# An Enhanced Approach for the Multiple Vehicle Routing Problem with Heterogeneous Vehicles and a Soft Time Window

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

_{2}emissions. An MIP was first proposed for small-sized problems, and an exact dynamic programming algorithm was presented next for large-sized problems. A hybrid GA, combined with the exact dynamic programming algorithm, was then developed as an efficient solution approach.

## 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

_{t,i,j,v}is a binary number, Q

_{t,v}, R

_{t,v}and S

_{t,j,v}are integer numbers.

## 4. Case Study

#### 4.1. Case Information

_{i}, ${p}_{i}$, and ${l}_{i}$. ${u}_{i}$ is the unload time (in minutes) required at supplier i, ${p}_{i}$ is the tardiness cost per minute charged by supplier i when a vehicle arrives after the latest soft time, and ${l}_{i}$ is the latest soft time (in minutes) to start the service at supplier i. Table 4 shows the travelling distance from one supplier (or depot) to the other. Table 5 shows the travelling time required from one supplier (or depot) to the other.

#### 4.2. Case Results and Analysis

_{t,}

_{i,}

_{j,}

_{v}equals one; otherwise, the value equals zero. In Period 1, the result from the GA shows X

_{1,0,3,1}= 1 and X

_{1,3,0,1}= 1 for Chromosome 1, X

_{1,0,1,2}= 1, X

_{1,1,4,2}= 1 and X

_{1,4,0,2}= 1 for Chromosome 2, and X

_{1,0,2,3}= 1, X

_{1,2,5,3}= 1 and X

_{1,5,0,3}= 1 for Chromosome 3. The routing can also be observed in Figure 3. For example, in Period 1, Vehicle 1 travels from the depot to supplier 3 (X

_{1,0,3,1}= 1), and then back to the depot (X

_{1,3,0,1}= 1).

## 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 |

$I$ | Set of suppliers $i\in \left\{0,1,2,\dots ,I\right\}$ |

$T$ | Set of periods $t\in \left\{1,2,3,\dots ,T\right\}$ |

$V$ | Set of vehicles $v\in \left\{1,2,3,\dots ,V\right\}$ |

${d}_{t,g}$ | Demand from supplier i in period t |

${\tau}_{v}$ | Fixed cost for assigning vehicle v |

${\pi}_{i,j}$ | Travelling distance from supplier i to supplier j |

${w}_{i,j}$ | Travelling time from supplier i to supplier j |

${u}_{i}$ | Unload time at supplier i |

${l}_{i}$ | Latest soft time to start the service at supplier i. |

${p}_{i}$ | Tardiness cost per unit time charged by supplier i when a vehicle arrives after the latest soft time. |

${q}_{v}$ | Maximum loading size for vehicle v |

${r}_{v}$ | Maximum travelling distance for vehicle v |

$M$ | Arbitrary large number |

${X}_{t,i,j,v}$ | 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 |

${L}_{t,i,v}$ | Tardiness time of vehicle v when arriving supplier i in period t. |

$S}_{t,i,v}^{$ | Service start time for supplier i served by vehicle v in period t. |

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**Figure 3.**Vehicle routing in Period 1. (

**a**) Vehicle v = 1 in Period 1; (

**b**) Vehicle v = 2 in Period 1; (

**c**) Vehicle v = 3 in Period 1.

Vehicle | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Vehicle type | S | M | M | L | L |

Assigning cost (${\tau}_{v}$) | $200 | $300 | $300 | $400 | $400 |

Unit travelling cost (${\rho}_{v}$) | $0.9 | $1 | $1 | $1.1 | $1.1 |

Vehicle | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Loading (${h}_{v}$) | 40 | 50 | 50 | 60 | 60 |

Travelling distance (${l}_{v}$) | 300 | 550 | 550 | 660 | 660 |

Supplier | i = 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

u_{i} | 11 | 10 | 12 | 21 | 10 | 13 | 15 | 12 | 8 | 16 | 13 | 11 |

p_{i} | $1.1 | $2.1 | $3.2 | $2.2 | $2.3 | $3 | $5.2 | $4.4 | $2.6 | $3.7 | $2.5 | $1.4 |

l_{i} | 120 | 260 | 310 | 160 | 295 | 340 | 358 | 415 | 320 | 340 | 375 | 455 |

_{i}: min; p

_{i}: $/min; l

_{i}: min.

π_{ij} | 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 |

w_{ij} | 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 |

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Kang, 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