VEHICLE ROUTING PROBLEM WITH MULTI-DEPOT
AND MULTI-TASK
Haoxiong Yang, Li Jing
School of Business, Beijing Technology and Business University, No.33 Fucheng Road, Beijing, China
Yongsheng Zhou, Mingke He
Department of Logistics Management, Beijing Technology and Business University, Beijing, China
Keywords: City distribution, Multi-depot, Multi-task, Vehicle routing problem, Self-adaptive, Polymorphic ant colony
algorithm (APACA).
Abstract: This paper presents a new mathematical model of the vehicle routing problem in the context of city
distribution, which considers multi-depot, multi-model vehicles and multi-task. The objective function
includes three parts: transport cost, deadheading cost and time cost. To solve this mathematical model, a
self-adaptive and polymorphic ant colony algorithm (APACA) has been introduced. Finally, a case study is
presented to compare the results based on APACA with that under stochastic condition. Simulation results
show that APACA is an effective and desirable algorithm for solving the mathematical model.
1 INTRODUCTION
Vehicle routing problem (VRP) is an important part
of city distribution activities. The optimization for
the vehicle routing problem contributes to improve
the efficiency of goods distribution, reduce total
distribution cost and increase customer satisfaction.
Take Keihanshin metropolitan region in Japan for
example, since the optimization strategy was taken
by local government in 1989, the next surveys
shown that average number of transport vehicles had
been reduced by 17%, total deliver time had been
reduced by 89%, and total travel distance had been
reduced by 72%.
In the context of city distribution, vehicle routing
problem has the following features: multi-depot,
multi-model vehicles and multi-task. These features
make vehicle routing problem more complicated
than that under the condition of single-depot, single-
model vehicle and single-task, but practical
significance. In previous studies, Koehler (1999)
described Kassel joint distribution system, and
pointed that this system can decrease total travel
time and the number of transport vehicles obviously;
T Yamada et al. (2001) proposed a decisive model
of co-operative vehicle routing, and analyzed the
impact of co-distribution center location on deliver
effectiveness; Renaud et al. proposed different
branch-and-bound algorithms for symmetrical multi-
depot vehicle routing problem and asymmetrical
multi-depot vehicle routing problem. Yaohuang Guo
(1995) and Shengce Hang et al. (1997) firstly
transformed multi-depot vehicle routing problem
into single-depot problem and then applied the
method for solving single-depot problem into multi-
depot problem; Mingshan Zhang (2002) discussed a
general distributing route problem under the
condition of multi-depot and full-load and proposed
a heuristic algorithm for optimizing distributing
route based on network flow model.
Take a broad view of previous studies, the
majority of the studies have been of help to
enterprises or governments; however for most of
them the following aspects can be improved in the
following aspects:
More constraints such as multi-depot, multi-
model vehicles, multi-task and capacity should be
considered together, rather than only one or two
constraints;
The target of mathematical model is not only to
minimize travel distance or cost in part distribution
activities but also to minimize total travel cost in all
650
Yang H., Jing L., Zhou Y. and He M..
VEHICLE ROUTING PROBLEM WITH MULTI-DEPOT AND MULTI-TASK.
DOI: 10.5220/0003573406500655
In Proceedings of the 13th International Conference on Enterprise Information Systems (MMLM-2011), pages 650-655
ISBN: 978-989-8425-56-0
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
distribution processes.
Since the vehicle routing problem in the context of
city distribution is a complex system in which
constraints including multi-depot, multi-model
vehicles and multi-task are often occurred and the
target to minimize total travel cost (includes
transport cost, deadheading cost and time cost) is
more meaningful than the target to minimize travel
distance. As a result, the previous studies are not
fully based on actual situations in the context of city
distribution.
Based on the above studies, we first describe a
practical city distribution problem whose constraints
involve multi-depot, multi-model vehicles and
multi-task, and establish its mathematical model
whose objective function includes three parts:
transport cost. Then, we state the design of an self-
adaptive and polymorphic ant colony algorithm
(APACA) for solving the problem. Finally the
computational results of the actual instance show the
effectiveness of the proposed method.
2 THE VEHICLE ROUTING
PROBLEM WITH
MULTI-DEPOT,
MULTI-MODEL VEHICLES
AND MULTI-TASK
2.1 Problem Description
In city distribution system, there are several depots.
Each of them has some different model vehicles.
Different model vehicle has different capacity, unit
transport cost, deadheading cost and unit time cost.
Each vehicle must set out from its depot to
distribution center for carrying goods. Then, the
vehicles deliver goods to all demand points. Each
demand point has a service soft time window which
means that if the arrival time at a demand point is
earlier than the beginning of the time window or
later than the end of the time window, the cost
function will be penalized by some amount. A
demand point is serviced exactly once by only one
vehicle. The problem that we propose is that in order
to minimize total travel cost (includes transport cost,
deadheading cost and time cost) how to decide right
vehicles and chose right distributing routes in the
above circumstances.
2.2 Mathematical Model
Let us consider a routing network which is compo-
sed of depots, distribution center and customer
points. In city distribution activities, the vehicle K
sets out from the depot S to the distribution center O
for loading goods, and then delivers goods to the
customer point N. Finally, the vehicle K needs to
return to its start point, namely depot S. The problem
is defined on a complete graph G(V,A) (shown in
Figure 1), where
V=(v
1
,v
2
,…,v
n
,v
n+1
,v
n+2
,v
n+3
,…,v
n+m+1
) is a vertex set
and E={v
i
,v
j
: v
i
, v
j
V , i j}is the arc set. V
includes three subsets: N={v
1
,v
2
,…,v
n
} is a customer
vertex set, O={ v
n+1
} is a distribution center vertex
set, S={ v
n+2
,v
n+3
,…,v
n+s+1
} is a depot vertex set. For
each v
i
N, it includes the demand of the customer;
for each v
i
S, it includes the number of the vehicle
model; K={1,2,…,k} is a vehicle model vertex set;
each customer v
i
has a service time window [ET
i
, LT
i
]
where
ET
i
is the earliest time that service can begin
and
LT
i
is the latest time that service can begin. If the
arrival time at a demand point is earlier than the
beginning of the time window or later than the end
of the time window, the cost function will be
penalized by some amount.
Figure 1: The illustration of the Vehicle Routing.
In order to simplify the problem, we define the
sets, parameters and variables used in the
mathematical model as follows:
We denote the customers by 1,2,…,N ;
distribution center by N+1; depots by N+2, N+3,…,
N+S+1 and the vehicle models by 1,2,…,K;
s
k
ij
r
=(1,0)If vehicle k in the depot s travel form
vertex i to vertex j the
s
k
ij
r
=1, otherwise
s
k
ij
r
=0
D
ij
: Distance between node i and node j
VEHICLE ROUTING PROBLEM WITH MULTI-DEPOT AND MULTI-TASK
651
N: Total number of customers
S: Total number of depots
C
k
:
Unit transportation cost of vehicle k
E
k
:
Unit deadheading cost of vehicle k
V
k
:Unit travel distance of vehicle k
Q
k
: Capacity of vehicle k
W
i
: Demand at customer i
T
i
: Arrival time at node i
t
ij
: Travel time from node i to node j
ET
i
: Earliest arrival time at node i
LT
i
: Latest arrival time at node i
A: Earliness penalty coefficient
B: Tardiness penalty coefficient
The target of this mathematical model is to minimize
total travel cost (includes transport cost,
deadheading cost and time cost) which is occurred in
the process of city distribution.
The formulation of the problem is as follows:
sk sk sk
ij ij k ij kk
ij ij ij
iONjNsSkKiSjOsSkKiNjssSkK
C
C
rD rDE rDE
min
  
  

max( , 0 ) max( , 0 )
sk
jj j j
jj
ij
iONjNsSkK
ab
rETT TLT



(1)
,,
s
k
j
ij
k
iO Nj N
Q
w
r
kKsS



(2)
1,
s
k
ij
iO NsSk K
r
j


N
(3)
0, ,
ksk
ij ji
jN jN
rr
kKsS



(4)
1, , ,
s
k
ij
js
r
iNkKsS

(5)
(),
s
k
ij
ij
ij
iN OsSkK
t
rT T
j
N



(6)
s
k
ij
iN OjNsSk K
r
N


(7)
0,
s
T
s
S
(8)
The objectives, formula (1) is to minimize the total
vehicle travel cost ((includes transport cost,
deadheading cost and time cost); Formula (2)
constraints the vehicle capacity; Formula (3) ensures
that a customer is serviced exactly once by only one
vehicle; Constraint (4) and (5) guarantee that each
vehicle sets out from the depot and returns to it;
Formula (6) is to compute the arrival time when the
vehicle arrives at customer j; Constraint (7) ensures
that each customer will be serviced; Formula (8)
denotes that the initial time starts form the depot,
and the initial time is equal to zero.
2.3 The Design of Self-adaptive and
Polymorphic Ant Colony
Algorithm (APACA) for Solving
Multi-depot, Multi-model Vehicles
and Multi-task Vehicle Routing
Problem
2.3.1 The Overall Idea of Self-adaptive and
Polymorphic ant Colony Algorithm
(APACA)
The design idea of self-adaptive and polymorphic
ant colony algorithm (APACA) in this paper is as
follows: Based on traditional colony algorithm for
the travelling salesman problem (TSP),we improve
the traditional colony algorithm in information
selective mechanism, information update mechanism
and information collaborative mechanism, and also
introduce self-adaptive migration rule into the
traditional colony. Because the division of labor in
ants is diverse, we allocate the tasks with many
constraints to detective ants. Then, the task to find
feasible solutions of the target function is completed
by searching ants. We combine information self-
adaptive rule and polymorphic ant colony algorithm
perfectly by the collaboration between different
kinds of ants. As a result, self-adaptive and
polymorphic ant colony algorithm (APACA) can
overcome the drawbacks such as long computing
time and precocity which happen in traditional ant
algorithm.
2.3.2 The Steps of Self-adaptive and
Polymorphic Ant Colony Algorithm
(APACA)
The heading of a subsection title should be in 12-
point bold with initial letters capitalized, aligned to
the left with a linespace exactly at 13-point, hanging
indent of 1,0-centimeter and with According to the
principle of self-adaptive and polymorphic ant
colony algorithm (APACA) proposed by Meijun
Chen et al. (2008), we apply this algorithm for
solving multi-depot, multi-model vehicles and multi-
task vehicle routing problem, Specific steps and
methods are as follows:
Step 1: Initialize Q, C and N
c,max
;
Step 2: Put n detective ants in n cities, and each
detective ant revolves around city i. Then, the
detective ant detects other n-1 cities. Calculate the
detective table and assign the calculating results to
S
ij
;
Step 3: Set the information value of every route
ICEIS 2011 - 13th International Conference on Enterprise Information Systems
652
at initial time;
Step 4: Set that the initial value of N
c
is equal to
zero;
Step 5:Randomly select the initial coordination
of every ant, and put the coordination into its tabu
k
table;
Step 6: Calculate the migrating coordination of
every searching ant. Suppose that the migrating
coordination is point j and the last coordination is
point i. Then, put the coordination of point j into
tabu
k
table of searching ant k. Until each searching
ant visits all coordinations, we can get a solution;
Step 7: Calculate all solutions of the target
function f(Z
k
) (k=1,2,…,n), and record all the
solutions;
Step 8: Lead 2-opt local optimization to all the
above solutions, if new solutions is better than initial
solutions, then we replace initial solutions with new
solutions; Otherwise, we have initial solutions;
Step 9: If cycle counts comes to N
c,max
, go to
step12. If cycle counts does not come to N
c,max
and
the solution isn’t improved in recent several
iterations, then we should change the value ρ, go to
step 11. Otherwise, go to step 10;
Step 10: Change the information value of every
route, reset
0
ij
and clear tabu
k
table;
Step 11: N
c
N
c
+1, go to step5;
Step 12: Output the optimal solution.
3 SIMULATION EXAMPLE
In simulation example, in order to get the optimal
solution, the algorithm is implemented under the
visual studio 2005. The computational experiment
assumed to have three delivery depots, one
distribution center and sixteen customers. All the
nodes are distributed in a 100×100 square region,
each depot has two kinds of model vehicle.
Customer data can be seen in Table 1; Depot and
distribution center data can be seen in Table 2;
Vehicle data can be seen in Table 3.
Table 1: Customer data.
Customer 1 2 3 4 5 6 7 8
Coordination
88,55 39,31 20,76 65,98 95,23 50,9 33,65 79,75
W
i
1.5 3 1 1.5 2.5 2 4 2
[ET
i
,LT
i
] [4,6] [4,7] [2,5] [1.5,4.5] [5,8] [1,4] [3,6] [5,7.5]
a 1 1 1 1 1 1 1 1
b 1 1 1 1 1 1 1 1
Customer 9 10 11 12 13 14 15 16
Coordination (55,35) (78,30) (11,20) (35,93) (73,85) (62,53) (33,13) (9,12)
W
i
5 4 1.5 2 4 2.5 45 3
[ET
i,
LT
i
] [2,4] [2,5] [3,5] [2,4] [3,7] [3,6] [1,2] [0.5,4]
a 1 1 1 1 1 1 1 1
b 1 1 1 1 1 1 1 1
Table 2: Depot and distribution center data.
Depot 18 19 20 17
Coordination 20,20 75,45 50,8 40,40
Number of model No.1 3 3 3
Number of model No.2 3 3 3
Table 3: Vehicle data.
Vehicle model No 1 2
Q
k
10 8
C
k
0.8 1
E
k
0.4 0.6
V
k
40 40
VEHICLE ROUTING PROBLEM WITH MULTI-DEPOT AND MULTI-TASK
653
Table 4: Vehicle scheduling situation after optimization.
Starting depotS Vehicle model NoK
Number of Vehicles Driving route
1 1 1 14191061
1 2 1 1420151
2 2 1 2418132
2 2 1 2414952
3 2 1 34817123
3 2 1 34117163
Table 5: Vehicle scheduling situation in in stochastic context.
Starting depotS Vehicle model NoK
Number of Vehicles Driving route
1 1 1 1419101
1 2 1 14152061
2 1 1 2459142
2 2 1 2413182
3 1 1 34817123
3 2 1 34167113
Experimental parameters used in simulation
example are as follows: α=1, β=2ρt
0
=1
n=100,Q=100,C=3,Max(P
c
)=10, iteration number is
equal to 50. Finally, we get the total cost that is
425.6. Vehicle scheduling situation can be seen in
Table 4.We also get a stochastic solution in
stochastic selection context, and assume that
stochastic vehicle scheduling situation can be seen in
Table 5. The total cost is 440.92.Simulation results
show that the total cost optimized by the self-
adaptive and polymorphic ant colony algorithm
(APACA) is 425.6, which is less than the total cost
in stochastic selection context that is 440.92. The
results also demonstrate that it is necessary to
optimize the multi-depot, multi-model vehicles and
multi-task vehicle routing problem and that APACA
is an effective and desirable algorithm for solving
the mathematical model.
4 CONCLUSIONS
In the process of city distribution, the optimal
selection of the vehicle route is a key issue to
improve the service quality, reduce the operation
cost and increase the profits. In this paper we present
a new mathematical model of the vehicle routing
problem in the context of city distribution, which
considers multi-depot, multi-model vehicles and
multi-task. The objective function includes three
parts: transport cost, deadheading cost and time cost.
A self-adaptive and polymorphic ant colony
algorithm (APACA) is introduced to solve this
mathematical model. Finally, the effectiveness of the
algorithm is demonstrated by the simulation results.
However, there are some aspects such as target
system, evaluating indicator of route and the
effectiveness of the algorithm that can be improved.
In future studies, more factors such as customer
priority and road condition should be considered
when we establish mathematical models. These will
make the solution more practical and perfect.
ACKNOWLEDGEMENTS
This work is supported by “Dynamic Allocation of
City Logistics Resource Based on the City
Sustainable Development Perspective”, a research
project of the humanities and social sciences of the
Ministry of Education of the People’s Republic of
China (No. 10YJC630324), and supported by
“Allocation of City Logistics Resource”, a project of
the Beijing Municipal Commission of Education
(No. PHR20110877).
REFERENCES
Koehler., 1999. City Logistics in Kassel. City Logistics J:
Letters, 1, 261-271.
Yamada, T., Taniguchi, E., Itoh, Y., 2001. Co-operative
Vehicle Routing Model with Optimal Location of
Logistics Terminals. City Logistics J: Letters. 2, 139-
153.
Renaud, J., Laporte, G., Boctor, F., 1996. A Tabu Search
Heuristic for the Multi-depot Vehicle Routing
Problem, Computers & Operations Research. 23:229–
235.
Yaohuang Guo, Jun li, 1995. A Survey of Logistics
Delivery Vehicle Scheduling. Journal of Xi’an Jiao-
ICEIS 2011 - 13th International Conference on Enterprise Information Systems
654
tong University J: Letters. 30(4), 376-382.
Shengce Hang, Huaizu Li, 1997. The Generalized
Assignment Model and Its Decomposition Algorithm of
Multiple Depot Vehicle Scheduling Problem (MDVSP).
Journal of Xi’an Jiaotong University J: Letters. 31(12),
111-116.
Mingshan Zhang, Xiaowo Tang, 2002. Full Load Vehicle
Routing with Multiple Depots: New Network Flow
Based Algorithm. Joural of Systems Engineering J:
Letters. 17(3), 216-220.
Meijun Chen, Zhisheng,Zhang, Jinfei Shi, 2008. Vehicle
Routing Problem with Multiple Constraints Using
Adaptive and Polymorphic Ant Colony Algorithm.
Journal of Southeast University (Natural Science
Edition). 38(1), 37-42.
Yuan Ding, Yinzhen Li2004. Greedy Sweep Algorithm
of Conveyance Route Optimization in Logistical
Delivery Management Systems. Journal of Lanzhou
Jiao Tong University J: Letters. 23(6), 140-142.
Nan Jia, Yongbo Lv, 2007. Mathematical Model and
Arithmetic of VRP in Physical Distribution Routing.
Logistics Techology J: Letters.26(4), 54-56.
Shiquan Zhong, Guoguang He, 2005. Study on Multi-
depot Vehicle Scheduling Problem with Time
Windows and Multi-type Vehicle Limits and Its Tabu
Search Algorithm. OR TRANSACTIONS J: Letters.
9(4), 67-70.
Meijun Chen, Zhisheng Zhang, Jinfei Shi, 2007. Study on
Intelligent and Polymorphic Ant Colony Algorithm
Based on Multi-depot Vehicle Routing Problem.
Proceedings of the 7th China ITS Annual Meeting, C:
Letters.
Lei Wen, FanHua Meng, 2008. An Improved PSO for the
Multi-depot Vehicle Routing Problem with Time
Windows. IEEE Pacific-Asia Workshop on
Computational Intelligence and Industrial Application .
852-856.
Kallehauge, Brian, 2008. Formulations and Exact
Algorithms for the Vehicle Routing Problem with
Time Windows. Computers and Operations Research
M: Letters. 2307-2330.
Nabila Azi, Michel Gendreau, 2007. An Exact Algorithm
for a Vehicle Routing Problem with Time Windows
and Multiple Use of Vehicles. Computers and
Operations Research M: Letters. 223-237.
VEHICLE ROUTING PROBLEM WITH MULTI-DEPOT AND MULTI-TASK
655