An Overview of Neighbourhood Search Metaheuristics
M.B. Wright
Management Science Department, Lancaster University Management School, LA1 4YX, UK.
email m.Wright@lancaster.ac.uk
Abstract
This paper gives an overview of the current state of research into neighbourhood search metaheuristics
for combinatorial optimization problems. The main variations are described and some pointers for future
research briefly discussed. Throughout there is extensive referencing to some of the most important
publications in the area.
Key Words: Metaheuristics, Neighbourhood Search, Combinatorial Optimization
Heuristics and Metaheuristics
A Heuristic has been defined as a procedure for solving problems by an intuitive approach in which the
structure of the problem can be interpreted and exploited intelligently to obtain a reasonable solution 1.
At the time of that paper's publication, most heuristics were very problem-dependent. Until the 1980s,
this was essential because of the limited computing power available at that time. It was only by taking
very careful account of the problem structure that good solutions could be produced in the time available.
However, with the increase of computing power came the use of more general approaches, which could
be successfully applied to a wide variety of problems. Such approaches came to be known as
Metaheuristics. These approaches have largely taken over from the more traditional forms of heuristics.
One definition is: A metaheuristic is an iterative generation process which guides a subordinate
heuristic by combining intelligently different concepts for exploring and exploiting the search spaces
using learning strategies to structure information in order to find efficiently near-optimal solutions2.
1
This definition may be rather more precise and restrictive than some would consider necessary, but it
does encapsulate the general principles involved in all well-known metaheuristics developed to date.
Neighbourhood Search, described below, encompasses a variety of successful metaheuristic approaches.
This paper will concentrate on Neighbourhood Search methods. However, other metaheuristics, such as
Genetic Algorithms3, Ant Systems4 and Neural Networks5 are in many ways similar and their study may
present a useful cross-fertilization of ideas and possibilities for hybridisation of approaches with
Neighbourhood Search approaches.
Neighbourhood Search
Neighbourhood search methods involve the guiding of progress through search space, where at each
stage a current point in search space is progressively replaced by a similar one, with the effect that
possibly a very poor initial position is eventually transformed into a good final position.
The following stages are required before a neighbourhood search technique can be used:
 Problem definition: the problem must be expressed quantitatively in terms of one or more
objectives and a set of constraints.
 Search Space definition: the method will involve moving between points (which may or may not
simply be solutions themselves) in search space. The set of these points must be defined.
 Neighbourhood definition: the points in search space must be linked in some way, such that the
search process will move along links. This involves the definition of the neighbourhood of any
given point in search space.
 Generation of an initial solution: an initial point in search space, from which the search
procedure will begin, must be generated.
 Neighbour generation: a method must be specified for determining which neighbour(s) of a
current point in search space are to be considered.
 Acceptance criterion: a decision criterion must be specified as to which, if any, of the generated
neighbours should be accepted. An accepted point in search space becomes the current point
from which a new neighbour or set of neighbours is generated.
 Stopping criterion: a rule must be specified for deciding when the search will end.
Problem definition
2
The problems we are considering here are combinatorial optimization problems, for which some
objective function needs to be near-optimised under a given set of constraints. In this supporting paper it
shall be assumed throughout that the objective function value is a cost (which may or may not be wholly
or partly financial) to be minimised, or near-minimised.
However, it is not necessarily the case that the objective function needs to be absolutely fixed before an
implementation of neighbourhood search can start. For example, Wright6 found that it can be very
effective in some circumstances to modify the objective function itself, by changing the weights of
different elements of this function and/or adding extra elements to it. Indeed, for many practical cases,
the objective function is to a large extent a subjective approximation anyway.
It can also prove effective to transform some constraints into elements of the objective function by means
of penalty costs for the infringement of these constraints, with the intention that the neighbourhood
search technique will ensure that all penalty costs are removed before the final solution is reached.
Again, in many practical cases, some constraints may be soft rather than hard, i.e. they may represent
preferences rather than absolute rules.
Search space
Even after a problem has been precisely specified (which may in itself be difficult for many ill-defined
types of problem) this does not of itself define the search space. Although it may frequently be
convenient to define the search space as being the same as either the total solution space (i.e. the set of
all solutions) or the feasible solution space (i.e. the set of all feasible solutions), there may be occasions
where it is better to define the search space somewhat differently.
For example, Wright7, when approaching a locomotive scheduling problem, defined the solution space as
the set of all locomotive schedules, but the search space as the set of all the ways that types of
locomotive can be allocated to the trains. There is a mapping from search space to solution space using
an exact algorithm. James and Buchanan8 proposed a similar approach, but with a heuristic mapping, for
a production scheduling problem, while Tsubakatani and Evans9 suggested a way of collapsing the search
space for a Travelling Salesman Problem into a two-dimensional search space.
Neighbourhood definition
3
The way in which neighbourhoods are defined will often play a great part in determining the success or
failure of a neighbourhood search method. A neighbourhood is defined by first defining a set of
allowable perturbations, i.e. means of amending one solution to form another. In most implementations
there are several plausible ways of defining these perturbations. Neighbouring solutions to a given
solution are those which can be reached by means of a single perturbation. Since the search progresses
by successive moves between neighbouring solutions, the definition of allowable perturbations is thus
critical to the effectiveness of the search procedure.
Often it can help to extend the definition of neighbourhood beyond the obvious formulation. This can
include the incorporation of Ejection Chains10,11, Kempe Chains12, etc., in which multiple moves can be
regarded as a single perturbation. However, this can be very difficult to do effectively, since there is a
danger that the neighbourhood may be so large as to render the search process very slow indeed.
In some approaches, the definition of neighbourhood is dynamic, i.e. it changes as the search progresses.
This is the principle behind Variable Neighbourhood Search13,14. Di Gaspero and Schaerf15 have taken
this a stage further by proposing the systematic use of combinations of neighbourhood functions.
Mautor16 proposes the use of special neighbourhood structures or meta-neighbourhoods.
Initial solution
The construction of an initial solution is usually either at least partly random or highly problemdependent. The quality of the initial solution is clearly important when very little time is available for
the search; otherwise most of the time may be wasted in moving from a very poor solution to a
reasonably good one, with insufficient time remaining to produce a very good one.
Otherwise it is unclear whether in general a higher-quality initial solution is more likely to lead to a highquality final solution than a lower-quality initial solution, though in some cases it has been found that a
good initial solution, constructed in a way taking account of specific problem characteristics, can help
ensure a high-quality final solution.
For example, this was found by Perttunen17 for Travelling Salesman problems and by Emden-Weinert
and Proksch18 in their study of the use of Simulated Annealing for airline crew scheduling problems.
4
When several runs of a method are being undertaken, each starting from a new initial solution, it may be
sensible to make intelligent use of the progress of the search to date when constructing the next initial
solution. This is the principle behind GRASP19,20.
Neighbour generation
In some techniques, at each iteration, a single neighbouring solution is generated by means of an
allowable perturbation. The decision that then needs to be made is whether this solution should replace
the current solution. If it does, the search continues from the new solution; otherwise it continues with a
different perturbation to the existing current solution.
The two main variants of this type of search are First-accept Local Improvement21 and Simulated
Annealing22,23,24.
In any method of this overall type, the neighbouring solution may be generated either randomly or
systematically. It is usual for neighbours in First-accept Local Improvement to be generated
systematically, so that a guaranteed local optimum is reached after a given number of iterations since the
last acceptance of a new solution. Most early Simulated Annealing applications used random neighbour
generation; however, as Dowsland24 reports, subsequent researchers often found it preferable to use a
systematic sampling method. It is also possible to regard some neighbours as tabu.
The other main type of neighbourhood search technique generates at each iteration several alternative
neighbouring solutions, each derived by means of a single perturbation to the current solution. The
decision to be made is which of these to accept. The two main variants of this type of search are Bestaccept Local Improvement21 and Tabu Search25,26.
In any technique of this type, all possible neighbours may be generated at each iteration; but, especially
for Tabu Search, and especially where neighbourhoods are large, it is more usual to generate a candidate
list of admissible neighbours, and the choice of which to accept is restricted to that list. A mechanism for
changing this list at each iteration must be included in order that all possible perturbations are tried from
time to time.
Acceptance criteria
5
For First-accept Local Improvement, the acceptance criterion is simple – the new solution is accepted if
and only if its cost (objective function value) is lower than that of the current solution. In Best-accept
Local Improvement, we have to choose one neighbour from several on the candidate list – the criterion is
again simple – accept the neighbour whose cost is lowest, as long as this cost is lower than that of the
current solution. If there is no such neighbour, the algorithm terminates.
The decision criterion for Simulated Annealing is rather more complex, involving a random number and
a dynamic parameter known as the temperature. At the start of Simulated Annealing, the temperature is
usually high (though not always – for example, see Connolly27, who proposes the use of a constant low
temperature). This has the effect that a very large proportion of generated neighbours are accepted. This
temperature gradually decreases until, near the end of the search, very few neighbours which increase
costs are accepted. However, there are many different ways in which this gradual decrease in
temperature can be achieved – moreover, some applications effectively use occasional reheating, where
the temperature is suddenly increased, within the overall pattern of gradual decrease. For an example of
this, see Dowsland28.
Other methods known as Threshold methods29, including the Great Deluge method30, have been proposed
which are very similar to Simulated Annealing, but without the random factor in the acceptance criterion.
Again, many neighbours are accepted early in the search, but few at the end. This ensures that the final
solution is very likely to be a local optimum.
In Tabu Search, the decision is more complex. In the simplest of implementations, the neighbour on the
candidate list whose cost is lowest (whether better or worse than that of the current solution) is chosen,
subject to tabu status (defining which neighbours can be considered) and aspiration criteria (defining
exceptions to these tabu rules). However, many implementations are more complex than this – see later.
Implementations of Tabu Search thus have to define the way in which tabu status is handled. Often there
are a fixed number of tabu perturbations (on a tabu list), though membership of this tabu list (and
sometimes the length of the list also) varies dynamically as the search progresses. The tabu moves will
often (though not always) consist of reversals of recently accepted perturbations, so as to avoid cycling.
The aspiration criterion is usually included in order to ensure that any perturbation leading to a better
solution than any found previously is accepted regardless of tabu status.
Stopping criteria
6
Some Neighbourhood Search methods have a natural stopping point. Best-accept Local Improvement
and First-accept Local Improvement terminate when it is clear that the current point in search space is a
local optimum, though there is then the option of restarting (see below) or redefining neighbourhoods via
Variable Neighbourhood Search13 or other methods. For two-phase methods (see later), the reaching of a
local optimum is the trigger for switching between phases of the search.
Other methods such as Simulated Annealing and Tabu Search have no natural stopping point, so this
must be specified in some way. Sometimes the search may stop after a prespecified number of iterations
or length of elapsed time. For Simulated Annealing, the trigger could perhaps be the reaching of a
prespecified ending temperature. In other cases, the decision to stop may depend on the progress of the
search; for example, when a prespecified number of iterations have occurred since the best solution
found to date by the search was encountered.
Main variants of Neighbourhood Search
Numerous variations on these basic ideas have been proposed and many have been found to be effective.
The acceptance decision and/or the neighbour generation mechanism may be influenced by memory
considerations, where the history of the search – e.g. which perturbations have been tried before, how
often and how recently, and/or details of previous solutions such as the values of specific variables, etc. –
is taken into account. Glover and Laguna26 review some of the more common methods for doing this for
implementations of Tabu Search.
Alternatively, as in Subcost-Guided Search31,32,33, the increase in overall cost may be modified according
to specific rules before being used in the acceptance criterion.
For any method, there is always the option of multiple restarts. Often this will simply mean restarting
from a new randomly (or part-randomly) generated initial solution. In particular, this is often done for
Local Improvement, where the search comes to a natural end at a local optimum. In methods such as
GRASP (Greedy Randomized Adaptive Search Procedure)19,20, the construction of the next initial
solution is influenced by the progress of the search to that point. Multiple restarts have the great
advantage that all the computer time available can be used.
All successful methods involve some element of intensification (trying to find the best nearby solution)
and diversification (moving to new areas of search space). For example, Simulated Annealing normally
7
starts with diversification being dominant, since the high temperature means that the random influence is
very high, but then the influence of intensification gradually increases as the temperature parameter
lowers until it is the dominant facet.
However, in some methods, intensification and diversification are explicitly separated into two separate
alternating phases of the solution procedure. Often the first phase is some form of local improvement,
with the second phase consisting of a small number of random or partly random perturbations, as in Tabu
Thresholding34,35 and Iterated Local Search36,37. Alternatively, as in Strategic Oscillation38,39, some
constraints may be relaxed in the second phase, allowing infeasible solutions to be accepted. These
infeasibilities are then removed at the start of the subsequent intensifying phase.
It can often prove fruitful to combine Neighbourhood Search approaches with each other and/or with
other techniques. Recent examples of the effective implementation of such hybrids can be found in
numerous publications. For example, Simulated Annealing has been hybridised with Tabu Search40,41,
with Genetic Algorithms42,43,44, and with Neural Networks45,46, while Tabu Search has been hybridised
with Genetic Algorithms47,48,49, with Strategic Oscillation50, with Memetic Algorithms51 and with exact
optimization approaches52,53,54.
For detailed reviews of neighbourhood search methods, see Reeves55, Osman and Kelly2 and Ahuja et
al.56.
Which is the best form of Neighbourhood Search?
This is a very difficult issue. Any Neighbourhood Search approach (indeed, any metaheuristic approach)
can be implemented in a variety of ways. For example, the choices of neighbourhood and parameters are
usually of crucial importance. Every method has numerous variations – indeed, it is likely that every
single researcher in the field implements Neighbourhood Search in a slightly different way.
Moreover, when comparing solution times, the running time strongly depends on the programming style,
on the compiler, on the compiling options etc. 57, as well as the type of computer. Data structures and
clever schemes for the avoidance of repetition of calculations may be more important than the precise
choice of technique.
8
Worse still, a method that works best on one set of data will not necessarily work best on another.
Moreover, since most methods involve random numbers, often a very large number of experimental runs
need to be carried out before statistically significant results can be achieved.
Some researchers have tried to compare methods, but without any kind of consistency in their
conclusions. For example, Cordreau et al. 58, say: "Tabu Search clearly stands out as the best
metaheuristic for the Vehicle Routing Problem". This in fact turns out to mean that the implementations
reported of Tabu Search have so far proved more successful than the implementations reported of other
metaheuristics – not the same thing, though it may be suggestive.
As another example, "Threshold Accepting: a general purpose optimization algorithm appearing superior
to Simulated Annealing" is the title of a paper29. Unfortunately, this conclusion is far from conclusively
backed up by the results reported in the paper.
Sinclair59 compared several methods, though his paper does not give a full description of how the
parameters and other implementation details were chosen, making it difficult to comment on his final
assertions, for example that: Tabu Search provides the best solutions, but at the cost of long solution
times.
Glass and Potts60, by contrast, claim that for some flow shop problems: "Based on the results of extensive
computational tests, simulated annealing is found to generate better quality solutions than the other
neighbourhood search methods".
Marett and Wright's61 tentative conclusions include: "Simulated Annealing is sometimes better,
sometimes worse, than … Tabu Search; the relative superiority of Simulated Annealing increases as the
complexity of the combinatorial problem increases and as the number of perturbations allowed
decreases." However, the authors stress that this conclusion is proposed tentatively for the reasons listed
above.
Even when authors go to great lengths to be fair in their comparisons, it is difficult for them to reach
robust conclusions. Rossi-Doria et al.62 have very clearly attempted to be as fair as possible in their
comparisons for timetabling problems, yet still their conclusions, for example that: "On the small
instances ILS (Iterated Local Search) generally performs best, followed closely by SA (Simulated
Annealing) and ACO (Ant Colony Optimization). GA (Genetic Algorithms) is definitely worse, but TS
(Tabu Search) shows the worst overall performance", must be treated with considerable caution.
9
One problem with published comparisons is that authors tend to be much more au fait with the best and
most up-to-date implementation details of their own preferred method than of other approaches. For
example, Glover and Laguna63, when comparing Tabu Search and Simulated Annealing refer to " … the
focus on exploiting memory in Tabu Search that is absent from Simulated Annealing", " … the Simulated
Annealing approach of randomly sampling … " and "the Simulated Annealing philosophy of adhering to
a temperature that only changes monotonically". As Dowsland24 explains in the previous chapter of the
very same volume, these features were true of the earliest applications of Simulated Annealing, but not of
many more recent implementations. Moreover, since the publication of that book in 1995, both Tabu
Search and Simulated Annealing (and other methods) have continued to develop, making sweeping
comparisons even more invalid.
While we have to accept that a clear comparison between general approaches is nigh on impossible, it is
clear that using intelligence in the implementation of any approach can make an enormous difference to
its effectiveness. For example, one very important issue is the balance between intensification and
diversification. If the first facet is underemphasized, excellent solutions may be missed because the
search space is not probed to sufficient depth. On the other hand, if the second facet is not given enough
scope, the search may be confined to mediocre areas of search space, never touching upon any of the
high-quality areas that may exist.
The holy grail of heuristics research is the provision of a technique which determines for itself how best
to solve any specific problem (rather than trying to find a technique that is best in all circumstances).
Some kind of preliminary analysis would determine the overall approach and any necessary parameters.
Such a method is known as a hyperheuristic. Preliminary work towards this end has been reported64,65,66.
However, much more research is required before such a proposition becomes realistic.
Applying Neighbourhood Search to problems with many objectives
Many real problems involve more than one type of objective. There are three main approaches to solving
such problems.
One such approach is a phased approach, where the objectives are considered in turn. In the first phase,
a solution is produced for the most important objective; then, in the second phase, it is modified with
regard to the next most important objective while ensuring that the solution is not made worse with
regard to the first objective, etc. For an example, see Lotfi and Cerveny67.
10
However, for many problems, such an ordering of objectives may not be appropriate. There may simply
be too many objectives, or the priorities may not be clear. For example, a high value of one objective
may be catastrophic, a middling value of the same objective may be important but not overriding, and a
low value of the same objective may be relatively trivial compared to other considerations.
Another approach has become known as multi-objective optimization, where all objectives are considered
together, but not combined in any way. This gives rise to the notions of Dominance and Paretooptimality. See Hansen68 and Viana and de Sousa69 for discussions of multi-objective metaheuristics.
However, the techniques so far developed have, to date, been used only on small, simple, artificial
problems. Very considerable progress is required if this approach is ever to have value for large,
complex, practical problems.
A third approach is the approach known as Subcost-guided Search31,32. The objectives are combined,
usually by means of linear weightings, into a single all-subsuming compound objective. Any
neighbourhood search method can then be used in the same way as for a genuinely single-objective
problem. However, when designing and selecting a neighbourhood search approach for solving such a
problem, it is still possible to take account of the way in which the single objective has been formed out
of all the subobjectives, in a way that can greatly improve the quality of final outcomes, despite the fact
that inevitably the relative weightings of the subobjectives have to be determined to some extent
subjectively.
Some directions for future research
This work can be developed in very many ways. In this section we concentrate on three linked contexts:
 the manipulation of objective functions so as to increase the effectiveness of neighbourhood search;
 the development and application of more intelligent diversification approaches for complex problems;
 the investigation of the nature of search space and its effect on the efficacy of heuristic search
methods.
Manipulation of objective functions
It can often be beneficial to manipulate the objective function in order to aid the progress of a heuristic
search method. Manipulation of the objective function can also be perceived as manipulation of the
11
search space. It is accepted that certain types of search space are spiky, making it difficult for good
solutions to be obtained, whereas others are smooth, making it easier.
One possible type of manipulation involves the adjustment of cost weightings in a counterintuitive way,
i.e. deliberately distorting the objective function away from an accurate representation of costs. Plumb et
al.70 have shown the value of this approach for an interesting pharmaceutical problem. Another type of
adjustment involves the addition of new elements of cost which are wholly unimportant in themselves,
but without which it may be very unlikely that other real elements of cost can be reduced to a satisfactory
level. Wright6 termed such elements "catalytic" in an analogy with chemical processes.
Dynamic adjustment of subcost weights, which would be varied according to the progress of the search
to date, is another area which could prove very fruitful. Rochat and Semet71 proposed a method which
dynamically updates penalties for infeasibilities within a solution, in order to aid the search for a feasible
solution; Storer et al.72 added random numbers to some of their data; while Voudouris and Tsang73
suggested adding penalty functions to local optima. However, these approaches are somewhat limited in
scope, though still useful and of interest.
Slightly more complex is the approach of Herault74, who has proposed a method called Rescaled
Simulated Annealing, where the value of the objective function is rescaled dynamically, depending on the
temperature. The principles behind Noising75 are very similar to this, but with the objective function
adjusted randomly.
There is great scope for extending the application of all these ideas. The first idea (modifying
weightings) has been systematically examined76 in three contexts - flow-shop, travelling salesman and
layout problems with many objectives. This work showed that such cost distortion can be very useful if
carried out intelligently (though it can of course be very damaging if done inappropriately). This work is
as yet unpublished beyond a PhD thesis77, and further work is required in order to provide robust
guidance as to when and how objective functions should be distorted.
In vehicle routeing, it is notoriously difficult to reduce the number of vehicles using heuristic search.
Progress towards this goal can be made by systematically reducing the number of customers served by a
particular vehicle, yet until the last customer is removed no cost will have been saved. One way of
encouraging a heuristic search method to move towards this goal would be to add one or more catalytic
cost elements to the objective function, perhaps relating to the smallest number of customers served by
the same vehicle. This would encourage the unbalancing of the vehicles' workloads, which is not
12
desirable in itself but could lead to an extremely desirable outcome, i.e. a reduction in the number of
vehicles required.
A similar position pertains to some types of personnel scheduling problem, for example train driver
scheduling. To reduce the total number of schedules requires that work be removed from one driver's
schedule until it disappears entirely. Without some kind of catalytic cost element encouraging this
process, it may be very difficult for any form of heuristic search to achieve this goal.
Influential diversification (e.g. Subcost-Guided Search)
There may be great value in using the progress of the subcosts to guide the search. The value of subcost
guidance is that it provides a focused, or influential form of diversification, which other methods have, to
date, been unable to provide effectively. Not only is the search moved away from well-visited areas of
search space, but it is also influenced to move to new areas that appear likely to produce good solutions.
There is very wide scope for such experimentation.
Hübscher and Glover78 described a simple way of achieving this aim to a limited extent. Their work has
been followed up by Armentano and Yamashita79. More complex and highly successful is the SubcostGuided Search approach, which grew out of two successful real-life applications80,81 and was backed up
by experimentation31,32. There is wide scope for investigating further ways of incorporating the overall
idea of subcost guidance.
The nature of search space
Some of the ideas described earlier concern the manipulation of the nature of the search space, yet this
nature has not yet been properly defined or described. Researchers will often characterise search space
as being smooth, spiky, etc., without being precise about the meanings of such terms. Since it is also
generally accepted that the nature of search space can have a very important effect on the efficacy of
heuristic search techniques, it would be very valuable to know more about the nature of search spaces.
Some research has been carried out in this area82,83,84,85,86. Most of this work has concentrated on the
solution landscapes near local optima. However, much more research is needed and other features could
also be usefully examined: the steepness of slopes, the existence of plateaux, etc. There is a need for
some generally accepted measures of search space which will be of value when determining how
13
effective particular search techniques will prove to be and will help in the setting of parameters for these
techniques.
Summary
Neighbourhood Search metaheuristics have come a very long way in quite a short time. Implementations
and applications are diverse and new approaches and variations are appearing all the time. This paper
has attempted to give an overview of the current position, though before long it will itself inevitably
become out of date.
14
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