3.1 Problem description

In this study, the VRPTW is framed as a critical routing challenge in the rescue process for highway incidents. The objective of the problem is to find optimal routes in terms of the response time. As previously mentioned, the constraints of hard and soft time windows are usually set for routing optimization. However, these constraints may face limitations for the practical application. The daily scheduling of EMS systems is faced with mass incidents of unknown causes and other sudden uncertainties. The ambulances might be occupied and unavailable for immediate emergency response during the peak hour. Therefore, the susceptibility of infeasible time window constraints to violation makes the constraints of hard time windows unsuitable for an EMS operation system. Motivated by the characteristics of the constraints of soft time windows, we design a programming model with the constraints of semi-soft time windows for emergency vehicle path optimization. Soft time windows provide a higher degree of flexibility without imposing penalties or negative consequences for deviations. Semi-soft time windows we used in this paper strike a balance between flexibility and adherence to certain constraints. The rescue time threshold of the prehospital emergency healthcare system restricts the pernicious effects caused by excessive deviation, so semi-soft time windows could satisfy the requirements of emergency management and it still allows flexibility for exceptional scenarios. In our paper, the model featuring semi-soft time window constraints incorporates distinct response time threshold values tailored to different priority levels of patients or events. Penalties are incurred when exceed specific time window frames, which ensures a practical and adaptive model for emergency vehicle path optimization.

Incidents occurring on highway networks significantly impact mortality and morbidity rates, primarily due to the high speeds involved [29]. Hospitals and ambulance bases are typically situated in urban or rural areas rather than within service zones along highway networks. The ambulance service operates on a comprehensive model, requiring departures from rescue bases or the site of the latest incident, navigating to the current incident location, and ultimately reaching the designated hospitals. In this intricate process, ambulance routes traverse both urban roads and highways. Consequently, it becomes imperative to consider the traffic conditions in both urban and highway environments when responding to incidents on the highway network.

During non-peak hours, ambulances and other emergency vehicles generally encounter minimal obstruction, as drivers are obligated to yield the right of way to these vehicles. However, exceptions occur during peak hours or particular holiday travel periods when heavy traffic congestion significantly contributes to delays for emergency vehicles on the road networks. Yielding to emergency vehicles may become challenging or even impossible, since it’s difficult for drivers to pull over or exit the road promptly. As a result, shoulders or emergency lanes are strategically designed to facilitate the passage of emergency vehicles. These lanes allow emergency vehicles to navigate around traffic congestion on busy multi-lane roads within highway networks, which ensures a smoother and more efficient response during peak hours.

Despite emergency vehicles having the right of way in the road network and the capability to automatically trigger changes in traffic light sequences, they must exercise caution in selecting routes to minimize potential delays. The emergency vehicles try to avoid road section with poor traffic reliability and high traffic impedance such as the areas near heavily congested schools and construction zones. Opting for paths that allow them to navigate around traffic congestion enhances the efficiency of emergency rescue efforts. However, the routing of emergency vehicles is inevitably susceptible to losses caused by unexpected traffic jams, particularly at two critical nodes: intersections within urban road networks and entrance/exit ramps on the highway networks. In instances where conditions at intersections are saturated or oversaturated, the resulting significant impacts on network performance—such as delays, queues, and reduced capacity—can impede the overall efficiency of emergency rescue operations [30]. Meanwhile, when ambulances enter or exit ramps, they must adjust their speed in accordance with the flow of upstream and downstream traffic around the ramp-weave sections. This careful consideration and strategic navigation are crucial for ensuring the smooth progress of emergency vehicles through these points on the road.

Our study is conducted on a complete directed graph G = (N, E) containing a set of nodes N and a set of arcs E. We develop an MIPSSTW model with the constraints of the semi-soft time windows considering the travel time of both nodes and arcs. The nodes refer to the intersections on the urban road networks and entrance/exit ramps on the highway networks, and arcs correspond to road sections within the study areas. The approach to the prediction of travel time and queueing time for the intersections is based on queuing theory. We employ an improved fixed detector method to estimate the travel time of emergency vehicles passing through ramp-weave sections. The improved Bureau of Public Roads (BPR) model serves as a valuable tool for predicting the travel time of road sections. This comprehensive approach enhances the precision and reliability of our study in modeling and optimizing emergency vehicle routes.

Assumptions and symbolic descriptions

Throughout this paper, the following assumptions are consistently applied:

1. The ambulance, origin, and destination are designated by the EMS system in the call center (or dispatch center), and thus, these pieces of information are considered known parameters.
2. If the paramedics in the ambulance can carry out the on-site treatment when the ambulance arrives at the accident scene, the rescue will be finished. If further treatment is required based on the severity of the patient, the ambulance will transport the patient to the hospital.
3. The urgency priority of the accident can be determined by the dispatch center.
4. The routing problem doesn’t consider the emergency call queue time, the paramedics’ preparation time, patient boarding and alighting time during the response process.
5. The travel time of emergency vehicles is also associated with the driver’s driving habits and personality. In our study, we assume each ambulance driver’s driving habits are consistent.
6. We assume that each ambulance is dedicated to serving only one incident.
7. This model doesn’t consider that the road network is disrupted due to natural disasters, such as hurricanes, earthquakes, tsunamis, etc.

Based on the above assumptions, we propose the MIPSSTW model. Table 1 summarizes all the notations we used in the MIPSSTW model.

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Table 1. Notation used in the model.

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3.2 Mathematical model

Based on the aforementioned assumptions and notation, we formulate a MIPSSTW model to optimize the routing problem of ambulances for highway incidents, outlined as follows: (1) (2) (3) (4) (5) (6)

The objective function (1) aims to minimize the total travel time and delay time of the rescue routing. Constraints (2) ensure that every incident receives a response from emergency vehicles. Constraints (3) guarantee the traffic flow balance for intermediate nodes. Constraints (4) ensure that emergency vehicles with different rescue abilities can meet the needs of each incident. Function (5) represents the constraints of semi-soft time windows. Where ψ₁,ψ₂ respectively denotes the weight of delay penalties for high- and low-priority requests. More precisely, we establish various time thresholds for the incident requests with different priorities. The semi-soft time windows penalize delays based on the variance between the actual travel time and the time threshold. This approach allows for a comprehensive consideration of the adverse impact of delays on events and patient conditions at different levels. Constraints (6) impose the attributes of binary decision variables.

It noted that τ_ij denotes the total travel time spent on nodes i,j and arcs (i,j). Therefore, it can be expressed as: (7) Where θ_ij denotes the node time consumption from node i to node j, and T_ij represents the travel time of road segment (i,j). We intend to use the improved BPR model to evaluate the travel time of road segments.

The classic BPR model was established by the U.S. Federal Highway Administration through regression analysis of highway historical highway data [31]. Due to its straightforward formulation for traffic impedance, this model is widely applied in transportation research. The expression of the classic BPR model is as follows: (8)

As the classic BPR model is developed from the highway traffic flow data, it is not suitable for saturated and oversaturated traffic conditions. In addition, the trend of vehicle speed typically shows an initial increase followed by a decrease with the increase of traffic flow. Obviously, the monotonic speed variations in the model doesn’t accurately reflect reality. The reason for this phenomenon is that the actual traffic flow is significantly higher than the design flow. The discrepancy arises because the actual traffic flow surpasses the design flow. Therefore, it is necessary to improve the classic BPR model to better suit real-life situations. The travel time of the road segment is expressed in Formula (9). (9) Where Q is the modified value of traffic volume: (10)

Furthermore, α and β are also recalculated by new approaches as follows: (11) (12) Where ω₁,ω₂ are the estimated parameters by the LSTM neural network [32,33].

In our paper, we emphasize the importance of node time consumption including intersections on the urban road networks and ramp-weave sections on the highway networks. We propose two distinct approaches to discuss these conditions separately.

1. When the node is an intersection on the urban road network

Despite emergency vehicles having priority of the way, delays may occur due to signal timing at intersections. When traffic volume is excessively high, the road network may reach a saturated or oversaturated status resulting in significant queuing. This might further contribute to emergency vehicle delays. In other words, the travel time of emergency vehicles is influenced by the traffic flow of the road network. Therefore, it is crucial to predict the travel time of intersections to prevent unnecessary delays. This node travel time θ_ij is divided into the queue waiting time and the travel time through the intersection: (13)

The vehicles at the intersection adhere to an queuing system. The vehicle arrival rate λ_ij(t) conforms to the Poisson distribution, and the traffic service rate μ_ij(t) follows a negative exponential distribution. We assume that the maximum number of vehicles m_ij can pass through the intersection. The queue system is designed to prevent the number of new arrival vehicles from exceeding this maximum capacity m_ij. Given the steady state of the traffic conditions, the transition probabilities of vehicles also remain steady and constant. We define this transition probability as . Meanwhile, the queuing time for other vehicles is associated with the timing of intersection signals. g_ij(t) denotes the green signal ratio of the intersection in the travel direction of road section (i,j), and the intersection’s effective service rate can be expressed as . can also be interpreted as the transition probability of the intersection state. According to the queueing theory, the probability of η queuing vehicles is: (14) When normalizing Eq (14), it can be expressed as: (15) (16) (17) Where ρ_ij(t) denotes the traffic intensity. Traffic intensity is a significant metric for traffic management and planning, providing a measure of the congestion level in the road networks. The expression of traffic intensity ρ_ij(t) is shown in Formula (18).

(18)

The number of queuing vehicles L_ij(t) is: (19)

Based on the Little’s law [34], the average queue waiting time is derived: (20)

The travel time for emergency vehicles to pass through the intersection : (21)

1. When the node is a ramp-weave section on the highway road network.

On- and off-ramps constitute typical weaving segments, resulting from the convergence and divergence of traffic streams. Weaving congestions easily occur and generate bottlenecks in the road networks due to the transfer behavior of drivers and the turbulence of traffic flow. Therefore, the average speed of emergency vehicles differs between weaving sections and freeways. In our study, the fixed detector method is employed to estimate the travel time of ambulances on highway entrance and exit ramps. The parameters indicative of the traffic operating conditions can be obtained through fixed detectors installed on the highway network, including vehicle velocity, traffic occupancy, and more. We calculate the travel time of emergency vehicles passing through ramp-weave sections using these traffic parameters obtained from fixed detectors.

In this study, we categorize two types of ramp-weave sections based on their geometric configuration for the placement of detectors. As shown in Fig 1, the first type of ramp-weave section includes a single entrance or exit ramp. Due to variations in traffic flow, the speed of ambulance changes only once. Given that q_on−ramp denotes the travel flow entering the on-ramp and D_i denotes the distance from the location of downstream detectors s_i to the junction of the entrance ramp and the highway segment. The total traffic flow on this road section can be expressed as q+q_on−ramp. The travel time Q_i(t) of emergency vehicles at node i is: (22)

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Fig 1. The first type of ramp-weave section.

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The second type incorporates adjacent weaving segments, which are displayed in Fig 2. Since node i and node j are in close proximity to each other, the emergency vehicle may be affected by merging traffic flow q+q_on−ramp when traveling on road section D_ij, and be affected by diverging traffic flow q+q_on−ramp−q_off−ramp when traveling on road section D_j. Therefore, the velocity of emergency vehicles may vary twice. The travel time at this node Q_ij(t) is: (23)

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Fig 2. The second type of ramp-weave section.

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4.1 Standard CS algorithm

The generic framework of the CS algorithm draws inspiration from the interesting feature of parasitism of cuckoo species and was initially proposed by Yang and Deb in 2009 [23]. Cuckoos exhibit brood parasitism and lack the behavior of hatching eggs by themselves. Thus, they need to search for high-quality nests and foster parents for incubation and chick-rearing [35]. Cuckoo birds have evolved a remarkable ability to lay eggs that closely resemble those of the host species in color, pattern, size, and shape. Their search process is implemented through Lévy flight. Lévy flight simulates the stochastic movement of insects and birds in mathematical modeling. As shown in Fig 3, the trajectory of Lévy flight represents the random movement of insects and birds, encompassing both short-distance turns and long-distance jumps. This characteristic enables the CS algorithm to undertake large-scale exploration and fine-tuned adjustments to find the optimal fitness value. Lévy flight is modeled as a probability density function with a power-law tail. According to the Central Limit Theorem, Lévy flight tends to reach a stable distribution after a sufficient number of random walks. The new solution for cuckoo i by Lévy flight is as follows: (24) Where step size α(α>0) is decided by the scale of the related research problem, and α = 1 is commonly applied in many cases. ⊕ denotes central multiplicative. Similar to the Particle Swarm Optimization (PSO) algorithm, the Central Limit Theorem is more effective in the CS algorithm, as Lévy flight exhibits longer step lengths in long distances. Levy(s,λ) denotes the random search path and its performance depends on the step size s and length randomly drawn from the Lévy distribution: (25) Where the parameter λ denotes the average value or expected value of events within a unit of time. In summary, Lévy flight focuses on searching around solutions and occasionally taking longer steps to minimize the probability of getting stuck in local optima.

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Fig 3. The trajectory of Lévy flight.

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CS algorithm is governed by three ideal rules: (1) Every cuckoo lays only one egg in a random nest where one egg represents one solution. (2) At the end of each iteration, the best nest with high-quality eggs is preserved for the next generation. (3) The number of host nests is fixed. Suppose the probability of discovering the cuckoo egg for a host bird is P_α∈(0,1). If the host bird finds a cuckoo egg, it will throw away this egg or abandon the nest. Hence, the egg will be hatched again, that is, this solution will be eliminated in the CS algorithm.

Compared to other heuristic algorithms, the CS algorithm could strike a good balance between local search strategies and overall space utilization, which has demonstrated excellent performance in terms of convergence speed and implementation. The CS algorithm has been proven successfully to solve a wide variety of multi-objective optimization problems. However, the standard CS algorithm has its limitations. It heavily relies on the random step size of Lévy flights during the searching and optimization process. The lack of communication with superior individuals leads to slow convergence and low precision in later iterations. Therefore, in order to improve the standard CS algorithm, researchers have paid more attention to exploring the variants of the CS algorithm for more complicated cases. In this paper, we develop an MCS algorithm to enhance the effectiveness of the standard CS algorithm via tuning parameters and diversity strategies.

4.2 MCS algorithm

This proposed variant of the CS algorithm adopts an effective multiple neighborhood structure and a path relink strategy to strengthen the exploration capability and facilitate escaping from local optima. Previous studies [36,37] have demonstrated that the neighborhood structures of 2-opt, 3-opt, OR-opt, and double bridge play a crucial role in providing benefits for a more effective random sequence in the search space. In addition, path relinking strategies (PRS) of Lévy flights are introduced to improve the diversity of exploration between initial solutions. The dynamic inertia weight strategy for the probability of eggs being discovered is designed to balance the performances of local search and global search. The implications of these key elements and improvements of the MCS algorithm are detailed below.

4.2.4 Path optimization of Lévy flight.

Lévy fight plays a vital role in modeling the navigational patterns of cuckoo birds during searching for the nests of other host birds. As shown in Formula (25), Lévy flight refers to both step size and direction for establishing new nests. In the standard CS algorithm, the long jumps of Lévy flight have the potential to skip over regions within the search space incorporating better solutions, which leads to premature convergence towards suboptimal solutions. Moreover, Lévy flight in the basic CS algorithm performs ineffectively in specific terrain types, particularly those characterized by narrow, deep valleys. It may have difficulties in local search around identified regions of interest. In order to address these limitations, a multi-neighborhood structure and path relinking strategies(PRS) are used to improve the diversity of exploration routes and facilitate the selection of the optimal route.

A multi-neighborhood structure consists of four intra-routes, as shown in Table 2. These intra-routes, namely 2-opt, 3-opt, OR-opt, and double-bridge moves, involve changes occurring within the same route. Each neighborhood structure is associated with specific step sizes of Lévy flight and generates a new solution by rearranging the sequence of visited nodes and arcs within another existing solution. This strategic integration aims to overcome the shortcomings of the standard CS algorithm and improve its performance across various landscapes.

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Table 2. Neighborhood structures [36,37].

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In order to enhance the survival chances of cuckoo bird eggs, these avian species are compelled to refine their evolution skills. The PRS for Lévy flights is adopted to optimize their trajectories between high-quality selected solutions. When the count of step sizes is below 2, the inter-route PRS involves the exchange of one node from one route with a node from another route. Conversely, if the number of step sizes exceeds 2, the cross-route PRS may engage in combining elements from one path with elements from another path to generate a new route and solutions. This strategy of path operations highlights more intensification and diversity in the search space to escape the MCS algorithm from the local optimum effectively.

4.3 Flowchart and pseudocode of MCS algorithm

The MCS algorithm amalgamates the strengths of the standard CS algorithm and enhances the search capability of cuckoo birds and the discovery ability of alien eggs of host birds. This improvement is achieved through the implementation of a dynamic inertia weight strategy and path optimization of Lévy flight. Fig 4 displays the flowchart of the MCS algorithm. The pseudocode of the MCS algorithm is shown in Fig 5.

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Fig 4. The flowchart of MCS algorithm.

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Fig 5. The pseudocode of the MCS algorithm.

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5.1 Experiment design

In this section, we design computational experiments to verify the efficacy and effectiveness of our MIPSSTW model and MCS algorithm in optimizing EMS ambulance routing. The research area contains three hospitals and three ambulance bases, as shown in Fig 6. It is noteworthy that the hospital and ambulance base in the northeast direction are considered the same location due to their close proximity. Excluding this particular node, the remaining ambulance bases and hospitals are situated in distinct locations. In addition, the research area is served by two intersecting highways, namely A1 and A2, both delineated by red lines on the map. The detailed experimental setup aims to provide a comprehensive assessment of the proposed MIPSSTW model and MCS algorithm in optimizing ambulance routing for EMS in a real-world scenario within the specified region in China.

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Fig 6. The map of the research area.

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Based on the actual road structure shown in Fig 6, we have undertaken simplifications to represent micro roads, residential roads, and some impassable roads in the specified area. The outcome of the topology road network is illustrated in Fig 7. The framework of this topology road network consists of a total of 71 nodes including intersections, highway ramps, hospitals, and ambulance bases. There are 123 road segments incorporating 109 urban road sections with various road grades and 14 highway segments. Nodes 20 and 45 serve as ambulance dispatch bases, and nodes 1 and 17 represent hospital locations. As mentioned above, the convergence node, denoting the shared location of the hospital and ambulance dispatch base, is marked as node 1. The refined representation of the road network provides a clear and simplified view, which enables a more focused analysis within the context of the EMS ambulance routing optimization.

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Fig 7. The topological diagram of the road network.

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We select two real-life incident scenarios to structure our experiments and validate the effectiveness of the MIPSSTW model. Based on actual traffic accident data, incidents on the highway network are annotated as nodes 35 and 57. Our goal is to assess the necessity of considering complex traffic conditions for optimizing emergency vehicle routing. Therefore, we compare ambulance routing optimization approaches with and without the consideration of time consumption for nodes and arcs in the experiments. Furthermore, the proposed MCS algorithm is coded in MATLAB 2020 using an Intel Core i7 processor, PCs with 3.07GHz and 8GB RAM memory. The performance of the MCS algorithm is evaluated through comparisons with the standard CS algorithm and other classic heuristic algorithms. Parameters for the MCS algorithm are set following the guidelines provided in literature [25], as detailed in Table 3.

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Table 3. Parameters setting in the MCS algorithm.

https://doi.org/10.1371/journal.pone.0301637.t003

5.2 Result and discussion

The results of the comparison between our MIPSSTW model and the approaches that neglect the time consumption of nodes and arcs are summarized in Table 4. We have considered four classic estimated indicators: travel time, average speed, distance, and average delay. Fig 8 shows the schematic diagram of emergency routing for both the traditional method and our model. The traditional method operates under the assumption that ambulances face no obstruction from other vehicles on the road network, that is, overlooking complex traffic conditions. However, our model takes the factors of time consumption of intersection, ramps, and other impedance for the highway incident into account. For accident 1, originating from node 45 and destined for node 17, the traditional approach yields an optimal path of 45-44-26-35-33-23-22-36-20-17. In contrast, our proposed model considering the traffic conditions, suggests a more efficient route of 45-44-26-35-33-23-37-21-20-17. Despite a 1.5km increase in distance, the travel time and average delay decreased by 3.7 mins and 3.2 mins, respectively. The average speed also experiences an improvement of 8.8km/h. The optimization ratios are 12%, 31%, and 21%, respectively. As for accident 2, where the ambulance originates from node 64 with the designated hospital at node 17., our model achieves a reduction of 2.2 mins in travel time and an enhancement of 7.9 km/h in average speed compared to the traditional method. The average delay is notably decreased by 28%. These findings highlight the superior performance of our MIPSSTW model in optimizing emergency routing by accounting for real-time traffic conditions and various impedances, which represents significant improvements in key performance indicators.

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Fig 8. The optimal paths of different methods for incident 1 and incident 2.

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Table 4. The comparison of routing optimization approaches.

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Table 4 unmistakably demonstrates the superiority of routing optimization that takes into account complex traffic conditions compared to traditional approaches. Traditional path search processes typically provide a rough evaluation of the travel time of emergency vehicles based on the traffic impedance of road segments, offering a limited reflection of actual traffic conditions. However, most of the stop- and go- conditions occurring at signalized intersections and ramp-weave sections, are not accurately captured by the traditional method. Especially when the traffic flow reaches or exceeds the capacity of the road section, the phenomenon of traffic flow breaking down incurs ambulance speed drops. The traditional method falls short in ensuring the optimal routing of emergency vehicles in the absence of considerations for complex traffic conditions. in contrast, our model comprehensively analyzes the characteristics of crucial compositions of the emergency vehicle path. We respectively apply an improved BPR function for road segments, a prediction method based on queueing theory for intersections on the urban road network, and an improved fixed detector method for ramp-weave sections on the highway network. Factors like signal light changes, queue lengths, and the speed of merging and diverging at ramps may lead to the randomness of the selection of rescue routing. The MIPSSTW model proves to be more effective in anticipating these challenges, consequently ensuring optimal routing with the shortest response time. This improvement is expected to significantly enhance the efficiency of emergency management for highway incidents.

In addition, we investigate the performance of dynamic inertia weight strategy and path optimization of Lévy fight on a randomly generated solution to validate the effectiveness of the MCS algorithm in the VRP. Several comparative experiments involving different solving algorithms were designed for this purpose. For the response routing for incidents 1 and 2, we explore the performance of both the standard CS algorithm and MCS algorithm within the context of our MIPSSTW model. The results of the CS algorithm and MCS algorithm are illustrated in Fig 9. As shown in Fig 9, it is observed that in the optimal path-solving for incidents 1 and 2, the MCS algorithm exhibits a faster convergence speed compared to the standard CS algorithm. Especially in the iteration process of incident 2, the CS algorithm initially converges faster than the MCS algorithm. However, the MCS algorithm ultimately achieves the optimal vehicle path with fewer iterations. This observation suggests that the neighborhood structures and PRS strategies in the MCS algorithm strengthen the diversity of exploration between initial solutions. The dynamic inertia weight strategy governing the probability of discovering eggs effectively balances the performances of local search and global search. Therefore, the MCS algorithm has been demonstrated that it has a higher likelihood of searching for the optimal solution, faster convergence speed, and more robust global search capabilities, which significantly improves the CS algorithm’s running efficiency.

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Fig 9. The comparative results between CS and MCS algorithms.

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Finally, we also conducted a comparative analysis of the performances of the MCS algorithm against three other commonly used swarm intelligence algorithms: Genetic Algorithm(GA), Particle Swarm Optimization(PSO), and Ant Colony Optimization(ACO). In order to ensure robustness, 50 different simulation experiments were randomly conducted based on historical incident data. We then selected 15 results shown in Table 5 and Fig 10. The evaluation criteria for the algorithms’ performances are the travel time and average speed associated with the optimal routing. Due to the relatively close distance and poor road conditions in incident 5, there is no significant difference in the travel time of the optimal routings among these four algorithms. However, the MCS algorithm comparatively performs well in other computational results and helps to reduce the travel time of ambulances during the rescue process. A similar situation also happens in incident 12, that is, the results represent the same trend among the four algorithms. This is because the incident occurred at night and the selected routing in the road network is limited leading to similar results at last. The ability of the MCS algorithm to consistently deliver favorable results in diverse scenarios indicates its potential as a robust optimization tool for emergency vehicle routing in various conditions.

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Fig 10. The comparative results of travel time and average speed of four algorithms.

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Table 5. The results of four algorithms.

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