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4OR
DOI 10.1007/s10288-008-0089-1
INVITED SURVEY
Variable neighbourhood search: methods
and applications
Pierre Hansen ·Nenad Mladenovi´c·
José A. Moreno Pérez
Received: 5 May 2008 / Revised: 20 October 2008
© Springer-Verlag 2008
Abstract Variable neighbourhood search (VNS) is a metaheuristic, or a framework
for building heuristics, based upon systematic changes of neighbourhoods both in
descent phase, to find a local minimum, and in perturbation phase to emerge from
the corresponding valley. It was first proposed in 1997 and has since then rapidly
developed both in its methods and its applications. In the present paper, these two
aspects are thoroughly reviewed and an extensive bibliography is provided. Moreover,
one section is devoted to newcomers. It consists of steps for developing a heuristic
for any particular problem. Those steps are common to the implementation of other
metaheuristics.
Keywords Variable neighbourhood search ·Metaheuristic ·Heuristic
MSC classification 90C59 ·68T20 ·68W25
1 Introduction
Variable neighbourhood search (VNS) is a metaheuristic, or framework for building
heuristics, aimed at solving combinatorial and global optimization problems. Its basic
P. Hansen
GERAD and HEC Montréal, Montréal, QC H3T 2A7, Canada
e-mail: pierre.hansen@gerad.ca
N. Mladenovi´c
GERAD and School of Mathematics, Brunel University, Uxbridge, UK
e-mail: nenad.mladenovic@brunel.ac.uk
J. A. Moreno Pérez (B
)
IUDR and Dep. Estadística, I.O. y Computación, University of La Laguna, 38271 La Laguna, Spain
e-mail: jamoreno@ull.es
123
P. Hansen et al.
idea consists in a systematic change of neighbourhood combined with a local search.
Since its inception, VNS has undergone many developments and been applied in
numerous fields. We review below the basic rules of VNS and of its main extensions.
In addition, some of the most successful applications are briefly summarized. Pointers
to many other applications are given in the reference list.
A deterministic optimization problem may be formulated as
min{f(x)|x∈X,X⊆S},(1)
where S,X,xand frespectively denote the solution space and feasible set,afeasible
solution and a real-valued objective function.IfSis a finite but large set, a combinato-
rial optimization problem is defined. If S=Rn, we refer to continuous optimization.
A solution x∗∈Xis optimal if
f(x∗)≤f(x), ∀x∈X.
An exact algorithm for problem (1), if one exists, finds an optimal solution x∗, together
with the proof of its optimality, or shows that there is no feasible solution, i.e., X=∅.
Moreover, in practice, the time needed to do so should be finite (and not too long).
When one deals with a continuous function, it is reasonable to allow for some degree
of tolerance, i.e., to stop when a feasible solution x∗has been found such that
f(x∗)< f(x)+ε, ∀x∈Xor f(x∗)−f(x)
f(x∗)<ε, ∀x∈X
for some small positive ε.
Many practical instances of problems of form (1), arising in Operations Research
and other fields, are too great for an exact solution to be found in reasonable time.
It is well-known from complexity theory (Garey and Johnson 1978;Papadimitriou
1994) that thousands of problems are NP-hard, such that no algorithm with a number
of steps polynomial in the size of the instances is known for solving any of them and
that if one were found it would be a solution for all. Moreover, in some cases where a
problem admits a polynomial algorithm, this algorithm may be such that realistic size
instances cannot be solved in reasonable time in the worst case, and sometimes also
in the average case or in most cases.
This explains the need to resort to heuristics which speedily yield an approximate
solution, or sometimes an optimal solution but one which has no proof of its optimality.
Some of these heuristics have a worst-case guarantee, i.e., the solution xhobtained
satisfies f(xh)−f(x)
f(xh)≤ε, ∀x∈X(2)
for some ε, though this is rarely small. Moreover, this upper bound εon the worst-
case error is usually much larger than the average error observed in practice and may
therefore be a bad guide in selecting a heuristic. In addition to avoiding excessive
computing time, heuristics address another problem: local optima. A local optimum
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Variable neighbourhood search: methods and applications
xLof problem (1) is such that
f(xL)≤f(x), ∀x∈N(xL)∩X(3)
where N(xL)denotes a neighbourhood of xL(ways to define such a neighbourhood
will be discussed below). If there are many local minima, the range of values which
they span may be large. Moreover, the globally optimum value f(x∗)may differ sub-
stantially from the average value of the local minima, or even from the best such value
among many, obtained by some simple heuristic such as multistart (a phenomenon
called the Tchebycheff catastrophe in Baum (1986)). There are, however, many ways
to escape from local optima or, more precisely, from the valleys which contain them.
Metaheuristics are general frameworks to build heuristics for combinatorial and
global optimization problems. For discussion of the best-known of them, the reader
is referred to the following survey books Reeves (1993), Glover and Kochenberger
(2003) and Burke and Kendall (2005). Some of the many successful applications of
metaheuristics are also mentioned there.
Variable Neighbourhood Search (VNS) (Mladenovi´c 1995;Mladenovi´c and Hansen
1997;Hansen and Mladenovi´c 1997,1999,2001a,2003,2005;Moreno-Pérez et al.
2005) is a metaheuristic which systematically exploits the idea of neighbourhood
change, both in descent to local minima and in escape from the valleys which contain
them. VNS heavily relies upon the following observations:
Fact 1 A local minimum with respect to one neighbourhood structure is not neces-
sarily a local minimum for another neighbourhood structure.
Fact 2 A global minimum is a local minimum with respect to all possible neighbour-
hood structures.
Fact 3 For many problems local minima with respect to one or several neighbour-
hoods are relatively close to each other.
This last observation is empirical. It implies that a local optimum often provides some
information about the global optimum. For instance, it may be the case that there are
several variables with the same value in both. However, it is not usually known which
ones are of this kind. An organized study of the neighbourhood of this local optimum
is therefore in order, until a better one is found.
Unlike many other metaheuristics, the basic schemes of VNS and its extensions
are simple and require few, and sometimes no parameters. Therefore, in addition to
providing very good solutions, often in simpler ways than other methods, VNS gives
insight into the reasons for such a performance, which, in turn, can lead to more
efficient and sophisticated implementations.
The paper is organized as follows. Background ideas, which in part inspired VNS,
are briefly discussed in Sect. 2. Basic schemes are reviewed in Sect. 3. Section 4is
devoted to newcomers. The steps for developing heuristics for any particular problem
are given. Most of those steps are common to the implementation of other meta-
heuristics. Then some tips which can help to improve the current VNS version are
123
P. Hansen et al.
listed. Various applications are classified and surveyed in Sect. 5. Section 6lists those
desirable properties of metaheuristics that are enjoyed by VNS.
The purpose of this paper is threefold: (i) to present to researchers the main ideas
and schemes of VNS; (ii) to provide an extensive list of successful applications and
(iii) to (gently) introduce newcomers into the metaheuristics area.
2 Background
VNS embeds a local search heuristic for solving combinatorial and global optimization
problems. This idea has had some predecessors. It allows a change of the neighbour-
hood structures within this search. In this section, we give a brief introduction to the
variable metric algorithm for solving continuous convex problems and local search
heuristics for solving combinatorial and global optimization problems.
2.1 Variable metric method
The variable metric method for solving unconstrained continuous optimization prob-
lem (1) has been suggested by Davidon (1959) and Fletcher and Powell (1963). The
idea is to change the metric (and thus the neighbourhood) at each iteration such that
the search direction (steepest descent with respect to the current metric) adapts better
to the local shape of the function. In the first iteration a Euclidean unit ball in the n
dimensional space is used and the steepest descent (anti-gradient) direction found. At
subsequent iterations, ellipsoidal balls are used and the steepest direction of descent
is obtained with respect to a new metric resulting from a linear transformation. The
purpose of such changes is to build up, iteratively, a good approximation to the inverse
of the Hessian matrix A−1of f, that is, to construct a sequence of matrices Hiwith
the property,
lim
i→∞ Hi=A−1.
In the convex quadratic programming case, the limit is achieved after niterations
instead of an infinity of them. In this way the so-called Newton search direction is
obtained. The advantages are that: (i) it is not necessary to find the inverse of the
Hessian (which requires O(n3)operations) at each iteration; (ii) the second order
information is not needed. Assume that the function f(x)is approximated by its
Taylor series
f(x)=1
2xTAx −bTx(4)
with positive definite matrix A(denoted by A>0). Applying the first order condition
∇f(x)=Ax −b=0wehave Axopt =b, where xopt is a minimum point. At
the current point we have Axi=∇f(xi)+b.We will not rigorously derive here
the Davidon–Fletcher–Powell (DFP) algorithm for transforming Hiinto Hi+1.Let
us mention only that subtracting one of these last two equations from the other and
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Variable neighbourhood search: methods and applications
multiplying (from the left) by the inverse matrix A−1,wehave
xopt −xi=−A−1∇f(xi).
Subtracting this last equation evaluated at xi+1from the same equation at xigives
xi+1−xi=−A−1(∇f(xi+1)−∇ f(xi)). (5)
Having made the step from xito xi+1, we might reasonably require that the new
approximation Hi+1satisfies (5) as if it were actually A−1; that is,
xi+1−xi=−Hi+1(∇f(xi+1)−∇f(xi)). (6)
We might also assume that the updating formula for matrix Hishould be of the form
Hi+1=Hi+U, where Uis a correction. It is possible to obtain different updating
formulas for Uand thus for Hi+1, keeping Hi+1positive definite (Hi+1>0). In fact,
there exists a whole family of updates, the Broyden family. From practical experience,
the so-called BFGS method seem to be the most popular (see, e.g., Gill et al. (1981)
for details). Steps are listed in Algorithm 1.
Function VarMetric(x);
let x∈Rnbe an initial solution;1
H←I;g←−∇f(x);2
for i=1to ndo3
α∗←arg minαf(x+α·Hg);4
x←x+α∗·Hg;5
g←−∇f(x);6
H←H+U;7
end
Algorithm 1: Variable metric algorithm
From the above one can conclude that even in solving a convex program, a change
of metric, and, thus, a change of the neighbourhoods induced by this metric, may
produce more efficient algorithms. Thus, using the idea of neighbourhood change for
solving NP-hard problems could well lead to even greater benefits.
2.2 Local search
Alocal search heuristic consists in choosing an initial solution x, finding a direction of
descent from x, within a neighbourhood N(x), and moving to the minimum of f(x)
within N(x)in the same direction. If there is no direction of descent, the heuristic
stops; otherwise, it is iterated. Usually the steepest direction of descent, also referred
to as best improvement, is used. This set of rules is summarized in Algorithm 2,
where we assume that an initial solution xis given. The output consists of a local
minimum, also denoted by x, and its value. Observe that a neighbourhood structure
123
P. Hansen et al.
N(x)is defined for all x∈X. In discrete optimization problems it usually consists
of all vectors obtained from xby some simple modification, e.g., in the case of 0-1
optimization, complementing one or two components of a 0-1 vector. Then, at each
step, the neighbourhood N(x)of xis explored completely. As this may be time-
consuming, an alternative is to use the first descent heuristic. Vectors xi∈N(x)are
then enumerated systematically and a move is made as soon as a direction for the
descent is found. This is summarized in Algorithm 3.
Function BestImprovement(x);
repeat
1
x←x;2
x←arg miny∈N(x)f(y)
3
until (f(x)≥f(x));
Algorithm 2: Best improvement (steepest descent) heuristic
Function FirstImprovement(x);
repeat
1
x←x;i←0;2
repeat3
i←i+1;4
x←arg min{f(x), f(xi)},xi∈N(x)5
until (f(x)< f(xi)or i=|N(x)|);
until (f(x)≥f(x));
Algorithm 3: First improvement heuristic
3 Basic schemes
Let us denote with Nk,(k=1,...,kmax), a finite set of pre-selected neighbourhood
structures, and with Nk(x)the set of solutions in the kth neighbourhood of x. We will
also use the notation N
k,k=1,...,k
max, when describing local descent. Neighbour-
hoods Nkor N
kmay be induced from one or more metric (or quasi-metric) functions
introduced into a solution space S.Anoptimal solution xopt (or global minimum) is a
feasible solution where a minimum of problem (1) is reached. We call x∈Xalocal
minimum of problem (1) with respect to Nk(w.r.t. Nkfor short), if there is no solution
x∈Nk(x)⊆Xsuch that f(x)< f(x).
In order to solve problem (1) by using several neighbourhoods, facts 1–3 can be
used in three different ways: (i) deterministic; (ii) stochastic; (iii) both deterministic
and stochastic. We first give in Algorithm 4the steps of the neighbourhood change
function which will be used later.
Function NeighbourhoodChange() compares the new value f(x)with the
incumbent value f(x)obtained in the neighbourhood k(line 1). If an improvement
is obtained, kis returned to its initial value and the new incumbent updated (line 2).
Otherwise, the next neighbourhood is considered (line 3).
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Variable neighbourhood search: methods and applications
Function NeighbourhoodChange (x,x,k);
if f(x)< f(x)then1
x←x;k←1/*Makeamove*/;2
else
k←k+1 /* Next neighbourhood */ ;
3
end
Algorithm 4: Neighbourhood change or Move or not function
3.1 Variable neighbourhood descent (VND)
The variable neighbourhood descent (VND) method is obtained if the change of neigh-
bourhoods is performed in a deterministic way. Itssteps are presented in Algorithm 5.
In the descriptions of all algorithms that follow, we assume that an initial solution x
is given. Most local search heuristics in their descent phase use very few neighbour-
Function VND (x,k
max);
repeat
1
k←1;2
repeat3
x←arg miny∈N
k(x)f(x)/* Find the best neighbor in Nk(x)*/ ;
4
NeighbourhoodChange (x,x,k) /* Change neighbourhood */ ;5
until k=k
max;
until no improvement is obtained;
Algorithm 5: Steps of the basic VND
hoods (usually one or two, i.e., k
Max ≤2). Note that the final solution should be a
local minimum with respect to all k
max neighbourhoods; hence the chances to reach
a global one are larger when using VND than with a single neighbourhood structure.
Moreover, using this sequential order of neighbourhood structures in VND above, one
can develop a nested strategy. Assume, for example, that k
max =3. Then a possible
nested strategy is: perform VND above for the first two neighbourhoods, in each point
xthat belongs to the third (x∈N3(x)). Such an approach is applied, e.g., in Brimberg
et al. (2000) and Hansen and Mladenovi´c(2001b).
3.2 Reduced VNS
The Reduced VNS (RVNS) method is obtained if random points are selected from
Nk(x)and no descent is made. Rather, the values of these new points are compared
with that of the incumbent and updating takes place in case of improvement. We
assume that a stopping condition has been chosen, among various possibilities, e.g.,
the maximum cpu time allowed tmax, or the maximum number of iterations between
two improvements. To simplify the description of the algorithms we always use tmax
123
P. Hansen et al.
below. Therefore, RVNS uses two parameters: tmax and kmax. Its steps are presented
in Algorithm 6. With the function Shake represented in line 4, we generate a point
xat random from the kth neighbourhood of x, i.e., x∈Nk(x).
Function RVNS (x,kmax,tmax );
repeat
1
k←1;2
repeat3
x←Shake(x,k);4
NeighborhoodChange (x,x,k);5
until k=kmax;
t←CpuTime()
6
until t>tmax;
Algorithm 6: Steps of the Reduced VNS
RVNS is useful in very large instances, for which local search is costly. It has been
observed that the best value for the parameter kmax is often 2. In addition, the max-
imum number of iterations between two improvements is usually used as a stopping
condition. RVNS is akin to a Monte-Carlo method, but is more systematic (see, for
example, Mladenovi´cetal.(2003b) where the results obtained by RVNS were 30%
better than those of the Monte-Carlo method in solving a continuous min–max prob-
lem). When applied to the p-Median problem, RVNS gave solutions as good as the
Fast Interchange heuristic of Whitaker (1983) while being 20–40 times faster (Hansen
et al. 2001).
3.3 Basic VNS
The Basic VNS (BVNS) method (Mladenovi´c and Hansen 1997) combines determin-
istic and stochastic changes of neighbourhood. Its steps are given in Algorithm 7.
Function VNS (x,kmax,tmax );
repeat
1
k←1;2
repeat3
x←Shake(x,k) /* Shaking */;4
x ←FirstImprovement(x) /* Local search */;5
NeighbourhoodChange(x,x,k) /* Change neighbourhood */;6
until k=kmax;
t←CpuTime()
7
until t>tmax;
Algorithm 7: Steps of the basic VNS
Often successive neighbourhoods Nkwill be nested. Observe that point xis gener-
ated at random in Step 4 in order to avoid cycling, which might occur if a deterministic
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Variable neighbourhood search: methods and applications
N1(x)
f
x
xí
f(x)
x
N (x)
k
Global minimum
Local minimum
Fig. 1 Basic VNS
Fig. 2 Four-cardinality tree
problem 2
1
3
4
5
6
7
9
8
11
10
12
1
26 6 23
25 15 16 24
17 20 18 16 916
86
16 99
17 9
rule were applied. In Step 5 the first improvement local search (Algorithm 3) is usually
adopted. However, it can be replaced with best improvement (Algorithm 2).
Example We illustrate the basic step on a minimum k-cardinality tree instance taken
from Jornsten and Lokketangen (1997)(seeFig.2). The minimum k-cardinality tree
problem on graph G(k-card for short) consists in finding a subtree of Gwith exactly
kedges whose sum of weights is minimum.
The steps of BVNS are given in Fig. 3. In Step 0 the objective function value, i.e.,
the sum of edge weights, is equal to 40; it is indicated in the right-hand bottom corner
of the figure. This first solution is a local minimum with respect to the edge-exchange
neighbourhood structure (one edge in, one out). After shaking, the objective function
is 60, and after another local search, we return to the same solution. Then, in Step 3,
we take out 2 edges and add another 2 at random, and, after a local search, an improved
solution is obtained with a value of 39, etc. In Step 8, we find the optimal solution
with an objective function value equal to 36.
3.4 General VNS
Note that the Local search Step 5 may also be replaced by VND (Algorithm 5).
Using this general VNS (VNS/VND) approach has led to the most successful applica-
tions reported (see, for example, (Andreatta and Ribeiro 2002;Brimberg et al. 2000;
123
P. Hansen et al.
40
9374
933493
49
4060
1 11
25 2525
8
88
8
886
66
6
6
6
26
17
16 16
1616
16
9
999
99
9
18
99
9
LS
Shake-2 Shake-1
LS
LS Shake-2 LS
LS
Shake-1
36
0
543
876
21
Fig. 3 Steps of the Basic VNS for solving four-card tree problem
Canuto et al. 2001;Caporossi and Hansen 2000,2004;Caporossi et al. 1999a,c;
Hansen and Mladenovi´c 2001b;Hansen et al. 2006;Ribeiro and de Souza 2002;
Ribeiro et al. 2002). Steps of the general VNS (GVNS) are given in Algorithm 8.
Function GVNS (x,k
max,kmax ,tmax );
repeat
1
k←1;2
repeat3
x←Shake(x,k);4
x ←VND(x,k
max);5
NeighborhoodChange(x,x,k);6
until k=kmax;
t←CpuTime()
7
until t>tmax;
Algorithm 8: Steps of the general VNS
3.5 Skewed VNS
The skewed VNS (SVNS) method (Hansen et al. 2000) addresses the problem of
exploring valleys far from the incumbent solution. Indeed, once the best solution in
a large region has been found, it is necessary to go some way to obtain an improved
one. Solutions drawn at random in distant neighbourhoods may differ substantially
from the incumbent and VNS can then degenerate, to some extent, into the Multistart
heuristic (in which descents are made iteratively from solutions generated at random, a
heuristic which is known not to be very efficient). Consequently, some compensation
for distance from the incumbent must be made and a scheme called Skewed VNS
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Variable neighbourhood search: methods and applications
is proposed for this purpose. Its steps are presented in Algorithms 10 and 11, where
the KeepBest(x,x) function simply keeps whichever is the better of xand x:if
f(x)< f(x)then x←x.
Function NeighbourhoodChangeS(x,x,k,α);
if f(x)−αρ ( x,x)< f(x)then1
x←x;k←12
else
k←k+1
3
end
Algorithm 9: Steps of neighbourhood change for the skewed VNS
Function SVNS (x,kmax,tmax ,α);
repeat
1
k←1; xbest ←x;2
repeat3
x←Shake(x,k);4
x ←FirstImprovement(x);5
KeepBest (xbest ,x);6
NeighbourhoodChangeS(x,x,k,α);7
until k=kmax;
x←xbest ;8
t←CpuTime();9
until t>tmax;
Algorithm 10: Steps of the Skewed VNS
SVNS makes use of a function ρ(x,x)to measure the distance between the incum-
bent solution xand the local optimum found x . The distance used to define the Nk,
as in the above examples, could be used also for this purpose. The parameter αmust
be chosen in order to accept the exploration of valleys far away from xwhen f(x )is
larger than f(x)but not too much larger (otherwise one will always leave x). A good
value is to be found experimentally in each case. Moreover, in order to avoid frequent
moves from xto a close solution, one may take a large value for αwhen ρ(x,x)is
small. More sophisticated choices for a function of αρ (x,x )could be made through
a learning process.
3.6 Some extensions of basic VNS
Several easy ways to extend the basic VNS are now discussed. The basic VNS is a first
improvement descent method with randomization. Without much additional effort it
can be transformed into a descent-ascent method: in NeighbourhoodChange()
function, replace also xby x with some probability, even if the solution is worse
than the incumbent. It can also be changed into a best improvement method: make
123
P. Hansen et al.
a move to the best neighbourhood k∗among all kmax of them. Its steps are given in
Algorithm 11.
Function BI-VNS (x,kmax,tmax );
repeat
1
k←1;2
xbest ←x;
repeat
3
x←Shake(x,k);4
x ←FirstImprovement(x);5
KeepBest(xbest ,x );6
k←k+1;7
until k=kmax;
x←xbest ;8
t←CpuTime()9
until t>tmax;
Algorithm 11: Steps of the basic Best Improvement VNS
Another variant of the basic VNS can be to find a solution xin the “Shaking”
step as the best among b(a parameter) randomly generated solutions from the kth
neighbourhood. There are two possible variants of this extension: (i) to perform only
one local search from the best among bpoints; (ii) to perform all blocal searches
and then choose the best. We now give an algorithm of a second type suggested by
Fleszar and Hindi (2004). There, the value of parameter bis set to k. In this way, no
new parameter is introduced (see Algorithm 12). It is also possible to introduce kmin
Function FH-VNS (x,kmax,tmax );
repeat
1
k←1;2
repeat3
for =1to k do4
x←Shake(x,k);5
x ←FirstImprovement(x);6
KeepBest(x,x);7
end
NeighbourhoodChange(x,x,k);8
until k=kmax;
t←CpuTime()
9
until t>tmax;
Algorithm 12: Steps of the Fleszar–Hindi extension of the basic VNS
and kstep, two parameters which control the change of neighbourhood process: in the
previous algorithms instead of k←1setk←kmin and instead of k←k+1set
k←k+kstep. The steps of Jump VNS are given in Algorithms 13 and 14.
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Variable neighbourhood search: methods and applications
Function J-VNS (x,kmin,kst ep,kmax ,tmax);
repeat
1
k←kmin;2
repeat3
x←Shake(x,k);4
x ←FirstImprovement(x);5
NeighbourhoodChangeJ(x,x,k,kmin,kstep);6
until k=kmax;
t←CpuTime()
7
until t>tmax;
Algorithm 13: Steps of the Jump VNS
Function NeighborhoodChangeJ (x,x,k,kmin,kst ep);
if f(x)< f(x)then1
x←x;k←kmin;2
else
k←k+kstep ;3
end
Algorithm 14: Neighbourhood change or Move or not function
3.7 Variable neighbourhood decomposition search
While the basic VNS is clearly useful for obtaining an approximate solution to many
combinatorial and global optimization problems, it remains a difficult or lengthy take
to solve very large instances. As often, the size of the problems considered is in practice
more limited by the tools available to solve them than by the needs of the potential
users of these tools. Hence, improvements appear to be highly desirable. Moreover,
when heuristics are applied to very large instances, their strengths and weaknesses
become clearly apparent. Three improvements of the basic VNS for solving large
instances are now considered.
The variable neighbourhood decomposition search (VNDS) method (Hansen et al.
2001) extends the basic VNS into a two-level VNS scheme based upon decomposition
of the problem. Its steps are presented in Algorithm 15, where tdis an additional
parameter and represents the running time given for solving decomposed (smaller
sized) problems by VNS.
For ease of presentation, but without loss of generality, we assume that the solution
xrepresents the set of some elements. In Step 4 we denote with ya set of ksolution
attributes present in xbut not in x(y=x\x). In Step 5 we find the local optimum
yin the space of y; then we denote with x the corresponding solution in the whole
space S(x =(x\y)∪y). We notice that exploiting some boundary effects in a
new solution can significantly improve the solution quality. This is why, in Step 6,
we find the local optimum x in the whole space Susing x as an initial solution.
If this becomes time-consuming, then at least a few local search iterations should be
performed.
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P. Hansen et al.
Function VNDS (x,kmax,tmax ,td);
repeat
1
k←2;2
repeat3
x←Shake (x,k); y←x\x;4
y←VNS(y,k,td); x =(x\y)∪y;5
x ←FirstImprovement(x);6
NeighborhoodChange(x,x,k);7
until k=kmax;
until t>tmax;
Algorithm 15: Steps of VNDS
VNDS can be viewed as embedding the classical successive approximation scheme
(which has been used in combinatorial optimization at least since the 1960s; see, for
example, Griffith and Stewart (1961) in the VNS framework.
3.8 Parallel VNS
Parallel VNS (PVNS) methods are another extension. Several ways of parallelizing
VNS have recently been proposed for solving the p-Median problem. In García-López
et al. (2002) three of them are tested: (i) parallelize local search; (ii) augment the
number of solutions drawn from the current neighbourhood and make a local search
in parallel from each of them and (iii) do the same as (ii) but update the information
about the best solution found. The second version gives the best results. It is shown
in Crainic et al. (2004) that assigning different neighbourhoods to each processor and
interrupting their work as soon as an improved solution is found gives very good
results. The best-known solutions have been found on several large instances taken
from TSP-LIB Reinelt (1991). Three Parallel VNS strategies are also suggested for
solving the Travelling purchaser problem in Ochi et al. (2001).
3.9 Primal-dual VNS
For most modern heuristics, the difference in value between the optimal solution
and the obtained one is completely unknown. Guaranteed performance of the pri-
mal heuristic may be determined if a lower bound on the objective function value is
known. To this end, the standard approach is to relax the integrality condition on the
primal variables, based on a mathematical programming formulation of the problem.
However, when the dimension of the problem is large, even the relaxed problem may
be impossible to solve exactly by standard commercial solvers. Therefore, it seems a
good idea to solve dual relaxed problems heuristically as well. In this way we obtain
guaranteed bounds on the primal heuristics performance. The next problem arises if
we want to reach an exact solution within a Branch and bound framework, since having
the approximate value of the relaxed dual does not allow us to branch easily, e.g., by
exploiting complementary slackness conditions. Thus, the exact value of the dual is
necessary.
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In Primal-dual VNS (PD-VNS) (Hansen et al. 2007a) one possible general way
to attain both the guaranteed bounds and the exact solution is proposed. Its steps are
given in Algorithm 16.
Function PD-VNS (x,k
max,kmax ,tmax );
BVNS (x,k
max,kmax ,tmax ) /* Solve primal by VNS */;1
DualFeasible(x,y) /* Find (infeasible) dual such that fP=fD*/ ;2
DualVNS(y) /* Use VNS do decrease infeasibility */;3
DualExact(y) /* Find exact (relaxed) dual */;4
BandB(x,y) /* Apply branch-and-bound method */ ;5
Algorithm 16: Steps of the basic PD-VNS
In the first stage an heuristic procedure based on VNS is used to obtain a near
optimal solution. In Hansen et al. (2007a) it is shown that VNS with decomposition is
a very powerful technique for large-scale simple plant location problems (SPLP) with
up to 15,000 facilities and 15,000 users. In the second phase, this approach is designed
to find an exact solution to the relaxed dual problem. Solving SPLP is accomplished
in three stages: (i) find an initial dual solution (generally infeasible), using the primal
heuristic solution and complementary slackness conditions; (ii) improve the solution
by applying VNS to the unconstrained nonlinear form of the dual; (iii) solve the dual
exactly using a customized “sliding simplex” algorithm which applies “windows” to
the dual variables, substantially reducing the size of the problem. In all the problems
tested, including instances much larger than previously reported in the literature, the
procedure was able to find the exact dual solution in reasonable computing time. In the
third and final phase armed with tight upper and lower bounds, obtained respectively
from the heuristic primal solution in phase one and the exact dual solution in phase two,
we apply a standard branch-and-bound algorithm to find an optimal solution of the
original problem. The lower bounds are updated with the dual sliding simplex method
and the upper bounds, whenever new integer solutions are obtained at the nodes of the
branching tree. In this way it is possible to solve exactly problem instances with up to
7,000 ×7,000 for uniform fixed costs and 15,000 ×15,000 otherwise.
3.10 Variable neighbourhood formulation space search
Traditional ways to tackle an optimization problem consider a given formulation and
search in some way through its feasible set X. The fact that the same problem may
often be formulated in different ways allows search paradigms to be extended to
include jumps from one formulation to another. Each formulation should lend itself
to some traditional search method, its “local search” which works totally within this
formulation, and yields a final solution when started from some initial solution. Any
solution found in one formulation should easily be translatable to its equivalent in
any other formulation. We may then move from one formulation to another, using the
solution resulting from the former’s local search as an initial solution for the latter’s
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P. Hansen et al.
local search. Such a strategy will, of course, be useful only in situations where local
searches in different formulations behave differently.
This idea was recently investigated in Mladenovi´cetal.(2005) using an approach
which systematically changes the formulations for solving circle packing problems
(CPP). It is shown there that the stationary point of a non-linear programming formu-
lation of CPP in Cartesian coordinates is not necessarily also a stationary point in a
polar coordinate system. A method Reformulation descent (RD) is suggested which
alternates between these two formulations until the final solution is stationary with
respect to both. The results obtained were comparable with the best known values,
but they were achieved some 150 times faster than by an alternative single formu-
lation approach. In the same paper, the idea suggested above of Formulation space
search (FSS) is also introduced, using more than two formulations. Some research
in this direction has been reported in Mladenovi´c(2005), Plastria et al. (2005) and
Hertz et al. (2008). One algorithm which uses the variable neighbourhood idea in
searching through the formulation space is given in Algorithms 17 and 18.
Function FormulationChange(x,x,φ,φ
,);
if f(φ,x)< f(φ , x)then1
φ←φ;x←x;←min
2
else
←+step;3
end
Algorithm 17: Formulation change function
Function VNFSS(x,φ,
max);
repeat
1
←1 /* Initialize formulation in F*/;2
while ≤max do3
ShakeFormulation(x,x,φ,φ ,)/*(φ,x)∈(N(φ),N(x)) at random */;4
FormulationChange(x,x,φ,φ ,) /* Change formulation */;5
end
until some stopping condition is met;
Algorithm 18: Reduced variable neighborhood FSS
In Fig. 4we consider the CPP case with n=50. The set consists entirely of mixed
formulations, in which some circle centres are given in Cartesian coordinates while
the others are given in polar coordinates. The distance between two formulations is
then the number of centres whose coordinates are expressed in different systems in
each formulation. FSS starts with the RD solution, i.e., with rcurr =0.121858. The
values of kmin and kstep are set to 3 and the value of kmax is set to n=50. We did not
gain any improvement with kcurr =3,6 and 9. The next improvement was obtained
for kcurr =12. This means that a “mixed” formulation with 12 polar and 38 Cartesian
coordinates is used. Then we turned again to the formulation with three randomly
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Variable neighbourhood search: methods and applications
Fig. 4 Reduced FSS for PCC problem and n=50
chosen circle centres, which was unsuccessful; but we obtained a better solution with
6, etc. After 11 improvements we ended with a solution with radius rmax =0.125798.
4 Developing VNS
4.1 Getting started
This section is devoted to newcomers. Its purpose is to help students in making a first
very simple version of VNS, which would not necessarily be competitive with later
more sophisticated versions. Most of the steps are common to the implementation of
other metaheuristics.
A step-by-step procedure
1. Getting familiar with the problem. Think about the problem at hand. In order to
understand it better, make a simple numerical example and spend some time in
trying to solve it by hand in your own way. Try to understand why the problem is
hard and why a heuristic is needed.
2. Read Literature. Read about the problem and the solution methods in the literature.
3. Test instances (read data). Use your numerical example as a first instance for
testing your future code, but if it is not large enough, take some instance from
the web, or make a routine for generating random instances. In the second case,
read how to generate events using uniformly distributed numbers from the (0,1)
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P. Hansen et al.
interval (since each programming language has statements for getting such random
numbers).
4. Data structure. Think about the way in which the solution of the problem will
be represented in the memory. Consider two or more representations of the same
solution to see if they can reduce the complexity of some routines, i.e., analyse
the advantages and disadvantages of each possible presentation.
5. Initial solution. Having established a routine for reading or generating instances
of the problem, the next step is to obtain an initial solution. For the simple version,
any random feasible solution may be used, but the usual way is to develop some
greedy constructive heuristic. This is normally not hard to do.
6. Objective value. Make a procedure for calculating the objective function value
for a given solution. Notice that at this stage, we already have all ingredients for
a Monte-Carlo method: the generation of a random solution and calculation of
the objective function value. Get a solution to your problem by the Monte Carlo
heuristic (i.e., repeat steps 5 and 6 many times and keep the best one).
7. Shaking. Make a procedure for Shaking. This is a key step of VNS. However, it
is easy to implement and usually requires only a few lines of computer code. For
example, in solving the multi-source Weber problem, the easiest perturbation of
the current solution is to re-allocate a randomly chosen entity from its cluster to
another one, also chosen at random. In fact, in this case, the shaking step (or jump,
in the kth neighbourhood) would need only three lines of the computer code:
For i=1tok
a(1+n·Rnd1)=1+m·Rnd2
EndFor
This solution is saved in an array a() ∈{1,...,m}which denotes the membership
or allocation of entity (=1,...,n); Rnd1 and Rnd2 denote random numbers
uniformly distributed from the (0,1) interval. Compare the results obtained by the
Reduced VNS (take kmax =2) with those of the Monte Carlo method.
8. Local search. Choose an off-the-shelf local search heuristic (or develop a new
one). In building a new local search, consider several usual moves which define
the neighbourhood of the solution, such as drop, add, swap, interchange,etc.In
addition, for the efficiency (speed) of the method, it is very important to pay special
attention to updating of the incumbent solution. In other words, it is usually not
necessary to use a procedure for calculating the objective function values for each
point in the neighbourhood, i.e., it is often possible to reach these values by very
simple calculations.
9. Comparison. Include the local search routine into RVNS to get the basic VNS,
and compare it with other methods from the literature.
4.2 More tips
Sometimes the basic VNS does not provide very good results and it must then be
refined in one of the following ways.
1. First vs. best improvement. Experimentally compare first and best improvement
strategies within a local search. Previous experience suggest the following: if your
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initial solution is chosen at random, use the first improvement rule, but if some
constructive heuristic is used, use the best improvement rule.
2. Reduce the neighbourhood. The reason for the bad behaviour of any local search
may be unnecessary visits to all the solutions in the neighbourhood. Try to identify
“promising” subsets of the neighbourhood and visit these only; ideally, find a rule
which automatically removes solutions from the neighbourhood solutions whose
objective values are no better than the current one.
3. Intensified shaking. In developing a more effective VNS, one must spend some
time in checking how sensitive is the objective function to small change (shaking)
of the solution. The trade-off between intensification and diversification of the
search in VNS is balanced in a Shaking procedure. For some problem instances,
a completely random jump in the kth neighbourhood is too diversified. In such
cases, some intensify shaking procedure is in order. For instance, a k-interchange
neighbourhood may be reduced by repeating ktimes random add followed by
best drop moves. A special case of intensified shaking is the so-called Large
neighbourhood search, where krandomly chosen attributes of the solutions are
destroyed (dropped), and then the solution is re-built in the best way (by some
constructive heuristic).
4. VND. Analyse several possible neighbourhood structures, estimate their sizes,
order them, try them out and keep the most efficient ones. In other words, develop
a VND and replace the local search routine with this VND to get a general VNS.
5. Experiments with parameter settings. The single central parameter of VNS is kmax,
which should be tuned experimentally. However, the procedure is not usually very
sensitive to kmax and, in order to obtain a parameter-free VNS, one can fix its value
at the value of some input parameter, e.g., for the p-median, kmax =p;forthe
minimum sum-of-square clustering kmax =m,etc.
5 Applications
Applications of VNS, or of hybrids of VNS combined with other metaheuristics, are
diverse and numerous. In this section, we review some of them.
5.1 Industrial applications
Regarding the first industrial applications, the oil industry has provided many prob-
lems. These include the design of an offshore pipeline network (Brimberg et al. 2003),
the pooling problem (Audet et al. 2004) and the scheduling of walkover rigs for Petro-
bras (Aloise et al. 2006).
5.2 Design problems in communication
Costa et al. (2002) apply a variable neighbourhood decomposition search (VNDS)
for the optimization of a power plant cable layout. Mladenovi´cetal.(2003b). Use
VNS for solving a spread spectrum radar polyphase code design problem.
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P. Hansen et al.
Degila and Sansò (2004) propose a VNS to deal with the topological design of a yotta-
bit-per-second (1 yotta =1024) multidimensional network based upon agile optical
cores which provides fully meshed connectivity with direct optical paths between
edge nodes which are electronically controlled. Lapierre et al. (2004) consider the
application of a Tabu Search/VNS hybrid for designing distribution networks with
transshipment centres. Meric et al. (2004) apply VNS for optical routing in networks
using latin routers. Dias et al. (2006) use a General VNS (GVNS) to improve the qual-
ity of the solution obtained with a Greedy Randomized Adaptive Search Procedure
(GRASP) for the ring star problem. In Loudni et al. (2006) a difficult real-life network
problem of France Telecom R&D, the on-line resources allocation for ATM networks
with rerouting is solved by VNS/LDS+ CP.
The application of VNS in the design of SDH/WDM networks is proposed in
Melián-Batista et al. (2008); it is improved with the use of an adaptive memory mech-
anism in Melián (2006) and by applying a pilot method in Höller et al. (2008). Taga wa
et al. (2007) deal with the robust design of Surface Acoustic Wave (SAW) filters.
Ribeiro et al. (2007) consider VNS and other metaheuristics for optimization prob-
lems in computer communications.
5.3 Location problems
Location problems have also attracted much attention from the VNS researchers
and practitioners. Among discrete models the p-median has been the most studied
and has played a central rule in the development of a VNS metaheuristic. Brimberg
and Mladenovi´c(1996) give the earliest applications of VNS. Hansen et al. (2001)
introduces a variable neighbourhood decomposition search solving the p-median
problem. García-López et al. (2002) is the first parallel version of the VNS. Hansen
and Mladenovi´c(2008) complete the comparative analysis in Alba and Domínguez
(2006) with a detailed comparison of several versions of VNS with other metaheuris-
tics for the p-median problem. See Mladenovi´cetal.(2007a) for the role of VNS in
solving the p-median problem.
Other discrete location problems solved with VNS are the p-centre problem
(Mladenovi´c et al. 2003a), the maximum capture problem (Benati and Hansen 2002)
and several variants of the p-median problem. Domínguez-Marín et al. (2005) deal
with solving the discrete ordered median problem, Fathali and Kakhki (2006) apply
VNS to the p-median problem with pos/neg weights, Fleszar and Hindi (2008)solve
the capacitated p-median problem and Pérez et al. (2007) propose a hybrid which com-
bines VNS with Path Relinking for the p-hub median problem. Osman and Ahmadi
(2007) investigate different search and selection strategies, including the VND for
the capacitated p-median problem with single source constraint. Moreno-Pérez et al.
(2003) propose a variable neighbourhood tabu search (VNTS) hybrid and consider its
application to the median cycle problem.
Among continuous models, the multi-source Weber problem is first addressed
in Brimberg et al. (2000) and in Brimberg et al. (2004) with constant opening costs.
Brimberg et al. (2006a) use VNS in a decomposition strategy for large-scale instances.
Brimberg et al. (2008a) apply VNS to the maximum return-on-investment plant
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Variable neighbourhood search: methods and applications
location problem with market share. Ljubic (2007) proposes a hybrid VNS for a con-
nected facility location problem which combines the facility location problem and the
Steiner tree problem in graphs. Hansen et al. (2007a) apply a primal-dual VNS for the
simple plant location problem. Finally, Bischoff and Dächert (2008) use VNS and other
heuristics for a generalized class of continuous location-allocation problems and Jabal-
ameli and Ghaderi (2008) propose hybrid algorithms which combine Genetic Algo-
rithm (GA) and VNS for the uncapacitated continuous location-allocation problem.
Drezner et al. (2005) analyse the difficulty in the instances of quadratic assign-
ment problems for metaheuristic approaches and Zhang et al. (2005) use a VNS with
permutation distance. Han et al. (2007) use a hybrid of VNS with Ant Colony Opti-
mization and Liu and Abraham (2007) a fuzzy hybrid of VNS with Particle Swarm
Optimization (PSO) method.
Yang et al. (2007) apply optimization strategies based on Simulated Annealing and
VNS for the base station location problem in a WCDMA (Wideband Code-Division
Multiple Access) network. Pacheco et al. (2008) use VNS to solve the classical max-
imum covering location problem for locating health resources.Wollenweber (2008)
uses several hybrids with VNS for a multi-stage facility location problem with stair-
case costs and splitting of commodities.
5.4 Data mining
VNS proved to be a very efficient tool in cluster analysis. In particular, the J-Means
heuristic combined with VNS appears to be state-of-the-art for the heuristic solution
of minimum sum-of-square clustering (Hansen and Mladenovi´c 2001b;Belacel et al.
2002,2004a). Combined with stabilized column generation (du Merle et al. 1999)it
leads to the most efficient exact algorithm at present for this problem (du Merle et al.
2000). Such an approach has also been applied by Hansen and Perron (2007)tosolve
the L1embeddability problem for data sets.
Belacel et al. (2004b) use VNS Metaheuristic for Fuzzy Clustering cDNA Microar-
ray Gene Expression Data. Negreiros and Palhano (2006) propose a constructive pro-
cedure followed by a VNS to solve the capacitated centred clustering problem. Brusco
and Steinley (2007a) compare a VNS method with the classical k-means for the clus-
tering of two-mode proximity binary matrices and Brusco and Steinley (2007b) com-
pare heuristic procedures for Minimum Within-Cluster Sums of Squares Partitioning.
Benati (2008) applies VNS to categorical data fuzzy clustering. Other clustering prob-
lem applications appear in Brusco et al. (2008).
Another important data mining task which has been managed with VNS is classi-
fication. Pacheco et al. (2007) use VNS in the variable selection and determination of
the linear discrimination function coefficients. Karam et al. (2007) perform arbitrary-
norm hyperplane separation by VNS. The same problem has also been attacked with
VNS in Plastria et al. (2007). Hansen et al. (2007b) apply VNS for colour image
quantization. Belacel et al. (2007) propose a VNS heuristic for learning the parame-
ters of the multiple criteria classification method PROAFTN from data. Carrizosa et
al. (2007) use VNS for the selection of the Globally Optimal Prototype Subset for
Nearest-Neighbour Classification.
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P. Hansen et al.
5.5 Graph problems
In addition to some design problems in communications and most of the location
problems, VNS has been applied to other combinatorial optimization problems on
graphs. A VNS is proposed for the max-cut problem in a graph and compared with
other metaheuristics in Festa et al. (2002) and an hybridization between a memetic
algorithm and VNS is proposed for the same problem by Duarte et al. (2005).
Moreno-Pérez et al. (2003) propose a VNTS hybrid for the median cycle problem.
Hansen et al. (2004) propose and test a basic VNS which combines greedy with the
simplicial vertex test in its descent step for the maximum clique problem. For the graph
colouring problem, Avanthay et al. (2003) propose an adaptation of the VNS meta-
heuristic, Galinier and Hertz (2006) present a survey of local search methods which
includes VNS and Hertz et al. (2008) analyse the variable space search methodology
which extends the FSS. Brimberg et al. (2008b) propose a new heuristic based on
VNS for the k-cardinality subgraph problem, in contrast with the constructive heuris-
tics proposed in the literature.
Several graph problems involving trees have also been tackled with VNS. VNS is
used in Canuto et al. (2001) as a post-optimization procedure for a multistart local
search algorithm for the prize-collecting Steiner tree problem, based on the generation
of initial solutions by a primal-dual algorithm using perturbed node prizes. Ribeiro
et al. (2002) use a hybrid VNS-GRASP with perturbations for the Steiner problem
in graphs. Mladenovi´c and Uroševi´c(2003) propose the use of a VNS for the edge
weighted k-cardinality tree problem Uroševi´cetal.(2004) propose a variable neigh-
bourhood decomposition search (VNDS) for the same problem and Brimberg et al.
(2006b) for the vertex weighted k-cardinality tree problem. Ribeiro and de Souza
(2002) propose the use of VNS for the degree constrained minimum spanning tree
problem and de Souza and Martins (2008) use a Skewed VNS enclosing a second
order algorithm for the same problem. Hu et al. (2008) propose a VNS approach which
uses three different neighbourhood types to solve the generalized minimum spanning
tree problem. Finally, a VNS is used to solve the minimum labelling spanning tree
problem by Consoli et al. (2008).
5.6 Knapsack and packing problems
Another important class of problems solved with VNS and its variants and hybrids
are the knapsack and packing problems. In Puchinger et al. (2006) a relaxation guided
VNS is applied to the multidimensional knapsack problem and to its core problems.
The paper by Puchinger and Raidl (2008) constitutes an excellent illustration of a
dynamic ordering of the neighbourhood structures embedded in a VND algorithm
which is used to solve also the multidimensional knapsack problem. VNS has also
been successfully applied to the bin packing problem (Fleszar and Hindi 2002) and to
the strip packing problem (Beltrán et al. 2004). Parreño et al. (2008) present a VNS
algorithm for the container loading problem.
Circle and sphere packing have also been approached with VNS. Mladenovi´cetal.
(2005) introduce the RD which is applied to CPPs and Mladenovi´cetal.(2007b)the
FSS for the same problems. Kucherenko et al. (2007) use VNS to solve the kissing
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Variable neighbourhood search: methods and applications
number problem, i.e., the problem of determining the maximum number of
D-dimensional spheres of radius rthat can be adjacent to a central sphere of radius r.
5.7 Mixed integer problems
Heuristics may help in finding a feasible solution or an improved and possibly optimal
solution to large and difficult mixed integer programs. The local branching method
of Fischetti and Lodi (2003) does this, in the spirit of VNS. For further developments
see Fischetti et al. (2004) and Hansen et al. (2006). Gutjahr et al. (2007) use the VNS
approach for noisy problems and its application to project portfolio analysis.
5.8 Time tabling
Timetabling and related manpower organization problems can be well solved with
VNS. They include the design of balanced MBA student teams (Desrosiers et al. 2005),
apportioning the European Parliament (Villa et al. 2006), detailed layout planning for
irregularly-shaped machines (Bock and Hoberg 2007) and nurse rostering (Burke
et al. 2004,2008). Cote et al. (2005) use a simplified VND in a hybrid multi-objective
evolutionary algorithm for the uncapacitated exam proximity problem. Sevkli and
Sevilgen (2006) propose a VNS approach for the orienteering problem and Archetti
et al. (2007) propose VNS to solve the team orienteering problem (TOP), that is, the
generalization to the case of multiple tours of the orienteering problem, known also
as the selective travelling salesman problem (TSP).
5.9 Scheduling
In recent years also several scheduling problems have been efficiently solved with
VNS approaches. They include single machine and parallel machines, multiobjective
scheduling, job shop scheduling, flow shop, resource-constrained project scheduling
and other scheduling problems.
5.9.1 Single machine scheduling
Gupta and Smith (2006) use a VNS algorithm for single machine total tardiness
scheduling with sequence-dependent setups. Lin and Ying (2008) propose a hybrid
Tabu-VNS meta-heuristic approach for single-machine tardiness problems with
sequence-dependent setup times. Liao and Cheng (2007) propose a VNS for minimiz-
ing single machine weighted earliness and tardiness with common due date. Tseng
et al. (2008) employ a VNS for large-size instances of the single machine total tar-
diness problem with controllable processing times. Wang and Tang (2008) propose a
population-based VNS for the single machine total weighted tardiness problem.
5.9.2 Parallel machine scheduling
Anghinolfi and Paolucci (2007) propose a hybrid metaheuristic approach which inte-
grates several features from tabu search, simulated annealing and VNS for a parallel
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P. Hansen et al.
machine total tardiness scheduling problem. De Paula et al. (2007) apply VNS for solv-
ing parallel machines scheduling problems with sequence-dependent setups. Chen and
Chen (2008) propose an approach which integrates the principles of the VND approach
and tabu search for the unrelated parallel-machine scheduling problem with sequence-
dependent setup times.
5.9.3 Multiobjective scheduling
Gagné et al. (2005) use compromise programming with Tabu-VNS metaheuristic for
the solution of multiple-objective scheduling problems. Qian et al. (2006) deal with
multi-objective flow shop scheduling, using differential evolution.
5.9.4 Job shop scheduling
Sevkli and Aydin (2006a,b) use VNS algorithms for job shop scheduling problems.
Sevkli and Aydin (2007) propose parallel VNS algorithms and Gao et al. (2008)
propose a hybrid GA/VND and Pan et al. (2007b) a PSO/VNS hybrid heuristic for
these problems. Liu et al. (2006) propose a variable neighbourhood particle swarm
optimization for multi-objective flexible job-shop scheduling problems. Aydin and
Sevkli (2008) consider sequential and parallel VNS algorithms for job shop scheduling.
5.9.5 Flow shop scheduling
Blazewicz et al. (2005) use VNS for late work minimization in a two-machine flow
shop with common due date. In Pan et al. (2007a) VNS and three other metaheuristic
approaches are proposed for a no-wait flow shop problem. In Blazewicz et al. (2008)
VNS and two other metaheuristics are presented for the two-machine flow shop prob-
lem with weighted late work criterion and common due date. Zobolas et al. (2008)
design a GA/VNS hybrid to minimize makespan in permutation flow shop scheduling
problems. In Tasgetiren et al. (2004) a simple but very efficient local search, based on
VNS, is embedded in the PSO algorithm in order to solve the permutation flow shop
sequencing problem. Liao et al. (2007) apply VNS for flow shop scheduling problems
and Tasgetiren et al. (2007) consider the makespan and total flow time minimization
in the permutation flow shop sequencing problem. Czogalla and Fink (2008) exam-
ine the application of a PSO with VND as an embedded local search procedure for
the continuous flow-shop scheduling problem. Rahimi-Vahed et al. (2008)devisea
hybrid multi-objective algorithm based on shuffled frog-leaping algorithm and VNS
for a bi-criteria permutation flow shop scheduling problem. Chyu and Chen (2008)
propose several VNS for a lump-sum payment model for the resource-constrained
project scheduling problem.
5.9.6 Resource-constrained project scheduling
Fleszar and Hindi (2004) propose a solution for the resource-constrained project
scheduling problem by a VNS and Kolisch and Hartmann (2006) include VNS in
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an experimental investigation of heuristics for resource-constrained project schedul-
ing. Bouffard and Ferland (2007) improve simulated annealing with VNS to solve the
resource-constrained scheduling problem.
5.9.7 Car sequencing
Prandtstetter and Raidl (2008) use a hybrid VNS for the car sequencing problem and
Gavranovi´c(2008) applies VNS to car-sequencing problems with colours. Ribeiro
et al. (2008a) propose a set of heuristics based on the paradigms of the VNS and ILS
metaheuristics for a multi-objective real-life car sequencing problem with painting
and assembly line constraints and Ribeiro et al. (2008b) provide an efficient imple-
mentation of the VNS/ILS heuristic for this real-life car sequencing problem. Joly
and Frein (2008) use VNS to tackle an industrial car sequencing problem considering
paint and assembly shop objectives.
5.9.8 Other scheduling problems
Davidovi´cetal.(2005) use VNS heuristics for multiprocessor scheduling with com-
munication delays. Higgins et al. (2006) apply VNS to the scheduling of brand pro-
duction and shipping within a sugar supply chain and Lejeune (2006) also consider
supply chain planning. Liang and Chen (2007) tackle the redundancy allocation of
series-parallel systems, using a VNS algorithm.
Remde et al. (2007) use reduced VNS and hyperheuristic approaches to tackle
subproblems in an Exact/Hybrid heuristic for Workforce Scheduling. Xhafa (2007)
considers a hybrid evolutionary metaheuristic based on memetic algorithms and VNS
to job scheduling on computational grids. Liang et al. (2007) apply VNS to redundancy
allocation problems.
Lusa and Potts (2008) use a VNS algorithm for the constrained task allocation
problem. Almada-Lobo et al. (2008) report the use of a VNS approach to production
planning and scheduling in the glass container industry. Dahal et al. (2008) apply
a constructive search and VNS to tackle a complex real world workforce scheduling
problem. Abraham et al. (2008) propose a VNS/PSO hybrid for the scheduling problem
in distributed data-intensive computing environments. Liao and Liao (2008) apply an
ACO algorithm which uses a VNS as the local search to make it more efficient and
effective for scheduling in agile manufacturing. Naderi et al. (2008) propose a VNS
which uses advanced neighbourhood search structures for flexible flow line problems
with sequence dependent setup times.
5.10 Vehicle routing problems
5.10.1 TSP and extensions
VNS is used for the TSP and its extensions. Hansen and Mladenovi´c(1999,2006)
consider basic VNS for the euclidean TSP. Burke et al. (2001) apply guided VNS
methods for the asymmetric TSP. VNS for the Pickup and Delivery TSP is considered
123
P. Hansen et al.
in Carrabs et al. (2007). Hu and Raidl (2008) study the effectiveness of neighbourhood
structures within a VNS approach for the Generalized TSP.
5.10.2 VRP and extensions
Standard versions of the vehicle routing problem (VRP) have been solved by VNS
or hybrids. A VND is applied to the VRP with backhauls in Crispim and Brandao
(2001). Rousseau et al. (2002) use a VND to take advantage of different neighbourhood
structures for the VRP. An interesting development of reactive VNS for the VRP
with time windows appears in Bräysy (2003). Polacek et al. (2004) use a VNS for
the multi depot VRP with time windows. A hybrid metaheuristic merging VNS and
Tabu Search applied to the location-routing problem with non-linear costs can be
found in Melechovsky et al. (2005). Repoussis et al. (2006) propose a reactive greedy
randomized VNTS for the VRP with time windows. Irnich et al. (2006) introduce
sequential search as a generic technique for the efficient exploration of local-search
neighbourhoods such as VNS and consider its application to vehicle-routing problems.
Kytöjoki et al. (2007) propose an efficient VNS heuristic for very large scale vehicle
routing problems. Geiger and Wenger (2007) use VNS within an interactive resolution
method for multi-objective vehicle routing problems. Fleszar et al. (2008) propose an
effective VNS for the open VRP.
5.10.3 Practical applications
VNS has also been useful for practical applications of routing problems. Cowling
and Keuthen (2005) examine iterated approaches of the Large-Step Markov Chain
and VNS type and investigate their performance when used in combination with an
embedded search heuristic for routing optimization. A VNS-based on-line method
is proposed and tested in Goel and Gruhn (2008) for the general VRP. The solution
methodology proposed by Repoussis et al. (2007) hybridizes in a reactive fashion
systematic diversification mechanisms of Greedy Randomized Adaptive Search Pro-
cedures with VNS for intensifying local searching regarding a real life VRP.
5.10.4 Arc routing and waste collection
Hertz and Mittaz (2001) use a VND algorithm for the undirected capacitated arc
routing problem. Ghiani et al. (2002) use VNS to solve the capacitated arc routing
problem. Polacek et al. (2008) develop a basic VNS algorithm to solve the capacitated
arc routing problem with intermediate facilities. Nuortio et al. (2006) use VNS in an
improved route planning and scheduling of waste collection and transport and Del
Pia and Filippi (2006) use a VND algorithm for a real waste collection problem with
mobile depots.
5.10.5 Fleet sheet problems
Yepes and Medina (2006) present a three-step local search algorithm based on a
probabilistic VNS for the VRP with a heterogeneous fleet of vehicles and soft time
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Variable neighbourhood search: methods and applications
windows. Paraskevopoulos et al. (2008) present a reactive VNTS for the heterogeneous
fleet VRP with time windows. Schmid et al. (2008) propose two hybrid procedures
based on a combination of an exact algorithm and a VNS approach for the distribution
of ready-mixed concrete using a heterogeneous fleet of vehicles. Imran et al. (2008)use
some adaptations of the VNS to solve a heterogeneous fleet vehicle routing problem.
5.10.6 Extended vehicle routing problems
Polacek et al. (2007) use VNS to assign customers to days and determine routes for a
travelling salesperson for scheduling periodic customer visits. Zhao et al. (2008) apply
a variable large neighbourhood search (VLNS) algorithm, which is a special case of
VNS for an inventory/routing problem in a three-echelon logistics system. Vogt et al.
(2007) present a heuristic for this problem based on a VNTS for the single vehicle
routing allocation problem. Hemmelmayr et al. (2008a) propose a VNS heuristic for
periodic routing problems. Liu and Chung (2008) propose a VNTS for the VRP with
backhauls and inventory and Liu et al. (2008) propose a modified VNS for solving
vehicle routing problems with backhauls and time windows. Subramanian et al. (2008)
present an iterated local search procedure, which uses a VND method to perform the
local search, for the VRP, with simultaneous pickup and delivery and a time limit.
5.11 Problems in biosciences and chemistry
VNS has been useful in recently emerging areas in Bioscience and Chemistry such as
Bio-informatics. Andreatta and Ribeiro (2002) propose VNS heuristics for the phy-
logeny problem and Ribeiro and Vianna (2005) use a GRASP with a VND heuristic
for this problem with a new neighbourhood structure. Kawashimo et al. (2006) apply
VNS to DNA Sequence Design and Liberti et al. (2008) propose a double VNS with
smoothing for the molecular distance geometry problem. Santana et al. (2008) illus-
trate another example of hybridization of metaheuristics through the combination of
VNS and Estimation Distribution Algorithms (EDAs). They present the first attempt
to combine these two methods testing it on the protein side chain placement prob-
lem. Belacel et al. (2004b) use VNS for Fuzzy Clustering of cDNA microarray gene
expression data and Draži´cetal.(2008) use a continuous VNS heuristic for finding the
three-dimensional structure of a molecule. Montemanni and Smith (2008) consider
the construction of constant GC-content DNA vodes via a VNS Algorithm.
A Multi-Start VNS hybrid (MSVNS) is applied, in Pelta et al. (2008), to the protein
structure comparison problem which is a very important problem in the bio-informatics
area. The Maximum Contact Map Overlap (Max-CMO) model of protein structure
comparison models the proteins as a graph of the contacts between the protein residues
to perform the comparison. The proposed MSVNS method is currently the best heuris-
tic algorithm for the Max-CMO model, both in terms of optimization and in terms of
the biological relevance of its results. The method is biologically relevant, since the
algorithm has proven to be good enough to detect similarities at SCOP’s family and
CATH’s architecture levels.
123
P. Hansen et al.
5.12 Continuous optimization
Several continuous optimization problems have also been successfully approached
with VNS. Mladenovi´cetal.(2008) propose a General VNS for continuous opti-
mization and Draži´cetal.(2006) a VNS-based software for Global Optimization.
Audet et al. (2008) deal with Nonsmooth optimization through Mesh Adaptive Direct
Search and VNS. Brimberg et al. (2006a) use VNS in a decomposition strategy for
large-scale continuous location-allocation problems. Solving the unconstrained opti-
mization problem by VNS has been successfully achieved in Toksari and Güner
(2007). Ling et al. (2008) use a modified VNS metaheuristic for max-bisection prob-
lems. Sevkli and Sevilgen (2008) consider the PSO hybridized with Reduced VNS for
continuous function optimization.
5.13 Other optimization problems
Some further optimization problems solved with VNS include the study of the
dynamics of handwriting (Caporossi et al. 2004), the problem of multi-item, single
level, capacitated, dynamic lot-sizing with set-up times (Hindi et al. 2003), the linear
ordering problem (García et al. 2006), the minimum cost berth allocation problem
(Hansen et al. 2008b) and the run orders problem in the presence of serial correla-
tion (Garroi et al. 2008).
Mori and Tsunokawa (2005) use a VNTS for capacitor placement in distribution
systems. Haugland (2007) develops a randomized search heuristic, which in some
sense resembles VNS, for the subspace selection problem. Hemmelmayr et al. (2008b)
apply solution approaches based on integer programming and VNS to organize the
delivery of blood products to Austrian hospitals for the blood bank of the Austrian
Red Cross for Eastern Austria. Claro and de Sousa (2008) propose a hybrid approach,
combining Tabu Search and VNS for a mean-risk multistage capacity investment
problem.
VNS is used to solve satisfiability problems. Hansen et al. (2000) use VNS
for the weighted maximum satisfiability problem. Ognjanovi´cetal.(2005), Jovanovi´c
et al. (2007) and Sevkli and Aydin (2007) use VNS for the probabilistic satisfiability
problem. Hansen and Perron (2008) use VNS to solve the subproblem in a column
generation approach which merges the local and global approaches to probabilistic
satisfiability. Loudni and Boizumault (2008) apply the (VNS/LDS+ CP) hybrid for
solving optimization problems in anytime contexts. The (VNS/LDS+ CP) procedure
combines a VNS scheme with Limited Discrepancy Search (LDS) using Constraint
Propagation (CP).
5.14 Discovery science
In all these applications VNS is used as an optimization tool. It can also lead to results
in “discovery science”, i.e., help in the development of theories. This has been done for
graph theory in a long series of papers with the common title “Variable neighbourhood
search for extremal graphs” and reporting on the development and applications of the
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Variable neighbourhood search: methods and applications
system AutoGraphiX (AGX) (Caporossi and Hansen 2000,2004;Aouchiche et al.
2005a). This system addresses the following problems:
– Find a graph satisfying given constraints;
– Find optimal or near optimal graphs for an invariant subject to constraints;
– Refute a conjecture;
– Suggest a conjecture (or repair or sharpen one);
– Provide a proof (in simple cases) or suggest an idea of proof.
A basic idea is then to consider all of these problems as parametric combinatorial
optimization problems on the infinite set of all graphs (or in practice some smaller
subset) with a generic heuristic. This is done by applying VNS to find extremal graphs,
with a given number nof vertices (and possibly also a given number of edges). Then
a VND with many neighbourhoods is used. Those neighbourhoods are defined by
modifications of the graphs such as the removal or addition of an edge, rotation of
an edge, and so forth. Once a set of extremal graphs, parameterized by their order, is
found, their properties are explored with various data mining techniques, leading to
conjectures, refutations and simple proofs or ideas of proof.
The current list of titles of papers in the series “VNS for extremal graphs” is given
in Table 1. Another list of papers, not included in this series is given in Table 2.
Papers in these two lists cover a variety of topics:
(i) Principles of the approach (1.1, 1.5) and its implementation (1.14);
(ii) Applications to spectral graph theory, e.g., bounds on the index for various
families of graphs, graphs maximizing the index subject to some conditions
(1.3, 1.11, 1.16, 1.17, 2.7);
(iii) Studies of classical graph parameters, e.g., independence, chromatic number,
clique number, average distance (1.13, 1.21, 1.22, 1.24, 1.25, 1.26, 2.8);
(iv) Studies of little known or new parameters of graphs, e.g., irregularity, proximity
and remoteness (1.9, 2.9)
(v) New families of graphs discovered by AGX, e.g., bags, which are obtained from
complete graphs by replacing an edge by a path, and bugs, which are obtained
by cutting the paths of a bag (1.15, 1.27);
(vi) Applications to mathematical chemistry, e.g., study of chemical graph energy,
and of the Randi´c index (1.4, 1.6, 1.7, 1.10, 1.18, 1.19, 2.2, 2.3, 2.6);
(vii) Results of a systematic study of 20 graph invariants, which led to almost 1500
new conjectures, more than half of which were proved by AGX and over 300
by various mathematicians (1.20);
(viii) Refutation or strengthening of conjectures from the literature (1.8, 2.1, 2.6);
(ix) Surveys and discussions about variousdiscovery systems in graph theory, assess-
ment of the state-of-the-art and the forms of interesting conjectures together with
proposals for the design of more powerful systems (2.4, 2.5).
6 Conclusions
The general schemes of VNS have been presented, discussed and illustrated by exam-
ples. In order to evaluate the VNS research program, one needs a list of the desirable
123
P. Hansen et al.
Tab l e 1 List of papers in the series “VNS for extremal graphs”
Author(s) Title
1.1 Caporossi and Hansen (2000) The AutoGraphiX System
1.2 Caporossi et al. (1999a) Finding graphs with extremal energy
1.3 Cvetkovic et al. (2001) On the largest eigenvalue of color-constrained trees
1.4 Caporossi et al. (1999c) Chemical trees with extremal connectivity index
1.5 Caporossi and Hansen (2004) Three ways to automate finding conjectures
1.6 Hansen and Mélot (2003) Analysing bounds for the connectivity index
1.7 Fowler et al. (2001) Polyenes with maximum HOMO-LUMO gap
1.8 Aouchiche et al. (2001) Variations on Graffiti 105
1.9 Hansen and Mélot (2005) Bounding the irregularity of a graph
1.10 Gutman et al. (2005) Comparison of irregularity indices for chemical trees
1.11 Belhaiza et al. (2007) Bounds on algebraic connectivity
1.12 Hansen et al. (2005b) A note on the variance of bounded degrees in graphs
1.13 Aouchiche and Hansen (2005) “A propos de la maille” (French)
1.14 Aouchiche et al. (2005a) The AutoGraphiX 2 system
1.15 Hansen and Stevanovi´c(2005) On bags and bugs
1.16 Aouchiche et al. (2008a) Some conjectures related to the largest eigenvalue of a graph
1.17 Aouchiche et al. (2009b) Further conjectures and results about the index
1.18 Aouchiche et al. (2006) Conjectures and results about the Randic index
1.19 Aouchiche et al. (2007d) Further conjectures and results about the Randic index
1.20 Aouchiche et al. (2007a) Automated comparison of graph invariants
1.21 Aouchiche et al. (2008b) Conjectures and results about the independence number
1.22 Aouchiche et al. (2009a) Extending bounds for independence to upper irredundance
1.23 Hansen and Vukiˇcevi´c (2009) On the Randic index and the chromatic number
1.24 Sedlar et al. (2007a) Conjectures and results about the clique number
1.25 Sedlar et al. (2007b) Products of connectivity and distance measures
1.26 Aouchiche et al. (2007c) “Nouveaux rèsultats sur la maille” (French)
1.27 Aouchiche et al. (2007b) Families of extremal graphs
properties of metaheuristics. The following eight of these are presented in Hansen and
Mladenovi´c (2003):
(i) Simplicity: the metaheuristic should be based on a simple and clear principle,
which should be widely applicable;
(ii) Precision: the steps of the metaheuristic should be formulated in precise
mathematical terms, independent of possible physical or biological analogies
which may have been the initial source of inspiration;
(iii) Coherence: all steps of the heuristics for particular problems should follow
naturally from the principle of the metaheuristic;
(iv) Efficiency: heuristics for particular problems should provide optimal or near-
optimal solutions for all or at least most realistic instances. Preferably, they
123
Variable neighbourhood search: methods and applications
Tab l e 2 A further list of papers on AGX
Author(s) Title
2.1 Caporossi et al. (1999b) Trees with Palindromic Hosoya Polynomials
2.2 Gutman et al. (1999) Alkanes with small and large Randi´c connectivity indices
2.3 Hansen (2002) Computers in Graph Theory
2.4 Hansen and Mélot (2002) Computers and Discovery in Algebraic Graph Theory
2.5 Caporossi et al. (2003) Graphs with maximum connectivity index
2.6 Hansen (2005) How far is, should and could be conjecture-making
in graph theory an automated process?
2.7 Hansen et al. (2005a) What forms do interesting conjectures have in graph theory?
2.8 Aouchiche et al. (2005b) AutoGraphiX: A Survey
2.9 Aouchiche and Hansen (2007a) Automated Results and Conjectures on Average Distance
in Graphs
2.10 Aouchiche and Hansen (2007b) On a Conjecture about the Randic Index
2.11 Stevanovic et al. (2008) On the Spectral Radius of Graphs with a Given
Domination Number
2.12 Aouchiche and Hansen (2009) Bounding Average Distance Using Minimum Degree
2.13 Aouchiche and Hansen (2008) Nordhaus-Gaddum Relations for Proximity and Remoteness
in Graphs
should find optimal solutions for most problems of benchmarks for which such
solutions are known, when available;
(v) Effectiveness: heuristics for particular problems should take a moderate com-
puting time to provide optimal or near-optimal solutions;
(vi) Robustness: the performance of heuristics should be consistent over a variety
of instances, i.e., not merely fine-tuned to some training set and less good
elsewhere;
(vii) User-friendliness: heuristics should be clearly expressed, easy to understand
and, most important, easy to use. This implies they should have as few parame-
ters as possible, ideally none;
(viii) Innovation: preferably, the principle of the metaheuristic and/or the efficiency
and effectiveness of the heuristics derived from it should lead to new types of
application.
This list has been completed with three more items added by one member of the
present team and his collaborators:
(ix) Generality: the metaheuristic should lead to good results for a wide variety of
problems;
(x) Interactivity: the metaheuristic should allow the user to incorporate his knowl-
edge to improve the resolution process;
(xi) Multiplicity: the metaheuristic should be able to present several near optimal
solutions from which the user can choose one.
123
P. Hansen et al.
As argued here and above, VNS possesses, to a great extent, all of the above proper-
ties. This has led to heuristics which are among the very best ones for many problems.
Interest in VNS is clearly growing at speed. This is evidenced by the increasing number
of papers published each year on this topic (ten years ago, only a few; five years ago,
about a dozen; and about 50 in 2007). Moreover, the 18th EURO Mini conference held
in Tenerife in November 2005 was entirely devoted to VNS. It led to special issues of
IMA Journal of Management Mathematics in 2007 (Melián and Mladenovi´c 2007),
and European Journal of Operational Research (Hansen et al. 2008a) and Journal of
Heuristics (Moreno-Vega and Melián 2008) in 2008. In retrospect, it appears that the
good shape of the VNS research program is due to the following decisions, strongly
influenced by Karl Popper’s philosophy of science (Popper 1959): (i) in devising
heuristics favour insight over efficiency (which comes later) and (ii) learn from the
heuristics mistakes.
Acknowledgments The first author was partly supported by NSERC grant number FQRNT. The second
author was partly supported by the Serbian Ministry of Science, grant number 144007. The third author was
partly supported by project TIN2005-08404-C04-03 of the Spanish Government (with financial support
from the E.U. under the FEDER project) and project PI042005/044 of the Canary Government.
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