Optimal Fleet Deployment Strategy: Model the Effect of Shared E-Bikes on Bike-Sharing System
Algorithm 1
Branch-and-bound algorithm.
Step 0: initialization and BSS/EBSS loading
(a)
Set up the parameters;
(b)
Load the integrated bike and e-bike sharing system;
(c)
The objective is to find the fleet deployment solution that maximises the earned profit in equation (8);
(d)
Let = 0 be the number of iteration.
Step 1: solve relaxed noninteger problem
The relaxed problem is solved without integer constraints based on dual-simplex algorithm [45] for linear programming. The results correspond to the first upper bound: .
Step 2: Find feasible solutions with heuristics
Heuristics is applied to generate a feasible point , such that the lower bound of the fleet deployment optimisation model can be determined based on :.
Step 3: iteration
The algorithm is terminated when meeting the stopping criteria as follows:
(a)
The difference between the lower bound and upper bound is less than the tolerance value, i.e. ;
(b)
The number of nodes explored exceeds the maximum number of nodes.
Otherwise, repeat steps 4-5.
Step 4: branching
Noninteger variable with value is chosen as the fractional variable to split;
;
Two subproblems are generated by adding constraints for the variable as and respectively;
Based on step 1 and step 2, determine the lower bounds (,) of the two subproblems with the corresponding decision variable vectors and respectively.
Step 5: update
;
ifthen
The corresponding region is eliminated from consideration as no branching is possible;
;
else ifthen
Additional constraint is added to the searching region;
;
Proceed to step 3;
else
Additional constraint is added to the searching region;
