Solving real world delivery problem using improved max-min ant system with local optimal solutions in wide area road network

International Journal of Artificial Intelligence & Applications (IJAIA), Vol. 5, No. 3, May 2014
DOI : 10.5121/ijaia.2014.5302 21
SOLVING REAL-WORLD DELIVERY PROBLEM
USING IMPROVED MAX-MIN ANT SYSTEM WITH
LOCAL OPTIMAL SOLUTIONS IN WIDE AREA ROAD
NETWORK
Junichi Ochiai1
and Hitoshi Kanoh2
1
Department of Computer Science, Graduate School of Systems and Information
Engineering, University of Tsukuba, Tsukuba, Ibaraki, Japan
2
Division of Information Engineering, Faculty of Engineering, Information and Systems,
University of Tsukuba, Tsukuba, Ibaraki, Japan
ABSTRACT
This paper presents a solution to real-world delivery problems (RWDPs) for home delivery services where
a large number of roads exist in cities and the traffic on the roads rapidly changes with time. The
methodology for finding the shortest-travel-time tour includes a hybrid meta-heuristic that combines ant
colony optimization (ACO) with Dijkstra’s algorithm, a search technique that uses both real-time traffic
and predicted traffic, and a way to use a real-world road map and measured traffic in Japan. We
previously proposed a hybrid ACO for RWDPs that used a MAX-MIN Ant System (MMAS) and proposed a
method to improve the search rate of MMAS. Since traffic on roads changes with time, the search rate is
important in RWDPs. In the current work, we combine the hybrid ACO method with the improved MMAS.
Experimental results using a map of central Tokyo and historical traffic data indicate that the proposed
method can find a better solution than conventional methods.
KEYWORDS
Ant Colony Optimization, Dijkstra’s Algorithm, Delivery Problem, Traffic, Real-World
1. INTRODUCTION
Ant colony optimization (ACO) is a stochastic search algorithm for problem solving that takes
inspiration from the foraging behaviors of ants. The main idea of ACO rests on the indirect
communication among individuals in an ant colony based on the pheromone trails that real ants
use for communication. ACO has been formalized into a meta-heuristic for combinatorial
optimization problems by Dorigo et al., and many applications are now available[1]-[3]. In
particular, many studies on ACO have been performed using the traveling salesman problem
(TSP[4]), and it has been shown that ACO is superior to other meta-heuristics [5],[6] for this type
of problem.
In this paper, we deal with real-world delivery problems (RWDPs) for home delivery services as
an extension of the TSP, where a large number of roads exist in cities and the traffic on the roads
rapidly changes with time. This scenario reflects the typical traffic congestion in a wide area
urban road network.
The problems that deal with finding optimal tours with time-dependent travel time have been
studied as TSPs, delivery problems [7], and vehicle routing problems [8]. Conventional problem-
solving methods using ACO repeat a search when the traffic flow changes during movement [9]-

22
[11]. However, finding the global optimal solution by this method is difficult because when the
traffic changes rapidly, the information obtained from an old search may not be helpful.
Furthermore, research based on real road maps and traffic information services in the real world
is seldom found.
In this paper, we propose a new method to solve RWDPs using ACO and Dijkstra’s algorithm
(DA). Search techniques based on only the predicted traffic have previously been presented for
real-world time-dependent TSPs using ACO [12] and real-world time-dependent vehicle routing
problems using ACO[13], an evolution strategy[14] , and a genetic algorithm[15]. When these
methods make mistakes with the prediction values, solution accuracy may deteriorate. We
previously proposed a hybrid ACO that combined real-time data with predicted traffic data for
RWDPs[16]. In that method, the calculation time was not discussed, even though traffic on roads
changes with time. The proposed method combines the hybrid ACO technique [16] with this
previous method in order to improve the search rate of the MAX-MIN Ant System[12].
In the following section, we start by describing the problem. Then, we detail the algorithm of the
proposed method. Finally, we present the results of experiments using a map of central Tokyo and
real traffic data.
2. OVERVIEW
In this paper, we regard the RWDP as an extension of the TSP: a vehicle starts from a depot,
visits all customers without any time constraint, and finally returns to the depot. This type of
problem is also called a one-to-many-to-one delivery problem[7]. Here, we first describe the
time-dependent TSP (TDTSP) [17] and the calculation of the tour travel time. Next, we explain
the RWDP and a traffic information service in the real world. Then, we give a brief description of
ACO and provide details about MAX-MIN Ant System (MMAS) as an example of typical ACO.
Finally, we present our method to improve the search rate of MMAS.
2.1. Time-Dependent Traveling Salesman Problem (TDTSP)
The TSP [4] can be represented by a complete graph G = (N, A), where N is a set of nodes, i.e.
cities, n = |N| is the number of nodes, and A is the set of arcs fully connecting the nodes. Each arc
(i, j) ∈ A is assigned a value dij (= dji), which represents the distance between nodes i and j. The
TSP then is the problem of finding the shortest closed tour that visits each of the nodes of G
exactly once. The TSP instances used in this paper are taken from the TSPLIB benchmark library:
http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/.
The TDTSP [17] extends the original TSP so that traffic congestion can be included. Let Tij (t) be
the travel time between nodes i and j at time t; Tij (0) means the original travel time of a given
TSP, i.e. Tij (0) = dij. Traffic congestion can be represented by a change in the travel time. Here,
this change is defined by the following formulas:
,)]1()([)( max
minjam
T
Tijij uRtTttT ×+×=∆+ (1)





<
>
=
,otherwise
,if
,if
][
x
bxb
axa
x a
b (2)

23
,uppermax BdT ij ×= (3)
,min ijdT = (4)
where ∆t is an updated interval of travel time, Rjam and Bupper are the parameters showing the rate
and the upper bound of traffic congestion, respectively, and u ∈ [−1, 1] is a uniform random
number. The time when a salesman leaves city 1 is set to t = 0, and he always starts from and
returns to city 1.
In the following, the time required to travel around a tour is called the tour travel time. The tour
travel time for solution S can be calculated by the following formulas:
,)()(
1
1,∑=
+=
n
i
iii tTST (5)



+
=
=
−−− ,otherwise)(
,1if0
1,11 iiii
i
tTt
i
t (6)
where ti is the time when a salesman reaches or leaves node i.
2.2. Real-World Delivery Problem
The RWDP can be represented by a quadruple GRW
= (NRW
, ARW
, MRW
, TRW
), where NRW
= {Ci | i
= 1,...n}, C1 is a depot and {C2,...Cn} is a set of customers, ARW
= {Pij (ti) | i, j = 1,...n (i ≠ j)} is a
set of optimal paths from Ci to Cj at time ti when a vehicle reaches Ci, and MRW
and TRW
are a
road map and a set of time-series traffic data in the real world, respectively. Each Pij (ti), generally
≠Pji (ti), can be calculated using TRW
on MRW
. It is necessary to calculate the optimal path between
customers in RWDPs, while the distance between cities is given in TSPs. The RWDP then is the
problem of finding a shortest-travel-time tour when a vehicle starts from C1 and returns to C1
visiting each of the customers {C2,...Cn} exactly once.
The road map MRW
used in this paper is the standard map database that is used in actual car
navigation systems. This map includes all drivable roads in Japan and its format was developed
and established by the Navigation System Researchers’ Association.
Historical time-series traffic data TRW
are also used to calculate the travel time of a vehicle. In
Japan, traffic meters are installed at more than 20,000 locations along principal roads throughout
the country. These meters measure the average travel time of cars passing through specific road
links at intervals of one or five minutes. The data so obtained is collected at a traffic information
center and provided to subscribers in real time. Figure 1 shows an example of time-series traffic
data. The vertical axis represents the average speed of cars on a link calculated directly from the
traffic data. In this figure, data in the range of 0:00 to 5:00, 5:00 to 8:00, 8:00 to 18:00, and 18:00
to 20:00 correspond to no congestion, outbreak of congestion, heavy congestion, and dissolution
of congestion, respectively. Thus, time-series traffic data is highly nonlinear, which makes it
difficult to perform accurate predictions. Application to real-world scenarios must take the
prediction error rate into consideration.
2.3. Ant Colony Optimization (ACO)
The generic ACO meta-heuristic [1] is shown in Fig. 2. After initialization, the meta-heuristic
iterates over two phases. First, a number of solutions are constructed by the ants, and second,

24
before the start of the next iteration, the pheromone trails are adapted to reflect the search
experience of the ants.
Figure1. Example of time-series traffic data.
procedure ACO ( )
Set parameters;
Initialize pheromone trails;
while ( terminal condition not met ) do
Construct ant solutions;
Update pheromone trails;
end-while
end-procedure
Figure 2. ACO meta-heuristic.
There have been many attempts to improve the performance of ACO. MAX-MIN Ant System
(MMAS) [18] has demonstrated an especially impressive performance[1], so we used the MMAS,
except for constructing the ant solution described in section 3.4, as the ACO in the proposed
method.
2.4. MAX-MIN Ant System (MMAS)
Here, we describe the MMAS for original TSPs. The MMAS constructs ant solutions and updates
pheromone by following (7) and (12), respectively. The MMAS has two main characteristics [2] :
only the best ant updates the pheromone trails, and the value of the pheromone is bound. Here,
the best means that the length of the tour is shortest. The pheromone τij, associated with the arc
joining cities i and j, is updated using (7):
,])1[( max
min
best τ
τττρτ ijijij ∆+×−← (7)



=∆
,otherwise0
best tour,thetobelongs),(if/1 bestbest
jiL
ijτ (8)
where ρ is the evaporation rate, τmax and τmin are respectively the upper and lower bounds imposed
on the pheromone, and Lbest is the length of the tour of the best ant.

25
Some guidelines have been provided for defining τmax, τmin, and Lbest on the basis of analytical and
empirical considerations. This paper defines these parameters by the following three formulas
[18] :
,
1
gb
max
L×
=
ρ
τ (9)
,
)12/(
1
max
best
best
min ττ ×
×−
−
=
n
n
pn
p
(10)
,ibbest LL = (11)
where Lgb is the length of the best tour from the beginning of the algorithm (global-best), Lib is the
length of the best tour found in the current iteration (iteration-best), and pbest is a parameter.
In the construction of a solution, ants select the next city to be visited through a stochastic
mechanism. When ant k is in city i and has so far constructed the partial solution, the probability
of going to city j is given by





∈
= ∑∈
otherwise,0
,if
][][
][][ k
Nl ilil
ijij
k
ij
Nj
p k
βα
βα
ητ
ητ
(12)
where Nk
is the set of cities not yet visited by the ant k. The parameters α and β control the
relative importance of the pheromone trail and the heuristic information ηij, which is given by
.
1
ij
ij
d
=η (13)
2.5. Effective Initialization of Pheromone Trail
An effective solution of TDTSP requires the early convergence for the reasons below.
・ It is necessary to calculate the optimal path between customers in practical problems, while
the distance between cities is given in TSPs. Consequently, the order of visiting customers is
calculated by ACO and the path between customers is calculated by Dijkstra’s algorithm, etc.
・ When traffic congestion changes during traveling, the path should be re-evaluated. Though
the traffic congestion information on main roads is provided every one or five minutes in
Japan, this calculation must be performed within one minute.
We previously proposed a method to improve the search rate of MMAS for TDTSPs[12]. The
strategies of this method are listed below.
・ In addition to the usual initial pheromone trails, additional pheromones are deposited on the
solution built before the first iteration. The method helps to guide the search and so might
reduce the number of points visited in the search space by giving deviations from the initial
pheromone trails.

26
・ This reduction is conducted using solutions built by a greedy algorithm. These solutions, i.e.,
initial solutions, can include candidate partial solutions of the global optimal solution.
Moreover, the computational time of the greedy algorithm is very short; it can be considered
to have no influence on the computational time of the method.
・ In TDTSPs, a salesman starts from city 1 and the travel time between cities might change. In
order to generate many tours using the greedy algorithm, we change the first visited city
from city 1 in the greedy algorithm.
The pheromone trail initialization is shown in Fig. 3 and below:
,
)(
1
best
0ST
ij
×
←
ρ
τ (14)
,
1
)1(
1
1
∑
−
=−
+×−←
n
k
k
ijijij
n
r
r δτττ (15)


 ∈
=
,otherwise0
,),(if)(/1 00 kkk
ij
SjiST
δτ (16)
where S0
best
is the best solution of the initial solutions, S0k (k = 1 to n –1) is the kth initial solution,
and r (0 ≤ r ≤ 1) is a parameter.
Here, the first term in the right side of (15) means the uniform distribution of pheromones, and
the second term means centralizing of the pheromones around the initial solutions. The parameter
r indicates the ratio of the decentralization of the initial pheromone trails and the centralization of
the initial pheromone trails. When r = 0, the method corresponds to the MMAS. In contrast, when
r = 1, the initial pheromone is distributed only on the arcs contained in the initial solutions.
3. PROPOSED METHOD
3.1. General Procedure
The general procedure of the proposed method is shown in Fig. 4. A target road map including
the depot and all customers is prepared along with the historical traffic data for principal roads on
the map. Index i in the outermost loop corresponds to the turn at which a vehicle visits customers.
When the vehicle is moving, the search is repeated. Real-time traffic data for the principal roads
are input at ∆t intervals, so updating the travel time, re-calculating the predicted traffic, and
planning the tour of the vehicle are performed at this interval. The prediction system, as we have
already reported [19] , [20] has an interpolation function as well as a prediction function. This
system can estimate traffic on roads not installed with detectors from the traffic on roads that are
installed with detectors. Constructing a tour by hybrid ACO is described in the
next section.
procedure Initialize-Pheromone ( )
Construct (n −1) solutions using greedy algorithm with (n −1) different second cities;
Initialize pheromone trails as well as original MMAS; // see Error! Reference source not
found.
Update τij individually using Error! Reference source not found.;
end-procedure
Figure.3. General procedure of pheromone trail initialization.

27
procedure main ( )
Input RWDP;
Cv = 1; // 1 is depot number
Nv = { 1 };
for ( i = 1 to n ) do
if ( ∆t passed or i = 1) then
Input TRW
( real-time );
Predict travel time for all roads on MRW
;
Execute Hybrid-ACO ( Cv, Nv ); // see Fig. 5
end-if
Move a vehicle to a next customer according to best tour;
Cv ← the number of current customer of the vehicle;
Nv ← Nv ∪ { Cv };
end-for
Move the vehicle to the depot according to best tour;
end-procedure
Figure4. General procedure of proposed method.
3.2. Constructing a Tour by Hybrid ACO
In the proposed method, the paths between customers are planned by Dijkstra’s algorithm (DA)
and the route of visiting customers is constructed by ACO. The DA is widely used as a path
planning method, and ACO is superior to other meta-heuristics including genetic algorithms [5]
and simulated annealing [6] in terms of constructing a tour.
Figure 5 shows the general procedure of hybrid ACO, where Cv is a current customer of a vehicle
and Nv is a set of customers visited by the vehicle. Ants start from the current location of the
vehicle and construct a route to the depot. When the ant k is at the customer Ci in constructing a
route, selection probabilities for all Cj ∈ Nk
are calculated by the following formulas:





∈
= ∑∈
otherwise,0
,if
)]([][
)]([][
)(
k
Nl iilil
iijij
i
k
ij
Nj
t
t
tp k
βα
βα
ητ
ητ
(17)
.
)(
1
)(
iij
iij
tT
t =η (18)
In conventional methods [13]-[15], paths connecting each pair of customers are calculated before
the algorithm, and the calculation time of the algorithm is not considered. In the proposed method,
we calculate these paths individually when they are required for the first time. The DA procedure
is shown in Fig. 6, where s is the starting node, g is the destination node, ts is the time at which a
vehicle leaves node s, and ncl is a parameter of the candidate lists[1]. These candidate lists
constitute a small set of promising neighbors of the current partial solution and their use enables
the ACO algorithms to focus on the more relevant components, thus significantly reducing the
dimensions of the search space. A sort algorithm is normally required in order to use candidate
lists, but since the DA already includes a sort function, the proposed method does not need any
additional sort algorithm for making the candidate lists.

28
procedure Hybrid-ACO ( Cv, Nv )
Input RWDP;
Execute Initialize-Pheromone ( ); // see Fig. 3
Sgb ← S0
best
;
while ( terminal condition not met ) do
for ( k = 1 to m ) do // m is the number of ants
Nk
← NRW
−Nv;
for ( j = |Nv| to n ) do // ants start from Cv
Move ant k to a next customer; // see Error! Reference source not found.
end-for
Move ant k to depot;
end-for
Sib ← the best tour in current iteration;
if ( T ( Sib ) < T ( Sgb ) ) then
Sgb ← Sib;
end-if
end-while
end-procedure
Figure 5. General procedure of hybrid ACO.
procedure DA ( s, g, ts, ncl )
Initialization for Dijkstra’s algorithm;
c = 0;
while ( c ≤ ncl ) do
Execute Dijkstra’s algorithm until an optimal path between customers is defined;
c += 1;
end-while
Create candidate list of s at time ts;
while ( an optimal path between s and g is not defined ) do
Execute Dijkstra’s algorithm until the optimal path between s and g is defined;
end-while
Store the optimal paths from s to other nodes in the memory;
Store the travel time from s to other nodes in the memory;
end-procedure
Figure 6. General procedure of path planning by DA.
4. EXPERIMENTS
4.1. Experiments with TSP Instances (TDTSP)
4.1.1 Experimental Methods
To evaluate how well the proposed method performs, we first conducted experiments using the
TSP instances eil51, eil76, kroA100, u159, and d198 from the TSPLIB. The number of cities
seems small in TSPLIB. However, considering correspondence with real-world problems, since
the number of cities (or customers) that a salesman (or a vehicle) can visit in one day is at most
200, this number should be suitable as a scale for benchmark problems.

29
TDTSPs were generated by the method described in section 2.1. Parameters Rjam and Bupper were
respectively set to 0.5 and 5 by reference to real-world traffic. Minutes and seconds are assumed
as units of travel time for instances {eil51, eil76} and {kroA100, u159, d198}, respectively. So,
the update interval of travel time ∆t is 5 and 300 for instances {eil51, eil76} and {kroA100, u159,
d198}, respectively.
We compared the minimal tour travel time obtained by the proposed method with those by the
conventional methods below.
・ Plain method: Search is conducted once before a vehicle starts using static traffic Tij (0).
・ Repeat method [9]-[11]: Search is repeated while a vehicle is moving using real-time traffic
Tij (t).
・ Prediction method [13] , [21] : Search is conducted once before a vehicle starts using
predicted traffic T′ij (t, ti).
・ Prediction and repeat method: The proposed method.
We assumed two kinds of prediction errors, i.e., 20% and 50% in Fig. 7. The predicted traffic
with error rate can be calculated as follows,
),)(1()(),( uttEtTttT iiijiij ×−+×=′ (19)
where t is the current time, ti is the departure time from Ci, E (ti −t) is the error rate given by Fig.
7, and u ∈ [−1, 1] is a uniform random number. This is based on experience in which short-term
prediction has a smaller error rate than long-term prediction. In the prediction method, since the
search is conducted only once before a vehicle starts, T′ij (t, ti) is calculated only once (t = 0). In
contrast, since the proposed method repeatedly searches for the best tour while a vehicle is
moving, T′ij (t, ti) is calculated every time real-time traffic is updated.
In this section, we assume that computational time is not limited and so we used original MMAS
(r = 0) for the tour planning and chose 3000 iterations as the terminal condition. The values of
parameters of the MMAS were as follows: α = 1.0, β = 5.0, ρ = 0.02, pbest = 0.05, and m is the
number of customers not yet visited by a vehicle.
Figure 7. Relationship between prediction error and time.
20%
50%

30
4.1.2 Experimental Results
First, since the optimal solutions of TDTSPs are not known, we obtain them by the prediction
method without error, which knows the exact travel time on all the links at all times. We assumed
the best solution of 30 trials to be the optimal solution. Table 1 lists the number of cities, the
optimal solution known in a static environment, the optimal solution in a dynamic environment,
the update interval of travel time in a dynamic environment, and the number of updates for each
instance. The number of updates ranged from 68 to 207—in other words, the traffic flow changed
very frequently.
Figure 8 shows the computational results of the prediction method without error. The plots on the
left show the development of the ratio of the tour travel time to the optimal tour travel time and
the plots on the right show the development of the average λ-branching factor, which measures
the distribution of the pheromone trail values[1]. Each plot in Fig. 8 is the average of 30 trails
using different random number sequences. These results demonstrate that the terminal condition,
3000 iterations, is sufficient to achieve convergence on every instance.
Table 2 lists the ratio of the tour travel time of each method to the optimal tour travel time. Each
value in Table 2 is the average of 30 trials using different random number sequences; the standard
deviation was about 5 to 10%. Table 2 reveals three main findings.
Table 1. TSP instances used in experiments, optimal tour travel time, and updated travel time.
Instance
Optimal
solution in TSP
Optimal
solution in
TDTSP
Update
interval
Number of
updates
eil51 426 574 5 114
eil76 538 730 5 146
kroA100 21282 29942 300 99
u159 42080 62355 300 207
d198 15780 20700 300 68
Figure 8. Computational results of prediction method without error: the ratio of tour travel time to optimal
tour travel time (left) and the average λ-branching factor, λ = 0.05 (right).

31
Table.2. Experimental results for TDTSP. Each value indicates the ratio of the tour travel time of each
method to the optimal tour travel time.
Instance Plain Repeat
Error rate 20% Error rate 50%
Prediction
Prediction
+ Repeat
Prediction
Prediction
+ Repeat
eil51 1.50 1.31 1.05 1.02 1.41 1.12
eil76 1.39 1.36 1.29 1.06 1.44 1.10
kroA100 1.27 1.18 1.16 1.03 1.17 1.05
u159 1.26 1.27 1.29 1.03 1.39 1.08
d198 1.39 1.27 1.17 1.03 1.26 1.08
(Mean) 1.39 1.27 1.17 1.03 1.33 1.08
• Compared with the plain method, the performances of the other methods are improved.
• When the prediction error rate becomes large (50%), the prediction method is inferior to
the repeat method.
• The proposed method is superior to the other methods, even when the error rate is large
(50%).
4.2. Experiments with Real-World Problems (RWDP)
4.2.1 Experimental Methods
Next, to evaluate the proposed method in a real-world environment, we applied it to the RWDP
described in section 2.2. Figure 9 shows a map of central Tokyo, which was the target area of this
experiment. This map is 9 km long by 11 km wide and includes 10056 links and 3542 nodes and
represents the most congested area in Japan. The positions of the depot and the customers
Figure 9. Road map of central Tokyo. Area is 9 km long by 11 km wide and there are 10,056 links and
3,542 nodes.
are randomly selected from the nodes. The travel time of a vehicle was calculated from historical
traffic data on June 17, 2003. An example of the data is shown in Fig. 1. We also assumed the

32
prediction error rate in Fig. 7. We performed nine experiments. The numbers of customers are 50,
100 and 200. The departure times from the depot are 6:00 (morning), 12:00 (noon), and 18:00
(night). The instances are described as “the number of customers–the departure time from the
depot,” i.e., 50–6, 50–12 ... 200–18.
In a real-world scenario, the calculation time of the tour planning is limited. In Japan, traffic
meters measure the average travel time of cars passing through specific road links at 1-minute
intervals. When a state-of-the-art prediction system [22] is used in the proposed method, the
calculation time of the prediction system is about 20 seconds. Therefore, we chose 10 seconds, 20
seconds, 40 seconds, and 3000 iterations (unlimited calculation time) as the terminal condition of
MMAS. The other parameter values of MMAS were set the same as in section 4.1. The code was
written in C++ and the experiments were executed on an Intel Core i7-870 (2.93 GHz).
4.2.2 Experimental Results
Figure 10 shows the best tours obtained by all experiments. The red and blue circles indicate a
depot and customers, respectively. As shown, the tour depends on the time period. Crosses and
returns appear in some parts of the tour in 2D, but in the 3D real-world, the tour is a complete
circuit.
Table 3 lists the best tour travel time (minutes) and the tour travel time (minutes) of each method.
The terminal condition of MMAS is 3000 iterations. Each value in Table 3 is the
Morning Noon Night
Figure10. Best tours in RWDP.
Table 3. Experimental results for RWDP. The terminal condition of MMAS is 3000 iterations. Each value
is shown in minutes.

33
Instance
Best
tour
Plain Repeat
Error rate 20% Error rate 50%
Prediction
Prediction
+ Repeat
Prediction
Prediction
+ Repeat
50–6 193 270 226 210 210 233 210
50–12 205 282 232 221 214 244 221
50–18 182 200 190 190 188 199 192
100–6 276 351 323 287 283 316 291
100–12 288 410 342 323 306 337 313
100–18 250 314 282 259 272 271 275
200–6 420 546 495 447 439 484 448
200–12 427 590 488 461 447 475 455
200–18 375 445 411 398 385 404 397
average of 30 trials using different random number sequences, and the standard deviation was
about 2 to 5%. Table 3 indicates the same results as the experiments for TDTSPs can be obtained
in a real-world environment.
When the calculation time of the tour planning is limited, the search rate of MMAS is quite
important. Here, we compare the performance of the proposed method using original MMAS (r =
0) with the performance of the proposed method using improved MMAS (r = 0.9). Table 4 lists
the tour travel time (minutes) of each method. Each value of each method in Table 4 is the
average and the standard deviation of 30 trials using different random number sequences. Table
4reveals four main findings.
・ The improved MMAS did not deteriorate compared to the original MMAS.
・ When the problem size was small (50 customers), 10 seconds was enough to obtain the best
tour.
・ When the problem size was large (200 customers), 40 seconds was not sufficient to achieve
convergence on the original MMAS.
・ The larger the problem size was, the more superior the improved MMAS was compared to
the original MMAS.
5. CONCLUSIONS
In this paper, we presented three techniques: a hybrid meta-heuristic that combines ACO with
Dijkstra’s algorithm, a search method that combines repeat and prediction, and a way to use a
real-world road map and measured traffic data in Japan. The experimental results suggest that the
proposed method is effective in a wide area road network. The results presented in this paper are
based on five benchmark problems and a real-world problem. Further investigation using other
maps and traffic from additional days is necessary. Although the proposed method is for delivery
problems, the basic idea can be used for other combinatorial optimization problems in networks.
In future work, we will combine the proposed method with traffic prediction systems.

34
Table 4. Experimental results for RWDP using the proposed method. The terminal condition of MMAS is
10 seconds, 20 seconds, and 40 seconds. Each value of each method is shown in minutes.
ACKNOWLEDGEMENTS
The authors wish to thank Mr. Yosuke Kameda for his considerable advice in carrying out this
research. This research was partly supported by a Grant-in Aid for Scientific Research (C) of the
Japan Society for the Promotion of Science (23500169).
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Authors
Junich Ochiai received M.E. degree and Ph.D. degree in computer science from
University of Tsukuba, Japan, in 2011 and 2014, respectively. He currently is a
postdoctoral researcher with the Center for Service Research, National Institute of
Advanced Industrial Science and Technology. His research interests include ant colony
optimization and evolutionary computation.
Dr. Ochiai is a member of the Information Processing Society of Japan. He received
the best presentation award from the Mathematical Modeling and Problem Solving Group, Information
Processing Society of Japan in 2012.

36
Hitoshi Kanoh received M.S. degree in physics and Ph.D. degree in computer science
from University of Tsukuba, Japan, in 1980 and 1992, respectively. In 1980, he joined
Hitachi Cable, Ltd. where he has been engaged in Research of expert systems, fuzzy
systems and neural networks as a section chief. In 1993 he moved to the University of
Tsukuba. He currently is a professor with the Division of Information Engineering,
Faculty of Engineering, Information and Systems, University of Tsukuba. His main
research interests include evolutionary computation, swarm intelligence, and soft
computing with special application to intelligent transportation systems.
Dr. Kanoh is a member of the IEEE, the Japanese Society for Artificial Intelligence, and Information
Processing Society of Japan. He received the best paper award from the Institute of electrical Engineers of
Japan in 1992, the best paper and presentation award from the World Conference on Soft Computing in
Industrial Applications in 1999, and the best paper and presentation award from the Intelligent Transport
Systems Group, Information Processing Society of Japan in 2003 and 2006.

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