## 2 A Review of Decision Model for the DGRP
\begin{table}
\begin{tabular}{c l} \hline
**Variable** & \multicolumn{1}{c}{**Meaning**} \\ \hline \(P\) & The number of passengers. \\ \(D\) & The number of drivers. \\ \(p\) & An integer index for a passenger, where \(p\in\{1,2,3,\ldots.P\}\). \\ \(d\) & An integer index for a driver, where \(d\in\{1,2,3,\ldots,D\}\). \\ \(K\) & The number of locations of passengers, where \(K\) is equal to \(P\). \\ \(k\) & An integer index for a location, where \(k\in\{1,2,\ldots.K\}\). \\ \(I_{d}\) & The number of bids submitted by driver \(d\), where \(d\in\{1,2,3,\ldots,D\}\). \\ \(j\) & The index of the \(j-th\) bid submitted by driver \(d\), where \(d\in\{1,2,3,\ldots,D\}\) and \(j\in\{1,2,\ldots,I_{d}\}\). \\  & The \(j-th\) bid submitted by driver \(d\) with \(DB_{dj}=(q_{dj1}^{1},q_{dj2}^{1},q_{dj3}^{1},\ldots,q_{djp}^{1},q_{dj1}^{2},q_ {dj2}^{2},q_{dj3}^{2},\ldots,q_{djp}^{2},o_{dj},c_{dj})\), where \\ \(DB_{dj}\) & \(q_{djk}^{1}\) The number of seats allocated to the pick-up location \(k\) in the \(j-th\) bid of driver \(d\), \\ \(q_{djk}^{2}\) The number of seats released at the drop-off location \(k\) in the \(j-th\) bid of driver \(d\), \\  & \(o_{dj}\): The original cost of driver \(d\in\{1,2,\ldots,D\}\) when the driver travels alone, \\  & \(c_{dj}\): The transport cost of the bid. \\  & The bid of passenger \\ \(PB_{p}\) & \(p\) with \(PB_{p}=(s_{p1}^{1}s_{p2}^{1},s_{p3}^{1},\ldots,s_{pp}^{1},s_{p1}^{2},s_{p2}^{2 },s_{p3}^{2},\ldots,s_{pp}^{2},f_{p})\), where \(p\in\{1,2,3,\ldots.P\}\), \\  & \(s_{pk}\): the number of seats requested by \(p\) at location \(k\) and \\  & \(f_{p}\): the passenger \(p^{\prime}\)’s original transport cost when he/she travels alone. \\ \(s_{pk}^{1}\) & No. of seats requested at the pick-up location \(k\), \(s_{pp}^{1}>0\) and \(s_{pk}^{1}=0\)\(\forall k\
eq p\). \\ \(s_{pk}^{2}\) & No. of seats released at the drop-off location \(k\), \(s_{pp}^{2}>0\) and \(s_{pk}^{2}=0\)\(\forall k\
eq p\). \\  & Driver \(d\)’s decision variable, where \(d\in\{1,2,\ldots,D\}\) and \(j\in\{1,2,\ldots,I_{d}\}\): \(x_{dj}=\) \\ \(x_{dj}\) & \(1\) if \(DB_{dj}\) is a winning bid and \(x_{dj}=0\) otherwise. \\ \(y_{p}\) & Passenger \(p^{\prime}\)’s decision variable, where \(p\in\{1,2,3,\ldots.P\}\): \(y_{p}=1\) if \(PB_{p}\) is a winning bid and \(y_{p}=0\) \\ \(r_{D}\) & Drivers’ minimal expected discount. \\ \(r_{P}\) & Passengers’ minimal expected discount. \\ \(r_{P}\) & The objective function, \\ \(F(x,y)\) & \(F(x,y)=\sum\limits_{p=1}^{p}y_{p}f_{p}+\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{ I_{d}}x_{dj}o_{dj}-\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{I_{d}}x_{dj}c_{dj}\). \\ \(\Gamma_{dj}\) & The set of passengers on the ride of the \(j-th\) bid \(DB_{dj}\) of driver \(d\), where \(d\in\{1,2,\ldots,D\}\). \\ \(F_{dj}(x,y)\) & Cost savings of the \(j-th\) bid \(DB_{dj}\) of driver \(d\), where \(d\in\{1,2,\ldots,D\}\). \\  & \(F_{dj}(x,y)=\sum\limits_{p\in\Gamma_{dj}}y_{p}f_{p}+x_{dj}o_{dj}-x_{dj}c_{dj}\). \\ \(cf_{pdj}\) & Transport cost for passenger \(p\in\Gamma_{dj}\) on the ride of the \(j-th\) bid \(DB_{dj}\), where \(d\in\{1,2,\ldots,D\}\). \\ \(U(0,1)\) & Uniform distribution. \\ \(N(\mu,\sigma^{2})\) & Gaussian distribution. \\ \hline \end{tabular}
\end{table} 
Table 1: Notation of parameters, variables and symbols. seats allocated to pick-up location \(k\), \(q^{2}_{djk}\) is the no. of seats released at the drop-off location \(k\), \(o_{dj}\) is the original cost of the driver when he/she travels alone and \(c_{dj}\) is the travel cost of the bid. To formulate the DGRP as an optimization problem, decision variables and an objective function must be defined.