the opposition-based harmony search algorithm

ORIGINAL CONTRIBUTION

The Opposition-based Harmony Search Algorithm

R. P. Singh • V. Mukherjee • S. P. Ghoshal

Received: 16 April 2012 / Accepted: 25 November 2013 / Published online: 8 January 2014

� The Institution of Engineers (India) 2014

Abstract This paper proposes a novel approach to accel-

erate the harmony search (HS) algorithm. The proposed

opposition-based HS of the present work employs opposition-

based learning for harmony memory initialization and also for

the generation jumping. In the present work, opposite numbers

have been utilized to improve the convergence rate of the HS.

The potential of the proposed algorithm is assessed by means

of an extensive comparative study of numerical results on

benchmark test functions. The results obtained confirm the

potential and effectiveness of the proposed algorithm com-

pared to some other algorithms surfaced in the recent state-of-

the art literatures. Additionally, the opposition concept has

been incorporated in an improved variant of HS such as local-

best HS algorithm with dynamic subpopulations and the

potential of incorporation of opposition concept in evolu-

tionary optimization algorithm is established.

Keywords Benchmark test function � Harmony search �Opposite numbers � Optimization

Introduction

The researchers, over the globe, are being inspired by

nature-inspired meta-heuristics [1] on a regular basis to

meet the demands of complex real-world optimization

problems. Thus, the computational costs of the algorithms

are being, dramatically, reduced for the recent past.

Being inspired by this tradition, Geem et al. [2] proposed

harmony search (HS) in 2001. It is a derivative-free meta-

heuristic algorithm. It is a new variant of meta-heuristic

algorithm inspired by the natural musical performance pro-

cess that occurs when a musician searches for a better state of

harmony. In the HS algorithm, the solution vector is analo-

gous to the harmony in music and the local and global search

schemes are analogous to the musician’s improvisations. In

comparison to other meta-heuristics in the literature, the HS

algorithm imposes fewer mathematical requirements and

can be easily adapted for solving various kinds of engi-

neering optimization problems. Furthermore, numerical

comparisons demonstrated that the evolution in the HS

algorithm is faster than genetic algorithm [3]. Therefore, HS

algorithm has captured much attention and has been, suc-

cessfully, applied to solve a wide range of practical optimi-

zation problems, such as structural optimization [4],

parameter estimation of nonlinear Muskingum model [5],

pipe network design [6], vehicle routing [7], design of water

distribution networks [8], scheduling of a multiple dam

system [9], and so on.

HS algorithm is good at identifying the high perfor-

mance regions of solution space within a reasonable time

[10]. Mahdavi et al. [3] presented an improved HS (IHS)

algorithm, by introducing a strategy to dynamically tune

the key parameters. Omran and Mahdavi [11] proposed a

global best HS (GHS) algorithm, by borrowing the concept

from the swarm intelligence. Pan et al. in [12], proposed a

self-adaptive global best HS (SGHS) algorithm for solving

continuous optimization problems.

Tizhoosh [13] introduced the concept of opposition-based

learning (OBL). This notion has been applied to accelerate

R. P. Singh

Department of Electrical Engineering, Asansol Engineering

College, Asansol, India

V. Mukherjee (&)

Department of Electrical Engineering, Indian School of Mines,

Dhanbad, India

e-mail: [email protected]

S. P. Ghoshal

Department of Electrical Engineering, National Institute of

Technology, Durgapur, India

123

J. Inst. Eng. India Ser. B (December 2013–February 2014) 94(4):247–256

DOI 10.1007/s40031-013-0069-5

the reinforcement learning and the backpropagation learning

in neural networks. The main idea behind OBL is the

simultaneous consideration of an estimate and its corre-

sponding opposite estimate (i.e., guess and opposite guess)

in order to achieve a better approximation for the current

candidate solution. In the recent literature, the concept of

opposite numbers has been utilized to speed up the conver-

gence rate of an optimization algorithm e.g. opposition-

based differential evolution (ODE) [14]. In this paper, OBL

has been utilized to accelerate the convergence rate of

the HS. Hence, our proposed approach has been called as

opposition-based HS (OHS). OHS uses opposite numbers

during HM initialization and also for generating the new

HM during the evolutionary process of HS. Additionally,

the concept of OBL is applied to an improved variant of

HS algorithm reported by Pan et al. [15] (termed as DLHS in

Ref. [15]) and this new algorithm is termed as ODLHS

in the present work. The potential of ODLHS is tested on a

suite of first fourteen CEC 2005 benchmark test functions

[16].

The objectives of the current article may be noted as

presented below.

• The proposed algorithm has been tested on a suite of

standard benchmark test functions.

• The obtained optimal results on benchmark test func-

tions are compared to other variants of HS reported in

the recent literatures.

• The comparative convergence profiles of fitness function

values for a few benchmark test functions are presented.

A Brief Description of HS Algorithm

In the basic HS algorithm, each solution is called a harmony.

It is represented by an n-dimension real vector. An initial

randomly generated population of harmony vectors is stored

in an HM. Then, a new candidate harmony is generated from

all of the solutions in the HM by adopting a memory con-

sideration rule, a pitch adjustment rule and a random re-

initialization. Finally, HM is updated by comparing the new

candidate harmony vector and the worst harmony vector in

the HM. The worst harmony vector is replaced by the new

candidate vector if it is better than the worst harmony vector

in the HM. The above process is repeated until a certain

termination criterion is met. Thus, the basic HS algorithm

consists of three basic phases. These are initialization,

improvisation of a harmony vector and updating the HM.

Sequentially, these phases are described below.

(i) Initialization of the Problem and the Parameters of

HS Algorithm

In general, a global optimization problem can be

enumerated as follows: min f ðxÞstxj 2 ½paraminj ;

paramaxj �; j ¼ 1; 2; . . .; n, where f ðxÞ is the

objective function; X ¼ ½x1; x2; . . .xn� is the set of

design variables; n is the number of design variables.

Here, paraminj ; paramax

j are the lower and upper bounds

for the design variable xj, respectively. The parameters

of the HS algorithm are harmony memory size (HMS)

(the number of solution vectors in HM), harmony

memory consideration rate (HMCR), pitch adjusting

rate (PAR), distance bandwidth (BW), and number of

improvisations (NI). The NI is the same as the total

number of fitness function evaluates (NFFEs). It may

be set as a stopping criterion.

(ii) Initialization of HM

HM consists of HMS harmony vectors. Let Xj ¼x

j1; x

j2; . . .; x j

n

� �represent the jth harmony vector

which is randomly generated within the parameter

limits [paraminj ; paramax

j ]. Then, the HM matrix is

filled with the HMS harmony vectors as in (1).

HM ¼

x11 x1

2 . . . x1n

x21 x2

2 . . . x2n

. . .xHMS

1 xHMS2 . . . xHMS

n

2

6664

3

7775ð1Þ

(iii) Improvisation of a New Harmony

A new harmony vector Xnew ¼ xnew1 ; xnew

2 ; . . .; xnewn

� �is

generated (called improvisation) by applying three

rules, namely, (i) a memory consideration, (ii) a pitch

adjustment, and (iii) a random selection. First of all, a

uniform random number r1 is generated in the range [0,

1]. If r1 is less than HMCR, the decision variable xnewj is

generated by the memory consideration, otherwise, xnewj

is obtained by a random selection (i.e., random re-

initialization between the search bounds). In the

memory consideration, xnewj is selected from any

harmony vector i in ½1; 2; . . .;HMS�. Secondly, each

decision variable xnewj will undergo a pitch adjustment

with a probability of PAR if it is updated by the memory

consideration. The pitch adjustment rule is given as

follows:

xnewj ¼ xnew

j � r3 � BW ð2Þ

where r3 is a uniform random number between 0 and 1.

(iv) Updating of HM

After a new harmony vector Xnewj is generated, HM

will be updated by the survival of the fittest vector

between Xnew and the worst harmony vector Xworst in

the HM. That is, Xnew will replace Xworst and become

a new member of the HM if the fitness value of Xnew

is better than the fitness value of Xworst.

248 J. Inst. Eng. India Ser. B (December 2013–February 2014) 94(4):247–256

123

(v) Process of Computation

The computational procedure of the basic HS

algorithm can be summarized as follows [2].

IHS Algorithm

The basic HS algorithm uses fixed values for PAR and BW

parameters. IHS algorithm, proposed by Mahdavi et al. [3],

applies the same memory consideration, pitch adjustment and

random selection on the basic HS algorithm but, dynamically,

updates the values of PAR and BW as in (3) and (4), respectively.

PARðgnÞ ¼ PARmin þ PARmax � PARmin

NI� gn ð3Þ

BWðgnÞ ¼ BWmax � e

ln BWmin

BWmax

� �

NI� gn

0

@

1

A

ð4Þ

In Eq. (3), PAR(gn) is the pitch adjustment rate in the current

generation (gn), PARmin and PARmax are the minimum and the

maximum adjustment rate, respectively. In (4), BW (gn) is the

distance bandwidth at generation (gn), BWmin and BWmax are

the minimum and the maximum bandwidths, respectively.

Opposition-based Learning: A Concept

Evolutionary optimization methods start with some initial

solutions (initial population) and try to improve them toward

some optimal solution(s). The process of searching terminates

when some predefined criteria are satisfied. In the absence of a

priori information about the solution, it is usually started with

random guesses. The computation time, among others, is related

to the distance of these initial guesses from the optimal solution.

The chance of starting with a closer (fitter) solution can be

improved by simultaneously checking the opposite solution [13].

By doing this, the fitter one (guess or opposite guess) can be

chosen as an initial solution. In fact, according to the theory of

probability, 50 % of the time a guess is further from the solution

than its opposite guess. Therefore, starting with the closer of the

two guesses (as judged by its fitness) has the potential to accel-

erate convergence. The same approach can be applied not only to

initial solutions but also continuously to each solution in the

current population.

(i) Definition of Opposite Number

Let x 2 ½ub; lb� be a real number. The opposite

number is defined as in (5).

x^ ¼ ubþ lb� x ð5Þ

Similarly, this definition can be extended to higher

dimensions [13] as stated next.

(ii) Definition of Opposite Point

Let X ¼ ðx1; x2; . . .; xnÞ be a point in n-dimensional

space, where ðx1; x2; . . .; xnÞ 2 R and xi 2 ½ubi; lbi�8i

2 f1; 2; . . .; ng. The opposite point X^

¼ ðx^1; x^

2; . . .;

x^

nÞ is completely defined by its components as in Eq. (6).

x^

i ¼ ubi þ lbi � xi ð6Þ

Now, by employing the opposite point definition, the

opposition-based optimization is defined next.

(iii) Opposition-based Optimization

Let X ¼ ðx1; x2; . . .; xnÞ be a point in n-dimensional

space (i.e., a candidate solution). Assume f ¼ ð�Þ is a

fitness function which is used to measure the candi-

date’s fitness. According to the definition of the

opposite point, X^

¼ ðx^1; x^

2; . . .; x^

nÞ is the opposite of

X ¼ ðx1; x2; . . .; xnÞ. Now, if f ðX^

Þ� f ðXÞ, then point X

can be replaced with X^

; otherwise, we continue with

X. Hence, the point and its opposite point are evaluated

simultaneously in order to continue with the fitter one.

Proposed Algorithms

(i) OHS

Similar to all population-based optimization algo-

rithms, two main steps are distinguishable for HS,

namely, HM initialization and producing new HM by

adopting the principle of HS. In the present work, the

strategy of the OBL [13] is incorporated in two steps.

The original HS is chosen as a parent algorithm and

opposition-based ideas are embedded in it with an

intention to exhibit accelerated convergence profile.

J. Inst. Eng. India Ser. B (December 2013–February 2014) 94(4):247–256 249

123

Corresponding pseudo code for the proposed OHS

approach can be summarized as follows:


123

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123

(ii) ODLHS

An improved version of HS such as DLHS reported

in Pan et al. [15] is chosen as the basic algorithm and

the concept of OBL described in this paper is blended

with it and the new variant of HS resulted is termed

as OHLHS.

Experimental Bench: Optimization of Benchmark Test

Function

OHS for Global optimization

(i) Benchmark Test Function

A suite of sixteen global optimization problems

(Table 1) are used to test the performance of the

proposed OHS algorithm. Among these sixteen

benchmark problems, sphere function, Schwefel’s

problem 2.22, step function, rotated hyper-ellipsoid

function, shifted sphere function and shifted Schwe-

fel’s problem 1.2 are unimodal. Step function is

discontinuous. Rosenbrock function, Schwefel’s

problem 2.26, Rastrigin function, Ackley function,

Griewank function, shifted Rosenbrock function,

shifted Rastrigin function, shifted rotated Griewank’s

function and shifted rotated Rastrigin function are

difficult multimodal problems where the number of

local optima increases with the problem dimension.

Six-hump Camel-back function is a low-dimensional

function with only a few local optima.

(ii) Parameter Setting

The best chosen variables for the proposed OHS are

Jr ¼ 0:3, HMCR ¼ 0:95, PARmin ¼ 0:35, PARmax ¼0:99, BWmin ¼ 1:00e� 06, BWmax ¼ 1=20 ðxmax�xminÞ.

(iii) Discussion on Benchmark Function Optimization

Each benchmark test function is run for 25

independent times. The average and standard

deviations over these 25 runs for 30 and 100

dimensions (except for the two-dimensional six-

hump Camel-back function) are presented in

Tables 1 and 2, respectively. Results of interest

are bold faced in the respective tables. The results

of the HS, IHS and GHS algorithms for these

problems are obtained from Omran and Mahdavi

[11] while those for SGHS are taken from Pan

et al. [12].

It can be observed from Table 1 that the OHS algorithm

generates nine best results out of sixteen functions and for

five test functions, OHS and SGHS yield the same results

(for dimension size of 30). For two functions like shiftedTa

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±25.5

76559

36.6

85585

±25.3

11496

509.0

66964

±45.1

83819

-317.2

25748

±2.7

32871

-319.2

01033

–2.4

36781

f 15

-180

1,8

85.1

00054

±12.4

99888

1,8

83.4

99365

±15.4

85959

1,8

29.6

69549

±33.5

04803

1,0

06.1

17891

±35.3

07793

1,0

04.2

31340

–34.1

72839

f 16

-330

341.6

76241

±48.3

72925

334.7

47556

±54.6

93700

763.8

18874

±43.6

13654

66.9

15779

±55.3

75297

46.3

10278

–53.1

12855


123

Rosenbrock function and shifted rotated Griewank’s

function, SGHS yields better results than OHS. It may also

be noted from Table 2 that with the increase in dimen-

sionality of the benchmark test functions, OHS offers sig-

nificantly better results than the compared results. Thus, as

the dimension, thereby, the complexity of the benchmark

test functions increases, OHS offers superior results.

The convergence profiles of the fitness function value

for the 30D—(a) Sphere function, and (b) Schwefel’s

problem 2.22 against the NFFEs are presented in Fig. 1a, b,

respectively. The HS-, IHS-, GHS-, SGHS-, and OHS-

based comparative convergence profiles of the fitness

function values for the 30D shifted Rastrigin function

against NFFEs are presented in Fig. 2. It can be observed

from these figures that the convergence profile of the

proposed OHS-based optimum value for this selected test

function descends much faster than the other compared

algorithms. It points out the fact that the proposed OHS-

based result for this benchmark test function is superior to

the compared methods.

ODLHS for CEC Benchmark Test Functions

(i) Benchmark Test Functions

The proposed ODLHS algorithm is tested and eval-

uated on CEC 2005 bench-mark functions [16]. The

CEC 2005 test problems include twenty-five functions

with different problems; five of them are unimodal

problems and other twenty are multimodal problems.

Out of these twenty-five test problems, first fourteen

test problems are taken in the present work and the

definitions of these test functions [16] are given in

Fig. 1 Comparative

convergence profiles of fitness

function values for 30D

a Sphere function b Schwefel’s

problem 2.22

Fig. 2 Comparative convergence profiles of fitness function values

for 30D shifted Rastrigin function

Table 3 Definition of first

fourteen CEC 2005 [16]

Benchmark test functions

Name Definition Range

f1 Shifted sphere function [-50, 100]

f2 Shifted Schwefel’s problem 1.2 [-50, 100]

f3 Shifted rotated high conditioned elliptic function [-50, 100]

f4 Shifted Schwefel’s problem 1.2 with noise in fitness [-50, 100]

f5 Schwefel’s problem 2.6 with global optimum on bounds [-50, 100]

f6 Shifted Rosenbrock’s function [-50, 100]

f7 Shifted rotated Griewank’s function without bounds [-300, 600]

f8 Shifted rotated Ackley’s function with global optimum on bounds [-16, 32]

f9 Shifted Rastrigin’s function [-2.5, 5]

f10 Shifted rotated Rastrigin’s function [-2.5, 5]

f11 Shifted rotated Weierstrass function [-0.25, 0.5]

f12 Schwefel’s problem 2.13 [-50,100]

f13 Expanded extended Griewank’s plus Rosenbrock’s function [-3, 1]

f14 Expanded rotated extended Scaffe’s F6 [-50,100]


123

Table 3. Sugantha [16] may be referred for detailed

mathematical formula.

(ii) Parameter Setting

The best chosen value of Jr is 0.3. The other

parameters of this algorithm are taken from Pan

et al. [15].

(iii) Discussion on CEC 2005 Benchmark Test Function

Optimization

The results obtained by adopting, the proposed

ODLHS algorithm of the present work on first

fourteen CEC 2005 benchmark test functions are

presented in Table 4 on sample basis. Each bench-

mark test function is run for 25 independent times.

The results yielded by the proposed ODLHS algo-

rithm are compared to SGHS [17], DLHS [17] and

IGHS [17]. In Table 4, an entry shown as ‘–’ means

that the result for this function was not reported in

the original reference. In this table, SGHS-, DLHS-

and IGHS results are taken directly from the

respective references. The results show that the

proposed opposition based strategy in DLHS algo-

rithm also performs well on unimodal and multi-

Table 4 Results of all the algorithms taken over 30 runs

Benchmark function Problem size SGHS [17] DLHS [17] IGHS [17] ODLHSProposed

f1 30 0 2.44e-07 1.19e-07 1.11e-07

0 1.33e-06 1.76e-08 1.04-08

f2 18.90 2.84e?03 1.85e-06 1.11e-06

17.25 1.77e?03 5.75e-07 5.18-07

f3 – 3.19e?06 2.21e?06 2.11e?06

1.72e?06 1.21e?06 1.01e?06

f6 2.12e?03 3.78e?03 1.67e?03 1.44e?03

3.97e?03 4.84e?03 3.58e?03 3.14e?03

f9 1.39e-01 1.58 6.66e-01 6.48e-01

3.50e-01 1.50 7.99e-01 7.89e-01

f10 97.86 – 62.02 62.00

30.03 15.56 15.14

f1 50 – 7.48e-10 3.36e-07 3.36e-07

8.04e-10 5.16e-08 5.11e-08

f2 – 7.98e?03 2.99e-04 2.41e-04

3.29e?03 2.77e-04 2.48e-04

f3 – 1.01e?07 2.70e?06 2.59e?06

3.09e?06 1.07e?06 1.01e?06

f6 – 3.08e?03 2.93e?03 2.78e?03

4.11e?03 3.79e?03 3.19e?03

f9 – 2.57 3.24e-01 3.11e-01

2.50 6.98e-01 6.48e-01

f1 100 2.00e-05 – 2.98e-06 2.49e-06

9.30e-05 3.87e-07 3.78e-07

f2 6.37e?04 – 5.81e?04 5.11e?04

1.24e?04 8.87e?03 8.41e?03

f6 391.51 – 138.91 138.14

293.23 62.59 62.59

f9 12.77 – 16.54 16.11

2.73 3.31 3.11

f10 396.92 – 371.98 370.17

55.37 57.28 57.14

f11 – – – 5.14e?0

f12 – – – 2.61e?4

f13 – – – 1.98e?0

f14 – – – 3.14e?0


123

modal functions. The SGHS-, DLHS-, IGHS- and the

proposed ODLHS-based comparative convergence

profiles of the fitness function values for 30D shifted

sphere function, shifted Schwefel’sproblem1.2 and

shifted rotated high conditioned elliptic function are

depicted in Figs. 3 to 5, in order. It may be observed

from these figures that blending of opposition

strategy in an improved variant of HS such as DLHS

(termed as ODLHS) offers faster convergence profile

of fitness function value for these selected test

functions as compared to other algorithms.

Conclusion

In this paper, the concept of OBL has been employed to

accelerate the basic HS algorithm as well as an improved

variant of HS such as DLHS algorithm. The notion of OBL

has been utilized to introduce opposition-based HM ini-

tialization and opposition-based generation jumping. By

embedding these two steps within the HS framework, OHS

algorithm and ODLHS are proposed in this paper. The

proposed algorithms are tested on benchmark test func-

tions. The simulation results demonstrate the effectiveness

and robustness of these proposed algorithms to solve

benchmark test functions. Moreover, the results yielded by

the proposed algorithms have been compared to those

surfaced in the recent state-of-the-art literatures. The

comparison of the numerical results and the convergence

profiles of the optimum objective function values confirm

the effectiveness and the superiority of the proposed two

approaches of the current article.

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the opposition-based harmony search algorithm

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