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.

Table 2 Table 2 Frimat coefficient variables Source: Frimat, 1988. [26]

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.

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.

Table 3 Table 3 Results for Frimat coefficient Source: Authors’ own creation.

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).

Figure 3 Figure 3 Results of the Frimat Coefficient related to crew members Source: Authors’ own creation.

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.

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).

Table 4 Table 4 Results for Frimat coefficient related to crew members Source: Authors’ own creation.

Figure 4 Figure 4 Results of Frimat Coefficient (FC) related to optimal assignment Source: Authors’ own creation.

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.

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.

Figure 5 Figure 5 Computational time for 46 instances (t= number of days; k=number of routes) Source: Authors’ own creation.

Table 5 Table 5 Computational results (TL = Time Limit) Source: Authors’ own creation.

FT: Obj. Function Value; CT (computational Time); AG (Absolute Gap); RG (Relative Gap)

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).

Figure 6 Figure 6 Computational time sectors for 46 instances Source: Authors’ own creation.

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.