Modelling the Shortest Path for Inner Warehouse Travelling Using the Floyd–Warshall Algorithm
Abstract
:1. Introduction
2. Literature Review
2.1. Warehouse Management
2.2. Floyd–Warshall Algorithm and Dynamic Programming Model
3. Methodology
3.1. Study Settings
3.1.1. Demand and Order Patterns
- All the order pickers are well experienced in the routes and the picking area;
- All the order pickers need to collect all the items in their picking list;
- Data collected only involve a normal working shift and normal working hours;
- Data collected do not involve holidays, special seasons, sales, or after 5 p.m.;
- The aisles in the zone are equal in length;
- The length of the aisle is measured as 1 cm = 10 m;
- All the items needed are available on the rack;
- Once the items reach 25, pickers need to place them in the packaging station, and continue to pick the remaining items again.
3.1.2. Current Practice in the Automotive Manufacturing Company
3.2. Phase I: Process of the Floyd–Warshall Algorithm
3.2.1. Step 1
3.2.2. Step 2
3.2.3. Step 3
3.2.4. Step 4
3.2.5. Step 5
3.3. Phase II: Process of DP Method
4. Results and Discussion
4.1. Phase I: Floyd–Warshall Algorithm
4.1.1. Result Using Floyd–Warshall for n = 9
4.1.2. Result Using Floyd–Warshall for n = 19
4.2. Phase II: DP Method
4.2.1. Result Using DP Method for n = 9
4.2.2. Result Using DP Method for n = 19
4.3. Comparison Analysis between Floyd–Warshall and DP
5. Conclusions and Recommendation
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Node | Items | Order Number | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Order 1 | Order 2 | Order 3 | Order 4 | Order 5 | Order 6 | Order 7 | Order 8 | Order 9 | ||
1 | A | 150 | ||||||||
2 | B | 90 | ||||||||
3 | C | 90 | ||||||||
4 | D | 60 | 60 | |||||||
5 | E | 60 | 60 | |||||||
6 | F | 60 | ||||||||
7 | G | 60 | 60 | |||||||
8 | H | 30 | 30 | 30 | ||||||
9 | I | 30 | 30 | 30 | ||||||
10 | J | 40 | 40 | 40 | 40 | |||||
11 | K | 40 | 40 | 40 | ||||||
12 | L | 250 | ||||||||
13 | M | 250 | ||||||||
14 | N | 30 | 30 | |||||||
15 | O | 30 | 30 | |||||||
16 | P | 90 | 120 | |||||||
17 | Q | 60 | 120 | |||||||
18 | R | 60 | 60 | 60 | ||||||
19 | S | 60 | 60 | 60 | ||||||
Total Daily Order | 100 | 730 | 340 | 430 | 100 | 210 | 200 | 340 | 130 |
Node | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|
1 | 385.3 | INF | INF | INF | INF | INF | INF | 360 | |
2 | 385.3 | 231.18 | INF | INF | INF | INF | INF | INF | |
3 | INF | 231.18 | 231.18 | INF | INF | INF | INF | INF | |
4 | INF | INF | 231.18 | 462.3 | INF | INF | INF | INF | |
5 | INF | INF | INF | 462.36 | 745.3 | INF | INF | INF | |
6 | INF | INF | INF | INF | 745.3 | 539.42 | INF | INF | |
7 | INF | INF | INF | INF | INF | 539.42 | 231.18 | INF | |
8 | INF | INF | INF | INF | INF | INF | 231.18 | 154.12 | |
9 | 360 | INF | INF | INF | INF | INF | INF | 154.12 |
Node | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|
1 | 0 | 385.3 | 616.48 | 847.66 | 1310 | 1284.75 | 745.3 | 514.12 | 360 |
2 | 385.3 | 0 | 231.18 | 462.36 | 924.72 | 1670.02 | 1130.6 | 899.42 | 745.3 |
3 | 616.48 | 231.18 | 0 | 231.18 | 693.54 | 1438.84 | 1361.78 | 1130.6 | 976.48 |
4 | 847.66 | 462.36 | 231.18 | 0 | 462.3 | 1207.66 | 1592.96 | 1361.78 | 1207.66 |
5 | 1310 | 924.72 | 693.54 | 462.36 | 0 | 745.3 | 1284.72 | 1515.9 | 1670 |
6 | 1284.75 | 1670.02 | 1438.84 | 1207.66 | 745.3 | 0 | 539.42 | 770.6 | 924.72 |
7 | 745.3 | 1130.6 | 1361.78 | 1592.96 | 1284.72 | 539.42 | 0 | 231.18 | 385.3 |
8 | 514.12 | 899.42 | 1130.6 | 1361.78 | 1515.9 | 770.6 | 231.18 | 0 | 154.12 |
9 | 360 | 745.5 | 976.48 | 1207.66 | 1670 | 924.72 | 385.3 | 154.12 | 0 |
Route | Distance (cm) | Number of Order Pickers |
---|---|---|
1-2-3-4-5 | 797.64 | 1 |
5-6-7-8 | 1091.17 | 1 |
8-9-10-11-12 | 1024.70 | 1 |
12-13-14-15-16-17-18 | 891.76 | 3 |
Total Distance | 4036.45 | 7 |
Nodes | No. of Pickers | Total Distance (m) | Percentage (%) Different | |
---|---|---|---|---|
Floyd–Warshall | DP Method | |||
n = 9 | 1 | 15,030.18 | 14,027.88 | 6.90 |
2 | 19,440 | 14,073.53 | 32.03 | |
3 | 10,096.02 | 17,491.68 | 53.62 | |
4 | 7964.8 | 16,426.92 | 69.39 | |
5 | 12,099.48 | 11,711.08 | 3.26 | |
6 | 15,026.6 | 8942.32 | 50.77 | |
7 | 11,690.14 | 12,126.96 | 3.67 | |
n = 19 | 1 | 34,579.28 | 42,526.9 | 20.61 |
2 | 34,503.38 | 45,171.08 | 26.78 | |
3 | 47,203.42 | 71,403.4 | 40.81 | |
4 | 61,644.62 | 62,127.74 | 0.78 | |
5 | 56,124.72 | 60,026.46 | 6.72 | |
6 | 62,018.34 | 60,848.8 | 1.92 | |
7 | 77,225.94 | 103,615 | 29.18 |
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Nordin, N.A.M.; Shariff, S.S.R.; Supadi, S.S.; Masudin, I. Modelling the Shortest Path for Inner Warehouse Travelling Using the Floyd–Warshall Algorithm. Mathematics 2024, 12, 2698. https://doi.org/10.3390/math12172698
Nordin NAM, Shariff SSR, Supadi SS, Masudin I. Modelling the Shortest Path for Inner Warehouse Travelling Using the Floyd–Warshall Algorithm. Mathematics. 2024; 12(17):2698. https://doi.org/10.3390/math12172698
Chicago/Turabian StyleNordin, Noraimi Azlin Mohd, S. Sarifah Radiah Shariff, Siti Suzlin Supadi, and Ilyas Masudin. 2024. "Modelling the Shortest Path for Inner Warehouse Travelling Using the Floyd–Warshall Algorithm" Mathematics 12, no. 17: 2698. https://doi.org/10.3390/math12172698