## 2 A Review of Decision Model for the DGRP
In our problem formulation, drivers' decision variables are denoted by \(x_{dj}\), where \(d\in\{1,2,\ldots,D\}\) and \(j\in\{1,2,\ldots,J_{d}\}\), \(x_{dj}\) is binary, \(DB_{dj}\) is a winning bid if \(x_{dj}\) is equal to one, \(DB_{dj}\) is not a winning bid if \(x_{dj}\) is equal to zero; passengers' decision variables are denoted by \(y_{p}\), where \(p\in\{1,2,3,\ldots P\}\), \(y_{p}\) is binary, \(PB_{p}\) is a winning bid if \(y_{p}\) is equal to one, \(PB_{p}\) is not a winning bid if \(y_{p}\) is equal to zero. The objective function is a function of \(x_{dj}\) and \(y_{p}\), where \(d\in\{1,2,\ldots,D\}\), \(j\in\{1,2,\ldots,J_{d}\}\) and \(p\in\{1,2,3,\ldots P\}\), used to characterize the goal to be achieved. Cost savings have been widely used as a performance index for ridesharing systems. For the DGRP, the cost savings objective function is described by \(F(x,y)=\sum\limits_{p=1}^{P}y_{p}f_{p}+\sum\limits_{d=1}^{D}\sum\limits_{j=1 }^{J_{d}}x_{dj}o_{dj}-\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{J_{d}}x_{dj}c_{ dj}\), where the first term \(\sum\limits_{p=1}^{P}y_{p}f_{p}\) represents the original travel cost of all winning passengers, the second term \(\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{J_{d}}x_{dj}o_{dj}\) represents the original travel cost of all winning drivers, and \(\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{J_{d}}x_{dj}c_{dj}\) represents the total travel cost of all winning passengers and winning drivers with ridesharing. The problem formulation aims to minimize the cost savings objective function \(F(x,y)\) to realize the benefits of ridesharing. In the DGRP, the demand and supply of seats must be balanced and this is described by Constraints (2) and (3). For the ridesharing participants to benefit from ridesharing, the cost savings must be non-negative and this is described by Constraint (4). Constraint (5) states that only one bid can be a winning bid for each driver even though a driver may submit multiple bids. In the DGRP, the discount for drivers and passengers must be satisfied to provide an incentive for drivers and passengers to accept ridesharing. This is described by Constraints (6) and (7). The decision variables of all drivers \(x_{dj}\in\{0,1\}\)\(\forall d\in\{1,\ldots,D\}\)\(\forall j\in\{1,\ldots,J_{d}\}\) and the decision variables of all passengers \(y_{p}\in\{0,1\}\)\(\forall p\in\{1,\ldots,P\}\) are binary. This is described by (8). \[\max\limits_{x,y}F(x,y) \tag{1}\]  
\[\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{J_{d}}x_{dj}q^{1}_{djk}=y_{p}s^{1}_{pk }\]  
\[\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{J_{d}}x_{dj}q^{2}_{djk}=y_{p}s^{2}_{pk }\]  
\[\sum\limits_{p=1}^{P}y_{p}f_{p}+\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{J_{d}} x_{dj}o_{dj}\geq\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{J_{d}}x_{dj}c_{dj}\]  
\[\sum\limits_{j=1}^{J_{d}}x_{dj}\leq 1\;\forall d\in\{1,\ldots,D\}\]  
\[x_{dj}(\frac{F_{dj}(x,y)}{\sum\limits_{p\in\Gamma_{dj}}y_{p}cf_{pdi}+x_{dj}c_{ dj}}-r_{D})\geq 0\]\[y_{p}(\frac{F_{dj}(x,y)}{\sum\limits_{p\in\Gamma_{dj}}y_{p}cf_{pdj}+x_{dj}c_{dj}}- r_{P})\geq 0 \tag{7}\]  
\[x_{dj}\in\{0,1\}\ \forall d\in\{1,\ldots,D\}\ \forall j\in\{1,\ldots,J_{d}\}\ \text{and}\ y_{p}\in\{0,1\}\ \forall p\in\{1,2,3,\ldots.P\} \tag{8}\]