Overview of Applications and Developments in the Harmony Search Algorithm - Music-Inspired Harmony Search Algorithm: Theory and Applications - page 22

using a modified HS method for several combinations of different types of design vari-
ables, including cross-sectional area, nodal position and topology (the presence or ab-
sence of a beam). Structures included 10- and 18-bar plane trusses, 25- and 39-bar space
trusses, 52- and 132-bar dome trusses, and a 160-bar three-dimensional transmission
tower. Lo's modified algorithm, HS-DLM, focussed on constraint handling using
Discrete Lagrange Multipliers (DLM).
Lee and Geem [4] reported on the geometrical optimisation of a pressure vessel and a
welded beam. These examples are also considered by Jang et al [54] using a new hybrid
Nelder-Mead (NM) simplex algorithm - HS technique, and by Mahdavi et al [18] using
an adaptive harmony search method (IHS). A modified algorithm combining gradient-
based Sequential Quadratic Programming (SQP) and HS was applied to three structural
problems, including the welded beam, in [55]. The design of an offshore mooring sys-
tem was considered in [56]. The sizes of the mooring system components were sought
to minimise the system cost, subject to constraints on the displacement of the moored
vessel, cable tension, and length of mooring chain lying on the seabed.
3.3 Benchmark Optimisation
Twelve benchmark continuous optimisation problems, including unconstrained and
constrained examples with 2-10 variables, were investigated using the original HS al-
gorithm in an extensive study in [4]. Benchmark problems are also briefly discussed
in [1, 7-9, 20, 30, 42].
Omran and Mahdavi [27] compared the performance of the original HS, IHS [18] and a
new Global-best HS (GHS) algorithm on ten continuous functions of up dimension 100,
and six integer programming problems with up to 30 variables. They also studied the sen-
sitivity to HS parameters and the effect of noise. Mukhopadhyay et al [21] compared
another modified HS method with IHS, GHS and Differential Evolution (DE) for five
continuous test functions. Gao et al [57] developed a modified HS method specifically for
multi-modal functions and applied it to three 2-dimensional multi-modal functions.
Optimisation of a selection of continuous functions with 2-30 variables using the
hybrid Harmony Annealing Algorithm (HAA) was reported in [58, 59]. The
Rastrigin, Griewank and Sphere functions were optimised for 30, 50 and 100 vari-
ables using a hybrid PSO-HS algorithm developed for high-dimension problems in
[26]. Results for the same three functions in 2-30 dimensions were reported in [60]
for a GA-HS algorithm. A hybrid HS-DE method for uni-modal problems was tested
on eight 50-dimensional benchmark functions in [57]. Jang et al [54] optimised two
unconstrained and three constrained bench-mark functions with 2-7 continuous vari-
ables using a hybrid NM-HS method. A hybrid algorithm that combined elements
from GA, HS, NM and the Tabu Search (TS) was tested on six continuous functions
of dimension 2 to 10 in [61]. Fesanghary et al [55] applied a modified HS-SQP
method to two constrained and two unconstrained benchmark problems.
3.4 Soil Stability Analysis
A body of soil with an inclined surface may become unstable and slip. The aim of
slope stability analysis is to predict the location of the surface inside the soil body
where slippage may occur (the critical slip surface) and to estimate the associated fac-
tor of safety, which is the ratio of inherent shear strength of the soil to the shear stress