# Improving Acceptability of Cost Savings Allocation in Ridesharing Systems Based on Analysis of Proportional Methods

## Abstract

**:**

## 1. Introduction

## 2. Proportional Methods to Divide Cost Savings among Ridesharing Participants

**Definition1.**

**Definition 2.**

## 3. Comparison with Proportional Allocation Methods

**Lemma 1.**

**Proof of Lemma 1.**

**Property 1.**

**Proof of Property 1.**

**Property 2.**

**Proof of Property 2.**

**Property 3.**

**Proof of Property 3.**

**Property 4.**

**Proof of Property 4.**

## 4. Results

## 5. Discussion

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof of Lemma 1.**

## Appendix B

**Proof of Property 2**

- (i)
- First, it is shown that the highest rewarding rate for shared rides under the Local Proportional Method is greater than that of the Global Proportional Method. Let the winning bid $\overline{j}$ of driver $\overline{d}$ be the shared ride with the highest rewarding rate for shared rides under the Local Proportional Method.

- (ii)
- If the Local Proportional Method is used, the rewarding rate for passenger $p$ is ${\sigma}_{p}^{P}{F}_{dj}(x,y)$, where ${\sigma}_{p}^{P}=\frac{{y}_{p}{f}_{p}(1-\alpha )}{\left[\left({\displaystyle \sum _{p\in {\mathrm{\Gamma}}_{dj}}^{}{y}_{p}{f}_{p}}\right)+{x}_{dj}{c}_{dj}\right]}$. The rewarding rate for passenger $p$ is $\frac{{\sigma}_{p}^{P}{F}_{dj}(x,y)}{{f}_{p}}=\frac{{y}_{p}(1-\alpha ){F}_{dj}(x,y)}{\left[\left({\displaystyle \sum _{p\in {\mathrm{\Gamma}}_{dj}}^{}{y}_{p}{f}_{p}}\right)+{x}_{dj}{c}_{dj}\right]}$. If the Local Proportional Method is used, the rewarding rate for driver $d$ is ${\sigma}_{d}^{D}{F}_{dj}(x,y)$, where ${\sigma}_{d}^{D}=\frac{{x}_{dj}{c}_{dj}(1-\alpha )}{\left[\left({\displaystyle \sum _{p\in {\mathrm{\Gamma}}_{dj}}^{}{y}_{p}{f}_{p}}\right)+{x}_{dj}{c}_{dj}\right]}$.

- (iii)
- In the Global Proportional Method, the rewarding rate for passenger $p$ is ${\beta}_{p}^{P}F(x,y)=\frac{(1-\alpha ){y}_{p}{f}_{p}^{}F(x,y)}{\left[\left({\displaystyle \sum _{p=1}^{P}{y}_{p}{f}_{p}^{}}\right)+\left({\displaystyle \sum _{d=1}^{D}{\displaystyle \sum _{j=1}^{{J}_{d}}}{x}_{dj}{c}_{dj}}\right)\right]}$. The rewarding rate for passenger $p$ is $\frac{{\beta}_{p}^{P}F(x,y)}{{f}_{p}}$ = $\frac{(1-\alpha ){y}_{p}F(x,y)}{\left[\left({\displaystyle \sum _{p=1}^{P}{y}_{p}{f}_{p}^{}}\right)+\left({\displaystyle \sum _{d=1}^{D}{\displaystyle \sum _{j=1}^{{J}_{d}}}{x}_{dj}{c}_{dj}}\right)\right]}$. In the Global Proportional Method, the rewarding rate for driver $d$ is ${\beta}_{d}^{D}F(x,y)=\frac{(1-\alpha ){x}_{dj}{c}_{dj}^{}F(x,y)}{\left[\left({\displaystyle \sum _{p=1}^{P}{y}_{p}{f}_{p}^{}}\right)+\left({\displaystyle \sum _{d=1}^{D}{\displaystyle \sum _{j=1}^{{J}_{d}}}{x}_{dj}{c}_{dj}}\right)\right]}$. The rewarding rate for driver $d$ is $\frac{{\beta}_{d}^{D}F(x,y)}{{c}_{dj}^{}}=\frac{(1-\alpha ){x}_{dj}F(x,y)}{\left[\left({\displaystyle \sum _{p=1}^{P}{y}_{p}{f}_{p}^{}}\right)+\left({\displaystyle \sum _{d=1}^{D}{\displaystyle \sum _{j=1}^{{J}_{d}}}{x}_{dj}{c}_{dj}}\right)\right]}$. Therefore, the rewarding rate for each passenger $p$ and each driver $d$ is the same.

## Appendix C

**Proof of Property 4**

- (i)
- If the Fifty-Fifty Method is used, the rewarding rate for passenger $p$ is ${\sigma}_{p}^{P}(1-\alpha ){F}_{dj}(x,y)$ with ${\sigma}_{p}^{P}=\frac{1}{2}$.

- (ii)
- In the Global Proportional Method, the rewarding rate for passenger $p$ is ${\beta}_{p}^{P}F(x,y)=\frac{(1-\alpha ){y}_{p}{f}_{p}^{}F(x,y)}{\left[\left({\displaystyle \sum _{p=1}^{P}{y}_{p}{f}_{p}^{}}\right)+\left({\displaystyle \sum _{d=1}^{D}{\displaystyle \sum _{j=1}^{{J}_{d}}}{x}_{dj}{c}_{dj}}\right)\right]}$.

- (iii)
- According to (i), the number of acceptable shared rides under the Fifty-Fifty Method is $\sum _{d\in {\Re}_{FF\_D}}^{}{\displaystyle \sum _{j=1}^{{J}_{d}}}{x}_{dj}}\ge 0$.

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**Figure 1.**Verification of Property 1 and Property 3 for total acceptable rides of Case 1 through Case 8 under $\alpha $ = 0.05.

**Figure 2.**Verification of Property 1 and Property 3 for total ridesharing participants of three schemes for Case 1 through Case 8 under $\alpha $ = 0.05.

**Figure 3.**Verification of Property 2 and Property 4 for total acceptable rides of three schemes under $\alpha $ = 0.2.

**Figure 4.**Verification of Property 2 and Property 4 for total acceptable rides of three schemes under $\alpha $ = 0.2.

Variable | Meaning |
---|---|

$p$ | a passenger, $p\in \{1,2,3,\dots ,P\}$ and $P$ is the number of passengers |

$d$ | a driver, $d\in \{1,2,3,\dots ,D\}$ and $D$ is the number of drivers |

$k$ | the index of a location, $k\in \{1,2,\dots ,P\}$ |

${J}_{d}$ | total bids submitted by driver $d\in \{1,2,\dots ,D\}$ |

$j$ | A bid of a driver, $j\in \{1,2,\dots ,{J}_{d}\}$ and ${J}_{d}$ is the number of bids submitted by driver $d\in \{1,2,\dots ,D\}$ driver $d$‘s $j$-th bid |

$BI{D}_{dj}$ | $BI{D}_{dj}$ = $({q}_{dj1}^{1},{q}_{dj2}^{1},{q}_{dj3}^{1},\dots ,{q}_{djP}^{1},{q}_{dj1}^{2},{q}_{dj2}^{2},{q}_{dj3}^{2},\dots ,{q}_{djP}^{2},{o}_{dj},{c}_{dj})$, where ${q}_{djk}^{1}$: seats allocated to pick-up location of passenger $k$, ${q}_{djk}^{2}$: seats released at drop-off location of passenger $k$,${o}_{dj}$ ${o}_{dj}$: original cost of the driver without ridesharing ${c}_{dj}$: the transport cost of driver $d$’s $j$-th bid. |

$BI{D}_{p}$ | passenger $p$’s bid $BI{D}_{p}$ = $({s}_{p1}^{1},{s}_{p2}^{1},{s}_{p3}^{1},\dots ,{s}_{pP}^{1},{s}_{p1}^{2},{s}_{p2}^{2},{s}_{p3}^{2}\dots ,{s}_{pP}^{2},{f}_{p})$, where ${s}_{pk}^{1}$: seats requested for pick-up location of passenger $k$, ${s}_{pk}^{2}$: seats released at drop-off location of passenger $k$, ${f}_{p}$: passenger $p$’s original cost without ridesharing. |

${x}_{dj}$ | driver $d$’s decision variable; if ${x}_{dj}$ = 1, the $j-th$ bid of driver $d$ is a winning bid and the $j-th$ bid of driver $d$ is not a winning bid otherwise (${x}_{dj}$ = 0) |

${y}_{p}$ | passenger $p$’s decision variable; if ${y}_{p}$ = 1, passenger $p$’s bid is a winning bid and passenger $p$’s bid is not a winning bid otherwise (${y}_{p}$ = 0) |

$\alpha $ | ridesharing information provider’s share value |

${\beta}_{p}^{P}$ | passenger $p$’s share value |

${\beta}_{d}^{D}$ | driver $d$’s share value |

${\mathrm{\Gamma}}_{dj}$ | the set of passengers on the ride of driver $d$’s $j$–th bid |

$F(x,y)$ | the cost savings function $F(x,y)=\left({\displaystyle \sum _{p=1}^{P}{y}_{p}\left({f}_{p}^{}\right)}\right)+\left({\displaystyle \sum _{d=1}^{D}{\displaystyle \sum _{j=1}^{{J}_{d}}}{x}_{dj}{o}_{dj}}\right)\u2013\left({\displaystyle \sum _{d=1}^{D}{\displaystyle \sum _{j=1}^{{J}_{d}}}{x}_{dj}{c}_{dj}}\right)$ |

${F}_{dj}(x,y)$ | the cost savings function of driver $d$’s $j$–th bid ${F}_{dj}(x,y)=\left[\left({\displaystyle \sum _{p\in {\mathrm{\Gamma}}_{dj}}^{}{y}_{p}{f}_{p}^{}}\right)+{x}_{dj}{o}_{dj}-\left({x}_{dj}{c}_{dj}\right)\right]$ |

Scheme | Stakeholder | Share Value | |
---|---|---|---|

Fifty-Fifty (FF) Method | information provider | $\alpha $ | (7) |

passenger | ${\beta}_{p}^{P}=\frac{{y}_{p}(1-\alpha ){F}_{dj}(x,y)}{2F(x,y)}$ | (8) | |

driver | ${\beta}_{d}^{D}=\frac{{\displaystyle \sum _{j=1}^{{J}_{d}}{x}_{dj}(1-\alpha ){F}_{dj}(x,y)}}{2F(x,y)}$ | (9) | |

Local Proportional (LP) Method | information provider | $\alpha $ | (10) |

passenger | ${\beta}_{p}^{P}=\frac{{\sigma}_{p}^{P}{y}_{p}(1-\alpha ){F}_{dj}(x,y)}{F(x,y)}$ | (11) | |

where, ${\sigma}_{p}^{P}=\frac{{f}_{p}}{\left[\left({\displaystyle \sum _{p\in {\mathrm{\Gamma}}_{dj}}^{}{y}_{p}{f}_{p}}\right)+{x}_{dj}{c}_{dj}\right]}$ | (12) | ||

driver | ${\beta}_{d}^{D}=\frac{{\displaystyle \sum _{j=1}^{{J}_{d}}{x}_{dj}{\sigma}_{d}^{D}(1-\alpha ){F}_{dj}(x,y)}}{F(x,y)}$ | (13) | |

where, ${\sigma}_{d}^{D}=\frac{{c}_{dj}}{\left[\left({\displaystyle \sum _{p\in {\mathrm{\Gamma}}_{dj}}^{}{y}_{p}{f}_{p}}\right)+{x}_{dj}{c}_{dj}\right]}$ | (14) | ||

Global Proportional (GP) Method | information provider | $\alpha $ | (15) |

passenger | ${\beta}_{p}^{P}=\frac{(1-\alpha ){y}_{p}{f}_{p}^{}}{\left[\left({\displaystyle \sum _{p=1}^{P}{y}_{p}{f}_{p}^{}}\right)+\left({\displaystyle \sum _{d=1}^{D}{\displaystyle \sum _{j=1}^{{J}_{d}}}{x}_{dj}{c}_{dj}}\right)\right]}$ | (16) | |

driver | ${\beta}_{d}^{D}=\frac{{\displaystyle \sum _{j=1}^{{J}_{d}}}(1-\alpha ){x}_{dj}{c}_{dj}^{}}{\left[\left({\displaystyle \sum _{p=1}^{P}{y}_{p}{f}_{p}^{}}\right)+\left({\displaystyle \sum _{d=1}^{D}{\displaystyle \sum _{j=1}^{{J}_{d}}}{x}_{dj}{c}_{dj}}\right)\right]}$ | (17) |

**Table 3.**Number of drivers and passengers accepted in Case 1 for $\alpha $ = 0.05, ${r}_{D}$ = 0.11 and ${r}_{P}$ = 0.1.

Case 1 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.114 | 1 | 0.114 | 1 | 0.069 | 0 |

P1 | 0.114 | 1 | 0.114 | 1 | 0.34 | 0 | |

Total rides | 1 | 1 | 0 | ||||

Participants on acceptable rides | 2 | 2 | 0 |

**Table 4.**Number of drivers and passengers accepted in Case 2 for $\alpha $ = 0.05, ${r}_{D}$ = 0.11 and ${r}_{P}$ = 0.1.

Case 2 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.21 | 1 | 0.119 | 1 | 0.068 | 0 |

P1 | 0.21 | 1 | 0.119 | 1 | 0.475 | 0 | |

Ride 2 | D2 | 0.21 | 1 | 0.218 | 1 | 0.142 | 1 |

P2 | 0.21 | 1 | 0.218 | 1 | 0.475 | 1 | |

Ride 3 | D3 | 0.21 | 1 | 0.277 | 1 | 0.196 | 1 |

P3 | 0.21 | 1 | 0.277 | 1 | 0.475 | 1 | |

Total rides | 3 | 3 | 2 | ||||

Participants on acceptable rides | 6 | 6 | 4 |

**Table 5.**Number of drivers and passengers accepted in Case 3 for $\alpha $ = 0.05, ${r}_{D}$ = 0.11 and ${r}_{P}$ = 0.1.

Case 3 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.166 | 1 | 0.189 | 1 | 0.121 | 1 |

P1 | 0.166 | 1 | 0.189 | 1 | 0.438 | 1 | |

Ride 2 | D2 | 0.166 | 1 | 0.098 | 0 | 0.06 | 0 |

P2 | 0.166 | 1 | 0.098 | 0 | 0.258 | 0 | |

Ride 3 | D3 | 0.166 | 1 | 0.193 | 1 | 0.121 | 1 |

P3 | 0.166 | 1 | 0.193 | 1 | 0.475 | 1 | |

Total rides | 3 | 2 | 2 | ||||

Participants on acceptable rides | 6 | 4 | 4 |

**Table 6.**Number of drivers and passengers accepted in Case 4 for $\alpha $ = 0.05, ${r}_{D}$ = 0.11 and ${r}_{P}$ = 0.1.

Case 4 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.209 | 1 | 0.232 | 1 | 0.154 | 1 |

P1 | 0.209 | 1 | 0.232 | 1 | 0.475 | 1 | |

Ride 2 | D2 | 0.209 | 1 | 0.054 | 0 | 0.034 | 0 |

P2 | 0.209 | 1 | 0.054 | 0 | 0.132 | 0 | |

Ride 3 | D3 | 0.209 | 1 | 0.353 | 1 | 0.28 | 1 |

P3 | 0.209 | 1 | 0.353 | 1 | 0.475 | 1 | |

Total rides | 3 | 2 | 2 | ||||

Participants on acceptable rides | 6 | 4 | 4 |

**Table 7.**Number of drivers and passengers accepted in Case 5 for $\alpha $ = 0.05, ${r}_{D}$ = 0.12 and ${r}_{P}$ = 0.1.

Case 5 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.207 | 1 | 0.265 | 1 | 0.184 | 1 |

P1 | 0.207 | 1 | 0.265 | 1 | 0.475 | 1 | |

Ride 2 | D2 | 0.207 | 1 | 0.138 | 1 | 0.081 | 0 |

P2 | 0.207 | 1 | 0.138 | 1 | 0.475 | 0 | |

Ride 3 | D3 | 0.207 | 1 | 0.223 | 1 | 0.145 | 1 |

P3 | 0.207 | 1 | 0.223 | 1 | 0.475 | 1 | |

Total rides | 3 | 3 | 2 | ||||

Participants on acceptable rides | 6 | 6 | 4 |

**Table 8.**Number of drivers and passengers accepted in Case 6 for $\alpha $ = 0.05, ${r}_{D}$ = 0.11 and ${r}_{P}$ = 0.1.

Case 6 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.2 | 1 | 0.192 | 1 | 0.123 | 1 |

P1 | 0.2 | 1 | 0.192 | 1 | 0.434 | 1 | |

Ride 2 | D2 | 0.2 | 1 | 0.12 | 1 | 0.069 | 0 |

P2 | 0.2 | 1 | 0.12 | 1 | 0.471 | 0 | |

Ride 3 | D3 | 0.2 | 1 | 0.253 | 1 | 0.172 | 1 |

P3 | 0.2 | 1 | 0.253 | 1 | 0.475 | 1 | |

Ride 4 | D4 | 0.2 | 1 | 0.254 | 1 | 0.174 | 1 |

P4 | 0.2 | 1 | 0.254 | 1 | 0.475 | 1 | |

Total rides | 4 | 4 | 3 | ||||

Participants on acceptable rides | 8 | 8 | 6 |

**Table 9.**Number of drivers and passengers accepted in Case 7 for $\alpha $ = 0.05, ${r}_{D}$ = 0.11 and ${r}_{P}$ = 0.1.

Case 7 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.234 | 1 | 0.241 | 1 | 0.172 | 1 |

P1 | 0.234 | 1 | 0.241 | 1 | 0.401 | 1 | |

Ride 2 | D2 | 0.234 | 1 | 0.238 | 1 | 0.159 | 1 |

P2 | 0.234 | 1 | 0.238 | 1 | 0.475 | 1 | |

Ride 3 | D3 | 0.234 | 1 | 0.144 | 1 | 0.085 | 0 |

P3 | 0.234 | 1 | 0.144 | 1 | 0.475 | 0 | |

Ride 4 | D4 | 0.234 | 1 | 0.153 | 1 | 0.091 | 0 |

P4 | 0.234 | 1 | 0.153 | 1 | 0.475 | 0 | |

Ride 5 | D5 | 0.234 | 1 | 0.331 | 1 | 0.254 | 1 |

P5 | 0.234 | 1 | 0.331 | 1 | 0.475 | 1 | |

Total rides | 5 | 5 | 3 | ||||

Participants on acceptable rides | 10 | 10 | 6 |

**Table 10.**Number of drivers and passengers accepted in Case 8 for $\alpha $ = 0.05, ${r}_{D}$ = 0.11 and ${r}_{P}$ = 0.1.

Case 8 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.181 | 1 | 0.222 | 1 | 0.155 | 1 |

P1 | 0.181 | 1 | 0.222 | 1 | 0.389 | 1 | |

Ride 2 | D2 | 0.181 | 1 | 0.223 | 1 | 0.146 | 1 |

P2 | 0.181 | 1 | 0.223 | 1 | 0.475 | 1 | |

Ride 3 | D3 | 0.181 | 1 | 0.275 | 1 | 0.194 | 1 |

P3 | 0.181 | 1 | 0.275 | 1 | 0.475 | 1 | |

Ride 4 | D4 | 0.181 | 1 | 0.21 | 1 | 0.135 | 1 |

P4 | 0.181 | 1 | 0.21 | 1 | 0.475 | 1 | |

Ride 5 | D5 | 0.181 | 1 | 0.145 | 1 | 0.085 | 0 |

P5 | 0.181 | 1 | 0.145 | 1 | 0.475 | 0 | |

Ride 6 | D6 | 0.181 | 1 | 0.029 | 0 | 0.019 | 0 |

P6 | 0.181 | 1 | 0.029 | 0 | 0.062 | 0 | |

Ride 7 | D7 | 0.181 | 1 | 0.2 | 1 | 0.127 | 1 |

P7 | 0.181 | 1 | 0.2 | 1 | 0.475 | 1 | |

Total rides | 7 | 6 | 5 | ||||

Participants on acceptable rides | 14 | 12 | 10 |

**Table 11.**Number of drivers and passengers accepted in Case 1 for $\alpha $ = 0.2, ${r}_{D}$ = 0.2 and ${r}_{P}$ = 0.1.

Case 1 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.096 | 0 | 0.096 | 0 | 0.058 | 0 |

P1 | 0.096 | 0 | 0.096 | 0 | 0.286 | 0 | |

Total rides | 0 | 0 | 0 | ||||

Participants on acceptable rides | 0 | 0 | 0 |

**Table 12.**Number of drivers and passengers accepted in Case 2 for $\alpha $ = 0.2, ${r}_{D}$ = 0.2 and ${r}_{P}$ = 0.1.

Case 2 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.176 | 0 | 0.1 | 0 | 0.057 | 0 |

P1 | 0.176 | 0 | 0.1 | 0 | 0.4 | 0 | |

Ride 2 | D2 | 0.176 | 0 | 0.184 | 0 | 0.119 | 0 |

P2 | 0.176 | 0 | 0.184 | 0 | 0.4 | 0 | |

Ride 3 | D3 | 0.176 | 0 | 0.234 | 1 | 0.165 | 0 |

P3 | 0.176 | 0 | 0.234 | 1 | 0.4 | 0 | |

Total rides | 0 | 1 | 0 | ||||

Participants on acceptable rides | 0 | 2 | 0 |

**Table 13.**Number of drivers and passengers accepted in Case 3 for $\alpha $ = 0.2, ${r}_{D}$ = 0.2 and ${r}_{P}$ = 0.1.

Case 3 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.14 | 0 | 0.159 | 0 | 0.102 | 0 |

P1 | 0.14 | 0 | 0.159 | 0 | 0.369 | 0 | |

Ride 2 | D2 | 0.14 | 0 | 0.082 | 0 | 0.051 | 0 |

P2 | 0.14 | 0 | 0.082 | 0 | 0.217 | 0 | |

Ride 3 | D3 | 0.14 | 0 | 0.163 | 0 | 0.102 | 0 |

P3 | 0.14 | 0 | 0.163 | 0 | 0.4 | 0 | |

Total rides | 0 | 0 | 0 | ||||

Participants on acceptable rides | 0 | 0 | 0 |

**Table 14.**Number of drivers and passengers accepted in Case 4 for $\alpha $ = 0.2, ${r}_{D}$ = 0.2 and ${r}_{P}$ = 0.1.

Case 4 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.176 | 0 | 0.196 | 0 | 0.13 | 0 |

P1 | 0.176 | 0 | 0.196 | 0 | 0.4 | 0 | |

Ride 2 | D2 | 0.176 | 0 | 0.045 | 0 | 0.029 | 0 |

P2 | 0.176 | 0 | 0.045 | 0 | 0.111 | 0 | |

Ride 3 | D3 | 0.176 | 0 | 0.297 | 1 | 0.236 | 1 |

P3 | 0.176 | 0 | 0.297 | 1 | 0.4 | 1 | |

Total rides | 0 | 1 | 1 | ||||

Participants on acceptable rides | 0 | 2 | 2 |

**Table 15.**Number of drivers and passengers accepted in Case 5 for $\alpha $ = 0.2, ${r}_{D}$ = 0.2 and ${r}_{P}$ = 0.1.

Case 5 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.174 | 0 | 0.224 | 1 | 0.155 | 0 |

P1 | 0.174 | 0 | 0.224 | 1 | 0.4 | 0 | |

Ride 2 | D2 | 0.174 | 0 | 0.116 | 0 | 0.068 | 0 |

P2 | 0.174 | 0 | 0.116 | 0 | 0.4 | 0 | |

Ride 3 | D3 | 0.174 | 0 | 0.187 | 0 | 0.122 | 0 |

P3 | 0.174 | 0 | 0.187 | 0 | 0.4 | 0 | |

Total rides | 0 | 1 | 0 | ||||

Participants on acceptable rides | 0 | 2 | 0 |

**Table 16.**Number of drivers and passengers accepted in Case 6 for $\alpha $ = 0.2, ${r}_{D}$ = 0.2 and ${r}_{P}$ = 0.1.

Case 6 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.168 | 0 | 0.161 | 0 | 0.103 | 0 |

P1 | 0.168 | 0 | 0.161 | 0 | 0.365 | 0 | |

Ride 2 | D2 | 0.168 | 0 | 0.101 | 0 | 0.058 | 0 |

P2 | 0.168 | 0 | 0.101 | 0 | 0.397 | 0 | |

Ride 3 | D3 | 0.168 | 0 | 0.213 | 1 | 0.145 | 0 |

P3 | 0.168 | 0 | 0.213 | 1 | 0.4 | 0 | |

Ride 4 | D4 | 0.168 | 0 | 0.214 | 1 | 0.146 | 0 |

P4 | 0.168 | 0 | 0.214 | 1 | 0.4 | 0 | |

Total rides | 0 | 2 | 0 | ||||

Participants on acceptable rides | 0 | 4 | 0 |

**Table 17.**Number of drivers and passengers accepted in Case 7 for $\alpha $ = 0.2, ${r}_{D}$ = 0.2 and ${r}_{P}$ = 0.1.

Case 7 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.197 | 0 | 0.203 | 1 | 0.145 | 0 |

P1 | 0.197 | 0 | 0.203 | 1 | 0.337 | 0 | |

Ride 2 | D2 | 0.197 | 0 | 0.201 | 1 | 0.134 | 0 |

P2 | 0.197 | 0 | 0.201 | 1 | 0.4 | 0 | |

Ride 3 | D3 | 0.197 | 0 | 0.121 | 0 | 0.071 | 0 |

P3 | 0.197 | 0 | 0.121 | 0 | 0.4 | 0 | |

Ride 4 | D4 | 0.197 | 0 | 0.129 | 0 | 0.077 | 0 |

P4 | 0.197 | 0 | 0.129 | 0 | 0.4 | 0 | |

Ride 5 | D5 | 0.197 | 0 | 0.279 | 1 | 0.214 | 1 |

P5 | 0.197 | 0 | 0.279 | 1 | 0.4 | 1 | |

Total rides | 0 | 3 | 1 | ||||

Participants on acceptable rides | 0 | 6 | 2 |

**Table 18.**Number of drivers and passengers accepted in Case 8 for $\alpha $ = 0.2, ${r}_{D}$ = 0.2 and ${r}_{P}$ = 0.1.

Case 8 | Participant | GP | GP Accepted | LP | LP Accepted | FF | FF Accepted |
---|---|---|---|---|---|---|---|

Ride 1 | D1 | 0.152 | 0 | 0.187 | 0 | 0.131 | 0 |

P1 | 0.152 | 0 | 0.187 | 0 | 0.328 | 0 | |

Ride 2 | D2 | 0.152 | 0 | 0.188 | 0 | 0.123 | 0 |

P2 | 0.152 | 0 | 0.188 | 0 | 0.4 | 0 | |

Ride 3 | D3 | 0.152 | 0 | 0.232 | 1 | 0.163 | 0 |

P3 | 0.152 | 0 | 0.232 | 1 | 0.4 | 0 | |

Ride 4 | D4 | 0.152 | 0 | 0.177 | 0 | 0.114 | 0 |

P4 | 0.152 | 0 | 0.177 | 0 | 0.4 | 0 | |

Ride 5 | D5 | 0.152 | 0 | 0.122 | 0 | 0.072 | 0 |

P5 | 0.152 | 0 | 0.122 | 0 | 0.4 | 0 | |

Ride 6 | D6 | 0.152 | 0 | 0.025 | 0 | 0.016 | 0 |

P6 | 0.152 | 0 | 0.025 | 0 | 0.052 | 0 | |

Ride 7 | D7 | 0.152 | 0 | 0.169 | 0 | 0.107 | 0 |

P7 | 0.152 | 0 | 0.169 | 0 | 0.4 | 0 | |

Total rides | 0 | 1 | 0 | ||||

Participants on acceptable rides | 0 | 2 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Hsieh, F.-S.
Improving Acceptability of Cost Savings Allocation in Ridesharing Systems Based on Analysis of Proportional Methods. *Systems* **2023**, *11*, 187.
https://doi.org/10.3390/systems11040187

**AMA Style**

Hsieh F-S.
Improving Acceptability of Cost Savings Allocation in Ridesharing Systems Based on Analysis of Proportional Methods. *Systems*. 2023; 11(4):187.
https://doi.org/10.3390/systems11040187

**Chicago/Turabian Style**

Hsieh, Fu-Shiung.
2023. "Improving Acceptability of Cost Savings Allocation in Ridesharing Systems Based on Analysis of Proportional Methods" *Systems* 11, no. 4: 187.
https://doi.org/10.3390/systems11040187