Ridesharing Problem with Flexible Pickup and Delivery Locations for App-Based Transportation Service: Mathematical Modeling and Decomposition Methods

Algorithm 5

LR algorithm for VRPSTW.

Step 1. Initialize:

(1) Lagrangian multipliers , , , ;

(2) Step parameters ;

(3) Iteration index ;

(4) Best upper bound ;

Step 2. Solve relaxed problem (25) with the solution approach described in Section 4.1, and then

  obtain the relaxed solution and the lower bound;

Step 3. If the lower bound does not improve in certain numbers of iterations, then update

   ();

Step 4. Adapt the relaxed solution obtained above to the relative feasible solution by using the

  implementing algorithm proposed in Section 4.2, and then update to the relative feasible

  objective if is greater than it;

Step 5. Calculate as equation (35);

Step 6. Update Lagrangian multipliers as equations (37) and (38);

Step 7. Terminate this algorithm if

(1) Optimality gap , where is a predefined error tolerance;

(2) Step parameters , where is the predefined lower threshold;

(3) Iteration index , where is the predefined upper threshold;

Otherwise, update and turn to Step 2;

Step 8. Return the current best upper bound as the final objective value and corresponding feasible

  solution and optimality gap.