# Implementing Horizontal Cooperation in Public Transport and Parcel Deliveries: The Cooperative Share-A-Ride Problem

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions and noise), they are not subject to accessibility restrictions, which usually affect trucks in some neighborhoods. Moreover, from the administration point of view, these transportation services stimulate the level of service, especially for regions characterized by a particular topography or by low demand.

## 2. Literature Review

## 3. Model Formulation

#### Properties

**Proposition**

**1**

**Proof.**

**Proposition**

**2**

- 1.
- ${X}^{coop-SARP\left({\alpha}_{1}\right)}\subseteq {X}^{coop-SARP\left({\alpha}_{2}\right)}$ and
- 2.
- ${\Theta}^{coop-SARP\left({\alpha}_{1}\right)}\le {\Theta}^{coop-SARP\left({\alpha}_{2}\right)}$.

**Proof.**

**Proposition**

**3**

- 1.
- ${X}^{coop-SARP(\alpha )}\subseteq {X}^{coop-SARP(-\infty )}$,
- 2.
- ${\Theta}^{coop-SARP(\alpha )}\le {\Theta}^{coop-SARP(-\infty )}$, and hence ${\Theta}^{coop-SARP(-\infty )}$ is an upper bound on the objective function value for any α value.

**Proof.**

## 4. Case Study

- The result of the model without cooperation
- The result of the cooperative model with $\alpha =100\%$
- The result of the cooperative model with $\alpha =80\%$
- The result of the cooperative model with $\alpha =-\infty $

## 5. Computational Results

#### 5.1. Instance Generation

- Group A: eight requests equally distributed between two service providers
- Group B: ten requests equally distributed between two service providers
- Group C: six requests equally distributed between three service providers
- Group D: nine requests equally distributed between three service providers

#### 5.2. Statistics

- Objective function value, computed as $\Theta (\alpha )$.
- Profits, computed as rewards minus routing costs, i.e.,$$\sum _{k\in \mathcal{K}}\sum _{c\in \mathcal{C}}{\eta}_{c}{y}_{c}^{k}-\sum _{k\in \mathcal{K}}\sum _{i\in \mathcal{V}}\sum _{j\in \mathcal{V}}{d}_{ij}{x}_{ij}^{k}.$$
- Delay costs, i.e., ${\sum}_{k\in \mathcal{K}}{\sum}_{c\in {\mathcal{C}}^{p}}\gamma ({l}_{{O}_{c}}^{k}+{l}_{{D}_{c}}^{k})$.
- Lost rewards, i.e., ${\sum}_{c\in \mathcal{C}}{\rho}_{c}(1-{\sum}_{k\in \mathcal{K}}{y}_{c}^{k})$.

- Percentage of increase in served passenger requests with respect to the noncooperative setting, where the percentage of served passenger requests for a service provider is defined as ${\sum}_{c\in {\mathcal{C}}_{m}^{p}}{\sum}_{k\in {\mathcal{K}}_{m}}{y}_{c}^{k}/\mid {\mathcal{C}}_{m}^{p}\mid $.
- Percentage of increase in served parcel requests with respect to the noncooperative setting, where the percentage of served parcel requests by a service provider is defined as ${\sum}_{c\in {\mathcal{C}}_{m}^{f}}{\sum}_{k\in {\mathcal{K}}_{m}}{y}_{c}^{k}/\mid {\mathcal{C}}_{m}^{f}\mid $.
- Average occupancy rate of vehicles by passengers, computed as the time in which at least one passenger is carried over the total routing time, where the occupancy rate for each vehicle is computed as ${\sum}_{i\in \mathcal{V}}{\sum}_{j\in \mathcal{V}}{x}_{ij}^{k}{d}_{ij}{\omega}_{i}^{k}/{\sum}_{i\in \mathcal{V}}{\sum}_{j\in \mathcal{V}}{x}_{ij}^{k}{d}_{ij}$.
- Average occupancy rate of vehicles by parcels, computed as the time in which at least one parcel is carried over the total routing time, where the occupancy rate of each vehicle is computed as ${\sum}_{i\in \mathcal{V}}{\sum}_{j\in \mathcal{V}}{x}_{ij}^{k}{d}_{ij}{\lambda}_{i}^{k}/{\sum}_{i\in \mathcal{V}}{\sum}_{j\in \mathcal{V}}{x}_{ij}^{k}{d}_{ij}$.
- Average, minimum, and maximum increases in the profit of each service provider with respect to the noncooperative setting, where the profit is computed as the difference between rewards and routing cost, i.e.,$$\sum _{k\in {\mathcal{K}}_{m}}\sum _{c\in \mathcal{C}}{\eta}_{c}{y}_{c}^{k}-\sum _{k\in {\mathcal{K}}_{m}}\sum _{i\in \mathcal{V}}\sum _{j\in \mathcal{V}}{d}_{ij}{x}_{ij}^{k}\phantom{\rule{1.em}{0ex}}m\in \mathcal{M}.$$
- Average, minimum, and maximum increases in travel times, where the travel time is computed as ${u}_{{h}_{m}^{{}^{\prime}}}^{k}-{u}_{{h}_{m}}^{k}$.

- Number of variables (continuous and integer)
- Number of constraints
- Runtime (in CPU seconds)

#### 5.3. The Coop-SARP($\alpha $) Model against the Ncoop-SARP Model

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Service assignment to service providers under different operating scenarios (pickup , delivery points , and depots ). For the sake of representation clarity, the connections between nodes are represented by segments, but in this work, map-based connections and distances have been considered.

**Figure 2.**Objective function values for instance groups A, B, C, and D and different cooperation levels.

**Figure 3.**Profits (rewards minus routing costs) for instance groups A, B, C, and D and different cooperation levels.

**Table 1.**Sets, parameters, and variables for the coop-share-a-ride problem (SARP) (2)–().

Sets | |

$\mathcal{M}$ | Set of service providers $\mathcal{M}=\{1,\cdots ,m,\cdots ,M\}$ |

${\mathcal{C}}_{m}$ | Set of requests for each service provider $m\in \mathcal{M}$ |

${\mathcal{C}}^{p}$ | Set of passenger requests |

${\mathcal{C}}^{f}$ | Set of parcel requests |

${\mathcal{C}}_{m}^{p}$ | Set of passenger requests for service provider $m\in \mathcal{M}$ |

${\mathcal{C}}_{m}^{f}$ | Set of parcel requests for service provider $m\in \mathcal{M}$ |

$\mathcal{C}$ | Set of requests, $\mathcal{C}\equiv {\cup}_{m\in \mathcal{M}}{C}_{m}$ or $\mathcal{C}\equiv {\mathcal{C}}^{p}\cup {\mathcal{C}}^{f}$ |

$\mathcal{V}$ | Set of vertices, $\mathcal{V}\equiv \mathcal{O}\cup \mathcal{D}\cup \mathcal{H}$ |

${\mathcal{V}}^{p}$ | Set of vertices that are either the origin or destination of a passenger request |

${\mathcal{V}}^{f}$ | Set of vertices that are either the origin or destination of a parcel request |

$\mathcal{O}$ | Set of request origins |

$\mathcal{D}$ | Set of request destinations |

$\mathcal{H}$ | Set of depots, $\mathcal{H}\subseteq \mathcal{V}$, where each depot ${h}_{m}\in \mathcal{H}$ has been doubled in ${h}_{m}$ and ${h}_{m}^{\prime}$ |

$\mathcal{A}$ | Set of arcs |

${\mathcal{K}}_{m}$ | Set of vehicles owned by service provider m |

$\mathcal{K}$ | Set of vehicles, $\mathcal{K}\equiv {\cup}_{m\in \mathcal{M}}{\mathcal{K}}_{m}$ |

Parameters | |

${Q}_{k}^{p}$ | Passenger capacity of vehicle k |

${Q}_{k}^{f}$ | Parcel capacity of vehicle k |

${T}_{k}$ | Maximal route duration for vehicle k |

${q}_{i}$ | Load of request i |

${s}_{i}$ | Service time of node i |

$[{e}_{i},{l}_{i}]$ | Time window for parcel request i |

${U}_{{O}_{c}}$ | Desired pickup ${O}_{c}$ time for passenger request $c\in {\mathcal{C}}^{p}$ |

${U}_{{D}_{c}}$ | Desired delivery ${D}_{c}$ time for passenger request $c\in {\mathcal{C}}^{p}$ |

${t}_{ij}$ | Travel time between stops i and j |

${\overline{r}}_{c}$ | Passenger request c’s maximum ride time |

${\eta}_{c}$ | Reward for each served passenger or parcel request c |

${d}_{ij}$ | Routing cost for arc $(i,j)$ |

$\gamma $ | Unit penalty cost for deviations from the passenger request desired pickup or delivery time |

${\rho}_{c}$ | Lost reward for not accepting request c (computed as percentage of the request reward) |

Variables | |

${x}_{ij}^{k}$ | Binary variable equal to 1 if arc $(i,j)\in \mathcal{A}$ is traversed by vehicle k, 0 otherwise |

${y}_{c}$ | Binary variable equal to 1 if passenger’s request $c\in {\mathcal{C}}^{p}$ is served and 0 otherwise |

${u}_{i}^{k}$ | Time at which vehicle k starts serving node i |

${l}_{{O}_{c}}^{k}$ | Discrepancy between the real serving time and desired serving time of origin ${O}_{c}$ by vehicle k |

${l}_{{D}_{c}}^{k}$ | Discrepancy between the real serving time and desired serving time of origin ${D}_{c}$ by vehicle k |

${w}_{i}^{k}$ | People load of vehicle k when leaving node i |

${\lambda}_{i}^{k}$ | Parcel load of vehicle k when leaving node i |

${r}_{c}^{k}$ | Ride time of passenger request $c\in C$ on vehicle k |

Parameter | Value | |
---|---|---|

$\mid {\mathcal{K}}_{m}\mid $ | 1 | |

${\eta}_{c}$ | $5*{d}_{ij}$ | |

$\gamma $ | $0.5$ | |

${\rho}_{c}$ | $5\%$${\eta}_{c}$ | |

${s}_{i}$ | 120 s | |

${\overline{r}}_{c}$ | 2000 s | |

${T}_{k}$ | 20,000 s | |

${Q}_{k}^{p}$ | 3 | |

${Q}_{k}^{f}$ | 300 |

**Table 3.**Model-based statistics for instance groups A, B, C, and D and different levels of cooperation $\alpha $.

Group | Cooperation ($\mathit{\alpha}$) | Avg. CPU Time (s) | Avg. # Variables | Avg. # Constraints |
---|---|---|---|---|

A | NC | 0.1 | 1040 | 2901.7 |

100% | 11.5 | 1040 | 2895.7 | |

90% | 9.5 | 1040 | 2895.7 | |

80% | 8.4 | 1040 | 2895.7 | |

B | NC | 0.5 | 1440 | 4064.7 |

100% | 2204.4 | 1440 | 4056.7 | |

90% | 1193.0 | 1440 | 4056.7 | |

80% | 274.2 | 1440 | 4056.7 | |

C | NC | 0.1 | 1296 | 3671.5 |

100% | 2.5 | 1296 | 3662.5 | |

90% | 2.8 | 1296 | 3662.5 | |

80% | 3.1 | 1296 | 3662.5 | |

D | NC | 0.4 | 2160 | 6214.5 |

100% | 969.5 | 2160 | 6199.5 | |

90% | 725.6 | 2160 | 6199.5 | |

80% | 1302.3 | 2160 | 6199.5 |

Group | Cooperation ($\mathit{\alpha}$) | Avg. Served Passengers (%) | Avg. Served Parcels (%) |
---|---|---|---|

A | 100% | +35.4 | +10.3 |

90% | +35.4 | +11.0 | |

80% | +31.1 | +8.3 | |

B | 100% | +20.5 | +5.9 |

90% | +21.15 | +4.1 | |

80% | +19.9 | +4.1 | |

C | 100% | +1.8 | +5.1 |

90% | +1.8 | +4.2 | |

80% | +1.8 | +4.2 | |

D | 100% | +65.6 | +13.2 |

90% | +65.6 | +13.8 | |

80% | +65.6 | +13.8 |

Group | Cooperation ($\mathit{\alpha}$) | Occupancy Passengers (%) | Occupancy Parcels (%) |
---|---|---|---|

A | NC | 35.5 | 56.2 |

100% | 48.2 | 62.3 | |

90% | 47.9 | 63.3 | |

80% | 48.1 | 60.7 | |

B | NC | 52.8 | 65.4 |

100% | 55.0 | 67.1 | |

90% | 61.1 | 67.7 | |

80% | 60.1 | 65.5 | |

C | NC | 29.2 | 34.8 |

100% | 36.1 | 36.1 | |

90% | 35.4 | 35.5 | |

80% | 36.9 | 36.1 | |

D | NC | 32.5 | 45.5 |

100% | 44.3 | 49.2 | |

90% | 43.1 | 52.4 | |

80% | 45.7 | 52.1 |

Group | Cooperation ($\mathit{\alpha}$) | Max. Increase (%) | Min. Increase (%) | Avg. Increase (%) |
---|---|---|---|---|

A | 100% | +24.3 | +5.1 | +14.7 |

90% | +25.8 | +2.8 | +14.3 | |

80% | +28.8 | +0.0 | + 14.4 | |

B | 100% | +41.5 | +5.7 | +23.6 |

90% | +45.8 | +1.5 | +23.6 | |

80% | +48.9 | +0.1 | +24.5 | |

C | 100% | +32.4 | +1.5 | +12.7 |

90% | +40.8 | −1.6 | +14.7 | |

80% | +52.1 | −5.1 | +17.5 | |

D | 100% | +76.1 | +0.8 | +32.1 |

90% | +81.8 | −0.8 | +32.7 | |

80% | +83.7 | −2.7 | + 32.7 |

**Table 7.**Increase in travel time with respect to Ncoop-SARP among all service providers (max, min, and avg).

Group | Cooperation ($\mathit{\alpha}$) | Max. (%) | Min. (%) | Avg. (%) |
---|---|---|---|---|

A | 100% | +16.0 | −12.3 | +1.8 |

90% | +16.8 | −14.6 | +1.1 | |

80% | +17.2 | −16.2 | +0.5 | |

B | 100% | +7.0 | −17.2 | −5.1 |

90% | +4.2 | −19.9 | −7.9 | |

80% | +5.8 | −20.9 | −7.5 | |

C | 100% | +6.7 | −14.6 | −3.9 |

90% | +10.6 | −23.1 | −6.5 | |

80% | +14.9 | −34.5 | −8.9 | |

D | 100% | +15.0 | −23.7 | −4.4 |

90% | +15.1 | −25.9 | −5.5 | |

80% | +16.6 | −27.5 | −5.2 |

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## Share and Cite

**MDPI and ACS Style**

Cavagnini, R.; Morandi, V.
Implementing Horizontal Cooperation in Public Transport and Parcel Deliveries: The Cooperative Share-A-Ride Problem. *Sustainability* **2021**, *13*, 4362.
https://doi.org/10.3390/su13084362

**AMA Style**

Cavagnini R, Morandi V.
Implementing Horizontal Cooperation in Public Transport and Parcel Deliveries: The Cooperative Share-A-Ride Problem. *Sustainability*. 2021; 13(8):4362.
https://doi.org/10.3390/su13084362

**Chicago/Turabian Style**

Cavagnini, Rossana, and Valentina Morandi.
2021. "Implementing Horizontal Cooperation in Public Transport and Parcel Deliveries: The Cooperative Share-A-Ride Problem" *Sustainability* 13, no. 8: 4362.
https://doi.org/10.3390/su13084362