Re-Supplying Autonomous Mobile Parcel Lockers in Last-Mile Distribution
Abstract
:1. Introduction
2. Literature Review
2.1. The Generalized Vehicle Routing Problem with Time Windows
2.2. The VRP with Multiple Time Windows
2.3. Mobile Parcel Lockers
3. Problem Description
Mixed-Integer Linear Programming Model
4. The Solution Approach
4.1. Cost Functions
4.1.1. Total Traveling Cost Formula
4.1.2. Capacity Violation and Penalty Cost Formula
4.1.3. Time Window Violation and Its Penalty Cost Formula
4.1.4. Parking Slot Cost Formula
4.2. A Cluster-Based Simulated Annealing Algorithm
Algorithm 1. The cluster-based simulated annealing algorithm (CSA) |
//Initialization phase creates an initial solution (instance) by calling Algorithm 3; ωbest ← ω; calculate the fitness value (ω); F(ωbest) ← F(ω); T ← T0; Tf ← 0; //improvement phase for externaliteration = 1 to EImax do if T > Tf then for internaliteration = 1 to IImax do ωnew ←createnighborhoodsolutions(ω); F(ωnew) ← calculatethefitnessvalue(ωnew); if F(ωnew) <= F(ω) then ω ← ωnew; else Δ = T0 − Tf; η = e(Δ/T); if random(0,1) <= η then ω← ωnew; end end end end if F(ω) <= F(ωbest) then ωnew ← ω; end T = εT; end |
Algorithm 2. Clustering algorithm. |
Input: Nodes’ coordinates, nodes’ labels |
Output: A set of H clusters, |
Step 1: Call data to extract horizontal and vertical coordinates. |
Step 2: Analyze the maximum number of distinct clusters by calling the silhouette method. |
Step 3: Select p based on the silhouette method. |
Step 4: Apply the K-means method for clustering. |
Step 5: Print zones (clusters) with memberships. |
Algorithm 3. Initial random solution algorithm. |
Input: An instance Output: An initial solution, ω function create initial solution (instance) N ← Ninstance; p ← pinstance; k ← kinstance; |ω| ← p + k − 1; ω ← 0; for g = 1 to p do cluster(g) ← random_sampling(g,1); end ω ← [permutation([p+1:p+k−1],[cluster(g) from 1 to p])]; |
Algorithm 4. Create neighborhoods algorithm. |
Input: An initial solution, ω Output: A new solution, ωnew function create neighborhoods (ω) r ← random_integer_number[1, 6] switch r do case 1 do ωnew ←2_opt(ω); end case 2 do ωnew ← remove_insert1(ω); end case 3 do ωnew ← shake_cluster(ω); end case 4 do ωnew ← 3_opt(ω); end case 5 do ωnew ← remove_insert_2(ω); end case 6 do ωnew ←reverse(ω); end end |
5. Computational Experiments
5.1. Dataset and Strategy Definition
5.2. Performance Metrics and Parameters
5.3. Results of the R-AMPLP and VRPTW
5.4. Comparison with CPLEX Solutions
5.5. Summary of Findings
5.6. Managarial Insights
6. Conclusions
Author Contributions
Funding
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Algorithm A1. 2_opt algorithm. |
Input: An initial solution, ω Output: A new solution, ωnew function 2_opt (ω) S ← |ω|; g(v) ← 0; g(u) ← 0; [list(1),list(2)] ← random sampling((1,S),2); g(v) ← list(1); g(u) ← list(2); update(ω) ← ω([g(u) g(v)]); ωnew ← ω; |
Algorithm A2. remove_insert_1 algorithm. |
Input: An initial solution, ω Output: A new solution, ωnew function remove_insert_1 (ω) S ← |ω|; g(v) ← 0; g(u) ← 0; [list(1),list(2)] ← random sampling((1,S),2); g(v) ← list(1); g(u) ← list(2); if g(v)<g(u) then update (ω) ← ω([[1:g(v)−1][g(v)+1:g(u)][g(v)][g(u)+1:|ω|])]); else update (ω) ← ω([[1:g(u)][g(v)][g(u)+1:g(v)−1][g(v)+1:|ω|])]); end ωnew ← ω; |
Algorithm A3. shake_cluster algorithm. |
Input: An initial solution, ω Output: A new solution, ωnew function shake-cluster (ω) S ← |ω|; L ← Linstance; C← Cinstance; for g(v) do if v\∈ L then update (ω) ← ω([random_sampling(g(g(v)).1)}); end end ωnew ← ω; |
Algorithm A4. 3_opt algorithm. |
Input: An initial solution, ω Output: A new solution, ωnew function 3_opt (ω) S ← |ω|; g(v) ← 0; g(u) ← 0; g(w) ← 0; [list(1),list(2),list(3)] ← random sampling((1,S),3); g(v) ← list(1); g(u) ← list(2); g(w) ← list(3); update (ω) ← ω([[g(u)] [g(w)] [g(v)]]); ωnew ← ω; |
Algorithm A5. remove_insert_2 algorithm. |
Input: An initial solution, ω Output: A new solution, ωnew function remove_insert_2 (ω) S ← |ω|; g(v) ← 0; g(u) ← 0; [list(v)] ← random sampling((1,S),1); [list(u)] ← find dummy(ω,1); g(v) ← list(v); g(u) ←list(u); if g(v) < g(u) then update (ω)←ω([[1:g(v)−1][g(v)+1:g(u)[g(v)][g(u)+1:|ω|]]); else update (ω)←ω([[g(v)][ 1:g(u)[g(u)+1:g(v)+1][g(v)+1:|ω|]]); end ωnew ← ω; |
Algorithm A6. reverse algorithm. |
Input: An initial solution, ω Output: A new solution, ωnew function reverse (ω) S ← |ω|; g(v) ← 0; g(u) ← 0; g(w) ← 0; g(y) ← 0; [list(1),list(2)] ← random sampling((1,S),2); g(v) ← list(1); g(u) ← list(2); g(w) ← find min(g(v),g(u)); g(y) ← find max(g(v),g(u)); update (ω) ← ω(reverse([[g(y)] : [g(w)]])); ωnew ← ω; |
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Reference | Setting | GVRPTW | VRPMTW | MPLs |
---|---|---|---|---|
[11] | Alternative locations | * | * | |
[24,25] | Alternative time windows | * | ||
[11,20] | Sharing delivery locations | * | ||
This paper | Alternative locations and alternative time windows | * | * | * |
Reference | Model | Solution | Roaming/ GVRPTW | Multiple Time Windows | Parking Slot Fee | AMPL | Sharing Locations |
---|---|---|---|---|---|---|---|
[5] | IP a | GRASP d-VNS e | * | ||||
[18] | - | TS f | * | ||||
[21] | MIP b | GRASP | * | ||||
[20] | MIP | LNS g | * | * | |||
[21] | SP c | BPC h | * | * | |||
This paper | MIP | CSA i | * | * | * | * |
Notation | Description |
---|---|
Sets | |
The set of nodes | |
The set of meeting points | |
The set of arcs | |
The set of clusters corresponding to meeting points | |
The complement set of clusters associated with AMPL paths | |
The set of vehicles | |
The depot | |
Leaving point in cluster h | |
Returning point in cluster h | |
Parameters | |
The traveling distance (cost) between meeting points and | |
The demand of node | |
The speed of vehicle | |
The capacity of vehicle | |
The earliest arriving time at a meeting point | |
The latest arriving time at a meeting point | |
The parking cost at a meeting point | |
Decision variables | |
The arrival time at a meeting point | |
If the traveling path between node to node is traveled by a vehicle , it is equal to , and otherwise | |
If the node in a cluster is visited by a vehicle , it is equal to , and otherwise |
Class | Scenario | Strategy | ID | NTWs | Average of Solutions | ||||
---|---|---|---|---|---|---|---|---|---|
C1 | c101—c103 | 25nps | c101-c103:25nps | 75 | 3 | 20 | 60 | 3 | 83.48 |
50nps | c101-c103:50nps | 150 | 6 | 20 | 60 | 3 | 231.58 | ||
75nps | c101-c103:75nps | 225 | 9 | 20 | 60 | 3 | 466.84 | ||
100nps | c101-c103:100nps | 300 | 10 | 20 | 60 | 3 | 478.41 | ||
c101—c106 | 25nps | c101-c106:25nps | 150 | 3 | 20 | 60 | 6 | 82.48 | |
50nps | c101-c106:50nps | 300 | 7 | 20 | 60 | 6 | 266.94 | ||
75nps | c101-c106:75nps | 450 | 10 | 20 | 60 | 6 | 495.03 | ||
100nps | c101-c106:100nps | 600 | 10 | 20 | 60 | 6 | 470.24 | ||
c101—c109 | 25nps | c101-c109:25nps | 225 | 3 | 20 | 60 | 9 | 71.12 | |
50nps | c101-c109:50nps | 450 | 6 | 20 | 60 | 9 | 243.29 | ||
75nps | c101-c109:75nps | 675 | 8 | 20 | 60 | 9 | 446.26 | ||
100nps | c101-c109:100nps | 900 | 10 | 20 | 60 | 9 | 488.05 | ||
Average | - | - | - | 375 | 8.5 | 20 | 60 | 6 | 318.64 |
Dataset | Subset | Optimal (Benchmark) | CSA Solution | Gap% | ||
---|---|---|---|---|---|---|
C1 | c101 | 25 | 25 | 191.3 | 191.81 | 0.27% |
c101 | 50 | 50 | 362.4 | 363.24 | 0.23% | |
c101 | 100 | 100 | 827.3 | 828.93 | 0.20% | |
c102 | 25 | 25 | 190.3 | 190.73 | 0.23% | |
c102 | 50 | 50 | 361.4 | 362.17 | 0.21% | |
c102 | 100 | 100 | 827.3 | 828.93 | 0.20% | |
c103 | 25 | 25 | 190.3 | 190.73 | 0.23% | |
c103 | 50 | 50 | 361.4 | 362.17 | 0.21% | |
c103 | 100 | 100 | 826.3 | 834.75 | 1.02% | |
c104 | 25 | 25 | 186.9 | 187.45 | 0.29% | |
c104 | 50 | 50 | 358 | 358.88 | 0.25% | |
c104 | 100 | 100 | 822.9 | 834.94 | 1.46% | |
c105 | 25 | 25 | 191.3 | 191.81 | 0.27% | |
c105 | 50 | 50 | 362.4 | 363.24 | 0.23% | |
c105 | 100 | 100 | 827.3 | 828.93 | 0.20% | |
c106 | 25 | 25 | 191.3 | 191.81 | 0.27% | |
c106 | 50 | 50 | 362.4 | 363.24 | 0.23% | |
c106 | 100 | 100 | 827.3 | 828.93 | 0.20% | |
c107 | 25 | 25 | 191.3 | 191.81 | 0.27% | |
c107 | 50 | 50 | 362.4 | 363.24 | 0.23% | |
c107 | 100 | 100 | 827.3 | 828.93 | 0.20% | |
c108 | 25 | 25 | 191.3 | 191.81 | 0.27% | |
c108 | 50 | 50 | 362.4 | 363.24 | 0.23% | |
c108 | 100 | 100 | 827.3 | 828.94 | 0.20% | |
c109 | 25 | 25 | 191.3 | 191.81 | 0.27% | |
c109 | 50 | 50 | 362.4 | 363.92 | 0.42% | |
c109 | 100 | 100 | 827.3 | 828.94 | 0.20% | |
Average | 459.65 | 463.45 | 0.36% |
Class | Scenario | Strategy | ID | NTWs | Average of Solutions | ||||
---|---|---|---|---|---|---|---|---|---|
C2 | c201—c203 | 25nps | c201-c203:25nps | 75 | 8 | 20 | 60 | 3 | 173.50 |
50nps | c201-c203:50nps | 150 | 9 | 20 | 60 | 3 | 213.34 | ||
75nps | c201-c203:75nps | 225 | 10 | 20 | 60 | 3 | 373.65 | ||
100nps | c201-c203:100nps | 300 | 8 | 20 | 60 | 3 | 294.02 | ||
c201—c206 | 25nps | c201-c206:25nps | 150 | 9 | 20 | 60 | 6 | 224.05 | |
50nps | c201-c206:50nps | 300 | 5 | 20 | 60 | 6 | 145.77 | ||
75nps | c201-c206:75nps | 450 | 8 | 20 | 60 | 6 | 221.37 | ||
100nps | c201-c206:100nps | 600 | 10 | 20 | 60 | 6 | 341.01 | ||
c201—c208 | 25nps | c201-c208:25nps | 200 | 10 | 20 | 60 | 8 | 266.68 | |
50nps | c201-c208:50nps | 400 | 7 | 20 | 60 | 8 | 204.00 | ||
75nps | c201-c208:75nps | 600 | 10 | 20 | 60 | 8 | 306.95 | ||
100nps | c201-c208:100nps | 800 | 7 | 20 | 60 | 8 | 216.77 | ||
Average | - | - | - | 354.16 | 8.41 | 20 | 60 | 5.66 | 248.42 |
Dataset | Subset | Optimal (Benchmark) | CSA Solution | Gap% | ||
---|---|---|---|---|---|---|
C2 | c201 | 25 | 25 | 214.7 | 215.54 | 0.39% |
c201 | 50 | 50 | 360.2 | 361.79 | 0.44% | |
c201 | 100 | 100 | 589.1 | 673.16 | 14.27% | |
c202 | 25 | 25 | 214.7 | 215.54 | 0.39% | |
c202 | 50 | 50 | 360.2 | 395.81 | 9.89% | |
c202 | 100 | 100 | 589.1 | 674.29 | 14.46% | |
c203 | 25 | 25 | 214.7 | 215.54 | 0.39% | |
c203 | 50 | 50 | 359.8 | 393.98 | 9.50% | |
c203 | 100 | 100 | 588.7 | 674.95 | 14.65% | |
c204 | 25 | 25 | 213.1 | 215.54 | 1.15% | |
c204 | 50 | 50 | 350.1 | 393.37 | 12.36% | |
c204 | 100 | 100 | 588.1 | 640.59 | 8.93% | |
c205 | 25 | 25 | 214.7 | 215.54 | 0.39% | |
c205 | 50 | 50 | 359.8 | 364.75 | 1.38% | |
c205 | 100 | 100 | 586.4 | 652.41 | 11.26% | |
c206 | 25 | 25 | 214.7 | 215.54 | 0.39% | |
c206 | 50 | 50 | 359.8 | 361.41 | 0.45% | |
c206 | 100 | 100 | 586 | 642.57 | 9.65% | |
c207 | 25 | 25 | 214.5 | 215.54 | 0.48% | |
c207 | 50 | 50 | 359.6 | 402.32 | 11.88% | |
c207 | 100 | 100 | 585.5 | 675.75 | 15.41% | |
c208 | 25 | 25 | 214.5 | 215.54 | 0.48% | |
c208 | 50 | 50 | 350.5 | 352.12 | 0.46% | |
c208 | 100 | 100 | 585.5 | 637.07 | 8.81% | |
Average | - | - | - | 386.41 | 417.53 | 8.05% |
Class | Scenario | Strategy | ID | NTWs | Average of Solutions | ||||
---|---|---|---|---|---|---|---|---|---|
R1 | r101–r103 | 25nps | r101-r103:25nps | 75 | 10 | 20 | 60 | 3 | 276.05 |
50nps | r101-r103:50nps | 150 | 4 | 20 | 60 | 3 | 88.46 | ||
75nps | r101-r103:75nps | 225 | 5 | 20 | 60 | 3 | 102.31 | ||
100nps | r101-r103:100nps | 300 | 4 | 20 | 60 | 3 | 78.92 | ||
r101–r106 | 25nps | r101-r106:25nps | 150 | 9 | 20 | 60 | 6 | 253.62 | |
50nps | r101-r106:50nps | 300 | 10 | 20 | 60 | 6 | 264.91 | ||
75nps | r101-r106:75nps | 450 | 4 | 20 | 60 | 6 | 78.84 | ||
100nps | r101-r106:100nps | 600 | 4 | 20 | 60 | 6 | 78.81 | ||
r101–r109 | 25nps | r101-r109:25nps | 225 | 9 | 20 | 60 | 9 | 284.7 | |
50nps | r101-r109:50nps | 450 | 8 | 20 | 60 | 9 | 240.79 | ||
75nps | r101-r109:75nps | 675 | 4 | 20 | 60 | 9 | 76.25 | ||
100nps | r101-r109:100nps | 900 | 4 | 20 | 60 | 9 | 65.59 | ||
r101–r112 | 25nps | r101-r112:25nps | 300 | 8 | 20 | 60 | 12 | 207.54 | |
50nps | r101-r112:50nps | 600 | 4 | 20 | 60 | 12 | 77.25 | ||
75nps | r101-r112:75nps | 900 | 3 | 20 | 60 | 12 | 34.26 | ||
100nps | r101-r12:100nps | 1200 | 4 | 20 | 60 | 12 | 68.69 | ||
Average | - | - | - | 468.75 | 5.87 | 20 | 60 | 7.5 | 142.31 |
Dataset | Subset | Optimal (Benchmark) | CSA Solution | Gap% | ||
---|---|---|---|---|---|---|
R1 | r101 | 25 | 25 | 617.1 | 627.13 | 1.63% |
r101 | 50 | 50 | 1044 | 1065.9 | 2.10% | |
r101 | 100 | 100 | 1637.7 | 1699.34 | 3.76% | |
r102 | 25 | 25 | 547.1 | 550.20 | 0.57% | |
r102 | 50 | 50 | 909 | 923.22 | 1.56% | |
r102 | 100 | 100 | 1466.6 | 1523.34 | 3.87% | |
r103 | 25 | 25 | 454.6 | 464.82 | 2.25% | |
r103 | 50 | 50 | 772.9 | 813.05 | 5.19% | |
r103 | 100 | 100 | 1208.7 | 1289.84 | 6.71% | |
r104 | 25 | 25 | 416.9 | 437.08 | 4.84% | |
r104 | 50 | 50 | 625.4 | 644.07 | 2.99% | |
r104 | 100 | 100 | 971.5 | 1043.67 | 7.43% | |
r105 | 25 | 25 | 530.5 | 531.53 | 0.19% | |
r105 | 50 | 50 | 899.3 | 948.36 | 5.46% | |
r105 | 100 | 100 | 1355.3 | 1462.89 | 7.94% | |
r106 | 25 | 25 | 465.4 | 466.48 | 0.23% | |
r106 | 50 | 50 | 793 | 830.78 | 4.76% | |
r106 | 100 | 100 | 1234.6 | 1299.89 | 5.29% | |
r107 | 25 | 25 | 424.3 | 437.67 | 3.15% | |
r107 | 50 | 50 | 711.1 | 746.12 | 4.92% | |
r107 | 100 | 100 | 1064.6 | 1127.83 | 5.94% | |
r108 | 25 | 25 | 379.3 | 398.29 | 5.01% | |
r108 | 50 | 50 | 617.87 | 630.53 | 2.05% | |
r108 | 100 | 100 | - | 1001.17 | - | |
r109 | 25 | 25 | 441.3 | 442.62 | 0.30% | |
r109 | 50 | 50 | 786.8 | 819.21 | 4.12% | |
r109 | 100 | 100 | 1146.9 | 1211.66 | 5.65% | |
r110 | 25 | 25 | 444.1 | 447.47 | 0.76% | |
r110 | 50 | 50 | 697 | 735.05 | 5.46% | |
r110 | 100 | 100 | 1068 | 1150.07 | 7.68% | |
r111 | 25 | 25 | 428.8 | 440.04 | 2.62% | |
r111 | 50 | 50 | 707.2 | 740.56 | 4.72% | |
r111 | 100 | 100 | 1048.7 | 1101.81 | 5.06% | |
r112 | 25 | 25 | 393 | 413.44 | 5.20% | |
r112 | 50 | 50 | 630.2 | 646.85 | 2.64% | |
r112 | 100 | 100 | - | 999.19 | - | |
Average | - | - | - | 792.31 * | 831.71 * | 4.97% |
Class | Scenario | Strategy | ID | NTWs | Average of Solutions | ||||
---|---|---|---|---|---|---|---|---|---|
R2 | r201—r203 | 25nps | r201-r203:25nps | 75 | 10 | 20 | 60 | 3 | 271.9 |
50nps | r201-r203:50nps | 150 | 4 | 20 | 60 | 3 | 85.06 | ||
75nps | r201-r203:75nps | 225 | 5 | 20 | 60 | 3 | 100.42 | ||
100nps | r201-r203:100nps | 300 | 4 | 20 | 60 | 3 | 84.01 | ||
r201—r206 | 25nps | r201-r206:25nps | 150 | 9 | 20 | 60 | 6 | 287.70 | |
50nps | r201-r206:50nps | 300 | 10 | 20 | 60 | 6 | 247.64 | ||
75nps | r201-r206:75nps | 450 | 4 | 20 | 60 | 6 | 79.52 | ||
100nps | r201-r206:100nps | 600 | 4 | 20 | 60 | 6 | 77.36 | ||
r201—r209 | 25nps | r201-r209:25nps | 225 | 9 | 20 | 60 | 9 | 285.15 | |
50nps | r201-r209:50nps | 450 | 8 | 20 | 60 | 9 | 232.81 | ||
75nps | r201-r209:75nps | 675 | 4 | 20 | 60 | 9 | 81.04 | ||
100nps | r201-r209:100nps | 900 | 4 | 20 | 60 | 9 | 70.90 | ||
r201—r211 | 25nps | r201-r211:25nps | 275 | 8 | 20 | 60 | 11 | 225.76 | |
50nps | r201-r211:50nps | 550 | 9 | 20 | 60 | 11 | 280.85 | ||
75nps | r201-r211:75nps | 825 | 3 | 20 | 60 | 11 | 34.30 | ||
100nps | r201-r211:100nps | 1100 | 4 | 20 | 60 | 11 | 72.23 | ||
Average | - | - | - | 453.12 | 6.18 | 20 | 60 | 7.25 | 157.29 |
Dataset | Subset | Optimal (Benchmark) | CSA Solution | Gap% | ||
---|---|---|---|---|---|---|
R2 | r201 | 25 | 25 | 463.3 | 475.51 | 2.64% |
r201 | 50 | 50 | 791.9 | 836.80 | 5.67% | |
r201 | 100 | 100 | 1143.2 | 1271.95 | 11.26% | |
r202 | 25 | 25 | 410.5 | 421.28 | 2.63% | |
r202 | 50 | 50 | 698.5 | 765.80 | 9.63% | |
r202 | 100 | 100 | - | 1265.5 | - | |
r203 | 25 | 25 | 391.4 | 399.67 | 2.11% | |
r203 | 50 | 50 | 605.3 | 618.98 | 2.26% | |
r203 | 100 | 100 | - | 977.39 | - | |
r204 | 25 | 25 | 355 | 358.57 | 1.01% | |
r204 | 50 | 50 | 506.4 | 530.84 | 4.83% | |
r204 | 100 | 100 | - | 824.16 | - | |
r205 | 25 | 25 | 393 | 407.00 | 3.56% | |
r205 | 50 | 50 | 690.1 | 723.90 | 4.90% | |
r205 | 100 | 100 | - | 1074.42 | - | |
r206 | 25 | 25 | 324.0 | 325.10 | 0.34% | |
r206 | 50 | 50 | 632.4 | 692.00 | 9.42% | |
r206 | 100 | 100 | - | 970.02 | - | |
r207 | 25 | 25 | 361.6 | 392.32 | 8.50% | |
r207 | 50 | 50 | - | 583.01 | - | |
r207 | 100 | 100 | - | 914.83 | - | |
r208 | 25 | 25 | 328.2 | 331.79 | 1.09% | |
r208 | 50 | 50 | - | 510.88 | - | |
r208 | 100 | 100 | - | 778.72 | - | |
r209 | 25 | 25 | 370.7 | 399.21 | 7.69% | |
r209 | 50 | 50 | 600.6 | 619.53 | 3.15% | |
r209 | 100 | 100 | - | 954.34 | - | |
r210 | 25 | 25 | 404.6 | 431.10 | 6.55% | |
r210 | 50 | 50 | 645.6 | 679.01 | 5.18% | |
r210 | 100 | 100 | - | 959.19 | - | |
r211 | 25 | 25 | 350.9 | 351.9 | 0.28% | |
r211 | 50 | 50 | 535.5 | 581.60 | 8.61% | |
r211 | 100 | 100 | - | 817.02 | - | |
Average | - | - | - | 526.33 * | 553.04 * | 5.07% * |
Class | Scenario | Strategy | ID | NTWs | Average of Solutions | ||||
---|---|---|---|---|---|---|---|---|---|
RC1 | rc101—rc103 | 25nps | rc101-rc103:25nps | 75 | 4 | 20 | 60 | 3 | 187.33 |
50nps | rc101-rc103:50nps | 150 | 5 | 20 | 60 | 3 | 235.24 | ||
75nps | rc101-rc103:75nps | 225 | 8 | 20 | 60 | 3 | 278.85 | ||
100nps | rc101-rc103:100nps | 300 | 10 | 20 | 60 | 3 | 350.11 | ||
rc101—rc106 | 25nps | rc101-rc106:25nps | 150 | 3 | 20 | 60 | 6 | 124.08 | |
50nps | rc101-rc106:50nps | 300 | 4 | 20 | 60 | 6 | 201.48 | ||
75nps | rc101-rc106:75nps | 450 | 9 | 20 | 60 | 6 | 288.80 | ||
100nps | rc101-rc106:100nps | 600 | 10 | 20 | 60 | 6 | 316.13 | ||
rc101—rc108 | 25nps | rc101-rc108:25nps | 200 | 7 | 20 | 60 | 8 | 235.24 | |
50nps | rc101-rc108:50nps | 400 | 5 | 20 | 60 | 8 | 255.85 | ||
75nps | rc101-rc108:75nps | 600 | 9 | 20 | 60 | 8 | 347.09 | ||
100nps | rc101-rc108:100nps | 800 | 9 | 20 | 60 | 8 | 284.67 | ||
Average | - | - | - | 354.16 | 6.91 | 20 | 60 | 5.66 | 258.74 |
Dataset | Subset | Optimal (Benchmark) | CSA Solution | Gap% | ||
---|---|---|---|---|---|---|
RC1 | rc101 | 25 | 25 | 461.1 | 464.57 | 0.75% |
rc101 | 50 | 50 | 944 | 968.80 | 2.63% | |
rc101 | 100 | 100 | 1619.8 | 1721.18 | 6.26% | |
rc102 | 25 | 25 | 351.8 | 352.74 | 0.27% | |
rc102 | 50 | 50 | 822.5 | 890.26 | 8.24% | |
rc102 | 100 | 100 | 1457.4 | 1539.48 | 5.63% | |
rc103 | 25 | 25 | 332.8 | 333.91 | 0.33% | |
rc103 | 50 | 50 | 710.9 | 753.52 | 6.00% | |
rc103 | 100 | 100 | 1258 | 1391.77 | 10.63% | |
rc104 | 25 | 25 | 306.6 | 307.14 | 0.18% | |
rc104 | 50 | 50 | 545.8 | 546.51 | 0.13% | |
rc104 | 100 | 100 | - | 1191.07 | - | |
rc105 | 25 | 25 | 411.3 | 434.29 | 5.59% | |
rc105 | 50 | 50 | 855.3 | 894.08 | 4.53% | |
rc105 | 100 | 100 | 1513.7 | 1649.14 | 8.95% | |
rc106 | 25 | 25 | 345.5 | 347.26 | 0.51% | |
rc106 | 50 | 50 | 723.2 | 814.49 | 12.62% | |
rc106 | 100 | 100 | - | 1475.35 | - | |
rc107 | 25 | 25 | 298.3 | 298.95 | 0.22% | |
rc107 | 50 | 50 | 642.7 | 671.77 | 4.52% | |
rc107 | 100 | 100 | 1207.8 | 1278.37 | 5.84% | |
rc108 | 25 | 25 | 294.5 | 295.74 | 0.42% | |
rc108 | 50 | 50 | 598.1 | 599.17 | 0.18% | |
rc108 | 100 | 100 | 1114.2 | 1217.62 | 9.28% | |
Average | - | - | - | 764.33 * | 807.76 * | 5.68% * |
Class | Scenario | Strategy | ID | NTWs | Average of Solutions | ||||
---|---|---|---|---|---|---|---|---|---|
RC2 | rc201—rc203 | 25nps | rc201-rc203:25nps | 75 | 4 | 20 | 60 | 3 | 187.33 |
50nps | rc201-rc203:50nps | 150 | 5 | 20 | 60 | 3 | 238.58 | ||
75nps | rc201-rc203:75nps | 225 | 8 | 20 | 60 | 3 | 297.67 | ||
100nps | rc201-rc203:100nps | 300 | 10 | 20 | 60 | 3 | 357.66 | ||
rc201—rc206 | 25nps | rc201-rc206:25nps | 150 | 3 | 20 | 60 | 6 | 143.4 | |
50nps | rc201-rc206:50nps | 300 | 4 | 20 | 60 | 6 | 209.03 | ||
75nps | rc201-rc206:75nps | 450 | 9 | 20 | 60 | 6 | 313.10 | ||
100nps | rc201-rc206:100nps | 600 | 10 | 20 | 60 | 6 | 347.58 | ||
rc201—rc208 | 25nps | rc201-rc208:25nps | 200 | 7 | 20 | 60 | 8 | 240.23 | |
50nps | rc201-rc208:50nps | 400 | 5 | 20 | 60 | 8 | 250.89 | ||
75nps | rc201-rc208:75nps | 600 | 9 | 20 | 60 | 8 | 359.58 | ||
100nps | rc201-rc208:100nps | 800 | 9 | 20 | 60 | 8 | 369.96 | ||
Average | - | - | - | 354.16 | 6.91 | 20 | 60 | 5.66 | 276.25 |
Class | Subset | Optimal (Benchmark) | CSA Solution | Gap% | ||
---|---|---|---|---|---|---|
RC2 | rc201 | 25 | 25 | 360.2 | 361.24 | 0.29% * |
rc201 | 50 | 50 | 684.8 | 686.31 | 0.22% | |
rc201 | 100 | 100 | 1261.8 | 1395.70 | 10.61% | |
rc202 | 25 | 25 | 338 | 338.82 | 0.24% | |
rc202 | 50 | 50 | 613.6 | 665.53 | 8.46% | |
rc202 | 100 | 100 | 1092.3 | 1233.72 | 12.95% | |
rc203 | 25 | 25 | 326.9 | 328.44 | 0.47% | |
rc203 | 50 | 50 | 555.3 | 625.60 | 12.66% | |
rc203 | 100 | 100 | - | 1082.26 | - | |
rc204 | 25 | 25 | 299.7 | 315.95 | 5.42% | |
rc204 | 50 | 50 | 444.2 | 462.98 | 4.23% | |
rc204 | 100 | 100 | - | 847.32 | - | |
rc205 | 25 | 25 | 338 | 338.93 | 0.28% | |
rc205 | 50 | 50 | 630.2 | 631.98 | 0.28% | |
rc205 | 100 | 100 | 1154 | 1241.2 | 7.56% | |
rc206 | 25 | 25 | 324 | 325.10 | 0.34% | |
rc206 | 50 | 50 | 610 | 611.75 | 0.29% | |
rc206 | 100 | 100 | - | 1190.45 | - | |
rc207 | 25 | 25 | 298.3 | 298.95 | 0.22% | |
rc207 | 50 | 50 | 558.6 | 561.42 | 0.50% | |
rc207 | 100 | 100 | - | 1081.70 | - | |
rc208 | 25 | 25 | 269.1 | 269.56 | 0.17% | |
rc208 | 50 | 50 | - | 491.46 | - | |
rc208 | 100 | 100 | - | 810.35 | - | |
Average | - | - | - | 564.38 * | 594.06 * | 5.26% * |
Class | Scenario | Strategy | ID | NTWs | MIP Solution | CSA Solution | Gap% | ||||
---|---|---|---|---|---|---|---|---|---|---|---|
C1 | c101-c103 | 25nps | c101-c103:25nps | 75 | 3 | 20 | 60 | 3 | 83.48 | 83.48 | 0.00% |
C1 | c101-c106 | 25nps | c101-c106:25nps | 150 | 3 | 20 | 60 | 6 | 82.48 | 82.48 | 0.00% |
C1 | c101-c109 | 25nps | c101-c109:25nps | 225 | 3 | 20 | 60 | 9 | 71.12 | 71.12 | 0.00% |
C2 | c201-c203 | 25nps | c201-c203:25nps | 75 | 8 | 20 | 60 | 3 | 172.81 | 173.5 | 0.40% |
C2 | c201-c206 | 25nps | c201-c206:25nps | 150 | 9 | 20 | 60 | 6 | 199.38 | 214.39 | 7.53% |
C2 | c201-c208 | 25nps | c201-c208:25nps | 200 | 10 | 20 | 60 | 8 | 236.03 | 243.67 | 3.24% |
R1 | r101-r103 | 25nps | r101-r103:25nps | 75 | 10 | 20 | 60 | 3 | 242.32 | 246.33 | 1.65% |
R1 | r101-r106 | 25nps | r101-r106:25nps | 150 | 9 | 20 | 60 | 6 | 207.00 | 234.7 | 13.38% |
R1 | r101-r109 | 25nps | r101-r109:25nps | 225 | 9 | 20 | 60 | 9 | 238.70 | 257.61 | 7.92% |
R1 | r101-r112 | 25nps | r101-r112:25nps | 300 | 8 | 20 | 60 | 12 | 197.00 | 207.54 | 5.35% |
R2 | r201-r203 | 25nps | r201-r203:25nps | 75 | 10 | 20 | 60 | 3 | 244.32 | 264.52 | 8.27% |
R2 | r201-r206 | 25nps | r201-r206:25nps | 150 | 9 | 20 | 60 | 6 | 209.32 | 233.73 | 11.66% |
R2 | r201-r209 | 25nps | r201-r209:25nps | 225 | 9 | 20 | 60 | 9 | 234.34 | 250.73 | 6.99% |
R2 | r201-r211 | 25nps | r201-r211:25nps | 275 | 8 | 20 | 60 | 11 | 198.63 | 209 | 5.22% |
RC1 | rc101-rc103 | 25nps | rc101-rc103:25nps | 75 | 4 | 20 | 60 | 3 | 187.33 | 187.33 | 0.00% |
RC1 | rc101-rc106 | 25nps | rc101-rc106:25nps | 150 | 3 | 20 | 60 | 6 | 124.08 | 124.08 | 0.00% |
RC1 | rc101-rc108 | 25nps | rc101-rc108:25nps | 200 | 7 | 20 | 60 | 8 | 231.23 | 235.24 | 1.73% |
RC2 | rc201-rc203 | 25nps | rc201-rc203:25nps | 75 | 4 | 20 | 60 | 3 | 187.33 | 187.33 | 0.00% |
RC2 | rc201-rc206 | 25nps | rc201-rc206:25nps | 150 | 3 | 20 | 60 | 6 | 143.39 | 143.39 | 0.00% |
RC2 | rc201-rc208 | 25nps | rc201-rc208:25nps | 200 | 7 | 20 | 60 | 8 | 231.23 | 240.23 | 3.89% |
Average | 186.07 | 194.52 | 4.54% |
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© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
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Hedayati, S.; Setak, M.; Van Woensel, T.; Demir, E. Re-Supplying Autonomous Mobile Parcel Lockers in Last-Mile Distribution. Future Transp. 2024, 4, 1266-1296. https://doi.org/10.3390/futuretransp4040061
Hedayati S, Setak M, Van Woensel T, Demir E. Re-Supplying Autonomous Mobile Parcel Lockers in Last-Mile Distribution. Future Transportation. 2024; 4(4):1266-1296. https://doi.org/10.3390/futuretransp4040061
Chicago/Turabian StyleHedayati, Sajjad, Mostafa Setak, Tom Van Woensel, and Emrah Demir. 2024. "Re-Supplying Autonomous Mobile Parcel Lockers in Last-Mile Distribution" Future Transportation 4, no. 4: 1266-1296. https://doi.org/10.3390/futuretransp4040061