Synthetic Optimization Model and Algorithm for Railway Freight Center Station Location and Wagon Flow Organization Problem

The railway freight center stations location and wagon flow organization in railway transport are interconnected, and each of them is complicated in a large-scale rail network. In this paper, a two-stage method is proposed to optimize railway freight center stations location and wagon flow organization together. The location model is present with the objective to minimize the operation cost and fixed construction cost. Then, the second model of wagon flow organization is proposed to decide the optimal train service between different freight center stations. The location of the stations is the output of the first model. A heuristic algorithm that combined tabu search (TS) with adaptive clonal selection algorithm (ACSA) is proposed to solve those two models. The numerical results show the proposed solution method is effective.


Introduction
In the past decade, China railway has invested many railway freight center stations, which are equipped comprehensive transportation facilities and logistics facilities for the purpose of centralized transportation.At the same time, most of the stations with small transport demand were closed.Those center stations can help to centralize transport demand and gain economic of scale, while the transport demand of closed stations may be lost, if the transport cost of the railway is higher than the other transport modes such as road and water transport.In order to attract the transport demand, railway must improve the wagon flow organization to provide service with high level.
The center station location and wagon flow organization are interacted and interconnected.In order to organize products with high level of service, the amount of goods must meet the train size limitation.Otherwise, long waiting time may be caused.At the same time, the efficiency of wagon flow organization can decide the operation efficiency of stations.For example, the limited storage capacity of station will be occupied, if there is no suitable train to serve the demand.
In order to solve the complicated station location and wagon flow organization problem, a two-stage method is proposed to optimize station location and wagon flow organization together.This paper is organized as follows.Section 2 briefly reviews the relevant literature.Section 3 introduces the optimal models in a two-stage method.In Section 4, a heuristic algorithm is proposed to solve the models.Finally, a numerical example is provided to illustrate the application of the models and algorithm.

Literature Review
The location of railway freight center stations is similar to hub location problem, which is among the most critical management decisions.Many models have been proposed such as set covering problem (SCP), p-center problem, and p-median problem [1][2][3][4][5].
The SCP model was presented by Caprara et al. [5] and solved by a heuristic method based on Lagrangian relaxation.Campbell [6] came up with formulations for the multiple and single allocation p-hub median problem with two heuristic methods.Besides, Campbell et al. [7] concluded that there are two types of hub location problem: single allocation and multiple allocation.Skorin-Kapov et al. [8] studied the uncapacitated multiple and single allocation phub median problems.O'Kelly [9] analyzed the models for both single-hub and two-hub problems.And O'Kelly [10] presented a quadratic integer program for the hub location.The model is linearized and solved by a heuristic algorithm.Furthermore, O'Kelly and Bryan [11] proposed a reliable model by considering the scale economies effect of traffic.Racunica and Wynter [12] studied the location problem in a hub-and-spoke network, aiming to increase the share of rail in intermodal transport.
Recently, fuzzy theory and dynamic environment were introduced into the location problem.Batanović et al. 's study [13] concerned the uncertain parameters in maximum covering location problem, which were modeled by fuzzy set.Sáez-Aguado and Trandafir [14] improved the p-median model by considering a coverage constraint.A dynamic uncapacitated hub location problem was present in Contreras et al. 's paper [15].Correia et al. [16] proposed four mixed-integer linear programming formulations in order to study the extension of classical capacitated single-allocation hub location problem.
For the wagon flow organization problem, Bodin et al. [17] developed a nonlinear, mixed-integer programming model for the railroad blocking problem.Martinelli and Teng [18] applied neural network to solve the railway operation problem.Some research described the railway blocking problem with network theory.Newton et al. [19] took the railroad blocking problem as a network design problem and developed a column generation and branch-and-bound algorithm to solve it.Similarly, Barnhart et al. [20] proposed a Lagrangian relaxation approach for optimizing railway blocking problem.Fukasawa et al. [21] considered both the loaded and empty cars in the network and proposed an integer multicommodity flow model for the problem.Woxenius [22] provided six principles of rail operation and applied them into intermodal rail freight transport network system.Jeong et al. [23] formulated a linear integer programming model in a hub-and-spoke network.Jha et al. [24] compared arcbased with path-based formulations of the block-to-train assignment problem and proved that the latter is smaller in scale.Keaton [25] presented a mixed-integer programming for railroad blocking problem.
In China, Wu [26] studied the organization scheme for through trains.Lin et al. [27] formulated a nonlinear 0-1 through train formation model considering the different cost parameters and block capacity constraint.Nonlinear 0-1 programming models were established in Cao et al. 's study [28,29] to determine the freight train scheduling plan based on analyzing the logistics system costs.
Most of the research in literature focused either on the location of stations or the wagon flow organization, while the influence of the location on wagon flow organization should be considered.In this paper, a two-stage method is used to solve this problem.

Model of Optimal Railway Freight Center Station Location
Problem.As the location of railway freight center stations is similar to the hub location problem, customer specifies the ordinary railway freight station, while service point specifies the candidate railway freight center station.

Decision Variables.
The objective of this model is to find the optimal location of service points and the assignment between customer and service point.The location decision and assignment are treated as decision variables.Those are   equals 1 if customer  is assigned to service point .Otherwise, it equals 0.
equals 1 if service point is located at candidate service point .Otherwise, it equals 0.

Objective Function.
Customers hope that the service points are located close to themselves, so as to send the goods to the station quickly and cheaply, while the planners want to maximize the railway coverage with limited center stations, so as to reduce the total investment.Both the transport cost from costumers to service points and construction costs of stations are considered by the following objective function: where  is unit cost to transport the goods from customer to service point;  is set of candidate service points,  ∈ .

Constraints.
The coverage constraint of a service point is considered.The whole distance which is greater than the preestablished coverage distance at a service point should not exceed a previously chosen value.This constraint is related to the maximum covering location problem (MCLP), whereas the investment of service point may change the transport demand.The model also takes this situation into account.
(i) Each customer must be assigned to one service point (ii) Candidate service point  cannot serve any customer, if  is not chosen as a service point (iii) The total number of chosen service points should be constrained is the maximum number of the stations.
(iv) Sum of the distance which is greater than coverage distance DC at a service point should not exceed .
Both DC and  are prespecified is defined as follows: (v) The goods serviced by point  cannot exceed its capacity Cap  .
is defined as follows: where  is a discount coefficient of demand at point  if  is not a service point, which describes the change of transport demand.  is the expected transport demand at point .

Model of Wagon Flow Organization Problem.
The outputs of M-I include the location of railway freight center stations and the transport demand of each station.Based on the average loaded weight of a wagon, the transport demand can be turned into wagon flows.Those are the input data of wagon flow organization.

An Example of Wagon Flow Organization Problem.
The wagon flow organization problem can be illustrated by a simple network (see Figure 1), which has three technical stations, two freight center stations, and three shipments ( 12 ,  14 , and  15 ).There are 12 combinations for routing all the shipments on potential train services (see Figure 2).Arcs in the network specify the available train connections.The OD pair will be reclassified at the technical stations in the itinerary if it is not served by through train.If transport demand between two adjacent stations is positive, the train service must be provided.So shipment  12 has only one service strategy that is served by the train service 1 → 2.
Strategy 2 (see Figure 2(b)).There are two through train services 1 → 4 and 1 → 5.In this case, both  15 and  14 are directly shipped to the destination.Strategy 3 (see Figure 2(c)).There are two through train services 1 → 3 and 1 → 4. In this case,  14 is directly shipped to the destination.While  15 is directly shipped to technical station 3, and  15 still needs to be reclassified at technical stations 3 and 4.
Strategy 4 (see Figure 2(d 2(i)).There is one through train service 1 → 3.In this case,  15 and  14 are directly consolidated together and shipped to the technical station 3 by train service 1 → 3.  14 is reclassified at technical station 3.  15 is reclassified at technical station 3 and 4.
Strategy 10 (see Figure 2(j)).There is one through train service 1 → 4. In this case,  15 and  14 are consolidated together and directly shipped to the technical station 4 by train service 1 → 4.  15 is reclassified there.
Strategy 11 (see Figure 2(k)).There are two through train services 1 → 3 and 1 → 5.In this case,  15 is directly shipped to destination by train service 1 → 5.  14 is directly shipped to the technical station 3 by train service 1 → 3 and reclassified there.
Strategy 12 (see Figure 2(l)).There is one through train service 1 → 5.In this case,  15 is directly shipped to destination by train service 1 → 5.  14 and  12 are consolidated in train service 1 → 2.  14 is reclassified at technical station 2 and 3.

Analysis of Wagon Flow Organization
Problem.Among the twelve strategies above, there are two extreme ones, that is, Strategy 1 and Strategy 2.
(1) In Strategy 1, train connection services are only provided between adjacent stations.In this way, the number of through train services reaches minimum; that is, the total waiting time at the dispatching station reaches minimum.However, each shipment will be reclassified at each technical station on its itinerary.Thus the reclassification fee and time consumption will increase significantly.The increasing number of reclassified wagons may cause congestion and time delay at some technical stations.
(2) In Strategy 2, transport demands are all delivered to their destinations without reclassification, which will reduce the reclassification fee and time consumption.However, this strategy may be unworkable.Such strategy requires enough classification tracks to store outbound trains, and single OD pair must be large enough to dispatch a through train service.Waiting time at the dispatching station will increase if the OD pair is small.
The solution of wagon flow organization problem is a tradeoff between Strategy 1 and Strategy 2, aiming at minimizing the total shipping and handling cost for all shipments.The time consumption is used to weigh the cost.

Decision Variables.
From the example above, it can be concluded that there are three train services for a wagon flow: the first one is shipped to the destination by nonstop shipping scheme at the origin station, the second one is served by nonthrough shipment, and the third one is shipped to a technical station in the itinerary by nonstop shipping scheme consolidated with other OD pairs.These variables are 1   is shipped by nonstop scheme 0 otherwise

Objective Function.
The optimal goal is to obtain the wagon flow organization plan with minimum time consumption, which means the level of service.The travel time is not considered for it is identical.In order to simplify the problem, the following assumptions are proposed: (1) the supply of empty wagons is not considered, (2) the storage capacity of the origin and destination station are not considered, (3) the shipping routes for all the service are given, and (4) the train formation plans (TFP) of the technical stations in the itinerary are known.This information can be obtained previously.
Time consumption of three train services for   is as follows.
(i) For the nonstop service, the service time equals the loading and unloading times As there is no intermediate service such as reclassification where   is loading and unloading time at the origin and destination for a wagon served by nonstop shipping scheme.(ii) Nonthrough shipment cost, which includes the reclassified time in the itinerary and loading and unloading time where   is loading and unloading time at the origin and destination for a wagon shipped by non-through shipment;   is reclassified time at technical station ; (, ) is set of technical stations in the itinerary after   has been reclassified at technical station , and  ∈ (, ).
(iii) The wagon flow is shipped to a technical station in the itinerary by nonstop shipping scheme.This item includes the reclassified time consumption in the itinerary and time consumption for loading and unloading where   is loading and unloading time at the origin and destination for a wagon shipped to a technical station in the itinerary by nonstop shipping scheme.
Then, the total time consumption for   can be formulated as The objective function of wagon flow organization can be stated as min where  is set of origin stations and  ∈ .() is set of destinations of the wagon flows that originate from .And () = { :  st ̸ = 0}. ∈  ().

Constraints. The wagon flow organization problem has two constrains.
(1) Loading and Unloading Capacity Constraint for Nonstop Train.The loading and unloading capacity must reach the minimum size of loading or unloading a whole nonstop train.If there are several wagon flows consolidated together to be shipped to a technical station in the itinerary by nonstop shipping scheme, the sum of their loading or unloading capacity must reach the average marshaling number of wagons in a nonstop shipping scheme where   is the smaller one between maximum loading capacity of station  and maximum unloading capacity of station  at certain period; is the average marshaling number of wagons in nonstop shipping scheme; (2) Each Wagon Flow Is Served by One and Only One Train Service.

Constraint Linearization. Constraint (16) can be linearized by introducing a variable 𝑦
Then constraint ( 16) can be formulated as follows: Constraint ( 20) is also introduced to ensure that   equals 1 if and only if where  is a large positive constant.

Solution Algorithm
Tabu search (TS) [30,31] uses the tabu list to record the information of local optimal solutions, which can help to enlarge the search space and avoid the locally optimal solutions.The adaptive clonal selection algorithm (ACSA) [32][33][34] is shown to be an evolutionary strategy capable of high convergence rate and diversified.In this section, TS and ACSA are combined into a new heuristics method, called T-ACSA, to solve the proposed models.

The Detail Techniques of T-ACSA
4.1.1.Affinity Measure.Affinity measure of the algorithm is the objective of the model; the smaller the better.In order to extend the space of solution, the algorithm accepts solutions which fail to satisfy the constraints.However, penalty coefficient will be added to the affinity measure.

The Design of Antibody. The antibody of model M-I is designed as follows.
The length of antibody is equal to the amount of customers in .The codes in antibody are in , and the amount of codes is .To better understand the antibody, a simple example consisting of seven customers and four service points is proposed. equals three (see Figure 3).Service point 2 is not included in the antibody, which means that it is not chosen as a service point.
The antibody of model M-II is designed as follows.
According to the optimal result of model M-I, the antibody of each service point can be designed.A piece of antibody is proposed (see Figure 4).
The number of    in each piece is decided by the amount of technical stations in () in the itinerary  → , which can be obtained when the route is set.The sum value of each piece is 1.The pieces are separated between each other.And their amount equals the number of wagon flows.The neighborhood search operation of model M-II is as follows: choose a piece in antibody randomly.Find a code whose value is 0 and assign it to 1. Change the other codes into 0. The other pieces of antibody remain the same.The neighborhood search operation of the antibody in Figure 4 is shown in Figure 6    four.If the amount of the chosen candidate points reaches the maximum, choose a code  in the antibody randomly.Change both  and the codes whose value is the same as  (see Figure 7 Step 1. Choose an antibody and set it as the current solution.Initialize the tabu list and the best solution.
Step 2. Use the neighborhood search operation to update the current solution and generate  candidate solutions.Then sort them by the affinity measure values.Choose the best solution.
Step 3. Judge whether the best solution is tabued.If it is tabued, go to Step 2. Else judge whether the length of tabu list reaches .If not, add the best solution into the tabu list and tabu it.Else remove the tabu information of the first solution Step 4. Judge whether the current best solution is the best one in the history.If not, go to Step 5. Else set it as the best one in the history.Go to Step 5.
Step 5. Judge whether the search times reach .If not, go to Step 2. Else update the antibody with the best solution in the history.

The Process of T-ACSA.
Based on the aforementioned detailed analysis, T-ACSA approach is designed as follows.
Step 1. Initialize the population of antibody.Generate  antibodies and constitute the species group .
Step 2. Count the affinities and sort the antibodies according to their affinities in an ascending order.
Step Where  is the coefficient,  is the average value of affinities in   and  min is the minimum value of affinities in   .
Step 7. If current status does not meet the terminal condition (the maximum searching times), go to Step 2. Otherwise, go to Step 8.
Step 8. Output the best solution, that is, the optimal location of service points or the wagon flow organization.

Numerical Experiments
The models and algorithm are tested by a railway network with 48 railway freight stations.Station 1 to station 16 is candidate freight center stations.The parameters of this physical network are provided in Tables 7 and 8, which include the distance information and the transport demand of each customer.Table 1 shows the investment to locate a center station at the candidate sites.Parameters of M-I model and T-ACSA are shown in Table 2.
For the second step, we can plan wagon flow organization with outputs of model M-I, such as the location of freight center stations, the transport volume between each center station.Assuming the average loaded weight of a rail wagon is 50 t, and the stations operate for 365 days per year.Then the total transport demand can converted into daily transport volume.The average volumes of railway freight center station 3, 5, 6, 13, and 14 are 95, 97, 84, 87, and 75, respectively.The railway network and the location of railway freight center stations are shown in Figure 9.The reclassified time  at technical stations is listed in Table 3.The wagon flows between railway freight center stations are shown in Table 4.
And the parameters of model M-II and T-ACSA are shown in Table 5.
We test the algorithm five times by different combination of parameters.The final result of wagon flow organization is shown in Table 6.The iterative process of the algorithm is shown in Figure 10.
To verify the algorithm, we solved the models M-I and M-II by ILOG Cplex at the same time.The final results of the present algorithms and Cplex are the same.The run time of models M-I and M-II is less than 2 s and 1 s.

Conclusion
A two-stage programming is proposed to describe and solve the location of railway freight center stations and wagon flow organization problem.The first stage determines the optimal location with the objective to minimize the total cost of service and investment.The second stage optimizes the wagon flow organization among different stations.Different from the research in literature, the first model considered the coverage distance constraint and the change of transport demand.The second model analyzed the cost of different schemes.A heuristic algorithm that combines TS with ACSA is designed.The numerical example of a network with 48 stations demonstrates that the method is workable.
While the microoperations in wagon flow organization, like wagon flow route decision and the supply of empty wagons, have not been considered in the scheduling process, these aspects can be considered in the future research.

Figure 2 :
Figure 2: Twelve combinations for routing all the shipments on potential train services.

1 𝑁
shipped to first technical station  in the itinerary and reclassified  is shipped to tcehnical station  and reclassified there 0 otherwise ∀ ∈  () , (10) where   is number of wagon flow from origin  to destination ;  is origin station;  is destination station; () is set of technical stations next to the origin station ,  ∈ (); () is set of technical stations in the itinerary of wagon flow   ; exclude the technical station next to the origin station, () ∩ () = 0,  ∈ ().

Figure 7 :
Figure 7: The mutation operation of model M-I.

Figure 8 :Figure 9 :
Figure 8: The iterative process of T-ACSA for M-I. .

4. 1 . 4 .
Mutation Operation.Mutation operation of model M-I is shown in Figure7: the maximum number of codes is (a)).Else choose a code  and change its value randomly (see Figure7(b)).The mutation operation of model M-II is the same as the neighborhood search operation.4.1.5.TS Operation.The tabu strategy is first in first out.The length of tabu list is ; the search times are .The number of the candidate solutions is .TS operation procedure is as follows.

Figure 10 :
Figure 10: The iterative process of T-ACSA for M-II.
)). Strategy 3 and Strategy 4 share the common train service set, but the train services are different.15isdirectly shipped to technical station 4 and reclassified there.14 is directly shipped to technical station 3 and reclassified there.Strategy 5 (see Figure2(e)).There is one through train service 1 → 4. In this case,  15 is directly shipped to the technical station 4 and reclassified there. 14 and  12 are consolidated in train service 1 → 2.  14 is reclassified at technical stations 2 and 3. Strategy 6 (see Figure 2(f)).Strategy 5 and Strategy 6 share the common train service set, but the train services are different. 14 is directly shipped to technical station 4 by train service 1 → 4.  15 and  12 are consolidated in train service 1 → 2.  15 is reclassified at technical stations 2, 3, and 4. Strategy 7 (see Figure 2(g)).There is one through train service 1 → 3.In this case,  15 is directly shipped to the technical station 3 and reclassified at technical stations 3 and 4.  14 and  12 are consolidated in train service 1 → 2.  14 is reclassified at technical stations 2 and 3. Strategy 8 (see Figure 2(h)).Strategy 7 and Strategy 8 share the common train service set, but the train services are different. 14 is directly shipped to technical station 3 by train service 1 → 3 and reclassified there. 15 and  12 are consolidated in train service 1 → 2.  15 is reclassified at technical stations 2, 3, and 4.

Table 1 :
The investment for each candidate station.

Table 2 :
The parameters of model M-I and T-ACSA.

Table 3 :
The reclassified time consumption at a technical station.

Table 4 :
The wagon flows between railway freight center stations.

Table 5 :
The parameters of model M-II and T-ACSA.

Table 6 :
The optimal result of wagon flow organization.
3. Clone each antibody in  then get a new species group .The number of clone is   = [ max (1−((−1)/))] and   ≥  min , where  is the order of the antibody after sorting. max is the maximum clone number and  min is the minimum clone number.Apply the TS operation to update each antibody in  and get the new species group   .Use mutation operation to update each antibody in   .And get the new species group   .The probability of mutation is inversely proportional to the evolution generation = [(1 − (/))].Where  is the coefficient,  is the current generation and  is the maximum generation.Step 6. Choose the first   antibodies in   and replace the worst   antibodies in  by them;   = [( −  min )(/)].

Table 8 :
The transport demand of each customer.