# Study on the Stochastic Chance-Constrained Fuzzy Programming Model and Algorithm for Wagon Flow Scheduling in Railway Bureau

Authors: Bin Liu
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The wagon flow scheduling plays a very important role in transportation activities in railway bureau. However, it is difficult to implement in the actual decision-making process of wagon flow scheduling that compiled under certain environment, because of the interferences of uncertain information, such as train arrival time, train classify time, train assemble time, and flexible train-size limitation. Based on existing research results, considering the stochasticity of all kinds of train operation time and fuzziness of train-size limitation of the departure train, aimed at maximizing the satisfaction of departure train-size limitation and minimizing the wagon residence time at railway station, a stochastic chance-constrained fuzzy multiobjective model for flexible wagon flow scheduling problem is established in this paper. Moreover, a hybrid intelligent algorithm based on ant colony optimization (ACO) and genetic algorithm (GA) is also provided to solve this model. Finally, the rationality and effectiveness of the model and algorithm are verified through a numerical example, and the results prove that the accuracy of the train work plan could be improved by the model and algorithm; consequently, it has a good robustness and operability.


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## Body

## 1. Introduction

The train work plan is the core of the daily work plan and the data hub of types of scheduling work in railway bureau. It plays an important role as the whole link between Railway Ministry of China and railway stations and depots. The main purpose of train work plan is to allocate wagons to departure trains. And the wagon flow has to match the time limitation of wagon operations on marshalling stations and the wagon flow direction and train-size limitation. Besides, the train work plan has to meet the demand of wagon loading plan and empty wagon reposition plan. All these train operations could be classified aswagon flow scheduling problem. The optimization objective of wagon flow scheduling is to accelerate the rolling stock turnover, to reduce the wagon residence time at railway stations, and to reduce the empty wagon running distance.For railway wagon flow scheduling problem, many scholars have done a lot of beneficial researches, especially for wagon flow scheduling in railway marshalling stations. Gulbroden studied the railway scheduling in marshalling station by using operations research [1]. Yager and his partner developed an efficient sequencing model for humping in railway marshalling stations [2].  Carey and Carville  developed scheduling heuristics analogous to those successfully adopted by train planners using “manual” methods [3].  Lentink et al.  discussed how to use network flow method to establish mathematical model to solve train scheduling problem [4]. The robust optimization in railway transportation is discussed by  Marton et al. in 2007 [5].In China, many railway transportation organization methods are fundamentally different from other countries. Wang presented a concept of “price” and used some techniques to transform the wagon flow allocating problem into a transportation problem model in operations research; the objective of the model is to minimize the total price so that the satisfactory solution can be attained by using the calculating method on table [6]. It is worthy of mentioning that the literature [7] is one of the most important literatures in this research field, and many successive studies derive from this.  He et al.  developed a fuzzy dispatching model for wagon flow scheduling in railway marshalling station and designed a genetic algorithm to obtain the satisfactory solution [8]. And  He et al.  developed an integrated dispatching model for railway station operations and a computer-aided decision support system [9].  Liu et al.  developed a chance-constrained programming model which aimed to reduce the residence time of wagons in the marshalling station and the average delay time of departure trains and designed an improved genetic algorithm to solve the problem [10].  Li et al.  addressed the problem of optimizing the marshalling station stage plan with the random train arrival time and developed a dependent-chance programming model and designed a hybrid intelligent algorithm based on stochastic simulation and tabu search [11]. And  Li et al.  put forwarded a brief survey of stage plans under certain and uncertain environments and with computer-aided dispatching methods and systems. He pointed out the existing and unresolved problems in application of the current theories and methods. What is more, he investigated the direction of future research of railway marshalling stations stage plan [12].In recent years, with the wide use of Train Dispatch and Management System (TDMS) and Synthetically Automatic Marshalling (SAM) station system in China, the wagon flow information between railway bureau and railway stations are shared completely, so we put forward a new transportation organization concept that is “integralization of railway bureau and railway stations.” Thus the wagon flow scheduling in railway bureau can replace the railway station wagon flow scheduling in great extent, and the accuracy of the wagon flow scheduling plan and the overall transportation organization efficiency can be improved dramatically.Based on above literatures and practical situation in China, considering all kinds of uncertain factors in wagon flow scheduling, such as the stochastic train arrival time and fuzzy train-size limitation of the departure trains, a stochastic chance-constrained fuzzy multiobjective model for the flexible wagon flow scheduling problem is set up. And a hybrid intelligent algorithm based on ant colony optimization (ACO) and genetic algorithm (GA) is also given in this paper.


## 2. Stochastic Chance-Constrained Fuzzy Programming Model

In this section, we aim at maximizing the satisfaction of departure train-size limitation and minimizing the wagon residence time at railway station to establish the multiobjective optimization model for wagon flow scheduling problem in railway bureau. First, we analyze and formulate the constraints of wagon flow scheduling by considering the some uncertain factor. Then we will summarize and formulate the objective function.


### 2.1. Start Time Constraints to Classify a Train

The train arrival time is stochastic. The lag between train actual and planed arrival time is a random variable with normal distribution, denoted byεi. Let Ti′′ be the planed arrival time of train i, so the actual arrival time of train i is Ti′=Ti′′+εi, Ti the earliest time after the inspection of train i,  Ti=Ti′+ai, where ai is the inspection time of train i. Assume that a train classifying process is in a time segment k. Let tk be the start time to classify a train in the kth time segment, Jki a boolean variable whose value is 1 if train i is classified in the kth time segment, otherwise the value is 0; let  n  be the total number of arriving trains in the stage [8]. So the start time constraints to classify a train are as follows:
(2.1)Pr(tk-∑i=1nTiJki≥0)≥α1,k=1,2,…,n.Expression (2.1) denotes that the start time to classify train i in the kth time segment must be after the end time of inspection train  i. Because of the stochasticity of train arrival time, the end time of train inspection is also stochastic. The expression (2.1) is chance constrained, and the probability of expression (2.1)’s holding is more than or equal to α1, where α1 is the given confidence level.


### 2.2. Start Time Constraints to Assemble a Train

Assume that a train assemble process is in a time segmentk′. Let bj be the process time of assemble train j; it is a random variable with normal distribution. Let n′ be the total number of departure trains in the stage.Lettk′′ be the start time to assemble a departure train in the kth time segment, Pikjk′ be a boolean variable whose value is 1 if train i which is classified in the kth time segment delivers wagons to departure train j which is assembled in the k′th time segment, otherwise the value is 0 [7]. So the start time constraints to assemble a departure train are as follows:
(2.2)Pr(tk+∑i=1njiJki-tk′′≤M(1-∑i=1n∑j=1n′Pikjk′))≥α2,k=1,2,…,n;k′=1,2,…,n′,
where ji is the random classifying time of train i with normal distribution. And M is an extremely big positive number. Expression (2.2) denotes that the start time to assemble train j in the k′th time segment must be after the end time to classify train i if inbound train i delivers wagons to departure train j. And the probability of expression (2.2)’s holding is more than or equal to  α2, where α2 is the given confidence level.


### 2.3. Wagon Flow Delivers Relationship Constraints

Letmir be the arrival wagon number in classified train i whose destination is direction r. And vj is the train-size limitation of departure train j, q is total number of destination direction in arrival train, xirj is decision variable which means the departure train j wagon number whose destination direction is r and from arrival train i, and Ω(j) is the total destination direction number of departure train j. Qj is a boolean variable, its value is 1 if the jth scheduled departure line is occupied by the train, otherwise the value is 0 [7]. So wagon flow delivers relationship constraints that are as follows:
(2.3)∑j=1n′xirj=mir,i=1,2,…,n;r=1,2,…q,(2.4)∑i=1n∑r=1qxirj=vj•Qj,j=1,2,…,n′.Expression (2.3) denotes that the wagon to direction r from arrival train i can be delivered to different departure train with the same direction. Expression (2.4) denotes the departure train-size limitation which will be further discussed in  Section 2.6.


### 2.4. Train Departure Time Constraints

Letdj′ be the scheduled departure time of train j in train timetable, hj the inspection time of departure train j, z the convoy time needed of the departure train from classification yard to departure yard, dj the latest time of the assembling of train j that should be completed, dj=dj′-hj-z, Bk′j a boolean variable whose value is 1 if train j is assembled in the kth time segment, otherwise the value is 0, and dj* the lag between train j scheduled departure time and actual departure time [9]. So train departure time constraints are as follows:
(2.5)Pr(tk′′+bjBk′j-dj≤M(1-Bk′j))≥α3,k′=1,2,…,n′;j=1,2,…,n′.Expression (2.5) denotes that the completed time to assemble train j must not exceed the latest time that is determined by the train timetable so that train j will depart on time. As the classifying time of a train is stochastic, the probability of expression (2.5)’s holding is more than or equal to α3, where α3 is the given confidence level. And M is an extremely big positive number.


### 2.5. Logic Constraints

In order to guarantee the logical relationships among variables in the model, the following logic constraints have to conform [8]:
(2.6)∑i=1nJki=1,k=1,2,…,n,∑k=1nJki=1,i=1,2…,n.Expression (2.6) denotes only one train can be classified in each time segment; one train can be classified only once:
(2.7)∑j=1n′Bk′j=1,k′=1,2…,n′,∑k′=1n′Bk′j=1,j=1,2…,n′.Expression (2.7) denotes only one train can be assembled in each time segment; one train can be assembled only once:
(2.8)∑j=1n′∑k′=1n′Pikjk′≤M·Jki,i=1,2,…,n;k=1,2,…,n.Expression (2.8) denotes if the arrival train delivers wagons to departure trainj; train i must be classified, where M is an extremely big positive number:
(2.9)∑i=1n∑k=1nPikjk′≤M·Bk′j,j=1,2,…,n′;k′=1,2…,n′.Expression (2.9) denotes if the departure train j will be assembled; it must have arrival trains deliver their wagons to the departure train j,  where M is an extremely big positive number:
(2.10)Jki∈{0,1},Pikjk′∈{0,1},Bk′j∈{0,1},Qj∈{0,1},i=1,2,…,n,j=1,2,…n′,k=1,2,…,n,j=1,2,…n′.


### 2.6. Objective Function

LetTe  be the end time of the wagon flow scheduling stage, then the objective function is as follows [8]:
(2.11)maxZ1=∑j=1n′(∑i=1nxij(Te-dj′-dj*)).Letλj(xj) be the satisfactory function of the actual train size xj compared to the expected train-size vj; it is a trapezoidal form fuzzy number as follows [8]:
(2.12)λj(xj)={0,xj<v_j1;xj-v_j1vj1-v_j1,v_j1≤xj<vj1;1,vj1≤xj≤vj2;v¯j2-xjv¯j2-vj2,vj2<xj≤v¯j2;0,xj>v¯j2,
where v_j1 is the minimum number, v¯j2 is the maximum number, and [vj1,vj2] is the expected interval of numbers.Considering the fuzzy train-size limitation of departure train, we formulated the second objective function:(2.13)maxZ2=∑j=1n′λj(∑i=1nxij).


## 2.1. Start Time Constraints to Classify a Train

The train arrival time is stochastic. The lag between train actual and planed arrival time is a random variable with normal distribution, denoted byεi. Let Ti′′ be the planed arrival time of train i, so the actual arrival time of train i is Ti′=Ti′′+εi, Ti the earliest time after the inspection of train i,  Ti=Ti′+ai, where ai is the inspection time of train i. Assume that a train classifying process is in a time segment k. Let tk be the start time to classify a train in the kth time segment, Jki a boolean variable whose value is 1 if train i is classified in the kth time segment, otherwise the value is 0; let  n  be the total number of arriving trains in the stage [8]. So the start time constraints to classify a train are as follows:
(2.1)Pr(tk-∑i=1nTiJki≥0)≥α1,k=1,2,…,n.Expression (2.1) denotes that the start time to classify train i in the kth time segment must be after the end time of inspection train  i. Because of the stochasticity of train arrival time, the end time of train inspection is also stochastic. The expression (2.1) is chance constrained, and the probability of expression (2.1)’s holding is more than or equal to α1, where α1 is the given confidence level.


## 2.2. Start Time Constraints to Assemble a Train

Assume that a train assemble process is in a time segmentk′. Let bj be the process time of assemble train j; it is a random variable with normal distribution. Let n′ be the total number of departure trains in the stage.Lettk′′ be the start time to assemble a departure train in the kth time segment, Pikjk′ be a boolean variable whose value is 1 if train i which is classified in the kth time segment delivers wagons to departure train j which is assembled in the k′th time segment, otherwise the value is 0 [7]. So the start time constraints to assemble a departure train are as follows:
(2.2)Pr(tk+∑i=1njiJki-tk′′≤M(1-∑i=1n∑j=1n′Pikjk′))≥α2,k=1,2,…,n;k′=1,2,…,n′,
where ji is the random classifying time of train i with normal distribution. And M is an extremely big positive number. Expression (2.2) denotes that the start time to assemble train j in the k′th time segment must be after the end time to classify train i if inbound train i delivers wagons to departure train j. And the probability of expression (2.2)’s holding is more than or equal to  α2, where α2 is the given confidence level.


## 2.3. Wagon Flow Delivers Relationship Constraints

Letmir be the arrival wagon number in classified train i whose destination is direction r. And vj is the train-size limitation of departure train j, q is total number of destination direction in arrival train, xirj is decision variable which means the departure train j wagon number whose destination direction is r and from arrival train i, and Ω(j) is the total destination direction number of departure train j. Qj is a boolean variable, its value is 1 if the jth scheduled departure line is occupied by the train, otherwise the value is 0 [7]. So wagon flow delivers relationship constraints that are as follows:
(2.3)∑j=1n′xirj=mir,i=1,2,…,n;r=1,2,…q,(2.4)∑i=1n∑r=1qxirj=vj•Qj,j=1,2,…,n′.Expression (2.3) denotes that the wagon to direction r from arrival train i can be delivered to different departure train with the same direction. Expression (2.4) denotes the departure train-size limitation which will be further discussed in  Section 2.6.


## 2.4. Train Departure Time Constraints

Letdj′ be the scheduled departure time of train j in train timetable, hj the inspection time of departure train j, z the convoy time needed of the departure train from classification yard to departure yard, dj the latest time of the assembling of train j that should be completed, dj=dj′-hj-z, Bk′j a boolean variable whose value is 1 if train j is assembled in the kth time segment, otherwise the value is 0, and dj* the lag between train j scheduled departure time and actual departure time [9]. So train departure time constraints are as follows:
(2.5)Pr(tk′′+bjBk′j-dj≤M(1-Bk′j))≥α3,k′=1,2,…,n′;j=1,2,…,n′.Expression (2.5) denotes that the completed time to assemble train j must not exceed the latest time that is determined by the train timetable so that train j will depart on time. As the classifying time of a train is stochastic, the probability of expression (2.5)’s holding is more than or equal to α3, where α3 is the given confidence level. And M is an extremely big positive number.


## 2.5. Logic Constraints

In order to guarantee the logical relationships among variables in the model, the following logic constraints have to conform [8]:
(2.6)∑i=1nJki=1,k=1,2,…,n,∑k=1nJki=1,i=1,2…,n.Expression (2.6) denotes only one train can be classified in each time segment; one train can be classified only once:
(2.7)∑j=1n′Bk′j=1,k′=1,2…,n′,∑k′=1n′Bk′j=1,j=1,2…,n′.Expression (2.7) denotes only one train can be assembled in each time segment; one train can be assembled only once:
(2.8)∑j=1n′∑k′=1n′Pikjk′≤M·Jki,i=1,2,…,n;k=1,2,…,n.Expression (2.8) denotes if the arrival train delivers wagons to departure trainj; train i must be classified, where M is an extremely big positive number:
(2.9)∑i=1n∑k=1nPikjk′≤M·Bk′j,j=1,2,…,n′;k′=1,2…,n′.Expression (2.9) denotes if the departure train j will be assembled; it must have arrival trains deliver their wagons to the departure train j,  where M is an extremely big positive number:
(2.10)Jki∈{0,1},Pikjk′∈{0,1},Bk′j∈{0,1},Qj∈{0,1},i=1,2,…,n,j=1,2,…n′,k=1,2,…,n,j=1,2,…n′.


## 2.6. Objective Function

LetTe  be the end time of the wagon flow scheduling stage, then the objective function is as follows [8]:
(2.11)maxZ1=∑j=1n′(∑i=1nxij(Te-dj′-dj*)).Letλj(xj) be the satisfactory function of the actual train size xj compared to the expected train-size vj; it is a trapezoidal form fuzzy number as follows [8]:
(2.12)λj(xj)={0,xj<v_j1;xj-v_j1vj1-v_j1,v_j1≤xj<vj1;1,vj1≤xj≤vj2;v¯j2-xjv¯j2-vj2,vj2<xj≤v¯j2;0,xj>v¯j2,
where v_j1 is the minimum number, v¯j2 is the maximum number, and [vj1,vj2] is the expected interval of numbers.Considering the fuzzy train-size limitation of departure train, we formulated the second objective function:(2.13)maxZ2=∑j=1n′λj(∑i=1nxij).


## 3. Chance Constraint Conversion

For a chance constraintPr{g(x,ξ)≤0}≥α,  where ξ is a random variable with distribution function Φ, if function g(x,ξ) has the form g(x,ξ)=h(x)-ξ, then  Pr{g(x,ξ)≤0}≥α if and only if h(x)≤Kα, where Kα=sup{K∣K=Φ-1(1-α)}. So the deterministic equivalent form of Pr{g(x,ξ)≤0}≥α is as follows [13]:
(3.1)h(x)≤Kα,Kα=sup{K∣K=Φ-1(1-α)}.The same goes forPr{g(x,ξ)≥0}≥α; the deterministic equivalent form is as follows [13]:
(3.2)h(x)≥Kα,Kα=inf{K∣K=Φ-1(α)}.We can convert all the chance constraints in the model above into deterministic equivalent form by expressions (3.1) and (3.2). Let ΦJ be the probability distribution function of ji, ΦD the probability distribution function of εi, and ΦB the probability distribution function of bj, and then we have the following deterministic equivalent form.Equivalence formula for expression (2.1) is
(3.3)tk-(Ti′′+ai+ΦD-1(α1))·∑i=1nJki≥0,k=1,2,…,n.Equivalence formula for expression (2.2) is
(3.4)M(1-∑i=1n∑j=1n′Pikjk′)+tk′-tk-ΦJ-1(α2)·∑i=1nJki≥0,k=1,2,…,n;k′=1,2,…,n′.Equivalence formula for expression (2.5) is
(3.5)M(1-Bk′j)-tk′′-ΦB-1(α3)·Bk′j+dj≥0,k′=1,2,…,n′;j=1,2,…,n′.


## 4. A Hybrid Algorithm Based on ACO and GA

The wagon flow scheduling problem is an NP-complete problem proved by  Dahlhaus et al.  [14]. In this section, we focus on the hybrid algorithm design based on ant colony optimization (ACO) and genetic algorithm (GA). ACO algorithms are the most successful and widely recognized algorithmic techniques based on ant behaviors, initially proposed by Dorigo in 1992 in his Ph.D. thesis [15]. Genetic algorithms are developed by Holland in 1975. It is a powerful and broadly applicable stochastic search and optimization techniques, inspired by natural evolution, such as inheritance, crossover, mutation, and selection [16]. In this paper, the hybrid algorithm is mainly based on ACO; the crossover and mutation operator of GA is used to avoid the “premature” or “stagnation” of ACO.Let the arrival train set beDDwhose element is dd1,dd2,…,ddm ordered by the train arrival time, and let the departure train set be CFwhose element is cf1,cf2,…,cfn ordered by the train departure time in this stage. These two sets are denoted by DD={dd1,dd2,…,ddm} and CF={cf1,cf2,…,cfn}, respectively. The train makeup destination direction is a set denoted by Ω={r1,r2,…,rq}  [6].From  Section2.1,  we know that Ti′=Ti′′+εi, Ti=Ti′+ai is the earliest start time to classify train i, so the actual classifying time cannot be earlier than Ti. Let tk be actual start time to classify train i in the time segment k. Assume that the departure train sequence is j1,j2…jm′ in which arrival train can deliver wagons to them and their departure time is dj1′,dj2′,…djm′′. From the train operating process, it is known that if the end time to classify the arrival train is later than djn′-hj-z-bj (where 1≤n≤m′), then the arrival train cannot deliver wagons to the departure train.Define the classifying time window [Ei,Di] for train i, where Ei is the earliest start time and Di the latest end time to classify train i. Thus the actual classifying time should be between Ei and Di. Let ωi be the penalty factor for the delay to classify train i, and in this paper ωi is the wagon number of arrival train i.


### 4.1. Initialization

A classifying sequence of arrival trains can be regarded as an ant’s travel path. For example,(3,1,5,…,i) represents the trains classified by the order 3,1,5,…i, and there are n nodes on the path which represents the arrive trains, respectively. If the ith ant passing node is j, it means putting the arrival train j in the position i to classify. In the process of ants travel, the passed nodes make up the train collection and it is the taboo list tabuk, so the every completion of ant’s travel makes a new solution.


### 4.2. Transition Probability

LetL={(i1,i2)∣i1,i2∈DD}, and we set up a network G=(DD,L), the purpose about this network is to search path that mostly satisfies the constraints of the departure train, such as train-size limitation, punctuality, and inviolate wagon flow direction. At first the pheromone on each edge is equal. And then every ant must make a choice to move to next node; it means that train will be classified in next step.Suppose that, at timet, the probability of ants s to transfer from train i1 to train i2 is [15, 17]
(4.1)Pi1,i2s(t)={[τi1,i2]α·[ηi1,i2]β∑z∉tabuk[τi1,z]α·[ηi1,z]β,i2∉tabuk0,i2∈tabuk,
in which ηi1,i2 is the heuristic information
(4.2)ηi1,i2=1di+ξ1[ωimax(0,Ci-Di)]+ξ2((Di-Ei)/bj).The tabuk is the tabu list that stands for the set of arrival train which has already been classified;Ci is the end time for the actual classifying time of train i; ξ1 and ξ2 are the weight coefficients; bj is the time of classifying operation; α and β are the parameters used to control the relative importance of pheromone and heuristic information.


### 4.3. Selection and Local Search Strategy

Letq0 be a constant, q∈(0,1) is a random number, if q≤q0, the next node the ants transfer to is a node that makes [τi1,i2]α·[ηi1,i2]β has the maximum value; otherwise, the node will be ensured upon the transition probability by taking the traditional roulette method.If a local optimal solution is found in the early iteration of ACO, it is easier to appear “premature” or “stagnation” phenomenon, and there is also a need to apply a local search strategy in order to adjust the obviously inappropriate classifying order. So the crossover and mutation strategy of GA is adopted in this paper.When an ant completes a tour, a train classifying sequence is obtained, then static wagon allocating method is adopted to calculate the “price” of the classifying sequence [6], and then two of the “minimum price” classifying sequences are selected to crossover.The crossover strategy [18]: for chromosomes P1 and P2, randomly generating two random numbers to determine the crossover position, exchange the classifying order between the crossover locations P1 and P2; if the gene is repeated between and outside of the crossover position, then delete the gene in this location, and then put the lacked gene to the chromosome by ascending order, and then the new chromosomes P1′′ and P2′′ are obtained. Then we calculate P1′′ and P2′′ classifying price and compare to the corresponding price of P1 and P2. We select the minimum price chromosome P and execute mutation operation, exchange the genes of the two positions which are determined by two random numbers, and calculate the price. Thus the current optimal classifying sequence is represented by the path where the price is the less one of P and P′.


### 4.4. Pheromone Updating Strategy

The pheromone can be updated as follows:(4.3)τi1,i2(t+1)=(1-ρ)τi1,i2(t)+Δτi1,i2,
where ρ is the parameter to control the pheromone evaporation rate between time t and t+1; 1-ρ is the retention of the pheromone in the current path. At the beginning, τi1,i2=c (c is a constant), and Δτi1,i2(t+1) is the residues pheromone on the passing edge. If the current path is the optimal one, then Δτi1,i2(t+1)=1/P*, where P* is the total price of the optimal sequence; otherwise, Δτi1,i2(t+1)=0.


### 4.5. The Steps of the Algorithm

Step 1.
Initialization. According to the train arrival information to calculate train the classifying time window, and initialize wagon allocating price table. Set the same amount of pheromone on each edge.Step 2.
Sort trains by their arrival time, and update the wagon allocating price table and calculate the price.Step 3.
Place each ant to each node inG, and set tabu list with the corresponding node.Step 4.
Take an ant, calculate the transition probability of selecting the next node to update the tabu list, and then calculate the transition probability, select the node, and update the tabu list again until traverse through all the nodes.Step 5.
Calculate the pheromone that the ant left to each edge, then the ant die.Step 6.
Repeat Steps3 and 4 until all the ants finish their tour.Step 7.
Calculate the prices of each path that ants choose.Step 8.
Choosing two of the smallest price paths (a path represents a chromosome)P1 and P2 to compare with path in Step 3, select the less one and make them crossover to obtain new paths P1′′ and P2′′.Step 9.
Calculate paths price ofP1′′ and P2′′, compare with the prices of P1 and P2, and select one of the smallest P.Step 10.
Execute mutation operation forP, and calculate the path price after the mutation compared with the price of P, and then select the less one as the optimal path so far.Step 11.
Update the current optimal path, and empty the tabu list tabuk.Step 12.
Judge whether the iterations hit the predetermined number, or whether there is stagnation. If it does, we terminate the algorithm and the output current optimal path; otherwise, go to Step3, execute the next loop of iteration.


## 4.1. Initialization

A classifying sequence of arrival trains can be regarded as an ant’s travel path. For example,(3,1,5,…,i) represents the trains classified by the order 3,1,5,…i, and there are n nodes on the path which represents the arrive trains, respectively. If the ith ant passing node is j, it means putting the arrival train j in the position i to classify. In the process of ants travel, the passed nodes make up the train collection and it is the taboo list tabuk, so the every completion of ant’s travel makes a new solution.


## 4.2. Transition Probability

LetL={(i1,i2)∣i1,i2∈DD}, and we set up a network G=(DD,L), the purpose about this network is to search path that mostly satisfies the constraints of the departure train, such as train-size limitation, punctuality, and inviolate wagon flow direction. At first the pheromone on each edge is equal. And then every ant must make a choice to move to next node; it means that train will be classified in next step.Suppose that, at timet, the probability of ants s to transfer from train i1 to train i2 is [15, 17]
(4.1)Pi1,i2s(t)={[τi1,i2]α·[ηi1,i2]β∑z∉tabuk[τi1,z]α·[ηi1,z]β,i2∉tabuk0,i2∈tabuk,
in which ηi1,i2 is the heuristic information
(4.2)ηi1,i2=1di+ξ1[ωimax(0,Ci-Di)]+ξ2((Di-Ei)/bj).The tabuk is the tabu list that stands for the set of arrival train which has already been classified;Ci is the end time for the actual classifying time of train i; ξ1 and ξ2 are the weight coefficients; bj is the time of classifying operation; α and β are the parameters used to control the relative importance of pheromone and heuristic information.


## 4.3. Selection and Local Search Strategy

Letq0 be a constant, q∈(0,1) is a random number, if q≤q0, the next node the ants transfer to is a node that makes [τi1,i2]α·[ηi1,i2]β has the maximum value; otherwise, the node will be ensured upon the transition probability by taking the traditional roulette method.If a local optimal solution is found in the early iteration of ACO, it is easier to appear “premature” or “stagnation” phenomenon, and there is also a need to apply a local search strategy in order to adjust the obviously inappropriate classifying order. So the crossover and mutation strategy of GA is adopted in this paper.When an ant completes a tour, a train classifying sequence is obtained, then static wagon allocating method is adopted to calculate the “price” of the classifying sequence [6], and then two of the “minimum price” classifying sequences are selected to crossover.The crossover strategy [18]: for chromosomes P1 and P2, randomly generating two random numbers to determine the crossover position, exchange the classifying order between the crossover locations P1 and P2; if the gene is repeated between and outside of the crossover position, then delete the gene in this location, and then put the lacked gene to the chromosome by ascending order, and then the new chromosomes P1′′ and P2′′ are obtained. Then we calculate P1′′ and P2′′ classifying price and compare to the corresponding price of P1 and P2. We select the minimum price chromosome P and execute mutation operation, exchange the genes of the two positions which are determined by two random numbers, and calculate the price. Thus the current optimal classifying sequence is represented by the path where the price is the less one of P and P′.


## 4.4. Pheromone Updating Strategy

The pheromone can be updated as follows:(4.3)τi1,i2(t+1)=(1-ρ)τi1,i2(t)+Δτi1,i2,
where ρ is the parameter to control the pheromone evaporation rate between time t and t+1; 1-ρ is the retention of the pheromone in the current path. At the beginning, τi1,i2=c (c is a constant), and Δτi1,i2(t+1) is the residues pheromone on the passing edge. If the current path is the optimal one, then Δτi1,i2(t+1)=1/P*, where P* is the total price of the optimal sequence; otherwise, Δτi1,i2(t+1)=0.


## 4.5. The Steps of the Algorithm

Step 1.
Initialization. According to the train arrival information to calculate train the classifying time window, and initialize wagon allocating price table. Set the same amount of pheromone on each edge.Step 2.
Sort trains by their arrival time, and update the wagon allocating price table and calculate the price.Step 3.
Place each ant to each node inG, and set tabu list with the corresponding node.Step 4.
Take an ant, calculate the transition probability of selecting the next node to update the tabu list, and then calculate the transition probability, select the node, and update the tabu list again until traverse through all the nodes.Step 5.
Calculate the pheromone that the ant left to each edge, then the ant die.Step 6.
Repeat Steps3 and 4 until all the ants finish their tour.Step 7.
Calculate the prices of each path that ants choose.Step 8.
Choosing two of the smallest price paths (a path represents a chromosome)P1 and P2 to compare with path in Step 3, select the less one and make them crossover to obtain new paths P1′′ and P2′′.Step 9.
Calculate paths price ofP1′′ and P2′′, compare with the prices of P1 and P2, and select one of the smallest P.Step 10.
Execute mutation operation forP, and calculate the path price after the mutation compared with the price of P, and then select the less one as the optimal path so far.Step 11.
Update the current optimal path, and empty the tabu list tabuk.Step 12.
Judge whether the iterations hit the predetermined number, or whether there is stagnation. If it does, we terminate the algorithm and the output current optimal path; otherwise, go to Step3, execute the next loop of iteration.


## 5. An Illustrative Example

We take a certain wagon flow scheduling platform as example in one of the railway bureaus in China. Assume that some technological standard operation time is as follows: convoy time is 10 min, arrival inspection time is 35 min, and departure inspection time is 25 min. The lag between train actual and planed arrival time is a normal distribution variable𝒩(0,5), train classify time obeys 𝒩(15,3), and train assemble time obeys 𝒩(15,3), and departure train-size limitation is a fuzzy trapezoidal variable with parameters of (40,45,50,52).Since train classify and assemble time conforms𝒩(μ,σ2), and the equivalence formula of them can convert to inf{K∣K=Φ-1(α)} by expression (3.4), so Φ-1(0.95)=1.6449 when the confidence level α=95%.The arrival train information is shown as in Table1. In the convenience of calculating, we set the start time stage is 0 and convert the train arrival time is an integer number which stands for the minutes that train arrival from the stage start time [19]. And assume that there are four train destination directions denoted by A, B, C, and D. The train 0 is a dummy train that represents the wagon flow in the beginning of this stage.Table 1
Information of arrival trains.
Train code
Arrival time
Train makeup
Train code
Arrival time
Train makeup
dd0
0
A/15, B/22, C/10, D/12
dd6
112
A/24, D/22
dd1
10
A/30, B/20
dd7
136
A/25, C/23
dd2
35
A/15, C/35
dd8
152
B/20,C/10, D/18
dd3
41
A/35, C/15
dd9
172
A/30, C/18
dd4
58
B/15, C/15, D/15
dd10
208
B/25, C/20
dd5
92
B/20, d/25
dd11
225
A/20, C/23Suppose that all the arrival train can be classified immediately. We can calculate the initial wagon allocating price table according to static wagon allocating problem [6]. In this paper, the Java programming language is used to implement the algorithm above with parameters α=1,β=1,ξ1=1,ξ2=2,q0=0.6,ρ=0.7. The satisfactory solution of wagon flow scheduling is shown in Table 2.Table 2
Information of departure trains.
Train code
Departure time
Train makeup: wagon source
Train code
Departure time
Train makeup: wagon source
cf1
112
B/50: dd0/22, dd1/28
cf6
235
A/50: dd3/15, dd6/24, dd7/11
cf2
142
A/50: dd0/15, dd2/15, dd3/20
cf7
266
D/42: dd5/2, dd6/22, dd8/18
cf3
150
C/50 : dd0/10, dd1/20, dd2/20
cf8
285
B/20, C/32:  dd8/20, dd2/2, dd3/15, dd4/15
cf4cf5
198216
D/50 : dd0/12, dd4/15, dd5/23
cf9
328
C/51: dd7/23, dd8/10, dd9/18
B/37, C13: dd1/2, dd4/15,
cf10
356
A/50: dd7/14, dd9/30, dd11/6
dd5/20, dd2/13From Table2, we know that all of the departure trains meet the train-size limitation from the point of the fuzzy constraint. And in this stage, there are 495 wagons scheduled to the departure trains. Since we consider the stochasticity and fuzziness in the model and algorithm, the robustness and operability of the work plan of railway bureau is prompted greatly.


## 6. Conclusions

In this paper, considering the stochasticity of train arrival time, train classify time, and train assemble time and fuzzy train-size limitation, a stochastic chance-constrained fuzzy multiobjective model for wagon flow scheduling is set up based on the uncertain programming theory. By analyzing the model in detail, a hybrid intelligent algorithm based on ACO and GA is given. Furthermore, a numerical example is also offered to verify the rationality and effectiveness of the model and algorithm. As we know, the China railway informatization is very fast in recent years. But the TDMS, SAM, and other management information system are separate and not intelligent in some extent. So it needs to integrate the related systems by optimizing the transportation business models. The model and algorithm proposed in this paper provide the theoretical basis for integrating and optimizing the related systems. We hope the TDMS and SAM will be more practical and intelligent by using our model and algorithm in this paper.In the future, we will study how to refine the basic wagon flow information, as well as the robust theory for wagon flow scheduling, and how to use synergetic theory in wagon flow and locomotive scheduling.



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*Source: 602153-2012-09-17.xml*