An Enhanced Approach for the Multiple Vehicle Routing Problem with Heterogeneous Vehicles and a Soft Time Window
Abstract
:1. Introduction
2. Literature Review
3. Proposed Model
3.1. Assumptions
- The decision model considers multiple periods, multiple suppliers, and multiple vehicles.
- The depot is both the shipping point and the final destination of each vehicle.
- The distance between each two suppliers is fixed and known.
- The travelling time between each two suppliers is fixed and known.
- Different types of vehicles have different unit travelling costs.
- The unloading time in each supplier is fixed and known.
- The assigning cost for each vehicle is fixed and known. Different assigning costs occur for different types of vehicles.
- Each vehicle has a limited loading capacity. Different types of vehicles have different loading capacities.
- Each vehicle has a limited travelling distance in a period. Different types of vehicles have different travelling distance limits.
- Each supplier has a soft time window. When a vehicle arrives to a supplier after the latest soft time, a tardiness cost will be charged by the supplier based on the tardiness time.
- Multiple vehicles can be used in a period. A supplier can only be visited by at most a vehicle in a period.
- No shortage of outsourced materials is allowed.
3.2. Construction of the MIP Model
3.3. Genetic Algorithm Model
4. Case Study
4.1. Case Information
4.2. Case Results and Analysis
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Notations for the MIP
i, j | Supplier node (i = 0,1,2,…, I, j = 0,1,2,…, I, depot = 0, supplier = 1,2,3,…, I) |
t | Period |
v | Number of vehicles |
Set of suppliers | |
Set of periods | |
Set of vehicles |
Demand from supplier i in period t | |
Fixed cost for assigning vehicle v | |
Travelling distance from supplier i to supplier j | |
Travelling time from supplier i to supplier j | |
Unload time at supplier i | |
Latest soft time to start the service at supplier i. | |
Tardiness cost per unit time charged by supplier i when a vehicle arrives after the latest soft time. | |
Maximum loading size for vehicle v | |
Maximum travelling distance for vehicle v | |
Arbitrary large number |
Binary variable, 1 indicates that vehicle v travels from supplier i to supplier j in period t, and 0 indicates that no travel is incurred | |
Tardiness time of vehicle v when arriving supplier i in period t. | |
Service start time for supplier i served by vehicle v in period t. |
References
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Vehicle | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
Vehicle type | S | M | M | L | L |
Assigning cost () | $200 | $300 | $300 | $400 | $400 |
Unit travelling cost () | $0.9 | $1 | $1 | $1.1 | $1.1 |
Vehicle | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
Loading () | 40 | 50 | 50 | 60 | 60 |
Travelling distance () | 300 | 550 | 550 | 660 | 660 |
Supplier | i = 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
ui | 11 | 10 | 12 | 21 | 10 | 13 | 15 | 12 | 8 | 16 | 13 | 11 |
pi | $1.1 | $2.1 | $3.2 | $2.2 | $2.3 | $3 | $5.2 | $4.4 | $2.6 | $3.7 | $2.5 | $1.4 |
li | 120 | 260 | 310 | 160 | 295 | 340 | 358 | 415 | 320 | 340 | 375 | 455 |
πij | j = 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
i = 0 | 0 | 150 | 100 | 120 | 210 | 100 | 135 | 150 | 127 | 85 | 175 | 120 | 115 |
1 | 150 | 0 | 160 | 310 | 120 | 195 | 140 | 158 | 215 | 227 | 40 | 175 | 255 |
2 | 100 | 160 | 0 | 150 | 145 | 65 | 55 | 250 | 245 | 190 | 147 | 40 | 173 |
3 | 120 | 310 | 150 | 0 | 285 | 95 | 200 | 256 | 178 | 82 | 265 | 124 | 35 |
4 | 210 | 120 | 145 | 285 | 0 | 205 | 70 | 270 | 310 | 290 | 32 | 169 | 309 |
5 | 100 | 195 | 65 | 95 | 205 | 0 | 115 | 270 | 235 | 160 | 207 | 61 | 125 |
6 | 135 | 140 | 55 | 200 | 70 | 115 | 0 | 260 | 275 | 225 | 100 | 80 | 225 |
7 | 150 | 158 | 250 | 256 | 270 | 270 | 260 | 0 | 111 | 195 | 215 | 283 | 254 |
8 | 127 | 215 | 245 | 178 | 310 | 235 | 275 | 111 | 0 | 80 | 305 | 265 | 143 |
9 | 85 | 227 | 190 | 82 | 290 | 160 | 225 | 195 | 80 | 0 | 290 | 198 | 55 |
10 | 175 | 40 | 147 | 265 | 32 | 207 | 100 | 215 | 305 | 290 | 0 | 180 | 289 |
11 | 120 | 175 | 40 | 124 | 169 | 61 | 80 | 283 | 265 | 198 | 180 | 0 | 168 |
12 | 115 | 255 | 173 | 35 | 309 | 125 | 225 | 254 | 143 | 55 | 289 | 168 | 0 |
wij | j = 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
i = 0 | 0 | 90 | 60 | 72 | 126 | 60 | 81 | 90 | 76 | 51 | 105 | 72 | 69 |
1 | 90 | 0 | 96 | 186 | 72 | 117 | 84 | 95 | 129 | 136 | 24 | 105 | 153 |
2 | 60 | 96 | 0 | 90 | 87 | 39 | 33 | 150 | 147 | 114 | 88 | 24 | 104 |
3 | 72 | 186 | 90 | 0 | 171 | 57 | 120 | 154 | 107 | 49 | 159 | 74 | 21 |
4 | 126 | 72 | 87 | 171 | 0 | 123 | 42 | 162 | 186 | 174 | 19 | 101 | 185 |
5 | 60 | 117 | 39 | 57 | 123 | 0 | 69 | 162 | 141 | 96 | 124 | 37 | 75 |
6 | 81 | 84 | 33 | 120 | 42 | 69 | 0 | 156 | 165 | 135 | 60 | 48 | 135 |
7 | 90 | 95 | 150 | 154 | 162 | 162 | 156 | 0 | 67 | 117 | 129 | 170 | 152 |
8 | 76 | 129 | 147 | 107 | 186 | 141 | 165 | 67 | 0 | 48 | 183 | 159 | 86 |
9 | 51 | 136 | 114 | 49 | 174 | 96 | 135 | 117 | 48 | 0 | 174 | 119 | 33 |
10 | 105 | 24 | 88 | 159 | 19 | 124 | 60 | 129 | 183 | 174 | 0 | 108 | 173 |
11 | 72 | 105 | 24 | 74 | 101 | 37 | 48 | 170 | 159 | 119 | 108 | 0 | 101 |
12 | 69 | 153 | 104 | 21 | 185 | 75 | 135 | 152 | 86 | 33 | 173 | 101 | 0 |
Case | Period Number | Supplier Number | Vehicle Number | Transportation Network | Variable Number for MIP | Constraint Number for MIP |
---|---|---|---|---|---|---|
I | 5 | 5 | 3 | 6 × 6 | 931 | 1171 |
II | 7 | 9 | 4 | 10 × 10 | 3903 | 3977 |
III | 9 | 12 | 5 | 13 × 13 | 9758 | 9919 |
Supplier | 1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|---|
Period | ||||||
1 | 21 | 22 | 40 | 13 | 26 | |
2 | 26 | 24 | 23 | 24 | 30 | |
3 | 28 | 20 | 31 | 9 | 17 | |
4 | 30 | 26 | 24 | 8 | 26 | |
5 | 22 | 12 | 32 | 28 | 23 |
Period | Vehicle | Vehicle Routing and Loading Size |
---|---|---|
1 | 1 | Depot → Supplier 3 (40) → Depot |
2 | Depot → Supplier 1 (21) → Supplier 4 (13) → Depot | |
3 | Depot → Supplier 2 (22) → Supplier 5 (26) → Depot | |
2 | 1 | Depot → Supplier 5 (30) → Depot |
2 | Depot → Supplier 3 (23) → Supplier 2 (24) → Depot | |
3 | Depot → Supplier 1 (26) → Supplier 4 (24) → Depot | |
3 | 1 | Depot → Supplier 3 (31) → Depot |
2 | Depot → Supplier 2 (20) → Supplier 5 (17) → Depot | |
3 | Depot → Supplier 1 (28) → Supplier 4 (9) → Depot | |
4 | 1 | Depot → Supplier 2 (26) → Depot |
2 | Depot → Supplier 3 (24) → Supplier 5 (26) → Depot | |
3 | Depot → Supplier 1 (30) → Supplier 4 (8) → Depot | |
5 | 1 | Depot → Supplier 3 (32) → Depot |
2 | Depot → Supplier 2 (12) → Supplier 5 (23) → Depot | |
3 | Depot → Supplier 1 (22) → Supplier 4 (28) → Depot |
Cost | Assigning Cost | Travelling Cost | Tardiness Cost | Total Transportation Cost |
---|---|---|---|---|
MIP | 4000 | 4888 | 143 | 9031 |
GA | 4000 | 4888 | 143 | 9031 |
Error = 0.00% |
Supplier | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|---|---|---|---|---|
Period | ||||||||||
1 | 22 | 19 | 20 | 13 | 21 | 23 | 22 | 24 | 23 | |
2 | 26 | 24 | 30 | 17 | 25 | 16 | 24 | 21 | 20 | |
3 | 24 | 26 | 19 | 26 | 17 | 28 | 20 | 9 | 21 | |
4 | 30 | 15 | 24 | 11 | 24 | 30 | 16 | 24 | 18 | |
5 | 22 | 20 | 25 | 28 | 23 | 22 | 13 | 21 | 19 | |
6 | 21 | 17 | 19 | 21 | 23 | 21 | 16 | 9 | 28 | |
7 | 22 | 13 | 30 | 28 | 13 | 22 | 12 | 21 | 24 |
Period | Vehicle | Vehicle Routing and Loading Size |
---|---|---|
1 | 1 | Depot → Supplier 2 (19) → Supplier 5 (21) → Depot |
2 | Depot → Supplier 1 (22) → Supplier 7 (22) → Depot | |
3 | Depot → Supplier 3 (20) → Supplier 8 (24) → Depot | |
4 | Depot → Supplier 6 (23) →Supplier 4 (13) →Supplier 9 (23) → Depot | |
2 | 1 | Depot → Supplier 3 (30) → Supplier 9 (10) → Depot |
2 | Depot → Supplier 1 (26) → Supplier 7 (24) → Depot | |
3 | Depot → Supplier 5 (25) → Supplier 8 (21) → Depot | |
4 | Depot → Supplier 4 (17) → Supplier 6 (16) → Supplier 2 (24) → Depot | |
3 | 1 | Depot → Supplier 3 (19) → Supplier 9 (21) → Depot |
2 | Depot → Supplier 1 (24) → Supplier 4 (26) → Depot | |
3 | Depot → Supplier 2 (16) → Supplier 6 (28) → Depot | |
4 | Depot → Supplier 5 (17) → Supplier 8 (19) → Supplier 7 (20) → Depot | |
4 | 1 | Depot → Supplier 2 (15) → Supplier 5 (24) → Depot |
2 | Depot → Supplier 1 (30) → Supplier 7 (16) → Depot | |
3 | Depot → Supplier 3 (24) → Supplier 8 (24) → Depot | |
4 | Depot → Supplier 6 (30) → Supplier 4 (11) → Supplier 9 (18) → Depot | |
5 | 1 | Depot → Supplier 8 (21) → Supplier 9 (19) → Depot |
2 | Depot → Supplier 3 (25) → Supplier 5 (23) → Depot | |
3 | Depot → Supplier 4 (28) → Supplier 6 (22) → Depot | |
4 | Depot → Supplier 2 (20) → Supplier 1 (22) → Supplier 7 (13) → Depot | |
6 | 1 | Depot → Supplier 2 (17) → Supplier 6 (21)→ Depot |
2 | Depot → Supplier 1 (21) → Supplier 4 (21) → Depot | |
3 | Depot → Supplier 3 (19) → Supplier 5 (23)→ Depot | |
4 | Depot → Supplier 7 (16) → Supplier 8 (9) → Supplier 9 (28) → Depot | |
7 | 1 | Depot → Supplier 3 (30) → Depot |
2 | Depot → Supplier 1 (22) → Supplier 4 (28) → Depot | |
3 | Depot → Supplier 5 (13) → Supplier 2 (13) → Supplier 6 (22) → Depot | |
4 | Depot → Supplier 7 (22) → Supplier 8 (21) → Supplier 9 (12) → Depot |
Period | Vehicle | Vehicle Routing and Loading Size |
---|---|---|
1 | 1 | Depot → Supplier 2 (19) → Supplier 5 (21)→ Depot |
2 | Depot → Supplier 3 (20) → Supplier 6 (23) → Depot | |
3 | Depot → Supplier 7 (22) → Supplier 8 (24)→ Depot | |
4 | Depot → Supplier 4 (13) → Supplier 1 (22) → Supplier 9 (23) → Depot | |
2 | 1 | Depot → Supplier 3 (30) → Supplier 9 (10)→ Depot |
2 | Depot → Supplier 1 (26) → Supplier 7 (24) → Depot | |
3 | Depot → Supplier 5 (25) → Supplier 8 (21) → Depot | |
4 | Depot → Supplier 4 (17) → Supplier 2 (24) → Supplier 6 (16) → Depot | |
3 | 1 | Depot → Supplier 3 (19) → Supplier 9 (21) → Depot |
2 | Depot → Supplier 1 (24) → Supplier 4 (26) → Depot | |
3 | Depot → Supplier 2 (16) → Supplier 6 (28) → Depot | |
4 | Depot → Supplier 5 (17) → Supplier 8 (19) → Supplier 7 (20) → Depot | |
4 | 1 | Depot → Supplier 2 (15) → Supplier 5 (24) → Depot |
2 | Depot → Supplier 3 (24) → Supplier 8 (24) → Depot | |
3 | Depot → Supplier 6 (30) → Supplier 7 (16) → Depot | |
4 | Depot → Supplier 4 (11) → Supplier 1 (30) → Supplier 9 (18) → Depot | |
5 | 1 | Depot → Supplier 8 (21) → Supplier 9 (19) → Depot |
2 | Depot → Supplier 1 (22) → Supplier 4 (28) → Depot | |
3 | Depot → Supplier 3 (25) → Supplier 5 (23) → Depot | |
4 | Depot → Supplier 6 (22) → Supplier 2 (12) → Supplier 7 (13) → Depot | |
6 | 1 | Depot → Supplier 2 (17) → Supplier 6 (21) → Depot |
2 | Depot → Supplier 1 (21) → Supplier 4 (21) → Depot | |
3 | Depot → Supplier 3 (19) → Supplier 5 (23) → Depot | |
4 | Depot → Supplier 7 (16) → Supplier 8 (9) →Supplier 9 (28) → Depot | |
7 | 1 | Depot → Supplier 8 (21) → Supplier 9 (12) → Depot |
2 | Depot → Supplier 3 (30) → Supplier 5 (13) → Depot | |
3 | Depot → Supplier 4 (28) → Supplier 6 (22) → Depot | |
4 | Depot → Supplier 2 (13) → Supplier 1 (22) → Supplier 7 (22) → Depot |
Cost | Assigning Cost | Travelling Cost | Tardiness Cost | Total Transportation Cost |
---|---|---|---|---|
MIP | 8400 | 11,521.5 | 193.6 | 20,115.1 |
GA | 8400 | 12,013.1 | 505.4 | 20,918.5 |
Error = 3.99% |
Supplier | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Period | |||||||||||||
1 | 22 | 19 | 20 | 13 | 21 | 23 | 22 | 24 | 13 | 26 | 24 | 24 | |
2 | 26 | 14 | 20 | 17 | 25 | 26 | 24 | 21 | 20 | 23 | 12 | 19 | |
3 | 24 | 16 | 9 | 26 | 17 | 28 | 20 | 9 | 31 | 22 | 25 | 24 | |
4 | 30 | 16 | 12 | 11 | 24 | 30 | 16 | 24 | 18 | 15 | 21 | 23 | |
5 | 22 | 12 | 15 | 28 | 23 | 22 | 13 | 25 | 25 | 23 | 22 | 17 | |
6 | 21 | 17 | 19 | 21 | 23 | 21 | 16 | 9 | 28 | 11 | 18 | 16 | |
7 | 22 | 13 | 30 | 28 | 13 | 22 | 12 | 21 | 20 | 22 | 27 | 11 | |
8 | 21 | 17 | 19 | 21 | 23 | 21 | 16 | 9 | 31 | 21 | 18 | 16 | |
9 | 22 | 13 | 20 | 28 | 13 | 22 | 12 | 11 | 29 | 22 | 27 | 11 |
Period | Vehicle | Vehicle Routing and Loading Size |
---|---|---|
1 | 1 | Depot → Supplier 2 (19) → Supplier 5 (21) → Depot |
2 | Depot → Supplier 6 (23) → Supplier 11 (24) → Depot | |
3 | Depot → Supplier 7 (22) → Supplier 8 (24) → Depot | |
4 | Depot → Supplier 10 (26) → Supplier 4 (13) → Supplier 3 (20) → Depot | |
5 | Depot → Supplier 1 (22) → Supplier 12 (24) → Supplier 9 (13) → Depot | |
2 | 1 | Depot → Supplier 3 (20) → Supplier 12 (19) → Depot |
2 | Depot → Supplier 8 (21) → Supplier 9 (20) → Depot | |
3 | Depot → Supplier 1 (26) → Supplier 7 (24) → Depot | |
4 | Depot → Supplier 4 (17) → Supplier 6 (26) → Supplier 2 (14) → Depot | |
5 | Depot → Supplier 5 (25) → Supplier 11 (12) → Supplier 10 (23) → Depot | |
3 | 1 | Depot → Supplier 3 (9) → Supplier 9 (31) → Depot |
2 | Depot → Supplier 1 (24) → Supplier 4 (26) → Depot | |
3 | Depot → Supplier 7 (20) → Supplier 12 (24) → Depot | |
4 | Depot → Supplier 2 (16) → Supplier 11 (25) → Supplier 5 (17) → Depot | |
5 | Depot → Supplier 6 (28) → Supplier 10 (22) → Supplier 8 (9) → Depot | |
4 | 1 | Depot → Supplier 2 (16) → Supplier 11 (21) → Depot |
2 | Depot → Supplier 1 (30) → Supplier 10 (15) → Depot | |
3 | Depot → Supplier 4 (11) → Supplier 6 (30) → Depot | |
4 | Depot → Supplier 5 (24) → Supplier 3 (12) → Supplier 12 (23) → Depot | |
5 | Depot → Supplier 9 (18) → Supplier 8 (24) → Supplier 7 (16) → Depot | |
5 | 1 | Depot → Supplier 3 (15) → Supplier 12 (17) → Depot |
2 | Depot → Supplier 8 (25) → Supplier 9 (25) → Depot | |
3 | Depot → Supplier 4 (28) → Supplier 6 (22) → Depot | |
4 | Depot → Supplier 2 (12) → Supplier 11 (22) → Supplier 5 (23) → Depot | |
5 | Depot → Supplier 10 (23) → Supplier 1 (22) → Supplier 7 (13) → Depot | |
6 | 1 | Depot → Supplier 2 (17) → Supplier 5 (23) → Depot |
2 | Depot → Supplier 3 (19) → Supplier 12 (16) → Depot | |
3 | Depot → Supplier 1 (21) → Supplier 10 (11) → Depot | |
4 | Depot → Supplier 4 (21) → Supplier 6 (21) → Supplier 11 (18) → Depot | |
5 | Depot → Supplier 7 (16) → Supplier 8 (9) → Supplier 9 (28) → Depot | |
7 | 1 | Depot → Supplier 8 (11) → Supplier 9 (20) → Depot |
2 | Depot → Supplier 3 (30) → Supplier 12 (11) → Depot | |
3 | Depot → Supplier 4 (28) → Supplier 6 (22) → Depot | |
4 | Depot → Supplier 10 (22) → Supplier 1 (22) → Supplier 7 (12) → Depot | |
5 | Depot → Supplier 2 (13) → Supplier 11 (27) → Supplier 5 (13) → Depot | |
8 | 1 | Depot → Supplier 2 (17) → Supplier 5 (23) → Depot |
2 | Depot → Supplier 3 (19) → Supplier 12 (16) → Depot | |
3 | Depot → Supplier 8 (9) → Supplier 9 (31) → Depot | |
4 | Depot → Supplier 4 (21) → Supplier 6 (21) → Supplier 11 (18) → Depot | |
5 | Depot → Supplier 10 (21) → Supplier 1 (21) → Supplier 7 (16)→ Depot | |
9 | 1 | Depot → Supplier 3 (20) → Supplier 12 (11) → Depot |
2 | Depot → Supplier 8 (21) → Supplier 9 (29) → Depot | |
3 | Depot → Supplier 4 (28) → Supplier 6 (22) → Depot | |
4 | Depot → Supplier 2 (13) → Supplier 11 (27) → Supplier 5 (13) → Depot | |
5 | Depot → Supplier 10 (22) → Supplier 1 (22) → Supplier 7 (12) → Depot |
Cost | Assigning Cost | Travelling Cost | Tardiness Cost | Total Transportation Cost |
---|---|---|---|---|
GA | 14,400 | 17,220 | 332.6 | 31,952.6 |
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Kang, H.-Y.; Lee, A.H.I. An Enhanced Approach for the Multiple Vehicle Routing Problem with Heterogeneous Vehicles and a Soft Time Window. Symmetry 2018, 10, 650. https://doi.org/10.3390/sym10110650
Kang H-Y, Lee AHI. An Enhanced Approach for the Multiple Vehicle Routing Problem with Heterogeneous Vehicles and a Soft Time Window. Symmetry. 2018; 10(11):650. https://doi.org/10.3390/sym10110650
Chicago/Turabian StyleKang, He-Yau, and Amy H. I. Lee. 2018. "An Enhanced Approach for the Multiple Vehicle Routing Problem with Heterogeneous Vehicles and a Soft Time Window" Symmetry 10, no. 11: 650. https://doi.org/10.3390/sym10110650
APA StyleKang, H.-Y., & Lee, A. H. I. (2018). An Enhanced Approach for the Multiple Vehicle Routing Problem with Heterogeneous Vehicles and a Soft Time Window. Symmetry, 10(11), 650. https://doi.org/10.3390/sym10110650