Optimal loading method of multi type railway flatcars based on improved genetic algorithm

2.1 Model construction of railway flatcar loading problem

Genetic algorithm is not only a classical heuristic algorithm, but also a nondeterministic quasi natural algorithm. Genetic algorithm refers to the biological genetic process and is used to solve the optimization problem. The basic process of genetic algorithm is as follows:

Step 1: Initialize the population P ( 0 ) ≔ ( p 1 , p 2 , ═ , p n ) randomly;

Step 2: Calculate the fitness value F of each individual in the group;

Step 3: Evaluate the fitness, and calculate the fitness F ( p i ) for each individual P i in the current population P ( t ) ;

Step 4: Apply the selection operator to generate the intermediate generation P r ( t ) according to a certain rule determined by the individual fitness value;

Step 5: Select individuals according to P c for crossover operation;

Step 6: Perform mutation operations on breeding individuals according to P m ;

Step 7: If a certain stop condition is not met, turn to step 2, otherwise go to step 8;

Step 8: Output the individual with the best fitness value in the population.

In genetic algorithm, all individuals with high hindering fitness value are generated, and the fitness value of the individual greatly exceeds the average fitness value of the population. When selecting according to the proportion of fitness value, the individual will soon occupy an absolute proportion in the population, resulting in the early convergence of the algorithm to a local optimum. This phenomenon is called premature convergence. Therefore, an improved genetic algorithm is used to map the range of fitness function, which is called the scale transformation of fitness function. Assuming that the original fitness function is f and the calibrated fitness function is f ′ , the linear transformation can be expressed as follows:

The coefficients a and b can be set in many ways, but two conditions must be met:

According to the derivation formula derived from the above conditions, the coefficients a and b of the linear transformation can be obtained by solving simple simultaneous equations. The linear transformation method changes the gaps in fitness and maintains the diversity of the population. If the fitness of some individuals in the population is far lower than the average value, or even negative, in order to meet the condition that the minimum fitness is not negative, the following transformation can be carried out:

The coefficients a and b under the condition of non-negative fitness value can be obtained through simple calculation. The difference from the basic condition is that the transformed fitness, which may be negative, is set to 0.

Railway transport flatcar loading mainly considers two factors: flatcar and equipment. The main considerations of flatcar are flatcar length, width, and flatcar load. In terms of equipment, the main considerations are the length, width, and height of the equipment, the weight of the equipment, and whether the equipment exceeds the limit [9]. When establishing the model, the related factors are simplified, and only the most important factors are considered, that is, the length of flatcar and the length of the equipment.

The loading problem of vehicles can be described as that there are m kinds of equipment B 1 , B 2 , ═ , B m need to be loaded and transported. The number of each kind of equipment is known as V 1 , V 2 , ═ , V m . The length of the flatcar is l 1 , l 2 , ═ , l i respectively. The distance between the center of gravity and the front end of the equipment is d 1 , d 2 , ═ , d m , respectively. The weight is W 1 , W 2 , ═ , W m , respectively. There are Q types of flatcars F 1 , F 2 , ═ , F Q available. The number of each type of flatcars is N 1 , N 2 , ═ , N Q , the length is L 1 , L 2 , ═ , L Q , the vehicle distance is S 1 , S 2 , ═ , S Q , and the carrying capacity is Z 1 , Z 2 , ═ , Z Q . n um is the number of flatcars used; x ij is the number of the i type equipment loaded on the j flatcar in the series of flatcars used; D is the minimum allowable distance between two adjacent equipment; α j is the longitudinal allowable displacement of the total center of gravity of the equipment on the j flatcar; Y j t is the distance from the center of gravity of the equipment on the j flatcar to the front end of the flatcar; W j τ is the weight of the equipment on the j flatcar; l j is the distance from the center of gravity of the equipment on the j flatcar to its front end; d j t is the length of the equipment on the j flatcar; and ε j is the distance between the front end of the first piece of equipment on the j flatcar and the front end of the vehicle [10]. The following relationships can be obtained by using the sequential loading method:

There are many feasible loading schemes to meet the above requirements, and the objective of optimization is to find a scheme with least resources of flatcars from all feasible schemes [11]. Due to the adoption of smooth loading, the number of equipment to be loaded on a flatcar mainly depends on the length of the flatcar and equipment [12]. Therefore, the utilization rate of the total length of flatcars can be used as an objective function:

Obviously, this is an integer programming problem, and the sum superscript in the above formula depends on the decision variable n m . Therefore, this problem is not a linear programming problem, and the traditional branch and bound method can not be used to solve the problem, so we must find another way to solve the problem [13].

2.2 Structure optimization of loading equipment for railway flatcars

For a long time, the floor of railway flatcar has been made of wood structure, which is the main part of loading cargo and reinforcement [14]. The reason is that the special properties of wood, such as high friction coefficient, strong nail holding force, initial strength meeting the design requirements, and low cost meet the special needs of the flatcar floor. Fiber reinforced composite floor for railway flatcar adopts three-layer structure: reinforcing layer, foaming layer, and protective layer [15]. The foaming layer is formed directly on the reinforced layer, and continuous production is realized through the transmission device. The protective layer is sprayed on the surface of the foaming layer with special equipment [16]. In order to ensure the loading effect of railway flatcar, the structure optimization process of railway flatcar loading equipment is optimized as Figure 1.

Figure 1

Structure optimization process of railway flatcar loading equipment.

Based on the process in Figure 1, the protective layer of damaged parts of railway flatcar loading equipment is removed, and the adhesive is applied on the surface of exposed foaming layer. After placing for 3 h, a protective layer is applied [17]. In actual loading, some equipment belongs to the towed equipment, and the corresponding traction equipment must be adjacent to and installed in front of it. Some equipment is allowed to be straddled, so as to carry out intelligent verification when the computer automatically generates the “loading scheme,” and filter out the “loading scheme” that does not meet the traction requirements on the same flatcar; for the flatcar that loads the towed equipment at the front of the flatcar, one flatcar needs to load the traction equipment at its rear as Figure 2.

Figure 2

Optimization of loading scheme for traction equipment.

Based on Figure 2, for each loading scheme, a constraint condition is added to the model: X j ∞ ⩾ X j k . Based on this, the correctness of the straddle mounting method is analyzed as Figure 3.

Figure 3

Correct loading method: no continuous straddle loading.

This study analyzes the wrong loading method, and a specific structure is shown inn Figure 4.

Figure 4

Wrong loading method: generate continuous straddle loading.

Based on the principle of the figure, according to the equipment database and the technical requirements of loading, we can calculate all the loading schemes of the unit through self programming, which is one of the key problems [18]. Branch and bound search and improved genetic algorithm are used to calculate the loading scheme array. In the branch and bound search process, infeasible loading schemes are eliminated, such as the towed equipment with no corresponding traction equipment, and some difficult loading schemes are obtained using improved genetic algorithm, such as overlapping length of equipment and relative crossing of gun barrel [19]. In addition, the front (back) cross loading schemes need to be generated in pairs to sort out the array of loading schemes. There are a large number of loading schemes of the same kind in the array that are automatically generated by computer, which need to be merged [20]. The so-called similar loading scheme means that for the same loading unit, although the order of equipment combination is different, the type and quantity of equipment are equal. After sorting, the number of loading schemes will be greatly reduced.

2.3 Selection of optimal loading scheme for multi type railway

In fact, the loading problem of railway flatcars can be regarded as a packing problem. They are essentially the same, that is, they all study how to load the same amount of goods and how to use the least flatcars or boxes [21,22]. A greedy strategy is adopted in most approximate algorithms of packing problem, that is, the overall planning for the optimization result is simplified at each step, that is, a local selection method is specified for each packing, as different greedy selection methods produce different strategies [23,24].

Improved genetic algorithm is a randomized search method derived from the evolutionary laws of the biological world (survival of the fittest genetic mechanism). Its main feature is to directly operate on structural objects without the limitation of derivation and function continuity [25,26]. It has inherent implicit parallelism and better global optimization capabilities. Using probabilistic optimization methods, it can automatically obtain and guide the optimized search space, and adjust the search direction adaptively, without the need for definite rules. First, establish the initial population, and then calculate the fitness, select individuals with high fitness, and disrupt the order of existing individuals to generate new individuals. The new individual is put back into the population, and after this process, a population is finally put out, and the optimal solution is obtained.

The control parameters (crossover rate and mutation rate) of genetic algorithm have an important impact on the performance of the algorithm. The control parameters of the basic genetic algorithm are determined in advance and remain unchanged in the process of genetic evolution. In the process of evolution, “good” individuals and “poor” individuals experience the same crossover and mutation rate, so it is difficult to ensure the recovery of gene information lost in the process of genetic evolution (that is, there is no gene in all individuals). For this reason, the crossover rate and mutation probability can be adjusted adaptively, and these excellent individuals can reproduce through the selective replication operation of survival of the fittest, which helps the algorithm gradually approach an optimal state. The crossover and variation rates of individuals with small fitness function values are high. Constantly updating these individuals with low fitness function values is helpful to restore the lost effective genes.

The implementation steps of the improved genetic algorithm are as follows:

Step 1: Initialize genetic algebra: G ← 0 ;

Step 2: Randomly generate the initial population P ( G ) , and calculate the fitness of each body of the population;

Step 3: Divide P ( G ) into multiple subgroups:

In Formula (13), n is the number of divided groups.

Step 4: Perform independent genetic operations on each P i ( G ) :

The copy operation is performed by the selection operator: select [ P i ( G ) ] → P i ′ ( G ) .

The crossover operation is performed by the crossover operator: cross [ P i ′ ( G ) ] → P i ″ ( G ) .

The mutation operation is performed by the mutation operator: mutation [ P i ″ ( G ) ] → P i ″ ′ ( G ) .

Step 5: Calculate the fitness of individuals in P i ″ ′ ( G ) by grouping;

Step 6: Exchange information between any combination of two groups to generate the next generation group:

Step 7: Judgment of termination conditions. If the termination condition is not met, ( G + 1 ) = G , skip back to step 4; if the termination condition is met, the optimal result is the output, and the algorithm ends.

Due to a large input length n of the bin packing problem in practice, the trade-off between the time cost of approximate algorithm and the optimization degree of solution is very important.

1. Define variables and arrays for loading;

2. Take out an equipment and prepare to load it. If there is no equipment, go to 6;

3. Check whether the remaining length of the existing flatcar is greater than or equal to the equipment length. If it is 4, it is not 6;

4. Load the equipment on the current inspection flatcar, and give x the value of the remaining length of the original flatcar minus the length of the equipment;

5. Compare x with the minimum value S. If x ≥ s, jump to 6, otherwise mark the current flatcar as the most suitable flatcar, and then jump to 6;

6. If the current flatcar is the last one, load the equipment on the most suitable flatcar. The remaining length of the most suitable flatcar is equal to the original remaining length minus the equipment length, and then jump to 2; if it is not the last flatcar, then change to the next flatcar and jump to 3.

There are m kinds of equipment to be transported. Among them, M ₁ is equipped with A ₁ pieces, M ₂ is equipped with A ₂ pieces M _i is equipped with A _i pieces M _m is equipped with A _m pieces. The railway flatcars used are K ₁, K ₂, …, K _f type. First, consider the case that one loading unit is one flatcar (the special loading scheme with two or more flatcars as one loading unit is supplemented later). All possible loading schemes are shown in Table 1 [27]. In the table, P(i,j) represents the number of loading equipment M _i under the j loading scheme for one loading unit. Based on this, parameter evaluation and standard treatment are carried out for all possible loading and cutting methods as Table 1.

There are many combination methods on a whole train, but the number of combinations on one flatcar will be greatly reduced (the “loading scheme” mentioned below is for one loading unit). In general, one loading unit is one flatcar; considering the existence of straddle loading or some fixed loading schemes, it can be more than two flatcars in succession). Practice has proved that one flatcar can only carry no more than three kinds of equipment, even if there are many kinds of equipment with very small length, so that it is possible to load four or more kinds of equipment on one flatcar. These equipment can also be combined into one kind of “equipment” for processing [28]. This can ensure that in any loading scheme, the number of equipment types does not exceed 3. After such treatment, the number of loading schemes is greatly reduced, and the linear programming model is established as follows: Let X j be the number of times that the j loading scheme is used (equal to the number of flatcars used), and C j the cost of flatcars corresponding to the j loading scheme. The objective of optimization is to use the least number of flatcars, and its objective function is

After further optimization, following results can be obtained:

When there is only one type of flatcar, the objective function is the least number of cars; when there is more than one type of flatcar, the unit freight (yuan/car kilometer) of different cars should be considered, and the objective function is the latter. Nonnegative constraint: X j > 0 , j = 1 , 2 , ═ n . The number of times used by each loading method must be an integer, so there are integer constraints: X j is an integer, j = 1 , 2 , ═ , n . The constraint condition for all equipment to be fully loaded is

The above algorithm can be abbreviated as

According to the loading standard, the fixed loading scheme for some equipment can adopt 2 cars with 3 pieces, 3 cars with 4 pieces, and 3 cars with 5 pieces. Then, the number of flatcars for each loading scheme should be multiplied by 2, or 3 (two or three flatcars are used as one loading unit). For each such loading scheme, a coefficient λ j is defined, λ j = 2 , 3 . Then the objective function is rewritten as

Further conversion can be obtained as follows:

Considering all the factors in flatcars and equipment, combined with mathematical modeling theory, this study establishes a model, analyzes and optimizes the model, and then uses a variety of approximate algorithms to calculate it, draws a conclusion, and compares and analyzes the conclusions, so as to select the optimal loading scheme and standby scheme, and ensure the loading effect of the railway flatcars.