V26136141

Shenglong YU, Xiaofei YANG, Yuming BO, Zhimin CHEN, Jie ZHANG / International Journal
of Engineering Research and Applications (IJERA) ISSN: 2248-9622
www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.136-141
A novel particle swarm Optimization algorithm based Fine
Adjustment for solution of VRP
1
Shenglong YU , 2 Xiaofei YANG , 2Yuming BO, 1 Zhimin CHEN, 1Jie
ZHANG
[1] School of Automation, Nanjing University of Science and Technology, Nanjing 2 10094,PR China,
[2] Shanghai Radio Equipment Research Institute, Shanghai 200090, China

Abstract
To solve the problem of easily trapped and electrical engineer Eberhart in 1995. It
into local optimization and instable is sourced from group behavior theory and
calculation results, a new fine-adjustment enlightened by the fact that bird group and fish
mechanism-based particle swarm optimized group will develop toward a correct and proper
algorithm applicable to solution seeking of direction as anticipated through the special
VRP model is presented in this paper. This information delivery among individuals. It is a
algorithm introduces the fine-adjustment self-adaption random optimization algorithm
mechanism so as to get adapt to the judgment based on population searching. Due to its
base of function directional derivative value. simplicit y, convenience in actualization and
By adjusting the optimal value and group high calculation speed, it has been widel y
value, the local searching ability of algorithm applied in route planning, multi-target
in the optimal area is improved. The tracking, data classification, flow planning and
experiment results indicate that the algorithm decision support etc.. However, PSO also has
presented here displays higher convergence some defects. If the initial state forces the
speed, precision and stability than PSO, and population to converge toward the local
is a effective solution to VRP. optimal area, then the population may fail t o
jump out of the local area and thus be trapped,
making application of PSO in VRP solution
Key words: particle swarm, fine adjustment, become difficult.
VRP, perturbation, convergence In this paper, a fine adjustment is
introduced to PSO, acquiring FT-PSO which
will conduce to enhancement of local
1. Introduction searching of population in the optimal area at
[1]
the end of searching. This method, taking the
Vehicle routing problem was firstl y optimal value of particles as the center, forms
put forward by Dantzing and Ramser in a fine-adjustment area based on the mode
[ 2] defined by the algorithm. In the fine
1959.VRP can be described as : for a series
adjustment area, a fine adjustment particle
of loading and unloading places, to plan
swarm will form to calculate the fitness
proper vehicle route will facilitate smooth pass
function of each fine adjustment particle. The
of vehicle and certain optimal goals can be
particle with the largest function value is
reached by satisfying specific constraints.
selected to replace the optimal value of
Generally the constraint include: goods
updated particles, thus the precision of the
demand, quantity, delivery time, deliver y
algorithm is enhanced.
vehicles, time constraint etc., while the
optimal goals may consist of shortest distance,
lowest cost, the least time and so on. VRP is a 2. Establishment of VRP model
[3] The problem of vehicle route is decried
very important content in logistics as follows: the distribution center is required to
distribution researches. A good VRP algorithm deliver goods to l clients (1, 2, , l ) , and the
and strategy can bring huge benefit to logistics
distribution. cargo volume of the i th client
Particle swarm optimization algorithm is gi (i  1, 2,  l ) , and the load weight of each
[ 4]
(PSO) is an intelligent optimization vehicle is q , at the same time g i  q . Here we
algorithm put forward by
need to find the shortest route which well
satisfies the transport requirements.
American social psychologist Kennedy is In the mathematical model established

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here, cij denotes the transportation cost from
point i to point j , including the distance, time yik  0or1， i  0,1, , l;k  1, 2, , m
 (9)
and expenses etc.. The distribution center is
number as 0, clients numbered i (i  1, 2, , l ) , 3. Particle swarm algorithm
[5]
vehicle numbered k , at most m vehicles can Particle swarm algorithm essentially
be used. After finishing the transportation, the is enlightened by the modeling and simulation
vehicle will return back to the distribution center. research results of many bird populations,
The variables are defining as follows: wherein the modeling and simulation algorithm
1vehiclekdriveto jfromi mainly utilizes the mode of biologist Hepper . It
xijk   (1) is an optimal algorithm based on group
0otherwise intelligent theory that will generate group
1Cargo transportation task of itobefinishedbyk intelligent to guide the optimal searching
yik   (2)
through cooperation and competition of particles,
0otherwise [6]
The vehicle route problem is dealing and has been widely applied now. PSO
with the integer planning with constraints, that is, algorithm can be expressed as: initializing a
the sum of the clients' task on each vehicle route particle swarm with quantity of m, wherein the
can not exceed the capacity of this vehicle. number of iteration n , and the position of the
The total minimal transportation cost after i th particle X i  ( xi1 , xi 2 ,  xin ) , the
finishing the service is:
speed Vi  (vi1 , vi 2 , vin ) . During each iteration,
l l m particles are keeping upgrade its speed and
m i n   c j xi k
Z i j position through individual
i 0 j  0 k 1
extremum P  ( P1 , P 2 , P )
i i i in and global
m l
R  max( gi yik  q, 0) (3) extremum G  ( g1 , g 2 ,  g n ) , accordingly
k 1 i 1 [7 ]
reaching optimization solution seeking . The
where R takes a very large positive number as upgrade equation is
the punishment coefficient.
Supposing all the vehicles used satisfying the
load requirements, then we have the following
vk 1  c0vk  c1 ( pbestk  xk )  c2 ( gbestk  xk ) (10)
expression: xk 1  xk  vk 1 (11)
l where v k denotes the particle speed, x k is the
g y
i 1
i ik  q,k  1, 2, , m (4) position of the current particles, pbestk is the
Giving that only one vehicle is assigned to serve position where particles find the optimal
each customer only once, that is, solution, gbestk is the position of the
optimal solution of the population, c 0 , c1 and
m
1i  1, 2, , l
 yik  mi  0 (5) c 2 denote the population recognition coefficient,
k 1  c 0 usually is a random number in interval (0,1),
In the model the vehicle reaching and leaving
each client are the same, which is expressed as c1 , c 2 is a random number [8] in (0,2).
follows:

l 4 Particle swarm algorithm improvement
x
i 0
ijk  ykj , j  0,1, 2, l ;k (6) 4.1 Mechanism of fine adjustment
According to the flow of PSO, after
calculation of all parties, pbest and a gbest
l

x
will be acquired, wherein the quantity and value
ijk  yki ,i  0,1, 2, l ;k (7)
j 0
of pbest depends on the total number of
Given the variable value constraint as 0 or 1, and population. In fine adjustment, gbest is
the expression is: adjusted as the object of fine adjustment with the
goal of searching the optimal solution in the
xijk  0or1， i, j  0,1, , l;k  1, 2, , m
 (8) neighboring area of gbest .

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Whereas the excellent performance of divides the area and range by the defined method
PSO in fine adjustment principle, we should and forms a fine-adjustment area based on the
make full use of the existing advantages in the mode defined by the algorithm. In the fine
adjustment. Fine adjustment is not necessarily adjustment area, a fine adjustment particle
implemented in each generation unless meeting swarm will form to calculate the fitness function
specific requirements. Directional derivative is of each fine adjustment particle. The particle
intended to be used here as the determinant with the optimal adjustment is selected to
benchmark. Determination methods are compare with Yg and replace the optimal value
introduced as follows:
of updated particles and becomes a new gbest
Given Y the function of X , if its fitness excels Yg .
X  [ x1 , x2 , , xn ] , the directional derivative of
Y at g direction is defined as： 5. Algorithm improvement and VRP
Y solving
Dg Y  (12)
|| X || 5.1 Model improvement
In a specific generation j , the target Particle swarm algorithm is a solution
[9]
function fitness Yg corresponding to the overall algorithm for continuous space, whereas VRP
involves integer planning, thus the model should
optimal value of the population shows the
be improved to construct particles in the
position of gbest in the design space of j algorithm so as to solve the constraint issue.
generation that can be expressed as: Letting X v of particle denotes the number of
gbest j  x1j ex1  x2j ex2 (13)
vehicle serving each task point, X r the
Supposing the population going through p delivery order of this node. Particle fitness
iterations, gbest arrives at the new position, function is the minimal cost value satisfying the
then the displacement vector of these two constraints.
positions are: The key of this paper lies in finding a
proper expression with which the particles
g  ( x1j  p  x1j )u  ( x2j  p  x2 )v (14)
correspond to the model. For this, the particle
The difference of the fitness of these two
objective functions is swarm optimized VRP issue is constructed as a
space with (l  m  1) dimensions
Y  Ygj  Ygj  p (15)
corresponding to l tasks. Numbering of clients
Using Equation A, the directional derivative
of function Yg in j  p th generation is: is represented by natural number, i denotes the
i th clients and 0 denotes the delivery center. For
Y the VRP with l clients and m vehicles,
Dg Y  (16)
|| g || m 1 of 0 are inserted into the client series
After calculation, if Dg Y acquired is which will be then divided into m sections,
larger, it means the population is performing each of which denotes the route of vehicles.
global searching and in a variable situation. Or
otherwise the distribution of population particles Each particle corresponds to a
are gathering in the area where the current l  m 1 -dimensional vector."
population locates for regional searching, while
the movement of population is also slow 5.2 Algorithm procedure
relatively. In this case, the algorithm determines The VRP solution by using the
whether to initiate fine adjustment mechanism algorithm presented here consists of the
by using Dg Y . following procedures:
Step 1: given the proper proportion of random
particle number q , execute the determined
4.2 Operation of fine adjustment
The core concept of fine adjustment is to values p of fine adjustment operation and
f
take gbest as the center. More specifically, it Dcr of directional derivative as well as the

138 | P a g e

particle number n of population.  8
(number 1, 2，，), and the cargo volume of
each task g (unit: ton), loading (or unloading)
Step 2: When c  p  1 , if the directional
time Ti (unit: hour) and the time range
derivative Dg Y of f g is less than or equal to
[ ETi ,LTi ] of each task are given in table 1.
f
Dcr , then implement step (3) and (4) for fine These tasks are to be finished by three vehicles
adjustment, or otherwise if Dg Y is more than with capacity of 8 ton respectively from parking
0. The Distance between the parking and each
f
Dcr , implement step (5). task point and the intervals between these tasks
points are given in Table 2. Supposing the
Step 3: Perform fine adjustment on gbesti , use driving time of vehicle is negatively
equation (17) to generate fine-adjustment proportional to distance, and the average driving
particles that are randomly generated and evenly speed of each vehicle is 50kh/h, than the driving
distributed in the fine-adjustment area centered time from i to j is tij  d ij / 50 . Taking the
by gbesti .
distance between each point as the expense
xtd  gbestd  l  [rand ()  0.5] (17) cij  dij (i, j  0,1, ,8) , and the unit
Step 4: For the fine-adjustment instance punishment of exceeding time window
generated, its fitness function values Yt are is c1  c2  50 , wherein n  80 ； w  1 ；
calculated and arranged in order to select the
optimal find-adjustment instance and the c1  1.5 ；c2  1.5 ，given the maximal iteration
corresponding optimal function fitness. If there number 200, each of the three algorithm runs
is an optimal function fitness in the original 1000 times and the operation results are given in
population, then it can be replaced by the Table 3.
optimal fin-adjustment particles and fitness
function values. Table 1 Cargo volume, loading and unloading
Step 5: Calculate the fitness function value Yi in time and time window at each distribution
point
accordance with the known target function;
Task 1 2 3 4 5 6 7 8
make a comparison between the fitness Yi of numb
each particle at the current position and the er
optimal solution Pbesti , if P  Pbesti , then
i
Freig 2 1, 4. 3 1. 4 2. 3
ht 5 5 5 5
new fitness function value can replace the
volum
previous optimal solution, and the new particles
e
replace the previous ones, e.g., P  Pbesti ,
i ( gi )
xi  xbesti , or otherwise implement step (7). Ti 1 2 1 3 2 2. 3 0.8
Step 6: Compare the optimal fitness Pbesti of 5
ETi ， 1, 4, 1, 4, 3, 2, 5, 1.5,
each particle with the optimal fitness gbesti of 4 6 2 7 5. 5 8 4
LTi 5
all particles, if Pbesti  gbesti , then replace the
optimal fitness of all particles with the one of
Table 2 Distance between the parking and
each particles, at the same time reserve the
each task point and the intervals between
current state of particles, e.g.,
these task points
gbesti  Pbesti ， xbesti  xbesti . 0 1 2 3 4 5 6 7 8
Step 7: use equation (10) and (11) to update the 0 0 40 60 75 90 20 10 16 80
speed and position of particles. 0 0 0
Step 8: determine the end condition, and exit if 1 40 0 65 40 10 50 75 11 10
the condition is satisfied, or otherwise transfer to 0 0 0
Step 2 for operation until the iterations are 2 60 65 0 75 10 10 75 75 75
finished or the preset precision is satisfied. 0 0
3 75 40 75 0 10 50 90 90 15
6. Simulation experiment 0 0
4 90 10 10 10 0 10 75 75 10
Experiment 1: Experimental analysis of
0 0 0 0 0
the example in literature[10] is performed. In
5 20 50 10 50 10 0 70 90 75
this example, there are 8 transportation tasks

139 | P a g e

0 0 0 Figure 1: Optimal solution curve of two
6 10 75 75 90 75 70 0 70 10 algorithms
0 0
7 16 11 75 90 75 90 70 0 10 Experimental results show that the
0 0 0 successful rate of the algorithm presented here is
8 80 10 75 15 10 75 10 10 0 higher than that of other algorithms. While
0 0 0 0 0 enhancing the precisions, it does not show any
significant increase in average searching time.
Taking time and precision into consideration, the
Table 3 Comparison of different results algorithm here is of more practical value.
gained by different algorithm
Algorithm Search Average
success rate search time Experiment 2: inspect the situation
/(%) /s with multiple dimensions. Model and parameter
Basic PSO 72.3 0.34 are referred to literature [8]. Clients are
2
Gen etic 54.7 0.28 distributed in the interval [0,100] and divided
algorithm into high, middle and low ends based on their
Predatory 93.8 0.55
needs, e.g., U (1, 9)  low ，
search
algorithm U (5,15)  middle ，
FA-PSO 96.1 0.39 U (10, 20)  high . Given the anticipation of
n
E[ Di ]
the cargo on the vehicle as f   ,
1250 gQ
i 1
PSO
where g denotes the quantities of vehicles.
1200
FA-PSO
8  10  15
1150 E ( Di )   11 . Letting
3
1100 f '  max{0, f  1} denotes the anticipation of
route failures. Q  11n / (1  f ) , where n
'
cost

1050
represents the number of client nodes to be
1000 served. In the experiment,
f '  0.7 and f '  0.9 . Taking
950
n  20,30, 40,50, 60 respectively, the
900 improvement rate is compared with greedy
algorithm, The simulation result is shown in the
850 following table:
0 20 40 60 80 100 120 140 160 180 200
iteration number
Table 2: Comparison of VRP solution performance by different algorithm
parameters Improvement rate /% Time/s
'
(n, Q, f ) PSO ACO FA-PSO CPSO PSO ACO FA-PSO CPSO
(20,129,0.7) 4.21 4.46 4.81 2.04 1.83 1.82
(20,116,0.9) 4.47 4.72 5.20 1.63 1.51 1.54
(30,194,0.7) 4.13 4.37 4.76 6.52 6.04 6.08
(30,174,0.9) 4.61 4.83 5.28 5.96 5.42 5.41
(40,259,0.7) 5.07 5.38 5.85 16.27 15.11 14.88
(40,232,0.9) 4.68 4.98 5.36 12.76 11.80 11.52
(50,324,0.7) 5.50 5.80 6.43 68.25 62.39 62.33
(50,289,0.9) 5.18 5.52 6.01 62.20 55.94 56.05
(60,388,0.7) 4.76 5.02 5.29 114.64 103.87 104.92
(60,347,0.9) 4.68 4.94 5.14 105.53 95.27 95.98
Average 4.72 5.03 5.40
Seeing from the simulation result, to

140 | P a g e

solve VRP by FA-PSO achieves better effect 507-519.
than ACO and PSO with an average [4] Ying Li, Bendu Bai, Yanning Zhang.
improvement rate of 5.40% by contrast with the Improved particle swarm optimization
primitive solution (greedy algorithm solution). algorithm for fuzzy multi-class SVM.
Besides, regarding the operation time, the Journal of Systems Engineering and
improved algorithm requires the least operation Electronics.2010,21(3): 509-513.
time, showing that the convergence of the [5] Z. Fang, G. F. Tong, X. H. Xu.
algorithm here has been improved in solving
VRP. Particle swarm optimized particle
filter. Control and Decision, 2007.
7. Conclusion
273-277.
Through analysis of the shortcomings
of PSO, a fine adjustment-based algorithm [6] ZHANG W , LIU Y T. Adaptive
FA-PSO, which is suitable for VRP solving, is particle swarm optimization for
proposed here. This algorithm enhances the
effectiveness of initial samples and at the same reactive power and voltage control in
time solve PSO's problem of being easily power systems. Lecture Note in
trapped into local optimization, thereby
improving the global searching ability. The Computer Science , 2005 , 449～452.
experimental result of VRP model indicates that
[7] Bergh F van den, Engelbrecht A P. A
the algorithm presented here features better
speed and precision than the previous algorithm Study of Particle Swarm Optimization
and thus is of good application value in solving
Particle Trajectories. Information
VRP.
Sciences. 2006, 176(8): 937-971.

9. Acknowledgement [8] Halil Karahan. Determining

This paper was supported by the National rainfall-intensity-duration-frequency

Nature Science Foundation of China (No. relationship using Particle Swarm

61104196), the National defense pre-research Optimization. KSCE Journal of Civil

fund of China (No. 40405020201), the Engineering.2012,16(4):667-675.

Specialized Research Fund for the Doctoral [9] Xudong Guo,Cheng Wang, Rongguo

Program of Higher Education of China (No. Yan. An electromagnetic localization

200802881017), the Special Plan of method for medical micro-devices

Independent research of Nanjing University of based on adaptive particle swarm

Science and Technology of China (No. optimization with neighborhood

2010ZYTS051). search.
Measurement.2011,44(5):852-858.
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