# Dynamic Demand-Responsive Feeder Bus Network Design for a Short Headway Trunk Line

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Last-Mile Transportation Problem

#### 2.2. Demand-Responsive Transit

#### 2.3. The Dial-a-Ride Problem

#### 2.4. Coordinated Feeder Bus Transit Systems (CFBT)

## 3. Methodology

#### 3.1. Mathematical Formulation

#### 3.2. Algorithm

Algorithm 1: Pseudocode for the developed SA algorithm to solve the model |

Step 0: Initialization: |

Set s=1, Best Cost=positive infinite, T=T_{0}, alpha=0.99, previous station help=0, next station help=0, vehicle (s; s: 1 to S)=4, min vehicle(s; s: 1 to S) = 0 |

Step 1: Clustering: Define passenger’s cluster |

Step 2: Create random solution |

Considering the length of trip (number of passengers (s) +vehicles(s)-1) |

Step 3: Sort the initial solution based of desired departure time of passengers for each feeder bus |

set x as a random solution |

Step 4: Find optimal solution: |

IF It1<It1max, THEN |

go to step 5, otherwise go to step 7 |

END IF |

Step 5: IF It2< It2max, THEN |

go to step 5.1, otherwise go to step 6 |

END IF |

Step 5.1: Creating neighborhood: |

set x_{new} = a neighborhood of x |

Step 5.2: Run Path Creator Algorithm |

Step 5.3: IF best cost for x< best cost for x_{new}, THEN |

set x=x_{new} and go to step 5.6, otherwise go to step 5.4 |

END IF |

Step 5.4: p= exp-(cost x_{new} – cost x)/T·Cost x |

Step 5.5: Accept x= x_{new} by p -probability and reject- and x= x_{new} by (1-p) and go to step 5.6 |

Step 5.6: Cost calculation for x_{new} |

Step 5.7: IF best cost for x_{new} > best cost, THEN |

set bestsol= x_{new} |

END IF |

Step 5.8: IF x_{new} is feasible (considering time ratio), and best cost for x_{new} > feasible_best cost, THEN |

set feasible_bestsol= x_{new} |

END IF |

Step 5.9: Reducing the temperature: |

set T = alpha·T0 (0<alpha<1) |

Step 5.10: set It2=It2+1 and go to step 5 |

Step 6: Set It1=It1+1 and go to step 4 |

Step 7: IF feasible_bestsol is empty, THEN |

min vehicle (s)= vehicle (s)+1 and go to step 8, otherwise go to step 15 |

END IF |

Step 8: Calculate the following proportion for stations s-1 and s+1: number of passengers (s)/vehicle(s) |

Step 9: IF s-1 exists and vehicle (s-1)> min vehicle (s-1), THEN |

go to step 10, otherwise go to step 12 |

END IF |

Step 10: IF proportion for station s is $\le $ the proportion for station s+1 or vehicle (s+1) $\le $ min vehicle (s+1) |

go to step 11, otherwise go to step 12 |

END IF |

Step 11: Set previous station help (s)= previous station help (s)+1 and vehicle (s-1) = vehicle (s-1)-1, s=s-1, and go to step 2 |

Step 12: IF s+1 exists and vehicle (s+1)> min vehicle (s+1) THEN |

go to step 13, otherwise go to step 14 |

END IF |

Step 13: Set next station help (s)= next station help(s)+1 and vehicle (s+1) = vehicle (s+1)-1 and go to step 2 |

Step 14: Show “The problem is not feasible; more vehicles is needed” |

Step 15: IF s<S, THEN |

set s=s+1 and go to step 2, otherwise go to step 16 |

END IF |

Step 17: END |

Algorithm 2: Pseudocode for the developed SA path creator algorithm |

Step 0: Initialization: |

Set used_capacity=0, maximum_used_capacity =0, c |

Step 1: Calculate updated latest pickup time of passengers based on the sequence of passengers |

Step 2: Add the first passenger to the path |

Step 3: Update total traveled distance |

Total traveled distance= Distance to current passenger |

Step 4: IFarrival time to current passenger > Earliest pick up time of the passenger |

Arrival time to the passenger= traveled distance/speed, |

else |

The vehicle should wait for the passenger |

Arrival time to the passenger = Earliest pick up time of the passenger |

END IF |

Step 5: Update time |

Time = Arrival time to the current passenger |

Step 6: Update used_capacity |

used_capacity = used_capacity + 1 |

Step 7: Update maximum_used_capacity |

IF used_capacity > maximum_used_capacity |

maximum_used_capacity= used_capacity |

END IF |

Step 8: For the next passengers: |

Step 7.1: Add the next passenger to the path |

Step 7.2: IF arrival time to current passenger > Earliest pick up time of the passenger THEN go to step 9, otherwise go to step 10 |

Step 9:Arrival time to the passenger= time + traveled distance/speed |

Time = Arrival time to the passenger |

Step 10: The vehicle should wait for the passenger |

Arrival time to the passenger = Earliest pick up time of the passenger |

Time = Arrival time to the passenger |

Step 11:Total traveled distance= Total traveled distance+ distance to current passenger |

Step 12:Check to see if the cost of coming back to the station is better than continuing to board passengers (for the last passenger) |

IF the cost of coming back to station is less than continuing to board passengers THEN go to step 13, otherwise go to step 15Step 13: Check to see coming back to station does not makes the path infeasible |

IF in case of coming back to the station the updated latest pick up time of the next passenger is accepted THEN go to step 14, otherwise go to step 15 |

Step 14:Add station as the next visited node |

Step 14.1: Time = Arrival time to the station |

Step 14.2: used_capacity = 0 |

Step 14.3: Update total traveled distance |

Step 14.4: Calculate waiting times of boarded passengers |

Step 15: Accept continue to board the next passenger |

Step 15.1: Update used_capacity |

used_capacity = used_capacity + 1 |

Step 15.2: Update maximum_used_capacity |

IF used_capacity > maximum_used_capacity |

maximum_used_capacity= used_capacity |

END IF |

Step 15.3: go to step 8 |

Step 16: Calculate waiting times of current boarded passengers |

Step 17: END |

## 4. Hypothetical Network

## 5. Results and Analysis

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Study | Type | Approach | Objective Function | Constraints | Relocation of Vehicles | Multiple Trains | Individual Passenger Travel Time | Short Headway |
---|---|---|---|---|---|---|---|---|

Horn [16] | DRT | Heuristic methods | Min. total vehicle travel time and max. ridership | Serving service, the time window | × | |||

Diana, Dessouky [15] | DRT | Analytical modeling | The optimal number of vehicles for DRT | Serving service, the time window | × | |||

Cordeau and Laporte [25] | DARP | Branch-and-cut algorithm | Min. total routing cost | Fleet size, vehicle capacity, the time window | ||||

Pavone, Frazzoli [34] | DARP | Heuristic methods | Min. average time demands spend | Passenger demand, fleet size | ||||

Arbex and da Cunha [14] | DRT | Genetic algorithm | Min. total operator and users’ costs | Route length, number of routes, fleet size, bus capacity | × | |||

Mahéo, Kilby [17] | DRT | Bender decomposition | Min. trips’ traveling cost and cost of opening the bus legs | Trip connectivity, flow conservation | × | |||

Wang [11] | DRT | Tabu search | Min. waiting and in-vehicle travel times of passengers | Unserved passengers, fleet size, bus capacity | × | |||

Dou, Gong [40] | CFBT | Genetic algorithm and Frank–Wolfe algorithms | Min. sum of passenger transfer and bus operating costs | Bus capacity, passenger demand | × | × | ||

Raghunathan, Bergman [12] | DARP | Constructive heuristic and local search procedure | Min. passengers’ transit time | Fleet availability, fleet size, time windows | ||||

Lee, Meskar [8] | DRTTW | Simulated annealing | Min. total vehicle and passenger travel time | Bus capacity, passengers’ demand, time window, fleet size, route length | × | × | × | |

Zhao, Sun [44] | DRT | Genetic algorithm | Min. total fleet size and passenger travel time | Bus capacity, passengers’ demand, time window, fleet size, overcrowding | × | × | ||

The current study | DRTTW | Simulated annealing | Min. total vehicle and passenger travel time | Bus capacity, passengers’ demand, time window, fleet size, route length | × | × | × | × |

Train/ Station | Station A | Station B | Station C | Station D | Average Total Direct Travel Distance (km) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

B ^{1} | A | Average Direct Travel Distance (km) ^{2} | B | A | Average Direct Travel Distance (km) | B | A | Average Direct Travel Distance (km) | B | A | Average Direct Travel Distance (km) | ||

Bus set 1 | 1 | 1 | 1.01 | 1 | 1 | 1.01 | 1 | 1 | 1.01 | 1 | 1 | 1.01 | 1.01 |

Bus set 2 | 3 | 1 | 1.01 | 4 | 2 | 2.02 | 6 | 1 | 1.01 | 2 | 3 | 3.03 | 1.77 |

Bus set 3 | 4 | 4 | 4.04 | 3 | 3 | 3.03 | 4 | 3 | 3.03 | 3 | 2 | 2.02 | 3.03 |

Bus set 4 | 3 | 2 | 2.02 | 2 | 2 | 2.02 | 4 | 2 | 2.02 | 2 | 2 | 2.02 | 2.02 |

Bus set 5 | 6 | 3 | 3.03 | 1 | 3 | 3.03 | 3 | 2 | 2.02 | 2 | 2 | 2.02 | 2.52 |

Bus set 6 | 2 | 3 | 3.03 | 3 | 1 | 1.01 | 3 | 3 | 3.03 | 2 | 2 | 2.02 | 2.27 |

Bus set 7 | 4 | 3 | 3.03 | 3 | 3 | 3.03 | 3 | 2 | 2.02 | 1 | 2 | 2.02 | 2.52 |

Bus set 8 | 3 | 2 | 2.02 | 2 | 2 | 2.02 | 4 | 2 | 2.02 | 1 | 7 | 7.07 | 3.28 |

Bus set 9 | 1 | 3 | 3.03 | 6 | 1 | 1.01 | 5 | 3 | 3.03 | 2 | 3 | 3.03 | 2.52 |

Bus set 10 | 3 | 3 | 3.03 | 1 | 1 | 1.01 | 2 | 1 | 1.01 | 1 | 1 | 1.01 | 1.51 |

Bus set 11 | 6 | 4 | 4.04 | 4 | 4 | 4.04 | 4 | 2 | 2.02 | 2 | 3 | 3.03 | 3.28 |

Bus set 12 | 0 | 1 | 1.01 | 0 | 7 | 7.07 | 1 | 3 | 3.03 | 1 | 3 | 3.03 | 3.53 |

Bus set 13 | 0 | 2 | 2.02 | 0 | 3 | 3.03 | 0 | 3 | 3.03 | 1 | 1 | 1.01 | 2.27 |

^{1}Boarding/alighting passengers (prs).

^{2}Average direct travel distance in kilometers (km).

Station | Total Vehicle Traveled Distance (km) | Total Passenger Travel Time (hour) | Average Passenger Distance Traveled to each Station (km) | Average Total Passenger Average Distance Traveled (km) | Average Total Passenger Travel Cost (USD) | Average Total Bus Operating Cost (USD) |
---|---|---|---|---|---|---|

#1 | 67.69 | 3.09 | 1.66 | 1.74 | 24.81 | 21.33 |

#2 | 48.31 | 2.51 | 1.47 | |||

#3 | 70.92 | 3.08 | 1.90 | |||

#4 | 64.34 | 3.13 | 1.50 |

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## Share and Cite

**MDPI and ACS Style**

Nickkar, A.; Lee, Y.-J.
Dynamic Demand-Responsive Feeder Bus Network Design for a Short Headway Trunk Line. *Algorithms* **2023**, *16*, 506.
https://doi.org/10.3390/a16110506

**AMA Style**

Nickkar A, Lee Y-J.
Dynamic Demand-Responsive Feeder Bus Network Design for a Short Headway Trunk Line. *Algorithms*. 2023; 16(11):506.
https://doi.org/10.3390/a16110506

**Chicago/Turabian Style**

Nickkar, Amirreza, and Young-Jae Lee.
2023. "Dynamic Demand-Responsive Feeder Bus Network Design for a Short Headway Trunk Line" *Algorithms* 16, no. 11: 506.
https://doi.org/10.3390/a16110506