Mixed Integer Linear Programming Models to Solve a Real-Life Vehicle Routing Problem with Pickup and Delivery
Abstract
:1. Introduction
2. Related Literature
- Delivery first and pickup second: vehicles perform a pickup operation after the delivery process. This category is called Vehicle Routing Problems with Backhauls (VRPB).
- Mixed pickup and delivery: vehicles deliver or pickup in any sequence along their routes. This category is called Mixed Vehicle Routing Problem with Backhauls (MVRPB).
- Simultaneous Pickup and delivery: vehicles simultaneously perform the delivery and pickup. This category is called Vehicle Routing Problems with Simultaneous Pickup and Delivery (VRPSPD).
3. Problem Ingredients and Approach of the Proposed MILP Models
- Time window:Includes the service time at each customer, loading time for each vehicle, travel time between each couple of customers, time necessary to park the vehicle, and the workday duration.
- Combined capacity; volume and weight:The majority of VRP formulations consider the physical capacity of vehicles, which is generally expressed in terms of weight. In our problem, goods are delivered using boxes. Therefore, beside the capacity in terms of weight, we have to make sure that each vehicle has the capacity in terms of number of boxes to be fitted into the vehicle. Thus, we defined a new volume capacity that includes both loaded weight and box numbers. We denote that we have considered the volume occupation in each vehicle to be discrete. Motivated by this assumption, which complicates the use of vehicle space, we assume that the combined capacity has not been addressed in the literature.
- Customer balance (stock) in terms of boxes:We suppose, initially, that each customer has boxes ready for pickup. These boxes are considered by our models before performing any delivery.
- Optimization choice:Three different assumptions are considered. The first one guarantees that the pickup and delivery are performed simultaneously, without leaving any empty boxes for any customer. The second assumption is more flexible. It provides two choices of pickup and delivery operation, separately and/or simultaneously. The third one ensures that the pickup and delivery operation are performed simultaneously. However, it provides a flexibility to drop and leave some boxes, which will be consequently penalized.
- Heterogeneity of the fleetMultiple vehicle types exist, having different sizes.
4. Mathematical Formulations
4.1. Simultaneous Pickup and Delivery Vehicle Routing Problem: MILP 1
- n: number of customers ().
- m: number of vehicles ().
- : customer i demand in .
- : volume demand of customer i in terms of number of boxes. This parameter is deduced from the following relation: .
- : average weight of a full loaded box dedicated for a customer i.
- : initial balance of boxes that exists at the store of a customer i.
- : weight of an empty box.
- : travel time between two customers i and j.
- : unitary time service expressed in minute per box.
- : fixed service time.
- : total service time at a customer i.
- : time necessary to load each vehicle at the depot.
- : unitary time needed to prepare the boxes expressed in terms of minute per box.
- : maximum duration of a tour.
- : weight capacity of a vehicle k expressed in .
- : volume capacity of a vehicle k expressed in .
- : travel cost from a customer i to another j.
- : weight of the vehicle load before leaving a customer j.
- : weight of goods at the depot for each vehicle k.
- : number of boxes before leaving a customer j.
- : number of boxes loaded at the depot for a vehicle k.
- : variable used for sub-tours elimination.
4.2. Flexible VRPPD: MILP 2
- : vehicle availability time at the depot. .
- : flow time at customer j.
- formulation is as follows:
4.3. Flexible VRPSPD: MILP 3
- : boxes left at customer j.
- : penalizing cost associated to the left boxes.
5. Computational Study
- The types of demands, i.e., the quantities of required goods (low/medium/high).
- The number of boxes for customers (delivery and initial balance) and their dispersion.
- The number of customers to be served.
- The geographic dispersion of customer locations (distance between customers).
- The number of vehicles.
- The vehicle capacities: we considered, small (S) with 1.4 tons, and big (B) with 3 tons.
6. Discussion of Limitations
- Due to a confidentiality issue, CHAHIA has fed our research work with limited data. Therefore, bigger data is required for a more relevant benchmark.
- CHAHIA has refused to reveal all serve stores, customers, and quantities, due to confidentiality.
- CHAHIA was looking for a prototype that could be enhanced in the future by its engineers.
- Due to lack of equipment, we were not able to perform any experiments with more than 23 customers for and , and 6 customers for .
- Exact methods considered in this work are incapable of solving large instances. Therefore, integrating heuristics is highly recommended.
7. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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n | m | Type | Instances | MILP1 | MILP2 | MILP3 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
OF (TND) | CPU (Sec) | Nodes | OF (TND) | CPU (Sec) | Nodes | OF (TND) | CPU (Sec) | Nodes | ||||
5 | 1 | S | Inst5.1.001 | 1.780 | 0.03 | 49 | 1.200 | 0.48 | 589 | 1.780 | 0.03 | 52 |
- | B | Inst5.1.002 | 1.780 | 0.08 | 55 | 1.200 | 4.32 | 5701 | 1.780 | 0.09 | 55 | |
2 | S | Inst5.1.003 | 1.780 | 0.1 | 128 | 1.200 | 39.17 | 16,901 | 1.780 | 0.1 | 133 | |
- | B | Inst5.1.004 | 2.660 | 0.3 | 528 | 1.200 | 53.83 | 26,082 | 2.660 | 0.3 | 687 | |
- | B | Inst5.1.005 | 1.780 | 0.1 | 127 | 1.200 | 67.72 | 19,848 | 1.780 | 0.1 | 145 | |
5 | 1 | S | Inst5.1.006 | 3.980 | 0.09 | 38 | 2.770 | 2729.4 | 5,099,447 | 3.980 | 0.09 | 37 |
- | B | Inst5.1.007 | 4.030 | 0.06 | 47 | 2.770 | 251.3 | 427,744 | 3.980 | 0.08 | 24 | |
2 | S | Inst5.1.008 | 3.980 | 0.05 | 32 | 2.770 | 309.8 | 410,657 | 3.980 | 0.09 | 32 | |
- | B | Inst5.1.009 | 4.700 | 0.2 | 260 | 2.770 | 3899.9 | 2,764,520 | 3.980 | 0.17 | 75 | |
6 | 1 | S | Inst6.1.001 | 2.200 | 0.1 | 155 | 1.510 | 267 | 232,811 | 2.200 | 0.2 | 173 |
- | B | Inst6.1.002 | 2.200 | 0.1 | 139 | 1.510 | 0.8 | 405 | 2.200 | 0.1 | 158 | |
2 | S | Inst6.2.003 | 2.600 | 1 | 1630 | 1.510 | 42.9 | 13,972 | 2.200 | 0.5 | 668 | |
- | B | Inst6.2.004 | 3.010 | 0.9 | 1263 | 1.510 | 499.4 | 230,267 | 2.600 | 1 | 1515 | |
- | B | Inst6.2.005 | 3.010 | 1.3 | 2401 | 1.510 | 255.6 | 101,203 | 2.200 | 0.3 | 366 |
n | m | Type | Instances | MILP1 | MILP2 | MILP3 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
OF (TND) | CPU (Sec) | Nodes | OF (TND) | CPU (Sec) | Nodes | OF (TND) | CPU (Sec) | Nodes | ||||
8 | 1 | S | Inst8.1.001 | 3.810 | 9.4 | 13,954 | - | - | - | 3.818 | 9.2 | 13,726 |
- | B | Inst8.1.002 | 3.810 | 1.3 | 1275 | - | - | - | 3.818 | 7 | 12.425 | |
2 | S | Inst8.2.003 | 5.770 | 81.6 | 92,173 | - | - | - | 5.580 | 145.7 | 187,467 | |
- | B | Inst8.2.004 | 10.750 | 120.7 | 137,284 | - | - | - | 7.540 | 515.7 | 506,652 | |
3 | S | Inst8.3.005 | 5.580 | 220.8 | 299,822 | - | - | - | 3.810 | 47.5 | 60,144 | |
- | B | Inst8.3.006 | 5.580 | 138.5 | 12,515.5 | - | - | - | 5.580 | 349 | 410,268 | |
10 | 2 | S | Inst10.2.001 | 2.870 | 17 | 1810 | - | - | - | 2.750 | 25.32 | 26,275 |
- | B | Inst10.2.002 | 3.170 | 394.5 | 418,597 | - | - | - | 3.160 | 2030.4 | 2,181,194 | |
3 | S | Inst10.3.003 | 3.170 | 6176.4 | 3,682,568 | - | - | - | 2.750 | 242.4 | 196,425 | |
- | B | Inst10.3.004 | 3.170 | 1619.2 | 1,134,364 | - | - | - | 3.160 | 5337.8 | 3,738,052 | |
12 | 2 | S | Inst12.2.001 | 3.270 | 3464 | 2,127,116 | - | - | - | 2.860 | 57.2 | 32,994 |
- | B | Inst12.2.002 | 4.020 | 9002.5 | 7,765,020 | - | - | - | 2.860 | 45 | 37,430 | |
3 | S | Inst12.3.003 | 3.270 | 5327.2 | 2,369,060 | - | - | - | 2.860 | 244.9 | 92,847 | |
- | B | Inst12.3.004 | 3.730 | 39,770.9 | 24,454,247 | - | - | - | 2.860 | 101.25 | 51,634 |
n | m | Type | Instances | MILP1 | MILP2 | MILP3 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
OF (TND) | CPU (Sec) | Nodes | OF (TND) | CPU (Sec) | Nodes | OF (TND) | CPU (Sec) | Nodes | ||||
15 | 2 | S | Inst5.2.001 | 5.380 | 24,453.7 | 12,171,800 | - | - | - | 5.060 | 1110.6 | 654,979 |
- | B | Inst5.2.001 | 5.5510 | 29,199.2 | 16,852,192 | - | - | - | 5.060 | 437 | 2,448,044 | |
3 | S | Inst5.3.003 | 5.060 | 2714.4 | 1,311,100 | - | - | - | 5.060 | 2548.8 | 835,035 | |
18 | 2 | S | Inst18.2.001 | 3.630 | 4959.2 | 1,834,815 | - | - | - | - | - | - |
- | B | Inst18.2.002 | - | - | - | - | - | - | 3.630 | 4475.7 | 1,830,280 | |
3 | S | Inst18.3.003 | 3.630 | 9512.2 | 3,381,239 | - | - | - | - | - | - | |
- | B | Inst18.3.004 | - | - | - | - | - | - | 3.630 | 6081.3 | 1,780,109 | |
20 | 2 | S | Inst20.2.001 | 3.580 | 916.1 | 254,457 | - | - | - | - | - | - |
- | B | Inst20.2.002 | - | - | - | - | - | - | - | - | - | |
3 | S | Inst20.3.003 | 3.580 | 4472.9 | 893,970 | - | - | - | 3.580 | 4208.4 | 1,128,435 | |
- | B | Inst20.3.004 | - | - | - | - | - | - | - | - | - | |
23 | 2 | S | Inst23.2.001 | 4.290 | 4396.2 | 851,800 | - | - | - | - | - | - |
- | B | Inst23.2.002 | - | - | - | - | - | - | 4.290 | 26,413.7 | 3,834,209 | |
3 | S | Inst23.3.003 | - | - | - | - | - | - | - | - | - | |
- | B | Inst23.3.004 | - | - | - | - | - | - | - | - | - |
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Louati, A.; Lahyani, R.; Aldaej, A.; Mellouli, R.; Nusir, M. Mixed Integer Linear Programming Models to Solve a Real-Life Vehicle Routing Problem with Pickup and Delivery. Appl. Sci. 2021, 11, 9551. https://doi.org/10.3390/app11209551
Louati A, Lahyani R, Aldaej A, Mellouli R, Nusir M. Mixed Integer Linear Programming Models to Solve a Real-Life Vehicle Routing Problem with Pickup and Delivery. Applied Sciences. 2021; 11(20):9551. https://doi.org/10.3390/app11209551
Chicago/Turabian StyleLouati, Ali, Rahma Lahyani, Abdulaziz Aldaej, Racem Mellouli, and Muneer Nusir. 2021. "Mixed Integer Linear Programming Models to Solve a Real-Life Vehicle Routing Problem with Pickup and Delivery" Applied Sciences 11, no. 20: 9551. https://doi.org/10.3390/app11209551
APA StyleLouati, A., Lahyani, R., Aldaej, A., Mellouli, R., & Nusir, M. (2021). Mixed Integer Linear Programming Models to Solve a Real-Life Vehicle Routing Problem with Pickup and Delivery. Applied Sciences, 11(20), 9551. https://doi.org/10.3390/app11209551