This paper investigates the freight empty cars allocation problem in railway networks with dynamic demands, in which the storage cost, unit transportation cost, and demand in each stage are taken into consideration. Under the constraints of capacity and demand, a stagebased optimization model for allocating freight empty cars in railway networks is formulated. The objective of this model is to minimize the total cost incurred by transferring and storing empty cars in different stages. Moreover, a genetic algorithm is designed to obtain the optimal empty cars distribution strategies in railway networks. Finally, numerical experiments are given to show the effectiveness of the proposed model and algorithm.
Freight empty cars allocation aims to distribute the available empty cars from origins to destinations in the railway networks so that the demands can be satisfied with the minimized shipment costs. Since the railway freight transportation plays an important role in modern society, reasonable allocation of the empty cars with systematic optimization is actually a crucial issue for the railway companies. In recent decades, a lot of researchers have investigated freight empty cars allocation problem and presented a variety of models and algorithms. In literature, Misra [
Besides the above mentioned researches, recent studies always formulated this problem based on the standard transportation problem and traffic flow problem. As for the transportation problem based methods, Liu [
More recently, some researchers developed network flow models and optimization methods which could be regarded as references for the freight empty car allocation problem. For instance, Wang et al. [
To understand the contributions of this research clearly, the detailed features will be summarized in Table
The detailed features of different studies.
Misra [ 
Kikuchi [ 
Lei et al. [ 
Joborn et al. [ 
This Paper  

Problem description  A simple transportation problem  A transshipment problem  A balanced transportation problem  A network flow problem  A timestage network flow problem 
Formulation  A linear programming model  An improved linear programming model  A stochastic programming model  A capacitated network design model  A dynamic network flow model 
Network  Physical network  Physical network  Physical network  Timespace network  Timespace network 
Characteristics  —  Singlestage demand  Singlestage demand  Singlestage demand  Multistage demand 
Algorithm  The simplex method  —  GA (genetic algorithm)  TS (tabu search)  GA (genetic algorithm) 
It is easy to see in the literature that the majority of researches investigate this problem in certain environments, in which all the parameters are assumed to be static. In practice, since railway traffic is a complicated system, the demands of empty freight cars are always dynamic in most circumstances. To clearly state this characteristic, in this study, we particularly treat the demand at each destination as a stagebased demand, in which one stage corresponds to a prespecified time period. Then, we formulate the problem as a mathematical optimization model with multistage demands. Also, a genetic algorithm is particularly designed for the problem to efficiently generate the approximate optimal solution of the model.
The rest of this paper is organized as follows. In Section
Allocating empty cars is an important operation for railway companies to guarantee the normal operations of transportation activity. In traditional operation modes, empty cars are in general required to be transferred to the destination according to the prespecified demand that is always treated as a fixed quantity. However, due to the uncertainty of decision environments, the demand of freight empty cars at the destination is always changeable in the realworld applications (namely, the needs vary in different time periods). Thus, the traditional model will potentially cause a large number of empty backlogs or supply shortage, leading to the transportation task being unable to be accomplished effectively. Aiming to provide a framework for practical decisions, in this research we are particularly interested in the formulation of the model with dynamic demands and effective algorithms for the freight empty cars allocation problem.
In dynamic case, the time horizon will be divided into a variety of time periods, denoted by stages. Then, some important parameters in the problem will be reconsidered as the stagebased quantities, including the demand, cost, capacity, and so forth. For different stages, all of these parameters can be different according to the practical requirements, while they are assumed to be fixed quantities within each given stage. To formulate the dynamic empty cars allocation problem, we need to specify the following three types of constraints. (1) The first constraint concerns the traffic flow volume from supply stations, which is determined by the predicted demands in organization plans and train schedules. In particular, the traffic flow volume is required to be consistent with the supply capacity of each supply station. (2) The second constraint refers to the transportation capacities of stations and railway links. The total traffic flow on these stations and links cannot exceed the corresponding capacities. (3) The last constraint is associated with the demand of traffic flow volume. That is, all the dynamic demands need to be satisfied in each stage.
To clearly state the considered problem, we shall give an illustration in Figure
An illustration of delivery plans.
Hereinafter, some assumptions will be given in order to formulate the mathematical model.
In order to characterize the considered railway network, an abstract graph, denoted by
The demand, storage cost, and unit transportation cost in the considered time horizon are all dynamic, while these parameters in each stage are treated as constants.
In the railway network, the total supply capacity in original stations is supposed to be greater than the total demand in destinations to guarantee the transportation activities.
The demand in each stage should be satisfied. Unused empty cars can be left for the next stage with an extra storage cost.
An illustration of a simple transportation network.
Here, to explicitly demonstrate the empty cars allocation process, an illustrative network, which consists of three node and three links, is given in Figure
An illustration of a spacetime network.
In this section, we shall formulate the freight empty cars allocation problem with dynamic demands as a stagebased network flow problem below.
To formulate the mathematical model for this problem, some relevant parameters are firstly listed in Table
Parameters used in formulation.
Parameters  Definition 


The set of links; 

The set of nodes; 

The index of nodes; 

The index of link from node 

The total number of links; 

The total number of nodes; 

The different stages, 

The unit transportation cost on link 

Origin station which supplies empty cars, 

The supply amount of empty cars in station 

Destination station which requires empty trains, 

The total demand of empty cars in station 

The link capacity on link 

The turnover capacity at station 

The unit storage cost for stage 

The demand of empty cars at station 
In this problem, the purpose is to generate an optimal transportation plan over the entire time horizon. As the multistage strategy will be considered, the decision variable can be treated as the amount of empty car flow corresponding to each link and stage. Then, we have the decision variable as
With this variable, the freight empty car allocation problem can be essentially treated as a traffic flow assignment process with the following constraint conditions.
To guarantee the feasibility of the allocation plans, we need to consider some system constraints in the process of allocating empty cars. Specifically, there are six types of system constraints in this model, which are presented as follows.
This problem aims to find the optimal allocation strategies with the minimized cost. As addressed in the illustration, the total cost in essence consists of two parts, namely, delivery cost in the network and storage cost in destinations.
The delivery cost refers to the expenses in the process of delivering empty cars. Since in stage
In this problem, the freight empty cars will be delivered in different stages, which produce the following two situations. (1) One is that the traffic flow volume just meets the demand. That is, the delivered total empty cars are equal to the number of cars required in this stage. Then, no storage cost will be produced in this case. (2) The other one is that the number of empty cars is larger than that required in this stage. For this case, the redundant cars will be used in the next stage, resulting in the extra storage cost, which can be formulated as
Consider the objective and constraints, the mathematical model of this problem can be formulated as follows:
In this model, the objective is to minimize the total cost in the railway transportation system. The first two constraints guarantee the rationality of the transportation activities. The third and fourth constraints ensure that the amount of traffic flow volume cannot exceed the capacities of each link and each node, respectively. The fifth constraint ensures that the stagebased demand can be satisfied with the demand constraints.
To discuss the complexity of the proposed formulation, the following discussion aims to analyze the characteristics of the proposed model. We firstly focus on the number of constraints in the model, which are displayed in Table
The amount of constraints in each constraint condition.
Constraints  Number 

( 

( 

( 

( 

( 



Total number 

An illustration will be given in the following to demonstrate the complexity of the proposed model. Consider a network with 20 nodes and 50 links. In this network, suppose that there are three origin stations and three destination stations. A total of three stages will be taken into consideration. With the assumption above, since a total of 9 paths need to be considered, the number of decision variables should be 150 over the entire network. Moreover, a total of 118 constraints need to be included in the model.
Note that when the network scale increases, the number of constraints and variables will increase to a great extent accordingly, which potentially leads to the difficulties in calculation. Then, in order to simplify the computational complexity in the process of allocating cars, in the next section, we shall transform the arcbased solutions into routebased solutions, in which constraint (
In practice, it is rather difficult to solve the largescale freight empty cars allocation problem efficiently. Because of this, we aim to design an approximation algorithm in this section, seeking a feasible solution which can replace the best solution approximatively.
The approximation algorithm is also called heuristic algorithms, which include tabu search algorithm, simulated annealing algorithm, genetic algorithm, neural network algorithm, and ant colony algorithm. As the genetic algorithm is a kind of direct searching method which does not depend on the specific characteristics of problems, it is efficient and effective in solving a variety of realworld problems. The principle of genetic algorithm is to simulate the mechanism of the living beings evolving and natural choosing. It was well developed by many researchers. Up to now, the genetic algorithm has been successfully used to solve practical optimization problems, such as transportation problem (Gen et al. [
In general, the procedure of genetic algorithm is as follows: (1) randomly generate a certain number of chromosomes; (2) evaluate the quality of those chromosomes with fitness function according to some criteria; (3) obtain fine chromosomes via selection, crossover, and mutation operations. Figure
The procedure of the genetic algorithm.
In solving the empty car allocation problem, we need firstly to determine the solution representation in the genetic algorithm. As described in the model, the solution is represented as linkbased flow in each link. To simplify the complexity of computation, we here particularly introduce the pathbased solution representation in the genetic algorithm. Since linkbased flow can be finally split into the flow on different paths between different OD pairs (here, “OD pair” is an abbreviation for the origin and destination terminals), in this method, potential paths should be determined firstly, and then we determine the flow volume on each path to satisfy the demand of each destination.
An illustration is given to show the difference between linkbased solution and pathbased solution, depicted in Figure
An illustration of a simple transportation network.
In Figure
In this procedure, a path, denoted by
In the genetic algorithm, each population consists of
In the genetic algorithm, the role of selection operation is to produce a new population for the following crossover and mutation operations. In the initial population, the chromosomes are first ranked from the good to the bad according to their objectives. Without loss of generality, the rearranged chromosomes are denoted by
Here, we give the detailed procedure to show the selection process below.
Crossover operations aim to generate a new population in the searching process. Before this operation, the parents should be specified according to the predetermined crossover probability
After the parent chromosomes are selected, any two chromosomes can be used for crossover operation. Suppose that the two parent chromosomes are
In order to avoid premature convergence, the mutation operation is necessary in the searching process of genetic algorithm. Like the crossover operation, we firstly need to determine the chromosomes for mutation operations. The number of chromosomes selected to perform mutation operation is also completely based on the predetermined mutation probability
For each selected parent chromosome
For the completeness of this paper, we shall give the detailed procedure of the algorithm in the following.
In this section, we give two numerical examples with different scales to illustrate the effectiveness of the proposed methods, in which the experiments are implemented by using C++ software in a personal computer with Intel(R) Core(TM) i53317U 1.70 GHz.
In the first set of experiments, we consider a smallscale network, shown in Figure
In this network, we consider twostage based demands. The detailed data are given as link
Besides, it is possible that the storage cost can occur in the allocation plan. Then, the unit storage cost for different stages are given in Table
Storage cost and demand in destination stations.
Station  Stage 1  Stage 2  

Storage cost  Demand  Storage cost  Demand  
4  2  30  1  50 
5  1  40  2  30 
It is easy to see in this problem that there are a total of 8 variables and 13 constraints in the allocation process. As we formulate this problem as a linear programming model, we firstly use the Lingo software to calculate this problem, where the optimal objective turns out to be 830 and the optimal solution is given in Table
The best solution solved by Lingo software.
Variable 










Value  65  25  55  5  80  0  40  30 
In the following, we shall investigate the performance of the genetic algorithm. To this end, we particularly list the best solutions encountered during the first 16 generations to analyze the convergence of the algorithm, shown in Table
The best solutions and objective values in different generations.
Generation 








Value 

1  44  56  44  8  47  33  41  31  883 
2  40  58  45  8  45  35  40  31  877 
3  40  58  45  7  45  35  40  30  873 
4  42  54  45  9  47  33  40  30  869 
5  52  42  39  18  51  29  40  29  867 
6  42  52  46  10  48  32  40  30  866 
7  49  49  46  6  55  25  40  30  863 
8  52  45  53  0  63  17  42  28  858 
9  52  44  54  0  64  16  42  28  856 
10  50  44  56  0  65  15  41  29  851 
11  52  40  56  2  66  14  42  28  850 
12  52  40  57  1  67  13  42  28  849 
13  47  44  58  1  65  15  40  30  846 
14  46  44  59  1  65  15  40  30  845 
15  47  43  60  0  65  15  40  30  843 
16  48  42  60  0  65  15  40  30  842 
In Table
The variation tendency of
It follows from Figure
The variation tendency of objective function.
In the following, we propose a computational experiment derived from reallife instances of the railway network in Beijing, Hebei, and Shanxi province of China. In this instance, a total of 103 stations are located on the network. In particular, we omit some intermediate points for displaying convenience in Figure
The considered paths and unit transportation cost on arcs.
Number  OD  Paths  Unit cost on arcs 

1 



2 



3 



4 



5 



6 



7 



8 



9 



10 



11 



The network of the reallife railway network.
An illustration of a path from station 1 to station 12.
In addition, we assume that supply capacities at stations 1 and 2 are 275 and 240, respectively. The empty cars allocating process are divided into three stages, and stagebased demands and storage costs are listed in Table
The demand and unit storage cost in different stages.
Stage  Time period  Node 12  Node 13  Node 14  Storage cost 

1  8:00~16:00  75  50  40  7 
2  16:00~24:00  90  85  50  4 
3  24:00~8:00  35  20  30  1 
This set of experiment is implemented by genetic algorithm in C++ software on a personal computer. To test the robustness of the proposed algorithm, we perform ten experiments with different parameters in the genetic algorithm. The results are given in Table
The comparison of the optimal objectives.
Pop_size 


Generation  Objective value  Error (%)  

1  10  0.4  0.4  1000  5062  1.52 
2  20  0.4  0.4  1000  5036  1.01 
3  20  0.6  0.4  1000  5036  1.01 
4  20  0.6  0.4  5000  5022  0.74 
5  30  0.6  0.4  1000  5010  0.50 
6  20  0.6  0.8  1000  5026  0.82 
7  20  0.6  0.8  5000  4994  0.18 
8  30  0.8  0.4  1000  5010  0.50 
9  30  0.6  0.8  2000  4985  0 
10  30  0.6  0.8  5000  4985  0 


Average value  0.628  
Variance  0.0022 
As shown in Table
To analyze the searching process of the proposed algorithm, we especially consider the computational results with parameters
The comparison of the optimal objectives for different scales of population.
It is easy to see from this figure that, with different scales of population, the approximate optimal solution can be quickly achieved almost within 4000 generations. As expected, when we set a larger population, the optimal solution can be obtained with less generation in the searching process.
Finally, we give a comparison with the singlestage model to further show the effectiveness of the proposed approaches in this study. Specifically, the sum of the stagebased demands given in Table
This paper investigated a railway freight empty car allocation problem in the dynamic decisionmaking environment, in which the stagebased demands are considered, and the cost is divided into transfer cost and storage cost at destinations. To characterize this problem mathematically, an integer linear programming model was formulated with the minimization of total involved cost based on the network flow optimization. To generate approximate optimal solutions, a genetic algorithm with pathbased solution representation is designed to seek the optimal empty car distribution strategies in railway networks. The numerical experiments are executed to show the performance of the proposed approaches. In particular, compared to the singlestage model, the total transportation cost in our model can be reduced by almost 30%, which implies the effectiveness of the proposed approaches.
The future research can be focused on the following two aspects. (1) The model can be generalized to the uncertainty environments within the framework of uncertain programming methods, since the reallife decision systems for railway operations are essentially in the state of uncertainty (see Yang et al. [
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was supported by the National Natural Science Foundation of China (no. 71271020), Research Foundation of State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University (no. RCS2014ZT02), the National Basic Research Program of China (no. 2012CB725400), and the Beijing Natural Science Foundation (no. 9144031).