Estimating the Railway Network Capacity Utilization with Mixed Train Routes and Stopping Patterns: A Multiobjective Optimization Approach
Algorithm 1
Solving the epsilon constraint multiobjective programming.
Input: The candidate train set , and the necessary timetabling parameters.
Output: a set of saturated timetables.
Step 1. Initialization
Generate the time-space network according to the conditions introduced in Section 3.2.
Step 2. Calculate utopian point
Calculate the utopian point: solving the following single-objective programming for by the Lagrangian relaxation algorithm introduced by Meng and Zhou [43].
With the utopian point , enumerate all possible , for all and .
Step 3. Lagrangian relaxation for train shortest path subproblems
Given , solve the Lagrangian relaxation problem iteratively by the shortest path algorithm introduced by Meng and Zhou [43]. The Lagrangian relaxation solution can be obtained from the Lagrangian relaxation dual problem.
Step 4. Heuristic method for fixing the Lagrangian relaxation solution
Execute the intensity-based train-by-train scheduling heuristic introduced in Meng and Zhou [43] to get a feasible solution from Step 3. During the train-by-train scheduling procedure, check whether satisfying the -constraint if the train is successfully scheduled before scheduling a train. If the -constraint is violated, abandon the train and turn to the next train.
Step 5. Update
Turn to the next , go to Step 3.
Step 6. Output timetables
Output the saturated timetables and the associated successfully scheduled train sets .
