Bi-objective green ride-sharing problem: Model and exact method

We investigate the bi-objective green ride-sharing problem (BGRSP) with consideration of the drivers' interests. The first objective is to minimize carbon emissions. The second objective is to maximize average ride profit so that every driver's interest can be satisfied. The average ride profit is the average profit of all used rides and it is non-linear due to the variable number of the used rides. The BGRSP is a nonlinear multi-objective problem. We develop an exact method with three steps to solve the BGRSP. The highlight of the exact method is to cut most of the non-Pareto-optimal solutions and use a decomposition method. First, we define the Pareto-optimal ride and prove that every Pareto-optimal solution of the BGRSP is composed of the Pareto-optimal rides; thus, the solution space is reduced by cutting the non-Pareto-optimal rides. Second, we define the partition (equivalent to the solution of BGRSP) based on the relationship matrix between customers and Pareto-optimal rides which is diagonalized into several submatrices, and prove that all partitions of the relationship matrix can be obtained by the partitions of the submatrices. Therefore, the larger-scale NP-hard problem is decomposed into several small-scale NP-hard problems, each of which produces partitions of each submatrix. Third, we define the Pareto-optimal partition and prove every Pareto-optimal solution of BGRSP is composed of the Pareto-optimal partitions of each submatrix. Thus, the solution space can be significantly reduced by cutting the non-Pareto-optimal partitions, even by (1-5.5E-42)*100%. The exact method is validated by solving a benchmark instance of pdp_100-lr101 from Li & Lim benchmark with 106 customers and 200 vehicle capacity. The proposed model and method can reduce carbon emissions and make every driver satisfied simultaneously.