ICS 2011
12th INFORMS Computing Society Conference
Computing Society
c 2011 INFORMS | isbn 978-0-9843378-1-1
doi 10.1287/ics.2011.0003
Fast Local Search and Insertion Heuristics for
Coordinated UGV Path Planning
Rich Kenefic
Raytheon, Fort Wayne, Indiana 46808, richard j kenefic@raytheon.com
Abstract
This paper presents an ant-colony approach for solving the vehicle routing problem
with time windows (VRPTW). The fast local search and insertion (FLS&I) methods
used are based on well known heuristics, were implemented in C as Matlab mexfunctions, and were tested against the well known Solomon benchmark for the VRPTW. It
is shown that solutions close to the best known results for the Solomon benchmark can
be reliably obtained within the re-planning horizon of a typical military UGV application. The FLS&I algorithm and similar methods significantly improve upon agent
based methods recently considered in the literature. It is also shown that, against the
type C2 Solomon instances, the FLS&I heuristics used here outperform some recent
methods that require much more processing time. The algorithm presented here has
a feature that determines when an acceptable solution has been found to facilitate
an early exit and save computer resources. With this feature, the average CPU time
required to find the best known solution for the clustered Solomon instances is under
one minute, well within the re-planning horizon.
Keywords vehicle routing; heuristics; path planning
Introduction
The importance of autonomous vehicles in future military operations is generally recognized
(Albus et al. [2], Deyst et al. [9], McLaughlin et al. [13]). These vehicles include unmanned
air vehicles (UAVs), unmanned ground vehicles (UGVs), and unmanned underwater vehicles
(UUVs) with varying levels of autonomy and mission complexity. Both UAVs and UGVs
have prominent roles in many planned military systems, and the control of multiple UAVs
has been proposed as an AFOSR grand challenge (McLaughlin et al. [13, p. 14]). Recent
work in this area has included the turn rate constraints of a UAV (Rathinam et al. [14]),
but treats the problem as a turn rate limited traveling salesman problem (kTSP) rather
than a vehicle routing problem (VRP) by omitting the UAV depot. Kinematic constraints
for UGVs are typically left out of vehicle routing problems, and many of the VRPs that
arise for UGVs in typical military applications can be considered variants of standard VRPs.
A comprehensive survey of UGV systems and algorithms has recently been published in
Anisi and Thunberg [3].
The VRPs of interest to military systems architects are known to be NP-hard, and in
the absence of kinematic constraints, there are many heuristic methods available for them.
Much research has focused on obtaining optimal solutions to standard instances, and the
impression left in the minds of many military system architects is that these methods are too
time consuming for use in a system that requires replanning on a relatively short time scale.
Some have suggested that artificial intelligence methods may be useful for solving problems
151
152
Kenefic: FLS&I Heuristics for Coordinated UGV Path Planning
c 2011 INFORMS
12th INFORMS Computing Society Conference, of this type,1 but the results reported for these methods on standard VRP instances have
been disappointing.
The purpose of this paper is two-fold: first, to demonstrate that near-optimal solutions
for important cases of the VRP relevant to mission planning can be computed within the
replanning horizon for a typical military application. The methods used here are based on
well known heuristics and were implemented in Matlab. Second, to show that an efficient
use of the cross-exchange and minimum delay heuristics can find near optimal solutions
more reliably and quickly than other methods recently employed on the type two Solomon
instances of the vehicle routing problem with time windows (VRPTW). In addition, the
algorithm employed here has a self-terminating feature that ends the search when it has
been determined that a good solution has been found or when the allotted computer time
has been exhausted. This feature saves computer resources on most of the clustered Solomon
instances, terminating the search long before the maximum allotted CPU time has been
reached.
Reference Ahmadzadeh et al. [1] was motivated by similar concerns, and used the hybrid
projected gradient evolutionary search (HPGES) algorithm for the VRPTW for UAV path
planning, but treated a limited set of standard instances and did not perform a Monte-Carlo
study of the algorithm’s performance. Reference Braysy et al. [7] makes use of heuristics similar to those used here for local search and insertion, and presents comparative performance
results against a comprehensive set of test cases, including the Solomon benchmark. This
reference also demonstrates that results close to the best known solutions can be obtained
within the replanning horizon, but doesn’t consider methods for recognizing when near
optimal solutions have been found, and doesn’t present detailed Monte-Carlo performance
results. In this study it is shown that the ant colony meta-heuristic, combined with a slight
variation of the cross-exchange for local search and minimum delay for insertion, quickly
and reliably finds good solutions for all of the Solomon instances. The method presented
here and the approach presented in Braysy et al. [7] both significantly outperform recently
proposed agent based methods that have been applied against the Solomon benchmark.
UGV Path Planning
Large UGVs would be expected to efficiently perform supply missions at the battalion level
over areas as large as 100 km by 100 km, with planning every 1–5 hours and replanning as
often as every 5 minutes (Albus et al. [2]) as shown in Figure 1.
Assuming that planning makes use of all information available at the highest level to
determine the tour order for every vehicle, the planning required for such missions resembles
the well studied vehicle routing problem with time windows (VRPTW). In the most general
case, where positions, demands, time windows, and travel times are not known apriori, or
when the vehicle on a route may become disabled, dynamic versions of VRPTW should be
considered.
In the VRPTW a single depot has a homogeneous fleet of vehicles, each of capacity C,
required to service a set of customers, each with demand di and with available times [ei , li ] in
such a way that the vehicle capacity is not exceeded and the vehicle arrives at every customer
on its route before time li . If a vehicle arrives before the customer opens at ei , it must wait.
When the customer opens, a service time si is required before the vehicle can leave for the
next customer. The vehicle must return to the depot before it closes. The objective is to first
minimize the number of vehicles required and then to minimize the total distance traveled.
This problem and its many variants is known to be NP-hard and many researchers have
investigated the performance of various meta-heuristic or evolutionary methods to obtain
1
“Research is also needed to combine artificial intelligence methods with operations research tools to overcome inefficiencies in solving some mission critical air-force problems (e.g., scheduling in a distributed
dynamic environment” (McLaughlin et al. [13], p. 33).
Kenefic: FLS&I Heuristics for Coordinated UGV Path Planning
c 2011 INFORMS
12th INFORMS Computing Society Conference, 153
Figure 1. 4D-RCS reference architecture for a typical military system (Albus et al. [2]).
24 hr plans
replan every 2 hr
Battalion HQ
Battalion
Artillary
Armor
Logistics
5 hr plans
replan every 25 mins
Platoon
IndirectFire
DirectFire
AntiAir
1 hr plans
replan every 5 mins
Section
UAV C2
Manned C2
UGS C2
10 mins plans
replan every 1 min
UARV
UGV Scout
Company
Vehicle
Subsystem
UAV
Communications
RSTA
Primitive Gaze
Servo Pan
Tilt
Iris
Mobility
Weapons
Gaze
Select
Focus
Sensors and actuators
1 min plans
replan every 5 s
Pan
Tilt
5 s plans
replan every 500 ms
Driver
Speed
500 ms plans
replan every
50 ms
Heading 50 ms plans
output every
5 ms
Note. Planning for the Battalion is assumed to take place at the support company level using information
available at the vehicle level.
solutions. One of the most popular sets of test cases was proposed by Solomon [15], and
is widely used in the literature as a basis for comparison. Much of the research emphasis
has been on finding the best solutions for each of the Solomon instances, and an enormous
amount of computing time has been spent attempting to find new best solutions.
A recent study of VRPTW (Bent and Van Hentenryck [4]) has resulted in several new
best solutions to the VRPTW, but the processing times used were 1,800 and 7,200 CPU
seconds, far beyond the replanning horizon for a typical military system. Results like these
have prompted some researchers to dismiss the use of evolutionary methods to obtain results
for real time systems.2 One objective of this paper is to demonstrate the near optimal
performance of an evolutionary approach to solving the VRPTW when the planning horizon
is limited to 5 minutes.
Vehicle Routing with Time Windows
The vehicle routing problem with time windows (VRPTW) has been well studied for over
20 years, and several heuristics have been discovered that lead to good solutions. Recent
attempts to deal with dynamic versions of VRPTW, where all of the information on the
customer locations, demands, time windows, and travel times isn’t known initially, have used
multiagent coalitions (Boudali et al. [5, 6], Gorodetski et al. [11], Zargayouna [17]), but the
performance on the Solomon benchmark when global information is available has been poor
compared with known heuristics.3 It is reasonable to expect heuristics of this type to yield
2
“Searching high dimensional spaces can be accomplished using evolutionary algorithms . . . . However, these
methods are typically too slow for real time use at levels where plans must be recomputed faster than once
every few minutes” (Albus et al. [2, p. 56]).
3
For example, Boudali et al. [6] considered the type 1 Solomon instances for 50 customers, and did not
obtain the best known solution for any of the C1 instances.
Kenefic: FLS&I Heuristics for Coordinated UGV Path Planning
c 2011 INFORMS
12th INFORMS Computing Society Conference, 154
near optimal performance when global information is available, especially when, as shown
here, the time to compute a good solution to the VRPTW is on the order of the travel time
between vertices. For the case of incomplete initial information, the “local approaches” to
solving dynamic VRPTW (Chen and Xu [8]) have been effective, and will yield near optimal
solutions when global information is available.
The algorithm used here is based on the multiple ant colony system, MACS-VRPTW
(Gambardella et al. [10]). Recently it has been shown that the ant colony meta-heuristic is
closely related to other recent model-based optimization methods like stochastic gradient
ascent and cross-entropy (Zlochin et al. [18]). Methods like simulated annealing and genetic
algorithms are characterized as instance-based optimization methods in Zlochin et al. [18].
The MACS-VRPTW used here makes use of a modification to Taillard’s cross exchange
(Taillard et al. [16]) that speeds up the computation by limiting the length of the chains to
examine for exchange and by finding the best disjoint set of exchanges to make at each step
of a local search, as shown in Figure 2. At every step of the local search the best exchange
within a route or between two routes is saved to a matrix and a greedy heuristic is used
to perform all possible exchanges. The length of the chains to examine is incremented at
every step and reinitialized if the number of vehicles is reduced by MACS-VEI or MACSTIME. When the maximum length is reached the chain length gets no longer. The algorithm
terminates and returns the best solution found when the maximum time has elapsed or
when the best solution has been rediscovered at least twice. If the elapsed time is less than
the maximum allowed before the start of any generation, then that generation is allowed to
run to completion. The time limit used here was 200 seconds, and since partial generations
are not allowed, the actual elapsed time can exceed this limit.
In addition to the cross exchange, which is effective in reducing the total length of the
routing plan, the insertion heuristic uses minimal delay in order to reduce the number of
vehicles (Homberger and Gehring [12]). The minimal delay heuristic was employed at the
insertion step of the MACS-VRPTW and operates only on the unrouted customers, as
shown in Figure 3. The heuristic first searches for feasible insertions of customers in routes
Figure 2. Modified cross-exchange.
K+1
I–1
• Given a route plan (ant)
• Run nested loops on I,J,K,L
— Save the largest distance saved by an exchange of
customers between two routes
— Save the largest distance saved by a 2-opt within a
route
— Limit the chain lengths searched
• Collect the results in a matrix
• Loop until no more feasible cross exchanges or 2-opts found
— Cover the row and columns for both routes with the
largest savings
— Continue until all rows and columns covered or no more
feasible found
Route 1
Route 2
Route 3
L
I
K
J –1
L+1
I –1
K +1
Route 4
Route 1
Route 2
Route 3
Route 4
Not feasible
J
J
L
I
K
Dist, I, J, K, L
J –1
L+1
Notes. The number of customers between I and K or J and L (chain length) is limited to speed up the
search. New steps shown in blue text.
Kenefic: FLS&I Heuristics for Coordinated UGV Path Planning
c 2011 INFORMS
12th INFORMS Computing Society Conference, 155
Figure 3. Modified minimum delay heuristic.
• Given a route plan (ant) and a set of unrouted
vertices
Insertion Route 1
Route 2
Route 3 Route 4
• Calculate the earliest departure for every vertex
vrtx a
in every route
— Earliest departure from vertex i,
vrtx b
i = max(i – + i – >i , ei ) + si
vrtx c
• Insert an unrouted vertex k between vertex i and
i+ in a route when the time shift in earliest
departure from i+ is minimized
— PTk (i , i +)
Route 1
Route 2
Route 3 Route 4
=[max(max(i + i >k , ei ) + sk + tk >i + , ei +) + si + ] – i + Exchange
vrtx a
• Exchange an unrouted vertex k with a vertex i in
a route when the time shift in the earliest
vrtx b
departure from i+ is minimized.
vrtx c
• Loop until no more unrouted or no more feasible
— Pick best insertion, cover row and column in both
matrices until no more insertions possible
Not feasible
(PT, i )
— Pick best exchange, cover row and column until no
more exchanges possible
Route 5
Route 5
Note. New steps shown in blue text.
and saves the insertion that results in the least delay to a matrix. If a customer cannot be
inserted in a route the heuristic looks for an exchange that will reduce the delay. Insertions
always have precedence over exchanges, and if an exchange was performed, the heuristic
will again look for insertions and exchanges until an iteration limit is reached. A tabu list is
maintained for exchanges within routes to avoid cycling. The MACS approach falls within
the hybrid methods discussed by Bent and Van Hentenryck [4].
Results for the Solomon Instances
The Solomon instances are divided into two types and three classes within each type with
100 customers and no more than 25 vehicles at a single depot. In type one the scheduling
times for the customers are tight and typically there will be few customers per vehicle and
many vehicles required to satisfy the schedule. In type two the time windows are loose and
vehicle capacity is large, so many customers can be assigned to each vehicle and few vehicles
are needed to satisfy the schedule. In both types the geographical distribution of customers
can be clustered, random, or mixed. The six categories are C1, R1, RC1, C2, R2, and RC2
where C1 is type one with clustered customers, etc.
In the figures presented below, 16 Monte-Carlo trials were performed for each instance
within each category. The fast local search and insertion (FLS&I) results for each instance
within a category are presented as a scatter plot (vehicles on the abscissa, distance on the
ordinate) with a modified range-finder box plot, all in blue. The best solution known is
presented with a magenta triangle, and the solutions found by Bent and Van Hentenryck
(B&V-H) [4] are presented in cyan. The B&V-H results place a cross at the mean number
of vehicles and mean distance with dashed lines from the minimum to maximum found
for each (distance or vehicles), all in cyan. These results were obtained from Bent and
Van Hentenryck [4] and through personal communication with the authors.
Type One Instances
Type one results for FLS&I are shown in Figures 4, 5, 6, and 7 for the clustered, random,
and mixed classes respectively. Note that, overall the B&V-H results are better, but require
1,800 seconds for each of the five trials in their study. For the C1 instances FLS&I finds
the best known solutions on every trial. A box plot of the CPU time required for the C1
instances is shown in Figure 5. Note that most trials require less than 50 seconds to recognize
that the best known solution has been found. Similar plots for R1 and RC1 are not included
Kenefic: FLS&I Heuristics for Coordinated UGV Path Planning
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Figure 4. Results for the type one clustered instances.
meanCPU = 49.01 sec.
C101
C102
C103
829.0
829.5
829.5
829.0
829.0
828.5
828.5
828.0
9
828.0
10
11
828.5
828.0
827.5
9
10
11
C105
C104
825.5
825.0
824.5
824.0
10
829.5
829.5
829.0
829.0
828.5
828.5
11
10
11
9
C108
829.5
829.5
829.0
829.0
829.0
828.5
828.5
828.5
828.0
10
11
10
11
C109
829.5
9
11
828.0
9
C107
828.0
10
C106
828.0
9
9
828.0
9
10
11
9
10
11
Notes. Results (Vehicles on the abscissa, distance on the ordinate) show that fast local search and insertion
(FLS&I) finds the optimal solution in every trial for every instance. Average time required over all trials and
instances is under 50 seconds (∆ = best known, + = B&V-H (Bent and Van Hentenryck [4]), -o- = FLS&I).
Figure 5. Box plots of CPU time (in seconds) required by FLS&I for the C1 instances.
C101
C102
C103
C104
C105
C106
C107
C108
C109
250
200
150
100
50
0
C101 C102 C103 C104 C105 C106 C107 C108 C109
Note. The time required to recognize optimality for most of the C1 instances is under one minute.
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Figure 6. Scatter plot and range-finder box plot for R1 with FLS&I.
meanCPU = 204.8 sec.
R101
R103
R102
1,350
1,680
1,500
1,300
1,660
1,250
1,480
18
19
20
16
18
17
1,450
1,000
1,400
10
12
13
11
13
15
14
1,360
1,340
1,320
1,300
1,280
1,260
11
16
15
12
13
14
R109
R108
R107
14
R106
1,500
1,020
9
19
R105
R104
1,250
1,120
980
1,100
1,200
1,080
10
960
11
9
12
11
11
11
12
10
12
13
R112
1,140
1,120
1,100
1,080
1,150
1,100
10
10
R111
R110
990
980
970
11
12
9
10
11
Note. Results are within one vehicle of the best known (∆) for every trial and respectable compared with
B&V-H (+ at the average).
because all of these cases required about 200 seconds per trial. For R1, the FLS&I results
are within one vehicle of the best known solution for every trial. For RC1, the results are
within one vehicle of the best known for all but a handful of trials (for RC105 and RC106).
Given the much shorter processing time, the R1 and RC1 results are respectable compared
to the results presented by B&V-H.
Type Two Instances
Type two results are shown in Figures 8, 9, 10, and 11 for the clustered, random, and mixed
classes respectively. Note that, for the C2 instances, the local search and insertion heuristic
used here significantly outperforms the results presented in Bent and Van Hentenryck [4]
using far less processing time. A box plot of the CPU time required for the C2 instances
is shown in Figure 9. There was one outlier at a local minimum in C202, all other trials
rediscovered the best known solution at 591.56 twice and terminated the processing early.
In C204, 11 of the 16 trials settled on the best known solution at 590.6: the remaining trials
rediscovered a local minimum at 593.93, which terminated the processing early. Also note
that FLS&I is competitive with the B&V-H results for many of the R2 and RC2 instances,
requiring, on average, about 200 seconds per trial.
Conclusions
Results indicate that MACS-VRPTW with the fast local search and insertion heuristic finds
solutions to the Solomon benchmark that are close to the best known solutions within the
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Figure 7. Scatter plot and range-finder box plot for RC1 with FLS&I.
meanCPU = 203.6 sec.
RC102
RC101
RC103
1,400
1,700
1,540
1,680
1,350
1,520
1,660
1,300
1,640
1,500
16
15
14
RC104
14
13
12
10
RC105
1,250
12
11
13
RC106
1,480
1,650
1,460
1,200
1,440
1,600
1,420
1,400
1,550
1,150
9
11
10
12
13
RC107
15
14
1,380
11
16
13
12
14
RC108
1,300
1,180
1,280
1,160
1,260
1,240
1,140
10
11
12
10
11
12
Note. Results are within one vehicle of the best known (∆) for most trials and respectable compared with
B&V-H (+ at the average).
Figure 8. Results for the type two clustered instances show that FLS&I finds the best known
solution in every trial for 6 of 8 instances with an average CPU time under one minute.
meanCPU = 54.42 sec.
C201
C202
592.5
C203
750
700
592.0
700
650
591.5
650
591.0
600
2
3
4
600
2
C204
3
4
2
C205
3
4
C206
680
660
589.5
640
620
600
660
589.0
640
588.5
620
600
588.0
2
3
4
2
C207
3
4
2
C208
680
589.0
660
588.5
640
588.0
620
600
587.5
2
3
4
2
3
4
Note. ∆ = best known, + = B&V-H, -o- = fast local search and insertion.
3
4
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Figure 9. Box plots of CPU time (in seconds) required for the C2 instances.
C201
C202
C203
C204
C205
C206
C207
C208
250
200
150
100
50
0
C201
C202
C203
C204
C205
C206
C207
C208
Note. The time required to recognize optimality for most of the C2 instances is under one minute.
Figure 10. Scatter plot and range-finder box plot for R2 with FLS&I.
meanCPU = 219.3 sec.
R202
R201
R203
1,250
1,300
1,000
1,200
1,280
1,150
1,260
3
5
4
3
950
4
900
1,000
3
2
2
4
4
1,000
980
960
940
920
850
800
3
R206
1,050
1
2
5
R205
R204
3
2
4
3
4
R209
R208
R207
980
950
960
760
900
940
740
850
1
3
2
920
1
4
2
3
2
3
4
R211
R210
1,000
900
950
800
2
3
4
2
3
4
Note. Results are within one vehicle of the best known (∆) for every trial and comparable with B&V-H
(+ at the average).
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Figure 11. Scatter plot and range-finder box plot for RC2 with FLS&I.
meanCPU = 217.8 sec.
RC201
RC203
RC202
1,520
1,500
1,120
1,400
1,480
1,100
1,460
1,300
1,080
1,440
1,420
1,060
1,200
3
5
4
3
4
2
5
860
4
RC206
RC205
RC204
3
1,250
1,380
1,360
840
1,200
1,340
820
1,320
1,150
1,300
800
2
3
4
3
4
5
2
3
4
RC208
RC207
1,150
900
880
1,100
860
840
1,050
2
3
4
5
2
3
4
Note. Results are within one vehicle of the best known (∆) for every trial and comparable with B&V-H
(+ at the average).
replanning horizon used here for a typical military application. Also, the method used to
terminate the search finds the best known solutions to the C1 and C2 instances with an
average CPU time under one minute.
The fast local search and insertion heuristics used here were both programmed in C as
Matlab mex-functions and the remainder of the MACS-VRPTW method was programmed
in Matlab. Elapsed time for each Monte-Carlo trial was determined using the tic and toc
operators. It is expected that a dedicated C++ implementation will perform better than
the one presented here.
Acknowledgements
This document does not contain technical data as defined by the International Traffic in
Arms Regulations, 22 CFR 120.10(a), and is therefore authorized for publication.
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