# 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.

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**ABSTRACT:**This first chapter intends to review and analyze the powerful new Harmony Search (HS) algorithm in the context of metaheuristic algorithms. I will first outline the fundamental steps of Harmony Search, and how it works. I then try to identify the characteristics of metaheuristics and analyze why HS is a good meta-heuristic algorithm. I then review briefly other popular metaheuristics such as par-ticle swarm optimization so as to find their similarities and differences from HS. Finally, I will discuss the ways to improve and develop new variants of HS, and make suggestions for further research including open questions. Comment: 14 pages in typeset publications03/2010; - 05/2008: pages 63 - 82; , ISBN: 9780470411353
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**ABSTRACT:**Most real world search and optimization problems naturally involve multiple responses. In this paper we investigate a multiple response problem within desirability function framework and try to determine values of input variables that achieve a target value for each response through three meta-heuristic algorithms such as genetic algorithm (GA), simulated annealing (SA) and tabu search (TS). Each algorithm has some parameters that need to be accurately calibrated to ensure the best performance. For this purpose, a robust calibration is applied to the parameters by means of Taguchi method. The computational results of these three algorithms are compared against each others. The superior performance of SA over TS and TS over GA is inferred from the obtained results in various situations.Journal of Computational and Applied Mathematics 01/2009; 230(2):463-476. · 0.99 Impact Factor

Page 1

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.

1 Introduction

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 [1], Kellar [2] and Gaiddon

C. A. Coello Coello et al. (Eds.): EMO 2005, LNCS 3410, pp. 490–504, 2005.

c ?Springer-Verlag Berlin Heidelberg 2005

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A Multi-objective Tabu Search Algorithm 491

et al. [3], 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 [4].

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 [1] 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 [5]. Similarly, multi-objective SA

methods have been developed [6]. 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 [7] 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.

1.1

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 [5]. 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.

Constraint Handling

Page 3

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.

1.2

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 [6].

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

[8] 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. [9] 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

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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” [10]. 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 [7] 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 [11] 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)

[12] coupled with short, medium and long term memories to implement search

intensification and diversification as prescribed by Glover and Laguna [10].

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 [11] 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.

2.1

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

Page 5

494D. Jaeggi et al.

1

2

3

4

Design Variable 1

Design Variable 2

11

22

3

3

4

4

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

x

Current Point

Candidate Point

Pareto Optimal Point

Intensification Memory

(Unselected Pareto Optimal

Points)

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 <).

2.2

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 [11]. 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

Page 6

A Multi-objective Tabu Search Algorithm495

Reduce step sizes and select

point from Medium Term

Memory

Select point from

Intensification

Memory

Select random point from

under-visited region of search

space using Long Term

Memory

Stop

i_local = 0

Start

Perform Hooke &

Jeeves Move

Stopping Criteria

met?

Addition to

Medium Term

Memory?

i_local++

i_local ==

diversify?

i_local ==

intensify?

i_local ==

restart?

Update Memories

(as required)

Yes

No

Yes

Yes

Yes

Yes

No

No

No

No

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.

2.3

In Connor’s single-objective TS [11], 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

Page 7

496D. Jaeggi et al.

2.4

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. [9].

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 [6]

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

2.5

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.

Constraint Handling

2.6

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 [13]) 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

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A Multi-objective Tabu Search Algorithm497

Table 1. Test Functions

Function Name Number of Number of Constraint Types

VariablesObjectives

SCH

FON

POL

KUR

ZDT1

ZDT2

ZDT3

ZDT4

ZDT6

CONSTR

SRN

TNK

WATER

1

3

2

3

30

30

30

10

10

2

2

2

3

2

2

2

2

2

2

2

2

2

2

2

2

5

None

None

None

None

None

None

None

None

None

2 Inequality

2 Inequality

2 Inequality

7 Inequality

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 [14].

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 [14] are also given.

In the studies that follow, the same test conditions as in [14] 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. [15]; 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.

Page 9

498D. Jaeggi et al.

Table 2. Tabu Search parameters

Parameter Initial Value Description

diversify

intensify

reduce

n stm

n regions

SS

SSRF

n sample

10

20

50

20

2

8%

0.5

6

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

0.9

0.95

1

1.05

1.1

1.15

1.2

5 10 15 20 25 30

Normalised convergence metric

n_stm

"SCH"

"FON"

"POL"

"KUR"

"ZDT1"

"ZDT2"

"ZDT3"

"ZDT4"

"ZDT6"

Fig.3. Normalised convergence metricˆΥ vs n stm over 9 test problems

3.1

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 [14], 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

Page 10

A Multi-objective Tabu Search Algorithm499

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

5 10 15 20 25 30 35 40

Normalised convergence metric

diversify

"SCH"

"FON"

"POL"

"KUR"

"ZDT1"

"ZDT2"

"ZDT3"

"ZDT4"

"ZDT6"

Fig.4. Normalised convergence metricˆΥ vs diversify over 9 test problems

to its value for n stm = 20, the value used in [14], 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 [14].

3.2

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 [14]. There appear to be two trends: for one group of

Variations in Intensify and Diversify

Page 11

500D. Jaeggi et al.

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

5

10 15 20 25 30 35 40

Normalised convergence metric

intensify

"SCH"

"FON"

"POL"

"KUR"

"ZDT1"

"ZDT2"

"ZDT3"

"ZDT4"

"ZDT6"

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. [16] used a

variant of this TS algorithm to perform a multi-objective optimisation of a gas-

Page 12

A Multi-objective Tabu Search Algorithm 501

-300

-250

-200

-150

-100

-50

0

50

100

150

0 50 100 150

f_1

200 250 300

f_2

0

2

4

6

8

10

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

f_2

f_1

Pareto Front for problem CONSTR Pareto Front for problem SRN

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6

f_1

0.8 1 1.2

f_2

Pareto Front for problem TNK

Mean

Standard Deviation

CONSTR

0.01523 0.06721 0.48625

0.00081 0.00378 0.29218

SRN TNK

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

problems.

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

[14]. 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

this problem.

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502D. Jaeggi et al.

4Constrained Test Problems

Deb et al. [15] 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

problem WATER

f1

f2

f3

f4

f5

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 [15]. 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.

5 Conclusions

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

Page 14

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

[17]); 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 [18] but were chosen to allow comparison with

previously published data [15]. Finally, determining optimal settings for this

algorithm requires further more detailed work.

Acknowledgements

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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