A Tutorial on Variable Neighborhood Search

Les Cahiers du GERAD G–2003–46 9
Due to the nestedness property the size of successive neighborhoods will be increasing.
Therefore one will explore more thoroughly close neighborhoods of x than farther ones,
but nevertheless search within these when no further improvements are observed within
the first, smaller ones.
Initialization. Select the set of neighborhood structures N k ,fork =1,...,k max ,
that will be used in the search; find an initial solution x; choose a stopping
condition;
Repeat the following sequence until the stopping condition is met:
(1) Set k ← 1;
(2) Repeat the following steps until k = k max :
(a) Shaking. Generate a point x ′ at random from the k th neighborhood
of x (x ′ ∈N k (x));
(b) Move or not. If this point is better than the incumbent, move
there (x ← x ′ ), and continue the search with N 1 (k ← 1); otherwise, set
k ← k +1;
Figure 4. Steps of the Reduced VNS.
Example 3. p-Median (see e.g. Labbé, Peeters and Thisse, 1995 for a survey). This is
a location problem very close to Simple Plant Location. The differences are that there are
no fixed costs, and that the number of facilities to be opened is set at a given value p. It
is expressed as follows:
subject to
min
m∑ n∑
c ij x ij (18)
i=1 j=1
m∑
x ij = 1, ∀j, (19)
i=1
y i − x ij ≥ 0, ∀i, j, (20)
m∑
y i = p, (21)
i=1
x ij ,y i ∈ {0, 1}. (22)
The Greedy and Interchange heuristics described above for Simple Plant Location are
easily adapted to the p-Median problem and, in fact, the latter was early proposed by Teitz
and Bart (1967).
Fast interchange, using Whitaker’s (1983) data structure applies here also (Hansen and
Mladenović, 1997). Refinements have recently been proposed by Resende and Werneck