_d^H$	Set	Set of high-intensity duties on date $d$
$\mathcal{W}$	Set	Set of all seven-day windows in $\mathcal{D}$
Model parameters
$A_{g,d}$	Binary	Take 1 if crew g is available on date d, otherwise 0
$M_{j_2,d_2}^{j_1,d_1}$	Binary	$\mathrm{Take} 1 \mathrm{if} \mathrm{assigning} \mathrm{duties} \left(j_1,d_1\right)$$\mathrm{and} \left(j_2,d_2\right)$ consecutively to the same crew would violate the minimum rest requirement, 0 otherwise.
$D_{\mathrm{Max}}$	Integer	Maximum number of working days in a month, units: days
$D_{\mathrm{Min}}$	Integer	Minimum rest days during the planning period, units: days
$N_C$	Integer	Maximum consecutive working days, units: days
$N_R$	Integer	Minimum required rest days after consecutive work days, units: days
$\delta$	Integer	Maximum total number of duty day for a crew during a seven-day time window, units: days
$Q$	Integer	Maximum high-intensity tasks (early departure or late arrival) for a crew during the planning period, units: tasks
$N_{\mathrm{Min}}^{\Delta }$	Integer	Minimum desired number of days on which a crew performs adjustment duties during the month, units: days
$N_{\mathrm{Max}}^{\Delta }$	Integer	Maximum allowable number of days on which a crew performs adjustment duties during the month, units: days
$L^{\Delta }$	Integer	Maximum allowed length of a consecutive sequence of days on which a crew performs adjustment duties, units: days
$h_{j,d}$	Continuous	Working hours of duty j on date d unit: hours
$R_{\mathrm{Min}}$	Continuous	Minimum rest time between two tasks, unit: hours
$H_{M\mathrm{ax}}$	Continuous	Maximum monthly total working hours, unit: hours
$H_{\mathrm{Min}}$	Continuous	Minimum monthly working hours, unit: hours
Decision variables
$X_{g,d}$	Binary	Take 1 if crew g works on date d, otherwise 0
$Y_{g,j,d}$	Binary	Take 1 if crew $g$ $\mathrm{is} \mathrm{assigned} \mathrm{to} \mathrm{task} j\in {\mathcal{J}}_d$, otherwise 0
$Z_{g,d}^{\Delta }$	Binary	Take 1 if crew g performs at least one adjustment duty on date d, otherwise 0
$T_g$	Continuous	Total working hours for crew g during the month, units: hours
$T_g^B$	Continuous	Total working hours of crew $g$ from baseline duties, units: hours
$T_g^{\Delta }$	Continuous	Total working hours of crew $g$ from adjustment duties, units: hours
$N_g^{\Delta }$	Integer	Number of days in the planning period on which crew $g$ performs at least one adjustment duty, units: days
$T_{\mathrm{Max}}$	Continuous	Maximum total working hours across all crews, units: hours
$T_{\mathrm{Min}}$	Continuous	Minimum total working hours across all crews, units: hours
$Z$	Continuous	Objective value: range of total monthly working hours, units: hours

.

3.2. Flexible Crew Scheduling Optimization Model

The optimization model balances labor intensity among high-speed railway crews under a daily-adjusted timetable. Unlike classical formulations built on a single periodic timetable, the model operates on a sequence of day-specific timetables and treats crews as a pooled resource that can be reallocated across days.

Let $\mathcal{D}$ be the set of operating dates in the planning horizon such the 31 days of a month. For each date $d\in \mathcal{D}$, ${\mathcal{J}}_d^B$ denotes the set of baseline duties derived from the reference periodic timetable, and ${\mathcal{J}}_d^{\Delta }$ denotes the set of adjustment duties created by the daily-adjusted timetable such as added services, coupled formations, short turns. The duty set on date $d$ is defined as ${\mathcal{J}}_d={\mathcal{J}}_d^B\cup {\mathcal{J}}_d^{\Delta }$ and the unified monthly duty set is $\mathcal{J}={\cup }_{d\in \mathcal{D}}{J}_d$.

Define $\mathcal{G}$ as the set of crews. For duty $j\in {\mathcal{J}}_d$ on date $d$, $h_{j,d}$ is its working time. Binary variable $Y_{g,j,d}=1$ is assigned to duty $j$ on date $d$, and 0 otherwise. The total monthly working hours of crew $g$ are:

$$T_g={\displaystyle \sum _{d\in \mathcal{D}}{\displaystyle \sum _{j\in {\mathcal{J}}_d}{h}_{j,d}{Y}_{g,j,d}}},\forall g\in \mathcal{G}$$

(1)

which can be decomposed into baseline and adjustment components:

$$T_g=T_g^B+T_g^{\Delta }$$

(2)

$$T_g^B={\displaystyle \sum _{d\in \mathcal{D}}{\displaystyle \sum _{j\in {\mathcal{J}}_d^B}{h}_{j,d}{Y}_{g,j,d}}},\forall g\in \mathcal{G}$$

(3)

$$T_g^{\Delta }={\displaystyle \sum _{d\in \mathcal{D}}{\displaystyle \sum _{j\in {\mathcal{J}}_d^{\Delta }}{h}_{j,d}{Y}_{g,j,d}}},\forall g\in \mathcal{G}$$

(4)

To measure fairness in monthly workload, continuous variables $T_{\mathrm{Max}}$ and $T_{\mathrm{Min}}$ denote the maximum and minimum total working hours among all crews. The primary objective is to minimize the range of monthly hours:

$$\min Z=T_{\mathrm{Max}}-T_{\mathrm{Min}}$$

(5)

The fairness-oriented objective can in principle be quantified by minimizing the variance of total working hours, minimizing the mean absolute deviation (MAD, i.e., the average absolute difference between each crew’s total working hours and the average value), or minimizing the range between the maximum and minimum workloads. We adopt the range-based formulation as the primary objective because it is linear and therefore easier to embed in a mixed-integer model, and because the resulting metric, the maximal difference in monthly hours between any two crews, is highly interpretable for practitioners. The variance- and MAD-based formulations are used in the numerical study as benchmarks for comparison. The objective subject to the following constraints:

$$T_{\mathrm{Min}}\le T_g,\forall g\in \mathcal{G}$$

(6)

$$T_{\mathrm{Max}}\ge T_g,\forall g\in \mathcal{G}$$

(7)

For task assignment and daily crew attendance constraints, every duty generated by the daily-adjusted timetable must be covered by exactly one crew:

$${\displaystyle \sum _{g\in \mathcal{G}}{Y}_{g,j,d}}=1,\forall d\in \mathcal{D},j\in {\mathcal{J}}_d$$

(8)

To represent daily participation, we introduce crew-date attendance variables $X_{g,d}$, equal to 1 if crew $g$ works on date $d$, and 0 otherwise. Parameter $A_{g,d}$ indicates whether crew $g$ is available on date $d$. Under the studied operational practice, each crew performs at most one duty per day, leading to

$${\displaystyle \sum _{j\in {\mathcal{J}}_d}{Y}_{g,j,d}}=X_{g,d},\forall g\in \mathcal{G},d\in \mathcal{D}$$

(9)

$$X_{g,d}\le A_{g,d},\forall g\in \mathcal{G},d\in \mathcal{D}$$

(10)

Fatigue regulation is modeled by minimum rest constraints between consecutive duties, as well as weekly and monthly work-rest patterns, all evaluated on the day-indexed assignment. Let $R_{\min }$ be the minimum required rest between two successive duties for the same crew. For any pair of duties $\left(j_1,d_1\right)$ and $\left(j_2,d_2\right)$ that may be consecutive in time, a binary parameter $M_{j_2,d_2}^{j_1,d_1}=1$ indicates that assigning both to the same crew would violate $R_{\min }$; otherwise $M_{j_2,d_2}^{j_1,d_1}=0$. The rest constraint is mathematically presented as:

$$M_{j_2,d_2}^{j_1,d_1}+Y_{g,j_1,d_1}+Y_{g,j_2,d_2}\le 2,\forall g\in \mathcal{G},\left(j_1,d_1\right),\left(j_2,d_2\right)\in J:\left(j_1,d_1\right)\ne \left(j_2,d_2\right)$$

(11)

This prevents any crew from being assigned two duties whose timing, in the daily-adjusted timetable, would lead to insufficient rest, including baseline-baseline, baseline-adjustment and adjustment–adjustment pairs.

Weekly work-rest rules are enforced using $X_{g,d}$. Let $\mathcal{W}$ be the set of all seven-day windows in $\mathcal{D}$, and let $\delta$ be the maximum allowed working days in any such window (typically five):

$${\displaystyle \sum _{d\in w}{X}_{g,d}}\le \delta ,\forall g\in \mathcal{G},w\in \mathcal{W}$$

(12)

Let $N_C$ be the maximum number of consecutive working days and $N_R$ the minimum number of rest days required after a full $N_C$-day consecutive working. For each crew $g$ and starting date $d$:

$${\displaystyle \sum _{k=0}^{N_C}{X}_{g,d+k}}\le N_C,\forall g\in \mathcal{G},d\in \mathcal{D}$$

(13)

and if the crew works all $N_C$ days in $[d,d+N_C-1]$, it must rest for at least $N_R$ days afterwards, encoded as:

$${\displaystyle \sum _{k=0}^{N_R-1}\left(1-X_{g,d+N_C+k}\right)}\ge N_R\left({\displaystyle \sum _{k=0}^{N_C-1}{X}_{g,d+k}}-N_C+1\right),\forall g\in \mathcal{G},d\in \mathcal{D}$$

(14)

It should be noted that, instead of simply resetting the count at the start of each planned month, the model employs a buffer period approach when enforcing the maximum consecutive workday limit. For each planned period (e.g., a given month), the recent work records of each crew within a short period prior to the start of the period are used as fixed input. Based on these pre-period records, the model initializes the consecutive workday count for each crew on the first day of the planned period and then updates it daily within the model. Therefore, if a crew has already worked several consecutive days at the end of the previous month, these days will be taken into account when evaluating the limit at the beginning of the current month, and the model may require them to rest for a few days before assigning subsequent tasks. This mechanism ensures that the maximum consecutive workday rule is adhered to within the boundaries of each planned period, not just within a single month.

Monthly workload intensity is further regulated by imposing upper and lower bounds on both total working hours and total working days per crew, as presented in Formulas (16)–(19).

$$T_g\le H_{\mathrm{Max}},\forall g\in \mathcal{G}$$

(15)

$$T_g\ge H_{\mathrm{Min}},\forall g\in \mathcal{G}$$

(16)

$${\displaystyle \sum _{d\in \mathcal{D}}{X}_{g,d}\le D_{\mathrm{Max}}},\forall g\in \mathcal{G}$$

(17)

$${\displaystyle \sum _{d\in \mathcal{D}}{X}_{g,d}\ge D_{\mathrm{Min}}},\forall g\in \mathcal{G}$$

(18)

The above constraints jointly bound the effect of baseline and adjustment duties on monthly workload. To control irregular duty timing, we limit the number of high-intensity tasks per crew. Let ${\mathcal{J}}_d^H\subseteq {\mathcal{J}}_d$ be duties on date $d$ that start before 07:00 or end after 21:00, and let $Q$ be the maximum number of such duties a crew may perform in the month:

$${\displaystyle \sum _{d\in \mathcal{D}}{\displaystyle \sum _{j\in {\mathcal{J}}_d^H}{Y}_{g,j,d}\le Q}},\forall g\in \mathcal{G}$$

(19)

High-intensity duties often arise from daily timetable adjustments, so (19) prevents late-added or irregular tasks from being concentrated on a few crews.

For fair allocation of adjustment duties, considering the adjustment duties ${\mathcal{J}}_d^{\Delta }$ are more volatile than baseline duties and can be perceived as less desirable. To reflect fairness not only in total hours but also in exposure to such duties, we introduce additional constraints controlling how adjustment duties are distributed across crews and across days.

Define a binary indicator $Z_{g,d}^{\Delta }$ that is 1 if crew $g$ performs at least one adjustment duty on date $d$, and 0 otherwise. It is linked to the daily assignment variables by:

$$Z_{g,d}^{\Delta }\ge Y_{g,i,d},\forall g\in \mathcal{G},d\in \mathcal{D},j\in {\mathcal{J}}_d^{\Delta }$$

(20)

$$Z_{g,d}^{\Delta }\le {\displaystyle \sum _{j\in {\mathcal{J}}_d^{\Delta }}{Y}_{g,j,d}},\forall g\in \mathcal{G},d\in \mathcal{D}$$

(21)

Thus, $Z_{g,d}^{\Delta }$ records, for each day, whether crew $g$ handles any adjustment duty under the daily-adjusted timetable. The total number of days on which crew $g$ performs adjustment duties over the month is:

$$N_g^{\Delta }={\displaystyle \sum _{d\in D}{Z}_{g,d}^{\Delta }},\forall g\in \mathcal{G}$$

(22)

To avoid concentrating adjustment duties on a small subset of crews, we bound $N_g^{\Delta }$ by operator-defined limits $N_{\mathrm{Min}}^{\Delta }$ and $N_{\mathrm{Max}}^{\Delta }$:

$$N_{\mathrm{Min}}^{\Delta }\le N_g^{\Delta }\le N_{\mathrm{Max}}^{\Delta },\forall g\in \mathcal{G}$$

(23)

These parameters can be calibrated to reflect acceptable ranges for the number of days with added or irregular services per crew. To prevent long streaks of days in which the same crew repeatedly performs adjustment duties, we limit the maximum length of consecutive adjustment-duty days. Let $L^{\Delta }$ be this maximum length. For each crew $g$ and each starting date $d$ such that $d+L^{\Delta }$ lies in $\mathcal{D}$:

$${\displaystyle \sum _{k=0}^{L^{\Delta }}{Z}_{g,d+k}^{\Delta }\le L^{\Delta }},\forall g\in \mathcal{G},d\in \mathcal{D}\wedge d+L^{\Delta }\in \mathcal{D}$$

(24)

In any window of $L^{\Delta }+1$ consecutive days, at least one day must be free of adjustment duties for crew $g$, ensuring that dynamic tasks arising from daily adjustments are rotated among crews.

Combining all constraints, the complete flexible high-speed railway crew scheduling optimization model forms a mixed-integer program that simultaneously enforces fairness, operational feasibility, and safety compliance under dynamically adjusted daily timetables, as shown in Formula (25):

$$\begin{array}{l}\min Z=T_{\mathrm{Max}}-T_{\mathrm{Min}}\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{l}\mathrm{Equations}\left(6\right)-\left(24\right)\\ X_{g,d}\in \left\{0,1\right\},\forall g\in \mathcal{G},d\in \mathcal{D}\\ Y_{g,d}\in \left\{0,1\right\},\forall g\in \mathcal{G},j\in \mathcal{J}\\ T_{\mathrm{Max}},T_{\mathrm{Min}},T_g\ge 0,\forall g\in \mathcal{G}\end{array}\right.\end{array}$$

(25)

3.3. Two-Stage Heuristic Algorithm

To address the low calculation efficiency of the high-speed railway crew scheduling optimization model in long-term, large-scale cases, this study proposes a two-stage heuristic algorithm, which follows a “construct-then-improve” strategy. In the first stage, a feasible scheduling plan is built by a deterministic greedy construction; in the second stage, this plan is refined using Iterative Local Search with Randomization (ILS-R) to improve the fairness objective. This approach significantly enhances both computational efficiency and scalability while maintaining high solution quality, which is crucial for multi-month planning.

The first stage is greedy initial solution construction, which aims to rapidly generate an initial scheduling plan $A_0$ that satisfies all necessary constraints. Given the complexity introduced by daily-adjusted timetables, the algorithm first imposes a priority ordering on tasks. Task allocation priority follows four decreasing principles: (i) task type (prioritizing baseline duties), (ii) start date (earlier dates first), (iii) duration (longer duration first), and (iv) total working hours (higher hours first). This process yields an ordered task list $J^{\prime }$.

The greedy construction proceeds by sequentially iterating through the ordered task list $J^{\prime }$. For each task $j$, the algorithm first identifies a feasible crew subset ${\mathcal{G}}_j^{\mathrm{fes}}\subseteq \mathcal{G}$. A crew $g$ belongs to ${\mathcal{G}}_j^{\mathrm{fes}}$ if assigning task $j$ to $g$ satisfying all operational and fatigue control constraints defined in the predefined optimization model. If ${\mathcal{G}}_j^{\mathrm{fes}}$ is empty, the task $j$ remains unassigned, recoded in set $J^{\mathrm{unassigned}}$ (In practice, uncovered tasks indicates that available crew capacity or operational constraints are too stringent to complete all tasks. However, in the high-speed rail real-world case studied in this paper, each task in the greedy stage had at least one feasible crew group, no tasks finally remained on the unassigned list, and the first stage always generated a fully feasible scheduling scheme. Therefore, the iterative local search (with randomization) in the second stage is purely a fairness enhancement procedure, not a mechanism for restoring feasibility), otherwise the algorithm applies a load-balancing rule and selects the crew with the minimum current accumulated total working hours. This selection criterion directly contributes to the primary objective of minimizing the workload range, promoting inherent fairness in the initial solution $A_0$.

Considering the initial solution often presents a non-optimal working hour range due to the localized nature of greedy decisions. The second stage utilize iterative improvement local search method to systematically refines $A_0$, minimizing the objective function in Formula (1). The neighborhood structure is defined by all possible single-task re-assignment tuple $\left(j,g_{\mathrm{ori}},g_{\mathrm{re}}\right)$, which presents that an assigned task is changed from crew $g_{\mathrm{ori}}$ to crew $g_{\mathrm{re}}$.

To maintain high search efficiency, the algorithm incorporates a critical search pruning strategy, restricting the search space to a feasible and beneficial neighborhood defined as $N_{\left(A\right)}^{\mathrm{fes}}$. A re-assignment operation is considered if two conditions are met: The target crew $g_{\mathrm{re}}$ must remain operationally feasible after accepting task $j$, and the total working hours of the origin crew $g_{\mathrm{ori}}$ must be strictly greater than the total working hours of the target crew $g_{\mathrm{re}}$. This pruning rule ensures that search efforts are concentrated only on re-assignments that reduce the overall disparity in crew workload.

In each iteration $\tau$, the algorithm first introduces randomization by shuffling the checking order of all currently assigned tasks. It then performs a best improvement search, systematically evaluating every feasible re-assignment to find the operation that provides the maximum reduction in the objective function $\Delta =Z\left(A_{\tau }\right)-Z\left(A_0\right)$. If an optimal re-assignment operation with $\Delta >0$ is found, the scheduling plan is permanently updated, and the algorithm proceeds to the next iteration. The search terminates when an iteration completes without any improving re-assignment or when a preset maximum number of iterations is reached. The pseudocode of the proposed two-stage heuristic algorithm is presented in Algorithm 1. Stage 1 builds a feasible plan by assigning duties in a prioritized order; for each duty, it selects a feasible crew that best supports fairness. Stage 2 iteratively improves fairness by attempting reassignment moves for selected duties, accepting a move only if it reduces the workload range while keeping all constraints satisfied.

Algorithm 1 Two-stage heuristic algorithm
Input: Crew set $\mathcal{G}$ , task set $\mathcal{J}$ , constraint set $\mathcal{C}$ , objective function $Z$ , maximum iteration $I$
Output: Optimized crew scheduling plan $A$
Stage-1 Greedy initial solution construction
01.	$\mathrm{Initialize} \mathrm{scheduling} \mathrm{plan} A_0\leftarrow \varnothing$ $\mathrm{and} \mathrm{unassigned} \mathrm{task} \mathrm{set} J^{\mathrm{unassigned}}\leftarrow \varnothing$.
02.	$\mathrm{Task} \mathrm{Ordering}: \mathrm{Sort} \mathrm{J}$ based on the priority $(\mathrm{Type} \to$ $\mathrm{Start} \mathrm{Date} \to$ $\mathrm{Duration} \to$ $\mathrm{Total} \mathrm{Hours}) \mathrm{to} \mathrm{obtain} \mathrm{ordered} \mathrm{list} {\mathrm{J}}^{\prime }$.
03.	$\mathrm{for} \mathrm{task} \mathrm{j}\in {\mathrm{J}}^{\prime }$ do
04.	Determine feasible crew subset         ${\mathrm{G}}^{\mathrm{f}\mathrm{e}\mathrm{s}}\leftarrow \mathrm{g}\in \mathrm{G}\mid \mathrm{Assignment} \left(\mathrm{g},\mathrm{j}\right) \mathrm{is} \mathrm{feasible} \mathrm{under} {\mathrm{A}}^0 \mathrm{according} \mathrm{to} \mathrm{C}]$.
05.	$\mathrm{if} {\mathrm{G}}^{\mathrm{f}\mathrm{e}\mathrm{s}}\ne \mathrm{⌀}$ then
06.	Crew selection:                 $\mathrm{Select} \mathrm{crew} {\mathrm{g}}^{*}\leftarrow \mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}\_\mathrm{g}\in {\mathrm{G}}^{\mathrm{f}\mathrm{e}\mathrm{s}}$.
07.	$\mathrm{Update} {\mathrm{A}}^0\leftarrow {\mathrm{A}}^0\cup \left[\left({\mathrm{g}}^{*},\mathrm{j}\right)\right]$.
08.	else
09.	${\mathrm{J}}^{\mathrm{u}\mathrm{n}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{e}\mathrm{d}}\leftarrow {\mathrm{J}}^{\mathrm{u}\mathrm{n}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{e}\mathrm{d}}\cup \left[\mathrm{j}\right]$.
10.	end if
11.	end for
12.	$\mathrm{A}\leftarrow {\mathrm{A}}^0$$; \mathrm{i}\leftarrow 0$.
Stage-2 Iterative Improvement Local Search (ILS-R)
13.	$\mathrm{while} \mathrm{i}<\mathrm{I}\_\mathrm{m}\mathrm{a}\mathrm{x}$ do:
14.	${\Delta }^{*}\leftarrow 0$$; {\mathrm{m}}^{*}\leftarrow \mathrm{Null} \mathrm{re}-\mathrm{assignment}$.
15.	$\mathrm{Randomly} \mathrm{shuffle} \mathrm{the} \mathrm{checking} \mathrm{order} \mathrm{of} \mathrm{assigned} \mathrm{task} \mathrm{set} \mathrm{J}\_\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{e}\mathrm{d}$.
16.	$\mathrm{for} \mathrm{task} \mathrm{j}\in \mathrm{J}\_\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{e}\mathrm{d}$ do
17.	$\mathrm{Let} \mathrm{g}\_\mathrm{o}\mathrm{r}\mathrm{i}$$\mathrm{be} \mathrm{the} \mathrm{current} \mathrm{assigned} \mathrm{crew} \mathrm{for} \mathrm{j}$.
18.	$\mathrm{for} \mathrm{target} \mathrm{crew} \mathrm{g}\_\mathrm{r}\mathrm{e}\in \mathrm{G}$ do
19.	$\mathrm{Pruning} \mathrm{Check}: \mathrm{if} \mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l} \mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k} \mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r} \mathrm{o}\mathrm{f}{\mathrm{g}}_{\mathrm{o}\mathrm{r}\mathrm{i}}>\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l} \mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k} \mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r} \mathrm{o}\mathrm{f} \mathrm{g}\_\mathrm{r}\mathrm{e}$ and               $\mathrm{g}\_\mathrm{r}\mathrm{e}$$\mathrm{is} \mathrm{feasible} \mathrm{after} \mathrm{receiving} \mathrm{j}$ $(\mathrm{satisfying} \mathrm{C}$) then
20.	$\mathrm{Generate} \mathrm{re}-\mathrm{assignment}$$\mathrm{m}\leftarrow (\mathrm{j},\mathrm{g}\_\mathrm{o}\mathrm{r}\mathrm{i},\mathrm{g}\_\mathrm{r}\mathrm{e})$$, \mathrm{get} \mathrm{tentative} \mathrm{schedule} {\mathrm{A}}^{\prime }$.
21.	$\mathrm{Calculate} \mathrm{improvement} \Delta \leftarrow \mathrm{Z}\left(\mathrm{A}\right)-\mathrm{Z}\left({\mathrm{A}}^{\prime }\right)$.
22.	$\mathrm{if} \Delta >{\Delta }^{*}$ then
23.	${\Delta }^{*}\leftarrow \Delta$$; {\mathrm{m}}^{*}\leftarrow \mathrm{m}$.
24.	end if
25.	end if
26.	end for
27.	end for
28.	$\mathrm{if} {\Delta }^{*}>0$ then
29.	$\mathrm{A}\leftarrow \mathrm{Apply} {\mathrm{m}}^{*} \mathrm{to} \mathrm{A}$.
30.	else
31.	$\mathrm{return} \mathrm{A}$ (Convergence termination).
32.	end if
33.	$\mathrm{i}\leftarrow \mathrm{i}+1$.
34.	end while
35.	$\mathrm{return} \mathrm{A}$ (Maximum iteration termination).

We also experimented with relaxed versions of this pruning rule by allowing task transfers between crews whose current workloads differ only by a small amount, instead of requiring a strictly larger workload for the origin crew. In the real-world instances studied, however, these relaxed variants significantly increased the neighborhood size and runtime while providing little or no additional improvement in the fairness objective and sometimes producing low-impact “back-and-forth” reallocations. Based on these observations, we retain the stricter pruning condition as the default setting, as it offers a better balance between search efficiency and fairness improvement.

The overall time complexity is approximated by the iterative neighborhood search in the second stage, which is $O(I\cdot \left|\mathcal{J}\right|\cdot \left|\mathcal{G}\right|\cdot (\left|\mathcal{D}\right|+\left|\mathcal{G}\right|))$, where $I$ represents the maximum iterations, $\left|\mathcal{J}\right|$ represents the number of tasks, $\left|\mathcal{G}\right|$ represents the number of crews, and $\left|\mathcal{D}\right|$ represents the number of days in the planning period. The complexity analysis indicates that this two-stage approach is a polynomial time heuristic.

In practice, the size of the candidate neighborhood in the local search stage is not controlled by an explicit tuning parameter, but is determined jointly by the feasibility constraints and the pruning rule. In theory, if one were to consider, for each assigned task, all other crews as potential targets, the number of candidate reassignments in a single iteration would scale with the product of the number of tasks and the number of crews, resulting in a very large search space. However, because the requirement that only moves from a higher-workload crew to a lower-workload crew are considered, each task typically has only a small number of feasible target crews in any given iteration. Empirical statistics on the real-world instances indicate that the effective neighborhood corresponds to only a small subset of all theoretically possible crew-task reassignments. Our experiments show that under this neighborhood definition, the algorithm achieves a good balance between runtime and fairness (as measured by the range and distribution of working hours). When we manually restricted the neighborhood further, iterations became slightly faster but the fairness results degraded.

4. Result Analysis

4.1. Case Study and Data Preparation

This study utilizes empirical operational data to assess the effectiveness of the proposed flexible crew scheduling optimization framework under daily-adjusted timetables.

The case study involves the monthly crew scheduling plan for August 2025 (31 days), aligned with the busy summer transportation season. The dataset comprises three components: (i) duty demand data covering both baseline service tasks and day-specific additional tasks, (ii) crew information, and (iii) high-speed railway operational regulations that ensure safety and compliance. The data were authorized by the railway operator and anonymized before analysis. Examples of baseline and temporary tasks for high-speed railway crew teams are presented in Appendix B 

Table A2. Example of baseline tasks for different high-speed railway crew teams.

Date	Crew A	Crew B	Crew C	Crew D
1 August	D3055 at Nanjing South (Stay overnight in Shanghai)	D3055 at Nanjing South (Stay overnight in Shanghai)	Rest	Rest
2 August	D3056/7 (Stay overnight in Chengdu East)	D3056/7 (Stay overnight in Chengdu East)	Rest	Rest
3 August	D954 to Nanjing South	D954 to Nanjing South	D3055 at Nanjing South (Stay overnight in Shanghai)	D3055 at Nanjing South (Stay overnight in Shanghai)
4 August	Rest	Rest	D3056/7 (Stay overnight in Chengdu East)	D3056/7 (Stay overnight in Chengdu East)
5 August	Rest	Rest	D954 to Nanjing South	D954 to Nanjing South

and

Table A3. Example of day-specific ad hoc duties for different high-speed railway crew teams.

Date	Crew A	Crew B	Crew C	Crew D
1 August	D2855	D5646	G7107	G7629 + G7796 + G2312 + G7179
2 August	D2855	G7103 + G8287	G9421	D2216 + D2218
3 August	D2855	D2167	G9421	G1605

Note: The “+” between train numbers indicates an overnight duty that continues to the next day.

.

Under the traditional high-speed railway crew management system, crew teams were permanently assigned to specific passenger routes. However, the growing transport demand and the adoption of daily-adjusted timetables have led to frequent adjustments in train operation plans, resulting in numerous temporary, unscheduled crew assignments such as those required for coupled train or additional passenger services. Since the original fixed model lacked inherent flexibility, these supplementary tasks were typically managed through extensive manual selection and temporary assignment from the existing crew teams. The description of the current high-speed railway crew management pattern for additional tasks in China is demonstrated in Appendix B 

Table A4. Current high-speed railway crew management pattern for additional tasks in China.

Pattern	Description	Train	Crew	Duration
Standby duty	Crews are pre-assigned to respond to immediate operational demand.	Standby train prepared	Pre-designated standby crew group	1 day
Extended duty	The same crew continues working beyond the baseline duty	The regular service train is already in operation	Continuation by the originally assigned crew	Less than 1 day
Rest-day duty	Crews scheduled for rest are temporarily recalled	Temporarily added or coupled train	Selected from the resting crew pool	1 day or more

Note: Table A4 summarizes three typical manual adjustment patterns currently used in practice when dispatchers modify a fixed crew plan in response to extra or irregular services. These patterns are included to illustrate existing operational practice and its limitations. In the proposed optimization model, we do not represent these patterns as separate decision variables; instead, all such services are treated as tasks within the daily-adjusted duty set and are assigned to crews through a unified assignment and constraint structure, so that fairness is optimized over the complete monthly horizon rather than through local, ad hoc adjustments.

.

For the case study, the minimum and maximum monthly working hours are taken directly from the China Railway Shanghai Group’s crew scheduling regulations and therefore reflect existing policy and regulatory limits, the values of which are 50 h and 240 h. This rigid scheduling structure and reliance on localized adjustment strategies resulted in significant drawbacks, primarily a severe imbalance in the distribution of monthly total work hours across crew groups. Building on the proposed flexible crew scheduling optimization model, we instead generate integrated plans that explicitly target workload fairness and can be solved efficiently by the two-stage heuristic.

4.2. Performance Comparison

To identify the most effective objective function for measuring fairness in the high-speed railway crew scheduling model, this study evaluated three quantification forms for assessing the balance of crew workload, including minimizing variance, minimizing the mean absolute deviation (the average absolute differences between each crew’s total working hours and the average value), and minimizing the range between the maximum and minimum total workload in Formula (1). A one-month instance with 1934 baseline tasks, 73 extra tasks, and 122 crew teams is used as a test case, and all three objectives are solved with the same Branch-and-Cut algorithm for comparability. The comparison results are presented in

Table 3. Optimization results for a one-month scheduling problem under different objectives.

Objective	Maximum Work Hours (h)	Minimum Work Hours (h)	Standard Deviation (h)	Range (h)	Solution Time (s)
Origin plan	219.17	127.18	17.63	91.99	/
Minimize variance	227.94	105.31	22.37	122.63	3761
Minimize mean absolute deviation	184.33	176.01	2.36	8.32	3753
Minimize the range	182.44	178.63	0.65	3.81	1826

.

Table 3 shows that all three objectives substantially reduce workload imbalance compared with the original fixed-assignment plan, but the range-based objective achieves the smallest maximum-minimum difference while keeping computation time moderate. Variance and MAD yield slightly smoother distributions but require longer runtimes. Given this trade-off, we adopt the range-based objective in the remaining experiments. From a computational perspective, the range objective function achieved the optimal solution in 1826 s, significantly outperforming both the variance and the minimizing mean absolute deviation solutions. Therefore, the range objective function is adopted in the subsequent experiments.

Although the exact Branch-and-Cut method is capable of achieving the optimal solution for the 1-month scheduling problem, the computational effort required is substantial. This inefficiency restricts the applicability of the exact method for long-term operational planning, which frequently spans several months. This study developed a two-stage heuristic algorithm to improve computational efficiency and scalability while maintaining high solution quality. The optimization results comparison among algorithms in Appendix C indicates that the proposed Two-stage heuristic algorithm achieves the smallest range among all comparison algorithms. Meanwhile, the experiment in

Table 4. Optimization results of the two-stage heuristic algorithm under different scheduling periods.

Scheduling Period	Maximum Work Hours (h)	Minimum Work Hours (h)	Standard Deviation (h)	Range (h)	Solution Time (s)
1 month	186.45	174.51	3.42	11.94	16.19
2 months	354.82	349.44	1.36	5.38	76.57
3 months	538.29	527.87	2.89	10.42	69.43
4 months	711.43	700.84	3.13	10.59	104.93
5 months	885.62	881.90	1.19	3.72	249.98

Note: The range does not exhibit a linear variation with the increase in the scheduling period, primarily due to the uneven distribution of task volumes and work durations across different months. Range results for all scaled examples are 12 h or less, performing as expected.

validates the robustness and scalability of the proposed heuristic framework when applied to instances ranging from 1 month to 5 months of crew scheduling. All experiments utilize the objective function focused on minimizing the range of total working hours.

While the range is reported as a numeric fairness indicator, it is also directly interpretable for planners and crew managers. For example, the 1-month instance yields a range of 11.94 h, meaning that the two most extreme crews differ by at most about half to two working shifts (depending on local duty-hour definitions) within the month. In contrast, the 5-month instance yields a range of 3.72 h, which indicates near-equal cumulative workload over a seasonal horizon, and can be perceived as a practically negligible disparity in long-horizon planning. From a governance perspective, these values can be used as auditable fairness thresholds (e.g., “keep monthly range within a workday”) and as a communication tool to justify that daily-adjusted duties are being rotated rather than repeatedly assigned to the same crews.

The experimental results demonstrate a significant computational advantage of the two-stage heuristic algorithm over the exact method. For the 1-month case, the heuristic converged in only 16.19 s, representing a 99.1% reduction in solution time compared to the 1826 s required by the exact Branch-and-Cut method. The resulting solution achieved a workload range of 11.94 h. Although this solution has not reached the optimal range, the solution quality still represents an 87.0% improvement over the unoptimized original plan (91.99 h). This outcome validates that the proposed heuristic achieves a balance between solution quality and computational efficiency, providing a rapid approach for generating high-quality approximate optimal schedules.

Figure 2a–e confirm that the differences in work hour distribution in all test cases were effectively controlled, effectively achieving the goal of fairness in crew scheduling. In addition to the quantified maximum-minimum range of work hours in the numerical tables, Figure 2a–e also shows the distribution characteristics of the optimized scheduling scheme. Throughout all planning periods, the monthly work hour histograms for crews are approximately symmetrical and exhibit a unimodal distribution, without significant skewness or heavy tails. There are no obvious cases where crew workloads are concentrated at extremely low or high levels. As the planning period lengthens, the algorithm also demonstrates excellent scalability and robust solution quality. For a 5-month scheduling case, the heuristic algorithm found a solution with a workload range of 3.72 h in 249.98 s, confirming that fairness is maintained even in large-scale scenarios. Figure 2f shows the rapid and stable convergence of the two-phase procedure.

4.3. Sensitivity Analysis

To assess the robustness of the fairness-oriented scheduling model to changes in key operating parameters and actual demand, we conducted sensitivity analyses on two regulatory parameters, supplemented by comparison case studies. The quantifiable metric was the standard deviation of total monthly work hours for each crew. For each parameter setting and planning period (1–5 months), we reported the percentage change in this standard deviation relative to the baseline configuration. To enhance statistical rigor, each experiment was repeated multiple times under the same settings, and the mean percentage change for each experiment and the empirical 95th percentile range (±%) are reported.

The first experiment varied the maximum number of working days per month, which directly impacts resource availability and scheduling flexibility. As shown in

Table 5. Sensitivity analysis of the maximum number of working days on the standard deviation of working hours (±%).

Change in Maximum Number of Working Days in a Month (%)	High-Speed Railway Crew Scheduling Period
	1 Month	2 Months	3 Months	4 Months	5 Months
+5%	−5.05% (±0.38%)	−4.05% (±0.30%)	−3.84% (±0.28%)	−3.57% (±0.25%)	−3.52% (±0.27%)
+10%	+3.82% (±0.27%)	+3.30% (±0.23%)	+2.85% (±0.20%)	+2.74% (±0.21%)	+2.50% (±0.18%)
+15%	+7.24% (±0.54%)	+6.29% (±0.47%)	+5.58% (±0.41%)	+5.29% (±0.40%)	+5.20% (±0.37%)
+20%	+12.47% (±0.91%)	+10.26% (±0.77%)	+9.82% (±0.72%)	+9.34% (±0.70%)	+8.23% (±0.61%)

, a moderate relaxation of this constraint by 5% reduced the standard deviation of work hours across all planning periods, with an average reduction of approximately 3.5% to 5.1%, and a very narrow 95% confidence interval (e.g., −5.05% ± 0.38% for a one-month planning period). Fairness statistically steadily improves when the allowed number of crew working days is slightly increased, as the additional flexibility allows the optimizer to smooth workload imbalances. However, when the maximum number of working days is further increased by 10%, 15%, and 20%, the impact becomes positive across all planning periods (up to +12.47% ± 0.91% in a one-month planning period). In these cases, 95% of the ranges are consistently strictly above zero, suggesting that excessively relaxing the monthly working day limit consistently reduces operational fairness, possibly because the model can concentrate more responsibilities on a subset of crew members without a constrained upper limit.

The second experiments varied the minimum rest time between two tasks, which determines the tightness of fatigue-related intervals between consecutive tasks.

Table 6. Sensitivity analysis of minimum rest time on the standard deviation of working hours.

Change in Minimum Rest Time Between Two Tasks (%)	High-Speed Railway Crew Scheduling Period
	1 Month	2 Months	3 Months	4 Months	5 Months
+5%	−3.94% (±0.29%)	−3.52% (±0.26%)	−3.29% (±0.24%)	−2.95% (±0.21%)	−2.69% (±0.20%)
+10%	+5.50% (±0.39%)	+5.29% (±0.37%)	+4.51% (±0.32%)	+4.18% (±0.31%)	+3.52% (±0.26%)
+15%	+9.08% (±0.68%)	+8.79% (±0.65%)	+7.71% (±0.56%)	+7.04% (±0.53%)	+6.39% (±0.46%)
+20%	+12.41% (±0.91%)	+11.39% (±0.85%)	+10.76% (±0.79%)	+9.86% (±0.74%)	+8.02% (±0.59%)

shows that a slight increase of 5% in the minimum rest time reduced the standard deviation of working hours over the planning period by approximately 2.7% to 3.9%, with all 95% confidence intervals below zero (e.g., −3.94% ± 0.29% for one month). Slightly stricter rest requirements help regulate work patterns and indirectly promote a more even distribution of working hours. Conversely, when the minimum rest time increased by 10%, 15%, or 20%, the standard deviation increased significantly, with an average increase of +12.41% ± 0.91% over a one-month period, and similar increases over longer periods. These results indicate that overly strict rest requirements significantly reduce the availability of effective personnel, making it difficult for optimizers to allocate tasks evenly, thus reducing fairness in practice.

In addition to the one-parameter analyses, we also performed exploratory experiments on the interaction between the maximum number of working days in a month and the minimum rest time between two tasks. In these tests, both parameters were scaled simultaneously by the same factor (for example, increasing the monthly working-day limit and the minimum rest requirement by 5%, 10%, and so on). The resulting changes in the range of monthly working hours across crews were very modest and did not exhibit a strong monotonic pattern. When both parameters are relaxed or tightened in a proportional way, the longer potential working stretches are largely offset by longer rest periods, so that the overall structure of feasible duty patterns and the fairness-driven allocation across crews remains broadly similar. Consequently, the fairness outcome is affected mainly when one parameter is varied while the other is kept fixed, as reported in the detailed single-parameter sensitivity tables.

Beyond parameter analysis, we also examined the robustness of the model under actual operational changes, comparing the summer peak season and normal months along the same high-speed rail corridor. In normal months, the case study included 1827 baseline train services and 55 adjusted services, totaling 1882 services. During the summer peak season, the corresponding number of services increased to 1999 baseline train services and 84 adjusted services, totaling 2083 services, representing a 10.68% increase. Despite this significant surge in demand, the range of monthly working hours for train crews generated by our method increased by only 0.53%, indicating that the fairness of the final schedule arrangement remained essentially unchanged. This comparison further demonstrates the robustness of the proposed fairness-oriented model and heuristic algorithm.

In practice, parameters that govern adjustment-duty exposure—especially the maximum consecutive adjustment-duty days, act as explicit policy levers. Constraint (24) enforces that within any day window, at least one day must be free of adjustment duties for each crew, thereby preventing long streaks of “less desirable” ad hoc work. Reducing this value strengthens perceived fairness and reduces the risk of repeated burden on the same crews, but it can also shrink the feasible assignment space and may slightly worsen the achievable total-hour range (because the optimizer has fewer freedom degrees to balance hours while respecting rest and continuity). Conversely, increasing this value expands flexibility and may help reduce the hour-range in some instances, but it can increase perceived inequity by allowing longer consecutive exposure to adjustment duties. Therefore, we recommend that operators calibrate this parameter (and related bounds on adjustment-duty days) based on labor agreements and acceptability considerations, and then use the model as a scenario-analysis engine to quantify the resulting fairness–flexibility trade-off.

5. Discussion

5.1. Decision-Support System for Crew Assignment Application

Daily timetable adjustments transform crew scheduling from responding to minor disruptions to predicting daily structural changes. In this context, treating crew members as a shared resource and optimizing monthly plans (which simultaneously allocate baseline tasks and tasks for specific dates) is not only a modeling choice but also an organizational one: it aligns plans with changes in daily service volume and translates fairness (balanced monthly hours) into a stable lever for execution and labor relations. To align with China’s “one plan per day” operational model, our solution directly optimizes one-off task allocation under operational rules, avoiding the traditional “CSP and CRP” framework.

From an implementation perspective, integrating a crew scheduling optimizer into a railway production system typically requires meeting several conditions: establishing a stable data interface with the daily timetable adjustments and the crew database; clearly coding regulatory and labor rules; ensuring the solution timeframe conforms to existing planning windows; and providing auditable and easily understandable outputs for planners. The proposed framework is designed based on these requirements. The model operates directly on the daily timetable adjustments and uses real operational data from a case study of a high-speed railway corridor to ensure consistency with actual task structures and regulatory constraints. As demonstrated by computational and sensitivity analyses, this two-stage heuristic algorithm can generate multi-monthly schedules within minutes and maintain fairness under conditions of significant workload variations, making it compatible with both monthly and seasonal scheduling cycles. While current validation was conducted offline rather than in a real-time scheduling environment, these characteristics indicate that the method meets the key technical and organizational prerequisites for deployment in a real-world high-speed rail crew scheduling system.

As shown in Figure 3, a prototype decision support system implements the method and integrates the model with scheduling practices. The system accepts daily adjusted timetables and crew data, normalizes baseline and ad hoc tasks into a unified task pattern, and provides two engines: an exact mixed-integer linear programming (MILP) engine for single-month time spans and a two-stage heuristic engine for multi-month time spans. This allows rail operators to select the optimal solution that meets audit requirements or quickly generate scenarios without altering workflows. The interface integrates multi-day timetables, making conflicts, uncovered tasks, and rest time violations readily apparent; planners can lock key crew groups or tasks, re-optimize while retaining these locks, and export authorized crew schedules downstream. The system’s diagnostic functions focus on management responsibility rather than the solver’s internal mechanisms (coverage, hourly dispersion metrics, compliance checks), which is beneficial for its application in the passenger transport sector, where fairness and rule compliance are equally important objectives.

Two implementation schemes deserve discussion because they impact governance rather than just computation. First is the choice of objective: assessing fairness with a maximum-minimum monthly work range, which is visible to managers and crew, easily auditable, and empirically more manageable and stable than a variance objective when embedded in long-term program execution. This makes a range objective a practical default option for monthly planning under daily adjusted schedules, while still allowing variance as a secondary diagnostic metric. Second is the choice of time range: precise models correspond to monthly approval cycles and compliance audits, while heuristics are suitable for seasonal time ranges and routine “hypothesis” experiments (e.g., adding extra work or coupling on specific dates) without relying on a commercial solver. These governance-oriented choices help integrate the tool into existing planning pattern rather than forcing changes to the process.

Furthermore, the limitations of large-scale application should be considered. The current deployment treats crew members as interchangeable within the same qualification level, without encoding fine-grained preferences or skill heterogeneity; a natural extension would be a preference-perceived allocation with qualification screening and soft penalties. Similarly, our focus is on monthly plans adjusted daily; the best approach to handling operational day emergencies (such as delays and absenteeism) is to overlay a lightweight re-scheduler on top of the monthly plan. In the medium term, tighter cross-layer integration, co-optimizing crew, vehicle utilization, and timetable templates, can translate fairness improvements into network-level robustness and fully leverage existing cross-disciplinary management platforms in Chinese railway practice. Our contribution lies not only in an optimization model but also in a practical planning capability: a fairness-first framework tailored for daily operational adjustments and integrated into a system that allows for human oversight, generates auditable diagnostic results, and conforms to monthly and seasonal planning cycles.

5.2. Operational Acceptability

While the proposed framework significantly improves the fairness of monthly working hours in a quantitative sense, its true value depends on whether the final schedule aligns with the actual daily operations of the railway. In reality, crew scheduling is not merely a mathematical problem, but a management process that must respect existing work habits, internal rules, and the experience of those responsible for scheduling and execution. For schedulers and dispatchers, any new tool must be integrated into their existing work methods, generating easily understandable and interpretable schedules, and allowing for manual adjustments when necessary. For crew members, their perception of fairness depends not only on total monthly working hours but also on rest patterns, the ratio of early to late tasks, familiarity with the line, and commuting time not included in working hours.

The current model assumes that crew members are homogeneous within a basic qualification level and focuses on balancing total monthly working hours. It does not yet account for subtle differences in individual preferences, skills, and certifications, as well as seniority rules. In practice, these factors often play a crucial role in the acceptability of a schedule. Some crew members prefer to work on routes or train types they are familiar with; many values reasonable rest day arrangements; and most want to avoid frequent long-distance transfers or repeated empty trips to remote areas. Senior crew members may expect certain types of work or more stable work patterns, while unions or employee representatives may need to review changes to scheduling policies. Ignoring these aspects means that even if a schedule appears fair and reasonable numerically, those actually implementing it may still find it unsatisfactory or unfair.

Therefore, qualitative validation with frontline staff is a crucial next step. In future work, the system should be tested and discussed with train timetable designers, crew scheduling personnel, and depot managers through workshops or pilot studies. These activities can be used to examine whether the generated schedule is easy to implement, conforms to basic planning habits and rules, and whether the tool assists planners rather than replacing their judgment. Simultaneously, meetings with crew representatives or focus group discussions can gather feedback on how the new schedule affects fatigue levels, satisfaction, and individual work-rest stability. This type of feedback helps identify which parts of the model work well in practice and which parts need adjustment.

Based on these improvements, the framework can be further refined to include richer descriptions of operational acceptability. Possible extensions include: adding eligibility and preference matrices for each crew; imposing soft constraints on rest day groupings; limiting the degree of deviation between the new schedule and the existing model; and providing functionality to explain why specific tasks are assigned to specific crews. Therefore, the fairness-oriented, daily-adjusted scheduling model developed in this paper should be considered a starting point and needs further refinement in collaboration with scheduling experts and crew representatives to ensure that future deployments guarantee both numerical fairness and crew well-being and stability.

5.3. Cost and Operational Efficiency

While this study focuses on achieving fairness in monthly work hours under daily adjusted timetables, actual crew scheduling must also consider cost and operational efficiency. In high-speed rail operations, crew schedules are not built from blank templates but follow long-established rules, such as keeping crew members geographically close to their bases to avoid unnecessary non-productive trips (empty runs). During the timetable and design phases, each schedule is assigned to one or a few stations based on the nearest station principle to ensure that the origin and destination of most crews are near the same crew base. Subsequently, when planners optimize these schedules manually or using large-scale schedule building tools, they strive to maintain consistency in origin and destination locations and reasonable daily continuity, intervening only when work hour standards or regulatory restrictions are violated.

In this operational context, our proposed “fairness-first” framework does not redesign the tasks but rather allocates existing tasks to crew teams. Crew assignments are typically performed in units, which are usually responsible for completing or near-completing vehicle scheduling diagrams. Significant adjustments to the scheduling chart structure are only made when the schedule is clearly unreasonable in terms of work hours. For example, a six-day scheduling chart would force crew members to work six consecutive days; therefore, it can be split into two three-day crew scheduling charts and assigned to different bases to conform to the longest consecutive work days and rest rules. Similarly, short scheduling charts with fewer total work hours can be merged into longer patterns, such as three-day cycles, while maintaining consistency in start and end points. In this case, our optimization primarily focuses on reallocating these pre-designed, operationally coherent tasks to crew members, thereby eliminating the remaining imbalance between scheduling charts with slightly higher and lower work hours, rather than creating many scattered small work units.

This design has a direct impact on cost and operational stability. Since the tasks already reflect geographical concentration, fairness optimization does not introduce new empty tasks or long-haul crew transfers; it reallocates all or almost all tasks to crew members at the relevant bases. Similarly, crew rotations also follow the principle of allocating as complete a task chart as possible, and are only split when the base work hours are clearly not up to standard. In other words, this model does not improve fairness by breaking tasks into many smaller segments or frequently changing crew members during task map execution; instead, it strives to balance workload while maintaining a basic task structure and similar to current practices.

Meanwhile, the current implementation does not incorporate explicit cost or efficiency metrics into the objective function or as separate constraints. For example, we do not track crew tasks, measure the distance of task locations from the base, or quantify the extent to which tasks can be broken down into shorter segments. These factors are indirectly controlled by how the task map is constructed and existing planning rules, but our experiments did not evaluate or optimize them. A beneficial extension of this work is to define and monitor metrics beyond fairness (e.g., tracking crew handovers, the proportion of tasks starting or ending outside the base, or non-productive travel volume), and then use them as secondary objectives or limiting constraints. This would facilitate a more systematic analysis of the trade-offs between fairness, cost, and operational stability, and help adjust flexible crew scheduling frameworks according to the specific efficiency priorities of different railway operators.

5.4. Improvement to Crew Homogeneity Assumption

Another limitation of the current framework lies in its simplistic assumption that all crew members in the planning pool are homogeneous, aside from basic availability and regulatory constraints. However, in actual high-speed rail operations, crew members’ qualifications, service quality requirements, and technical certifications vary considerably; some crew members may even be limited to operating specific types of trains, specific routes, or express train services. From a crew management perspective, while the proposed method has reduced manual workload and improved the fairness of monthly working hours, it has not yet reflected these more nuanced differences. From the crew members’ own perspective, factors such as the concentrated scheduling of rest days, reducing unpaid empty runs, and preference for familiar routes also significantly influence their perceptions of fairness and timetable acceptability factors that are only indirectly reflected in the current model.

Therefore, a natural extension would be to incorporate a multi-skill or qualification matrix into the model, which could encode each crew member’s qualifications and suitability across different routes, train types, and service levels. In practice, this can be achieved by assigning skill and service rating labels to crews (e.g., the minimum rating threshold for operating important lines like the Beijing-Shanghai High-Speed Railway) and introducing binary parameters to indicate whether a particular crew holds a qualification certificate for a specific train type or is familiar with a specific line. Hard qualification constraints ensure that only qualified crew members are assigned to specific tasks, while soft preference mechanisms and penalty clauses can be added to prioritize crews with high line familiarity, low empty mileage, and those who prioritize seniority or service quality. Embedding this multi-skill/qualification layer into a daily adjustment and fairness-first framework and calibrating it based on operator feedback are important directions for future work, allowing the model to more closely reflect the complexities of real-world crew management.

5.5. Stress-Testing on Extreme Scenarios

This study’s numerical experiments focus on typical and moderate-stress operating conditions, including comparisons of operations during normal months and peak summer months on the same high-speed rail corridor. However, in actual operation, crew scheduling must also cope with more extreme situations, such as surges in demand, numerous temporary trains, simultaneous absences of multiple crew members, and severe timetable asymmetry during holidays. In these cases, trains and services are often unevenly distributed across different directions and dates, potentially leading to increased empty runs, more cross-station deployments, and more frequent use of substitute crew members at intermediate stations. Therefore, understanding the performance of the proposed heuristic algorithm in these environments is an important task, beyond the scope of this experiment.

In current practice, when train timetables are highly asymmetrical during peak hours, railway operators still strive to adhere to the basic principle of balanced working hours, while also placing great emphasis on limiting non-local services to avoid unnecessary relocations. The initial allocation of the vehicle dispatching map fully considers geographical proximity; during manual adjustments, dispatchers tend to prioritize assigning tasks to crew members stationed at or near relevant terminating stations. Non-local crew members or cross-station deployments will only be used as alternatives when local capacity is significantly insufficient. These practical rules indicate that any attempt to extend this model to extreme scenarios should incorporate fairness considerations and include explicit controls on uncounted trip segments, non-local deployments, and redeployment.

Future stress tests of this heuristic algorithm will require constructing a series of scenarios to reflect the most challenging situations encountered in practice. These include, for example, composite or historical timetables with a large number of extra trains concentrated during peak hours, situations where multiple crews cannot be on duty within a short period, and holiday timetables resulting in empty runs and overnight stops due to significant directional differences. For each scenario type, not only can the final fairness metrics be evaluated, but also operational metrics such as the number of non-local task assignments, empty runs, and the extent to which the heuristic algorithm still adheres to planning principles such as base concentration and reasonable rest patterns.

A promising extension is to adjust the current model to explicitly prioritize local crews during peak and asymmetric periods, for example, by adding priority rules or penalties for off-site missions and long-distance ferry segments, while still maintaining range-based fairness as the primary monthly driver. These parameters can then be fine-tuned in collaboration with operators through stress testing under extreme scenarios, verifying the stability, interpretability, and efficiency of the heuristic algorithm under significant system stress. In this sense, the current work should be considered a first step, and stress testing the system under extreme demand, heavy temporary workloads, multiple crew absences, and peak holiday periods is an important direction for future research.

6. Conclusions

This research aims to address a key challenge in daily timetable adjustments for high-speed railway crew scheduling: the significant workload imbalance resulting from traditional fixed-route allocation models. We propose a novel ensemble optimization framework that prioritizes fairness by minimizing the total range of crew working hours. Based on a series of daily timetable operations, this framework co-allocates baseline and adjustment tasks to shared crew resources while enforcing safety, work hours, and rest regulations on a monthly scale. For longer time spans, the heuristic solves a five-month instance with a workload range of 3.72 h in less than 250 s, demonstrating its applicability to large-scale planning.

From an algorithmic perspective, the contribution of this research lies in applying a two-stage heuristic combining greedy construction and local search to a fairness-oriented crew-sharing scenario with daily adjustments, rather than proposing a novel metaheuristic. This heuristic leverages the problem structure, directly optimizing for the reduction in the monthly working hour range. Empirical results demonstrate that the design achieves near-equilibrium resource allocation in real-world scenarios while keeping runtime within acceptable limits. The algorithm’s runtime exhibits a polynomial relationship with crew size, task quantity, and number of working days, and its runtime moderates with increasing problem size, supporting its potential for daily application in planning departments.

The transition from a fixed-route system to this flexible resource-sharing model has significant management implications. It provides a science-based decision support tool that automates scheduling processes, reduces manual workload, and enhances the robustness and adaptability of crew management under daily adjusted timetables. By introducing fairness and flexibility into the planning process, this approach can improve crew morale and stability, reduce management costs associated with ad hoc adjustments, and ultimately contribute to providing more stable and reliable passenger service.

Despite these improvements, the current model still has two main limitations. First, the model does not incorporate details such as individual crew preferences or skill differences, which could affect the acceptability of scheduling schemes in actual operation. Second, the framework is designed for planned daily adjustments, rather than real-time responses to unforeseen events. Therefore, future work will extend the model to include preference- and skill-based allocations and develop a lightweight rescheduling module to address unforeseen events during the operating day. Furthermore, while we adopted range minimization as the single primary fairness objective for reasons of modeling transparency and computational tractability, in practice, considering multi-objective extensions that can simultaneously balance range, discrete metrics based on variance or mean absolute deviation (MAD), and cost-oriented criteria is reasonable. Designing and calibrating such multi-objective models and solving them in collaboration with operators using Pareto-based methods is an important direction for future research. Finally, this case study is based on a high-speed rail network with specific labor rules and daily adjustment patterns. Applying this framework to other scenarios, such as urban rail transit, mixed passenger and freight systems, or international high-speed rail corridors, may require adjustments to fairness indicators, regulatory parameters, and tasks restrictions. These aspects constitute a broader research agenda for promoting fairness-oriented daily crew scheduling methods.