A Proactive Approach to Extended Vehicle Routing Problem with Drones (EVRPD)
Abstract
:1. Introduction
2. Vehicle Routing Problem with Drones-Literature Review
- (a)
- Area coverage where sensor or camera drones should monitor (cover) an area that may take various shapes. This problem occurs in disaster management (assessment of the damage after an earthquake, flood, tornado, etc.), agriculture (observing vegetation indexes and creating digital maps of the crop area), aerial archeology, etc. [15]
- (b)
- Search operations where drones need to find a stationary or moving object. In a drone search, this problem is easily observed in wildlife monitoring and search and rescue applications. You need to define a search path for one or more drones to find an object in an unknown location.
- (c)
- Routing for a set of locations where drones must visit a specific and finite set of addresses. In many surveillance and delivery applications, drones must tour a certain set of locations starting and ending at the depot. The resulting planning problems can be modeled as generalized versions of one of the underlying routing problems, such as a TSP [16,17], multiple TSPs [18,19,20,21,22,23,24,25], or a VRP [26,27,28,29,30].
- (d)
- Data collection and charging on wireless sensor networks (WSNs) where drones must collect information from a specific or given set of locations while considering the communication schedule and limited memory capacity.
- (e)
- Allocating communication links and computing power to mobile devices with set (or redirected) drones to provide communication links with mobile devices of sufficient quality.
3. Problem Statement and Mathematical Model
3.1. Problem Statement
- Recipients (customers) were defined (the location of each recipient was known, which translated into the determination of the distance–cost grid (network) between recipients and the travel time from recipient to recipient);
- Shipments (parcels) must be delivered or picked up from the customer;
- Several shipments (parcels) could be addressed to a customer (parcels should be delivered at once; if there are too many of them, e.g., the capacity of the drone was reached, we replaced one customer with two or more (we introduced the so-called virtual customers who had the same location)), which created significant novelty in relation to the VRPD;
- Various types of shipment were considered;
- Mobile points (mobile hubs—MHs) were used for drone take-offs;
- Drones and vehicles (i.e., trucks) were used to transport shipments (trucks to deliver drones to MHs and drones to deliver shipments to customers);
- The number of MHs was limited;
- Locations for MHs were known;
- The selection of MHs was optimized for a given set of shipments which created novelty in relation to VRPD;
- The number of drones and trucks was limited;
- Each drone was characterized by:
- ○
- Load capacity (which determined the number of shipments (parcels) it could carry);
- ○
- The maximum range (which determined the time in which the parcels must be delivered);
- In general, a drone could visit several clients, which was especially useful for a network of clients, e.g., a network of pharmacies, a network of car repair shops, etc.;
- The delivery route for all drones was minimized;
- The allocation of shipments to drones and drones to vehicles was determined, which introduced novelty in relation to the VRPD;
- If it was impossible to deliver a set of shipments, the shipments preventing delivery were found;
- Situations where a specific drone dx was damaged or unavailable and could not take part in deliveries were also considered.
3.2. Mathematical Model
4. Materials and Methods—Implementation of the EVRPD Model
4.1. AMPL and Mathematical Programming
4.2. Dedicated Genetic Algorithm
4.2.1. Dedicated Representation of Individuals for the EVRPD
4.2.2. Initiation—Selection of the Initial Population
4.2.3. Evaluation Procedure
- Step 1: The traveling salesman problem (TSP) was solved for each MH starting point and each drone. The route was determined in such a way that the cost of delivery was the lowest, and the delivery time was within the set time.
- Step 2: The MH starting points were selected in such a way that the total cost of deliveries was as low as possible and that the number of allowed starting points was not exceeded. This step also ensured that constraint (14) was satisfied.
4.2.4. Constraint Handling Procedure
4.2.5. Genetic Operators
5. Computational Examples
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. AMPL Model for the EVRPD (Files Ready to Use for the Gurobi Solver)
Appendix B. Data for EVRPD
k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Vkk | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
UAk | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
k | 1 | 2 | 3 | 4 |
VCd | 10 | 10 | 10 | 10 |
WAk,o | o | |||||||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||
k | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
3 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
4 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | |
5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | |
6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | |
7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | |
8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | |
9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | |
10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
FAo,j | j | |||||||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||
o | 1 | 0 | 2 | 4 | 3 | 6 | 7 | 5 | 3 | 1 | 5 | 3 | 4 | 3 |
2 | 2 | 0 | 5 | 4 | 5 | 9 | 6 | 5 | 1 | 7 | 4 | 2 | 3 | |
3 | 4 | 5 | 0 | 4 | 7 | 3 | 5 | 4 | 5 | 2 | 2 | 7 | 6 | |
4 | 3 | 4 | 4 | 0 | 8 | 7 | 2 | 1 | 3 | 4 | 5 | 5 | 3 | |
5 | 6 | 5 | 7 | 8 | 0 | 8 | 10 | 8 | 6 | 9 | 4 | 7 | 8 | |
6 | 7 | 9 | 3 | 7 | 8 | 0 | 8 | 7 | 8 | 3 | 4 | 10 | 9 | |
7 | 5 | 6 | 5 | 2 | 10 | 8 | 0 | 2 | 5 | 5 | 7 | 7 | 4 | |
8 | 3 | 5 | 4 | 1 | 8 | 7 | 2 | 0 | 4 | 4 | 5 | 5 | 3 | |
9 | 1 | 1 | 5 | 3 | 6 | 8 | 5 | 4 | 0 | 6 | 5 | 2 | 2 | |
10 | 5 | 7 | 2 | 4 | 9 | 3 | 5 | 4 | 6 | 0 | 4 | 9 | 7 | |
11 | 3 | 4 | 2 | 5 | 4 | 4 | 7 | 5 | 5 | 4 | 0 | 7 | 6 | |
12 | 4 | 2 | 7 | 5 | 7 | 10 | 7 | 5 | 2 | 9 | 7 | 0 | 3 | |
13 | 3 | 3 | 6 | 3 | 8 | 9 | 4 | 3 | 2 | 7 | 6 | 3 | 0 |
TAo,j | o | |||||||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | ||
j | 1 | 0 | 1 | 2 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 2 | 2 | 1 |
2 | 1 | 0 | 3 | 2 | 2 | 4 | 3 | 2 | 1 | 4 | 2 | 1 | 2 | |
3 | 2 | 3 | 0 | 2 | 3 | 2 | 3 | 2 | 2 | 1 | 1 | 4 | 3 | |
4 | 1 | 2 | 2 | 0 | 4 | 4 | 1 | 0 | 2 | 2 | 3 | 2 | 1 | |
5 | 3 | 2 | 3 | 4 | 0 | 4 | 5 | 4 | 3 | 4 | 2 | 3 | 4 | |
6 | 3 | 4 | 2 | 4 | 4 | 0 | 4 | 3 | 4 | 1 | 2 | 5 | 5 | |
7 | 2 | 3 | 3 | 1 | 5 | 4 | 0 | 1 | 3 | 2 | 3 | 3 | 2 | |
8 | 1 | 2 | 2 | 0 | 4 | 3 | 1 | 0 | 2 | 2 | 2 | 3 | 2 | |
9 | 1 | 1 | 2 | 2 | 3 | 4 | 3 | 2 | 0 | 3 | 2 | 1 | 1 | |
10 | 3 | 4 | 1 | 2 | 4 | 1 | 2 | 2 | 3 | 0 | 2 | 4 | 4 | |
11 | 2 | 2 | 1 | 3 | 2 | 2 | 3 | 2 | 2 | 2 | 0 | 3 | 3 | |
12 | 2 | 1 | 4 | 2 | 3 | 5 | 3 | 3 | 1 | 4 | 3 | 0 | 1 | |
13 | 1 | 2 | 3 | 1 | 4 | 5 | 2 | 2 | 1 | 4 | 3 | 1 | 0 |
Appendix C. Obtained Routes for Experiments E01, E05, E07, E09, E12 and E16
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(A) | ||||
Symbol | Description | P/R | ||
Q1 | Is it possible to make deliveries? | P/R | ||
Q2 | What is the minimum cost of deliveries? | P/R | ||
Q3 | Is it possible to make deliveries within the given time T? | P/R | ||
Q4 | What is the minimum cost of deliveries in the given time T? | P/R | ||
Q5 | What is the minimum number of mobile points (mobile hubs—MHs) required to make deliveries? | P | ||
Q6 | What is the minimum number of drones needed to make deliveries? | P | ||
Q7 | Can deliveries be made if the drone dx is damaged? | P | ||
Q8 | Is it possible to make deliveries if it is not possible to use the mobile point mx? | P | ||
P/R—proactive/reactive | ||||
(B) | ||||
Question | Set of constraints | Problem type | Model type | Decision Variable |
Q1 | (2)–(8) and (10)–(15) | feasible | CSP | Rs |
Q2 | (1a) and (2)–(8) and (10)–(15) | optimal | BIP | Xh,d,c,j, Rs |
Q3 | (2)–(15) | feasible | CSP | Rs |
Q4 | (1a) and (2)–(15) | optimal | BIP | Xh,d,c,j, Rs |
Q5 | (2)–(8) and (10)–(15) and min MS | optimal | BIP | Zh |
Q6 | (1b) and (2)–(8) and (10)–(15) | optimal | BIP | Yh,d,c,j,s |
Q7 | (2)–(8) and (10)–(15) and VCdi = 0 | feasible | CSP | Rs, Yh,d,c,j,s |
Q8 | (2)–(8) and (10)–(15) and VKmx = 0 | feasible | CSP | Zh, Xh,d,c,j |
Symbol | Description |
---|---|
Indices and Sets | |
C | The set of customers |
S | The set of all items (shipments, parcels) |
D | The set of all drones |
H | A set of possible locations for mobile points (mobile hubs—MHs) |
c, j | Delivery point (customer) index (c, j ∈ C) |
s | Shipment index (s∈S) |
d | Drone index (d∈D) |
h | Mobile point index (h∈K) |
Parameters | |
VSs | Parcel volume (volumetric weight) s (s∈S) |
VCd | Drone’s d payload (d∈D) |
RAs,c | If parcel k should be delivered to customer o, then RAs,c = 1, otherwise RAs,c = 0 (s∈S, c∈C) |
FAc,j | Cost of travel between customers c and j (c, j ∈ C ∪ M)—average values include 2D and 3D |
TAc,j | Transfer time between customers c and j (c, j ∈ C ∪ M)—average values include 2D and 3D |
TX | Time within which the delivery should be made |
MS | Maximum number of mobile points (mobile hubs—MHs) |
ST | A large constant |
UAs | If parcel s was delivered from the mobile point (MH) to the customer, then UAs = 1; otherwise, UAs = 0 (s∈S). The parameter value determined the delivery direction. |
Decision variables | |
Xh,d,c,j | If a drone d traveled from customer c to customer j and took off from MH, h then Xh,d,c,j = 1, otherwise Xh,d,c,j = 0, (h∈H, d∈D, c,j∈C∪M). |
Yh,d,c,j,s | If a drone d traveled from customer c to customer j and took off from MH, h carrying parcel s, then Yh,d,c,j,s = 1, otherwise Yh,d,c,j,s = 0, (h∈H, d∈D, c, j∈C∪M). |
Rs | If parcel s could not be delivered, then Rs = 1 otherwise Rs = 0 (s∈S). (The introduction of this decision variable meant that the model will always be solvable.) |
Zh | If there were take-offs from mobile point h, then Zh = 1; otherwise, Zh = 0. |
Constraints | Description |
---|---|
(1a) | Delivery cost (for Q2 and Q4, it is an objective function (FC1) and is minimized) |
(1b) | Number of drones used (for Q6, it is an objective function (FC2) and is minimized) |
(2) | Determined the arrival and departure of the drone at and from the delivery point (customer) |
(3) | If no items were to be carried on the route, then a drone did not travel that route. |
(4) | If a drone did not travel along a route, no items were to be carried on that route. |
(5) | In no route segment could a drone carry more parcels than its payload. |
(6) | Items were delivered to and from mobile points, h∈H. |
(7) | Ensured a single run of the drone |
(8) | Items were picked up or delivered to customers. This constraint prevented sub-tours. |
(9) | Runs were executed within the required time. |
(10) | Each drone picked up or delivered parcels from or to a source (MH); the first component of the constraint was responsible for the delivery of parcels to the customer, while the second was responsible for the pick up from the customer. |
(11) | Ensured the drone was at a point only once (no returns). |
(12) | Ensured the drone took off and returned to the MH point. (We excluded the possibility that the drone left the parcel with a given customer and another took it to the next customer). |
(13) | Determined the value of the decision variable Zh. This variable determined mobile point h∈H from the drone take-off point (in this version of the model, we did not introduce capacity limits for trucks, so Zh = 1 determined both the MH point and the truck). |
(14) | Drone take-offs could not take place from more mobile points h∈H than the number of vehicles (trucks). |
(15) | Binarity |
E | Q | N_MH | ST | MI (MO) | SC | TR | AMPL+Gurobi | DGA | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
TX | V | T | FC | TX | T | FC | |||||||
E01 | Q1 | 1 | 1 | 10 (10) | 1 | 30 | 26 | 1211 | 1 | 56 | 22 | 2 | 48 |
E02 | Q2 | 1 | 1 | 10 (10) | 1 | 30 | 16 | 1211 | 1 | 34 | 16 | 25 | 34 |
E03 | Q3 | 1 | 1 | 10 (10) | 1 | 16 | 16 | 1221 | 1 | 42 | 16 | 4 | 35 |
E04 | Q4 | 1 | 1 | 10 (10) | 1 | 16 | 16 | 1211 | 1 | 34 | 16 | 25 | 34 |
E05 | Q1 | 1 | 1 | 10 (10) | 4 | 30 | 15 | 4841 | 2 | 55 | 18 | 1 | 51 |
E06 | Q2 | 1 | 1 | 10 (10) | 4 | 30 | 16 | 4841 | 2 | 34 | 16 | 24 | 34 |
E07 | Q3 | 1 | 1 | 10 (10) | 4 | 10 | 7 | 4841 | 15 | 52 | 7 | 3 | 52 |
E08 | Q4 | 1 | 1 | 10 (10) | 4 | 10 | 9 | 4841 | 35 | 40 | 9 | 25 | 40 |
E09 | Q1 | 3 | 2 | 10 (10) | 4 | 30 | 19 | 17,427 | 3 | 75 | 18 | 4 | 66 |
E10 | Q2 | 3 | 2 | 10 (10) | 4 | 30 | 15 | 17,427 | 3 | 33 | 15 | 25 | 33 |
E11 | Q3 | 3 | 2 | 10 (10) | 4 | 10 | 6 | 17,427 | 67 | 41 | 6 | 8 | 41 |
E12 | Q4 | 3 | 2 | 10 (10) | 4 | 10 | 7 | 17,427 | 85 | 37 | 7 | 25 | 37 |
E13 | Q1 | 3 | 3 | 10 (10) | 4 | 30 | 18 | 17,427 | 2 | 42 | 15 | 10 | 38 |
E14 | Q2 | 3 | 3 | 10 (10) | 4 | 30 | 15 | 17,427 | 7 | 33 | 15 | 25 | 33 |
E15 | Q3 | 3 | 3 | 10 (10) | 4 | 8 | 7 | 17,427 | 8 | 52 | 5 | 12 | 50 |
E16 | Q4 | 3 | 3 | 10 (10) | 4 | 8 | 5 | 17,427 | 12 | 40 | 5 | 25 | 40 |
E17 | Q5 | 1 | 1 | 10 (10) | 1 | 30 | --- | 1211 | 2 | 1 | --- | 2 | 1 |
E18 | Q6 | 1 | 1 | 10 (10) | 1 | 30 | --- | 1211 | 2 | 1 | --- | 2 | 1 |
E19 | Q5 | 1 | 1 | 10 (10) | 1 | 16 | --- | 1211 | 2 | 1 | --- | 2 | 1 |
E20 | Q6 | 1 | 1 | 10 (10) | 1 | 16 | --- | 1211 | 2 | 1 | --- | 2 | 1 |
E21 | Q5 | 1 | 1 | 10 (10) | 4 | 30 | --- | 4841 | 2 | 1 | --- | 2 | 1 |
E22 | Q6 | 1 | 1 | 10 (10) | 4 | 30 | --- | 4841 | 2 | 1 | --- | 2 | 1 |
E23 | Q5 | 1 | 1 | 10 (10) | 4 | 16 | --- | 4841 | 4 | 1 | --- | 4 | 1 |
E24 | Q6 | 1 | 1 | 10 (10) | 4 | 16 | --- | 4841 | 34 | 2 | --- | 27 | 2 |
E25 | Q5 | 3 | 2 | 10 (10) | 4 | 30 | --- | 17,427 | 3 | 1 | --- | 3 | 1 |
E26 | Q6 | 3 | 2 | 10 (10) | 4 | 30 | --- | 17,427 | 4 | 1 | --- | 4 | 1 |
E27 | Q5 | 3 | 2 | 10 (10) | 4 | 30 | --- | 17,427 | 48 | 2 | --- | 23 | 2 |
E28 | Q6 | 3 | 2 | 10 (10) | 4 | 30 | --- | 17,427 | 92 | 3 | --- | 25 | 3 |
E29 | Q5 | 3 | 3 | 10 (10) | 4 | 30 | --- | 17,427 | 3 | 1 | --- | 3 | 1 |
E30 | Q6 | 3 | 3 | 10 (10) | 4 | 30 | --- | 17,427 | 1 | 1 | --- | 1 | 1 |
E31 | Q5 | 3 | 3 | 10 (10) | 4 | 30 | --- | 17,427 | 34 | 2 | --- | 24 | 2 |
E32 | Q6 | 3 | 3 | 10 (10) | 4 | 30 | --- | 17,427 | 67 | 4 | --- | 28 | 4 |
F (Instance) | Q | MI (MO) | SC | TR | AMPL+Gurobi | AMPL+LINGO | DGA | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
V | TX | T | FC | T | FC | TX | T | FC | |||||
F1 (10.10.1.txt) | Q2 | 10 (10) | 4 | 30 | 17,427 | 15 | 3 | 33 | 34 | 33 | 15 | 25 | 33 |
F2 (10.10.1.txt) | Q4 | 10 (10) | 4 | 8 | 17,427 | 7 | 67 | 37 | 134 | 37 | 7 | 25 | 37 |
F3 (10.10.1.txt) | Q5 | 10 (10) | 4 | 8 | 17,427 | --- | 4 | 1 | 35 | 1 | --- | 8 | 1 |
F4 (10.10.1.txt) | Q6 | 10 (10) | 4 | 8 | 17,427 | --- | 54 | 1 | 145 | 1 | --- | 14 | 1 |
F5 (10.10.3.txt) | Q2 | 10 (10) | 4 | 30 | 17,427 | 14 | 6 | 29 | 41 | 29 | 14 | 24 | 29 |
F6 (10.10.3.txt) | Q4 | 10 (10) | 4 | 10 | 17,427 | 7 | 68 | 36 | 145 | 36 | 7 | 25 | 36 |
F7 (10.10.3.txt) | Q5 | 10 (10) | 4 | 10 | 17,427 | --- | 6 | 1 | 45 | 1 | --- | 10 | 1 |
F8 (10.10.3.txt) | Q6 | 10 (10) | 4 | 10 | 17,427 | --- | 3 | 1 | 189 | 1 | --- | 13 | 1 |
F9 (12.10.1.txt) | Q2 | 12 (12) | 4 | 30 | 28,395 | 16 | 145 | 32 | 456 | 32 | 16 | 34 | 32 |
F10 (12.10.1.txt) | Q4 | 12 (12) | 4 | 10 | 28,395 | 10 | 345 | 35 | 678 | 35 | 10 | 33 | 35 |
F11 (12.10.1.txt) | Q5 | 12 (12) | 4 | 10 | 28,395 | --- | 67 | 2 | 189 | 2 | --- | 23 | 2 |
F12 (12.10.1.txt) | Q6 | 12 (12) | 4 | 10 | 28,395 | --- | 345 | 2 | 867 | 2 | --- | 34 | 2 |
F13 (20.10.1.txt) | Q2 | 20 (20) | 4 | 30 | 116,424 | 18 | 567 | 67 | 956 | 67 | 18 | 61 | 67 |
F14 (20.10.1.txt) | Q4 | 20 (20) | 4 | 12 | 116,424 | 10 | 754 | 82 | 1256 | 82 | 10 | 62 | 82 |
F15 (20.10.1.txt) | Q5 | 20 (20) | 4 | 12 | 116,424 | --- | 134 | 2 | 565 | 2 | --- | 55 | 2 |
F16 (20.10.1.txt) | Q6 | 20 (20) | 4 | 12 | 116,424 | --- | 534 | 3 | 987 | 3 | --- | 57 | 3 |
F17 (50.10.1.txt) | Q2 | 50 (50) | 4 | 30 | 1,727,485 | 28 | 1800 * | 132 | 1800 * | NFSF | 26 | 78 | 128 |
F18 (50.10.1.txt) | Q4 | 50 (50) | 4 | 25 | 1,727,485 | 24 | 1800 * | 156 | 1800 * | NFSF | 24 | 76 | 141 |
F19 (50.10.1.txt) | Q5 | 50 (50) | 4 | 25 | 1,727,485 | --- | 1800 * | 2 | 1800 * | NFSF | --- | 77 | 2 |
F20 (50.10.1.txt) | Q6 | 50 (50) | 4 | 25 | 1,727,485 | --- | 1800 * | 3 | 1800 * | NFSF | --- | 79 | 3 |
F21 (100.10.1.txt) | Q2 | 100 (100) | 6 | 35 | 19,318,893 | - | 1800 * | NFSF | 1800 * | NFSF | 34 | 87 | 194 |
F22 (100.10.1.txt) | Q4 | 100 (100) | 6 | 30 | 19,318,893 | - | 1800 * | NFSF | 1800 * | NFSF | 28 | 89 | 210 |
F23 (100.10.1.txt) | Q5 | 100 (100) | 6 | 30 | 19,318,893 | --- | 1800 * | NFSF | 1800 * | NFSF | --- | 86 | 1 |
F24 (100.10.1.txt) | Q6 | 100 (100) | 6 | 30 | 19,318,893 | --- | 1800 * | NFSF | 1800 * | NFSF | --- | 85 | 2 |
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Sitek, P.; Wikarek, J.; Jagodziński, M. A Proactive Approach to Extended Vehicle Routing Problem with Drones (EVRPD). Appl. Sci. 2022, 12, 8255. https://doi.org/10.3390/app12168255
Sitek P, Wikarek J, Jagodziński M. A Proactive Approach to Extended Vehicle Routing Problem with Drones (EVRPD). Applied Sciences. 2022; 12(16):8255. https://doi.org/10.3390/app12168255
Chicago/Turabian StyleSitek, Paweł, Jarosław Wikarek, and Mieczysław Jagodziński. 2022. "A Proactive Approach to Extended Vehicle Routing Problem with Drones (EVRPD)" Applied Sciences 12, no. 16: 8255. https://doi.org/10.3390/app12168255
APA StyleSitek, P., Wikarek, J., & Jagodziński, M. (2022). A Proactive Approach to Extended Vehicle Routing Problem with Drones (EVRPD). Applied Sciences, 12(16), 8255. https://doi.org/10.3390/app12168255