Fuzzy Decision Adjustment of Train Operation Plan for High-Speed Rail Network Based on Multi-Objective Optimization

The TOP adjustment has long been treated as an optimization problem. For example, Gustavsson [1] constructed a 0-1 mixed integer programming (MIP) model to adjust the TOP in a single section of railway that handles both passenger and cargo traffic. For TOP adjustment, Wen and Li [2] determined the train sequence through tabu search (TS). Chen et al. [3] adjusted the TOP under the full failure of railway section capacity: Considering the uncertainty of failure duration, a two-stage stochastic expectation model with compensation and an incomplete continuous multi-stage decision-making model were established, and solved by branch and bound (BB) algorithm.

Zhang et al. [4] adjusted the TOP under the full failure of HSR section capacity: From the macro level, the train operations in the HSR network were regarded as an event-activity network, a mixed integer route planning model was constructed for TOP adjustment, and the two-stage solution algorithm was adopted to solve the model. Zhang et al. [5] summarized the features of interferences in railway operation, and designed a TOP adjustment strategy through sequence optimization. Focusing on the TOP adjustment of HSR, Wakisaka and Masuyama [6] refined the arrival and departure time of train at the station, and built a HSR TOP adjustment model that simultaneously optimizes arrival time and route; the model aims to minimize the time of train spent in the delayed section under the constraint of the minimum train interval. Cacchiani et al. [7] developed an optimization strategy and plan for the adjustment of the periodic TOP.

Krasemann [8] rescheduled the train sequence for TOP adjustment, created a MIP model to minimize the total delay, and verified the reliability of the model using four heuristic algorithms. Huo and Wu [9] adjusted the TOP for single-track railway with both passenger and cargo traffic, with the aim to minimize the weighted delay of trains. Considering the uncertain conflicts between trains, Zhang et al. [10] adjusted the TOP to minimize the delay of all trains at the terminal station. Nie et al. [11] minimized the deviation of the adjusted TOP from the original TOP in HSR network. Sun et al. [12] optimized punctuality, interval operation time, and freight train operation rate through TOP adjustment. Zhang et al. [13] attempted to minimize the weighted total delay and total number of delayed trains through TOP adjustment.

Xu et al. [14] plotted the discrete event topology of single-track railway, developed a discrete-event TOP adjustment model with maximal overall satisfaction, and proposed the iterative repair algorithm to solve the model. Based on graph theory, Choi et al. [15] constructed a train state diagram for TOP adjustment, expressed train sate in station as a quintuple, and put forward two heuristic search algorithms: node priority search and differential-operator initial time selection based on maximum delay. Zhuang et al. [16] improved the difference algorithm to solve the HSR TOP adjustment model, and proved that the improved algorithm can prevent the premature convergence of the original algorithm. Drawing on adaptive dynamic planning, Chen and Zhou [17] solved the TOP adjustment problem by replacing the cost function with backpropagation neural network (BPNN), and calling the double heuristic dynamic planning algorithm.

The above studies on TOP adjustment have improved the efficiency of railway transport to varied degrees, but most of them focus on a single optimization objective. This paper mainly designs a multi-objective optimization model to adjust the TOP in a single section. In addition to train flow, the optimization objectives also cover multiple indices related to passenger satisfaction.

4.1 Constraints

For single-section TOP adjustment, the multi-objective optimization is subjected to the following constraints:

(1) The stop time of the train at the station should not be shorter than the minimum stop time of the train: $T_{r_{i}, u_{j}}^{d}-T_{r_{i}, u_{j}}^{a} \geq t_{r_{i}, u_{j}}^{\min }$.

where, $T_{r_{i}, u_{j}}^{d}$ and $T_{r_{i}}^{a} u_{j}$ are the departure time and arrival time of train line $r_{i}$ at station $u_{j},$ respectively; $t_{r i, u j}^{m i n}$ is the minimum stop time of the train.

(2) The time interval of train arrival at station should satisfy the following constraint:

If If $B_{r_{i}, u_{j}}^{A} \cap B_{{{r}_{i'}},{{u}_{j}}}^{A} \neq \emptyset,$ then $\left|T_{r_{i} u_{j}}^{A}-T_{{{r}_{i'}},{{u}_{j}}}^{A}, u_{j}\right| \geq t_{u_{j}}^{A A}$.

where, $B_{r_{i}, u_{j}}^{A}$ and $T_{r_{i}, u_{j}}^{A}$ are the arriving route and arriving time of train line $r_{i}$ at station $u_{j}$, respectively.

(3) The time interval of train departure at station should satisfy the following constraint:

If $B_{r_{i}, u_{j}}^{D} \cap B_{{{r}_{i'}},{{u}_{j}}}^{D} \neq \emptyset,$ then $\left|T_{r_{i}, u_{j}}^{D}-T_{r_{i}^{\prime}}^{D}, u_{j}\right| \geq t_{u_{j}}^{D D}$.

where, $B_{r_{i}, u_{j}}^{D}$ and $T_{r_{i}, u_{j}}^{D}$ are the departure route and departure time of train line $r_{i}$ at station $u_{j}$, respectively.

(4) The time intervals of train arrival and departure at station should satisfy the following constraints:

If $B_{r_{i}, u_{j}}^{D} \cap B_{{{r}_{i'}},{{u}_{j}}}^{A} \neq \varnothing$, then $T_{r_{i}, u_{j}}^{D}-T_{{{r}_{i'}},{{u}_{j}}}^{A}, \geq t_{u_{j}}^{A D}$ or $T_{{{r}_{i'}},{{u}_{j}}}^{A}-T_{r_{i}, u_{j}}^{D}, \geq t_{u_{j}}^{D A}$.

If $B_{r_{i}, u_{j}}^{A} \cap B_{{{r}_{i'}},{{u}_{j}}}^{D} \neq \varnothing$, then $T_{{{r}_{i'}},{{u}_{j}}}^{D} \geq t_{u_{j}}^{A D}$ or $T_{r_{i}, u_{j}}^{A}-T_{{{r}_{i'}},{{u}_{j}}}^{D} \geq t_{u_{j}}^{D A}$.

(5) If the train line $r_{i}$ has a train connection at station $u_{j}$, the minimum connection time should satisfy the following constraint:

If $r_{i} \in c_{k}, c_{k}=\left\{r_{i}, r_{i'}, u_{j}\right\}$, then $T_{{{r}_{i'}},{{u}_{j}}}^{D}-T_{r_{i}, u_{j}}^{A} \geq t_{c_{k}}^{\min }$.

If $r_{i} \in c_{k}, c_{k}=\left\{r_{i'}, r_{i}, u_{j}\right\}$, then $T_{r_{i}, u_{j}}^{D}-T_{{{r}_{i'}},{{u}_{j}}}^{A}, u_{j} \geq t_{c_{k}}^{\min }$.

(6) The arrival time of train at departure station should satisfy the following constraint:

If $u_{j}=u_{r_{i}}^{0},$ then $T_{r_{i}, u_{j}}^{A}=T_{r_{i}, u_{j}}^{D}$.

(7) The departure time of train at terminal station should satisfy the following constraint:

If $u_{j}=u_{r_{i}}^{e},$ then $T_{r_{i}, u_{j}}^{D}=T_{r_{i}, u_{j}}^{A}$.

4.2 Objective functions

Two sets of objective functions were established for single-section TOP adjustment: those based on train flow features and those based on passenger satisfaction.

There are four objective functions based on train flow features:

(1) Minimum delay at station

$o b_{1}^{T}=\min \sum_{i=1}^{s} \sum_{j=1}^{t}\left(w_{r_{i}, u_{j}}^{d-A} \cdot t_{r_{i}, u_{j}}^{d-A}+w_{r_{i}, u_{j}}^{d-D} \cdot t_{r_{i}, u_{j}}^{d-D}\right)$

where, $w_{r_{i}, u_{j}}^{d-A}$ is the weight of train line $r_{i}$ arriving delay at station $u_{j} ; w_{r_{i}, u_{j}}^{d-D}$ is the weight of train line $r_{i}$ departure delay at station $u_{j}$

(2) Minimum departure delay of the departure train in the current section at the departure station:

$o b_{2}^{T}=\min \sum_{i=1}^{s} t_{r_{i}, w_{r_{i}}^{o}}^{d-A}\cdot t_{r_{i}, u_{r_{i}}^{o}}^{d-A}$

(3) Minimum arrival time of the terminal train in the current section at the terminal station:

$o b_{3}^{T}=\min \sum_{i=1}^{s} w_{r_{i}, u_{r_{i}}^{e}}^{d-A} \cdot t_{r_{i}, u_{r_{i}}^{e}}^{d-A}$

(4) Minimum number of delayed trains

$o b_{4}^{T}=\min \sum_{i=1}^{s} T_{r_{i}}^{d e l a y}$

There are two objective functions based on passenger satisfaction:

(1) Minimum delay at station for passenger satisfaction

$o b_{1}^{P}=\sum_{i=1}^{s} \sum_{j=1}^{t}\left(w_{r_{i}, u_{j}}^{S 1} \cdot\left(t_{r_{i}, u_{j}}^{d-A}+t_{r_{i}, u_{j}}^{d-D}\right)\right.$

(2) Minimum delay at station for transfer passengers’ satisfaction

$o b_{2}^{P}=\sum_{i=1}^{s} \sum_{j=1}^{t} w_{r_{i}, u_{j}}^{S 2} \cdot t_{r_{i}, u_{j}}^{d-A}$

The robustness of the overall objective function depends on the weight of each objective function. The weighting of objective functions is a multi-attribute decision-making problem with high fuzziness and uncertainty. The fuzzy, uncertain problem can be solved satisfactorily with intuitionistic fuzzy sets [19]. Therefore, the stochastic intuitionistic fuzzy decision-making algorithm was adopted to determine the weight of each index in the overall objective function.

The two sets of weighted objective function were integrated into an overall objective function for the multi-objective optimization:

$o b=\rho_{1}^{T} o b_{1}^{T}+\rho_{2}^{T} o b_{2}^{T}+\rho_{3}^{T} o b_{3}^{T}+\rho_{4}^{T} o b_{4}^{T}+\rho_{5}^{T} o b_{1}^{P}$

$+\rho_{6}^{T} o b_{2}^{P}$ 

4.3 Model solving with the CFA

To solve our TOP adjustment model, the basic firefly algorithm is improved into the CFA through the logical self-mapping below:

$y_{n+1}=f\left(\mu, y_{n}\right)=1-\mu y_{n}^{2}$       (1)

where, $n=0,1,2, \ldots, \infty,-1<y_{n}<1 ; \mu$ is the control parameter.

Based on logical self-mapping, the CFA optimization process is as follows:

(1) The solution of firefly algorithm is mapped to [-1, 1]:

$L_{d}=2 * \frac{x_{d}-a_{d}}{b_{d}-a_{d}}-1$        (2)

where, $x_{d}$ is the position of firefly i in the d-dimensional space; $a_{d}$ and $b_{d}$ are the lower and upper search bounds of each variable, respectively.

(2) The generated chaotic variable is introduced into the optimization model, and the new chaotic variable is obtained using the chaos operator:

$L_{n+1, d}=1-2 L_{n, d}^{2}$     (3)

where, $d=1,2, \ldots, D ; n=0,1,2, \ldots, \infty ;-1<L_{n, d}<1$.

(3) The obtained chaotic variable sequence is converted to the original solution space:

$y_{d}^{\prime}=\frac{\left(b_{d}-a_{d}\right) * L_{d}+\left(b_{d}+a_{d}\right)}{2}$        (4)

where, $d=1,2, \ldots, D ; a_{d}$ and $b_{d}$ are the lower and upper search bounds of each variable, respectively.

If the current position of a firefly is better than the original position, then the original position of that firefly should be replaced by the current position. Otherwise, the algorithm continues until meeting the termination condition.

In each round, if all fireflies are optimized by the chaos operator, the algorithm will become more accurate. However, the improved accuracy comes at the cost of a longer runtime. To improve the algorithm efficiency, a few fireflies should be selected for chaotic optimization. Here, the search space is adjusted dynamically to change the upper and lower search bounds in each dimension. The upper and lower search bounds can be defined respectively as:

$y_{\min , d}=\max \left\{y_{\min , d}, y_{g, d}-\rho *\left(y_{\max , d}-y_{\min , d}\right)\right\}$                         (5)

$\begin{aligned} y_{\max , d}=\min \left\{y_{\max , d}, y_{g, d}+\rho\right.& \\\left.*\left(y_{\max , d}-y_{\min , d}\right)\right\} \end{aligned}$                     (6)

where, $y_{g, d}$ is the value of the current optimal solution in the d-dimensional space; $\rho$ is the transform factor.

The workflow of the CFA based on logical self-mapping is explained as follows:

Step 1. Initialize the number and positions of fireflies based on the arrival time and departure time of the target train at the station.

Step 2. Derive the maximum brightness from the initial positions of fireflies.

Step 3. Calculate the relative brightness and attractiveness of each firefly.

Step 4. Update the position of each firefly.

Step 5. Select the best s% of fireflies for chaotic optimization with logical self-mapping and dynamic transform of the search space, and replace the worst s% fireflies with the s% newly generated fireflies.

Step 6. Recalculate the brightness of each firefly after the position update.

Step 7. Output the optimal solution if the maximum number of iterations is reached or the search accuracy is as required; otherwise, jump to Step 3 for iterative search.