Urban traffic flow forecasting using Gauss–SVR with cat mapping, cloud model and PSO hybrid algorithm

Neurocomputing 99 (2013) 230–240

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

Urban traffic flow forecasting using Gauss–SVR with cat mapping, cloud model and PSO hybrid algorithm Ming-Wei Li a,n, Wei-Chiang Hong b, Hai-Gui Kang a a b

Faculty of infrastructure Engineering, Dalian University of Technology, Dalian, Liaoning116024, China Department of Information Management, Oriental Institute of Technology, 58, Sec. 2 Sichuan Rd., Panchiao, 220, Taipei, Taiwan

a r t i c l e i n f o

a b s t r a c t

Article history: Received 13 February 2012 Received in revised form 6 June 2012 Accepted 1 August 2012 Communicated by T. Heskes Available online 9 August 2012

In order to improve forecasting accuracy of urban traffic flow, this paper applies support vector regression (SVR) model with Gauss loss function (namely Gauss–SVR) to forecast urban traffic flow. By using the input historical flow data as the validation data, the Gauss–SVR model is dedicated to reduce the random error of the traffic flow data sequence. The chaotic cloud particle swarm optimization algorithm (CCPSO) is then proposed, based on cat chaotic mapping and cloud model, to optimize the hyper parameters of the Gauss–SVR model. Finally, the Gauss–SVR model with CCPSO is established to conduct the urban traffic flow forecasting. Numerical example results have proved that the proposed model has received better forecasting performance compared to existing alternative models. Thus, the proposed model has the feasibility and the availability in urban traffic flow forecasting fields. & 2012 Elsevier B.V. All rights reserved.

Keywords: Traffic flow forecasting Support vector regression Cat mapping Particle Swarm Optimization Chaos theory Cloud model

1. Introduction Congestion of urban transport network has caused pedestrian time waste, fuel consumption and environment contamination, particularly during daily peak periods. Precise urban traffic flow forecasting is a critical technology to make proper traffic control decision. It may also effectively avoid congestion (saving travel time) and increase efficiency of limited traffic resources (reducing environment contamination) during peak periods. Therefore, urban traffic flow forecasting has received lots of intentions in traffic control operations. The current traffic flow forecasting approaches can be divided into two categories. The first one is based on mathematical determination approach, such as auto regressive moving average model (ARMA) [1] and its improved models (more complicate and with high-accurate-capability) [2], and Kalman filtering model [3]. These models are simple in calculation and fast in speed, however, it fails to reflect uncertainty and nonlinearity in traffic flow process and is unable to handle the rapid variational and complicated process changes underlying of traffic flow [4]. In addition, traffic flow data are often collected at specific locations with constant intervals of time, i.e., the data are often in the

n

Corresponding author. E-mail addresses: [email protected] (M.-W. Li), [email protected] (W.-C. Hong), [email protected] (H.-G. Kang). 0925-2312/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.08.002

form of spatial time series. The empirical results of mentioned approaches have indicated that the problem of traffic flow forecasting in an inter-urban motorway network is multi-dimensional at different times and geographical sites. Due to the multidimensional pattern recognition requests, such as intervals of time, geographical sites, and the relationships between dependent variable and independent variables, non-parametric regression models [5,6] have also successfully been employed to forecast motorway traffic flow. These models are suffered from a dilemma that employed socio-economic factors have insignificant coefficients but strongly affect traffic flow, i.e., eliminating these redundant variables may raise the explanation ability (denoted by R2), but raises the co-linearity problem. This represents a major restriction of econometric models. The second one is the knowledge-based intelligent model, including fuzzy theory model [7], wavelet theory model [8], chaos theory model [9], and artificial neural network (ANN) model [10–12]. The classical representative is ANN model due to its superior performance to approximate any degree of complexity and without prior knowledge of problem solving. ANN model is based on a model of emulating the processes of the human neurological system to determine the numbers of vehicle and temporal characteristics from the historical traffic flow patterns, especially for nonlinear and dynamic evolutions. However, the superiority of the ANN model is based on experienced risk minimization, thus, it may be with insuperable defect when the samples are limited. On the contrary, it will fail into over-fitting

M.-W. Li et al. / Neurocomputing 99 (2013) 230–240

situation when there are more samples, thus, it is also difficult to guarantee the high-precision. In addition, another limitation of the ANN model is difficult to explain the operations, and the training errors of any ANN models are non-convex to difficultly find out the global optimum. To overcome the latter drawback, an up-to-date research tendency is to employ novel evolutionary algorithms or wavelet function with ANN to explore the global optimum in the non-convex solution ranges [13–15]. With the structure risk minimization criterion, support vector regression (SVR) has overcome the inherent defects of the ANN model [16]. It possesses not only greater nonlinear modeling ability, but also several superior advantages, such as theoretically ensuring global optimum, simple modeling structure and processes, and small sample popularization requirement. It has been applied to solve lots of practical problems to receive better solutions, such as small sample, nonlinearity, high dimensionality, and local minima, etc. In addition, it also has successfully been applied to urban traffic flow forecasting [16–19]. However, the original SVR model fails to effectively deal with the error generated by the effect of random factor in traffic flow data series, Zhu et al. [20] propose the urban traffic flow forecasting model based on SVR model with wavelet analysis, they have obtained better forecasting performance in terms of forecasting accuracy. Although wavelet theory has superiority in noise processing, however, the transforming process of wavelet theory is not only complicated to implement, but also time consuming. In consideration of the very characteristic of Gauss function that it can effectively deal with the normally distributed random error (which is also the very feature of normally distributed noise), Wu [21] proposes the SVR model with Gauss loss function (Gauss– SVR) to forecast the product sales data, and has received better de-noised effects, i.e., more accurate forecasting performance. Therefore, this paper will also employ the Gauss–SVR model to conduct urban traffic flow forecasting problem, and use the input historical flow data as the validation data to reduce the random error of the traffic flow data sequence. The practical results indicate that poor forecasting accuracy is suffered from the lack of knowledge of the selection of the hyper parameters in a SVR based model [22]. However, the Gauss–SVR model, even the original SVR model, does not give any useful recommendation to the determination of these parameters, in addition, the common cross validation method also has its certain cross error [23]. Especially in complex forecasting problem, it cannot guarantee high forecasting accuracy level, which seriously affects the generalizations and applications of the SVR model. Therefore, it is deserved to provide more alternative evolutionary algorithms to see each performance to improve the forecasting accuracy. Authors have conducted a series of research by employing different hybrid evolutionary algorithms with chaos theory to determine well the parameter combination. Particle swarm optimization (PSO) algorithm is firstly proposed by Kennedy and Eberhart in simulating the foraging behavior of the natural biotic populations [24–26]. Due to its simple implementation (including parameters determination and easiness-to-realize), parallel processing, and good robustness, the PSO algorithm has received wide attentions and explorations, it also has already been successfully applied in many fields, such as system identification design [27], communication system design [28], constrained optimization [29], process optimization [30], economic management [31], traffic control [32], FIR digital filter design [33], and so on. The PSO algorithm is suitable for those monotonic and unimodal problems, the optimal solution can be found out rapidly. However, the PSO algorithm greatly depends on its initial values, and the swarm diversity is dropped rapidly along with the increasing of the iteration times which makes it been trapped in the local optimum, i.e., premature convergence,

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accordingly, the global search capacity has also been affected. Particularly, as for the high-dimensional multi-modal problems, premature convergence may be appeared easily. Chaos is a ubiquitous phenomenon in the nonlinear system, chaotic behaviors has such characteristics as high sensitivity for initial value, ergodicity, and randomness of motion trail and it can traverse each trail within a certain range according to its rule. Therefore, chaotic variable may be adopted by utilizing these characteristics of chaotic phenomenon for global search and optimizing to increase the particle diversity. Sangwine [34] firstly introduces the chaotic concept to the PSO algorithm; Lv et al. [35], Sheng et al. [36], and Dong and Guo [37] respectively employ the Logistic mapping, the Tent mapping and the An mapping to generate chaotic variable, modified PSO algorithm, and proposed Chaotic PSO (CPSO). The use of chaotic strategy for increasing the swarm diversity improves the deficiency that the PSO algorithm falls into a premature convergence. Proposed by Li et al. [38], the cloud model is an uncertainty conversion model between qualitative knowledge description and quantitative value expression. The cloud model has the characteristics of uncertainty with certainty and stability with variation in knowledge expression, which embodies the basic principle of evolution of natural species. The cloud model has been successfully applied to such fields as intelligent control [39] and data mining [40]. Recently, the cloud model has also been introduced to the PSO algorithm, for example, Wei et al. [41] and Liu and Zhou [42] adjust the inertia weight of the particle with the cloud model and proposed cloud adaptive particle swarm optimization (CAPSO); Zhang and Shao [43] execute variation for the particle with the cloud model, designed new complete cloud particle swarm optimization (NCCPSO) and cloud hyper-mutation particle swarm optimization (CHPSO), and eventually, all receive better performances. To quickly and accurately look for the optimal parameter combination of the Gauss–SVR model, based on the advantages of the ergodicity from the cat mapping function, the superiority of randomness and stability of the cloud model, and with respect to the deficiency of the PSO algorithm, this investigation proposes the chaotic cloud particle swarm optimization (namely CCPSO), which combines the cat mapping function, the cloud model, and the PSO algorithm, with the Gauss–SVR model, namely, the Gauss–SVR–CCPSO urban traffic flow forecasting model. The measured traffic flow data of some road section of Xi’an Road, Dalian City, China, is employed to conduct simulation forecasting, and conduct some necessary comparative analysis with the existed forecasting models. The rest of this paper is organized as follows. Section 2 describes the SVR with Gauss loss function. Section 3 provides the CCPSO hybrid algorithm based on the Cat mapping, the Cloud model and PSO algorithm. Section 4 introduces the proposed Gauss–SVR–CCPSO urban traffic flow forecasting model. Section 5 illustrates a numerical example that reveals the forecasting performance of the proposed Gauss–SVR–CCPSO urban traffic flow forecasting model. The conclusions are given in Section 6.

2. v-support vector regression model with Gauss loss function 2.1. v-support vector regression model with e-loss function Suppose the training set T ¼ ðx1 ,y1 Þ,. . .,ðxi ,yi Þ,. . .,ðxl ,yl Þ, where xi A Rd is d-dimensional input variable, and yi A R is the corresponding output value, i ¼ 1,2,. . .,l. Through a nonlinear mapping function, fðxÞ ¼ fðx1 Þ, fðx2 Þ,. . ., fðxi Þ, the SVR model maps the sample into a high dimensional feature space, Rdf

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+ε 0

ζi*

∑

-ε

a1 *- a1

ai *- ai

K (x1,x )

Φ (x)

K (xi,x )

K (xn,x )

ζi x1

Input space

an *- an

x2

Feature space

xi

xn

Basic structure

Fig.1. Transformation process illustration and basic structure of SVR.

(Fig. 1(a) and (b)), in which the optimal decision function is constructed as follows: f ðxÞ ¼ oT fðxÞ þ b,

o A Rdf , b A R

ð1Þ

where o is weight vector, b is bias value, and fitting function f(x) minimizes the following objective function (structural risk): :o: þ CRemp 2

ð2Þ 2

where the first item 99o99 =2 is the expression for the complexity of the decision function; the second item, empirical risk Remp is for the training errors; C is a regulatory factor used to adjust the ratio 2 between the model complexity 99o99 =2 and the training error    Pl   Remp . The training error Remp ¼ 1=l i ¼ 1 yi f ðxi Þ can be measured with e (Fig.1(b)), the insensitive loss function defined by     cðxi ,yi ,f ðxi ÞÞ ¼ Max 0, yi f ðxi Þe . In the original support vector regression (e-SVR) model, e insensitive factor controls the sparsity of the solutions and the generalization of models. However it is very difficult to reason¨ ably determine the value of e in advance. Therefore, Scholkopf et al. [44] present v-SVR by introducing a parameter v into the eSVR model. At this point, the v-SVR model with the e-insensitive loss function based on are as follows: Min tðo, e, BðnÞ Þ ¼

o,b, e, BðnÞ

8 > < ðoxi þbÞyi s:t: yi ðoxi þ bÞ > : BðnÞ Z0,

l 1 1X 1 2 2 :o: þ Cðve þ ðB þ Bni 2 ÞÞ 2 l i¼12 i

Min tðo, e, BðnÞ Þ ¼

2

Min

establishment of the relationship between slack variables and the loss function and normally distributed noise under the condition that the inequality constraints are not changed, and slack variable is also squared [45]. At this point, the Gauss–SVR model optimization problem using Gauss loss function is as follows:

l 1 1X 2 :o: þCðve þ ðB þ Bni ÞÞ 2 l i¼1 i

o,b, e, BðnÞ

8 > < ðoxi þ bÞyi s:t: yi ðoxi þbÞ > : BðnÞ Z 0, ðnÞ

ð3Þ

n

Min oða, an Þ ¼ n a, a

n

l 1 X ðan ai Þðanj aÞKðxi ,xi Þ 2 i,j ¼ 1 i



where BðnÞ ¼ ðB1 , Bn1 , B2 , Bn2 ,. . ., Bl , Bnl Þ (Fig. 1(b)) is a slack variable, o is the d-dimensional row vector, CðC Z0Þ is a penalty coefficient, deciding the balance between confidence risk and experience risk; v A ½0,1 is the upper bound of the proportion of error samples in the total number of training samples and the lower bound of the proportion of support vectors in the total number of training samples; unlike standard SVR, e is present as the variable of optimization problem, and its value will be given as part of the solution.

The original SVR model with e-insensitive loss function cannot deal with the random error (white noise) in normal distributed prediction sequences; so the original SVR theoretically does not guarantee the accuracy of time series prediction problems containing white noise. The Gauss function is in accord with the characteristics of normally distributed noise, thus, it can minimize the effects of normally distributed noise as the loss function of SVR to a certain extent. LSSVR uses the e-insensitive function as the loss function, uses the sum of the squares of the slack variables and changes the inequality constraints into equality constraints, which aims to simplify the solution of the SVR model. The Gauss–SVR model is the

n

 Step 1: Suppose the known training set T ¼ ðx1 ,y1 Þ,. . ., d ðx1 ,y1 Þ,. . .,ðx1 ,y1 Þg, where xi A R ,yi A R, i ¼ 1,2,. . .,l. Step 2: Select the appropriate positive v and C, and the kernel function Kðxi ,xi Þ. Step 3: Construct and solve the optimization problem, and the basic structure is shown in Fig.1(c);

e Z0

2.2. v-support vector regression model with Gauss loss function

ð4Þ

eZ0

where z ¼ ðz1 , z1 , z2 , z2 ,. . ., z1 , z1 Þ is the slack variable; o is the ddimensional row vector; CðC Z0Þ is the penalty coefficient; v value range is [0,1]; e is present as optimization variable; its value will be given as part of the solution. Yan and Xu [23] make a detailed proof on the existence and uniqueness of the Gauss–SVR model solution. The calculation steps for the Gauss–SVR model are as the follows:

r e þ Bi r e þ Bni

r e þ Bi r e þ Bni

l X

ðani ai Þyi þ

i¼1

l 1 X ðan2 þ a2i Þ 2C i ¼ 1 i

8 l X > > > ðani ai Þ ¼ 0 > > > > i ¼ 1 > < l s:t: X > ðani þ ai Þ r vC > > > > i ¼ 1 > > > : 0 r ai , ani r Cl , i ¼ 1,2,. . .,l

ð5Þ

Get the optimal solution aiðnÞ ¼ ða1 , a2 ,. . ., al ; an1 , an2 ,. . ., anl Þ. Step 4: Construct decision function f ðxÞ ¼

l X

ðani ai ÞKðxi ,xÞ þ b

ð6Þ

i¼1

where b is calculated as follows; select the two components aj or

ak in the open interval ð0,C=lÞ, then

b¼

" !# l l X X 1 yj þ yk  ðani ai ÞKðxi ,xj Þ þ ðani ai ÞKðxi ,xk Þ 2 i¼1 i¼1

ð7Þ

M.-W. Li et al. / Neurocomputing 99 (2013) 230–240

Parameter e can be calculated by the following two equations:

e¼

l X

ðani ai ÞKðxi ,xj Þ þ byi or e ¼ yk 

i¼1

l X

ðani ai ÞKðxi ,xk Þb

ð8Þ

i¼1

The parameter selection in the Gauss–SVR forecasting model determines the generalization performance of the model, but there is cross error in traditional cross validation [23]. In view of this, in order to select the optimal parameter combination of the Gauss–SVR model precisely, this paper combines the cat mapping function, the cloud model and the PSO algorithm to present the hybridized CCPSO algorithm. 3. The modified Particle Swarm Optimization

233

optimization process is as follows, when the PSO algorithm operates a certain generations mix_gen*gen, the individual optimal position, PGg , of each particle will form a new swarm and sequence according to the fitness value of PGg ; the new P Gg swarm is divided into pop_distrnpop_size better individuals and (1 pop_distr)npop_size poorer individuals; for pop_distrnpop_size better individuals, execute local search with the cloud model to accelerate the algorithm search speed; for (1 pop_distr)npop_size poorer individuals, implement global chaotic disturbance with the cat mapping function to increase the swarm diversity, and finally mix the obtained new pop_distrn pop_size best individuals after local search and new (1 pop_distr) npop_size best individuals after chaos disturbance to form a new swarm, and continue to execute evolution operation of the PSO algorithm.

3.1. The standard Particle Swarm Optimization 3.2.1. Global disturbance with the cat mapping The PSO algorithm is originated in the studies of bird or fish group behaviors. Similar to evolutionary computational concepts, the PSO algorithm uses a set of particles, representing potential solutions to the considered problem. Each particle has its position,     xi ¼ xi1 ,xi2 ,. . .,xiQ , a velocity vi ¼ vi1 ,vi2 ,. . .,viQ , where i ¼ 1, 2,. . .,N, and moves through a Q-dimensional search space. Denote that the individual optimal position of the ith particle as P Gi ¼   Pi1 ,P i2 ,. . .,P iQ , and the global optimal position of the swarm as   P Gg ¼ Pg1 ,Pg2 ,. . .,PgQ , G is denoted as the iteration number. According to the global variant of the PSO algorithm, each particle moves towards its individual optimal position PGi and towards the global optimal position P Gg in the swarm. The change of position of each particle from some iteration to another can be computed according to the distance between the current position and its individual optimal position and the P Gg distance between the current position and the global optimal position PGg of swarm. Then, the updating of velocity and particle position can be obtained by using the following equations: vGidþ 1 ¼ wvGid þ c1 r 1 ðP Gid xGid Þ þ c2 r 2 ðP Gid xGid Þ Gþ1 xGidþ 1 ¼ xGid þvid

3.2.1.1. Analysis on chaotic characteristics of the cat mapping. Chaos optimization method is an optimization technique which presents in the last few years and applies such properties as chaotic ergodicity and initial value sensitivity as a global optimization mechanism. However, the current PSO algorithm based on chaos optimization method mostly adopts the Logistic mapping function, the Tent mapping function and the An mapping function as the chaotic sequence generator. By introducing the analysis on chaotic characteristics of these three different mapping functions and the cat mapping function, this paper will determine to employ the cat mapping function, with good ergodic uniformity and not easily to fall into minor cycle, to the PSO algorithm. Analysis on chaotic characteristics of these four mapping functions is shown as follows. For two-dimensional Cat mapping function [46] it is shown as follows: ( xn þ 1 ¼ ðxn þ yn Þmod 1 ð11Þ yn þ 1 ¼ ðxn þ 2yn Þmod 1

ð9Þ ð10Þ

where w is called inertia weight and is employed to control the impact of the previous history of velocities on the current one. Accordingly, the parameter w regulates the trade-off between the global and the local exploration abilities of the swarm. A large inertia weight facilitates global exploration, while a small one tends to facilitate local exploration. A suitable value of the inertia weight w usually provides balance between global and local exploration abilities, and consequently, results in a reduction of the number of iterations required to locate the optimum solution, c1 and c2 are the learning factors, usually take c1 ¼ c2 ¼2, r1 and r2 are random numbers uniformly distributed in the range [0,1]. The swarm initialization process in the PSO algorithm is random, even though the random process can mostly guarantee initial solution population distributed uniformly, however, it cannot guarantee the initial swarm includes optimal individual, particularly in the late evolution period. Each particle updates its own speed and position via the individual optimal position, PGg , and the global optimal position, P Gg . If some particle searches some local optimal solution, each particle is easy and rapid to aggregate nearby due to the attraction of this optimal solution. ‘‘Aggregation’’ will make the population lose diversity, fall into local optimal solution, and affect the global search of the algorithm. 3.2. Chaos Cloud Particle Swarm Optimization The CCPSO algorithm is proposed with respect to the deficiency that the particle diversity weakens and the convergent speed slows in the late convergence of the PSO algorithm. The hybrid

where x mod1 ¼ x½x. For the Logistic mapping function [35] it is shown as follows: xn þ 1 ¼ mxn ð1xn Þ

ð12Þ

where xn is the iteration value of the variable x in the nth time; m is a control parameter; when m ¼4, the system will be completely in a chaos state, and x0 may take any initial value in (0,1) except 0.25, 0.5 and 0.75. For the Tent mapping function [36], it is shown as follows: ( x A ½0,0:5 2xn xn þ 1 ¼ ð13Þ 2ð1xn Þ x A ð0:5,1 where xn is the iteration value of the variable x in the nth time and n is the iteration times. For the An mapping function [37], it is shown as follows: (3 1 2 xn þ 4 x A ½0,0:5Þ xn þ 1 ¼ 1 ð14Þ 1 x  x A ½0:5,1 2 n 4 where xn is the iteration value of the variable x in the nth time and n is the iteration times. Set the initial value of these four mapping functions as 0.1, respectively, and set the iteration times as 50,000. Then, record the occurrence number of the obtained chaotic variable for each mapping function. The statistical results for each mapping function are as shown in Fig. 2: From Fig. 2, the value of the probability density for the chaotic sequence generated by the Logistic mapping function is mostly distributed at both ends due to complying with Chebyshev distribution of more at both ends and less in middle; the chaotic sequence generated by the Tent mapping function is affected by

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limited word length and limited precision of the computer and is rapid to fall into a minor cycle or a fixed point; the number of variables generated by the An mapping function gradually reduces along with variable values changing from small to large; however, different from above three mapping functions, the distribution of the Cat mapping function is relatively uniform in the interval [0,1], and has no cyclic phenomenon during its iteration process. Therefore, the Cat mapping function has better chaotic distribution characteristic, and its application in chaos disturbance of the PSO algorithm can more excellently strengthen the swarm diversity. 3.2.1.2. Global disturbance with the cat mapping function. Employ the cat mapping function to implement global disturbance, generate (1 pop_distr)*pop_size new particles and sequence according to the fitness value, meanwhile, mix with (1pop_distr)*pop_size poorer particles. Then, select the first mixed (1pop_distr)*pop_size best particles to replace the original (1pop_distr)*pop_size poorer particles of the new PGi swarm, and complete the global chaos disturbance, with the specific procedure as follows: Step 1: Employ the Cat mapping function to generate (1 pop_distr)*pop_size chaotic variables with different trails xi ¼ xi1 ,xi2 ,. . .,xiQ , i¼1,2,ab(1  pop_distr)*pop_size. Step 2: Map all the components of xi according to Eq. (15) to the value interval ½g jmin ,g jmax  to obtain gi. g i ¼ g jmin þðg jmax g jmin Þxi

for some iteration time during the searching process of the PSO algorithm, based on the social statistics principle [43], there are more advantages available around the local optimal points, this implies that it has more opportunity to find the optimal solution around them. Therefore, in order to improve the convergent speed of the PSO algorithm, this paper employs the normal cloud model to complete local search of the pop_distr*pop_size better individuals of the new pGi swarm. Suppose T be the language value in the domain u, mapping CTðxÞ : u-½0,1, 8x A u, x-CTðxÞ, then the distribution of CT(x) on u is called the membership cloud under T. In case of obeying the normal distribution, CT(x) is known as the normal cloud model [38]. The overall characteristics of the cloud model can be represented by the three digital features including desired E, entropy S, and hyper entropy H. The schematic for the normal cloud model is shown in Fig. 3. E is the expectation of spatial distribution of cloud droplet in the domain as well as the point that is the most able to represent the qualitative concept; S represents the measurable granularity of the qualitative concept, and the greater the entropy S is, usually the more macro the concept is. H is the uncertain measurement of entropy, and is jointly determined by the randomness and fuzziness of entropy S. In case of knowledge representation, the cloud model has the characteristics of the uncertainty with

ð15Þ

E=10,H=0.1,S=1,n=1500

where g jmin and g jmax are, respectively, the minimum value and the maximum value of the jth vector of gi, j ¼1,2,y,Q. Step 3: Calculate the value of the objective function, f(gi), according to gi; mix the (1 pop_distr)*pop_size new particles with (1 pop_distr)*pop_size original poorer particles and sequence according to the fitness value. Then, replace the (1 pop_distr)*pop_size original poorer particles with (1 pop_distr)*pop_size best particles ranked ahead in the fitness value.

1.0 0.8

0.4 0.2 0.0

3.2.2. Local search with the cloud model

7

8

9

10

11

1400 1200 1000 800 600 400 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

Logistic Map value

0.4 0.6 0.8 Tent Map value

1.0

560

900 800 700 600 500 400 300

Times

540 520 500 480 460 440 0.0

0.2

0.4

0.6

0.8

1.0

12

Fig. 3. Schematic of the normal cloud model.

Times

Times

3500 3000 2500 2000 1500 1000 500 0

E 6

3.2.2.1. The cloud model. In consideration of the relative distance between the current value and the optimal solution after evolving

Times

H

3S

0.6

0.0

0.2

An Map value Fig. 2. Iterative distribution of four mapping.

0.4

0.6

Cat Map value

0.8

1.0

13

14

M.-W. Li et al. / Neurocomputing 99 (2013) 230–240

235

certainty and the stability with change, and reflects the basic principle of the evolution of species in nature. For the cloud model parameters, E represents the parent’s good individual genetic characteristics, and is the offspring’s inheritance from the parent; entropy S and hyper entropy H indicates the uncertainty and fuzziness of inheritance process, showing the mutation characteristics of species in the evolutionary process. The algorithm or hardware for the generation of cloud droplets is called cloud droplet generator [38]. This paper applies normal cloud generator to realize local search of the better individuals. 3.2.2.2. Calculation steps of local search with the cloud model Step 1: Assume the individual optimal position of the ith particle in the swarm is P Gi ¼ P i1 ,P i2 ,. . .,Pij ,. . .,P iQ , i¼ 1,2,y,pop_distr* pop_size; j¼1,2, is Q. Step 2: Set i¼1 and j ¼1, let Ej ¼ PGi , Sj ¼ Hj ¼ ðg jmax g jmin Þ=G. Then, generate Hnj , which is the normal random number with the expectation value of Sj and the standard deviation of Hj . Step 3: Generate PGi n , which is the normal random number with the expectation value of Ej and the standard deviation of Hnj , if PGi n A ½g jmin ,g jmax , go to Step 4, otherwise, repeat Step 3, until satisfaction. Step 4: Set j¼j þ1, until the local search of Q-dimensional space for the current optimal position pGi of the ith particle is completed. Step 5: Calculate the objective function f ðPGi n Þ, if f ðpGi n Þ is superior to f ðP Gi Þ, then, let P Gi ¼ P Gi n . Step 6: Set i¼iþ1, repeat Steps 2–6, until the local search of the pop_distr*pop_size better particles of the new PGi swarm completes. During the evolution of particle swarm, with the escalation of iteration number, the individual optimal position PGi of each particle will gradually go closed to the global optimal position, P Gg . This paper enables niche search space to be gradually decreased with the escalation of iteration number by introducing the current iteration number G to Si and Hi, and thereby realizes the automatic control for the size of niche search space and conducts the local search better.

4. The forecasting processes of the Gauss–SVR–CCPSO model The mean absolute percentage error (MAPE) of the Gauss– SVR–CCPSO model is used as the fitness function and expressed as    N ^ 100 X f i ðxÞ-f i ðxÞ ð16Þ Fitness ¼ MAPEð%Þ ¼    f i ðxÞ  N i¼1

where N is the number of forecasting samples, f i ðxÞ is the actual value of the ith period; and f^ i ðxÞ is the forecasting value of the ith period. Considering the good performance of the radial basis in the application of SVR [16,19,22], this paper employs the radial basis function as the kernel function of the Gauss–SVR model, the radial basis function is expressed as ! 2 :xi xj : Kðxi ,xj Þ ¼ EXP  ð17Þ 2s2 The procedure of the Gauss–SVR–CCPSO urban traffic flow forecasting model is illustrated as the followings and the flowchart is shown in Fig. 4. Step 1: Normalize the data sets according to Eq. (18), and set the size of particle swarm, pop_size, the acceleration parameters, c1 and c2, the maximum evolution generation, gen, the

Fig. 4. The flowchart of the Gauss–SVR–CCPSO model.

population distribution coefficient, pop_distr, the hybrid control parameter, mix_gen, and the minimum fitting residual error emin . xðiÞ ¼

xðiÞxmin xmax xmin

ð18Þ

x(i) is the value of the ith parameter; xmin and xmax are the minimum and the maximum value of the parameters, respectively. Step 2: Randomly generate three pop_size particles, xi ¼ C i ,vi , di , i¼1,2,y,pop_size, in the feasible intervals, [Cmin, Cmax], [vmin, vmax], and [dmin, dmax]. In addition, randomly initialize the speed and update the individual optimal position, P Gi and the global optimal position, P Gi . Step 3: Set each particle xi ¼ C i ,vi , d as the parameter combination of the Gauss–SVR model, for all i¼1,2,y,pop_size, input the parameter combination to formulate the Gauss–SVR model, then, calculate the fitness function value, f(xi). Step 4: Set G ¼1, and GG ¼1. Step 5: If the current swarm satisfies the stopping criteria R, go to Step 9, otherwise go to Step 6, the evolution stopping criteria R adopts the combination of maximum evolution generation, gen, and the minimum fitting residual error, emin. Step 6: If GG rmix_genngen, go to Step 7, otherwise go to Step 8. Step 7: Calculate the adaptive inertia weight factor $ according to Eq. (19). Update the speed and position of each particle in particle swarm according to Eqs. (9) and (10). Then, calculate the fitness values of all the particles, and update the individual optimal position P Gi and the global optimal position P Gi of pop_size particles, and set G ¼G þ1, GG ¼GGþ1, go back to Step 5.

$ ¼ $max Gn

$max $min gen

ð19Þ

where $ is the updated inertia weight, $min and $max are the minimum value and the maximum value of inertia

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M.-W. Li et al. / Neurocomputing 99 (2013) 230–240

specially, and the traffic flow of the road section is necessary to be subject to the effect of traffic flow of the upstream road section. Therefore, the traffic flow of the first N periods of the road section and the traffic flow of the current period of M upstream road sections may be used to conduct rolling forecasting for the traffic flow of the next period of this road section. Let t express the current period, Y(tþ 1) be the traffic flow of (tþ1) period, then, Y(tNþ1),y, Y(t 1), Y(t) represent respectively the traffic flows of the first N periods of this road section in traffic flow sequence, and Y1(t), Y2(t),y,YM(t) are respectively the traffic flows of t period of M upstream road sections. The specific rolling training traffic flow sequence is shown as Fig. 5. This paper employs the traffic flow data from Xi’an Road, Dalian City in north direction as the forecasting example. There are three upstream relevant road sub-sections in the forecasting road section. Therefore, this paper adopts four speed spy (SI3 K band) radar traffic flow detectors to obtain traffic flow data. The data collection time periods include the morning peak period (from 6:30 to 9:30) and the evening peak period (from 17:00 to 20:00), from 02 May 2011 to 12 June 2011. The unit of data sampling time period is 1 h, and there are 156 (hrs) effective traffic flow data from the morning peak period and the evening peak period, respectively. For convenience, the traffic flow data are converted to equivalent of passengers (EOP). In addition, traffic flow data are divided into three parts: training data,

weight, respectively. In general, they are set as 0.4 and 0.9, correspondingly. Step 8: Form the individual optimal position PGi of each particle to a new swarm and sequence according to the fitness value of the particle. Divide the whole swarm into pop_distr*pop_size better individuals and (1 pop_distr)*pop_size poorer individuals, in which, the global chaos disturbance of (1 pop_distr) npop_size poorer individuals are also executed to obtain new (1 pop_distr)*pop_size best individuals according to Section 3.2.1; the local search of pop_distr*pop_size better individuals is completed to obtain new pop_distr*pop_size best individuals according to Section 3.2.2. Then, form an elite swarm with the swarm quantity of pop_size, and calculate the fitness value of pop_size elite individuals, set G¼Gþ1, GG¼0, go back to Step 5. Step 9: Output the global optimal position PGg ¼ C,v, d, set P Gg as the parameter combination of the Gauss–SVR model, input P Gg , output the forecasting value.

5. Numerical examples Urban traffic flow is fluid and continuous in time distribution. The traffic flow of the next moment of road section and that of the previous several periods on this road section have inevitable relationships. Meanwhile, the road section is a part of road traffic network

Fed-out value

Fed-in value First Y(t-N+1) Y(t-N+2) Y(t-N+3) … Y(t-2) Y(t-1) Y(t) rolling N Y1(t) Y2(t) Feed into Gauss-SVR-CCPSO … YM(t) Second Y(t-N+1) Y(t-N+2) Y(t-N+3) … rolling

Y(t+3)

Y(t+1)

Y(t+2)

Y(t+3)

Feed into Gauss-SVR-CCPSO

Y(t+2) forecast

Y(t-1)

Y(t)

Y(t+1) forecast

N

Y1(t+1) Y2(t+1) … YM(t+1) Third rolling Y(t-N+1) Y(t-N+2) Y(t-N+3) … Y1(t+2) Y2(t+2) … YM(t+2)

Y(t+2)

Y(t-2)

Y(t+1)

Y(t-2)

Y(t-1) N

Y(t)

Y(t+1)

Y(t+2)

Feed into Gauss-SVR-CCPSO

Fig. 5. The rolling-based forecasting procedure (training).

Fig. 6. Data grouping among the three data subsets.

Y(t+3)

Y(t+3) forecast

M.-W. Li et al. / Neurocomputing 99 (2013) 230–240

5.1. Optimization performance analysis of the CCPSO algorithm In order to test the optimization capacity of the proposed CCPSO algorithm in determining the parameter combination of the Gauss–SVR model, this paper employs the morning peak period traffic flow data as example, by taking the first 5 periods flow data in the road section and 3 upstream relevant road sections’ flow data of each current period as influencing factor set. Meanwhile, four PSO-series improved algorithms, including PSO [32], CPSO [35], CHPSO [43], and CCPSO are applied to optimize the parameter combination of the Gauss–SVR model. To avoid the inter-affects of algorithm’s parameters, all parameters except that the parameter is required to be set specially in the four employed algorithms should be treated as the same criteria. The MAPE of the Gauss–SVR model, as mentioned above, is used as the fitness function in comparing the superiority among these four algorithms. The ranges of these three parameters in the Gauss–SVR model are set as, C A ½0:01,10000, v A ½0:01,1, and d A ½0:01,1. The same parameters in the four algorithms are set as, pop_size ¼50, gen¼200, c1 ¼ c2 ¼2.0. For specially set parameters, mix_gen and pop_distr, in the CCPSO algorithm, the analysis indicates that the performance of CCPSO algorithm is the best when mix_gen¼0.2 and pop_distr¼0.5. Finally, the mean optimal solution curves, for 50 times randomly, in determining the parameter combination of the Gauss–SVR model by these four algorithms, are shown as Fig. 7. Fig. 7 implies that, firstly, the mean optimal solution of the initial swarm of the CPSO algorithm is superior to other compared alternative algorithms but rapidly falls into the local extremum point with the increase of iteration number; secondly, during the search process, even though CHPSO algorithm can continuously generate better individuals according to cloud local search, it still also rapidly fall into the local extremum point due to diversity deficiency; thirdly, the evolution curve of CCPSO algorithm in initial evolution is consistent with that of PSO algorithm, but with frequent mixing of the cat mapping function and the cloud model, the swarm diversity is increased and the convergence speed is accelerated, CCPSO has obvious advantage in convergence curve shape.

5.2. Predictive effect analysis of the Gauss–SVR–CCPSO model In order to determine the optimal input number, n, of the historical relevant flow, this paper respectively takes n¼ 5, 10, 15, 20, 30, 45, 65, 95, 110 to consist of influencing factor set with the 3 upstream relevant road sections’ flow data of each current period, to respectively optimize the parameter combination of the Gauss–SVR model. So as to obtain the optimal parameter combination with respect to different n values, utilize the obtained parameter combination to forecast validation data, calculate the MAPE of validation data, and finally obtain the parameter combination with respect to different n values and its corresponding MAPE of validation data, as shown in Table 1. From Table 1, the optimal input number n of historical relevant flow of the morning peak period is 20, and the corresponding optimal parameter combination is C ¼256.4, v ¼0.84, and d ¼ 0.63; the optimal input number n of historical relevant flow of the evening peak period is 30, and the corresponding optimal parameter combination is C¼ 282.6, v ¼0.87, and d ¼0.49. To analyze the forecasting performance of the Gauss–SVR– CCPSO model, the ARMA model, the BPNN model, and the SVR– CCPSO model are also selected to forecast the morning peak and the evening peak traffic flow. Considering the fact that the increase of modeling time may also improve the optimization results, the maximum modeling time of each model should be guaranteed as the same. To evaluate the forecasting performances of each algorithm, except MAPE mentioned above, the root mean square error (RMSE) is also employed as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X RMSE ¼ t ðf ðxÞf^ i ðxÞÞ2 Ni¼1 i

ð20Þ

8.0 7.5

PSO

CPSO

CHPSO

CCPSO

7.0 6.5 MAPE(%)

validation data and testing data. For the ratio of validation data to training data, it is recommended by Schalkoff [47] to be approximately one to four, therefore, for both peak periods, the training data set, validation data set and testing data set are set as 112 (4 weeks), 28 (1 week), and 16 (Monday, Wednesday, Friday, Sunday), respectively. These three divided data subsets are shown as Fig. 6. Numerical experiments have been implemented in Matlab 7.1 programming language with a 1.80 GHz Core(TM)2 CPU personal computer (PC) and 2.0G memory under Microsoft Windows XP professional.

237

6.0 5.5 5.0 4.5 4.0 3.5 3.0 0

50

100

150

200

Evolution algebra Fig. 7. The mean optimal solution curves of the four algorithms.

Table 1 Forecasting results and associated parameter combination of the Gauss–SVR–CCPSO models. Morning peak period No. of input data

5 10 15 20 30 40 50 60 70

Evening peak period Parameters

MAPE (%)

C

v

d

193.8 267.5 612.5 256.4 365.2 423.5 189.2 657.8 368.1

0.93 0.98 0.62 0.84 0.87 0.92 0.82 0.91 0.84

0.56 0.48 0.37 0.63 0.47 0.82 0.76 0.83 0.72

6.75 6.38 5.67 5.18 5.76 5.32 6.04 6.41 6.48

No. of input data

5 10 15 20 30 40 50 60 70

Parameters

MAPE (%)

C

v

d

256.3 367.1 420.6 397.5 282.6 367.5 521.9 416.5 358.4

0.56 0.68 0.76 0.42 0.87 0.92 0.79 0.64 0.88

0.42 0.51 0.65 0.72 0.49 0.46 0.61 0.63 0.50

4.78 5.96 6.02 5.31 4.93 5.98 6.87 6.56 5.82

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M.-W. Li et al. / Neurocomputing 99 (2013) 230–240

where N is the number of forecasting samples, f i ðxÞ is the actual value of the ith period; and f^ i ðxÞ is the forecasting value of the ith period. After well training, ARMA, BPNN, SVR–CCPSO and Gauss–SVR– CCPSO models are applied to traffic flow forecasting during the morning and the evening peaks. Tables 2 and 3, respectively, show the actual values, the forecast values of these four models, and the forecasting accuracy indices in terms of RMSE and MAPE in the morning and the evening peaks. Figs. 8 and 9 illustrate the forecasting results of different models in the morning and the evening peaks, respectively. As shown in Tables 2 and 3, the prediction performance of the ARMA model is the worst among these models, this also indirectly implies that the sample data are of nonlinear characteristic. The forecasting methods based on SVR are all with higher precision than the BPNN model; moreover, the proposed Table 2 The morning peak period traffic flow forecasting results (unit: EOP). Peak periods

606630(Mon) 606730(Mon) 606830(Mon) 606930(Mon) 608630(Wed) 608730(Wed) 608830(Wed) 608930(Wed) 610630(Fri) 610730(Fri) 610830(Fri) 610930(Fri) 612630(Sun) 612730(Sun) 612830(Sun) 612930(Sun) RMSE MAPE (%)

Actual value

4389.7 6709.8 7707.5 5460.2 5764.3 7703.3 7048.6 4395.1 5010.1 6568.1 7428.6 4536.8 5405.9 6686.9 7005.4 4978.6

Forecasting value ARMA

BPNN

SVRCCPSO

Gauss-SVRCCPSO

9537.8 6872.6 2062.0 5901.8 4771.9 10,621.8 6878.0 9015.1 2566.3 6677.3 5932.1 9119.3 7057.7 7353.4 11,019.5 3169.3

5281.2 7426.4 9497.6 5981.9 6228.8 8059.8 6485.8 5442.6 6686.3 7362.1 8708.5 3772.3 4244.8 7061.6 5878.9 3871.9

5243.2 6949.2 8041.9 5616.4 5207.9 7329.8 7406.6 4458.7 5328.8 6675.6 7150.1 4092.4 4750.2 6822.3 6593.1 5465.1

4765.8 6938.5 7412.4 5506.5 6066.4 8043.8 6846.0 3675.5 5088.7 6595.6 7505.6 4832.9 5360.5 6560.9 6931.8 4928.9

2971.2 1005.8 41.84 15.83

414.3 6.41

271.1 4.24

‘‘606630’’ denotes the 6:30 on 6 June 2011, and so on.

Gauss–SVR–CCPSO model has smaller RMSE and MAPE values than the SVR–CCPSO model.

6. Conclusions Real-time and accurate forecasting of the urban traffic flow can effectively save travel time, ease road congestion, minimize environmental contamination and conserve limited energy. This paper proposes the Gauss–SVR–CCPSO urban traffic flow forecasting model, based on CCPSO for parameter determination, conducts an example analysis for optimization capability of CCPSO algorithm, and provides the superior forecasting performance in urban traffic flow management. The comparison result shows that, (1) the hybrid CCPSO algorithm, based on the cat mapping function, the cloud model and the PSO algorithm, enhances its global exploration and local development capacity, in the meanwhile, it also improves the deficiency that causes the PSO algorithm easily to fall into local extremum and slow convergence speed when optimizing multi-modal complex problems, and finally, it reveals a novel and effective approach for solving high-modal complex problems; (2) the Gauss–SVR–CCPSO model, based on the hybrid CCPSO algorithm for parameter optimization, overcomes the difficulty of parameter combination determination in the SVR-based models, and improves the generalization capacity and self-study capacity of the SVR-based models; (3) the SVR-based models can well solve nonlinearity and wave property (fluctuation) problems and can overcome the deficiency of poor global search and easy to converge to local minima in the BPNN model; the introduction of the Gauss loss function enhances the capacity in dealing with normally distributed random error from traffic flow sequence and hence improves the forecasting accuracy; (4) in the forecasting process, this paper utilizes validation data to determine the suitable input number n and its corresponding parameters, which effectively reflects the periodicity in historical relevant traffic flow; (5) forecasting accuracy of the Gauss–SVR–CCPSO model is superior to other compared alternative models, it significantly improves the urban traffic flow forecasting accurate level, in addition, the model parameters can also be updated continuously according to real-time traffic flow data. Finally, the certification on CCPSO

Table 3 The evening peak period traffic flow forecasting results (unit: EOP). Peak periods

60617(Mon) 60618(Mon) 60619(Mon) 60620(Mon) 60817(Wed) 60818(Wed) 60819(Wed) 60820(Wed) 61017(Fri) 61018(Fri) 61019(Fri) 61020(Fri) 61217(Sun) 61218(Sun) 61219(Sun) 61220(Sun)

Actual value

5778.6 7990.9 6024.4 4501.1 4567.3 6193.4 7752.5 5646.3 4917.7 7986.9 7748.6 5710.8 4579.7 7857.2 6563.5 5575.3

RMSE MAPE (%) ‘‘60617’’ denotes the 17 o’clock on 6 June 2011, and so on.

Forecasting value ARMA

BPNN

SVR-CCPSO

Gauss-SVR-CCPSO

2627.1 13,893.4 3162.5 9066.8 3720.5 1504.1 4084.9 8289.2 8672.3 12,410.6 3080.5 8615.6 1866.1 3907.1 13,667.9 1362.1

3089.3 9558.5 6197.4 5968.9 3751.9 6968.3 8495.2 6951.2 5147.6 7305.6 8200.4 7429.7 3855.5 6432.6 8175.1 6007.9

6252.2 7969.2 5472.5 4302.6 4782.8 7496.9 7532.2 5753.9 5568.4 7398.3 7887.4 6133.8 4336.3 8147.2 6401.5 5438.9

5574.8 7369.7 6266.1 4859.8 4103.3 6457.2 7889.9 5487.1 5119.5 8325.3 8042.1 6018.9 5042.3 8470.9 6522.2 5367.6

4123.4 62.62

1167.6 17.51

469.8 6.02

346.0 5.13

M.-W. Li et al. / Neurocomputing 99 (2013) 230–240

Forecasting value

12000

Actual

ARMA

BPNN

SVR-CCPSO

239

Gauss-SVR-CCPSO

10000 8000 6000 4000 2000 606630(Mon)

608630(Wed)

610630(Fri)

612630(Sun)

612930(Sun)

Time Fig. 8. The morning peak period traffic flow forecasting results.

16000 Actual

ARMA

SVR-CCPSO

BPNN

Gauss-SVR-CCPSO

Forecasting value

14000 12000 10000 8000 6000 4000 2000 0 60617(Mon)

60817(Wed)

61017(Fri)

61217(Sun)

61220(Sun)

Time Fig. 9. The evening peak period traffic flow forecasting results.

algorithm and the selection of influencing factor set in this paper need to be further researched.

Acknowledgments The work is supported by the National Natural Science Foundation of China (71101014; 50679008), Specialized Research Fund for the Doctoral Program of Higher Education (200801411105), Department of Communications Science and Technology for the Project of Henan Province (2010D107-4), and National Science Council, Taiwan (NSC 100-2628-H-161-001-MY4, NSC 100-2811-H-161-001, and NSC 101-2410-H-161-001). References [1] D. Huang, J. Song, D. Wang, J. Cao, W. Li, Forecasting model of traffic flow based on ARMA and wavelet transform, Comput. Eng. Appl. 6 (36) (2006) 191–194. [2] C. Han, S. Song, C. Wang, A Real-time short-term traffic flow adaptive forecasting method based on ARIMA model, J. Syst. Simulat. 7 (16) (2004) 1530–1534. [3] Y. Wang, M. Papageorgiou, A. Messmer, Real-time freeway traffic state estimation based on extended Kalman filter: a general approach, Transport. Res. 41 (2) (2007) 167–181. [4] P.J. Angeline, G.M. Saunders, J.B. Pollack, An evolutionary algorithm that constructs recurrent neural networks, IEEE Trans. Neural Networks 5 (1994) 54–65. [5] M. Guo, J. Lan, X. Xiao, H. Lu, Forecasting short-time traffic flow for Beijing 2nd ring road using chaos theory, J. Transport. Syst. Eng. Inform. Technol. 10 (2010) 106–111. [6] B.L. Smith, B.M. Williams, R.K. Oswald, Comparison of parametric and nonparametric models for traffic flow forecasting, Transport. Res. Part C 10 (2002) 303–321. [7] H. Lan, Short-term traffic flow prediction for highway tunnel based on fuzzy clustering analysis, Comput. Appl. Softw. 27 (1) (2010) 151–153. [8] D. Hu, J. Xiao, C. Che, Lifting wavelet support vector machine for traffic flow prediction, Appl. Res. Comput. 24 (8) (2007) 276–278.

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Mingwei Li was born in 1984. He received his master degree of applied engineering from Dalian University of Technology in 2007. He is working toward a Ph.D. degree in School of Dalian University of Technology. His research interests are intelligent algorithm, the city transportation optimization and logistics optimization.

WeiChiang Hong received his Ph.D. degree in Management from Da-Yeh University, Taiwan, in 2008. Since September 2006, he has been with the Department of Information Management of the Oriental Institute of Technology, where he is currently an associate professor. His research interests mainly include applications of forecasting technology, computational intelligence and tourism competitiveness evaluation and management. He is currently appointed as the Editor-in-Chief of the International Journal of Applied Evolutionary Computation, he is also on the Editorial Board of several journals, including Neurocomputing, Applied Soft computing, Energy Sources Part B: Economics, Planning, and Policy, and Mathematical Problems in Engineering.

Haigui Kang was born in 1945. He is a professor in the School of Dalian University of Technology, China. His research interests are structural optimization of offshore platforms, transportation systems, intelligent optimization, and ocean energy research.