Introduction

Crew assignment is a classical problem in Operations Research (OR), with broad applicability across industries such as aviation [1], public transportation [2], last-mile logistics [3], private security [4], home healthcare [5], and emergency medical services [6]. Integrating worker risk conditions into scheduling and workload balancing represents a critical intersection between OR and Human Factors and Ergonomics (HFE). Both disciplines aim to optimize performance while safeguarding worker well-being. This study focuses on cash-in-transit (CIT) operations, characterized by complex logistical and ergonomic constraints. Although the methodology is applied to CIT, it is generalizable to other industries where physical and cognitive demands are significant.

CIT services are executed by crews, and personnel assignment must comply with strict security and operational requirements, making it a highly constrained decision-making problem [7]. The central challenge lies in achieving equitable workload distribution without compromising efficiency [8]. CIT personnel perform diverse tasks, including money counting, carrying bags, and handling location-specific operations [9]. These activities involve substantial physical exertion, particularly affecting the spine and upper limbs [10] and impose cognitive demands due to long driving hours and heightened responsibility [11]. Additionally, crews face exposure to violence, robbery threats, and high-stress scenarios that contribute to emotional fatigue [12][13].

To address these multidimensional demands, structured and fair assignment processes are required. This includes rotation, balanced shifts, and equitable workload distribution [14]. However, deriving optimal solutions that minimize operational costs while incorporating worker preferences and well-being remains challenging. Integrated systems that combine mathematical modeling, optimization algorithms, and ergonomic quantification methods are therefore essential for supporting decision-making in CIT environments.

The literature on personnel scheduling in transport systems is extensive. Kasirzadeh et al. [15] provides a comprehensive overview of airline crew scheduling, a domain studied for over six decades. Road-based freight transport presents distinct challenges [16][17], such as variable arrival times and frequent rest breaks, which differ from the strictly timed operations of air or rail transport[18]. CIT operations further increase complexity due to their specialized and high-risk nature.

Previous studies have addressed components such as driver routing, crew scheduling, and vehicle assignment. For example, Goel and Vidal [11] applied a hybrid genetic algorithm for efficient schedule generation, while Wen et al. [19]used a variable neighborhood metaheuristic focused on cost minimization. Archetti and Savelsbergh [20]investigated how labor constraints affect operational performance. Integrated approaches have also been proposed. Drexl et al. [21] addressed simultaneous vehicle and crew scheduling, and Goel and Irnich [22] introduced the Vehicle Routing and Truck Driver Scheduling Problem (VRTDSP), combining time windows and working hours constraints. Ciancio et al.[23] developed a framework linking the Multiple Depot Vehicle Scheduling Problem (MDVSP) with the Crew Scheduling Problem (CSP). In the CIT context, most studies focus on vehicle routing to minimize risk rather than personnel scheduling. Talarico et al. [10] introduced the Risk-Constrained Cash-In-Transit Vehicle Routing Problem (RCTVRP), incorporating risk thresholds in route selection. More recently, Tikani et al. [24] proposed models considering stochastic, time-varying traffic conditions and risk exposure. Bowden and Ragsdale [14] extended the Truck Driver Scheduling Problem by integrating the Three-Process Model of Alertness (TPMA) to monitor fatigue via circadian rhythms (See Table 1).

Despite these advances, most approaches emphasize administrative constraints (e.g., rest periods or shift durations) rather than directly quantifying human fatigue. Saavedra-Robinson and Quintana [25] evaluated drivers’ physiological performance, but their scope was limited to driving tasks. Simulation-based approaches often fail to capture real physiological responses. This study contributes by integrating direct fatigue quantification into crew scheduling for CIT operations, supporting equitable assignments that balance logistical efficiency with employee welfare.

Materials and Methods

The methodological framework integrates ergonomic assessment with mathematical optimization. Two complementary components are outlined (see Figure 1):

• Ergonomics-based fatigue quantification, using physiological measurements to estimate workload demands.

• Optimization-based crew assignment, employing mathematical programming to distribute workloads equitably while satisfying operational constraints.

Fatigue Estimation via Physiological Measurement

Heart rate monitoring was selected as the primary indicator of physical effort due to its established linear correlation with energy expenditure. Compared to oxygen consumption, heart rate monitoring is non-intrusive, cost-effective, and feasible in operational environments.

Crew members wore a chest-mounted heart rate monitor beneath their safety vests during daily tasks. Data were collected from the moment crews entered the vehicle until the end of their shift, using Bluetooth-enabled devices and mobile applications. Physiological workload for each route was quantified via the Penosity Coefficient, derived from effort-related variables (Table 2). This coefficient was interpreted using the Frimat Criteria [26], enabling the classification of activities by difficulty level.

Model Approach

The cumulative coefficients yield a final value proportional to the required effort, which is then evaluated using the Frimat Criteria. The formulation addresses the crew assignment problem in the CIT logistics context (Figure 2).

Let 𝐶𝐿 be the set of crew leaders; 𝐶𝐺 the set of cash guard; 𝑅 the set of routes to be scheduled; 𝑉 the set of homogeneous vehicles, and 𝜏 the set of periods (for example, weeks); 𝛼𝑖𝑘, 𝑖∈𝐶𝐿, 𝑘∈𝑅, the FC of the crew leader 𝑖 in the route 𝑘; 𝛽𝑗𝑘, 𝑗∈𝐶𝐺, 𝑘∈𝑅, the FC of the cash guard 𝑗 in the route 𝑘; 𝑊𝑘𝑙, 𝑖∈𝐶𝐿, 𝑘∈𝑅, the FC of the route 𝑘 in the vehicle 𝑙. It is assumed that at the end of the planning term, there will be a Cumulative Frimat Coefficient (CFC) which should be close to an established goal 𝜌 for each set of workers (For instance, crew leader, cash guard, and driver). The parameters Ω and π are used so that crew member assignments within the planning horizon T have a frequency limitation.

Index

i: Index identifying the crew leader

j: Index identifying the cash guards

k: Index identifying the route

l: Index identifying the vehicle

m: Index identifying the number of crew leaders

n: Index identifying the number of cash guards

p: Index identifying the number of vehicle

Free Variable

FT: Total Fatigue

Decision Variables

Binary assignment variables:

1. 𝑋𝑖𝑘𝑡: 1 if crew leader 𝑖 is assigned to route 𝑘 in period 𝑡 (1), (0) otherwise.

2. 𝑌𝑗𝑘𝑡: 1 if cash guard 𝑗 is assigned to a route 𝑘 in period 𝑡 (1), (0) otherwise.

3. 𝑍𝑙𝑘𝑡: 1 if vehicle 𝑙 is assigned to a route 𝑘 in period 𝑡 (1), (0) otherwise.

Deviation variables (positive and negative deviations from target fatigue)

Objective Function

The objective is to minimize the total deviation from the target fatigue levels:

Constraints

Assignment constraints (Equations 6–8) ensure that each route is assigned exactly one crew leader, one cash guard, and one vehicle per period. Constraints (9–11) ensure no crew member is assigned to more than one route per day. Equations (12–13) limit the number of assignments per worker over the planning horizon. Balance constraints (14–16) ensure cumulative fatigue is balanced around the targets:

This model assumes that there will always be enough cars and staff to complete the routes.

Results

Case Study for Cash-in-Transit Logistics

The CIT process encompasses cash collection, transportation, processing, storage, distribution, and ATM replenishment. The case study was conducted within a Colombian CIT company operating 15 established routes, each with varying distances, cash volumes, number of stops, and occasionally serving different cities. These differences result in unequal workloads across routes, influenced by the role of each crew member—driver, crew leader, and cash guard. Drivers remain inside the vehicle, functioning primarily as stationary escorts. Cash guards handle cash directly, exiting the vehicle at each destination while carrying both a cash bag and a firearm. Crew leaders are responsible for route knowledge and frequently accompany cash guards, also handling money and security equipment.

Physiological workload (penosity) was measured using wearable heart rate monitors with integrated GPS. Data were collected across 14 routes. Penosity metrics are presented in Table 3, while corresponding Frimat Coefficients (FC) are illustrated in Figure 3 (also see Appendix A).

To enhance operational security, the study proposes a rotation system where crew members rotate among routes and roles. Under the proposed scheme, no operator may perform the same route more than twice per week (Ω=𝜋=2).

• Rotation of staff across different routes is mandatory.

• Assignments must be as equitable as possible.

• Each vehicle is consistently operated by the same driver.

The optimization model successfully generated equitable crew assignments that distribute workload uniformly while adhering to all company security constraints. Upon solving the case study instance, the resulting assignments satisfy all operational restrictions. The model utilizes target goals of 𝜌1=90, 𝜌2=90, and 𝜌3=100 in Equations (14), (15), and (16), respectively.

Table 4 presents the optimized Frimat Coefficients per crew member, along with their absolute deviation from the target. The last two rows summarize the total deviation and the standard deviation of the assignments.

A comparison between the current state (Figure 3) and the optimized solution (Figure 4) reveals a significant reduction in FC dispersion across crew members. In the current scenario, FC values fluctuate widely around the target, highlighting inequitable workloads (see Appendix B, Figure B.1). The optimized allocation results in FC values more concentrated around the targets, particularly among crew leaders and cash guards (see Appendix B, Figure B.2).

Crew leaders generally exhibit FCs very close to the target, except for CL7, whose FC is 12 units below the target. This deviation implies reduced fatigue and is not considered operationally problematic. For cash guards, optimized assignments yield small deviations from the target. Only CG9 exhibits a 10-unit shortfall, which also poses no practical issue.

Drivers, however, show more substantial deviations. For example, driver D6 is expected to experience relatively high fatigue levels, while drivers D10 and D12 display FC values below the target. This variability reflects the limitations in combinatorial possibilities for small instances. Overall, the optimized assignments result in significantly lower variability in workload for crew leaders and cash guards. In contrast, drivers exhibit higher variability, likely due to the limited number of feasible assignment combinations. Despite these constraints, the model delivers near-optimal results, maintaining equitable workload distribution wherever possible. The model’s validity is further confirmed by improvements over the company’s current manual planning process. Presently, weekly planning requires approximately six hours and still leads to suboptimal and inequitable assignments. The proposed optimization model reduces the planning time to a matter of seconds for small to medium instances, though execution time increases with problem size. This behavior is examined in detail in the following section.

Computational Analysis

A computational experiment was conducted to evaluate the scalability and performance of the proposed model, emphasizing its relevance to short-term operational planning in logistics. CIT crew assignment problems do not require real-time execution but must be solved efficiently within limited horizons.

Problem instances were generated with varying numbers of routes and planning periods. Initial test cases included 12, 18, 25, and 30 routes, each assessed over 6- and 12-day horizons. Larger instances were constructed with 50, 100, and 175 routes, using an 18-day horizon. All instances were implemented in GAMS v24.7.4 and executed on a 3.4 GHz Intel Core i7 processor.

Figure 5 reports total computational times (in seconds) for all 46 test instances. Solver stopping criteria were defined based on gap thresholds. For small instances, solutions were required to achieve zero absolute and relative gaps (“no gap”). For larger instances, time limits of 1,000, 3,600, and 7,200 seconds were applied, while the most challenging cases were allowed up to 10,800 and 14,400 seconds. Table 5 summarizes the computational results.

The results indicate that the model achieves optimal solutions within seconds for small instances (≤12 routes), but computational times increase substantially with problem size. For example, instances with 25 or more routes required execution times approaching or exceeding one hour. In large-scale scenarios (≥100 routes), solver performance was constrained by time limits, yielding feasible but not necessarily optimal solutions. This behavior reflects the problem's combinatorial complexity and highlights the need for scalable solution strategies (see Figure 6).

The computational experiment provided evidence of the model’s capacity to handle small- and medium-scale CIT crew assignment problems within acceptable time limits. However, scalability remains a critical concern when addressing large-scale instances, where exact optimization may become impractical due to exponential growth in computational requirements. As the number of routes and planning horizons increases, solution times exhibit a superlinear trend, reflecting the combinatorial complexity inherent to crew assignment problems. This behavior is consistent with findings in related logistics domains, where exact solvers often struggle to deliver optimal solutions under stringent time constraints [27], [28].

To enhance scalability, future research may incorporate metaheuristic approaches that balance solution quality with computational efficiency. Techniques such as Greedy Randomized Adaptive Search Procedure (GRASP) and Tabu Search have demonstrated effectiveness in vehicle routing and crew scheduling, providing near-optimal solutions within reduced computational times [29]. Similarly, Simulated Annealing and Genetic Algorithms offer robust exploration of the solution space, particularly valuable in large-scale scenarios where exact methods are computationally prohibitive [30]. Hybrid frameworks (matheuristics) that combine mathematical programming for critical subproblems with metaheuristic exploration for global search represent a promising avenue, enabling efficient handling of complex constraints while maintaining scalability. Such approaches have been successfully applied in transportation and logistics optimization, suggesting their applicability to CIT crew assignment problems [31].

In summary, while the current computational experiment validates the feasibility of the proposed model for short-term operational planning, integrating metaheuristic strategies could significantly improve scalability. This would allow the model to address larger and more complex instances, thereby strengthening its practical relevance in real-world logistics operations.

Conclusions

This study integrated human factors/ergonomics techniques with OR techniques to jointly solve the problem of scheduling crews to CIT routes. The methods presented in this paper focused on a route-dependent assignment. This means that the balanced fatigue is dependent on the difficulty of vehicle routes. The goal-based programming approach allowed us to find assignments that balance the fatigue levels of the crew members. For the case study, these assignments have considered a previous investigation of the difficulty levels of the routes and measurements of the Frimat coefficient. At the end, after a process of computational experimentation, those instances where the problem can be solved accurately in reasonable times are identified. In the same way, large-scale instances were identified that demand methods that allow finding solutions close to the optimal ones.

Based on mathematical representation, we conclude that when applying different instances in the proposed model, we find that the problem is intractable when overcoming more than 50 routes. Hence, it is necessary to use other heuristic solutions, such as Greedy procedures [32], a Simulated Annealing [33], or a Tabu Search. Column generation algorithms can also offer an acceptable solution approach for large-scale problems [34]. For this kind of problem, evolutionary algorithms can also provide good performance, for example, a genetic algorithm [32]. On the other hand, it would be a question to project how a population meta-heuristic would behave, such as Ant Colony Optimization.

Limitations and Future Work

This study adopts the simplifying assumption that each staff member is assigned to exactly one route, and that sufficient personnel and vehicles are available to complete all scheduled tasks. While this assumption facilitates tractability and allows us to isolate the effects of physical fatigue on scheduling decisions, we acknowledge that it may not fully reflect the complexity of real-world operations. In practice, staff shortages, overlapping demands, or flexible work arrangements often require that a single worker cover multiple routes within a given period.

The current formulation should therefore be regarded as a baseline model that highlights the fundamental trade-offs between scheduling efficiency and fatigue accumulation. A natural extension of this work is to relax the one-to-one assignment constraint and explicitly incorporate scenarios in which staff members may cover multiple routes. Such an extension would require modeling cumulative fatigue, recovery periods, and operational feasibility constraints, thereby offering a more realistic representation of workforce scheduling. Comparing the outcomes of this extended formulation with those obtained under the current assumption will provide valuable insights into the robustness of our findings and the practical implications of fatigue-aware scheduling policies.

References

Notes

* Research paper

Author notes

^aCorresponding author. E-mail: l.saavedra@javeriana.edu.co

Additional information

How to cite this article: N Clavijo-Buritica, L A Saavedra-Robinson, J Posada-Galvez, D Carrero-Soto, “Crew Assignment Problem for Routes Scheduling: A Fatigue Balancing Approach for Cash-In-Transit Logistics” Ing. Univ. vol. 30, 2026. https://doi.org/10.11144/Javeriana.iued30.capr

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