harmony search algorithm for continuous network design problem with link capacity expansions

KSCE Journal of Civil Engineering (0000) 00(0):1-11

DOI 10.1007/s12205-013-0122-6

− 1 −

www.springer.com/12205

Transportation Engineering

Harmony Search Algorithm for Continuous Network Design Problem

with Link Capacity Expansions

Ozgur Baskan*

Received March 6, 2012/Accepted April 4, 2013/Published Online July 26, 2013

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Abstract

The Continuous Network Design Problem (CNDP) deals with determining the set of link capacity expansions and the correspondingequilibrium link flows which minimizes the system performance index defined as the sum of total travel times and investment costs oflink capacity expansions. In general, the CNDP is characterized by a bilevel programming model, in which the upper level problem isgenerally to minimize the total system cost under limited expenditure, while at the lower level problem, the User Equilibrium (UE)link flows are determined by Wardrop’s first principle. It is well known that bilevel model is nonconvex and algorithms for findingglobal or near global optimum solutions are preferable to be used in solving it. Furthermore, the computation time is tremendous forsolving the CNDP because the algorithms implemented on real sized networks require solving traffic assignment model many times.Therefore, an efficient algorithm, which is capable of finding the global or near global optimum solution of the CNDP with lessnumber of traffic assignments, is still needed. In this study, the Harmony Search (HS) algorithm is used to solve the upper levelobjective function and numerical calculations are performed on eighteen link and Sioux Falls networks. The lower level problem isformulated as user equilibrium traffic assignment model and Frank-Wolfe method is used to solve it. It has been observed that the HSalgorithm is more effective than many other compared algorithms on both example networks to solve the CNDP in terms of theobjective function value and UE traffic assignment number.

Keywords: harmony search, continuous network design, traffic assignment, link capacity expansion, bilevel programming

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1. Introduction

The Continuous Network Design Problem (CNDP) is at the

core of transportation problems and has been extensively studied

in the literature aiming to determine the optimal capacity

expansions for a set of selected links in a given transportation

network. The measure of network performance can be described

as the sum of total travel times and investment cost of link

capacity expansions. Determining the global or near global

optimum solution is of great importance in the CNDP. Due to the

non-convex and non-smooth characteristics of the CNDP, it is

formulated as bilevel programming problem. Bilevel problems

generally are difficult to solve, because the evaluation of the

upper level objective involves solving the lower level problem

for every feasible set of upper level decisions. In the CNDP,

upper level could be a model with objective function defined as

the sum of total travel time and total investment cost of link

capacity expansions in a given network whilst the lower level is

formulated traffic assignment model, such as user equilibrium

assignment (Suwansirikul et al., 1987; Friesz et al., 1992; Chiou,

2005; Ban et al., 2006; Xu et al., 2009), variable demand

equilibrium (Chen and Chou, 2006), stochastic user equilibrium

assignment (Davis, 1994; Chen et al., 2006) and dynamic traffic

assignment (Karoonsoontawong and Waller, 2006).

Bilevel problems are known to be one of the most attractive

mathematical problems in the optimization field because of non-

convexity of feasible region that it has multiple local optima. The

difficulty with solving the bilevel formulation of the CNDP is that

the objective function may not be optimized at the upper level that

is defined as the sum of total travel times and investment costs of

link capacity expansions without considering the reactions of the

road users at the lower level. Even when considering both the upper

and lower level problems that consist of convex programming

problems, the bilevel solution of the CNDP may be non-convex

due to both traffic assignment constraints and nonlinear travel time

function. Due to the non-convexity, multiple local optima may

exist. This non-convexity also implies a serious problem for

deterministic algorithms (Bell and Iida, 1997) and they may not be

effective since the decision space is highly convoluted.

Abdulaal and LeBlanc (1979) were the first ones to solve the

CNDP problem for medium-sized realistic network using

Hooke-Jeeves (HJ) algorithm. Suwansirikul et al. (1987) proposed

Equilibrium Decomposition Optimization (EDO) for finding an

approximate solution to the CNDP and tested this heuristic on

*Research Assistant, Dept. of Civil Engineering, Faculty of Engineering, Pamukkale University, Denizli 20070, Turkey (Corresponding Author, E-mail: obaskan@

pau.edu.tr)

Ozgur Baskan

− 2 − KSCE Journal of Civil Engineering

several networks. Their results showed that the proposed heuristic

is more efficient than the HJ algorithm. The efficiency of the

method is from decomposition of the original problem into a set

of interacting optimization sub problems. Marcotte (1983) and

Marcotte and Marquis (1992) presented efficient implementations

of heuristic procedures in small-sized networks for solving the

CNDP where network users behave according to Wardrop's first

principle of traffic equilibrium. In addition, a number of

sensitivity-based heuristic algorithms were developed for the

CNDP (Friesz et al., 1990; Cho, 1998; Yang, 1995; Yang, 1997).

Furthermore, Friesz et al. (1992) used Simulated Annealing (SA)

approach to solve the CNDP and found that the proposed

heuristic is more efficient than Iterative Optimization Assignment

(IOA) algorithm, HJ algorithm, and EDO approach for finding

the global optimum solution. Friesz et al. (1993) presented a

model for continuous multiobjective optimal design of a

transportation network. The results showed that SA is ideally

suited for solving multi objective versions of the equilibrium

network design problem. Davis (1994) used the generalized

reduced gradient method and sequential quadratic programming

to solve the CNDP. Meng et al. (2001) converted the bilevel program

of the CNDP to a single level continuously differentiable

optimization problem. Chiou (2005) used a bilevel programming

technique to formulate the CNDP. Four variants of gradient-

based methods are presented and numerical comparisons are

made with several test networks to solve the CNDP.

Similarly, Ban et al. (2006) proposed a relaxation method to solve

the CNDP when the lower level is a nonlinear complementary

problem and obtained good results. Karoonsoontawong and Waller

(2006) presented SA, Genetic Algorithm (GA), and random search

techniques to solve the CNDP. Their study showed that GA is

performed better than the others on the test networks. Xu et al.

(2009) used SA and GA methods to find optimal solutions of the

CNDP. They found that when demand is large, SA is more efficient

than GA in solving the CNDP, and much more computational time

is needed for GA to achieve the same optimal solution as SA.

The methods developed so far to solve the CNDP are generally

based on the heuristic approaches. Although the proposed

algorithms are capable of solving the CNDP for a given road

network, an efficient algorithm, which is capable of finding the

global or near global optimum of the upper level decision

variables of the CNDP, is still needed. Additionally, it is

important to compare the efficiency of the available heuristic

methods in literature and to find the most efficient ones because

the computation time for solving traffic assignment problem in

lower level of a real road network is considerably large and the

traffic assignment problem has to be solved in order to evaluate

the upper level objective function value. Therefore, this paper

deals with the finding optimal link capacity expansions in a

given road network using Harmony Search (HS) algorithm. For

this purpose, a bilevel model has been proposed, in which the

lower level problem is formulated as user equilibrium traffic

assignment model and Frank-Wolfe (FW) method is used to

solve it. This method is especially useful for determining the

equilibrium flows for transportation networks since the direction

finding step can be executed relatively efficiently (Sheffi, 1985).

The rest of this paper is organized as follows. In Section 2,

basic notations are defined. Problem formulation for the

CNDP subject to link capacity expansions is given in Section

3. In the next section, HS approach and its solution procedure

are presented for solving upper level problem. In Section 5,

numerical calculations are conducted both on eighteen link

network and on a real data Sioux Falls network where good

results are obtained to show the performance and robustness

of the HS algorithm. Finally, concluding remarks and future

study directions are given in Section 6.

2. Problem Formulation

2.1 The Basic Idea of the Bilevel Programming Model

The CNDP can be represented within the framework of a leader-

follower or Stackelberg game, where the supplier is the leader and

the user is the follower (Fisk, 1984). It is assumed that the leader as

transportation planning manager can influence the users’ path

choosing behavior but cannot control it. The users make their

decision in a user optimal manner under the given service level of

transportation networks (Gao et al., 2007). This interaction can be

formulated as bilevel programming model. It is clear that the bilevel

model consists of two levels, which are defined as Upper Level

(UL) and Lower Level (LL) problems as following:

(UL) (1)

s.t.

where, is implicitly defined as:

(LL) (2)

s.t.

where Z and x are the objective function and decision vector of upper

level decision-makers, H and h are the constraint sets of the upper

level and lower level decision vectors, z and y are the objective

function and decision vector of lower level decision-makers and

is usually called the reaction or response function.

The upper level defines leader problem to manage the

transportation network and the lower level represents user’s

behavioral problem to this action. In the upper level, the road

capacity expansions are determined to minimize the total system

travel time and the associated capacity expansion cost. The

lower-level program determines the User Equilibrium (UE) flow

pattern given a particular capacity expansion plan. In this study,

the Wardrop’s first principle is followed, in which, at

equilibrium, the link-flow pattern is such that the travel times on

all used paths connecting any given Origin-Destination (O-D)

pair will be equal; the travel times on these used paths will also

be less than or equal to the travel time on any of the unused

paths. At this point, the network is in user equilibrium and no

user can experience a lower travel time by unilaterally changing

routes (Wardrop, 1952).

Z x y,( )x

limmin

H x y,( ) 0≤

y y x( )=

Z x y,( )y

limmin

h x y,( ) 0≤

y y x( )=

Harmony Search Algorithm for Continuous Network Design Problem with Link Capacity Expansions

Vol. 00, No. 00 / 000 0000 − 3 −

2.2 The Upper Level Optimization Model

In the upper level optimization problem is aimed to determine

the optimal link capacity expansions for a set of selected links in

a given transportation network by minimizing the total system

cost as well as considering the route choice behavior of

individual network users. The upper level objective function for

the CNDP can be formulated as follows:

(3)

s.t.

where x is the implicitly function of the y which may be obtained

by solving the lower level problem; is the investment

function of link ; and are the capacity expansion and

the upper bound of capacity expansion of link , respectively,

ρ the conversion factor from investment cost to travel times. The

network planners of the upper level are assumed to make the

decisions about the improvement of link capacities and

investments to minimize the total system cost. The constraint of

Eq. (3) ensures that the investment cost of link will not

exceed the related budget. It is also the non-negativity constraint

of the decision variables.

2.3 The Lower Level Optimization Model

In general, the CNDP models assume that the demand is given

and fixed, and the user’s route choice is characterized by the UE

assignment. The UE assignment problem is to find the link

flows, x, which satisfies the user-equilibrium criterion when all

the O-D entries have been appropriately assigned. In general,

improvement of transportation road network characteristics may

be induced changes in traffic flow over the whole network.

Addition of a new road segment to the network, or capacity

expansion, without considering the response of network users

may actually increase network-wide congestion. Due to this

well-known phenomenon Braess' paradox, prediction of traffic

patterns is essential to the network design process. The UE

assignment problem can be represented as follows:

(4)

s.t.

where the constraints of Eq. (4) are definitional, conservation of

the flow constraints and non-negativity, respectively.

3. Solution Algorithm

The HS algorithm proposed by Geem et al. (2001) is a meta-

heuristic method and based on the musical process of searching

for a perfect state of harmony, such as jazz improvisation. In this

improvisation process, members of the music group try to find

the best harmony as determined by an aesthetic standard, just as

the optimization algorithm tries to find the global optimum as

determined by the objective function. The notes and the pitches

getting played by the individual instruments determine the

aesthetic quality, just as the objective function value is determined

by the values assigned to design variables. The harmony quality

is enhanced practice after practice, just as the solution quality is

enhanced iteration by iteration. The HS algorithm is simple in

concept, few in parameters and easy in implementation and it has

been successfully applied to various optimization problems. In

the HS, four algorithm parameters are used to control the solution

procedure; Harmony Memory Size (HMS) that represents the

number of solution vectors in the harmony memory; Harmony

Memory Considering Rate (HMCR) that is the probability of

assigning the values to the variables from harmony memory;

Pitch Adjusting Rate (PAR) sets the rate of adjustment for the

pitch chosen from the harmony memory; and the Number of

Improvisations (NI) that represents the number of iterations to

be used during the solution process. NI also can be assumed as

the termination criterion (Ceylan et al., 2008). Harmony

Memory (HM) is a memory location where all the solution

vectors and corresponding objective function values are stored.

The function values are used to evaluate the quality of solution

vectors. The HS algorithm also considers several solution

vectors simultaneously, in a manner similar to the GA.

However, the major difference between the two heuristic

algorithms is that the HS generates a new vector from all the

existing vectors, whereas the GA produces a new vector from

only two of the existing vectors. In addition, iterations in the

HS algorithm are faster than that in GA (Lee et al., 2005). The

procedure of the HS algorithm is composed as:

Let the basic optimization problem is as follows:

Min Z = f(x) subject to (5)

where f(x) is the objective function to be minimized, x is the set

(so called orchestra) of decision variables xi; N is the number of

decision variables (music instruments); Xi is the set of the

possible range of values for each decision variable (the pitch

range of each instrument).

As first step, the HM is generated as memory location where

all the solution vectors and corresponding objective function

values are stored. In this step, the HM matrix is filled with

randomly generated solution vectors as the HMS and their

corresponding fitness function values are shown in Eq. (6):

(6)

Z x y,( )x

lim ta xa y( ) ya,( )xa y( ) ρga ya( )+( )a A∈

∑=min

0 ya ua≤ ≤ , a A∈∀

ga ya( )a A∈ ya ua

a A∈

a A∈

z x y,( )x

lim ta w ya,( ) wd

0

xa

∫a A∈

∑=min

f k

rs

k K∈

∑ Drs= r R∈∀ s S∈ k Krs∈, ,

xa f k

rsδ a k,

rs

k Krs

∈

∑rs

∑= r R s R a A k Krs∈,∈,∈,∈∀

f k

rs0≥ r R s S k Krs∈,∈,∈∀

xi Xi∈ i, 1 2 … N, , ,=

HM

x1

1x2

1 … xN 1–

1xN

1

x1

2x2

2 … xN 1–

2xN

2

… … … … …

x1

HMS-1x2

HMS-1 … xN 1–

HMS-1xN

HMS-1

x1

HMSx2

HMS … xN 1–

HMSxN

HMS

f x1( )

f x2( )

f xHMS-1( )

f xHMS( )

⇒=

…

Ozgur Baskan


In improvisation step, a new harmony vector =

is generated based on three rules. These rules

are the memory consideration, pitch adjustment, and random

selection. In the memory consideration, the value of the first

decision variable ( ) for the new vector is selected from any

value in the specified HM range ( ). Values of the other

decision variables are selected in the same manner.

The HMCR parameter varies between 0 and 1 and represents the

rate of choosing one value from HM whereas (1-HMCR) is the

rate of randomly selecting a value from the possible range. Each

decision variable is selected with the procedure that is given in

Eq. (7):

(7)

The next step under the improvisation process is to check

whether the pitch adjustment is necessary or not. After the

memory consideration, pitch adjustment probability is evaluated

with parameter of PAR which represents the pitch adjusting and

varies between 0 and 1 as follows:

(8)

where bw is an arbitrary bandwidth and Rnd(0; 1) is a random

number between a value range of 0-1. The pitch adjusting

process is performed only after a value has been chosen from the

HM. The value (1-PAR) sets the rate of doing nothing. Note that

the HMCR and PAR parameters introduced in the HS help the

algorithm to find globally and locally improved solutions (Lee et

al., 2005).

In past decade, HS based algorithms have been applied to the

different optimization problems that are known to be extremely

difficult to find an optimal solution using traditional mathematical

methods. Lee and Geem (2004) described a new structural

optimization method based on the HS algorithm for solving

structural engineering problems, which was conceptualized

using the musical process of searching for a perfect state of

harmony. Lee et al. (2005) presented a discrete search strategy

using the HS algorithm and its effectiveness and robustness, as

compared to current discrete optimization methods, are

demonstrated through several standard examples. Geem (2009)

proposed an improved HS algorithm for the optimal design of

water distribution networks by incorporating particle-swarm

concept. It was found that the HS algorithm performed well in

designing water distribution networks, especially for small-scale

and medium-scale networks. Degertekin and Hayalioglu (2010)

developed HS based algorithm to determine the minimum cost

design of steel frames. Ayvaz (2010) proposed a linked

simulation-optimization model based on the HS algorithm for

solving the unknown groundwater pollution source identification

problems. Sivasubramani and Swarup (2011) presented a new

multi-objective HS algorithm for environmental/economic dispatch

problem. Kayhan et al. (2011) proposed a solution model to

obtain input ground motion datasets compatible with given

design spectra based on HS algorithm. A hybrid modified

global-best harmony search algorithm for solving the blocking

permutation flow shop scheduling problem with the makespan

criterion was proposed by Wang et al. (2011).

Similarly, Askarzadeh and Rezazadeh (2011) studied accurate

identification of voltage versus current characteristics of proton

exchange membrane fuel cell using HS algorithm. Erdal et al.

(2011) formulated the design problem of cellular beams as

optimum design problem using HS and particle swarm optimization

methods. Khorram and Jaberipour (2011) presented a solution

method based on the HS approach for combined heat and power

economic dispatch problem.

Although the HS based algorithms can be used to solve

different optimization problems, their applications are still

limited on the transportation area such as Ceylan et al. (2008);

Suh et al. (2010); Mun and Lee (2011). Moreover, there is no

study based on the HS algorithm for solving the CNDP on the

available literature. The presented study, to the best of my

knowledge, is the first attempt at applying the HS algorithm to

find optimal capacity expansions for given set of links on a

transportation road network. The algorithm of the HS on the

CNDP is summarized as:

Step 0 :Set the user-specified HS parameters.

Step 1 : Generate the HM of capacity expansions ya by giving

the upper and lower bounds.

Step 2 : A new harmony vector is generated based on three rules

that are memory consideration, pitch adjustment and random

selection. Choosing the solution parameters (i.e., HMCR, PAR and

HMS) for the HS algorithm is a very important step for the

algorithm to find optimal or near-optimal solutions. Geem (2000)

has recommended parameter values ranged between 0.70 and

0.95 for HMCR, 0.20 and 0.50 for PAR, and 10 and 50 for HMS

to produce good performance of the HS algorithm. Hence,

HMCR, PAR and HMS are selected as 0.90, 0.40 and 10 for all

numerical experiments on the basis of the empirical findings by

Geem (2000). The effect of HS parameters on the CNDP is not

taken into account. This is out of the scope of this study.

Step 3 : Solve the lower level problem by way of the Frank-

Wolfe method using populated capacity expansions in HM. This

procedure gives UE link flows for each link .

Step 4 : Find the values of the upper level objective function for

resulting capacity expansions at Steps 1-2 and the corresponding

equilibrium link flows resulting in Step 3.

Step 5 : All of the objective function values in HM are set in

order from the best to the worst and the new harmony vector is

compared with the vector giving the worst objective function

value in this step. If new harmony vector gives a better functional

value than the worst one, the new harmony vector is included to

the HM and the worst harmony is excluded from the HM.

Step 6 : Check the stopping criterion. If the difference between

the average and best objective function values in HM is less than a

predetermined value, the algorithm is terminated. Else go to Step 7.

x′x1′ x2′ … xN

′, , ,( )

x1′x1

1x1

HMS–

x2′ … xN′, ,( )

xi′xi′ xi

1

xi

2 … xi

HMS, , ,{ } with problility HMCR∈

xi′ Xi with probability (1-HMCR)∈⎩⎨⎧

←

xi′xi′ Ran 0 1;( ) bw× with probability PAR±

xi′ with Probability 1 PAR–( )⎩⎨⎧

=

a A∈


Vol. 00, No. 00 / 000 0000 − 5 −

Step 7 : Terminate the algorithm if maximum number of

improvisations is reached. Else go to Step 2.

4. Numerical Examples

4.1 Eighteen Link Network

In order to demonstrate the capability of the HS algorithm in

solving the CNDP, it is firstly applied to the eighteen link

network adopted from Xu et al. (2009). This network consists of

18 links and 6 nodes as shown in Fig. 1. The travel demand for

this network includes three cases and is shown in Table 1. Case 1

describes the condition of light demand while case 2 and 3 are

for heavy demand condition. The results obtained through the

HS algorithm for eighteen link network are compared with the

results of Simulated Annealing (SA) and Genetic Algorithm

(GA) taken from Xu et al. (2009) on the same network.

The link travel time function is defined as shown in Eq. (9) and

its corresponding parameters for the eighteen link network are

given in Table 2.

(9)

where α, β are the parameters and θ is the link capacity for each

link . The upper level objective function for the eighteen

link network is defined as:

(10)

s.t. ,

where da is the cost coefficient, ua is upper bound for capacity

expansion for link a , and is set to 20 for this example

network.

The application of the HS algorithm for finding optimal link

capacity expansions is tested on eighteen link network, where the

upper level optimization problem is carried out based on the HS

algorithm, and UE traffic assignment is performed by way of

FW method at the lower-level. The HS algorithm has been

executed in MATLAB programming, and performed on PC with

Intel Core2 2.00 GHz, RAM 2 GB. Each function evaluation

took about 3.5 seconds of CPU seconds for case 1. Comparison

of computation times for the HS algorithm against the other

algorithms for the CNDP is difficult because each author uses a

different computer, and in many cases, a different test network.

However, one useful way of comparing computation times is in

terms of UE assignment number performed, since it is the most

time consuming part of the algorithms. Therefore, this performance

measure is used to compare the computation time of the HS

algorithm with the other algorithms in this study.

The HS algorithm was initially structured with randomly

generated solution vectors with HMS. In the improvisation step,

a new harmony vector is generated based on three rules such as

memory consideration, pitch adjustment and random selection,

respectively. In memory consideration, the value of first link

capacity expansion for the new vector is selected from any value

in the specified HM range using the HMCR. Values of the other

capacity expansions are selected in the same manner. The next

step under the improvization process is to check whether the

pitch adjustment is necessary or not. If a decision variable attains

its value from HM, it is checked whether this value should be

pitch-adjusted or not using the parameter of PAR. Pitch

adjustment simply means sampling the variable’s one of the

neighboring values, obtained by adding or subtracting one from

its current value. If not activated by parameter of PAR, the value

of the decison variable does not change. After a new harmony

vector is generated based on the above mentioned rules, UE link

flows are obtained by way of the FW method and the Objective

Function Values (OFVs) in HM are determined using generated

link capacity expansions and corresponding UE link flows.

ta xa ya,( ) αa βa

xa

θa ya+--------------⎝ ⎠⎛ ⎞

4

+=

a∀ A∈

minZ x y,( ) ta xa ya,( )xa daya+( )a A∈

∑=

0 ya ua≤ ≤ a A∈∀

a A∈∀

Fig. 1. Eighteen Link Network

Table 1. Travel Demand for the Eighteen Link Network

Case D16 D61 Total flow

1 5 10 15

2 10 20 30

3 15 25 40

Table 2. Parameters for the Eighteen Link Network

Link a αa

βa

θa

da

1 1 10 3 2

2 2 5 10 3

3 3 3 9 5

4 4 20 4 4

5 5 50 3 9

6 2 20 2 1

7 1 10 1 4

8 1 1 10 3

9 2 8 45 2

10 3 3 3 5

11 9 2 2 6

12 4 10 6 8

13 4 25 44 5

14 2 33 20 3

15 5 5 1 6

16 6 1 4.6 1

17 5 9 45 2

18 5 9 45 2

Ozgur Baskan


Then, all of the OFVs in HM are set in order from the best to the

worst and a new harmony vector is compared with the vector,

giving the worst OFV. If the new harmony vector gives a better

OFV than the worst one, the new harmony vector is included to

the HM and the existing worst harmony is excluded from the

HM. In general, when the difference between average and best

OFVs is less than a predetermined threshold value then it is

assumed that the HS algorithm has found optimum or near-

optimum solution. However, in eighteen link network, the HS

algorithm is terminated when the given maximum number of

improvisations is reached, to compare with the results taken from

Xu et al. (2009).

The plots of OFVs versus UE assignment number (UE*) on the

eighteen link network for case 1, 2 and 3 are given in Figs. 2(a),

2(b) and 2(c), respectively. Table 3 shows comparison of the

value of link capacity expansions generated by using the HS, SA

and GA. The HS algorithm shows steady convergence towards

the optimum or near optimum for all cases.

In case 1, the best OFV using the HS algorithm resulting after

50000 UE assignments is found as 189.59 while SA and GA

reach optimal values of 205.89 and 191.26, respectively, and

15000 and 50000 UE assignments need to be made to reach this

optimal values as shown in Table 3. Additionally, the HS

algorithm reaches an optimal value of 191.14 after 15000 UE

assignments while SA and GA methods produce the values of

about 206.00 and 214.00 for the same UE assignment number

(see for details Xu et al., 2009). The HS algorithm initially starts

with a random generated HM and picked up the best OFV which

is about 761. It easily locates the best OFV after a few thousand

UE assignments, as can be seen in Fig. 2(a).

In case 2 and 3, the HS reaches optimal values of 488.80 and

726.84 after 50000 UE assignments as shown in Table 3. SA reaches

the values of 505.39 and 739.54 and the corresponding UE

assignment numbers are 42500 and 22500 while GA reaches the

values of 515.09 and 744.39 and 50000 UE assignments are needed.

On the other hand, the HS reaches below the value of 495 after only

10000 UE assignments for case 2 as can be seen in Fig. 2(b) while the

SA and GA reach the values of about 511 and 549 for the same UE

assignment number (see for details Xu et al., 2009). Similarly, the HS

algorithm reaches below the value of 738 after only 2500 UE

assignments for case 3 as shown in Fig. 2(c) while the SA and GA

reach the values of about 748 and 788 although 7500 UE

assignments are needed to reach these optimal values (see for details

Xu et al., 2009). The results show that the HS algorithm is much

more efficient and effective method than SA and GA in solving the

CNDP based on the results of all cases for eighteen link network in

terms of the best OFV and required UE assignment number.

4.2 Sioux Falls Network

To show the effectiveness of the HS algorithm on the realistic test

network, the city of Sioux Falls is used. It consists of 24 nodes and

76 links. The parameters of the network and the travel demands

between the 552 O-D pairs are adopted from Suwansirikul et al.

(1987), and they are given in Tables 4 and 5. The Sioux Falls is the

network used by Suwansirikul et al. (1987); Chiou (2005); Abdulaal

and LeBlanc (1979); LeBlanc (1975) for studying various aspects

of network design problems.

The 10 dashed links 16, 17, 19, 20, 25, 26, 29, 39, 48 and 74 of

Sioux Falls network are candidates for capacity expansion as shown

in Fig. 3. At the lower level, UE traffic assignment is performed by

Table 3. Comparison of HS, SA and GA Methods on Eighteen Link Network

Case 1 Case 2 Case 3

HS SA GA HS SA GA HS SA GA

y1 0.00 0.00 0.00 0.74 0.00 0.00 1.68 0.00 0.00

y2 0.00 0.47 0.00 1.74 1.73 2.20 8.16 9.12 11.98

y3 0.00 0.65 0.00 9.83 11.77 10.61 15.45 18.12 16.24

y4 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

y5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

y6 4.30 6.53 4.47 8.52 4.75 6.68 8.60 4.98 5.40

y7 0.00 0.80 0.00 0.00 0.14 0.00 0.00 0.11 0.00

y8 0.00 0.25 0.00 0.00 0.78 0.00 3.23 1.58 6.04

y9 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

y10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

y11 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

y12 0.00 0.00 0.00 0.10 0.00 0.00 0.10 0.00 0.00

y13 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

y14 0.00 0.84 0.00 1.31 5.94 1.22 12.00 11.66 12.28

y15 0.02 0.14 0.00 0.05 1.51 6.30 0.07 2.97 0.82

y16 7.32 7.34 7.54 19.95 18.45 11.93 19.99 19.71 19.99

y17 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

y18 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

OFV 189.59 205.89 191.26 488.80 505.39 515.09 726.84 739.54 744.39

UE* 50000 15000 50000 50000 42500 50000 50000 22500 50000


Vol. 00, No. 00 / 000 0000 − 7 −

way of the FW method. The upper level objective function for the

Sioux Falls network is formulated as in Eq. (11). The HS algorithm

run is terminated when the difference between average and best

OFVs is less than the value of 0.03 for Sioux Falls network.

(11)

s.t. ,

The compared algorithms with the HS and comparison results

are given in Table 6 and 7, respectively. To investigate of the HS

algorithm’s stochastic nature, the Sioux Falls network has been

solved 100 times, and the best and worst OFVs are given in

Table 7. In addition, the average number of UE assignments has

been reported as the cost of solving the CNDP, which may be

considered as the CPU time of computer.

Although the difference between OFVs for all algorithms was

not significant, the HS produced better solution than the EDO,

SAB, QNEW and PT algorithms in terms of OFV as it can be

seen in Table 7. In addition, the optimal link capacity expansions

produced by all algorithms were different from each other. The

reason is that each method leads to a different solution to the

CNDP since it has multiple local optima due to the non-

convexity of the bilevel formulation of the CNDP. The HS

algorithm reaches best OFV of 81.83 and 895 UE assignments

minZ x y,( ) ta xa ya,( )xa 0.001da ya

2+( )

a A∈

∑=

0 ya ua≤ ≤ a A∈∀

Fig. 2. Performance of the HS Algorithm on Eighteen Link Net-

work for: (a) Case 1, (b) Case 2, (c) Case 3

Table 4. Parameters for Sioux Falls Network

Linksα

a

(hours)βa

(hours)

θa

(thousand vehicles)

da

(thousand dollars)

1-3 0.06 0.0090 25.9002

2-5 0.04 0.0060 23.4035

4-14 0.05 0.0075 4.9582

6-8 0.04 0.0060 17.1105

7-35 0.04 0.0060 23.4035

9-11 0.02 0.0030 17.7828

10-31 0.06 0.0090 4.9088

12-15 0.04 0.0060 4.9480

13-23 0.05 0.0075 10.0000

16-19 0.02 0.0030 4.8986 26.00

17-20 0.03 0.0045 7.8418 40.00

18-54 0.02 0.0030 23.4035

21-24 0.10 0.0150 5.0502

22-47 0.05 0.0075 5.0458

25-26 0.03 0.0045 13.9158 25.00

27-32 0.05 0.0075 10.0000

28-43 0.06 0.0090 13.5120

29-48 0.05 0.0075 5.1335 48.00

30-51 0.08 0.0120 4.9935

33-36 0.06 0.0090 4.9088

34-40 0.04 0.0060 4.8765

37-38 0.03 0.0045 25.9002

39-74 0.04 0.0060 5.0913 34.00

41-44 0.05 0.0075 5.1275

42-71 0.04 0.0060 4.9248

45-57 0.04 0.0060 15.6508

46-67 0.04 0.0060 10.3150

49-52 0.02 0.0030 5.2299

50-55 0.03 0.0045 19.6799

53-58 0.02 0.0030 4.8240

56-60 0.04 0.0060 23.4035

59-61 0.04 0.0060 5.0026

62-64 0.06 0.0090 5.0599

63-68 0.05 0.0075 5.0757

65-69 0.02 0.0030 5.2299

66-75 0.03 0.0045 4.8854

70-72 0.04 0.0060 5.0000

73-76 0.02 0.0030 5.0785

Ozgur Baskan


need to be made to reach this value, as it shown in Table 7. On

the other hand, the value of required UE assignment numbers to

reach optimal OFVs varies according to the used algorithms, but the

SA and AL need much more UE assignments, namely 3900 and

2700, than the other compared algorithms to reach corresponding

OFVs. In comparison with the SA and AL algorithms, HJ, EDO,

SAB, QNEW, PT and HS algorithms reached to the corresponding

optimal solutions with less number of UE assignments. Although

SA and AL slightly outperformed than the other algorithms in

terms of the OFV, they need much more UE assignments to

reach these optimal values. As it seen in Table 7, HJ, EDO, SAB,

QNEW and PT algorithms need less number of UE assignments

than the HS algorithm to reach the best OFVs, but the HS

Table 5. Travel Demand Matrix for Sioux Falls Network (Thousands of Vehicles/peak-hour)

11 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.11 0.11 0.55 0.22 0.33 0.55 0.88 0.55 1.43 0.55 0.22 0.55 0.33 0.55 0.55 0.44 0.11 0.33 0.33 0.11 0.44 0.33 0.11

2 0.11 0.11 0.22 0.11 0.44 0.22 0.44 0.22 0.66 0.22 0.11 0.33 0.11 0.11 0.44 0.22 0.00 0.11 0.11 0.00 0.11 0.00 0.00

3 0.11 0.11 0.22 0.11 0.33 0.11 0.22 0.11 0.33 0.33 0.22 0.11 0.11 0.11 0.22 0.11 0.00 0.00 0.00 0.00 0.11 0.11 0.00

4 0.55 0.22 0.22 0.55 0.44 0.44 0.77 0.77 1.32 1.54 0.66 0.66 0.55 0.55 0.88 0.55 0.11 0.22 0.33 0.22 0.44 0.55 0.22

5 0.22 0.11 0.11 0.55 0.22 0.22 0.55 0.88 1.10 0.55 0.22 0.22 0.11 0.22 0.55 0.22 0.00 0.11 0.11 0.11 0.22 0.11 0.00

6 0.33 0.44 0.33 0.44 0.22 0.44 0.88 0.44 0.88 0.44 0.22 0.22 0.11 0.22 0.99 0.55 0.11 0.22 0.33 0.11 0.22 0.11 0.11

7 0.55 0.22 0.11 0.44 0.22 0.44 1.10 0.66 2.09 0.55 0.77 0.44 0.22 0.55 1.54 1.10 0.22 0.44 0.55 0.22 0.55 0.22 0.11

8 0.88 0.44 0.22 0.77 0.55 0.88 1.10 0.88 1.76 0.88 0.66 0.66 0.44 0.66 2.42 1.54 0.33 0.77 0.99 0.44 0.55 0.33 0.22

9 0.55 0.22 0.11 0.77 0.88 0.44 0.66 0.88 3.08 1.54 0.66 0.66 0.66 0.99 1.54 0.99 0.22 0.44 0.66 0.33 0.77 0.55 0.22

10 1.43 0.66 0.33 1.32 1.10 0.88 2.09 1.76 3.08 4.4 2.20 2.09 2.31 4.4 4.84 4.29 0.77 1.98 2.75 1.32 2.86 1.98 0.88

11 0.55 0.22 0.33 1.65 0.55 0.44 0.55 0.88 1.54 4.29 1.54 1.10 1.76 1.54 1.54 1.10 0.11 0.44 0.66 0.44 1.21 1.43 0.66

12 0.22 0.11 0.22 0.66 0.22 0.22 0.77 0.66 0.66 2.2 1.54 1.43 0.77 0.77 0.77 0.66 0.22 0.33 0.44 0.33 0.77 0.77 0.55

13 0.55 0.33 0.11 0.66 0.22 0.22 0.44 0.66 0.66 2.09 1.1 1.43 0.66 0.77 0.66 0.55 0.11 0.33 0.66 0.66 1.43 0.88 0.88

14 0.33 0.11 0.11 0.55 0.11 0.11 0.22 0.44 0.66 2.31 1.76 0.77 0.66 1.43 0.77 0.77 0.11 0.33 0.55 0.44 1.32 1.21 0.44

15 0.55 0.11 0.11 0.55 0.22 0.22 0.55 0.66 1.1 4.4 1.54 0.77 0.77 1.43 1.32 1.65 0.22 0.88 1.21 0.88 2.86 1.10 0.44

16 0.55 0.44 0.22 0.88 0.55 0.99 1.54 2.42 1.64 4.84 1.54 0.77 0.66 0.77 1.32 3.08 0.55 1.43 1.76 0.66 1.32 0.55 0.33

17 0.44 0.22 0.11 0.55 0.22 0.55 1.1 1.54 0.99 4.29 1.1 0.66 0.55 0.77 1.65 3.08 0.66 1.87 1.87 0.66 1.87 0.66 0.33

18 0.11 0.00 0.00 0.11 0.00 0.11 0.22 0.33 0.22 0.77 0.22 0.22 0.11 0.11 0.22 0.55 0.66 0.33 0.44 0.11 0.33 0.11 0.00

19 0.33 0.11 0.00 0.22 0.11 0.22 0.44 0.77 0.44 1.98 0.44 0.33 0.33 0.33 0.88 1.43 1.87 0.33 1.32 0.44 1.32 0.33 0.11

20 0.33 0.11 0.00 0.33 0.11 0.33 0.55 0.99 0.66 2.75 0.66 0.55 0.66 0.55 1.21 1.76 1.87 0.44 1.32 1.32 2.64 0.77 0.44

21 0.11 0.00 0.00 0.22 0.11 0.11 0.22 0.44 0.33 1.32 0.44 0.33 0.66 0.44 0.88 0.66 0.66 0.11 0.44 1.32 1.98 0.77 0.55

22 0.44 0.11 0.11 0.44 0.22 0.22 0.55 0.55 0.77 2.86 1.21 0.77 1.43 1.32 2.86 1.32 1.87 0.33 1.32 2.64 1.98 2.31 1.21

23 0.33 0.00 0.11 0.55 0.11 0.11 0.22 0.33 0.55 1.98 1.43 0.77 0.88 1.21 1.1 0.55 0.66 0.11 0.33 0.77 0.77 2.31 0.77

24 0.11 0.00 0.00 0.22 0.00 0.11 0.11 0.22 0.22 0.88 0.66 0.55 0.77 0.44 0.44 0.33 0.33 0.00 0.11 0.44 0.55 1.21 0.77

Fig. 3. Sioux Falls Network

Table 6. The Compared Algorithms with the HS on Sioux Falls Net-

work

Methods Sources

Hooke-Jeeves (HJ)Abdulaal and LeBlanc(1979)

Equilibrium Decomposed Optimization(EDO)

Suwansirikul et al. (1987)

Simulated Annealing (SA) Friesz et al. (1992)

Sensitivity Analysis Based algorithm (SAB) Yang and Yagar (1995)

Augmented Lagrangian algorithm (AL) Meng et al. (2001)

Quasi-NEWton projection method (QNEW) Chiou (2005)

PARATAN version of gradient projectionmethod (PT)

Chiou (2005)


Vol. 00, No. 00 / 000 0000 − 9 −

algorithm produced better solution than those generated by other

algorithms in terms of the OFV except HJ. The HJ algorithm was

slightly better than the HS in terms of the OFV, and it required

less number of UE assignments. To show the effectiveness of the

HS and its robustness, the HS algorithm was solved 100 times

and the best and worst OFVs, and average number of UE

assignments are also given as shown in Table 7. The average

number of UE assignments was found as 910.45 for 100 runs

while the best and worst OFVs are obtained as 81.83 and 84.67,

respectively. The best and worst OFVs show also the range of

non-optimality of the best solution found by the HS algorithm.

5. Conclusions

In this paper, the HS algorithm was employed to solve the

CNDP problem with link capacity expansions. The CNDP is

modeled as a bilevel programming model which is nonconvex

and nondifferentiable. The upper level objective function is

defined as the sum of the total travel time and total investment

costs of link capacity expansions on the network while the lower

level problem is formulated as user equilibrium traffic

assignment model. The Frank-Wolfe method was used to solve

the traffic assignment problem at the lower level. Numerical

computations and comparisons are conducted on eighteen link

and Sioux Falls networks. Firstly, for the eighteen link network,

the HS approach produced better results than SA and GA

methods in terms of the OFV and required UE assignment

number. The HS algorithm shows steady convergence towards

the global or near global optimum for all cases for the eighteen

link network. Secondly, the network of Sioux Falls is used to

show the effectiveness and robustness of the HS algorithm on the

realistic test network. In comparison with the results obtained by

EDO, SAB, QNEW, and PT algorithms in solving the CNDP, the

HS algorithm yielded slightly better performance in terms of the

OFV. The HJ, SA and AL algorithms slightly outperformed than

the HS in terms of the OFV, but they need much more UE

assignments to reach corresponding best OFVs except HJ.

This research can be extended in multiple directions from the

practical perspective for future studies. The HS proposed

algorithm would help to manage traffic states in a real time

traffic control since updating traffic states dynamically is a

crucial importance for real time traffic control with link capacity

expansions in urban road networks. Thus, this research would

further be improved by taking the dynamic traffic characteristics

into account.

6. Notations

A = The set of links,

crs = The set of minimum path travel times between O-D pair

rs

D = The vector of O-D pair demands, ,

f = The vector of path flows, , ,

g = The vector of investment costs,

h = The constraint set of the lower level decision vector

H = The constraint set of the upper level decision vector

Krs = The set of paths between O-D pair ,

S = The set of destinations

R = The sset of origins

t = The vector of link travel times,

u = The vector of upper bound for link capacity expansions,

x = The vector of equilibrium link flows, ,

y = The vector of link capacity expansions, ,

Z= Upper level objective function

a∀ A∈

r R s S∈,∈∀D Drs[ ] r R∈∀= s S∈

f f k

rs[ ] r R∈∀,= s S∈ k Krs∈

g ga ya( )[ ]= a A∈∀

rs r R∈∀ s S∈

t ta xa ya,( )[ ]= a A∈∀

u ua[ ] a∀ A∈,=

x xa[ ]= a A∈∀y ya[ ]= a A∈∀

Table 7. Comparison of Results for All Algorithms on Sioux Falls Network

HJ EDO SA SAB AL QNEW PT HS

Upper bound of ya

25.0 25.0 25.0 25.0 25.0 25.0 25.0

Initial value of ya

1.0 12.5 6.25 12.5 12.5 6.25 6.25

y16 3.8 4.59 5.38 5.7392 5.5728 4.9776 4.7921 4.4482

y17 3.6 1.52 2.26 5.7182 1.6343 5.0287 5.0827 1.2926

y19 3.8 5.45 5.50 4.9591 5.6228 1.9412 2.0046 5.4675

y20 2.4 2.33 2.01 4.9612 1.6443 2.1617 1.3947 2.3064

y25 2.8 1.27 2.64 5.5066 3.1437 2.6333 2.6430 0.6453

y26 1.4 2.33 2.47 5.5199 3.2837 2.7923 2.8031 2.7100

y29 3.2 0.41 4.54 5.8024 7.6519 5.7462 5.3823 4.1596

y39 4.0 4.59 4.45 5.5902 3.8035 5.6519 5.4699 3.6761

y48 4.0 2.71 4.21 5.8439 7.3820 4.5738 5.0102 4.9047

y74 4.0 2.71 4.67 5.8662 3.6935 4.1747 4.4771 4.3878

Z 81.77 83.47 80.87 84.21 81.75 83.08 82.72 81.83

UE* 108 12 3900 11 2700 5 9 895

Best OFV 81.83a

Worst OFV 84.67a

Average UE* 910.45a

aThe values were obtained for 100 runs.

Ozgur Baskan


z = Lower level objective function

αa, βa=The parameters of link cost function,

= The link/path incidence matrix variables, , ,

, . . if route k between O-D pair rs

uses link a, and otherwise

θa = The link capacity,

ρ= The conversion factor from investment cost to travel times

Acknowledgements

The author would like to thank the anonymous referees for

their constructive and useful comments during the development

stage of this paper.

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harmony search algorithm for continuous network design problem with link capacity expansions

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