A Multi-objective Tabu Search Algorithm for Constrained Optimisation Problems.
ABSTRACT Real-world engineering optimisation problems are typically multi-objective and highly constrained, and constraints may be
both costly to evaluate and binary in nature. In addition, objective functions may be computationally expensive and, in the
commercial design cycle, there is a premium placed on rapid initial progress in the optimisation run. In these circumstances,
evolutionary algorithms may not be the best choice; we have developed a multi-objective Tabu Search algorithm, designed to
perform well under these conditions. Here we present the algorithm along with the constraint handling approach, and test it
on a number of benchmark constrained test problems. In addition, we perform a parametric study on a variety of unconstrained
test problems in order to determine the optimal parameter settings. Our algorithm performs well compared to a leading multi-objective
Genetic Algorithm, and we find that its performance is robust to parameter settings.
- SourceAvailable from: Timoleon KipourosAIAA Journal 01/2008; 46(3):701-711. · 1.08 Impact Factor
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ABSTRACT: While there have been many adaptations of some of the more popular meta-heuristics for continuous multi-objective optimisation problems, Tabu Search has received relatively little attention, despite its suitability and effectiveness on a number of real-world design optimisation problems. In this paper we present an adaptation of a single-objective Tabu Search algorithm for multiple objectives. Further, inspired by path relinking strategies common in discrete optimisation problems, we enhance our algorithm to allow it to handle problems with large numbers of design variables. This is achieved by a novel parameter selection strategy that, unlike a full parametric analysis, avoids the use of objective function evaluations, thus keeping the overall computational cost of the procedure to a minimum. We assess the performance of our two Tabu Search variants on a range of standard test functions and compare it to a leading multi-objective Genetic Algorithm, NSGA-II. The path relinking-inspired parameter selection scheme gives a clear performance improvement over the basic multi-objective Tabu Search adaptation and both variants perform comparably with the NSGA-II.European Journal of Operational Research 03/2008; 185(3):1192–1212. · 2.04 Impact Factor
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A Multi-objective Tabu Search Algorithm for
Constrained Optimisation Problems
Daniel Jaeggi, Geoff Parks, Timoleon Kipouros, and John Clarkson
Engineering Design Centre, Department of Engineering,
University of Cambridge, Trumpington Street,
Cambridge CB2 1PZ, United Kingdom
Abstract. Real-world engineering optimisation problems are typically
multi-objective and highly constrained, and constraints may be both
costly to evaluate and binary in nature. In addition, objective functions
may be computationally expensive and, in the commercial design cycle,
there is a premium placed on rapid initial progress in the optimisation
run. In these circumstances, evolutionary algorithms may not be the
best choice; we have developed a multi-objective Tabu Search algorithm,
designed to perform well under these conditions. Here we present the
algorithm along with the constraint handling approach, and test it on a
number of benchmark constrained test problems. In addition, we perform
a parametric study on a variety of unconstrained test problems in order
to determine the optimal parameter settings. Our algorithm performs
well compared to a leading multi-objective Genetic Algorithm, and we
find that its performance is robust to parameter settings.
Real-world optimisation problems have a number of characteristics which must
be taken into account when developing optimisation algorithms. Real-world
problems are typically multi-objective; trade-offs between risk and reward, and
cost and benefit exist at a fundamental level throughout the natural world and
are a deep-seated part of human consciousness. These trade-offs carry over di-
rectly to the business world, and thus into any form of design activity. Any
optimisation method which is to have any serious benefit to the design process
must be able to handle multiple objectives.
Real-world problems also tend to be highly constrained. The nature of these
constraints and their effect on the optimisation landscape varies from problem to
problem. However, optimisation problems in a number of fields have constraints
with similar characteristics, and this is discussed further in Section 1.1 below.
The optimisation landscape – regions of feasible, highly constrained design space
and the variations of objective function values within that space – is strongly
influenced by the parameterisation scheme, for any one given problem. Good
parameterisation schemes for aerodynamic shape optimisation problems – the
particular focus of our work – as shown by Harvey , Kellar  and Gaiddon
C. A. Coello Coello et al. (Eds.): EMO 2005, LNCS 3410, pp. 490–504, 2005.
c ?Springer-Verlag Berlin Heidelberg 2005
A Multi-objective Tabu Search Algorithm 491
et al. , tend to produce optimisation landscapes that are highly constrained,
have many variables, and many local minima. Thus, the optimisation algorithm
must be chosen to perform well in these circumstances .
These characteristics quickly rule out the use of traditional gradient-based op-
timisation methods: notwithstanding their requirement of gradient information
(which may be difficult, expensive, or impossible to obtain), these algorithms
perform poorly in problems which are highly constrained and contain local min-
ima. Harvey  tested a number of meta-heuristic methods on a representative
aerodynamic design optimisation problem and found Tabu Search (TS) to be su-
perior to the Genetic Algorithm (GA) and Simulated Annealing (SA) methods.
Numerous of multi-objective GAs exist . Similarly, multi-objective SA
methods have been developed . However, despite its popularity in single-
objective optimisation problems, very few attempts have been made at develop-
ing a multi-objective version of TS. Jones  reviewed the literature on multi-
objective meta-heuristics and found only 6% of 124 papers concerned with TS.
Given that it may well perform better than a GA or SA method (assuming
Harvey’s results carry over into multi-objective optimisation) on aerodynamic
design optimisation problems, and there is a strong real-world requirement to
perform multi-objective optimisation, there appears to be both a need and an
opportunity to develop a new multi-objective TS algorithm.
It is important that the constraint handling method of an optimisation algorithm
is able to deal with constraints which are binary, as happens in a number of real-
world engineering problems. The constraint handling in many multi-objective
evolutionary algorithms requires the ability to assign some kind of constraint
violation distance to points in design space which violate constraints, and points
are then ranked accordingly . In the presence of binary constraints, such an
approach cannot be used.
Such constraints occur typically in shape optimisation problems, especially
when the cost function is evaluated using a finite difference type of method
(including finite element and finite volume methods) which solves a system of
equations over a finite mesh. The constraints for these problems arise from three
sources, amongst others:
1. Geometric considerations. The parameterisation scheme may give rise to
shapes which are physically impossible (i.e. negative volumes). Conceptually,
a distance measure in design space may be formulated by considering an
offset vector ∆¯ x which can be added to the design vector ¯ x to make the
design feasible; in practice this may be too costly due to the interdependence
between design variables.
2. Mesh considerations. Given a valid geometry, it may be impossible to fit a
mesh that satisfies certain criteria relevant to the numerical solution of the
problem (i.e. skewed cells). Again, finding an offset vector ∆¯ x is conceptually
possible, but is in practice even more costly than finding a ∆¯ x that satisfies
just the geometric considerations.
492D. Jaeggi et al.
3. Numerical solution considerations. Given a valid mesh and geometry, it may
still be impossible to reach a numerical solution of the system of equations
for that geometry (i.e. lack of convergence in a finite difference method). Yet
again, finding a ∆¯ x that makes the solution converge is possible but totally
unrealistic, given the massive computational cost of a solution.
The extent to which such constraint violations are encountered is strongly
dependent on the parameterisation scheme used. For any problem, it is probably
possible to employ a parameterisation scheme that gives a continuous design
space and constraint violations for which ∆¯ x may be easily calculated. However,
it is the authors’ belief that the parameterisation scheme should be chosen first
and then the optimisation algorithm, not the other way round. Of course, there
is a trade-off: a parameterisation scheme that produces a design space that is
almost impossible to search is of virtually no use. Yet we can clearly enhance
our choice of parameterisation scheme by easing the restrictions placed on it by
the optimiser’s constraint handling method.
There are two approaches to solving a multi-objective optimisation problem. The
first reduces the multiple objectives to a single objective by generating a com-
posite objective function, usually from a weighted sum of the objectives. This
composite objective function can be optimised using existing single-objective op-
timisers. However, the weights must be pre-set, and the solution to this problem
will be a single vector of design variables rather than the entire Pareto-optimal
(PO) set. This can have undesirable consequences: setting the weights implicitly
introduces the designer’s preconceptions about the relative trade-off between
objectives. Real-world problems can produce surprising PO sets which may pro-
foundly affect design decisions, and the potential to generate novel designs is a
key benefit of optimisation .
The second approach to solving the multi-objective problem is to search di-
rectly for the entire PO set. This can be achieved in a number of ways and
requires modification to existing single-objective algorithms.
The authors know of only two attempts to produce a multi-objective TS
algorithm which finds multiple PO solutions in a single run. Hansen’s algorithm
 is an extension of the composite objective approach: his algorithm performs
a number of composite objective Tabu searches in parallel. Each search has a
different and dynamically updated set of weights, and in this way the search
can be driven to explore the entire Pareto front. This algorithm, although a
good implementation of TS, suffers the problems common to all weighted-sum
approaches: for problems with concave Pareto fronts, there may be regions of
the front that are not defined by a combination of weights and conversely certain
combinations of weights represent two points on the front. Thus, this algorithm
may not adequately locate the entire PO set.
Baykasoglu et al.  developed a TS algorithm combining a downhill local
search with an intensification memory (IM) to store non-dominated points that
Existing Multi-objective Tabu Search Algorithms
A Multi-objective Tabu Search Algorithm 493
were not selected in the search. When the search fails to find a downhill move, a
point from the IM is selected instead. When the IM is empty and all search paths
exhausted, the algorithm stops. This cannot be considered a true TS algorithm:
in restricting the search to only downhill moves its originators reject one of the
basic tenets of TS, that “a bad strategic choice can yield more information than
a good random choice” . Also, the lack of any diversification strategy renders
the algorithm incomplete and merely an elaborate local search algorithm.
The other TS algorithms reviewed by Jones  either use a composite ob-
jective function or are little more than local search algorithms similar to the
algorithm of Baykasoglu et al.
2 Multi-objective Tabu Search Adaptation
The single-objective TS implementation of Connor and Tilley  is used as
a starting point for our multi-objective variant. This uses a Hooke and Jeeves
(H&J) local search algorithm (designed for continuous optimisation problems)
 coupled with short, medium and long term memories to implement search
intensification and diversification as prescribed by Glover and Laguna .
TS operates in a sequential, iterative manner: the search starts at a given
point and the algorithm selects a new point in the search space to be the next
current point. The basic search pattern is the H&J search.
Recently visited points are stored in the short term memory (STM) and
are tabu – the search is not allowed to revisit these points. Optimal or near-
optimal points are stored in the medium term memory (MTM) and are used
for intensification, focusing the search on areas of the search space with good
objective function values. The long term memory (LTM) records the regions
of the search space which have been explored, and is used on diversification,
directing the search to regions which are under-explored. This is achieved by
dividing each control variable into a certain number of regions and counting
the number of solutions evaluated in those regions. A local iteration counter
i local is used and reset upon a successful addition to the MTM. When i local
reaches user-specified values, the algorithm will diversify or intensify the search,
or reduce the search step size and restart the search from the best solution found.
Thus, TS combines an exhaustive, systematic local search with a stochas-
tic element and an intelligent coverage of the entire search space. Our multi-
objective TS implementation of  is modified in the following areas: search
point comparison; the H&J move; optimal point archiving and the MTM; search
intensification and restart strategy. These modifications are described briefly
below, along with some further improvements.
In a single-objective optimisation problem, points may be compared using the
operators ==, > and < acting on the objective function values for those points.
Similarly, points in a multi-objective problem can be compared in the same way
Search Point Comparison
494D. Jaeggi et al.
Design Variable 1
Design Variable 2
Medium Term Memory
(Pareto Optimal Points)
Short Term Memory
(Recently Visited Points)
Long Term Memory
(All Generated Points)
Objective Function 1
Objective Function 2
Point Selection at the Hooke & Jeeves Step:
Points in design variable space (below)
Points in objective function space (right)
Point 4 is selected as the next point and is
added to the STM at the next step
Pareto Optimal Point
(Unselected Pareto Optimal
Fig.1. Point Selection for the Hooke & Jeeves move and Tabu Search Memories
(thus preserving the logic of the single-objective algorithm) by using the concepts
of Pareto equivalence (==) and dominance (> and <).
At each iteration, a H&J move is made. 2n var new points are generated by
incrementing and decrementing each design variable by a given step around the
current point. The objective functions for each new point are evaluated and, as
long as the point is neither tabu (i.e. not a member of the STM) nor violates
any constraints, it is considered as a candidate for the next point in the search.
In the single-objective TS algorithm, these candidates are sorted and the
point with the lowest objective is chosen as the next point. A similar logic can
be applied to the multi-objective case: however, the possibility of multiple points
being Pareto equivalent (PE) and optimal must be allowed for.
This is achieved by classifying each candidate point according to its domina-
tion or Pareto equivalence to the current point. If there is a single dominating
point, it is automatically accepted as the next point. If there are multiple dom-
inating points, the dominated points within that group are removed and one
is selected at random from those remaining. The other points become candi-
dates for intensification (discussed below). If there are no dominating points,
the same procedure is applied to those candidate points which are PE to the
current point. If there are no PE points, a dominated point is selected in the
same fashion. Thus, our strategy accepts both downhill and uphill moves – the
next point is simply the “best” point (or one of the PE best points) selected
from the candidate solutions. This logic is shown in Fig. 1 for clarity.
In addition, a pattern move strategy is implemented in the same way as
Connor and Tilley . Before every second H&J move, the previous move is
repeated. This new point is compared to the current point, and, if it dominates
The Hooke and Jeeves Move
A Multi-objective Tabu Search Algorithm495
Reduce step sizes and select
point from Medium Term
Select point from
Select random point from
under-visited region of search
space using Long Term
i_local = 0
Perform Hooke &
i_local = 0
Fig.2. Flow Diagram of the Multi-objective Tabu Search Algorithm
it, is accepted as the next point; if not, the standard H&J move is made. In this
way, the search may be accelerated along known downhill directions.
In Connor’s single-objective TS , the MTM is a bounded, sorted list of near-
optimal solutions. As the concept of a single optimal point does not exist in
multi-objective optimisation (see Section 1.2), we replace the MTM in our multi-
objective TS variant by an unbounded set of non-dominated solutions produced
by the search. As new points are evaluated, they become candidates for addition
to this set. Thus, the MTM represents the PO set for the problem at that stage
in the search.
Optimal Point Archiving and the Medium Term Memory
496D. Jaeggi et al.
The original single-objective TS produced intensification points by using the
MTM to generate points in the neighbourhood of good solutions. Although the
replacement of the MTM by a PO set of solutions allows us to use a variant of
this strategy, a feature of multi-objective optimisation suggests an alternative
strategy, similar to that used by Baykasoglu et al. .
A multi-objective H&J iteration may produce multiple PO points (see Fig.
1). As only one point may be selected as the next point, it seems wasteful to
discard the other points. Therefore, we incorporate an intensification memory
into our algorithm. This is a set of PE points; at each H&J step, points which
dominate the current solution, but are not selected as the next point (of which
there can be only one), are considered as candidates for addition to the set.
At search intensification, a point is chosen randomly from the IM. The IM is
continuously updated and points which become dominated by the addition of a
new point are removed. Thus, the IM should always contain points which are
on, or near to, the current PO front (stored in the MTM).
The single-objective TS restart strategy returns the search to the current best
point in the MTM. As the MTM is now a set of PO points, we simply select one
point at random from the set. More intelligent restart strategies are possible 
and are under investigation. Fig. 1 gives an overview of the various memories
used, and Fig. 2 a flow diagram for our multi-objective TS implementation.
Intensification and Restart Strategy
We employ a very simple constraint handling strategy: any point which vio-
lates any constraint is deemed to be tabu and the search is not allowed to visit
that point. Thus, accepted solutions are limited to feasible space. On search
diversification, we allow the algorithm to loop until a feasible point has been
found. Depending on the problem, it would also be possible to introduce penalty
functions to handle certain constraints.
The H&J local search strategy requires roughly 2n var solution evaluations (al-
lowing for points that are tabu or violate constraints) at each step, where n var
is the number of design variables. A real-world problem may contain a large
number of variables (the shape optimisation of a Boeing 747 wing required 90
variables ) and this strategy could become prohibitively expensive. One so-
lution to this is to incorporate an element of random sampling in the H&J step.
We generate the 2n var new points, remove those that are tabu, and only
evaluate n sample ≤ 2n var points from those that remain, selecting randomly
to avoid introducing any directional bias. If one of these points dominates the
current point, it is automatically accepted as the next point. If more than one
point dominates the current point, a non-dominated point from these is randomly
selected. If no points dominate the current point, a further n sample points are
sampled and the comparison is repeated. If all the feasible, non-tabu points have
Improving Local Search Efficiency and Parallelisation Strategy
A Multi-objective Tabu Search Algorithm497
Table 1. Test Functions
Function Name Number of Number of Constraint Types
been sampled without finding a point that dominates the current solution, the
standard selection procedure is employed.
Any optimisation procedure that forms part of a real-world design cycle must
be able to complete in a reasonable time-frame. Parallel processing offers a large
potential speed-up; any serious optimisation algorithm should be designed with
this in mind. Our multi-objective Tabu Search algorithm is parallelised by means
of functional decomposition. At each H&J move, the required objective function
evaluations are computed in parallel.
3 Tabu Search Parameter Investigation
The performance of this algorithm has already been tested on nine standard
unconstrained test problems and compared to the performance of NSGA-II .
Over these nine problems, the algorithm performed comparably with NSGA-II.
In that initial study, no attempt was made to find optimal TS parameter settings;
they were set to reasonable values, based on experience. Here, we conduct a
more systematic parameter setting investigation. The parameters that may be
set in the algorithm are shown in Table 2. The parameter settings used in the
benchmarking in  are also given.
In the studies that follow, the same test conditions as in  were used.
Parameters not being varied were kept fixed at the values given in Table 2.
Performance was assessed using the convergence metric Υ, described by Deb
et al. ; the mean and standard deviation for the results of 45 runs were
calculated, and the same set of random number generator seeds used. Each run
was stopped and the results calculated after 25000 function evaluations.
498D. Jaeggi et al.
Table 2. Tabu Search parameters
Parameter Initial Value Description
Diversify search when i local == diversify
Intensify search when i local == intensify
Reduce step sizes and restart when i local == reduce
STM size – the last n stm visited points are tabu
In the LTM each variable is divided into n regions regions
Initial H&J step size as percentage of variable range
Factor by which step sizes are reduced on restart
Number of points randomly sampled at each H&J move
5 10 15 20 25 30
Normalised convergence metric
Fig.3. Normalised convergence metricˆΥ vs n stm over 9 test problems
One of the distinguishing features of TS algorithms in general is the use of a
short term memory to define points that are tabu and may not be revisited. This
gives the search algorithm a means by which it can climb out of local minima.
We might expect that the size of the STM may affect the performance of the
algorithm: a STM of zero size would reduce the algorithm to a mere local search
algorithm coupled with a global random search; a STM of infinite size would
prevent the search from refining solutions in known good regions of the search
space. There are also algorithm run time considerations: the computational cost
of the algorithm increases with STM size, but for most real-world problems this
cost is negligible compared to the cost of function evaluations.
We consider variations in the STM size in the range 5 ≤ n stm ≤ 30 for the
nine test problems used in , keeping all other parameter values constant as
given in Table 2. The results are shown in Fig. 3; Υ is normalised with respect
Variation in Short Term Memory Size
A Multi-objective Tabu Search Algorithm499
5 10 15 20 25 30 35 40
Normalised convergence metric
Fig.4. Normalised convergence metricˆΥ vs diversify over 9 test problems
to its value for n stm = 20, the value used in , and is plotted against n stm.
Performance is improved by increasing n stm from 5 up to around 15 on most
test problems. Increases beyond that improve performance but only marginally,
and there does not appear to be much benefit in this. On problems with low
numbers of design variables, there is hardly any variation in performance; this
is most likely due to the lower potential for “cycling” (the search repeating a
recent search path) as there are fewer paths that can be taken. The exception is
Poloni’s problem which shows quite a large improvement in performance between
n stm = 5 and n stm = 15 despite having only two design variables. Problem
ZDT3 also displays a large improvement in performance over this range of n stm;
on this problem, it appears that a larger STM is required to prevent cycling.
The other test problem that exhibits sensitivity to changes in n stm is prob-
lem ZDT4; this shows continually improving performance with increases in n stm.
However, because the presented results have been normalised to show relative
performance the incredibly poor absolute performance of our algorithm on ZDT4
has been masked. This is commented on in .
The parameters intensify and diversify control the balance between a local search
of the design space and a global one. Therefore, they are critical in governing per-
formance: certain problems, particularly multi-modal ones, will benefit strongly
from a strategy which favours diversification; other problems, such as those with
clearly defined local regions containing many near PO points, will be better
searched by a strategy favouring intensification.
Fig. 4 shows the effect of varying diversify on the normalised convergence
metricˆΥ; as in Section 3.1, Υ is normalised against its value for diversify = 10,
used in the original study . There appear to be two trends: for one group of
Variations in Intensify and Diversify
500D. Jaeggi et al.
10 15 20 25 30 35 40
Normalised convergence metric
Fig.5. Normalised convergence metricˆΥ vs intensify over 9 test problems
problems, the point at which diversification takes place has little effect on overall
performance; another group definitely favours early diversification. In general,
early diversification brings performance benefits for these problems.
It also is worth noting two points. First, intensify was fixed at 20: thus,
over the range of values for diversify used in this study, the order in which
intensification and diversification takes place changes. Some of the performance
variation is attributable to this. In particular, on problem ZDT4, which contains
a large number of false Pareto fronts, intensifying before diversifying traps the
optimiser in these false fronts and hinders the location of the true Pareto front.
Second, early diversification tends to increase the total number of diversifica-
tion moves performed during an optimisation run. This behaviour is beneficial
on problems such as SCH; good performance on this problem depends on how
fast the region of design space in which the Pareto front is located is found. Thus,
more random behaviour in the search speeds its discovery; once this region is
found, the local search component in TS effectively finds the rest of the front.
Similarly, Fig. 5 shows the effect onˆΥ of varying intensify; Υ is normalised
against its value for intensify = 20.
For the majority of problems, the absolute value of intensify appears rela-
tively unimportant; of more importance is whether intensify is less or greater
than diversify. The results suggest that for good performance on these test func-
tions in general, diversification must occur first and its value primarily governs
performance. However, the results also show that there is some benefit in reduc-
ing the gap between diversification and intensification; a value of intensify = 15
gives, on average, better performance. This would prevent the algorithm from
needlessly searching poor areas of the design space.
These results raise a concern about the use of test functions in determin-
ing algorithm performance on real-world problems. Kipouros et al.  used a
variant of this TS algorithm to perform a multi-objective optimisation of a gas-
A Multi-objective Tabu Search Algorithm 501
0 50 100 150
200 250 300
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Pareto Front for problem CONSTR Pareto Front for problem SRN
0 0.2 0.4 0.6
0.8 1 1.2
Pareto Front for problem TNK
0.01523 0.06721 0.48625
0.00081 0.00378 0.29218
Mean & Standard Deviation of the conver-
gence metric for the three test problems.
Convergence was measured to an ideal
Pareto front consisting of 500 points.
Fig.6. Pareto fronts for constrained test problems CONSTR, SRN & TNK and
convergence metric results
turbine compressor blade. For that particular application, intensification was
found to be particularly beneficial to overall performance; the optimiser was
able to make steady progress (indicated by rate of addition to the MTM) by
performing many intensification steps. This suggests that the problem contains
regions with many locally PO points and TS is able to effectively search all these
through its intensification strategy. In contrast, this characteristic does not ap-
pear to be present in this set of test functions. There appears to be an urgent
need to devise test problems that accurately reflect characteristics of real-world
For both these sets of results, error bars based on the standard deviation ofˆΥ
were not plotted, for reasons of clarity. Although there was some variation in the
standard deviation with parameter setting, for the most part, these variations
were small and the values remained close to the nominal values reported in
. There are two main exceptions to this: with diversify = 5 seven problems
showed an increase in the standard deviation; the standard deviations for ZDT4
varied greatly, which is probably due to the algorithm’s poor performance on
502D. Jaeggi et al.
4Constrained Test Problems
Deb et al.  also tested NSGA-II on four constrained test problems. Although
no quantitative performance results were published, plots showed that NSGA-II
was capable of finding a good spread of results along the Pareto front for each.
We have tested our algorithm on those four constrained problems: the results for
the problems CONSTR, SRN and TNK are shown in Fig. 6; results for problem
WATER are presented as the range of values found for all five objectives and are
given in Table 3. We used the lower limit of 20000 function evaluations prescribed
by Deb et al. The parameter settings used were the same as given in Table 2,
with the exception that intensify was reduced to 15 as a result of the parameter
study presented above.
Table 3. Lower and upper bounds of objective function values on the Pareto front for
Multi-objective TS 0.804-0.918 0.022-0.857 0.104-0.962 0.056-1.320 0.129-3.121
NSGA-II0.798-0.920 0.027-0.900 0.095-0.951 0.031-1.110 0.001-3.124
On problem CONSTR, coverage of the Pareto front is excellent, except in
a small region near the tail where f 1 approaches 1.0. The range of solutions
found is comparable to NSGA-II and far superior to Ray-Tai-Seow’s algorithm
presented in . Similar performance is achieved on problem SRN – convergence
to and coverage of the Pareto front are good and comparable to NSGA-II.
Problem TNK is slightly harder – the Pareto front is discontinuous and, as
shown by Deb et al., some algorithms have difficulty finding the entire central
continuous region. Although the spread of solutions that our algorithm finds in
this region is slightly worse than on the rest of the Pareto front, it succeeds in
locating the continuous region correctly.
Problem WATER is a five-objective problem; due to the difficulty of visual-
ising the Pareto front, figures are presented in Table 3 for the minimum and
maximum values of the objective functions found on the Pareto front. Deb
et al. presented similar figures for NSGA-II on this problem. Of the 10 mini-
mum/maximum values, NSGA-II finds better values on 7, although the differ-
ences in most cases are small.
In this paper, we have presented a multi-objective TS algorithm with features
that make it particularly attractive to real-world optimisation problems, in par-
ticular with regard to its constraint handling. In previous work, we benchmarked
A Multi-objective Tabu Search Algorithm503
this algorithm against NSGA-II on a number of test functions and found that it
performed comparably. Here, we performed a study on the effect of varying the
TS parameters on the algorithm performance on the same nine test functions.
As regards the STM size, the results suggest that a value of n stm ≥ 15 gives
good performance on a range of test functions. Increasing n stm beyond 25 does
not appear to give much performance benefit on the functions tested. Although
the computational cost of the algorithm increases with n stm, this cost is usually
negligible for real-world problems where function evaluations are expensive.
Results for varying intensify and diversify show that algorithm perfor-
mance is, in general, dominated by diversification; performance is improved by
increasing the diversification element, and intensification has relatively little ef-
fect. This is slightly at odds with experience of using this algorithm on aero-
dynamic shape optimisation problems, where intensification is a more effective
means of advancing the search. This raises a concern about the use of test func-
tions to derive algorithm performance information for use in real-world problems.
Over the majority of test functions, performance is reasonably independent
of the TS parameter settings; this suggests that the power of TS comes from
its fundamental elements – the combination of a local search algorithm with a
variety of search memories – rather than particular, carefully chosen parameter
settings. This should ease its application to new problems.
Finally, our TS algorithm was tested on four constrained test problems. On
all the problems, the algorithm was able to find a good spread of solutions along
the Pareto front, despite our strict constraint handling approach. We believe
our multi-objective TS algorithm is well suited to use on real-world optimisation
problems where constraints can be binary in nature and such an approach is
required to allow the optimiser to run. Indeed the only way to show this is to
actually apply it to real-world problems (as we have done in a companion paper
); test problems that exist in the literature do not share these characteristics,
and it is hard to draw meaningful conclusions from tests using these problems.
However, future work should include a more rigorous performance comparison
with other leading MO optimisation algorithms. The performance metrics used
in this study are not optimal  but were chosen to allow comparison with
previously published data . Finally, determining optimal settings for this
algorithm requires further more detailed work.
This research is supported by the UK Engineering and Physical Sciences Re-
search Council (EPSRC) under grant number GR/R64100/01. The authors would
also like to thank Prof. Bill Dawes for his support and encouragement.
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A simulated annealing