Parameters Sensitive Analyses for Using Genetic Algorithm to Solve Continuous Network Design Problems

Procedia - Social and Behavioral Sciences 43 ( 2012 ) 435 – 444

1877-0428 © 2012 Published by Elsevier B.V. Selection and/or peer review under responsibility of Beijing Jiaotong University [BJU], Systems Engineering Society of China (SESC)

doi: 10.1016/j.sbspro.2012.04.117

8

th

International Conference on Traffic and Transportation Studies

Changsha, China, August 1–3, 2012

Parameters Sensitive Analyses for Using Genetic Algorithm

to Solve Continuous Network Design Problems

Meng Xu

a,

*, Jin Yang

a

, Ziyou Gao

a

a _{Institute of Systems Science, School of Traffic and Transportation, Beijing Jiaotong University, Beijing, 100044, China}

Abstract

In this paper, we will discuss the parameters settings using genetic algorithm to solve continuous network design problems (CNDP). The CNDP is formulated as a bi-level programming model. A sensitive analysis method, one-at-a-time designs, is used to analyze the effects of parameters. The analyses demonstrated that the setting of population size has clear effects to the solution; the effects of crossover probability and mutation probability are less than the effects of their combinations. The fields of these parameters are also given in this paper, which avoid to set them blindly in algorithm designs.

© 2012 Published by Elsevier Ltd. Selection and peer-review under responsibility of Beijing Jiaotong University (BJU) and Systems Engineering Society of China (SESC).

Keywords: Sensitive analysis, continuous network design, genetic algorithm

1. Introduction

The network design problem (NDP) involves the optimal decision on the improvement of a transportation system in response to a growing travel demand while considering the route choice behavior of network users. Continuous network design problem (CNDP) is concerned with the optimal capacity expansion of existing links in a given network by minimizing the total system cost as well as considering the route choice behavior of individual users. In the transportation area, CNDP was first proposed by Morlok et al. (Morlok et al., 1973). Due to multiple objectives for formulating CNDP, it was modeled as a

*_{Corresponding author. Tel.: +86-10-5168-7070; fax: +86-10-5168-7127.}

E-mail address: [email protected]

© 2012 Published by Elsevier B.V. Selection and/or peer review under responsibility of Beijing Jiaotong University [BJU], Systems Engineering Society of China (SESC) Open access under CC BY-NC-ND license.

bilevel programming problem, where the upper level is a nonlinear programming problem to minimize the system cost and the lower level is user equilibrium (UE) problem to assume drivers’ route choice behavior. There are many researchers focus on the studies of CNDP. Early detailed reviews on CNDP can be found on Boyce (1984), Yang and Bell (1998).

For solution approaches, several algorithms have been proposed in the literature for solving the CNDP formulated as the bilevel model, which has the characteristics of non-convex and non-smooth. Gao et al. (2004) discussed a bilevel programming model for transit network design problem, and designed a heuristic solution algorithm based on sensitivity analysis. Chiou (2005) proposed four gradient-based methods to solve the CNDP, which include gradient projection method, conjugate gradient projection method, quasi-Newton projection method, and PARATAN version of gradient projection method.

Besides the optimal methods, heuristic approaches are also important ways for solving the bilevel model. There are many important studies such as simulated annealing (SA) (Friesz et al., 1992, 1993; Lee and Yang, 1994; Yang et al., 2009), genetic algorithm (GA) (Cree and Maher, 1998; Yin, 2000; Chen et al. 2006; Fan, et al. 2006; Zhang, et al, 2007, 2008; Xu, et al., 2009; Duan, et al., 2011; Ding, et al., 2011), GA, SA, and tabu search (TS) (Cantarella et al., 2002), ant algorithm (Poorzahedy and Abulghasemi, 2005), and hybrid meta-heuristic algorithm (Poorzahedy and Rouhani, 2007).

Using GA to solve CNDP, different choices of parameters affect clearly the convergence level of solution and its implementation speed. How to choose the parameters is an important problem. However, the importance was probably underestimated because there are not further investigations in the literature. In the contexts of numerical modeling, sensitivity analyses (SA) studies the relationships between information flowing in and out of model (Saltelli, et al., 2000). SA is widely used in model development, verification, calibration, model identification and mechanism reduction. It can assist the modeler to determine whether the parameters are sufficiently precise for the model to give reliable predictions. In this paper, we introduce sensitivity analyses (SA) to appraise the estimated parameters of GA to the CNDP. One-at-a-time design (OATD) is used to analyze the effects of parameters, which include the population size, crossover probability, and mutation probability.

This paper is organized as follows. In Section 2, the bi-level programming formulation for CNDP and the SA method OATD is summarized. In Section 3, the GA algorithm details for CNDP are introduced. A typical transportation networks are introduced in Section 4, the GA implementations are presented and SA methods are carried for the network. Conclusions are given in the last section.

2. Summary of bi-level programming formulation for CNDP and SA methods

This section provides a summary of bi-level programming formulation for CNDP, and the sensitivity analysis method OATD. The presentation in this summary section follows Gao et al. (2004), Chiou (2005) and Saltelli, et al. (2000). Notation is provided firstly for convenience.

2.1. Notation

Considering a transportation network G(N,A), we give the following notations.

A Set of links, aA

R Set of origins, rR

S Set of destinations, sS

rs

P Set of routes between OD pair

 

r,s , rR, sS

a

f flow on link

a

,aA, f (,fa,)

rs p

h flow on route

p

between OD pair

 

r,s , pP_rs, (, rs,)

p

h h

rs

d travel demand between OD pair

 

r,s , d (drs)_RuS rs

p a,

G

indicator variable: if link

a

is on route p between OD pair

 

r,s ,Gars,p 1; , 0

rs p a G , otherwise.

 

rs p a p a rs , , G ' ,



, rs,



' ǻ . a

l capacity expansion on link aA, l (,la,), y_a

>

l_a,la

@

) (_a

a l

G investment costs function on link

a

, G

 

y



,G_a

 

l_a ,



)

, ( _{a a} a f l

t travel time on link aA, t



f,l





,t_a



f_a,l_a



,



I

conversion factor from investment cost to travel cost

2.2. CNDP model

In general, a bi-level programming problem is defined as follows:

 

U

_min



_x,y(x)



x F

s.t. G



x,y

 

x



d0 where y(x) is defined by

 

L



x y



y ,

min f

s.t. g



x,y



d0

Obviously, the bi-level programming model consists of two sub-models,

 

U which is defined as an upper-level problem and

 

L which is a lower-level. F(x, y(x)) is the objective function of upper-level decision-makers or system managers, x is the decision vector of the upper-level decision-makers;

 



x yx



G , is the constraint set of the upper-level decision vector. f(x, y) is the objective function of lower-level decision-makers; y is the decision vector of the lower-level decision-makers; and g(x, y) is the constraint set of the lower-level decision vector. It is noted that the decision variable of the lower-level problem is expressed as a function of the decision variable of the upper-level v(x) which is usually called reaction or response function. The maker at the upper level influences the lower-level decision-maker by setting x, thus restricting the feasible constraints set for the lower-level decision maker. The

upper-level decision-maker also interacts with the lower-level decision-maker via the objective function of the lower-level decision maker.

Thus CNDP can be formulated in terms of the bi-level programming model as follows:

 

U

¦

¦

   A A l l a a a a a a a a f l f G l t Z ( (), ) ( ) ( ) min

I

(1) s.t. ladladla,aA (2) where f=f(l) is the equilibrium flow defined by the following UE problem.

 

L z t vla dv a f a a ) , ( ) , ( min ) ( 0

¦ ³

_A l l f (3) s.t. _{R S} P   

¦

 s r d h rs p rs rs p , , (4) rs rs p r s p h t0, R, S, P (5) A  

¦¦¦

h a f rs p a r s p rs p a G , , (6)

In this bi-level model, the upper level is non-convex and non-differentiable in

l

which is defined by the lower level model. In the next section, we will introduce GA to solve the bi-level formulation of CNDP.

2.3. One-at-a-time designs (OATD)

In statistical designs and analysis for screening basing on computer experiments, OATD is an important method, which evaluates the impact of changing the values of each parameter in turn (Saltelli, et al., 2000). The advantage of OATD is that it does not make simplifying assumptions. Moreover, the computational cost is linear.

With OATD to parameter analysis, it uses the ‘nominal’ or ‘standard’ value per parameter, which is often taken from the literature. Two extreme values are usually proposed to represent the range of likely values for each parameter. Normally, the ‘standard’ value of a parameter is midway between the two extremes. The magnitudes of the differences between the outputs for extreme inputs and the ‘control’ are then compared to find those parameters that significantly affect the model. For further information about OATD, it can be refer to Kleijnen (1998).

3. Algorithm design

GA is a global search heuristic technique used to find true or approximate solutions to optimization problems. It consists of techniques inheritance, mutation, selection and crossover, which are inspired by evolutionary biology. GA is implemented as a computer simulation in which a population of chromosomes of candidate solutions to an optimization problem evolves toward better solutions. Solutions can be represented in binary or real-coded. The evolution usually starts from a population of randomly generated individuals and happens in generations. In each generation, the fitness value of every individual in the population is evaluated, multiple individuals are randomly selected from the current population (based on their fitness value) and modified (recombined and possibly randomly mutated) to form a new population. The new population is then used in the next iteration of the algorithm. Commonly, the algorithm terminates when either a maximum number of generations arrived, or a satisfactory fitness value reached for the population.

Before using GA to solve CNDP, it needs define the fitness value, individual and GA operators that used in the following sections.

Fitness value: the fitness value can use the upper object function in the bi-level programming model, that is,

¦

¦

   A A l l a a a a a a a a f l f G l t Z ( (), ) () I ( )

Individual: for the CNDP, coding the individual with real number, the benefit of using real number is it can show the individual character directly. The

i

-th gene represent the improve capacity of the

i

-th route. The length of the individual is equal to the number of links that need to improve the capacity.

GA operators: The sets of GA operators include choice strategy, crossover strategy and mutation strategy.

Choice strategy: order the fitness value of each individual in the population size, and the individual with the least fitness value has the largest choice probability, which can guarantee the convergence level of the solution. The fixed choice probability is defined as

) 1 , 0 ( , ) 1 ( 1   ˜ _D  _D D i i p

From above choice probability, it can calculate the sum probability q_i(i=1,,N), let q0=0,

ě

1 = = i j j i p q .

Circle N times, and generate ręU(0, 1), choosing individual

i

, when q_i1İrİq_i.

Crossover strategy: randomly choose two individuals from the population size. Generate randomly ) 1 , 0 ( ęU

r , if r <q_c, carry crossover operate to the two individuals, where

q

c is the crossover probability.

Mutation strategy: generate randomly ręU(0, 1), if r <q_m, it will take the mutation operate to one of the gene in the individual, and generate new individual.

The overall real-coded GA on CNDP is summarized as follows: Step 1: Initialization.

1.1 Define the parameters for the GA. Including the population size N, crossover rate q_c, mutation rate

q

_m, and the maximum number of generations M .

1.2 Generate an initial random population of chromosomes. Initialize the individual solutions, which are represented by chromosomes of the population. That is, choosing randomly the capacity expansion

schemes lia i N

i _{=( , , ), =1, ,}

, 0

0   

l in the fields of y , l_aİl_aİl_a, ൿaęA. Set the iteration number

0

t _.

Step 2: Evaluate the fitness value of each chromosome i t

l in the population. For given i

t

y , solve the lower-level problem, and find the user equilibrium link flows

N i fi a t i t =(, , ,), =1,,

f , Calculate the upper object function value ( , i)

t i t i t Z f l , which is regard as fitness scale of each chromosome i

t l .

Step 3: Order the fitness value ( , i)

t i t i t

Z f l from the least to the largest.

3.1 The least fitness scale corresponds to the maximal choice probability, and the largest fitness scale corresponds to the least choice probability, where the choice probability is fixed. The local optimal solution Zbest ( the minimize fitness value) can be identify from current population, the corresponding individual is ybest .

3.2 Generate randomly ręU(0, 1), choose individual

i

when q_i1 İrİq_i. Generate N _individuals

with N times loop.

Step 4: Crossover. Generate new offspring by exchange of information between the individuals, which generate in Step 3.2. Generate randomly ręU(0, 1), carry the crossover process if r <qc.

Step 5: Mutation. Generate randomly ręU(0, 1), carry the mutation process if r <qm.

Step 6: Stop if t ıM; return to Step 2 otherwise, and sett= t+1.

In solving SCNP, It is noted that we do not need solve the lower level problem if the change of y is small, which can reduce the implementation time.

4. GA sensitivity analysis

In this section, we first introduce the classical network used in this paper, numerical experiments are demonstrated to use GA to solve CNDP in Section 4.1. SA with OATD for the example is also given.

4.1. A simple example

Example 1 is from Suwansirikul et al. (1987), which includes one OD pair, four nodes and five links as shown in Fig. 1. The network information can refer to Table 3.

Fig. 1 Test network for Example l

Table 1 Input data for links’ time function in Example l

Links Aa Ba Ka ga 1 4.0 0.60 40.0 2.0 2 6.0 0.90 40.0 2.0 3 2.0 0.30 60.0 1.0 4 5.0 0.75 40.0 2.0 5 3.0 0.45 40.0 2.0 Time function ( , ) ( /( )4) a a a a a a a a f l A B f K l t   Object function

¦

  A a a a a a a a f l f g l t Z ( ( , ) 1.5 ( )2)

The parameters in example 1 are setting as following. Population size N 30, crossover probability 95

. 0

c

q , mutation probability qm 0.05, decision variable la

>

la,la

@

>

0,30

@

.

Under different demands, the optimal solution Zbest_{, the implementation time of GA time(s) and the} iteration times solve the lower level problem UEt _{are shown in Table 4.}

4.2. Parameters sensitivity analysis of GA

For example 1, we set the benchmark of population size N 20, crossover probability q_c 0.95, and

mutation probability q_m 0.05, other settings see Section 4.1. We study the variations of these parameters in the following.

Fixed crossover probability q_c =0.95, and mutation probability qm=0.05. The population size is change in the field of Nę

[

2,40

]

. The optimal solution Zbest, the implementation time time(s) and total iteration number to solve the lower level problem UEt with the variation of population size are shown in Fig. 2.

From Fig. 2(a), when the population size less than 10, the GA is easy to converge a local solution; when the population size is between 10 and 16, the GA can converge to a solution near the optimal solution. However, it still easy to arrive a local optimal solution; when the population size is larger than 16, the algorithm can converge to the optimal solution. From Fig. 2 (b) and (c), time(s) and UEt increase almost linearly with the increase of population size. To balance the convergence level and implementation time, it can be decided the population size between 16 and 30 for example 1.

Fixed population size N=20 , mutation probability qm=0.05 , and crossover probability

[

0.80,0.99

]

ę

c

q . The variation of Zbest, time(s) and UEt about crossover probability is shown in Fig. 6.

From Fig. 3(a), when q_cę

[

0.80,0.99

]

, the crossover probability do not clear effect to the optimal solution. the difference is in the field of 3, and the optimal solution decrease with the increase of crossover probability; From Fig. 3(b) and (c), the changes of crossover probability will not clear effect to the implementation time (time(s)) and the total iteration times of the lower problems (UEt). The

implementation time time(s) is keep on about 25 seconds and the UEt is keep on about 2700 times.

Therefore, in the solving the CNDP with GA, it can set the crossover probability over 0.9.

2 10 20 30 40 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 Population size Zb e s t a 2 10 20 30 40 0 10 20 30 40 Population size Tim e s b 2 10 20 30 40 0 1000 2000 3000 4000 5000 Population size UE t c

Fig. 2. Results of GA under different pop-size

0.8 0.85 0.9 0.95 0.99 1200.5 1201 1201.5 1202 1202.5 1203 Crossover probability Z bes t a 0.8 0.85 0.9 0.95 0.99 0 10 20 30 Crossover probability Ti m e s b 0.8 0.85 0.9 0.95 0.99 0 1000 2000 3000 Crossover probability UE t c

Fig. 3. Results of GA under different crossover

Fixed population size N=20, crossover probability _{qc 0.95}, and mutation probability qm

_>

0.01,0.2

_@

, The variation of Zbest, time(s) and UEt about mutation probability is shown in Fig. 4.

0.01 0.05 0.1 0.15 0.2 1200 1220 1240 1260 1280 1300 1320 1340 1360 1380 1400 Mutation probability Z bes t a 0.010 0.05 0.1 0.15 0.2 10 20 30 40 Mutation probability Ti m e s b 0.01 0.05 0.1 0.15 0.2 0 1000 2000 3000 4000 Mutation probability UE t c

Fig. 4. Results of GA under different mutation

Mutation probability control the ratio of new individuals load to population size, that is, the mutation can provide different individuals from the population size, also, it can find back some lost individuals. When setting the mutation probability, if it is too low, some useful individuals are lost; if it is too high, the uncertainty facts increase, and some good properties of the individuals maybe lost. From Fig. 4(a), when the mutation probability qm!0.1, the solution changes greatly; when qm <0.05, the solution is

near the optimal solution; when 0.05qm0.1, it can be find the optimal solution. From Fig. 4(b) and (c), the GA implementation time time(s) increase when 0.01qm0.1, and also the total iteration times of the lower problems UEt; when 0.1qm 0.2, time(s) and UEt have no big change. The time difference is mainly caused by searching the solution space with different mutation probability, and if the mutation probability is larger (for example, q_m 0.2), the solution may converge into local solution, which reduce the GA implementation time time(s).

5. Conclusions

In this paper, with the OATD sensitivity analysis method, we analyzed the characteristics of parameters choices when using GA to solve CNDP, and presented rulers to set parameters. Experiments demonstrate that the convergence level of lower level problems have a clear effects to the solution of bi-level programming models; however, the implementation time increase rapidly with the increase of convergence level of lower level problems. In general, when the convergence level of lower level problems is less than 0.01, the GA can converge to the optimal solution. From the sensitivity analysis of OATD, population size has important effects to the GA to solve CNDP. When setting the population size, which is no less than 3 times of the network scale (size of decision variable l), GA can converge to the

optimal solution; the values of crossover probability and mutation probability can keep fixed in the GA, that is, both the values are not very much sensitive to the network scale. The crossover probability can be set in the field of

_>

0.9,0.99

_@

, and the mutation probability can be set in the field of

_>

0.05,0.1

_@

.

How to set the parameters in using GA is an important issue. This paper gives a special discussion for CNDP. For further studies, more SA methods are need to check since most of existing studies only focus

on the GA designs to solve network design problem. More large-scale network needs to be analyzed, although it needs much time. Furthermore, other heuristic algorithms need to be analyzed for solving the CNDP in the future.

Acknowledgements

The study is supported by the National Natural Science Foundation of China (Grant No. 71071014), the Program for New Century Excellent Talents in University (NCET-11-0569) and the Key Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji University(201103).

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