Multi-Hive Artificial Bee Colony Algorithm for
Constrained Multi-Objective Optimization
Hao Zhang
1,2
, Yunlong Zhu
1
, Xiaohui Yan
1,2
Key Laboratory of Industrial Informatics
1
Shenyang Institute of Automation of Chinese Academy of Sciences
110016, Shenyang, China
2
Graduate School of the Chinese Academy of Sciences,
100039, Beijing, China
{zhanghao, ylzhu, yanxiaohui}@sia.cn
Abstract—This paper presents a general cooperative coevolution
model inspired by the concept and main ideas of the coevolution
of symbiotic species in natural ecosystems. A novel approach
called “multi-hive artificial bee colony” for constrained multi-
objective optimization (MHABC-CMO) is proposed based on this
model. A novel information transfer strategy among multiple
swarms and division operator are proposed in MHABC-CMO to
tie it closer to natural evolution, as well as improve the
robustness of the algorithm. Simulation experiment of MHABC-
CMO on a set of benchmark test functions are compared with
other nature inspired techniques which includes multi-objective
artificial bee colony (MOABC), nondominated sorting genetic
algorithm II (NSGAⅡ) and multi-objective particle swarm
optimization (MOPSO). The numerical results demonstrate
MHABC-CMO approach is a powerful search and optimization
technique for constrained multi-objective optimization.
Keywords-Multi-Hive; ABC algorithm; Constraint; Multi-
objective Optimization; symbiosis theory
I. INTRODUCTION
Many real world optimization problems in engineering,
finance, and science involve simultaneous optimization of
several objective functions. Generally, these objective
functions are noncommensurable and often competing and
conflicting. Multi-objective optimization having such objective
functions gives rise to a set of optimal solutions, instead of one
optimal solution because no solution can be considered to be
better than any other with respect to all objectives. A solution
x
1
of the multi-objective problem is said to be Pareto optimal
iff there does not exist another solution x
2
, such that f (x
1
)
dominates f (x
2
). These optimal solutions are called Pareto-
optimal solutions. There are often many constraint conditions
in multi-objective problems. Generally, constrained multi-
objective optimizations (CMO) are more difficult to solve, as
finding a feasible solution may require substantial
computational resources. Constrained multi-objective
optimization is important from the point of view of practical
problem solving, but not much attention has been paid so far in
this respect among the researchers.
Generally, multi-objective optimization problems with
constrained conditions consist of n decision variables, m
objective functions, p equality constraints and q inequality
constraints. It can be formulated as follows:
12
Minimize () [ (), (), ()]
m
yfx fxfx fx== (1)
( ) 0, 1, 2, ,
Subject to:
( ) 0, 1, 2, ,
i
i
xi p
hx i q
≤=
⎧
⎨
==
⎩
(2)
where x = (x
1
, x
2
,…, x
n
)∈D is a decision vector that represents
a solution, y = (f
1
, f
2
,…, f
m
)∈Y represent objective functions,
D is a n-dimensional search space for decision vectors, and Y
is a m-dimensional search space for objective vectors.
The set of optimal trade-offs forms the solution set which is
called the Pareto set and it is denoted by P
*
. The set PF
*
= { f
(x) | x∈P
*
} is called the Pareto front.
Multi-objective optimization problems introduce challenges
to researchers, since the traditional techniques like linear and
non-linear programming are unable to solve them efficiently.
In pursuit of finding Pareto-optimal solutions to these multi-
objective optimization problems many researchers have been
drawing inspiration from the nature [1]. Over the past two
decade, a lot of successful multi-objective algorithms based on
such biologically inspired algorithms to optimize multi-
objective problems were proposed in literature, such as Pareto-
archived evolution strategy (PAES) [2], Pareto envelope-based
selection algorithm (PESA)-II [3], nondominated sorting
genetic algorithm II (NSGAII) [4], strength Pareto evolutionary
algorithm (SPEA2) [5], indicator-based evolutionary algorithm
(IBEA) [6], multi-objective particle swarm optimization
(MOPSO) [7], multi-objective evolutionary algorithm based on
Decomposition (MOEA/D) [8], two lbests multi-objective
particle swarm optimization (2LB-MOPSO) [9], multi-
objective differential evolution (MODE) based on summation
of normalized objective values and diversified selection
(SNOV-DS) [10] and so on. The primary reason for this is their
ability to find multiple Pareto-optimal solutions in one single
simulation run.
How to deal with the infeasible individuals throughout the
search process is one of the major issues of constrained multi-
objective optimization. Some researches have been conducted
in the area of constrained multi-objective problems. Different
constraint handling techniques have been proposed in different
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WCCI 2012 IEEE World Congress on Computational Intelligence
June, 10-15, 2012 - Brisbane, Australia