# Integrated Optimization of Stop Location and Route Design for Community Shuttle Service

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Route Design

#### 2.2. Stop Location

#### 2.3. Integrated Optimization of Stop Location and Route Design

## 3. Mathematical Formulation

#### 3.1. Problem Description and Assumptions

- The locations of all demand points and candidate stops are known.
- The travel demand for the community shuttle service of each demand point is given.
- The distances between different candidate stops and the distances between demand points and candidate stops are known.

#### 3.2. Mathematical Formulation

## 4. Solution Method

_{0}of size Z is created. Then, all solutions in the initial population P

_{0}are sorted into different non-dominated levels by the fast non-dominated sorting procedure. Each non-dominated front is assigned a rank which is equal to its non-dominated level and represents the fitness of solutions in this non-dominated front. Thereafter, the selection operator, crossover operator, and mutation operator are applied to obtain a child population Q

_{0}of the same size Z. After the child population Q

_{0}is generated, P

_{0}and Q

_{0}are associated to create a new population R

_{0}of size 2Z. The best Z solutions are chosen from R

_{0}to fill a new population P

_{1}. Then, the procedure continues, iterating until the number of iterations I is reached. Finally, the Pareto optimal front is obtained.

#### 4.1. Chromosome Coding

#### 4.2. Solution Evaluation

**M1**, shown as follows.

**M1**) min C

_{2}

**M1**is a linear programing formulation and can be solved by ILOG CPLEX.

#### 4.3. Infeasible Solution Processing Method

**M1**has no solution, the route length of the infeasible solution is set to a large constant. Infeasible solutions of these two cases will be treated as feasible solutions in the algorithm. Then, the initial population is created under the premise that all solutions in the initial population are feasible. Algorithm 1 can be used to generate the initial population.

Algorithm 1 |

Step 1For $i\in {P}_{0}$, set i = 0. P _{0} is the initial population whose size is Z.Step 2For solution i, set j = 0 and $j\in D$. Step 3Define a set L _{j}, and put each candidate stop whose distance to demand point j does not exceed the maximum tolerable walking distance into L_{j}.Step4 Choose a candidate stop from L _{j} randomly to serve demand point j. Set j = j + 1, go to Step 3, until j = |D|, then solution i can be created.Step 5For solution i, solve model M1, if the optimal solution of M1 exists, set objective value of this optimal solution as the route length of solution i. Otherwise, go to Step 4 to create solution i again. Step 6Calculate the total walking distance of passengers for solution i. Set i = i + 1, go to Step 2, until i = Z. Output the initial population. |

#### 4.4. NSGA-II Based Algorithm

Algorithm 2 |

Step 1Initialization. Set i = 0, and determine population size Z, crossover probability p _{c}, mutation probability p_{m}, and the number of iterations I. Create the initial population P_{0} using Algorithm 1.Step 2Apply the fast non-dominated sorting procedure to P _{i} to obtain a series of non-dominated fronts.Step 3Perform selection, crossover, and mutation operators for P _{i}, and then the child population Q_{i} with size Z can be obtained. Calculate each solution’s route length and total walking distance of passengers in Q_{i}.Step 4Combine P _{i} and Q_{i} to generate a new population ${R}_{i}={P}_{i}\mathrm{U}{Q}_{i}$ with size 2Z. Implement the fast non-dominated sorting procedure for R_{i}. Then, a series of non-dominated fronts can be obtained and solutions whose non-dominated rank is r are in Pareto front F_{r}.Step 5Elitism. Set ${P}_{i+1}=\varnothing $, r = 1. Execute Step 5.1–5.3: Step 5.1 If the total number of solutions in F _{r} and P_{i}_{+1} does not exceed Z, put all solutions of F_{r} in P_{i}_{+1}.Step 5.2If the total number of solutions in F _{r} and P_{i}_{+1} exceeds Z, calculate crowding distances of solutions in F_{r}. Then, these solutions are put in P_{i}_{+1} according to the crowding distances from large to small, until the number of solutions in P_{i}_{+1} is Z.Step 5.3If the number of solutions in P _{i+1} does not exceed Z, r = r + 1, go to Step 5.1. Otherwise, go to Step 6.Step 6Apply the fast non-dominated sorting procedure to P _{i} _{+ 1} to obtain a set of non-dominated fronts.Step 7Update the Pareto optimal front. Solutions in the first non-dominated front are the latest Pareto optimal front. Step 8Judge whether stop. If i = I, output the Pareto optimal front and stop the algorithm. Otherwise, set i = i + 1, go to Step 3. |

## 5. Computational Experiments

#### 5.1. Network Configuration

_{min}is 3 km, and the maximum route length l

_{max}is 12 km. The minimum and maximum stop spacing (i.e., s

_{min}and s

_{max}) values are 300 m and 840 m, respectively. The maximum tolerable walking distance d

_{max}is 400 m.

#### 5.2. Parameters Setting

_{c}and mutation probability p

_{m}are 0.9 and 0.5, respectively. The number of iterations I is set to 500. The NSGA-II-based algorithm is implemented in C# language and ILOG CPLEX 12.4. All tests are executed on an Intel Core (TM) i5 processor at 2.27 GHz under Windows 7 using 4 GB of RAM.

#### 5.3. Computational Results

#### 5.4. Effect of the Maximum Tolerable Walking Distance

## 6. Case Study

_{1}) near a metro station, and 12 other candidate stops in the community. The connection between the demand points and the candidate stops and the connection between the candidate stops are shown by the topological diagram in Figure 6. The travel demand for the community shuttle service of each demand point is shown in Table 5 (refer to Wu [36]). Actual data, including the distances between candidate stops and demand points and the distances between candidate stops, are shown as Table A3 and Table A4 in Appendix A. The minimum route length is 2 km and the maximum route length is 8 km. The minimum and maximum stop spacing values are 150 m and 1000 m, respectively. The maximum tolerable walking distance of passengers is 500 m. The key parameters used in the algorithm are the same as those in Section 5.2.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Distances between demand points and candidate stops for the computational experiments (m).

H_{1} | H_{2} | H_{3} | H_{4} | H_{5} | H_{6} | H_{7} | H_{8} | H_{9} | H_{10} | H_{11} | H_{12} | H_{13} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

D_{1} | 1140 | 1140 | 540 | 1140 | 540 | 1080 | 480 | 720 | 120 | 660 | 60 | 600 | 1140 |

D_{2} | 900 | 900 | 300 | 600 | 1140 | 540 | 1080 | 480 | 1020 | 420 | 960 | 360 | 900 |

D_{3} | 540 | 360 | 1140 | 540 | 1080 | 480 | 1020 | 480 | 1020 | 420 | 960 | 360 | 900 |

D_{4} | 600 | 60 | 600 | 1140 | 540 | 1080 | 540 | 1080 | 480 | 1020 | 420 | 960 | 360 |

D_{5} | 720 | 720 | 420 | 960 | 360 | 900 | 300 | 840 | 240 | 780 | 180 | 720 | 120 |

D_{6} | 780 | 780 | 180 | 720 | 120 | 660 | 60 | 600 | 1140 | 300 | 840 | 240 | 780 |

D_{7} | 600 | 60 | 840 | 240 | 540 | 1080 | 480 | 1020 | 420 | 960 | 600 | 1140 | 540 |

D_{8} | 600 | 60 | 300 | 1140 | 540 | 1080 | 480 | 1020 | 420 | 960 | 360 | 900 | 300 |

D_{9} | 540 | 420 | 60 | 600 | 1140 | 840 | 240 | 780 | 180 | 720 | 120 | 660 | 60 |

D_{10} | 540 | 420 | 1020 | 120 | 660 | 60 | 600 | 1140 | 780 | 240 | 780 | 180 | 720 |

D_{11} | 1080 | 1080 | 480 | 180 | 420 | 960 | 360 | 900 | 300 | 840 | 240 | 780 | 180 |

D_{12} | 600 | 600 | 1140 | 780 | 180 | 720 | 1020 | 420 | 960 | 360 | 900 | 300 | 840 |

D_{13} | 540 | 300 | 840 | 480 | 1080 | 480 | 1020 | 420 | 960 | 360 | 900 | 300 | 840 |

D_{14} | 1140 | 1140 | 540 | 240 | 780 | 180 | 720 | 120 | 660 | 900 | 300 | 900 | 300 |

D_{15} | 900 | 900 | 300 | 1080 | 480 | 1020 | 420 | 960 | 360 | 960 | 360 | 900 | 1140 |

D_{16} | 900 | 900 | 300 | 540 | 1080 | 780 | 180 | 720 | 120 | 660 | 600 | 600 | 1140 |

D_{17} | 1020 | 1020 | 420 | 660 | 1140 | 540 | 1080 | 660 | 60 | 600 | 480 | 1080 | 480 |

D_{18} | 960 | 960 | 360 | 900 | 1140 | 540 | 1140 | 540 | 180 | 720 | 120 | 660 | 60 |

D_{19} | 900 | 900 | 1140 | 780 | 180 | 720 | 120 | 660 | 60 | 660 | 60 | 600 | 1140 |

D_{20} | 780 | 780 | 180 | 420 | 960 | 660 | 60 | 600 | 1140 | 540 | 1080 | 480 | 1020 |

H_{14} | H_{15} | H_{16} | H_{17} | H_{18} | H_{19} | H_{20} | H_{21} | H_{22} | H_{23} | H_{24} | H_{25} | ||

D_{1} | 540 | 1140 | 780 | 180 | 720 | 120 | 360 | 900 | 360 | 900 | 300 | 840 | |

D_{2} | 300 | 840 | 540 | 1080 | 480 | 1020 | 420 | 960 | 360 | 900 | 300 | 840 | |

D_{3} | 300 | 540 | 1080 | 480 | 1020 | 480 | 1020 | 420 | 960 | 360 | 900 | 300 | |

D_{4} | 1140 | 540 | 1080 | 540 | 1080 | 480 | 1020 | 420 | 960 | 360 | 900 | 300 | |

D_{5} | 660 | 120 | 660 | 60 | 600 | 840 | 240 | 780 | 180 | 720 | 120 | 660 | |

D_{6} | 180 | 720 | 120 | 660 | 60 | 600 | 1140 | 840 | 240 | 780 | 180 | 720 | |

D_{7} | 1080 | 540 | 1080 | 480 | 1020 | 420 | 960 | 360 | 900 | 1140 | 540 | 1080 | |

D_{8} | 840 | 240 | 780 | 240 | 480 | 1020 | 420 | 960 | 360 | 900 | 300 | 840 | |

D_{9} | 600 | 1140 | 540 | 1080 | 540 | 1080 | 480 | 1020 | 420 | 660 | 60 | 600 | |

D_{10} | 120 | 660 | 60 | 600 | 1140 | 540 | 1080 | 480 | 1020 | 420 | 720 | 120 | |

D_{11} | 720 | 180 | 720 | 120 | 660 | 60 | 600 | 1140 | 540 | 180 | 720 | 180 | |

D_{12} | 240 | 780 | 180 | 720 | 120 | 660 | 60 | 660 | 60 | 840 | 240 | 780 | |

D_{13} | 240 | 780 | 180 | 480 | 1020 | 420 | 960 | 360 | 900 | 300 | 840 | 240 | |

D_{14} | 840 | 240 | 780 | 180 | 720 | 120 | 660 | 300 | 900 | 300 | 840 | 240 | |

D_{15} | 540 | 1080 | 480 | 1020 | 420 | 960 | 360 | 960 | 360 | 900 | 300 | 840 | |

D_{16} | 540 | 1080 | 480 | 1020 | 420 | 720 | 120 | 660 | 60 | 600 | 1140 | 540 | |

D_{17} | 1020 | 420 | 960 | 60 | 600 | 1140 | 540 | 1080 | 480 | 1080 | 480 | 1020 | |

D_{18} | 600 | 1140 | 540 | 1140 | 540 | 1080 | 480 | 1020 | 660 | 60 | 600 | 1140 | |

D_{19} | 540 | 1080 | 480 | 720 | 120 | 660 | 60 | 660 | 60 | 600 | 1140 | 540 | |

D_{20} | 420 | 960 | 420 | 960 | 360 | 600 | 1140 | 540 | 1080 | 480 | 1020 | 420 |

H_{1} | H_{2} | H_{3} | H_{4} | H_{5} | H_{6} | H_{7} | H_{8} | H_{9} | H_{10} | H_{11} | H_{12} | H_{13} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

H_{1} | 0 | 420 | 960 | 660 | 60 | 600 | 1140 | 780 | 180 | 780 | 180 | 720 | 120 |

H_{2} | 420 | 0 | 960 | 360 | 900 | 300 | 840 | 240 | 780 | 180 | 720 | 120 | 660 |

H_{3} | 960 | 960 | 0 | 300 | 540 | 1140 | 540 | 1080 | 480 | 1020 | 420 | 960 | 360 |

H_{4} | 660 | 360 | 300 | 0 | 300 | 840 | 300 | 1080 | 480 | 720 | 120 | 660 | 60 |

H_{5} | 60 | 900 | 540 | 300 | 0 | 540 | 1080 | 480 | 1020 | 480 | 1020 | 420 | 960 |

H_{6} | 600 | 300 | 1140 | 840 | 540 | 0 | 1080 | 480 | 1020 | 420 | 960 | 360 | 900 |

H_{7} | 1140 | 840 | 540 | 300 | 1080 | 1080 | 0 | 1080 | 480 | 120 | 720 | 120 | 660 |

H_{8} | 780 | 240 | 1080 | 1080 | 480 | 480 | 1080 | 0 | 480 | 1020 | 420 | 960 | 420 |

H_{9} | 180 | 780 | 480 | 480 | 1020 | 1020 | 480 | 480 | 0 | 480 | 1020 | 480 | 1020 |

H_{10} | 780 | 180 | 1020 | 720 | 480 | 420 | 120 | 1020 | 480 | 0 | 780 | 180 | 720 |

H_{11} | 180 | 720 | 420 | 120 | 1020 | 960 | 720 | 420 | 1020 | 780 | 0 | 1080 | 480 |

H_{12} | 720 | 120 | 960 | 660 | 420 | 360 | 120 | 960 | 480 | 180 | 1080 | 0 | 240 |

H_{13} | 120 | 660 | 360 | 60 | 960 | 900 | 660 | 420 | 1020 | 720 | 480 | 240 | 0 |

H_{14} | 660 | 60 | 900 | 600 | 360 | 300 | 60 | 960 | 420 | 120 | 1020 | 780 | 240 |

H_{15} | 60 | 600 | 300 | 60 | 900 | 840 | 600 | 360 | 960 | 660 | 420 | 180 | 780 |

H_{16} | 600 | 300 | 840 | 600 | 300 | 300 | 1140 | 600 | 360 | 60 | 960 | 720 | 180 |

H_{17} | 1140 | 840 | 240 | 240 | 840 | 840 | 540 | 1140 | 900 | 600 | 360 | 120 | 960 |

H_{18} | 240 | 240 | 1080 | 780 | 240 | 1080 | 1080 | 540 | 300 | 1140 | 900 | 660 | 360 |

H_{19} | 780 | 780 | 480 | 180 | 780 | 480 | 480 | 1080 | 1080 | 540 | 600 | 60 | 900 |

H_{20} | 180 | 180 | 1020 | 720 | 180 | 1020 | 1020 | 480 | 480 | 240 | 1140 | 300 | 360 |

H_{21} | 780 | 720 | 420 | 120 | 720 | 420 | 420 | 1020 | 1020 | 780 | 540 | 840 | 900 |

H_{22} | 180 | 120 | 960 | 660 | 420 | 960 | 720 | 420 | 480 | 180 | 1080 | 300 | 300 |

H_{23} | 720 | 660 | 360 | 60 | 960 | 360 | 120 | 960 | 1020 | 720 | 480 | 840 | 840 |

H_{24} | 120 | 60 | 900 | 600 | 360 | 900 | 660 | 420 | 420 | 120 | 1020 | 240 | 240 |

H_{25} | 660 | 600 | 300 | 60 | 900 | 300 | 60 | 960 | 960 | 660 | 420 | 780 | 780 |

H_{14} | H_{15} | H_{16} | H_{17} | H_{18} | H_{19} | H_{20} | H_{21} | H_{22} | H_{23} | H_{24} | H_{25} | ||

H_{1} | 660 | 60 | 600 | 1140 | 240 | 780 | 180 | 780 | 180 | 720 | 120 | 660 | |

H_{2} | 60 | 600 | 300 | 840 | 240 | 780 | 180 | 720 | 120 | 660 | 60 | 600 | |

H_{3} | 900 | 300 | 840 | 240 | 1080 | 480 | 1020 | 420 | 960 | 360 | 900 | 300 | |

H_{4} | 600 | 60 | 600 | 240 | 780 | 180 | 720 | 120 | 660 | 60 | 600 | 60 | |

H_{5} | 360 | 900 | 300 | 840 | 240 | 780 | 180 | 720 | 420 | 960 | 360 | 900 | |

H_{6} | 300 | 840 | 300 | 840 | 1080 | 480 | 1020 | 420 | 960 | 360 | 900 | 300 | |

H_{7} | 60 | 600 | 1140 | 540 | 1080 | 480 | 1020 | 420 | 720 | 120 | 660 | 60 | |

H_{8} | 960 | 360 | 600 | 1140 | 540 | 1080 | 480 | 1020 | 420 | 960 | 420 | 960 | |

H_{9} | 420 | 960 | 360 | 900 | 300 | 1080 | 480 | 1020 | 480 | 1020 | 420 | 960 | |

H_{10} | 120 | 660 | 60 | 600 | 1140 | 540 | 240 | 780 | 180 | 720 | 120 | 660 | |

H_{11} | 1020 | 420 | 960 | 360 | 900 | 600 | 1140 | 540 | 1080 | 480 | 1020 | 420 | |

H_{12} | 780 | 180 | 720 | 120 | 660 | 60 | 300 | 840 | 300 | 840 | 240 | 780 | |

H_{13} | 240 | 780 | 180 | 960 | 360 | 900 | 360 | 900 | 300 | 840 | 240 | 780 | |

H_{14} | 0 | 780 | 180 | 720 | 120 | 660 | 60 | 600 | 900 | 300 | 840 | 240 | |

H_{15} | 780 | 0 | 1080 | 480 | 720 | 120 | 660 | 120 | 660 | 60 | 600 | 1140 | |

H_{16} | 180 | 1080 | 0 | 840 | 240 | 780 | 180 | 720 | 120 | 960 | 360 | 900 | |

H_{17} | 720 | 480 | 840 | 0 | 1080 | 480 | 1020 | 420 | 1020 | 420 | 960 | 360 | |

H_{18} | 120 | 720 | 240 | 1080 | 0 | 240 | 780 | 480 | 1020 | 420 | 960 | 360 | |

H_{19} | 660 | 120 | 780 | 480 | 240 | 0 | 540 | 1080 | 480 | 1020 | 660 | 60 | |

H_{20} | 60 | 660 | 180 | 1020 | 780 | 540 | 0 | 1080 | 480 | 1020 | 420 | 960 | |

H_{21} | 600 | 120 | 720 | 420 | 480 | 1080 | 1080 | 0 | 1080 | 480 | 1020 | 660 | |

H_{22} | 900 | 660 | 120 | 1020 | 1020 | 480 | 480 | 1080 | 0 | 480 | 1020 | 420 | |

H_{23} | 300 | 60 | 960 | 420 | 420 | 1020 | 1020 | 480 | 480 | 0 | 780 | 180 | |

H_{24} | 840 | 600 | 360 | 960 | 960 | 660 | 420 | 1020 | 1020 | 780 | 0 | 780 | |

H_{25} | 240 | 1140 | 900 | 360 | 360 | 60 | 960 | 660 | 420 | 180 | 780 | 0 |

H_{1} | H_{2} | H_{3} | H_{4} | H_{5} | H_{6} | H_{7} | H_{8} | H_{9} | H_{10} | H_{11} | H_{12} | H_{13} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

D_{1} | 2370 | 2510 | 2250 | 1920 | 1800 | 1590 | 1270 | 1370 | 930 | 1730 | 830 | 330 | 150 |

D_{2} | 2480 | 2620 | 2360 | 2030 | 1910 | 1700 | 1380 | 1480 | 1040 | 1260 | 360 | 440 | 620 |

D_{3} | 1950 | 2090 | 1830 | 1500 | 1380 | 1170 | 850 | 950 | 510 | 1350 | 890 | 90 | 570 |

D_{4} | 2390 | 2530 | 2270 | 1940 | 1820 | 1610 | 1290 | 1270 | 1310 | 470 | 430 | 1230 | 1410 |

D_{5} | 1030 | 1170 | 1130 | 800 | 460 | 250 | 150 | 520 | 490 | 970 | 1870 | 1090 | 1570 |

D_{6} | 1020 | 1160 | 1220 | 890 | 450 | 240 | 560 | 930 | 900 | 1380 | 2280 | 1500 | 1980 |

D_{7} | 1530 | 1670 | 1070 | 740 | 960 | 750 | 430 | 800 | 770 | 1250 | 1890 | 1090 | 1570 |

D_{8} | 880 | 1020 | 1080 | 750 | 310 | 100 | 220 | 590 | 560 | 1040 | 1940 | 1160 | 1640 |

D_{9} | 1630 | 1770 | 1510 | 1180 | 1060 | 850 | 530 | 630 | 190 | 1030 | 1210 | 410 | 890 |

H_{1} | H_{2} | H_{3} | H_{4} | H_{5} | H_{6} | H_{7} | H_{8} | H_{9} | H_{10} | H_{11} | H_{12} | H_{13} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

H_{1} | 0 | 140 | 1020 | 1350 | 570 | 780 | 1100 | 1120 | 1440 | 1920 | 2820 | 2040 | 2520 |

H_{2} | 140 | 0 | 880 | 1210 | 710 | 920 | 1240 | 1260 | 1580 | 2060 | 2960 | 2180 | 2660 |

H_{3} | 1020 | 880 | 0 | 330 | 1190 | 980 | 980 | 1350 | 1320 | 1800 | 2700 | 1920 | 2400 |

H_{4} | 1350 | 1210 | 330 | 0 | 860 | 650 | 650 | 1020 | 990 | 1470 | 2370 | 1590 | 2070 |

H_{5} | 570 | 710 | 1190 | 860 | 0 | 210 | 530 | 550 | 870 | 1350 | 2250 | 1470 | 1950 |

H_{6} | 780 | 920 | 980 | 650 | 210 | 0 | 320 | 690 | 660 | 1140 | 2040 | 1260 | 1740 |

H_{7} | 1100 | 1240 | 980 | 650 | 530 | 320 | 0 | 370 | 340 | 820 | 1720 | 940 | 1420 |

H_{8} | 1120 | 1260 | 1350 | 1020 | 550 | 690 | 370 | 0 | 440 | 800 | 1700 | 1040 | 1520 |

H_{9} | 1440 | 1580 | 1320 | 990 | 870 | 660 | 340 | 440 | 0 | 840 | 1400 | 600 | 1080 |

H_{10} | 1920 | 2060 | 1800 | 1470 | 1350 | 1140 | 820 | 800 | 840 | 0 | 900 | 1440 | 1880 |

H_{11} | 2820 | 2960 | 2700 | 2370 | 2250 | 2040 | 1720 | 1700 | 1400 | 900 | 0 | 800 | 980 |

H_{12} | 2040 | 2180 | 1920 | 1590 | 1470 | 1260 | 940 | 1040 | 600 | 1440 | 800 | 0 | 480 |

H_{13} | 2520 | 2660 | 2400 | 2070 | 1950 | 1740 | 1420 | 1520 | 1080 | 1880 | 980 | 480 | 0 |

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**Figure 4.**Relation between the maximum tolerable walking distance, total walking distance of passengers, and route length.

Sets | Definition |

D | set of demand points, $i,j\in D$ |

H | set of candidate stops, $k,l\in H$ |

Parameters | Definition |

${\overline{d}}_{ik}$ | distance between demand point i and candidate stop k, $i\in D,k\in H$ |

${\underset{\_}{d}}_{ik}$ | distance between candidate stop k and candidate stop l, |

q_{i} | travel demand of demand point i |

l_{max} | the maximum length of the community shuttle route |

l_{min} | the minimum length of the community shuttle route |

s_{max} | the maximum stop spacing of the community shuttle route |

s_{min} | the minimum stop spacing of the community shuttle route |

d_{max} | the maximum tolerable walking distance of passengers |

Decision Variables | Definition |

x_{ik} | binary variable (1 if demand point i is served by candidate stop k and 0 otherwise) |

y_{kl} | binary variable (1 if candidate stop k and l are adjacent stops and 0 otherwise) |

u_{k} | nonnegative integer assistant variable to eliminate sub-tour |

D_{1} | D_{2} | D_{3} | D_{4} | D_{5} | D_{6} | D_{7} | D_{8} | D_{9} | D_{10} | |

Travel demand | 673 | 553 | 572 | 582 | 542 | 627 | 605 | 614 | 590 | 586 |

D_{11} | D_{12} | D_{13} | D_{14} | D_{15} | D_{16} | D_{17} | D_{18} | D_{19} | D_{20} | |

Travel demand | 514 | 637 | 636 | 610 | 513 | 564 | 544 | 609 | 591 | 589 |

No. | Route | Total Walking Distance of Passengers (km) | Route Length (km) |
---|---|---|---|

1 | H_{1}-H_{2}-H_{16}-H_{9}-H_{3}-H_{13}-H_{22}-H_{12}-H_{6}-H_{1} | 1632.00 | 3.48 |

2 | H_{1}-H_{2}-H_{6}-H_{14}-H_{23}-H_{3}-H_{13}-H_{22}-H_{9}-H_{16}-H_{1} | 1597.68 | 3.78 |

3 | H_{1}-H_{2}-H_{6}-H_{12}-H_{22}-H_{13}-H_{3}-H_{7}-H_{9}-H_{16}-H_{1} | 1561.32 | 4.02 |

4 | H_{1}-H_{14}-H_{6}-H_{12}-H_{22}-H_{13}-H_{3}-H_{7}-H_{9}-H_{16}-H_{2}-H_{1} | 1527.00 | 4.38 |

Solution | D_{1} | D_{2} | D_{3} | D_{4} | D_{5} | D_{6} | D_{7} | D_{8} | D_{9} | D_{10} |

1 | H_{9} | H_{3} | H_{12} | H_{2} | H_{13} | H_{16} | H_{2} | H_{2} | H_{3} | H_{6} |

2 | H_{9} | H_{3} | H_{14} | H_{2} | H_{13} | H_{16} | H_{2} | H_{2} | H_{3} | H_{6} |

3 | H_{9} | H_{3} | H_{12} | H_{2} | H_{13} | H_{16} | H_{2} | H_{2} | H_{3} | H_{6} |

4 | H_{9} | H_{3} | H_{14} | H_{2} | H_{13} | H_{16} | H_{2} | H_{2} | H_{3} | H_{6} |

Solution | D_{11} | D_{12} | D_{13} | D_{14} | D_{15} | D_{16} | D_{17} | D_{18} | D_{19} | D_{20} |

1 | H_{13} | H_{22} | H_{12} | H_{6} | H_{22} | H_{22} | H_{9} | H_{13} | H_{9} | H_{3} |

2 | H_{13} | H_{22} | H_{23} | H_{6} | H_{22} | H_{22} | H_{9} | H_{13} | H_{9} | H_{3} |

3 | H_{13} | H_{22} | H_{12} | H_{6} | H_{22} | H_{22} | H_{9} | H_{13} | H_{9} | H_{7} |

4 | H_{13} | H_{22} | H_{12} | H_{6} | H_{22} | H_{22} | H_{9} | H_{13} | H_{9} | H_{7} |

No. | D_{1} | D_{2} | D_{3} | D_{4} | D_{5} | D_{6} | D_{7} | D_{8} | D_{9} |
---|---|---|---|---|---|---|---|---|---|

Travel demand | 673 | 553 | 572 | 582 | 542 | 627 | 605 | 614 | 590 |

No. | Route | Total Walking Distance of Passengers (km) | Route Length (km) |
---|---|---|---|

1 | H_{1}-H_{5}-H_{7}-H_{10}-H_{11}-H_{12}-H_{9}-H_{6}-H_{1} | 2300.13 | 5.66 |

2 | H_{1}-H_{5}-H_{9}-H_{12}-H_{13}-H_{11}-H_{10}-H_{7}-H_{6}-H_{1} | 1998.81 | 6.32 |

Demand Points | D_{1} | D_{2} | D_{3} | D_{4} | D_{5} | D_{6} | D_{7} | D_{8} | D_{9} |
---|---|---|---|---|---|---|---|---|---|

Solution 1 | H_{12} | H_{11} | H_{12} | H_{10} | H_{7} | H_{5} | H_{7} | H_{6} | H_{9} |

Solution 2 | H_{13} | H_{11} | H_{12} | H_{10} | H_{7} | H_{5} | H_{7} | H_{6} | H_{9} |

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## Share and Cite

**MDPI and ACS Style**

Guo, X.; Song, R.; He, S.; Bi, M.; Jin, G.
Integrated Optimization of Stop Location and Route Design for Community Shuttle Service. *Symmetry* **2018**, *10*, 678.
https://doi.org/10.3390/sym10120678

**AMA Style**

Guo X, Song R, He S, Bi M, Jin G.
Integrated Optimization of Stop Location and Route Design for Community Shuttle Service. *Symmetry*. 2018; 10(12):678.
https://doi.org/10.3390/sym10120678

**Chicago/Turabian Style**

Guo, Xiaole, Rui Song, Shiwei He, Mingkai Bi, and Guowei Jin.
2018. "Integrated Optimization of Stop Location and Route Design for Community Shuttle Service" *Symmetry* 10, no. 12: 678.
https://doi.org/10.3390/sym10120678