Overview of Applications and Developments in the Harmony Search Algorithm - Music-Inspired Harmony Search Algorithm: Theory and Applications - page 31

In some modified algorithms only a single, simple change to the original HS
method is made; in others several changes are combined in a complex way. Many in-
dividual innovations were summarised in [6].

4.2 Survey of Modified Algorithms

In this section, the innovations used in the current range of modified HS algorithms
are outlined. We distinguish between hybrid and non-hybrid algorithms. We have de-
fined a hybrid algorithm loosely as one whose structure is essentially no longer HS.
Table 2 presents an algorithm-innovation matrix for modified, non-hybrid HS meth-
ods. Table 3 lists hybrid HS methods, and the reader is guided to the references to
explore those methods further.

Table 3. Hybrid HS Methods

Author's Algorithm Name Type of Hybrid Reference

GA incorporated with Harmony Procedure GA + HS [11]

Hybrid Algorithm (HA) GA + HS + NM + TS [61]

Improved GA GA + HS [65, 66]

Mixed Search Algorithm (MSA) PSO + HS [28]

Novel Hybrid Real-Valued GA (NHRVGA) GA + HS [60]

Novel Hybrid PSO (NHPSO) PSO + HS [26]

Improved PSO (IPSO), Heuristic PSO (HPSO) PSO + HS [51-53]

New Hybrid Metaheuristic Algorithm GA + SA + HS [75]

HSCLUST / HClust, HKClust, IHKClust HS / IHS + K -means [15, 71]

Fusion of HS and DE (HS-DE) HS + DE [57]

Simplex-Harmony Search (SHS) HS + NM [54]

New version of PSO PSO + HS [68]

HS-DLM Algorithm HS + DLM [29]

Hybrid Optimization Method CSA + HS [72]

4.3 Theoretical Analyses of the Original HS Algorithm

A few publications attempt a theoretical analysis of the HS algorithm, although many pre-
sent limited sensitivity analyses using the algorithm parameters HMS, HMCR and PAR.
In brief early work, Geem [1, 7] calculated the probability of finding the optimal
solution in HM for

PAR . More recently, he derived the 'stochastic partial deriva-
tive' [20], which in fact estimates the probability that a particular value will be chosen
for a discrete decision variable given the current contents of HM. A benchmark func-
tion and a water distribution problem were used to show how the 'derivative' evolves
to favour the optimal solution. Mukhopadhyay and colleagues [21] present an infor-
mative population variance analysis for HS. They develop an expression for the ex-
pected variance of the solution vectors stored in HM, and use it to show how the HS
algorithm can be modified to maintain diversification during its iterations.

= 0