Methodology to Solve a Special Case of the Vehicle Routing Problem: A Case Study in the Raw Milk Transportation System
Abstract
:1. Introduction
2. Literature Review
3. Problem Description and Mathematical Formulation
s (time) | Shifting of the production line (s = 1..S) |
z (timezone) | Time zone (z = 1..Z) |
i | Raw milk farm (i = 1..I) |
j | Raw milk farm (j = 1..J) |
t | Truck label (t = 1..T) |
c (compartment) | Label of compartment of truck (c = 1..C) |
r (round) | Round of Truck (r = 1..R) |
P | Production rate (ton/hours) |
MX | Capacity of the production tank (ton) |
WC | Waiting cost of the truck before it can load (bath/hours) |
CT | Cleaning cost of the production tank (bath/time) |
CTI | Cleaning time of the production tank (hours) |
CTC | Cleaning time of a compartment in truck (hours) |
Capacity of compartment c in truck t | |
Cleaning cost of compartment c in truck t | |
Amount of milk available at milk raw milk farm k | |
Start time of windows z | |
End time of windows z | |
LRT | Loading rate of milk into the truck (ton/hours) |
LRA | Loading rate of milk into the production tank (ton/hours) |
FR | Driving speed of truck (km/hours) |
Distance of raw milk raw milk farm i to j | |
SPT1 | Start time of the first loading in a day |
MN | Maximum number of milk raw milk farms that can mix the milk into the same compartment |
Consume rate of truck t (bath/km) |
1 if round r of truck t traveling from i to j 0 otherwise | |
1 if compartment c is used in round r truck t 0 otherwise | |
1 if shifting s need to clean the tank 0 otherwise | |
1 if amount of milk that is delivered in round r truck t shifting s greater than zero 0 otherwise | |
1 if round r truck t using time windows zone z 0 otherwise | |
1 if amount of milk that is delivered from raw milk farm k in round r truck t compartment c greater than zero 0 otherwise | |
1 if amount of milk from raw milk farm that is delivered in round r truck t is greater than zero 0 otherwise | |
Dummy variable of round r truck t milk raw milk farm k for subtour elimination constraint | |
Waiting time of operation round t truck t shifting s | |
Amount of milk that is delivered in round r truck t to shifting s | |
Amount of milk that is delivered from milk raw milk farm k in round r truck t compartment c | |
Start traveling time of round r truck t | |
Waiting time to start round r truck t | |
End time of the time windows of round r truck t | |
Processing time of shifting s | |
Start loading time of shifting s | |
End processing time of shifting s | |
Arrive time of round r truck t shifting s | |
End traveling time of round r truck t | |
Final end of tour and production of round r truck t | |
Arrival time of round r truck t at milk raw milk farm k | |
Finish time at milk raw milk farm k of round r truck t | |
Large number | |
Start time of the time windows of round r truck t |
4. The Proposed Heuristics
4.1. Generate a Set of Initial Vectors
- Discover the raw milk farms’ order by sorting the value in the position of a vector in increasing order (see Figure 3).
- Discover the order of the truck which is randomly generated.
- Assign and construct the route of the raw milk farms that will be serviced by the truck according to the order obtained from step 1. The assigning of the raw milk farm to the truck needs to keep the following conditions satisfied.
- (3.1)
- There are two time zones. The first time zone runs from 06:00–9:00, and the second time zone runs from 17:00–21:00. The truck needs to arrive at the farm within the time zone times.
- (3.2)
- Each truck has 480 min to collect the raw milk from the farm and deliver it to the production line.
- (3.3)
- There are 6 trucks available to use in each time zone. The trucks have capacities of 12, 12, 9, 9, 6 and 6 respectively. Each truck is divided into 3 equal compartments.
- (3.4)
- All customers must be visited.
4.2. Mutation
4.3. Recombination
4.3.1. Vector Transition Process
4.3.2. Vector Exchange Process
4.3.3. Vector Insertion Process
4.4. Selection
5. Computational Framework and Result
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Raw Milk Farms | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|
Target Vector 1 | 0.45 | 0.57 | 0.12 | 0.45 | 0.63 | 0.14 | 0.65 |
Target Vector 2 | 0.03 | 0.11 | 0.93 | 0.12 | 0.73 | 0.41 | 0.42 |
Target Vector 3 | 0.63 | 0.08 | 0.08 | 0.33 | 0.66 | 0.79 | 0.09 |
Target Vector 4 | 0.99 | 0.28 | 0.14 | 0.57 | 0.91 | 0.97 | 0.73 |
Target Vector 5 | 0.66 | 0.00 | 0.77 | 0.73 | 0.94 | 0.09 | 0.45 |
Raw Milk Farm | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|
Raw milk (Ton) | 4.1 | 5.6 | 2.4 | 4.5 | 3.5 | 5.6 | 5.8 |
From/To | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|
0 | 0 | 54.4 | 54.88 | 23.91 | 62.64 | 15.46 | 14.06 | 47.18 |
1 | 54.4 | 0 | 72.06 | 46.45 | 45.55 | 22.87 | 48.42 | 58.1 |
2 | 54.88 | 72.06 | 0 | 82.4 | 77.37 | 32.01 | 37.45 | 65.44 |
3 | 23.91 | 46.45 | 82.4 | 0 | 85.08 | 11.63 | 14.84 | 15.54 |
4 | 62.64 | 45.55 | 77.37 | 85.08 | 0 | 28.12 | 15.88 | 86.63 |
5 | 15.46 | 22.87 | 32.01 | 11.63 | 28.12 | 0 | 69.76 | 75.63 |
6 | 14.06 | 48.42 | 37.45 | 14.84 | 15.88 | 69.76 | 0 | 27.52 |
7 | 47.18 | 58.1 | 65.44 | 15.54 | 86.63 | 75.63 | 27.52 | 0 |
Maximum capacity of raw milk tank | 17 ton |
Waiting cost of the truck before it can load | 90 bath/hours |
Cleaning cost of the raw milk tank of the factory | 2000 bath |
Capacity of compartment c in truck t | A compartment 4,3,2 ton |
Cleaning cost of raw milk compartment c of truck t | 400,300,200 bath/compartment |
Fuel consumption rate of truck | 4.5,4.2,3.9 bath/km |
Time windows z | 8 h |
Loading time of raw milk to truck | 10 ton/hours |
Loading time of raw milk to Tank | 10 ton/hours |
Truck (Ton)/Time Zone | Route Sequence | Operating Time | Remaining Time of the Route | Distance (km) | Total Traveling Cost (Baht) | Cleaning of Compartment Cost (Baht) | Cleaning of Tanks Cost (Baht) | Total Cost (Baht) |
---|---|---|---|---|---|---|---|---|
12/1 | 0-3-6-4-1-0 | 298.58 | 181.42 | 154.58 | 695.61 | 1200 | 4000 | 1895.61 |
6/1 | 0-1-0 | 164 | 17.42 | 108.8 | 424.32 | 600 | 1024.32 | |
9/2 | 0-2-5-0 | 210.35 | 269.65 | 102.35 | 429.87 | 900 | 1329.87 | |
6/2 | 0-5-7-0 | 101.72 | 167.93 | 30.92 | 120.59 | 600 | 720.59 | |
Total | 1670.39 | 3300 | 4000 | 8970.39 |
DE-1 | Original Differential Evolution |
DE-2 | Original Differential evolution and vector transition process |
DE-3 | Original Differential evolution and vector exchange process |
DE-4 | Original Differential evolution and vector insertion process |
DE-5 | Modified Differential evolution algorithm (Original DE and Disturb Selection) |
DE-6 | Modified Differential evolution algorithm and vector transition process |
DE-7 | Modified Differential evolution algorithm and vector exchange process |
DE-8 | Modified Differential evolution algorithm and vector insertion process |
Instance No. | # Of Clients | Lower Bound (BAHT) | % Diff | |||||||
---|---|---|---|---|---|---|---|---|---|---|
DE-1 | DE-2 | DE-3 | DE-4 | DE-5 | DE-6 | DE-7 | DE-8 | |||
1 | 5 | 6,012.70 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
2 | 10 | 10239.4 | 11.84 | 9.71 | 9.30 | 12.23 | 4.25 | 3.81 | 6.16 | 2.67 |
3 | 15 | 15161.1 | 12.59 | 10.60 | 11.87 | 8.93 | 4.07 | 0.29 | 0.48 | 2.04 |
4 | 15 | 15830.5 | 9.71 | 10.70 | 6.90 | 8.24 | 6.88 | 2.84 | 0.71 | 1.87 |
5 | 20 | 20566.1 | 10.80 | 6.26 | 5.69 | 10.14 | 3.34 | 3.38 | 2.67 | 2.99 |
6 | 20 | 20539.7 | 12.50 | 8.49 | 10.45 | 10.68 | 4.87 | 3.31 | 6.26 | 2.90 |
7 | 25 | 26407.1 | 10.85 | 11.63 | 10.04 | 11.68 | 3.86 | 3.48 | 3.59 | 2.96 |
8 | 25 | 27737.1 | 6.02 | 2.72 | 4.92 | 3.01 | 1.78 | 2.09 | 1.40 | 1.69 |
9 | 30 | 32618.5 | 8.29 | 6.26 | 4.88 | 7.65 | 3.67 | 3.12 | 3.25 | 2.36 |
10 | 30 | 32045.6 | 12.71 | 9.34 | 5.38 | 7.66 | 2.81 | 3.94 | 1.75 | 4.67 |
11 | 35 | 38076.0 | 8.29 | 10.16 | 5.91 | 5.55 | 5.47 | 1.73 | 5.30 | 2.89 |
12 | 35 | 37517.6 | 8.90 | 8.04 | 8.86 | 10.67 | 4.62 | 5.72 | 4.12 | 2.96 |
13 | 40 | 43807.1 | 6.72 | 6.61 | 5.07 | 4.70 | 3.59 | 3.11 | 2.77 | 3.47 |
14 | 40 | 40773.2 | 0.99 | 2.45 | 2.38 | 0.55 | 9.54 | 7.62 | 2.00 | 9.38 |
15 | 45 | 65931.3 | 4.50 | 3.75 | 3.38 | 3.27 | 1.49 | 0.48 | 0.94 | 0.60 |
16 | 45 | 65728.4 | 5.20 | 4.61 | 4.45 | 2.17 | 1.33 | 1.46 | 1.10 | 0.58 |
average | 8.12 | 6.96 | 6.22 | 6.70 | 3.85 | 2.90 | 2.66 | 2.75 |
DE-2 | DE-3 | DE-4 | DE-5 | DE-6 | DE-7 | DE-8 | |
---|---|---|---|---|---|---|---|
DE-1 | 0.034 | 0.009 | 0.012 | 0.009 | 0.007 | 0.008 | 0.009 |
DE-2 | - | 0.568 | 0.952 | 0.041 | 0.030 | 0.002 | 0.026 |
DE-3 | - | 0.818 | 0.012 | 0.009 | 0.0006 | 0.009 | |
DE-4 | - | 0.03 | 0.019 | 0.003 | 0.014 | ||
DE-5 | - | 0.035 | 0.010 | 0.005 | |||
DE-6 | - | 0.192 | 0.912 | ||||
DE-7 | - | 0.652 |
Time/Min | ZMIN | % Diff | |||||||
---|---|---|---|---|---|---|---|---|---|
DE-1 | DE-2 | DE-3 | DE-4 | DE-5 | DE-6 | DE-7 | DE-8 | ||
5 | 66653.4 | 4.02 | 3.38 | 3.20 | 0.80 | 0.87 | 0.46 | 0.02 | 0.00 |
10 | 66640.4 | 4.04 | 3.40 | 3.23 | 0.82 | 0.89 | 0.48 | 0.00 | 0.02 |
20 | 66615.6 | 4.08 | 3.44 | 3.26 | 0.86 | 0.00 | 0.52 | 0.04 | 0.02 |
30 | 66572.9 | 4.15 | 3.50 | 3.33 | 0.92 | 0.06 | 0.32 | 0.00 | 0.08 |
40 | 61486.3 | 12.77 | 12.06 | 11.88 | 9.27 | 4.34 | 4.62 | 0.00 | 3.53 |
50 | 60164.1 | 15.24 | 14.53 | 14.34 | 11.68 | 6.72 | 5.86 | 0.00 | 4.89 |
60 | 59767.42 | 16.01 | 15.29 | 15.10 | 12.42 | 7.46 | 6.60 | 0.00 | 5.62 |
average | 8.62 | 7.94 | 7.76 | 5.25 | 2.91 | 2.69 | 0.01 | 2.02 |
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Chokanat, P.; Pitakaso, R.; Sethanan, K. Methodology to Solve a Special Case of the Vehicle Routing Problem: A Case Study in the Raw Milk Transportation System. AgriEngineering 2019, 1, 75-93. https://doi.org/10.3390/agriengineering1010006
Chokanat P, Pitakaso R, Sethanan K. Methodology to Solve a Special Case of the Vehicle Routing Problem: A Case Study in the Raw Milk Transportation System. AgriEngineering. 2019; 1(1):75-93. https://doi.org/10.3390/agriengineering1010006
Chicago/Turabian StyleChokanat, Peerawat, Rapeepan Pitakaso, and Kanchana Sethanan. 2019. "Methodology to Solve a Special Case of the Vehicle Routing Problem: A Case Study in the Raw Milk Transportation System" AgriEngineering 1, no. 1: 75-93. https://doi.org/10.3390/agriengineering1010006