4.1 Railway passenger flow assignment overall process

Railway passenger flow assignment is the process of assign passenger flow to different trains according to certain principles and methods. It is an important link in the planning and design of passenger transport products.

Through passenger flow assignment, the train matching relationship under given conditions can be obtained, and then the passenger transport products can be evaluated and optimized, which can effectively improve the utilization level of railway resources and service level, and has important theoretical and practical significance.

The high-speed railway passenger flow assignment process is mainly composed of three parts, respectively route selection, model selection, passenger-train match, as shown in Fig 1.

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Fig 1. High-speed railway passenger flow assignment process.

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4.2 Construction of train service network based on timetable

The whole travel process of railway passengers mainly consists of several stages: waiting on stations, getting on trains, moving in trains, transferring if needed, getting off trains and going out of stations. The 2nd to 5th stages are selected to construct the service network which contained a sequence of service nodes and arcs.

Service nodes comprise of one or several stations in the same city or special area where trains depart, pass or arrive and passengers can transfer from one train to another. The arcs are used to describe all the actions of passengers including getting on/off the trains, moving by the trains and transfer between the trains. A small service network comprise of 3 trains and 5 nodes (A-E) are shown in Fig 2 where the horizontal direction indicates time and the vertical direction indicates space.

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Fig 2. Sample of train service network.

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The service network is represented by G = {Node, Arc}, where Node is the set of nodes and Arc = {e_ft,st,n} is the set of all kinds of arcs. The symbol e_ft,st,n represents 4 types of arcs:

1. Getting on arc by which passengers get on train ft from node n when st = 0;
2. Getting off arc by which passengers get off train st from node n when ft = 0;
3. Moving arc by which passengers pass through the st^th section of train ft when n = 0;
4. Transfer arc by which passengers transfer from train ft to st in node n.

Then several sample routes in Fig 2 from node A to node E can be presented as following.

1. Two Direct routes: {e_2,0,a, e_2,1,0, e_2,2,0, e_2,3,0, e_2,4,0, e_0,2,e}, {e_3,0,a, e_3,1,0, e_3,2,0, e_3,3,0, e_3,4,0, e_0,3,e};
2. Eight one-time transfer route: {e_2,0,a, e_2,1,0, e_2,3,b, e_3,2,0, e_3,3,0, e_3,4,0, e_0,3,e}, {e_2,0,a, e_2,1,0, e_2,2,0, e_2,3,c, e_3,3,0, e_3,4,0, e_0,3,e}, {e_2,0,a, e_2,1,0, e_2,2,0, e_2,3,0, e_2,3,d, e_3,4,0, e_0,3,e},etc;

In summary, there are 14 travel routes for the passengers from node A to E including another 4 two-time transfer routes. The quantity of service routes in such a simple service network is evidently a little much, and it will go up very sharply with the increase of the number of trains in the network.

In order to reduce the scale of train service network search, trains are converted to nodes and the main part of abstracted service network is comprised of trains nodes and transfer arcs. In addition, passengers enter or exit the network by passenger nodes and getting on/off arcs, as shown in Fig 3. Figs 2 and 3 are intended to illustrate the different forms of train service network, and there is no correspondence between them.

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Fig 3. Abstracted train service network.

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TransferArc = {e_ft,st} is the set of transfer arcs, where ft represents the first train, st represents the second train. OnArc = {e_n,ontrain} is the set of getting on arcs, where n represents the getting on passenger node and ontrain represents the getting on train. OffArc = {e_n,offtrain} is the set of getting off arcs, where n represents the getting off passenger node and offtrain represents the getting off train. The time attributes of passenger getting on/off and transfer are included in the sub-attributes of trains and are not represented in the abstracted train service network.

4.3 Generalized cost formula of reasonable service routes

In the space-time service network, there are a large number of available routes for same OD at the same time, but in the actual travel process, some service routes will not be chosen. Therefore, the reasonable service route is defined as the OD service route that meets the passenger travel demand and conforms to the relevant business rules and common sense.

The passenger choice of service route affected by several factors including total travel time, total ticket price, convenience of the journey, departure and arrival time, etc. Passengers would generally choose the routes with the minimum cost in their minds.

[9] noted that the path over-lapping problem violates the IIA irrelevant independence hypothesis of MNL model and further affects the passenger route selection behavior. Therefore, PSL model is used to replace MNL model, in which the route generalized cost is divided into fixed cost, route correlation cost and random cost.

From the perspective of passengers, the generalized cost formula of reasonable service routes is defined as follows.

(1)

(2)

(3)

(4)

(5)

(6)

In terms of calculating values, each subitem of the fixed cost FC_w,k,s are uniformly transformed into time to calculate.

Time cost TC_w,k is related to the travel time in transit and convenience of the journey which can be expressed as transfer time, as shown in Eq (3). The transfer penalty coefficient ψ_w,k takes 0.5 hours per time.

Money cost MC_w,k is related to the total ticket price that can be converted into time cost according to the passenger time value, as shown in Eqs (4) and (5).

Since the Shandong circular high-speed railway not only includes the passenger flow in Shandong province, in order to ensure the availability of data, the passenger time value of China and Shandong province were calculated respectively. Table 3 is the calculation of passenger time value.

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Table 3. Calculation of passenger time value.

https://doi.org/10.1371/journal.pone.0314713.t003

Table 4 is the Passenger flow of Jinan Railway Bureau in 2019. The local passengers and export passengers correspond to the passenger time value of Shandong province. The import passengers and carrying passengers correspond to the passenger time value of China.

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Table 4. Passenger flow of Jinan Railway Bureau in 2019.

https://doi.org/10.1371/journal.pone.0314713.t004

According to the proportion of different types of passenger flow, the passenger time value was 49.18 yuan/h, which was rounded up to 50 yuan/h.

Seat comfort cost SC_w,k,s is related to the fatigue recovery time, as shown in Eq (6). For high-speed railway trains [35], defined the values of M, τ and δ as 15, 59 and 0.28 respectively. Based on the actual ticket price approximate ratio of the same OD and same train, the passenger fare sensitivity coefficient η for business seat, first class seat and second class seat are 3, 1.5, 1, respectively.

[9] deduced three formulas of route choice correlation and pointed out that the original formulation should be preferred, which has a theoretical foundation.

(7)

On the basis of Eq (1), the passenger travel utility formula is defined as follows. The purpose of arcosh (2) is to guarantee U_w,k,s ∈ [0,1].

(8)

When , the relationship between passenger travel utility and generalized cost of reasonable service routes is shown in Fig 4. As can be seen from the slope of the curve in Fig 4, the lower the generalized cost value, the more difficult it is to change the passenger travel utility.

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Fig 4. Chart of changes in passenger travel utility.

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5.1 Objective function

The first optimization objective is to maximize the corporate revenue. Let M denote the total revenue of high-speed railway enterprises and Z₁ denote the revenue-related optimization objective.

(9)

(10)

Let N denote the passenger travel choice benefit and Z₂ denote the passenger-related optimization objective.

(11)

(12)

The methods for solving multi-objective programming problems include transformation constraint method, evaluation function method and hierarchical sequence method. In the absence of historical data, it is difficult for either Z₁ or Z₂ to set the initial constraint range, so one of Z₁ or Z₂ cannot be converted into a constraint for solving. Considering that total revenue represented by Z₁ and passenger travel choice benefit represented by Z₂ are difficult to be treated as the same evaluation function.

To sum up, the hierarchical sequence method is selected. Firstly, the ticket price in Z₁ is dynamic by heuristic algorithm (Large Neighborhood Search). Secondly, the precise algorithm (Path Size Logit Passenger Flow Assignment) is used to simulate passenger travel choice behavior and solve Z₂ passenger flow assignment. Finally, the Z₂ result is substituted into Z₁ to solve the total revenue.

5.2 Decision variable and constraints

As the decision variable of dynamic pricing problem, the ticket price p is related to the line price rate per unit distance p₀ and the distance of current OD. p changes with the change of p₀, and then changes the reasonable service route utilities c and passenger flow assignment results f. It should be noted that since the reasonable service routes contains direct routes and transfer routes, the subscript of p uses route (i.e. subscript k) instead of train.

Z₁ considers multiple constraints which are related to ticket price, including dynamic pricing constraint, price not inverted constraint, as shown in Eqs (13) and (14).

1. Dynamic pricing constraint. In terms of ticket price related constraints, the dynamic range of price is restricted. Too high a price is not conducive passenger travel, and too low a price is not conducive corporate revenue. (13)
2. Price not inverted constraint. Ticket prices should not be reversed, which means the price of a short-distance train ticket is lower than the price of a long-distance train ticket. If the total distance is equal, the combined value of two short-distance train ticket is not less than that of one long-distance train ticket. (14)
   Z₂ considers multiple constraints which are related to passenger flow, including passenger choice route capacity constraint, route passenger flow non-negative constraint, OD passenger flow identically equal constraint, transfer times constraint, as shown in Eqs (15)–(18).
3. Passenger choice route capacity constraint. Each reasonable service route has a maximum passenger flow capacity, which is limited by the capacity of the trains serving the space-time service route. (15)
4. Route passenger flow non-negative constraint. In terms of passenger flow related constraints, the passenger flow of any seat level on each route must remain non-negative be negative at any time in any OD. (16)
5. OD passenger flow identically equal constraint. According to common sense, the sum of passenger flow on all routes at any time is a fixed value. (17)
6. Transfer times constraint. When the capacity of direct service routes is insufficient, passengers will tend to choose transfer service routes. However, too many transfer times will against the travel experience of passengers. According to the investigation of the maximum passenger acceptable transfer times in relevant areas and the passenger flow analysis of Shandong circular high-speed railway, the maximum acceptable transfer times is set to be no more than 1. (18)

5.3 Large neighborhood search-path size logit passenger flow assignment algorithm

In this paper, the problem of dynamic pricing and passenger flow assignment of high-speed railway is transformed into a combination optimization problem of maximizing passenger travel choice benefit under the condition of maximizing corporate revenue.

The path size logit passenger flow assignment accurate algorithm is embedded into the structure of large neighborhood search heuristic algorithm. The overall algorithm structure and calculation process are shown in Fig 5. The pseudo-code of large neighborhood search (LNS) algorithm is shown in Table 5.

1. Step 1: Input initial data, including line price rate R₀, dynamic pricing range [LowerRate, UpperRate], total iterations times T, convergence value Δc, OD passenger flow data, train timetable data.
2. Step 2: Establish the classification of trains based on multiple factors (such as number or proportion of different level stations, etc.). Single or multiple factors are considered, depending on the availability of relevant data.
3. Step 3: Establish the subdivision ticket price adjustment strategy based on train class.
   • Step 3.1: Set different price gaps. The positions of different gaps are different. The practical significance of the narrow gap is greater, and the research significance of the wide gap is greater.
   • Step 3.2: The ticket price adjust ranges of different class trains under different gaps are determined. The higher the train class, the higher the ticket price rate.
4. Step 4: Carry out the large neighborhood search (LNS) algorithm.
   • Step 4.1: Select P percent of the trains randomly for Destroy and Repair operations. The value of P can be deterministic or random.
   • Step 4.2: Destroy means the selected trains (P percent quantity and reserved integer) are removed from the train alternative set. The ticket price rate of the selected trains will be adjusted randomly within the range by the above strategy. The ticket price rate of other trains in the alternative set remains unchanged.
   • Step 4.3: Repair means the ticket price rate of the selected trains has been adjusted. And then they are inserted into the train alternative set.
5. Step 5: Carry out the path size logit (PSL) assignment algorithm based on capacity constraint-passenger flow increment (CCPFI).
   • Step 5.1: Search for direct routes and transfer routes for different ODs based on train timetable data.
   • Step 5.2: For different routes of different ODs, calculate route utility and path size logit choice probability based on train price rate (i.e. the route selection ratio).
   • Step 5.3: Considering OD classes, OD passenger flow demand and train capacity, simulate passenger travel choice behavior. If the train capacity is sufficient, the number of passengers on board is the product of route selection ratio and OD passenger flow demand. Otherwise, the number of passengers on board is the train surplus capacity.
   • Step 5.4: Output enterprise income x and passenger flow assignment results.
6. Step 6: Evaluate the output result against the current global optimal solution x^b. Determine whether the iteration times T is reached. If true, go Step 9. Otherwise, go Step 7.
7. Step 7: If | x—x^b |≤Δc, go Step 9. Otherwise, go Step 8.
8. Step 8: If x≥x^b, update the current global optimal solution x^b = x. Then, go Step 4.
9. Step 9: End the loop and output max{x, x^b}.

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Fig 5. The overall algorithm structure.

https://doi.org/10.1371/journal.pone.0314713.g005

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Table 5. Pseudo-code of large neighborhood search (LNS) algorithm.

https://doi.org/10.1371/journal.pone.0314713.t005

According to the random utility theory of passenger choice behavior, path size logit assignment algorithm based on capacity constraint-passenger flow increments calculated using Eq (19) where U_w,k,s, U_w,m,s are given in Eq (8). The specific process is shown in Fig 6.

(19)

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Fig 6. Path size logit assignment based on capacity constraint-passenger flow increment algorithm.

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6.1 Basic information of Shandong circular high-speed railway

The Shandong circular high-speed railway consists of 5 railway lines, namely the Jiqing HSR(East-West direction), Jiaoji HSR (East-West direction), Jinghu HSR (North-South direction), Rilan HSR (East-West direction), and Qingyan HSR (Northeast-Southwest direction). The schematic diagram of Shandong circular high-speed railway based on station levels is shown in Fig 7.

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Fig 7. The schematic diagram of Shandong circular high-speed railway.

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Based on the number of trains served, the number of passengers served and the city administrative level, the stations are divided into three levels, including large station, medium station and small station. For Shandong circular high-speed railway, there are 33 stations, including 7 large stations, 8 medium stations and 18 small stations, as shown in Fig 7.

It is worth noting that Jinan and Qingdao are the key traffic nodes of Shandong circular high-speed railway, both of which have multiple large stations. Although Qingdao West belongs to Qingdao administratively, the passenger flow is relatively small, so it is classified as a medium station.

Linyi North is located in the center of southern part of Shandong province, so it is classified as a large station. Rizhao West is a transit node connecting Shandong province and Jiangsu province. Although the passenger flow is lower than Jinan node and Qingdao node, it is classified as a large station considering its special geographical location. In addition, due to the close distance between Jiaozhou North and Qingdao Airport, they are merged into a small station.

Based on the historical ticket data, the average basic ticket price rate of Shangdong circular high-speed railway is set at 0.4 yuan/km. In fact, the ticket price rate of different sections of the same line may be different, and high-speed railway has the characteristics of decreasing rate with increasing distance. However, the above points do not affect case design and validation of the model and algorithms.

The passenger flow data and train timetable data on October 25 in 2019 were provided by the Jinan Railway Bureau.

Due to the availability of data, only second-class seats are considered in case. It should be noted that the research approach for different classes of seating is the same, with the emphasis on distinguishing the heterogeneity of the target passengers. The acceptable range of dynamic pricing varies among different passengers. Therefore, the second-class seat scenario case can still verify the applicability of the model and algorithms which can still be applied to the scenarios considering different seats.

6.2 Train classification and price adjustment strategy

The process of railway transportation organization is very complex, involving railway network planning, line plan, train timetable, EMU plan, crew plan, station organization, marketing organization, etc. Each link is nested and affects each other.

In order to maintain the stability of the railway system and improve the social and economic benefits of high-speed railway, the classification process of differentiated train products should consider both scientific and practical, which means that the classification criteria should be scientifically grounded but not overly complex.

[36] described a route synthesis layering approach that defined stations and trains according to two classes, which contained the idea of classification. On this basis, the train classification operation is as follows: Firstly, the station level is divided based on the historical average passenger flow and the city administrative level. Secondly, differentiated train products are divided into Class I, Class II and Class III based on the number of stations and station level. Finally, the train classification is linked to ticket price. Different classes of trains have different dynamic pricing range.

Generally speaking, there are two optional criteria for the establishment of train classification. First, it is divided by the station levels of train service, such as the large station train only serves the large station, the medium station train only serves the large station and the medium station, and the small station train must serve the small station. Second, it is divided by the proportion of stations at all levels of train service, such as the proportion of x stations in all stations p^x, where x∈{large, medium, small}.

Due to the large number of small stations along the Shandong circular high-speed railway, all trains serve the above three levels of station. It is impossible to construct the train classification according to the first criteria.

As can be seen from Fig 7, the proportion of large stations p₁≈21%, that of medium stations p₂≈24%, and that of small stations p₃≈55%. Based on the above ratio and p^x, the matrix [p^large -p₁, p^medium-p₂, p^small-p₃] is used as the train classification standard, where each item is a 0–1 variable. If the positive value is 1, otherwise 0. Since p^large+ p^medium +p^small = 100%, there is no case of [1,1,1] or [0,0,0]. The train classification determination table is shown in Table 6. Based on the decision criteria in Table 6, the different classes of trains is shown in Table 7.

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Table 6. Train classification decision table.

https://doi.org/10.1371/journal.pone.0314713.t006

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Table 7. Train classification table of Shandong circular high-speed railway.

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Different price adjustment ranges are set based on different train class. When the train class is same and the decision matrix is different, the price adjustment ranges should be slightly adjusted. Taking the Class I as an example, due to the higher proportion of medium stations, the lower limit of the price adjustment range with the judgment matrix [1,1,0] is supposed to be higher than the lower limit of the price adjustment range with the judgment matrix [1,0,0], and the increase proportion ought to be related to the price adjustment gap.

The national conditions determine that high-speed railway enterprises must assume political tasks and social missions while pursuing the maximization of profits with the goal of marketization. The fixed ticket price mechanism formed under the government guidance for a long time involves many factors in railway transportation management, resulting in the current dynamic pricing mechanism is relatively strict control. The actual maximum price fluctuation gap in the eastern developed region is about 20%.

Based on this, we set price gap to a maximum of 40% and a minimum of 20%, with the same upper and lower bounds. The overall range based on train classification is half of the price gap. According to the station decision matrix, the subdivide range is half or all of the overall range. The specific ticket price adjustment strategies of Shandong circular high-speed railway are shown in Table 8.

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Table 8. Specific ticket price adjustment strategy of shandong circular high-speed railway.

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Considering that there are differences in the output results of the heuristic algorithm each time, the large neighborhood search (LNS)-path size logit (PSL) assignment algorithm is set to cycle 100 times to 1000 times, and the average is taken as the result. In addition, the maximum number of iterations in a single loop to 20, and the convergence threshold Δc = 1‰.

6.3 Total revenue analysis

In order to verify the effectiveness of the model and algorithm under different price adjustment ranges (Series1[-10%,10%], Series2[-15%,15%], Series3[-20%,20%]), the Destroy and Repair operations ratio in the large neighborhood search is set to 50%, and the income curves under different price adjustment ranges are drawn respectively, as shown in Fig 8.

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Fig 8. Revenue comparison under different price adjustment ranges.

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In terms of the revenue curve, with the increase of the ticket price adjustment range, the absolute income and the fluctuating range increases. The reason is that the upper and lower bounds of the ticket price floating range are the same, and they are gradually increasing.

In terms of average income, compared with original price, the average income increase rate of Series1, Series2 and Series3 is 3.63%, 5.9% and 7.92% respectively.

6.4 Ticket price comparison analysis

The basis of enterprise ticketing and passenger ticket purchase is the ticket price with ODs as the unit. Select “Linyi North-Qingdao North”, “Qingdao West-Qufu East”, “Dongjiakou-Mengshan” as the key ODs of different levels.

When the price adjustment range is set to [-20%,20%], the Destroy and Repair operations ratio in the large neighborhood search is set to 40%, remove unreasonable detour trains and calculate the price fluctuation range of different trains under various ODs. The average iterated ticket prices for key ODs of different classes are shown in Table 9. The box plots are drawn, as shown in Fig 9.

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Fig 9. Ticket price fluctuation for different classes of trains under different OD classes.

https://doi.org/10.1371/journal.pone.0314713.g009

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Table 9. Average iterated ticket prices for key ODs of different classes.

https://doi.org/10.1371/journal.pone.0314713.t009

In the box plots, the data above the box indicates that the size is sorted in the first 25%, the data below the box indicates that the size is sorted in the last 25%, the data box indicates that the size is sorted between 25%-75%. The upper boundary of the box represents the upper quartile and the lower boundary represents the lower quartile. The whiskers are lines extending from the box and cover 99.3% of the data volume. The upper and lower boundaries of the whiskers represent the maximum and minimum values of the data, respectively. MEAN value is the literal meaning. Outliers represent data which exceeds the whiskers.

The ticket price fluctuation of different trains(Class I G5526 and Class III G5536) under large OD (Linyi North-Qingdao North) is calculated, as shown on the left side of Fig 9. In terms of ticket price dynamic range, the box range of G5536 is obviously narrower than that of G5526, indicating that the ticket price of G5536 is relatively stable and according with the basic situation that the ticket price fluctuation range of Class I is larger than that of Class III. In terms of MEAN value, the average ticket price of G5536 is lower than that of G5526. In terms of whiskers, G5526 has only upper whisker and G5536 has only lower whisker. The reason is that Class I only consider ticket price rate increase while Class III only consider ticket price rate decrease Table 8.

The ticket price fluctuation of different trains(Class I G5565 and Class IIG6975) under medium OD (Qingdao West-Qufu East) is calculated, as shown in the middle of Fig 9. In terms of ticket price dynamic range, the box range of G5565 is wider than that of G6975, matching the basic situation that the ticket price fluctuation range of Class I is larger than that of ClassII. In terms of MEAN value, the average ticket price of G5565 is higher than that of G6975. The reason why the MEAN value of G6975 close to the box is that some outliers are small, which lowers the average ticket price. Generaly speaking, as long as the MEAN value is within two whiskers, the result can be reasonably analyzed. In terms of whiskers, G5556 lacks lower whisker, which is consistent with the ticket price adjustment range in Table 8.

The ticket price fluctuation of different trains(ClassII G6975 and Class III G6955) under small OD (Dongjiakou-Mengshan) is calculated, as shown on the right of Fig 9. In terms of price dynamic range, the box range of G6975 and G6955 is similar. In terms of MEAN value, the average ticket price of G6975 is higher than that of G6955. In terms of whiskers, G6955 lacks upper whisker, aligning with the ticket price adjustment range in Table 8.

6.5 Algorithm core parameter sensitivity analysis

In the LNS algorithm, the destroy and repair parameters are key parameters that together define the neighborhood structure of LNS. The destroy method increases the diversity of the search space by breaking the current solution, while the repair method attempts to find a better solution within the broken solution. Generally speaking, the higher the ratio of destroy and repair operations, the more trains change price rate in each cycle.

In large-scale neighborhood search, if the parameter setting is too small, only a small part of the solutions is destroyed and repaired, then the heuristic search is difficult to search the whole neighborhood. If the parameter setting is too large, most of the solutions are broken, then the heuristic algorithm almost degenerates into repeated re-optimization.

In the scenario where the price adjustment range is set as [-20%,20%], destroy and repair parameters are set at 10% intervals. The average income value and calculation solving time curve are shown in Fig 10.

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Fig 10. Revenue comparison under different heuristic algorithm parameters.

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In the context of average revenue, a steady increase is observed as the parameter ratio escalates. With respect to calculation time, an exponential growth is evident as the parameter ratio is augmented.

Particularly, upon incrementing the parameter ratio from 50% to 60%, there is a modest acceleration in calculation time, concomitant with a marginal decline in average revenue. This observation underscores the significance of the 60% threshold as a critical reference point.

As the scale of data intensifies, it becomes imperative for railway staff to achieve a delicate equilibrium between optimizing total revenue and managing calculation time.