Heuristics for Two Depot Heterogeneous Unmanned Vehicle Path Planning to Minimize Maximum Travel Cost
Abstract
:1. Introduction
2. Problem Statement
a set of UVs | |
a set of depots | |
a set of targets | |
a set of vertices for | |
a set of edges that connect all vertices in | |
a set of travel costs of all edges in | |
the subset of the edges of that have one end in S and the other end in | |
the subset of the edges such that and |
the decision variable that represents whether edge is used for the tour of | |
the decision variable that represents partition of targets in T | |
t | the maximum travel cost |
the final tour cost of within a given partition of the targets | |
the final tour of within a given partition of the targets | |
a set of edges in the forest corresponding to | |
a set of connected components in | |
the dual variable of a set C in , which keeps track of |
the dual variable of a set C where , which keeps track of | |
the dual variable of a set C in , which keeps track of | |
the activeness of set . only if C is active. | |
the subsets of set in |
3. TDHTSP with Symmetric Travel Costs
3.1. Problem Formulation
3.2. A Heuristic for the TDHTSP
Algorithm 1 A heuristic for min-max TDHTSP | |
1: | ; |
2: | Determine a heterogeneous spanning forest using the proposed primal-dual heuristic. |
3: | fordo |
4: | Let the connected targets reachable from the depot be a partition and label it as . |
5: | end for |
6: | For , derive and for within (using an existing routing algorithm). |
7: | |
8: | ifthen |
9: | while do |
10: | |
11: | Redetermine a target assignment using the proposed primal-dual heuristic and obtain the partitions . |
12: | for do |
13: | Derive and within . |
14: | end for |
15: | if is infeasible then |
16: | break, {comment: for some } |
17: | else |
18: | |
19: | if then |
20: | |
21: | for do |
22: | |
23: | end for |
24: | end if |
25: | end if |
26: | end while |
27: | else |
28: | while do |
29: | |
30: | Redetermine a target assignment using the proposed primal-dual heuristic and obtain the partitions . |
31: | for do |
32: | Derive and within . |
33: | end for |
34: | |
35: | if then |
36: | |
37: | for do |
38: | |
39: | end for |
40: | end if |
41: | end while |
42: | end if |
43: | return |
Algorithm 2 Primal-dual heuristic for finding an HSF | |
1: | Initialization |
2: | , for |
3: | All vertices are unmarked. |
4: | All dual variables are set to zero. |
5: | , , for |
6: | , for |
7: | Main loop |
8: | while there exists any active component in do |
9: | for do |
10: | Find an edge with where that minimizes . |
11: | end for |
12: | Let . |
13: | Find that minimizes |
14: | |
15: | for do |
16: | for do |
17: | |
18: | |
19: | if then |
20: | |
21: | end if |
22: | end for |
23: | end for |
24: | if for or 2 then |
25: | |
26: | |
27: | |
28: | if then |
29: | |
30: | end if |
31: | if then |
32: | |
33: | if then |
34: | |
35: | end if |
36: | else |
37: | |
38: | end if |
39: | else |
40: | |
41: | Mark all of the vertices of with the label . |
42: | end if |
43: | end while |
44: | Pruning |
45: | |
46: | |
47: | Let be the minimum spanning tree of . |
48: | Let be the sum of edge costs present in |
49: | while is not empty do |
50: | Find the shortest edge that connects a vertex in and a vertex in for each k. |
51: | if then |
52: | Add to , remove the corresponding vertex from , and add it to . |
53: | else |
54: | Add to , remove the corresponding vertex from and add it to . |
55: | end if |
56: | end while |
4. TDHTSP with Asymmetric Travel Costs
4.1. Problem Formulation
4.2. A Heuristic for the TDHATSP
Algorithm 3 Primal-dual heuristic for finding an HDSF | |
1: | Initialization |
2: | , for |
3: | All the vertices are unmarked. |
4: | All the dual variables are set to zero. |
5: | , , for |
6: | , for |
7: | Main loop |
8: | while there exists any active component in do |
9: | for k = 1, 2 do |
10: | Find an edge with , where that minimizes |
11: | end for |
12: | Let the corresponding be , while satisfies and are active. |
13: | |
14: | Increase the dual variables of by . |
15: | if forms a new strongly connected component and the component is not reachable from then |
16: | Let the strongly connected component be an active component. |
17: | else if makes any vertex reachable from then |
18: | Let the depot and all vertices that are reachable from the depot be an inactive component. |
19: | if then |
20: | Deactivate all subsets of this component in . |
21: | else |
22: | Mark all vertices in the supersets of this component in . Deactivate it if the corresponding component consists of all marked vertices. |
23: | end if |
24: | else |
25: | Deactivate . |
26: | end if |
27: | if there exists no that can be chosen that satisfies the given conditions and there exists any inactive set without an incoming edge that is not connected to the depot then |
28: | Pick an inactive component for each k that consists of marked vertices that have incoming or outgoing edges. Combine those connected components until the new component does not have any incoming edges. |
29: | end if |
30: | end while |
31: | Pruning |
32: | |
33: | |
34: | Let be the minimum directed spanning tree of . |
35: | Let be the sum of the edge costs present in . |
36: | while is not empty do |
37: | Find the shortest edge that makes a vertex in reachable from the vertices in for each k. |
38: | if then |
39: | Add to , remove the corresponding vertex from , and add it to . |
40: | else |
41: | Add to , remove the corresponding vertex from , and add it to . |
42: | end if |
43: | end while |
5. Implementation
Discussion of Implementation Results
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Depots | Heterogeneity(s) | Heterogeneity(f) | Objective | Methodology | Cost | |
---|---|---|---|---|---|---|
This Paper | Two | Yes | No | Min-Max | Primal-Dual | Symmetric and Asymmetric |
[18] | two | Yes | No | min-sum | primal-dual | symmetric |
[19] | multiple | Yes | No | min-max | gossip algorithm | symmetric |
[20] | single | No | Yes | min-max | 5-approx algo. | symmetric |
[21] | multiple | No | No | min-max | ant colony opt. | symmetric |
[22] | multiple | No | No | min-max | LP based heuristic | symmetric |
[23] | multiple | Yes | Yes | min-sum | branch and cut | asymmetric |
[24] | multiple | Yes | No | min-sum | transformation | asymmetric |
[25] | multiple | No | Yes | min-sum | hybrid GA | symmetric |
Average | Worst | |||
---|---|---|---|---|
No. of Jobs | LP Rounding | Primal-Dual | LP Rounding | Primal-Dual |
20 | 2.11 | 0.81 | 2.91 | 1.61 |
30 | 31.13 | 1.79 | 41.23 | 2.35 |
40 | 429.61 | 3.30 | 867.66 | 5.26 |
60 | 5623.48 | 8.62 | 8470.95 | 11.72 |
80 | 39,891.46 | 19.65 | 63,239.30 | 31.02 |
Average | Worst | |||
---|---|---|---|---|
No. of Jobs | LP Rounding | Primal-Dual | LP Rounding | Primal-Dual |
20 | 1.59 | 1.23 | 2.38 | 3.28 |
30 | 17.70 | 4.80 | 22.11 | 12.31 |
40 | 237.06 | 4.86 | 345.81 | 12.79 |
60 | 3634.76 | 16.00 | 10006.30 | 46.08 |
80 | 30,329.19 | 36.21 | 41,330.20 | 99.87 |
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Bae, J.; Chung, W. Heuristics for Two Depot Heterogeneous Unmanned Vehicle Path Planning to Minimize Maximum Travel Cost. Sensors 2019, 19, 2461. https://doi.org/10.3390/s19112461
Bae J, Chung W. Heuristics for Two Depot Heterogeneous Unmanned Vehicle Path Planning to Minimize Maximum Travel Cost. Sensors. 2019; 19(11):2461. https://doi.org/10.3390/s19112461
Chicago/Turabian StyleBae, Jungyun, and Woojin Chung. 2019. "Heuristics for Two Depot Heterogeneous Unmanned Vehicle Path Planning to Minimize Maximum Travel Cost" Sensors 19, no. 11: 2461. https://doi.org/10.3390/s19112461
APA StyleBae, J., & Chung, W. (2019). Heuristics for Two Depot Heterogeneous Unmanned Vehicle Path Planning to Minimize Maximum Travel Cost. Sensors, 19(11), 2461. https://doi.org/10.3390/s19112461