Sequential order vs random order in operators of variable neighborhood descent method

TELKOMNIKA, Vol.17, No.2, April 2019, pp.801~808
ISSN: 1693-6930, accredited First Grade by Kemenristekdikti, Decree No: 21/E/KPT/2018
DOI: 10.12928/TELKOMNIKA.v17i2.11789  801
Received October 24, 2018; Revised November 10, 2018; Accepted December 2, 2018
Sequential order vs random order in operators of
variable neighborhood descent method
Darmawan Satyananda*1
, Sapti Wahyuningsih2
Mathematics Department, Universitas Negeri Malang, Semarang St. No. 5 Malang, Indonesia,
telp: (0341) 552182
*Corresponding author, email: darmawan.satyananda.fmipa@um.ac.id1
,
sapti.wahyuningsih.fmipa@um.ac.id2
Abstract
Many optimization problems require heuristic methods to solve the problem. Variable
Neighborhood Search (VNS) is a metaheuristic form that systematically changes its “neighborhood” in
search of solutions. One method in VNS is Variable Neighborhood Descent (VND), which performs a
deterministic neighborhood change. The change of the neighborhood in VND can be done in a random
and sequential order. This paper compares sequential and random neighborhood selection methods in
solving Capacitated Vehicle Routing Problem (CVRP) problems. There are 6 intra-route neighborhood
structures and 4 inter-route structures used. CVRP problems are taken from several dataset providers.
The initial solution is formed by Sequential Insertion method. The experimental results show that the
random selection of neighborhood operators can provide a more optimal route length (in 10 of 13 datasets
used) than that of sequential selection (only better in 3 dataset). However, the random selection takes
more iterations to reach convergent state than the sequential one. For sequential selection, determination
of the neighborhood structure’s order affects the speed to the convergent state. Hence, a random selection
in VND method is more preferable than sequential selection.
Keywords: CVRP, local search, neighborhood selection methods, VND, VNS
Copyright © 2019 Universitas Ahmad Dahlan. All rights reserved.
1. Introduction
Vehicle Routing Problem (VRP) have long been known as "difficult" problems where no
algorithm in polynomial time proportional to the size of the instance can solve all the problems
(belongs to NP-hard family). The exact method may solve a “big” problem, however, it takes a
very long execution time. By using the exhaustive search (brute force), it would have obtained a
solution (it’s optimal), but this would require a very large computation time (it isn’t efficient).
Algorithms that are significantly faster than exhaustive search is still possible to be designed for
some problems, however, still not in polynomial time [1]. Hence, a heuristic algorithm is
introduced that can immediately provide an approximate solution which is sometimes
considered as the optimal solution. Researchers focused on approximate algorithm that
provides near-optimal solution in quick time [2].
The problem found in heuristics is local optimum (local minima), i.e. a solution that
never reaches a feasible state (global optimum value). In this instance, the algorithm is
"trapped" in certain circumstances where the operation only generates values around the local
optimum area (optimum at within a neighborhood is not necessarily optimum in another one).
The global optimum is a proven optimum solution for all locals. To overcome the problems that
arise in the heuristics, metaheuristic is raised; it is a framework to build heuristics for
combinatorial problems and global optimization [3].
One of the frameworks offered to generate solutions using metaheuristic is Variable
Neighborhood Search (VNS) metaheuristic which exploits systematically the idea of
neighborhood change, both in descent to local minima and in escape from the valleys which
contain them [3]. A variant of VNS is Variable Neighborhood Descent (VND) which explore
neighbohood in a deterministic way. They both belong to a local search metaheuristic group,
applying iteratively to various neighborhood structures, modifying an initial solution [4]. The use
of various neighborhood structures in the local search will result in a better solution than the
original solution.

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TELKOMNIKA Vol.17, No.2, April 2019: 801-808
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Types and number of neighborhood operator in VND affect performance of VND [4], but
in other hand it will increase operation time as iteration will increase as well. Sequence of
operator is important since it influences how fast convergence will be reached. Operators can
be examined in sequential or random [5, 6], moreover, in arbitrary way. It is necessary to study
about method of the selection, to find better configuration of operators needed in finding best
optimal solution of VRP.
This paper discusses the comparison of a selecting VND operator in sequential and
random order, in CVRP. The paper is organized as follows. In Introduction Section, the literature
related to VNS, VND, and neighborhood operators are reviewed. Next section is about
methodology. Finally, test results for some instances and some computational results are
discussed in Results and Discussion Section.
Variable Neighborhood Search (VNS)
Metaheuristic is a common framework for building heuristics for combinatorial issues
and global optimization. One of the most widely used metaheuristic is Variable Neighborhood
Search (VNS) developed by Mladenović since 1995 [3]. VNS systematically changes the
neighborhood, either in the gradual form leaving local optimum or "jumping" from the valley that
loads the local optimum by using perturbation. Simply put, "neighborhood" is a set of solution
results from the application of a particular operation. From an initial solution, an operation can
generate various alternative solutions; those are “neighborhoods”. Figure 1 shows the local
optima and perturbation conditions for jumping to other expected circumstances.
Basically, VNS consists of stages:
 Shaking (perturbation): to determine a solution x' to be fixed. Shaking is done with the initial
solution x using operations taken from a particular neighborhood.
 Repairing x' with local search using a particular neighborhood operator, generating x''
 Change Neighborhood by comparing local search results x'' with initial solution x. If x’’ is
better than x, then x’’ is considered as a new x and the iteration is reset back to the first
neighborhood operator. Otherwise next operator is applied to x’.
The problem that arises is to determine what operators can be included in the
neighborhood structure, and how the sequence, so as to produce global or near-optimal.
Variable Neighborhood Descent (VND). VND is obtained when the neighborhood change is
determined deterministically, by eliminating the random shaking stage. VND explores the
solution space using various neighborhood structures to ensure that there are optimal solutions
in all different neighborhoods thus it can be considered as global solutions. Algorithm of VND
can be seen in Figure 2 [3].
Nℓ is a neighborhood structure as a set of operators applied to a neighborhood, x is an
initial solution, and Nℓ(x) is set of solutions using ℓth neighborhood of x. At the exploration of
neighborhood stage, a local search for a neighborhood that gives the best x' of x is performed,
started from the first neighborhood. At the next stage (“move or not” stage), a neighborhood
index is set to the first neighborhood when the solution x' is better than x, and to the next
neighborhood when x' equal or even worse than x. The iteration stops after all neighborhoods
are examined and no improvement can be found. This is to ensure that x' are optimum for all
neighborhoods, starting from the first neighborhood.
VND (even VNS) is based on principles: a local optimum in a certain neighborhood
structure does not correspond to local optimum in other neighborhood structure (but in many
problems, they are close each other), and global optimum corresponds to all local optimum in all
neighborhood structures [6]. Choice of the neighborhoods at each iteration is main factor form
the performance of the VND. The choice of structure (operator to find a “best” solution in certain
neighborhood) can be done sequentially, nested, or a combination of the two [6, 7]. Moreover,
random neighborhood selection gives better results than deterministic selection [8], and does
not require a neighborhood setting [9]. The repetition to the first neighborhood by [7] is referred
as Basic VND. There are other variants, such as Pipe VND (continue searching for the same
neighborhood), Cyclic VND (proceed to the next neighborhood), and Union VND
(a neighborhood that is a composite of several neighborhoods, so the search is continued in the
combined neighborhood).

TELKOMNIKA ISSN: 1693-6930 
Sequential order vs random order in operators of variable.. (Darmawan Satyananda)
803
Figure 1. Local optima condition and perturbation to jump from lofrical optima
Figure 2. Stages of VND
Some things noticed in choosing neighborhood are: (1) Number of neighborhood
operators used, (2) The type of neighborhood operators used, (3) The order of the
neighborhood, and (4) Whether the use of the neighborhood ensure the most optimal results.
Mistakes in selection can result in enormous computing time, remain trapped in a certain
neighborhood, and less optimal results.
To the best of author’s knowledge, the basic structure of VND (and VNS as well), is still
relatively remain unchanged. Variations is in addition of other method (as a hybrid VND),
variations in number and type neighborhood operators, and area of implementation. Other
interesting feature is implementation parallel programming due to a great number of iterations in
local search process and to take advantage of multi-core computer processors [8].
VNS and VND are reported have been successfully implemented in many cases such
as Traveling Salesman Problem [7], and Vehicle Routing Problem [2, 5], Location Routing
Problem [10], Flowshop Scheduling Problem [11], Redundancy Allocation Problem [12], Order
Batching Problem [13], Dial-a-ride problem (special case of VRP for patient transportation or

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804
disabled people) [14], and in minimum labelling Steiner tree problem [15], and Course Time
Table Problem [16]. Those references reported that VNS/VND could improve the objective
function compared to other method.
VND in many cases is combined (hybrided) with other method. Genetic Algorithm is
hybrided with VND for flexible job shop scheduling problems [2], [17] used Iterated Local Search
(ILS) VND in VRPTW, [11] used ILS VND in flowshop scheduling problem, [18] used ILS VND in
VRP with Multiple Trip, [6] used Intensified ILS Method with Random VND in VRPTW, and [19]
that used Iterated VND for CVRP problems. Those articles proved that the combination could
give better result, compared to other method and other best results known. A careful
consideration must take into account, since ILS uses VND in two steps: to refine initial result,
and in another iteration to meet acceptance criterion, as in [2]. Another interesting hybridization
are in [20, 21] that used GRASP (Greedy Randomized Adaptive Search Procedure) in VNS and
VND to control randomness in greedy heuristic.
Neighborhood Operators
Neighborhood operator can be classified into two classes: intra-neighborhood and
inter-neighborhood. Intra-neighborhood operations are performed on a single route with the goal
of minimizing the overall distance, and inter-neighborhood operations are performed on two or
more different routes with the goal of minimizing the overall distance and even minimizing the
number of vehicles [5].
Examples of inter-neighborhood are [9]:
 Shift(1,0) or 1-0 Exchange Neighborhood: One customer i is moved from route r1 to r2.
 Shift(2,0) or 2-0 Relocate Neighborhood: Two adjacent customers i and j (or an arc (i, j)) are
moved from route r1 to route r2. The transfer of the opposite arc(j, i) is also considered.
 Swap(1,1) or 1-1 Exchange Neighborhood: Interchange between customer i from route r1
and customer j from route r2.
 Swap(2,1) or 1-2 Exchange Neighborhood: Interchange between adjacent customers i and j
from route r1 and customer k from route r2. The opposite arc (j, i) is also considered as well.
 Swap(2,2) or 2-2 Exchange Neighborhood: Interchange between adjacent customers i and j
from route r1 and two other adjacent customers k and l, from route r2. The opposite arcs (j, i)
and (l, k) are also considered, yielding 4 possible combinations.
 Cross Exchange: exchange of two arcs (two strings of consecutive nodes) from two routes,
preserving the order of customers in each string [7], in swapping two paths belonging to two
different routes [5].
Examples of intra-neighborhood are [8]:
 Or-opt: one to three consecutive nodes are moved into other position in the route.
 Two-opt: two nonadjacent edges are removed, and two new edges are formed then to make
a new route.
 Exchange: exchanging two nodes position. This is another version of swap(1,1), but it
applied to the same route.
 Reinsertion: a node is removed, and inserted into other position in the route.
2. Research Method
This paper compares selecting neighborhood structure, in sequential and random order,
during process of repairing the initial CVRP route using VND Metaheuristic. The initial route was
generated based on the Sequential Insertion algorithm based on [22]. There were 10
neighborhood operators used for local search: 1-0 Exchange neighborhood (N1), 1-1 Exchange
neighborhood (N2), 1-2 Exchange neighborhood (N3), 2-2 Exchange neighborhood (N4), 2-0
Relocate neighborhood (N5), Cross-Exchange neighborhood (N6); and (b) intra route operator:
Or-opt neighborhood (N7), Two-opt neighborhood (N8), Exchange (N9), Reinsertion (N10). The
initial route establishment and repair process was implemented in a computer program created
with Borland Delphi, executed in computer with Intel i-7 microprocessors, with 32 GB memory
and operating system Microsoft Windows 7.
To test the comparation, 13 data tests were used. Those were taken from http://vrp.
atd-lab.inf.puc-rio.br/index.php/en/ of various publisher: Augerat, Christofides and Eilon,
Christofides-Mingozzi-Toth, Fisher, Taillard, and Uchoa. Number of customers ranged from 16
to 200 customers. Properties of each data set are presented in Table 1.

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Table 1. Properties of Each Dataset
Dataset Publisher
Number of depot &
customer
Capacity of
vehicle
Best Record
E-n13-k4 Christofides and Eilon 13 6000 247
P-n16-k8 Augerat set P 16 35 450
B-n31-k5 Augerat set B 31 100 672
A-n32-k5 Augerat set A 32 100 784
F-n45-k4 Fisher 45 2010 724
A-n60-k9 Augerat set A 60 100 1354
P-n60-k15 Augerat set P 60 80 968
tai75a Taillard 76 1445 1618,36
M-n101-k10 Christofides-Mingozzi-Toth 101 200 820
X-n101-k25 Uchoa [23] 101 206 27591
M-n200-k17 Christofides-Mingozzi-Toth 200 200 1373
X-n200-k36 Uchoa [23] 200 402 58578
For random neighborhood selection, 5 executions were performed and its average
result was taken then. The selection of neighborhood in sequential order was only done in 1 run
time because it did not involve random number generation, hence the algorithm always gave the
same result at every running for each data set. Objective for each running is to find the route
with shortest length, that is route that started from depot, passes every customer, and back to
the depot as final destination. There is no requirement on restrictions of route length for all
datasets.
3. Results and Analysis
The computer program made was then executed using test data prepared. Table 2
summarizes the results of program execution. The best results from this experiment are then
compared to best record for each dataset. Table 2 presents result of each dataeset. For each
case was recorded best value, worst value, average value, number of iterations, and ratio of
best case to best record known, both for sequential and random selection. The table presents
fact that selecting operators in random order can provide a more optimal route length than the
sequential order on 10 of of 13 datasets. This is in accordance with the [12] which states that
random selection can provide better results. In general, number of iterations in random selection
is less than iteration in sequential selection. Number of iterations get higher as the number of
customers increases, since more local search are performed (in “exploration of neighborhood”
stage), and more comparations to check optimality of a new modified route (in “move or not”
stage).
Table 2. The Result of Experiments in Worst and Best Cases
Figure 3 is graphs showing the results obtained per iteration for E-n13-k4 as an
example of dataset. X-axis states number of iteration, and Y-axis states length of route found.
Data of these graphics are taken from the best result on each iteration. For each picture, leftside

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graphic is for random selection, and rightside graphic is for sequential selection. Selection of
operators in sequential order provides a more stable output graph than a random selection at an
equivalent level, but there are operations that produce "jumping" results, making it the worst
route as shown in Figure 3 (b). For example, in Figure 3 (b), the spikes are about in length of
3000 unit, and mostly other results are about below 500 unit. Meanwhile, in Figure 3 (a), the
highest results are about in level of 625 unit, and in average, results in the end of iteration are
about 300 unit. This pattern is also found in other datasets. In Table 2, column of “Worst” stated
the worst case, with value much higher than the average length.
The graphs on the figures also showed that an operator could generate a completely
"new" route, which is not just an improvement over the previous route. Such a route was noted
as a “spike”, which had much worse length than previous route, and surely it would not be
chosen because a route must be minimal on all structure (N1-N10) to be said as a global
optimum. From observation, this maximum-length route occurs in two-opt (N8) operation. The
existence of this "leap" can be used as a consideration for not using of this operator on local
search.
From all datasets, unfortunately VND method still could not give better result than best
result known in the reference (except for E-n13-k4), but still in acceptable level. Ratio best value
to best record for all dataset ranged from -2.07% to 6.6% for random selection, and ranged from
-4.22% to 8,29% for sequential selection). In average, ratio of best results in random order is
2.13% which is better than average for selection order, that is 3.56%.
(a) (b)
Figure 3. The Result of execution for E-n13-k4
Route length relates to customer’s demand and capacity of vehicle (CVRP assumes
that number of vehicles is unlimited [24]). If the total demand of all customers is far beyond the
capacity of the vehicle, the more routes are generated. The number of routes will affect the
number of iterations for local search and for checking the optimization of a route. This number of
iterations is also directly proportional to the number of customers on a route. For example, the
Shift (1,0) operation on a route that contains n customers will be performed n times to find the
best improvement. If there exists m route, there will be m x (m-1) x n displacement. If there a
route x' which is better than x, the neighborhood will be re-examined to ensure that the result is
the global optimum with respect to all neigborhoods. The more iterations occur, the longer the
execution time would be. The dataset that has the longest execution time is M-n200-k17, which
had 19 million iterations that took 6 minutes to complete.
Maximizing number of customers in a route depends on demand of each customer. If
the demand is relatively large compared to capacity of vehicle, then there will be relatively large
unoccupied space in a vehicle. Demand for each customer in CVRP cannot be splitted to
maximizing vehicle capacity, contrary to VRP with Split Demand (VRPSD). In [3] stated that
some considerations that influence the result: whether to use first or best improvement,

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reducing the neighborhood, and distribution of neighborhoods. Number of operators used needs
to be considered in reducing the neighborhood. There are several operators that are similar in
exchange and relocate type, hence it can be considered to choose only one of them. Usage of
similar operator does not much influence optimum result [25].
Reduced operators will have an impact on processing time and the number of iterations
required, as fewer "move or not" steps are performed. Specifically, for sequential selection,
operator sequences should be considered by combining intra-route and inter-route
neighborhoods or choosing the most advantageous operator. Other ways of improvement can
be considered also, as in implementation of parallel programming [8], and global vector (global
memory) to store all generated neighbor solutions to avoid any duplication during the
execution [26]
4. Conclusion
The use of the VND framework may be considered in the improvement process,
particularly on VRP issues. The selection of operators in random order have proven provide
better results on the data test used than that of the sequential selection. The types and
sequence of operators can be considered further in other experiments in order to provide better
results, as well as implementation of parallel programming to shorten execution time in cases of
great number of customers.
Acknowledgements
This article is part of research entitled “Utilization of Google Map Services for the
Application of VRP Problem Solving” (year 2017, first year of two years planned). Special
thanks addressed to Directorate of Research and Community Services (DRPM), Indonesian
Ministry of Research, Technology, and Higher Education (Kemenristekdikti) as funding this
research.
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