2018 algorithms for the minmax regret path problem with interval data

Accepted Manuscript
Algorithms for the Minmax Regret Path Problem with Interval Data
Francisco Pérez-Galarce, Alfredo Candia-Véjar, César Astudillo,
Matthew Bardeen
PII: S0020-0255(18)30456-0
DOI: 10.1016/j.ins.2018.06.016
Reference: INS 13709
To appear in: Information Sciences
Received date: 12 September 2017
Revised date: 5 June 2018
Accepted date: 7 June 2018
Please cite this article as: Francisco Pérez-Galarce, Alfredo Candia-Véjar, César Astudillo,
Matthew Bardeen, Algorithms for the Minmax Regret Path Problem with Interval Data, Information
Sciences (2018), doi: 10.1016/j.ins.2018.06.016
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Algorithms for the Minmax Regret Path Problem with Interval
Data
Francisco Pérez-Galarce1
, Alfredo Candia-Véjar2∗
, César Astudillo3
, Matthew Bardeen3
1
Computer Science Department, Pontificia Universidad Católica de Chile, Santiago, Chile
2
Departamento de Ingenier´ıa Industrial, Universidad de Talca, Camino Los Niches km. 1, Curicó, Chile
3
Departamento de Ciencias de la Computación, Universidad de Talca, Camino Los Niches km. 1, Curicó, Chile
Abstract
The Shortest Path in networks is an important problem in Combinatorial Optimization and
has many applications in areas like Telecommunications and Transportation. It is known that this
problem is easy to solve in its classic deterministic version, but it is also known that it is an NP-Hard
problem for several generalizations. The Shortest Path Problem consists in finding a simple path
connecting a source node and a terminal node in an arc-weighted directed network. In some real-
world situations the weights are not completely known and then this problem is transformed into an
optimization one under uncertainty. It is assumed that an interval estimate is given for each arc length
and no further information about the statistical distribution of the weights is known. Uncertainty
has been modeled in different ways in Optimization. Our aim in this paper is to study the Minmax
Regret Path with Interval Data problem by presenting a new exact branch and cut algorithm and,
additionally, new heuristics. A set of difficult and large size instances are defined and computational
experiments are conducted for the analysis of the different approaches designed to solve the problem.
The main contribution of our paper is to provide an assessment of the performance of the proposed
algorithms and an empirical evidence of the superiority of a simulated annealing approach based on
a new neighborhood over the other heuristics proposed.
Keywords: Minmax Regret Model with Interval Data; Simulated Annealing; Shortest Path Prob-
lem; Branch and Cut; Neighbourhoods for path problems
1 Introduction1
We study a variant of the well known Shortest Path (SP) problem called the Minmax Regret2
Path (MMR-P) Problem. In the classic SP problem, a digraph G = (V, A), where V is the set of3
nodes and A is the set of arcs, with non-negative lengths associated to each arc and two special nodes4
s and t belonging to V are considered. The SP problem consists of finding a path between s and5
t (s-t-path) with the minimum total length. Efficient algorithms for the original SP problem have6
been known since [14], in which the authors proposed a polynomial time algorithm and from that7
study, multiple approaches have been proposed. Some SP variants, algorithms and applications are8
discussed in [2].9
In this research the focus is on SP problems where there is uncertainty in the objective function10
parameters (the length function). In this SP variant, for each arc we have a closed interval that11
defines the possibilities for the arc length. The uncertainty model used here is the minmax regret12
approach (MMR), sometimes named robust deviation. In this approach the aim is to make decisions13
that will have a good objective value under any likely input data scenario included in the decision14
model. Three criteria are known to select among robust decisions, they are: absolute, MMR and15
relative MMR [27]. We use MMR, where the regret associated with each combination of decisions16
and input data scenario is defined as the difference between the resulting cost to the decision maker17
and the cost from the decision taken if it had been known prior to the time of the decision which18
scenario of data input would have occurred. In the context of Optimization with Uncertainty an19
important alternative model is the Fuzzy model, where several papers have studied the SP problem,20
see [20, 36, 17].21
The MMR Model has been increasingly studied in combinatorial optimization, see the books by22
[27], and [23], as well as the reviews by [4] and [8]. Most research on Minmax Regret Combinatorial23
Optimization (MMR-CO) has been focused on mono objective problems and recently, a paper has24
proposed robust multiobjective CO problems [15] and, in the last years, several papers have extended25
∗Corresponding author.
E-mail addresses: fjperez10@uc.cl (Francisco Pérez-Galarce), Alfredo Candia-Véjar (acandia@utalca.cl), César As-
tudillo (castudillo@utalca.cl), Matthew Bardeen (mbardeen@utalca.cl)
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the concepts of robustness to Multiobjective CO problems [45, 9]. Moreover, SP has been studied in26
the context of multi-objective uncertain problems,[44].27
It is known that MMR-CO problems with interval data are usually NP-hard, even when the28
underlying classic problem is easy to solve; this is the case of the minimum spanning tree problem,29
SP problem, assignment problem and others, see [4] and [23] for a detailed analysis. Several efforts30
have been made to obtain exact solutions using a broad set of exact methods, frequently formulating31
an MMR problem like a Mixed Integer Linear Programming (MILP) problem and then using a32
commercial code or applying branch and bound, branch and cut or Benders decomposition approaches33
in a dedicated scheme. Some problems that have been studied are: MMR Spanning Trees [30, 42],34
MMR Paths [22, 23, 31, 32], MMR Assignment [39], MMR Set Covering [40], and MMR Traveling35
Salesman [34].36
Particularly, for MMR-P, [47] proved that the problem is NP-Hard even when a graph is restricted37
to be directed acyclic planar and regular of degree three and [46] proved that the problem is NP-Hard38
even in the case of a restricted class of Layered networks. Additional results about the complexity of39
MMR-P for some classes of networks is given in [23] and [5]. Exact algorithms for MMR-P have been40
proposed by [23, 31, 32], which show the application of several algorithmic approaches. However,41
most of these papers had computational experiments using small instances or instances with a special42
structure like real road networks. In fact, [32] compared several exact algorithms and concluded that43
an algorithm able to clearly outperform the others does not exist. Moreover, they established some44
recommendations depending on the type of instances to be solved. [16] presented some results about45
some classes of networks of MMR-P for which polynomial or pseudopolynomial approaches exist. The46
authors of [38] addressed the MMR-P on a finite multi-scenario model and they proposed three new47
approaches for algorithmic purposes. Numerical experiments using randomly generated instances48
showed that some of the proposed algorithms were able to obtain solutions in reasonable times for49
network instances up to 750 nodes. Very recently, [18] have proposed a new procedure to obtain a50
lower bound for the optimal value of instances of MMR-P. This value is part of a branch and bound51
algorithm that outperforms existing exact algorithms in the literature when it is applied to some52
classes of MMR-P instances.53
With respect to heuristic approaches, only a few methods are available. A basic heuristic based on54
the definition of a particular scenario (the midpoint of the intervals) was designed as an approximation55
algorithm for general MMR-CO problems [24, 23]. A new basic heuristic, HMU, solves an MMR-CO56
problem for two scenarios: the midpoint scenario and the scenario in which all the weights are set57
to their upper bounds, then the HMU returns the better of these two solutions. HMU achieves a58
good performance for several MMR-CO problems [24, 23]. [21] proposed a heuristic for MMR-P but59
only small instances were tested for comparison with other approaches. A new lower bound for the60
optimal value of MMR-CO problems was proposed in [10]. In particular, for MMR-P, [23] showed61
that for networks with a number of nodes under 1 000, HMU obtained solutions with gaps under62
6% (relative deviation from the reported optimum) for several classes of directed and undirected63
networks.64
A problem related to MMR-P, the minmax relative regret robust shortest path problem (MMRR-65
P), was studied in [11]. They proposed a mixed integer linear programming formulation and also66
developed several heuristics with emphasis on providing efficient and scalable methods for solving67
large instances for the MMR-P, based on pilot method and random-key genetic algorithms. The68
CPLEX branch-and-bound algorithm based on this formulation found optimal solutions for most69
of the small Layered and Grid instances with up to 200 nodes. However, gaps of 10% or higher70
were found for some instances. The Grid instances proposed in this paper were much harder to71
solve than the Layered instances found in the literature. Other heuristic approaches for MMR-CO72
problems are the Simulated Annealing approach for MMR-Spanning Tree by [35], the heuristic based73
on a bounding process for MMR- spanning Arborescences by [12], the metaheuristic approach for74
MMR-Assignment problem [39] and the Tabu Search for the MMR-Spanning Tree by [25].75
Our main contributions in this paper are: i) an efficient Branch and Cut algorithm was able to76
find exact solutions for some classes of large size instances and outperformed other exact algorithms77
for several of these instances, ii) a local search heuristic and a simulated annealing metaheuristic that78
uses a novel neighborhood to find good solutions for large sized instances that exact algorithms could79
not and iii) an extensive experimental analysis using several classes of network instances showing the80
performance of the different algorithms and highlighting the particular conditions when they could81
be used.82
In Section 2 the problem is formally defined and known results about the computational complex-83
ity of the problem are presented; in Section 3 a new Branch & Cut exact algorithm for MMR-P is84
introduced; in Section 4, various heuristics are analyzed including well-known basic heuristics, then85
a local search and simulated annealing approaches based on a new neighborhood for the problem are86
also presented; in Section 5, benchmark instances are presented and an implementation description87
is given. In Section 6 experiments are conducted with exact approaches, determining the perfor-88
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mance of the algorithms when applied in several types of instances. The computational results of the89
heuristic and their analysis for hard instances are presented in Section 7, finally, in Section 8 some90
conclusions are discussed.91
2 Definition of MMR-P and Computational Complexity92
First of all, in 2.1 basic notation and the formal definition of MMR-P are presented. Then, in93
2.2 important known results about the computational complexity of the problem are presented.94
2.1 Notation for MMR-P95
We use a standard notation for MMR-CO problems, specially we follow the notation used in [39].96
We considered a digraph G = (V, A) where V is the set of nodes, | V |= n and | A |= m the set of97
arcs. For each arc e ∈ A, two non negative numbers c−
ij and c+
ij are given and c−
ij ≤ c+
ij. The length98
can take on any real number from its uncertainty interval c−
ij, c+
ij , regardless of the values taken by99
the costs of other arcs. The Cartesian product of the uncertainty intervals c−
ij, c+
ij , (i, j) ∈ A, is100
denoted as S and any element s of S is called a scenario; S is the vector of all possible realizations101
of the costs of arcs. cs
ij, (i, j) ∈ A denotes the cost of the arc (i, j) corresponding to scenario s.102
Let Φ the set of all s-t paths in G. For each X ∈ Φ and s ∈ S, let F(s, X) be the cost of the s-t103
path X in the scenario s.104
F(s, X) =
(i,j)∈X
cs
ij (CP)
The classical s-t SP problem for a fixed scenario s ∈ S is:105
min {F(s, X) : X ∈ Φ} (CSP)
Let F∗
(s) be the optimum objective value for problem (CSP). For any X ∈ Φ and s ∈ S, the value106
R(s, X) = F(s, X) − F∗
(s) is called the regret for X under scenario s. For any X ∈ Φ, the value107
Z(X) is called the maximum (or worst-case) regret for X.108
Z(X) = max
s∈S
R(s, X) (MR-Path)
The MMR version of Problem (CSP) is:109
min {Z(X) : X ∈ Φ} = min
X∈Φ
max
s∈S
R(s, X) (MMR-Path)
Let Z∗
denotes the optimum objective value for Problem MMR-P. Further, Z∗
is called a worst-110
case scenario for X. For any X ∈ Φ, the scenario induced by X, s(X), for each (i, j) ∈ A is defined111
by112
c
s(X)
ij =
c+
ij, (i, j) ∈ X
c−
ij, otherwise.
(1)
Property 1: For each s-t path X in Φ it is verified,113
Z(X) = Fs(X)
(X) − Fs(X)
(P1)
It is clear from the above definitions that the worst-case regret can be computed by solving just114
two classic SP problems.115
2.2 Computational Complexity of MMR-P116
Several works analyzing the computational complexity of MMR-P have shown that the problem is117
NP-Hard even for several classes of special networks. In the following two classes of directed graphs118
(digraphs) are defined. More details about the classes of digraphs and computational complexity119
results can be found in [23].120
Layered digraphs: In a layered digraph G = (V, A), set V can be partitioned into disjoint subsets121
V1, V2, ..., Vk called layers and arcs exist only between nodes from Vi and Vi+1 for i = 1, ..., k − 1.122
The maximal value of |Vi| for i = 1, ..., k is called a width of G. In every layered digraph all paths123
between two specified nodes s and t have the same number of arcs.124
Edge series-parallel multidigraphs: An edge series-parallel multidigraph (ESP) is recursively de-125
fined as follows. A digraph consisting of two nodes joined by a single arc is ESP. If G1 and G2 are126
ESP, so are the multidigraphs constructed by each of the operations:127
• Parallel composition p(G1, G2): identify the source of G1 with the source of G2 and the sink of128
G1 with the sink of G2.129
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• Series composition s(G1, G2): identify the sink of G1 with the source of G2.130
In the following some computational complexity results are summarized:131
- MMR-P is strongly NP-hard for acyclic directed layered graphs, even if the bounds of weight132
intervals are 0 or 1.133
- MMR-P is strongly NP-hard for undirected graphs, even if the bounds of weight intervals are134
0 or 1.135
- MMR-P is NP-hard for edge series-parallel digraphs with a maximal node degree at most 3.136
- MMR-P is NP-hard for layered digraphs of width 3 and for layered multidigraphs of width 2.137
- MMR-P for ESP admits an FPTAS, that is an algorithm that for a given ESP computes path138
P such that ZG(P) ≤ (1 + )OPT in time O |A|3
/ 2
.139
The above results show that MMR-P is a very difficult problem still for some special classes of140
graphs. From the algorithmic point of view this represents a challenge when the objective is to141
develop efficient algorithms for its resolution.142
3 Exact Algorithms for MMR-P Problem143
In this section the proposed branch and cut (B&C) algorithm and a known MILP formulation144
for MMR-P are presented.145
3.1 A MILP Formulation for the MMR-P Problem146
We consider a digraph G = (V, A) with two distinguished nodes s and t and according the previous147
section each arc (i, j) ∈ A has associated an interval length c−
ij, c+
ij . We use Kasperski’s MILP148
formulation of the MMR-P Problem [23], this formulation is obtained using the duality properties.149
The problem MMR-P is formulated using the general formulation MMR-P defined in the previous150
section, by introducing both, the property P1 and the particular definitions of (CSP) and (CP) for151
SP. In this formulation each arc (i, j) in A has associated a binary variable xij expressing if the arc152
(i, j) is part of the solution X ∈ Φ. The constraints yij ∈ {0, 1} have been replaced by yij ≥ 0153
because the matrix A associated to the typical constraints of s-t paths is totally unimodular and154
yij ≤ 1 in every optimal solution of the above relaxed formulation.155
min
(i,j)∈A
(c+
ijxij + c−
ij(1 − xij))yij (2)
156
{i:(j,i)∈A}
yji −
{k:(k,j)∈A}
ykj =



1, j = s
0, j ∈ V {s, t}
−1, j = t
(3)
157
yij ≥ 0, ∀ (i, j) ∈ A (4)
The dual for this problem (2-4) is presented in (5-6).158
max λs − λt (5)
159
λi ≤ λj + c+
ijxij + c−
ij(1 − xij), (i, j) ∈ A (6)
Then we can use these results and tackle the MMR-P problem with the integer programming for-160
mulation showed in (7-10). This formulation can be numerically solved by a software like CPLEX.161
min
(i,j)∈A
c+
ijxij − λs + λt (7)
162
λi ≤ λj + c+
ijxij + c−
ij(1 − xij), (i, j) ∈ A (8)
163
{i:(j,i)∈A}
xji −
{k:(k,j)∈A}
xkj =



1, j = s
0, j ∈ V {s, t}
−1, j = t
(9)
xij ∈ {0, 1} , ∀ (i, j) ∈ A (10)
It is important to comment that we use this approach for evaluating the performance of both the164
B&C algorithm described next and the heuristics proposed in Section 4.165
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3.2 Branch and Cut Approach166
We implemented a B&C over CPLEX framework using the formulation presented in equations167
(11), (12), and (13) where the constraints are separated by robust constraints in Equation (12) and168
topology in Equation (13). This formulation has an exponential number of robust constraints (one169
per each path s-t ∈ Φ) and it is based on [42].170
The topology constraints consider the flow formulation for the shortest path problem 3 and they171
are represented for X ∈ Φ in Equation (13), these constraints are added at the beginning of the172
algorithm. The robust constraints are the cuts in our B&C and they are added when a new feasible173
solution is found in each node of the branching process.174
Z∗
MMR = min
e∈E(X)
c+
e − θ (11)
s.t. θ ≤
e∈E(Y)
c−
e +
e∈E(Y)∩E(X)
(c+
e − c−
e ), ∀Y ∈ Φ (12)
θ ∈ IR≥0 and X ∈ Φ. (13)
Additionally, if a fractional solution ( ˜X) is found, we find a valid cut by rounding this fractional175
solution to a feasible one; to do so, we find a near integer vector ˜X by solving the SP on G with edge176
costs defined by Equation (14), using the obtained vector ˜X , an induced solution ˜Y is calculated177
and the corresponding cut is added to the model if the cut is violated.178
˜ce = (c−
e + c+
e ) min{1 − ˜xij, 1 − ˜xji}, ∀e : {i, j} ∈ E; (14)
Moreover, using ˜X (feasible or not) we apply a local-search in order to find still more violated179
robust constraints and add them to the model. We have also embedded into the B&C a primal180
heuristic which attempts to provide better upper bounds using the information of the fractional181
solution ˜X; a feasible vector ˜X is calculated by solving the SP on G with edge costs defined by(14).182
4 Heuristics for MMR-P183
In this section we present the proposed heuristic approaches for solving MMR-P. It contains184
(i) Two simple and known heuristics based on the definition of specific scenarios (ii) A Simulated185
Annealing and a Local Search approaches based on a novel definition of a neighborhood of feasible186
s-t paths and (iii) a Simulated Annealing approach based on a traditional k-opt type neighborhood187
for combinatorial optimization problems.188
4.1 Basic Heuristics for MMR-P189
Two basic heuristics for MMR-P are known; in fact the heuristics are applicable to any MMR-CO190
problem. These heuristics are based on the idea of specifying a particular scenario and then solving191
the classic problem using this scenario. The output of these heuristics are feasible solutions for the192
MMR-CO problem, for more details see [8, 12, 23], [34] and [40].193
First we mention the midpoint scenario, sM
, defined for each edge e ∈ A as sM
= c+
e + c−
e /2 .194
We refer to the heuristic based on the midpoint scenario as HM. The other heuristic based on the195
upper limit scenario will be denoted by HU. The computation of the output solution for each one196
of these heuristics implies to solve only twice the corresponding classic problem. The first of these197
problems is the computation of the solution Y in the specific scenario, sM
for HM or sU
for HU,198
and the second one is the computation of Z(Y ). These heuristics have been integrated in the new199
heuristic HMU by the sequential computing of the solutions given by HM and HU and getting the200
best. In the evaluation of heuristics for MMR problems several experiments have shown that if these201
heuristics are considered as an initial solution, the performance of more sophisticated heuristics is202
improved. For an in-depth discussion, please refer to [34, 39, 40] and [8].203
4.2 Local Search for MMR-P204
Local Search (LS), described in Algorithm 1, is a traditional search method for a CO problem205
P with feasible space S. The method starts from an initial solution and iteratively improves it by206
replacing the current solution with a new candidate, which is only marginally different. During this207
initialization phase, the method selects an initial solution s from the search space S. This selection208
may be at random or may take advantage of some a priori knowledge about the problem.209
An essential step in the algorithm is the acceptance criterion, i.e., a neighbor is identified as the210
new solution if its cost is strictly less in comparison to the current solution. This cost is a function211
assumed to be known and is dependent on the particular problem. The algorithm terminates when no212
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improvements are possible, which happens when all the neighbors have a higher (or equal) cost when213
compared to the current solution. The method outputs the current solution as the best candidate.214
Observe that, at all iteration steps, the current solution is the best solution found so far. LS is a215
sub-optimal mechanism, and it is not unusual that the output will be far from the optimum. The216
literature reports many algorithms that attempt to overcome the hurdles encountered in the original217
LS strategy.218
Algorithm 1 Local Search
Input: Search space (S), cost function (f(·)), neighborhood function (N(·)).
Output: best solution founded Y , cost f(Y ).
Y ← s // s ∈ S
while Termination Criterion = TRUE do
Y ← N(S, Y )
if f(Y ) ≤ f(Y ) then
Y ← Y
end if
end while
219
4.3 A Simulated Annealing Approach for MMR-P Problem220
Simulated Annealing (SA) is a well known probabilistic metaheuristic proposed by Kirkpatrick221
et al. in the 80’s for solving hard combinatorial optimization [26, 6]. SA seeks to avoid being222
trapped in local optimum as would normally occur in algorithms using local search methods. A223
key characteristic of SA is the possible acceptation of worse solutions than the current during the224
exploration of the local neighborhood. Accordingly with the physical analogy of SA with metallurgy,225
several parameters must be tuned in order to find good solutions. Typical parameters are associated226
to concepts like neighborhood, cooling schedule, size of internal loop and termination criterion. These227
parameters are usually adjusted through experimentation and testing (see Algorithm 2).228
Algorithm 2 Simulated Annealing (SA)
Input: Search space (S), cost function (f (·)), neighborhood function (N(·)),
initial and final temperature (ti, tf ), number of internal loops (K), cooling
programming (β), acceptance function (g(·)).
Output: best solution founded Y ∗, cost f(Y ∗).
t ← ti
Y ← s // s ∈ S
while t ≥ tf do
k ← 0
while k ≤ K do
Y ← N(S, Y )
if f(Y ) ≤ f(Y ) then
Y ← Y
if f(Y ) ≤ f(Y ∗) then
Y ∗ ← Y
end if
else
if g(Y, Y ) == TRUE then
Y ← Y
end if
end if
k ← k + 1
end while
t ← βt
end while
229
Within the context of the MMR-P problem, we shall now describe the main concepts and param-230
eters generally used in SA.231
Search Space: A subgraph S of the original graph G is defined such that this subgraph contains a s-t232
path. In S a classical s-t shortest path subproblem is solved, where the arc lengths are chosen taking233
the upper limit arc costs. Then, the optimum solution of this problem is evaluated for acceptation.234
Next Subsection details this part.235
Initial Solution: The initial solution s is obtained by applying the heuristic HMU to the original236
network.237
Cooling Programming: A geometric descent of the temperature is used according to parameter β.238
Internal Loop: Next subsection describes in detail about this parameter.239
Neighborhood Search Moves: Next subsection describes in detail the structure of the neigbourhood240
used.241
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Acceptation Criterion: A standard probabilistic function is used for managing the acceptation of242
new solutions.243
Termination Criterion: A fixed value of temperature (final temperature Tf ) is used as termination244
criterion.245
4.4 Neighborhood Structure for the MMR-P problem246
Two fundamental concepts in LS are the Search space and Neighborhood structure. The Search247
space, denoted as S, is defined as the set of all feasible solutions for the problem. At each iteration248
of LS, a slight modification of the current solution leads to a neighbor, which on a more critical249
inspection, can be seen as a function which corresponds to a local transformation on the current250
solution. This function induces a set of possible neighbors to a current solution, concept know as251
the neighborhood set, and which is denoted by N(Y ). In particular N(Y ) ⊆ S. Many different252
neighborhood structures can be defined for the same problem, yielding the challenge of selecting253
the most suitable. It is important to note that depending on the context, small modifications of254
the neighborhood structure may lead to strongly different cost for the best solution found by the255
algorithm.256
In the classic SP problem the determination of neighborhood is more complex than in other257
problems, such as the TSP [28]. In [37] is presented a LS heuristic for the multicriteria SP problem.258
The mechanism to obtain a new path p from an existing path p is described as follows: first, a259
subpath starting from node s is obtained by cutting the path p at node i. Next, an arc emanating260
from node i and connected to the node j is attached to the new solution. Finally, the algorithm261
searches for a path from j to the terminal node t. This entire process is repeated for every node in262
the original path, and for every node j adjacent to node i, which, from our perspective, is prohibitive263
for many applications of the SP.264
A traditional neighborhood used in designing heuristics for CO problems is the family of k-opt.265
The idea in this scheme is to eliminate k arcs (in the network problems context) and add new arcs to266
complete a feasible solution. Typically, in problems where the cardinality of the arcs in the solution267
is fixed, (like the TSP or the Minimum Spanning tree problems) k eliminated arcs are replaced by k268
new arcs. In paths optimization problems, if k arcs are eliminated from a feasible solution, a different269
number of arcs added could generate a feasible solution. Some papers [19, 43, 29] have considered270
this strategy. For our problem, k-opt strategy is used by considering the values k = 2 and k = 3.271
Given the importance of the new neighborhood structure in our proposed method, we have272
dedicated this section to explain it in detail. We start by defining the LS mechanism. Subsequently273
we detail the concepts of neighborhood structure and Search space. After that, we explicitly describe274
an architectural model for obtaining a new candidate solution by restricting the original search space.275
Typically, in LS, several types of neighborhood structures are analogous to the k-opt method276
explained above, in the sense that a candidate solution is obtained by applying a slight modification to277
the previous candidate, see [3] for an analysis of several types of large neighborhoods for combinatorial278
optimization problems. A fundamentally different philosophy is the one of using subspaces to induce279
candidate solutions. In this model, the new candidate is not obtained directly from a previous280
solution. Rather the candidate is generated by an indirect step, which consists in perturbing a281
subspace in a LS fashion so as to obtain a new subspace which is marginally different in comparison282
to the former. Finally, the new subspace is employed to derive the new candidate solution. This283
concept adds an extra layer in the architectural model for defining the neighborhood structure. The284
method is detailed in Algorithm 3, which generalizes the method presented in [35] for solving minmax285
regret spanning tree problem. [35], in the first step, applied local transformations to a connected286
graph (subspace) to obtain a new graph which is also connected (new subspace). In the second step,287
the differences in the regret between the original and the modified candidate solutions are evaluated.288
Algorithm 3 Neighbor induction (R)
Input: R, a subspace of original search space S.
Output: Y , the new candidate solution.
1: R’← subspace-perturbation (R).
2: Y ← generate-candidate (R’).
289
Our proposed solution for the implementation of the MMR-P Neighborhood retains the idea of290
using bitmap strings to represent (and restrict) the search space. We start by defining a bitmap291
string with cardinality |A|, such that π (j) = 1 if edge aj belongs to the current subset, and π (j) =292
0 otherwise. Further, π (j) denotes the bit j of the bitmap vector. The full process for creating a293
new search space is detailed in Algorithm 4.294
At each iteration, a predetermined fraction of arcs from the original subspace are modified, i.e.,295
they are set to 1 (added) if they were not present in π or set to 0 (deleted) otherwise. This fraction296
is controlled by the parameter γ, and directly relates the concept of exploration and exploitation297
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as detailed as follows. Small values for γ lead to slight perturbations of the current subspace, i.e.,298
the resultant subspace will be only marginally different from the subspace currently being examined.299
This configuration favors the exploitation of the current solution. In contrast, large values for γ300
produce strong perturbations of the subspace, producing subspaces which are expected to be much301
different from the subspace currently being perturbed, which favors the exploration of unvisited302
regions in the original search space. Exploratory test on a variety of datasets have show evidence303
that a suitable value for depends on the dataset being tested and particularly its size.304
Once the subspace is determined, the algorithm ensures that there exists a path between s and305
t. If so π is accepted, otherwise we reject it and randomly generate a new version of π following306
the same scheme. The overall algorithm starts with the entire search space by setting all the bits of307
the vector π to 1.308
Observe that, in our definition of neighborhood, a subspace is not restricted to connected graphs,309
i.e., a subspace may (or may not) possesses disconnected components. For this reason, we must check310
at all iterations that possess at least a single s-t path. Note that the disconnected components may311
become connected depending on the stochastic properties of the environment. Once the auxiliary312
graph is determined, we obtain a new candidate solution from it. When the node t is reachable from313
the node s, the new candidate solution is processed using Algorithm 5. In our proposition, the new314
candidate solution, i.e., a new s-t path, is obtained by a heuristic criterion.315
We decided to apply the HMU method mentioned earlier. We then calculate the regret of this316
path with a classical SP algorithm over the original graph, then using it to determine whether or not317
to accept the new subspace.318
With this method, we are able to tailor the percentage of arcs we flip when generating a neighbor319
candidate, enabling us to find the correct balance between exploration and exploitation. The result320
of this, however, is that we can no longer use the delta between the regrets as our acceptation321
criteria. Instead we have calculate the regret via a heuristic method. For MMR-P this compromise322
is acceptable, as we know of linear time algorithms for calculating the two SP required for the323
calculation of the HU and HM heuristics.324
Algorithm 4 Algorithm MMR-P for subspace perturbation (π, γ)
Input:
- π, a bitmap string with cardinality |A|, such that π (j)=1 if edge ej belongs
to the current subset, and π (j)=0 otherwise.
- γ, the fraction of arcs from the original subspace which are to be flipped
(Γ = γ ∗ n , where n is the number of arcs).
Output:
- π’, a bitmap string with cardinality |A|, such that π (j)=1 if edge ej belongs
to the current subset, and π (j)=0 otherwise.
π ← π
for k = 0 → Γ do
j ← RANDOM(0, |π |)
if π (j) = 0 then
π (j) ← 1
else
π (j) ← 0
end if
end for
325
Algorithm 5 Algorithm MMR-P for generate candidate
Input:
- π, a bitmap string with cardinality |A|, such that π (j)=1 if edge ej
belongs to the current subset, and π (j) = 0 otherwise.
- f(·), a cost function.
Output:
- Y ’, a new candidate solution.
1: YHU ← HU(π)
2: YHM ← HM(π)
3: if f(YHU ) < f(YHM ) then
4: Y ← YHU
5: else
6: Y ← YHM
7: end if
326
5 Benchmark Instances327
In the literature, several classes of instances have been considered in computational experiments328
for evaluating the performance of algorithms proposed for MMR-P. Among them we found the329
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following, Random networks [33, 31, 32] and [41], Road networks located in some European cities330
[33, 31, 32] and Layered networks [33, 31, 32, 41]. Extensive experiments on random networks [41]331
showed that instances from 1 000 to up 20 000 nodes were solved, in short times, by an implementation332
in CPLEX and thus this class of instances were not considered at the present research. Road networks333
from European cities are not available and therefore only Layered networks, from this traditional334
group of instances, is considered here. A new particular class of networks, Grid instances (which335
could be interpreted as a type of road networks) was defined in [11] when they studied the relative336
robust version of MMR-P. In the present paper this class of instances is considered in the experiments337
and defined below.338
Layered networks were introduced in the paper of [46] in the study of the computational com-339
plexity of MMR-P problem. In [32] it is mentioned that Layered networks simulate some class of340
telecommunication networks. Layered networks are named as K-n-c-d-w, where n is the number341
of nodes, each cost interval has form c−
ij, c+
ij where a random number cij ∈ [1, c] is generated and342
c−
ij ∈ [(1 − d)cij, (1 + d)cij], c+
ij ∈ c−
ij + 1, (1 + d)cij ( 0 < d < 1) and w is the number of layers343
[31]. In Figure 1 an example of a Layered instance (K-12-c-d-3) is presented. Two groups of Layered344
instances were created. The group L1 contains eight subgroups of instances where for each subgroup345
only the width of the uncertainty interval is variable. The number of nodes is 1 000 for the first346
subgroup and 10 000 for the last. The number of layers at each subgroup is fixed as the 10% of347
n. The second group of Layered instances, L2, contains four subgroups of instances where for each348
subgroup is varied the width of the uncertainty interval and the number of layers. The number of349
nodes is 250 for the first subgroup and 2 000 for the last. Both group of instances are described in350
detail in Tables 1 and 4, for L1 and L2, respectively.351
A Grid network is related to a matrix with n rows and m columns. Each matrix cell corresponds352
to a node and two arcs with different directions connecting each pair of nodes whose respective353
matrix cell are adjacent. Therefore, the resulting directed graph has nxm nodes and 2(2mn−n−m)354
arcs. The node s is assumed located in the position (1,1) of the matrix and the node t in the position355
(m, n), an example is given in Figure 2 with n = 3 and m = 4. The interval costs were generated the356
same way as for Layered instances. The instances are named as G-n-m-c-d, where G identifies the357
instance type, n is the number of rows and m is the number of columns. We consider c = 200 and358
d = 0.5 for all instances in this group. For grid group, G, instances of different sizes were considered.359
2x{20, 40, 80, 160, 320} with {40, 80, 160, 320} nodes respectively and {116, 236, 476, 956, 1916} arcs,360
4x40 with 160 nodes and 552 arcs, 8x80 with 640 nodes and 2 384 arcs, 16x160 with 2 560 nodes361
and 9 888 arcs and 32x320 with 10 240 nodes and 40 256 arcs.362
Figure 1: Example of a Layered instance K-12-c-d-3 Figure 2: Example of a Grid instance G-3-4-c-d
Implementation of Algorithms: The exact approaches were implemented using CPLEX 12.5 and363
Concert Technology. The heuristic approaches were implemented in C++. All CPLEX parameters364
were set to their default values, except in B&C approach where the following parameters were set:365
(i) CPLEX cuts were turned off, (ii) CPLEX heuristics were turned off, (iii) the time limit was set366
to 900 seconds. All the experiments were performed on a Intel Core i7-3610QM machine with 16 GB367
RAM, where each execution was run on a single processor.368
Instances and best known solutions can be found at https://github.com/frperezga/MinmaxRegretPath
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6 Exact Results and Analysis369
We know of four papers that propose exact algorithms and conduct experiments for MMR-P. In370
[32], according to the authors, outperformed previous approaches by the same group of researchers371
[33, 31], therefore we focus on the first paper. Other experimental research appears in a chapter of372
the book [23]. A general drawback of the experiments conducted using these approaches is the size of373
the instances tested. Only instances with small sizes were tested and then was very difficult to outline374
some conclusions. Even so, in [32] the performance of the algorithms was analyzed when applied on375
random instances, Layered instances and three instances from real road networks, and the authors376
concluded that Benders approach had a better performance than a branch and bound algorithm and377
a MILP formulation given in [22] and implemented by CPLEX. Very recently, [18] proposed a B&C378
procedure which considers an improved lower bound for the problem. They considers several classes379
of graph instances, including two real large size instances.380
Group L1. Our effort in this paper is to try to gain more information about the performance of381
algorithms when applied to instances of both greater size and different structure. In the case of the382
group L1 of Layered instances, Table 2 shows the results of MILP considering a time limit of 900383
seconds. It is clear that from 4 000 nodes and up, the algorithm’s performance degrades dramatically,384
so that for 5 000 nodes no optimum solution was achieved and worse yet, no feasible solutions were385
found. For the same group of instances, B&C algorithm was always able to find optimal solutions386
in no more than 250 seconds on average over ten runs, except for n =10 000 where the algorithm387
begins to be affected by the combinatorial explosion.388
Group L2. In Table 3 and Table 4 the performance of MILP and B&C algorithms for the second389
group of instances L2 is illustrated. These instances contain 250, 500, 1 000 and 2 000 nodes and each390
one contains two, four and six layers. In Table 3 is shown that MILP is able to get optimal solutions391
for all combinations of number of nodes when the number of layers is equal to six. However, its392
performance clearly diminished when the number of nodes increased and the number of layers is two393
or four. For example, for 2 000 nodes and two layers, MILP achieved 8% gap on average. In Table 4394
is shown that the performance of B&C is clearly inferior to MILP, achieving large gaps (about 30%)395
for 250 nodes and two layers. Clearly MILP outperforms B&C for this class of instances.396
In conclusion, after the experimentation with the exact algorithms MILP and B&C applied to397
Layered instances, the group L1 of large instances can be rapidly solved by B&C. With respect to398
group L2, the performance of MILP is better than B&C but loses efficiency from 1 000 nodes and399
two layers. It is clear that heuristic approaches are necessary for solving the large size L2 instances.400
Group G. MILP provides better solutions than B&C. However, as the size of the instances is401
increased, gaps also increase (see Table 1). For two combinations of the parameters m and n, both402
exact algorithms generate high gaps. It is also noted that the time limit was exhausted for the403
instances. Considering that the size of these instances is relatively small, it is clear that heuristics404
are necessary for solving large instances with this structure.405
Table 1: Running times and gaps for B&C and MILP in G instances. n and m represent the rows and columns
in the grid.
class gap (%) time (sec.)
n m min av max min av max
B&C
2 20 0 0 0 0.02 0.03 0.05
2 40 0 0 0 0.02 0.03 0.05
2 80 0 0 0 0.19 0.52 0.77
2 160 0 5.32 13.91 412.00 818.47 900.16
2 320 26.32 32.13 36.89 900.05 900.12 900.20
4 40 0 0 0 0.062 0.089 0.141
8 80 0 0 0 1.16 2.33 4.25
16 160 0 0 0 10.16 33.69 65,36
32 320 3.80 7.00 14.50 900.20 900.90 900.90
MILP
2 20 0 0 0 0.03 0.04 0.06
2 40 0 0 0 0.03 0.05 0.08
2 80 0 0 0 0.16 2.31 5.00
2 160 0 0 0 3.10 7.82 15.20
2 320 5.49 9.19 13.04 900.14 900.15 900.16
4 40 0 0 0 0.11 0.14 0.19
8 80 0 0 0 1.13 2.28 5.66
16 160 0 0 0 13.94 105.48 240.83
32 320 1.60 3.10 5.10 900.10 900.60 900.90
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Table 2: Running times and gaps for MILP and B&C in L1 instances. * very large gap (UB and / or LB very
low quality). n is the number of nodes, nk is the number of nodes in each layer, d manages the interval length
and #optimum is the number of instances that achieve the optimal solution.
instance gap (%) time (sec.) #optimum
n d nk min av. max min av. max
MILP
1 000 0.15 100 0 0 0 20.94 25.34 27.80 10
0.50 0 0 0 22.67 26.85 30.36 10
0.85 0 0 0 23.64 28.05 32.53 10
2 000 0.15 200 0 0 0 118.47 139.26 159.84 10
0.50 0 0 0 137.30 158.06 176.95 10
0.85 0 0 0 133.31 162.69 187.33 10
3 000 0.15 300 0 0 0 358.11 407.35 485.86 10
0.50 0 0 0 400.77 455.92 519.02 10
0.85 0 0 0 408.13 464.93 522.63 10
4 000 0.15 400 0 *(3) * 675.55 845.94 900.00 7
0.50 0 *(7) * 864.91 900.00 900.00 3
0.85 0 *(7) * 818.83 900.00 900.00 3
5 000 0.15 500 * *(10) * 900.00 900.00 900.00 0
0.50 * *(10) * 900.00 900.00 900.00 0
0.85 * *(10) * 900.00 900.00 900.00 0
6 000 0.15 600 * *(10) * 900.00 900.00 900.00 0
0.50 * *(10) * 900.00 900.00 900.00 0
0.85 * *(10) * 900.00 900.00 900.00 0
7 000 0.15 700 * *(10) * 900.00 900.00 900.00 0
0.50 * *(10) * 900.00 900.00 900.00 0
0.85 * *(10) * 900.00 900.00 900.00 0
10 000 0.15 1 000 * *(10) * 900.00 900.00 900.00 0
0.50 * *(10) * 900.00 900.00 900.00 0
0.85 * *(10) * 900.00 900.00 900.00 0
B&C
1 000 0.15 100 0 0 0 0.94 1.41 2.31 10
0.50 0 0 0 1.08 1.75 2.13 10
0.85 0 0 0 1.16 1.53 1.75 10
2 000 0.15 200 0 0 0 5.47 5.93 6.30 10
0.50 0 0 0 5.52 7.91 10.25 10
0.85 0 0 0 4.50 7.18 8.63 10
3 000 0.15 300 0 0 0 17.72 19.44 21.81 10
0.50 0 0 0 19.50 23.02 27.49 10
0.85 0 0 0 12.50 19.85 23.69 10
4 000 0.15 400 0 0 0 38.31 41.20 45.00 10
0.50 0 0 0 42.92 49.46 67.98 10
0.85 0 0 0 31.27 49.74 95.67 10
5 000 0.15 500 0 0 0 67.41 76.30 80.09 10
0.50 0 0 0 67.28 84.86 120.67 10
0.85 0 0 0 52.02 67.04 108.77 10
6 000 0.15 600 0 0 0 126.95 137.67 152.44 10
0.50 0 0 0 135.84 145.01 168.50 10
0.85 0 0 0 95.67 126.02 247.25 10
7 000 0.15 700 0 0 0 193.64 228.99 327.39 10
0.50 0 0 0 207.67 241.70 341.64 10
0.85 0 0 0 149.44 213.20 363.34 10
10 000 0.15 1 000 0 **(4) 0 570.89 693.81 860.27 6
0.50 0 **(3) 0 542.10 719.92 900.00 7
0.85 0 **(2) 0 386.67 508.32 900.00 8
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Table 3: Running times and gaps for MILP in L2 instances. n is the number of nodes, nk is the number of nodes
in each layer, d manages the interval length and #optimum is the number of instances that achieve the optimal
solution.
gap (%) time (sec.) #optimum
n nk p min av max min av max
250 2 0.15 0.01 0.01 0.01 1.95 11.95 34.48 10
0.50 0.01 0.01 0.01 2.83 15.67 58.66 10
0.85 0.00 0.01 0.01 2.28 30.65 126.11 10
4 0.15 0.00 0.00 0.01 0.42 1.33 3.13 10
0.50 0.00 0.00 0.01 0.45 1.56 3.20 10
0.85 0.00 0.00 0.01 0.38 1.38 2.81 10
6 0.15 0.00 0.00 0.00 0.27 0.43 0.81 10
0.50 0.00 0.00 0.00 0.28 0.43 0.64 10
0.85 0.00 0.00 0.00 0.30 0.43 0.64 10
500 2 0.15 0.62 2.03 3.39 900.08 900.10 900.11 0
0.50 0.56 2.38 3.26 900.08 900.10 900.25 0
0.85 0.90 2.69 3.80 900.06 900.09 900.11 0
4 0.15 0.00 0.00 0.01 4.91 6.52 9.58 10
0.50 0.00 0.01 0.01 4.77 7.27 14.64 10
0.85 0.00 0.01 0.01 5.27 6.88 12.56 10
6 0.15 0.00 0.00 0.00 1.06 3.27 6.36 10
0.50 0.00 0.00 0.00 1.14 3.25 6.44 10
0.85 0.00 0.00 0.00 1.02 3.10 6.78 10
1 000 2 0.15 4.30 5.26 6.40 900.23 900.25 900.30 0
0.50 4.66 5.80 6.68 900.23 900.25 900.27 0
0.85 5.23 6.05 7.38 900.23 900.25 900.27 0
4 0.15 0.01 0.06 0.51 46.44 284.59 900.28 9
0.50 0.01 0.03 0.26 40.03 372.91 900.28 9
0.85 0.01 0.12 0.62 59.02 397.96 900.30 8
6 0.15 0.00 0.00 0.00 13.64 18.85 23.81 10
0.50 0.00 0.00 0.00 13.58 19.61 23.97 10
0.85 0.00 0.00 0.00 17.19 19.73 24.73 10
2 000 2 0.15 6.43 7.45 7.96 900.86 901.02 901.50 0
0.50 7.24 7.98 8.85 900.83 900.88 900.98 0
0.85 7.49 8.31 9.31 900.86 900.97 900.38 0
4 0.15 0.62 1.55 2.18 900.86 901.19 902.61 0
0.50 0.95 1.65 2.14 900.88 900.91 900.99 0
0.85 0.90 1.56 1.96 900.88 900.97 901.33 0
6 0.15 0.00 0.00 0.00 58.81 183.86 303.00 10
0.50 0.00 0.00 0.00 56.20 357.61 901.00 7
0.85 0.00 0.00 0.00 69.38 517.14 901.09 5
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Table 4: Running times and gaps for B&C in L2 instances. n is the number of nodes, nk is the number of nodes
in each layer, d manages the interval length and #optimum is the number of instances that achieve the optimal
solution.
gap (%) time (sec.) #optimum
n nk p min av max min av max
250 2 0.15 24.36 27.66 31.49 900.02 900.07 900.16 0
0.50 24.74 27.59 30.74 900.03 900.13 900.63 0
0.85 24.55 27.83 31.70 900.03 900.06 900.11 0
4 0.15 0.00 0.01 0.01 3.67 206.71 717.99 10
0.50 0.00 0.15 1.40 5.17 275.76 900.06 10
0.85 0.00 0.19 1.86 4.27 270.76 900.06 10
6 0.15 0.00 0.00 0.01 0.36 1.34 4.16 10
0.50 0.00 0.00 0.01 0.64 1.33 3.19 10
0.85 0.00 0.00 0.00 0.55 1.34 2.45 10
500 2 0.15 33.82 36.10 38.24 900.05 900.10 900.14 0
0.50 33.97 35.60 37.25 900.03 900.07 900.14 0
0.85 33.89 35.72 37.23 900.03 900.12 900.30 0
4 0.15 7.06 9.88 12.57 900.03 900.07 900.14 0
0.50 6.48 10.08 13.08 900.03 900.07 900.23 0
0.85 7.05 10.71 14.13 900.05 900.11 900.33 0
6 0.15 0.00 0.01 0.01 10.34 156.85 524.13 10
0.50 0.01 0.01 0.01 8.36 151.48 522.14 10
0.85 0.01 0.01 0.01 9.52 183.92 744.63 10
1 000 2 0.15 35.50 36.85 37.77 900.06 900.10 900.16 0
0.50 35.63 37.39 38.56 900.06 900.08 900.13 0
0.85 35.03 37.18 37.12 900.00 900.00 900.00 0
4 0.15 17.43 19.69 22.96 900.05 900.08 900.17 0
0.50 24.36 27.66 31.49 900.02 900.07 900.16 0
0.85 18.56 20.37 24.94 900.00 900.00 900.00 0
6 0.15 3.81 5.57 7.37 900.05 900.08 900.16 0
0.50 4.74 5.82 7.51 900.06 900.10 900.16 0
0.85 4.06 6.84 8.73 900.00 900.00 900.00 0
2 000 2 0.15 36.55 37.61 43.15 900.00 900.00 900.00 0
0.50 36.46 38.87 42.94 900.00 900.00 900.00 0
0.85 36.15 39.03 43.06 900.00 900.00 900.00 0
4 0.15 22.21 24.72 28.30 900.00 900.00 900.00 0
0.50 22.89 25.82 28.78 900.00 900.00 900.00 0
0.85 22.22 25.32 28.38 900.00 900.00 900.00 0
6 0.15 8.27 12.37 15.59 900.00 900.00 900.00 0
0.50 9.27 11.59 13.42 900.00 900.00 900.00 0
0.85 9.25 12.35 13.81 900.00 900.00 900.00 0
7 Performance of the Heuristic Approaches406
Taking into account the conclusion related to hard instances in both topologies (Layered and407
Grid), we have considered appropriate to apply heuristics only to hard instances. Specifically, we408
consider six groups of L2 instances and two groups of G instances (shown in bold in tables 1, 3 and409
4). Our heuristic approaches are based on the neighborhood (Nγ) defined in Subsection 4.4, Nγ410
is embedded in two SA settings and in a local search setting, both metaheuristic frameworks were411
explained in Section 4. Additionally, as pointed out in Subsection 4.4, a SA approach using the412
neighborhood Nk-opt based on the traditional heuristic k-opt was implemented here using k = 2 and413
k = 3.414
7.1 Algorithm parameters and measure of performance415
An important drawback of metaheuristic approaches is the step related to the selection of the416
best set of parameters. This task can be time-consuming and it is always necessary to deal with the417
tradeoff between time and solution quality. Good discusions can be found in [13, 1] and [7].418
The selected parameters were obtained through a mixed process based on a brute-force search419
over a grid and a trial-and-error procedure. The search over the grid allows a good exploration420
in the parameter space and trial-and-error was used in order to intensify the search near good421
solutions. After the experiments, we defined the settings shown in Table 5. Note that we chose one422
configuration for Nk-opt and three configurations for Nγ in order to represent the trade-off between423
time-consumption and solution quality in our neighborhood. In the case of SA using Nk-opt, more424
demanding parameters were tested but the results had a very marginal improvement.425
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Table 5: Parameters selected for heuristic algorithms. ti is the initial temperature, tf is the final temperature
and N is the neighborhood structure for each metaheuristic.
Algorithm id ti tf cooling factor loops N
Simulated Annealing SA0 50 0.1 0.9 800 Nk-opt
Simulated Annealing SA1 5 0.01 0.9 800 Nγ
Simulated Annealing SA2 5 0.1 0.88 500 Nγ
Local Search LS - - - 20 000 Nγ
The parameter γ must be regulated depending on the density, size and topology of the graph.426
The selection must consider the trade-off between exploration and the probability of obtaining a427
disconnected graph. We have estimated γ according to γ ≈ k
|A|
, where |A| is the total number of428
arcs in G and k ∈ [2, 10] is the number of modified edges in each iteration. Table 6 shows the final429
value of γ in each group of instances.430
Table 6: Selected values for the parameter γ, considering different groups of instances.
Group γ Group γ
L2 - 1 000 0.004 G - 2 - 320 0.004
L2 - 2 000 0.001 G - 32 - 320 0.0001
To measure performance, we use basic statistics (minimum, average and maximum) for the gaps431
and execution times from 50 runs for each instance. The results presented for the gaps are relative432
to the best solution found by the best exact algorithm in each instance ((S − Sbest) /Sbest).433
7.2 Performance comparison of the algorithms434
As we mentioned above, few papers have tackled the MMR-P problem using heuristics, therefore435
ad-hoc neighborhood structures that consider the nested structure in the problem formulation MMR-436
Path defined in Subsection 2.1 do not exist. As a natural strategy we use the neighborhood (Nk-opt)437
mentioned in Subsection 4.4 in a SA scheme (SA0 algorithm). This implementation had a better438
performance than another approach based on Ant Colony Optimization algorithm (ACO) that we439
designed for the problem. So, ACO was discarded and SA0 was compared with the heuristic HMU,440
since the literature has shown that it obtains moderate gaps for several classes of MMR-P instances441
and it is a fast algorithm that only needs to solve four classic problems [23, 41].442
As detailed in Table 7, HMU achieved gaps between 2.37% and 4.33% for most L2 instances.443
However, in G instances its performance is irregular. In the G-2-320 instances, the gaps are 11.47%444
on average and in the other group of instances, G-32-320, they do not exceed 1.53%. To the best of445
our knowledge, the performance of HMU over the G-2-320 instances is its worst performance over446
all classes of instances reported in the literature. SA0+2-opt and SA0+3-opt outperform HMU in447
the majority of L2 instances and SA0+3-opt outperforms SA0+2-opt in most of the L2 instances448
(except the last) but it achieves worse gaps in G instances. Note that for instances with smaller449
interval (d = 0.15) the performance of SA0+2-opt is worse. For detailed results, see the Tables 9 10450
12 11 in Appendix 10.451
In summary, k-opt neighborhood in SA framework obtained interesting results, it is able to452
improve the solutions reached by HMU heuristics in the majority of instances.453
Regarding run times, in Table 7, we highlight the difference observed between the two classes454
of G instances. Both variants of SA0 took much more run time in instances G-32-320 than the455
instances G-2-320. This is due to the difficulty in rebuilding a path in G-32-320 class using the k-opt456
framework.457
From the previous analysis it is clear that SA0 (using both variants) outperforms HMU but over458
most instances it does not reach the best known solutions BKS (they can be accessed in the link at459
the footnote of page 9). Therefore the task of the SA approach using the new neighborhood Nγ is460
to compete with the BKS values. In this context the performance of the LS and SA using a set of461
different parameters is analyzed (SA1 and SA2). The objective in including the performance of LS462
using the proposed neighborhood is to analyze to what extent the mechanism of SA to escape from463
the local optimum found in LS is effective.464
Table 8 shows the results of LS and SA approaches using Nγ. LS clearly achieved better gaps465
than HMU and SA0 for all instances, running at similar times to SA0. From the same Table, it is466
clear that, respect to L2 instances, SA1 and SA2 outperform LS noting that SA2 is able to obtain467
better results than LS in less time. Additionally, it can be also noted that the performance of SA1 is468
slightly better than SA2 as it was expected since the parameters used by SA1 are computationally469
more expensive than those used by SA2. These results are detailed in Tables 16 17 and 18. For470
example, in L2 instances with 2 000 nodes, the statistics related to gap (minimum, average and471
maximum) are 0.76, 1.06, 1.39 for LS and 0.71, 0.93, 1.22 for SA2. At the same time, when the472
variant SA1 is applied, more run time is necessary, but the results are better than the obtained by473
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Table 7: Gaps (%) and running times obtained by SA0 and HMU for each class of instances. Each class contains
10 instances and we run 50 experiments for each one in SA0 approach.
SA0+2opt SA0+3opt HMU
Class min av max min av max min av max
gap (%)
L2 - 1 000 - 0.15 1.45 2.97 3.99 0.41 1.37 2.28 2.97 3.51 4.03
L2 - 1 000 - 0.50 0.54 1.52 2.80 0.35 1.17 2.13 2.70 3.19 3.83
L2 - 1 000 - 0.85 0.34 1.11 2.13 0.24 1.16 2.17 2.74 3.24 3.88
L2 - 2 000 - 0.15 3.06 3.53 4.33 0.86 1.65 2.32 3.06 3.54 4.33
L2 - 2 000 - 0.50 1.22 2.08 3.19 0.73 1.46 2.10 2.78 3.28 4.19
L2 - 2 000 - 0.85 0.50 1.30 2.16 0.66 1.29 1.93 2.37 3.19 4.00
G - 2 - 320 1.92 8.68 15.15 6.27 11.47 15.04 6.70 11.57 15.20
G - 32 - 320 0.00 0.53 1.53 -0.18 0.39 1.53 0.00 0.53 1.53
time (seconds)
L2 - 1 000 - 0.15 36.63 37.81 40.08 36.61 37.65 39.62
L2 - 1 000 - 0.50 36.47 37.14 38.42 36.63 37.48 39.24
L2 - 1 000 - 0.85 36.36 36.82 37.56 36.64 39.69 39.58
L2 - 2 000 - 0.15 73.42 74.56 77.97 73.38 74.18 75.69
L2 - 2 000 - 0.50 73.58 75.60 78.89 73.30 74.75 77.50
L2 - 2 000 - 0.85 70.44 71.52 88.33 73.58 75.41 78.74
G - 2 - 320 30.31 32.04 34.47 30.22 31.28 33.08
G - 32 - 320 776.77 894.10 932.64 710.54 889.89 938.68
SA2. These results confirm the effectivity of SA using Nγ when a group of difficult instances is474
investigated.475
The performance of heuristics applied to G instances is very different depending on the type of476
the instances used, G-2-320 or G-32-320. LS, SA1 and SA2 are not able to improve the quality of477
the solutions provided by exact algorithms nor the quality of the solutions provided by HMU for478
the instances (32,320). Considering that the best gap is 1.53% from MILP, these instances could be479
well solved for the corresponding size.480
The situation for the G-2-320 instances is different. The heuristics are able to largely improve the481
gaps of HMU and SA0 and are almost able to equal the best known value of the exact algorithms.482
In particular, SA1 is able, in one instance, to improve the solution given by exact approaches. It is483
clear that HMU finds solutions with large gaps, over 15% in some instances. Considering that the484
best gap from MILP is 5%, these instances tend to be difficult to solve when the size of the instances485
increases.486
As previously mentioned, two versions with different parameters of SA algorithm were tested487
with our novel neigbourhood. The degradation in the quality of the obtained solutions when more488
relaxed parameters were considered was small but significant. This allows the priorization of either489
time or quality of the solution. However, even the more relaxed version of the Simulated Annealing490
algorithm found better solutions than the implemented Local Search. For detailed results, see the491
tables 13 14 15 16 17 and 18 in Appendix 10.492
Table 8: Gaps (%) and running times obtained by LS, SA1 and SA2 for each class of instances. Each class
contains 10 instances and considers 50 runs.
LS SA1 SA2
class min av max min av max min av max
gap (%)
L2 - 1 000 - 0.15 0.07 1.70 3.56 0.00 1.05 1.53 0.00 1.35 3.56
L2 - 1 000 - 0.50 0.02 1.38 3.3 -0.04 0.93 3.31 -0.04 1.11 3.31
L2 - 1 000 - 0.85 0.00 1.44 3.54 0.00 1.11 3.54 0.00 1.24 3.54
L2 - 2 000 - 0.15 0.00 0.97 3.39 -0.07 0.64 3.39 0.01 0.83 3.39
L2 - 2 000 - 0.50 0.04 1.15 3.10 -0.11 0.88 3.10 -0.05 1.01 3.10
L2 - 2 000 - 0.85 -0.43 1.07 3.08 -0.45 0.83 3.08 -0.44 0.96 3.14
G - 2 - 320 -0.05 2.15 7.72 -0.12 1.62 6.97 0.00 2.23 8.18
G - 32 - 320 0.00 0.53 1.53 0.00 0.53 1.53 0.00 0.53 1.53
time (seconds)
L2 - 1 000 - 0.15 33.05 36.05 41.56 81.34 85.82 97.66 26.39 27.43 29.33
L2 - 1 000 - 0.50 34.03 35.94 39.63 80.74 85.00 89.70 26.44 27.61 30.28
L2 - 1 000 - 0.85 34.11 35.59 38.20 81.03 84.20 89.39 26.53 27.59 29.20
L2 - 2 000 - 0.15 69.97 72.96 76.57 167.48 175.00 190.40 54.99 56.97 61.22
L2 - 2 000 - 0.50 71.33 72.82 76.08 162.89 173.68 182.99 54.88 56.55 58.34
L2 - 2 000 - 0.85 70.42 72.65 79.83 157.30 163.90 178.73 55.33 57.20 60.12
G - 2 - 320 19.97 36.93 39.52 85.47 89.24 93.52 27.17 19.11 30.83
G - 32 - 320 425.49 537.62 690.44 418.83 439.19 579.79 203.92 227.89 273.21
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8 Conclusions and final comments493
Both exact and heuristic algorithms were proposed for solving the MMR-P problem, a NP-Hard494
combinatorial optimization problem with uncertainty. The problem has been used as an effective way495
to formulate a version of the very known shortest path problem in a network when the arc weights496
are not completely known.497
A B&C exact algorithm has been proposed here for solving MMR-P. A broad set of instances498
from telecommunication networks, the Layered instances, whose size range from 100 to 10 000 nodes499
were analyzed. The algorithm has proven to outperform another traditional exact approach based500
on a MILP formulation and implemented by the CPLEX solver when applied to the set of Layered501
instances. Additionally, a class of Layered networks with special structure is investigated because502
exact algorithms have great difficulty finding their exact solutions. For these instances MILP outper-503
formed the B&C approach. However, the MILP approach loses efficiency as the size of the instance504
grows.505
Another class of test instances was introduced for the problem in our research, the Grid instances,506
which resembles road networks. For these networks, MILP approach outperformed B&C approach507
but is unable to solve instances with more than 5 000 nodes.508
A new and sophisticated neighborhood was designed for MMR-P and Local Search and Simulated509
Annealing algorithms based on this neighborhood were proposed. These heuristics were able to510
outperform a traditional basic heuristic, HMU, a metaheuristic ACO and another SA approach511
using the neighborhood k-opt, when they were tested on the sets of instances considered. More512
important, the Simulated Annealing algorithm was able to obtain feasible solutions with a similar513
quality to the solutions found by the two developed exact algorithms for the Grid instances. For514
larger Grid instances, both exact algorithms generate larger gaps or are unable to obtain feasible515
solutions in reasonable times. In this context, Simulated Annealing was able to find good feasible516
solutions in relatively short times. Since the SP problem and its variants have many important517
applications in several fields, the study of new efficient heuristics for large instances is necessary.518
Future research should consider to exploit the novel neighborhood applying it to different MMR519
Problems.520
9 Acknowledgements521
Alfredo Candia-Véjar was supported by CONICYT, FONDECYT project N◦
1121095.522
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10 Appendix635
Table 9: Running times and gaps for Simulated Annealing considering SA0 parameter and 2-opt neigbourhood
in G instances
n m #
gap(%) time (sec.)
HMUmin av max min av max
2 320 0 3.13 10.07 15.04 30.68 32.09 32.92 15.04
2 320 1 6.37 10.31 13.08 31.42 32.04 32.86 13.50
2 320 2 4.20 7.30 10.29 30.31 31.46 31.86 10.29
2 320 3 3.24 7.33 8.16 31.31 31.6 32.2 8.16
2 320 4 6.17 9.81 10.58 31.52 32.02 33.31 10.58
2 320 5 3.73 10.70 13.74 31.44 31.76 32.22 13.74
2 320 6 1.92 5.74 6.72 31.48 31.81 32.34 6.72
2 320 7 2.20 6.85 9.74 31.58 32.22 33.53 9.74
2 320 8 2.23 8.20 15.15 31.52 32.68 34.24 15.15
2 320 9 7.49 10.52 12.82 31.7 32.68 34.47 12.82
¯x 4.07 8.68 11.53 31.30 32.04 33.00 11.60
32 320 0 1.53 1.53 1.53 896.62 910.8 927.51 1.53
32 320 1 0.72 1.02 1.05 898.16 906.56 914.49 1.02
32 320 2 0.13 0.49 0.50 888.18 900.92 917.93 0.50
32 320 3 0.00 0.00 0.00 879.52 892.47 913.02 0.00
32 320 4 0.02 0.02 0.02 899.52 913.54 932.64 0.02
32 320 5 1.14 1.14 1.14 887.7 904.76 918.02 1.14
32 320 6 0.30 0.30 0.30 891.9 909.44 925.54 0.30
32 320 7 0.40 0.66 0.68 886.23 900.94 915.43 0.68
32 320 8 0.00 0.00 0.00 884.54 898.33 913.98 0.00
32 320 9 0.12 0.12 0.12 776.77 803.21 910.18 0.12
¯x 0.44 0.53 0.54 878.91 894.10 918.87 0.50
Table 10: Running times and gaps for Simulated Annealing considering SA0 parameter and 3-opt neigbourhood
in G instances
n m #
gap(%) time (sec.)
min av max min av max
2 32 0 15.04 15.04 15.04 31.5 32.1 32.91
2 32 1 13.08 13.08 13.08 30.26 31.79 32.72
2 32 2 10.29 10.29 10.29 30.25 30.31 30.42
2 32 3 8.16 8.16 8.16 30.25 30.61 31.55
2 32 4 10.58 10.58 10.58 30.28 30.63 31.56
2 32 5 13.74 13.74 13.74 30.22 30.96 31.84
2 32 6 6.27 6.40 6.40 31.38 31.67 32.24
2 32 7 9.74 9.74 9.74 31.42 31.95 33.08
2 32 8 14.86 14.86 14.86 31.33 31.71 32.16
2 32 9 12.82 12.82 12.82 30.3 31.09 32.12
¯x 11.46 11.47 11.47 30.72 31.28 32.06
32 320 0 1.53 1.53 1.53 710.54 758.06 929.12
32 320 1 0.43 0.53 0.70 896.01 912.94 938.68
32 320 2 -0.12 0.07 0.13 888.93 909.13 919.4
32 320 3 0.00 0.00 0.00 877.52 894.36 911.28
32 320 4 -0.18 -0.13 0.02 881.04 897.88 917.06
32 320 5 1.07 1.14 1.14 894.51 911.62 929.2
32 320 6 0.30 0.30 0.30 890.59 907.34 924.43
32 320 7 0.40 0.40 0.40 889.29 908.33 928.76
32 320 8 0.00 0.00 0.00 885.76 898.22 919.31
32 320 9 -0.02 0.08 0.12 887.29 901.05 917.87
¯x 0.34 0.39 0.43 870.15 889.89 923.51
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Table 11: Running times and gaps for SA considering SA0 parameters and 2-opt neigbourhood in L2 instances.
n is the number of nodes, d manages the interval length and i is an identiﬁer for each instance.
gap(%) time (sec.)
n d i min av max min av max HMU
1 000 0.15 0 1.97 2.93 3.14 36.69 37.62 38.92 3.14
1 000 0.15 1 1.80 2.82 3.99 36.63 37.77 39.3 4.03
1 000 0.15 2 2.08 3.20 3.43 37.02 37.73 39.48 3.43
1 000 0.15 3 1.57 2.77 2.97 36.74 37.82 39.38 2.97
1 000 0.15 4 2.00 2.94 3.56 36.92 37.83 39.64 3.56
1 000 0.15 5 1.82 2.63 3.84 36.81 37.97 39.39 3.84
1 000 0.15 6 2.23 3.33 3.74 36.8 38.01 39.27 3.74
1 000 0.15 7 2.23 3.29 3.60 36.77 38.01 40.08 3.60
1 000 0.15 8 1.82 2.89 3.22 36.7 37.83 40.03 3.22
1 000 0.15 9 1.45 2.89 3.55 36.66 37.54 38.56 3.55
1 000 0.50 0 0.79 1.47 2.52 36.75 37.4 38.42 3.09
1 000 0.50 1 0.54 1.17 2.11 36.67 37.26 38.19 3.53
1 000 0.50 2 0.86 1.54 2.16 36.67 37.23 38.09 3.31
1 000 0.50 3 0.72 1.25 1.92 36.67 37.12 37.92 2.74
1 000 0.50 4 0.92 1.49 2.46 36.66 37.09 37.95 3.03
1 000 0.50 5 0.82 1.50 2.41 36.64 37.14 38.02 3.31
1 000 0.50 6 1.29 1.97 2.80 36.47 37.07 37.98 3.83
1 000 0.50 7 0.98 1.87 2.73 36.63 37.07 37.78 3.27
1 000 0.50 8 0.63 1.19 1.87 36.58 37 37.67 2.70
1 000 0.50 9 1.12 1.78 2.73 36.55 36.97 37.56 3.04
1 000 0.85 0 0.46 1.06 1.91 36.47 36.94 37.56 3.11
1 000 0.85 1 0.34 0.81 1.46 36.56 36.9 37.48 3.88
1 000 0.85 2 0.57 1.08 1.63 36.55 36.88 37.42 3.54
1 000 0.85 3 0.43 0.82 1.28 36.5 36.87 37.39 2.74
1 000 0.85 4 0.64 1.18 1.75 36.52 36.87 37.5 3.08
1 000 0.85 5 0.86 1.26 1.93 36.44 36.83 37.56 3.45
1 000 0.85 6 0.66 1.28 2.13 36.5 36.82 37.31 3.52
1 000 0.85 7 0.51 1.39 2.13 36.45 36.79 37.08 3.24
1 000 0.85 8 0.36 0.91 1.46 36.5 36.82 37.39 2.83
1 000 0.85 9 0.83 1.34 1.86 36.36 36.8 37.27 3.01
¯x 1.11 1.87 2.54 36.63 37.27 38.25 3.31
2 000 0.15 0 3.46 3.53 3.53 73.44 74 74.86 3.53
2 000 0.15 1 3.06 3.06 3.06 73.56 74.19 75.3 3.06
2 000 0.15 2 3.72 3.75 3.75 73.42 74.4 75.53 3.75
2 000 0.15 3 3.54 3.54 3.54 73.44 74.28 75.42 3.54
2 000 0.15 4 3.39 3.39 3.39 73.44 74.3 75.63 3.39
2 000 0.15 5 3.71 4.30 4.33 73.47 74.43 75.66 4.33
2 000 0.15 6 3.99 4.03 4.03 73.69 74.48 75.5 4.03
2 000 0.15 7 3.37 3.37 3.37 73.98 74.81 77.02 3.37
2 000 0.15 8 3.21 3.23 3.23 73.81 75.22 77.74 3.23
2 000 0.15 9 3.12 3.12 3.12 73.84 75.49 77.94 3.12
2 000 0.50 0 1.22 1.96 2.73 74.19 75.91 78.5 3.26
2 000 0.50 1 1.36 1.98 2.84 74.48 76.1 78.7 2.94
2 000 0.50 2 1.58 2.06 2.79 74.89 76.42 78.38 3.29
2 000 0.50 3 1.65 2.36 3.19 74.13 76.28 78.89 3.47
2 000 0.50 4 1.56 2.20 2.86 74.11 75.84 77.83 2.92
2 000 0.50 5 1.52 2.01 2.52 74.02 75.57 77.63 4.19
2 000 0.50 6 1.30 1.89 2.71 73.66 75.28 77.6 3.79
2 000 0.50 7 1.49 2.07 2.83 73.67 75.04 77.19 3.10
2 000 0.50 8 1.34 2.09 2.78 73.58 74.83 76.28 2.78
2 000 0.50 9 1.34 2.14 3.01 73.58 74.73 76.06 3.01
2 000 0.85 0 0.85 1.25 1.67 73.5 74.62 75.91 3.40
2 000 0.85 1 0.82 1.18 1.69 70.63 74.04 88.33 3.07
2 000 0.85 2 1.15 1.49 2.03 70.58 70.93 72.88 3.38
2 000 0.85 3 1.09 1.47 2.07 70.53 70.79 71.11 3.23
2 000 0.85 4 0.87 1.29 1.86 70.52 70.81 71.08 2.80
2 000 0.85 5 1.22 1.52 1.93 70.52 70.84 71.14 4.00
2 000 0.85 6 0.71 1.14 1.55 70.48 70.8 71.16 3.77
2 000 0.85 7 1.06 1.45 2.16 70.5 70.81 70.97 3.08
2 000 0.85 8 0.50 0.89 1.27 70.55 70.81 71.1 2.37
2 000 0.85 9 0.88 1.28 1.68 70.44 70.78 71.08 2.81
¯x 1.94 2.30 2.72 72.82 73.89 75.75 3.33
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Table 12: Running times and gaps for SA considering SA0 parameters and 3-opt neigbourhood in L2 instances.
n is the number of nodes, d manages the interval length and i is an identiﬁer for each instance.
gap (%) time (sec.)
n d i min av max min av max
1 000 0.15 0 1.04 1.50 1.93 36.75 37.65 39.20
1 000 0.15 1 0.91 1.17 1.66 36.75 37.69 39.25
1 000 0.15 2 0.70 0.97 1.44 36.66 37.70 39.28
1 000 0.15 3 0.41 0.97 1.48 36.77 37.58 39.36
1 000 0.15 4 1.17 1.57 1.89 36.72 37.74 39.06
1 000 0.15 5 1.20 1.77 2.28 36.66 37.69 39.00
1 000 0.15 6 0.79 1.13 1.63 36.61 37.62 39.30
1 000 0.15 7 1.37 1.72 2.13 36.77 37.61 38.91
1 000 0.15 8 0.73 1.11 1.40 36.69 37.57 38.97
1 000 0.15 9 1.39 1.75 2.23 36.70 37.60 39.62
1 000 0.50 0 1.21 1.50 1.78 36.64 37.48 38.95
1 000 0.50 1 0.44 0.64 1.03 36.75 37.49 39.00
1 000 0.50 2 0.48 0.69 0.86 36.77 37.52 38.84
1 000 0.50 3 0.35 0.75 1.13 36.75 37.36 38.72
1 000 0.50 4 0.93 1.18 1.46 36.74 37.29 38.56
1 000 0.50 5 1.46 1.86 2.13 36.72 37.22 38.24
1 000 0.50 6 1.00 1.27 1.75 36.74 37.43 38.55
1 000 0.50 7 1.35 1.57 1.76 36.63 37.59 39.03
1 000 0.50 8 0.62 0.79 1.00 36.7 37.71 39.09
1 000 0.50 9 1.18 1.42 1.61 36.74 37.72 39.24
1 000 0.85 0 1.09 1.37 1.70 36.88 37.77 39.02
1 000 0.85 1 0.48 0.83 1.30 36.67 37.72 38.95
1 000 0.85 2 0.51 0.82 1.03 36.91 37.97 39.27
1 000 0.85 3 0.24 0.53 0.89 37.05 37.96 39.23
1 000 0.85 4 0.96 1.14 1.33 36.70 37.95 39.58
1 000 0.85 5 1.70 1.89 2.17 36.97 38.01 39.53
1 000 0.85 6 0.87 1.12 1.58 36.64 37.51 38.22
1 000 0.85 7 1.39 1.64 1.90 36.94 37.48 38.20
1 000 0.85 8 0.72 0.92 1.24 36.67 37.29 38.06
1 000 0.85 9 1.26 1.35 1.48 36.67 37.23 37.92
¯x 0.93 1.23 1.57 36.75 37.61 38.94
2 000 0.15 0 1.43 1.80 2.19 73.84 74.60 75.55
2 000 0.15 1 0.86 1.16 1.41 73.86 74.53 75.69
2 000 0.15 2 1.48 1.82 2.12 73.8 74.44 75.17
2 000 0.15 3 1.39 1.68 2.08 73.66 74.27 75.08
2 000 0.15 4 1.46 1.70 2.00 73.66 74.13 74.86
2 000 0.15 5 1.61 1.93 2.32 73.48 74.09 74.88
2 000 0.15 6 1.18 1.39 1.80 73.38 73.97 74.64
2 000 0.15 7 1.20 1.65 2.06 73.56 73.94 74.61
2 000 0.15 8 1.44 1.72 2.04 73.42 73.91 74.52
2 000 0.15 9 1.30 1.62 1.93 73.41 73.89 74.84
2 000 0.50 0 1.12 1.38 1.58 73.52 74.28 75.63
2 000 0.50 1 0.73 0.95 1.17 73.3 74.23 75.31
2 000 0.50 2 1.47 1.64 1.75 73.56 74.22 75.31
2 000 0.50 3 1.31 1.52 1.68 73.41 74.24 75.39
2 000 0.50 4 1.17 1.39 1.65 73.6 74.40 75.64
2 000 0.50 5 1.48 1.83 2.10 73.88 74.54 75.27
2 000 0.50 6 1.11 1.22 1.42 73.7 74.86 76.48
2 000 0.50 7 1.23 1.55 1.84 74.08 75.20 76.92
2 000 0.50 8 1.24 1.55 1.88 74.31 75.64 77.13
2 000 0.50 9 1.29 1.52 1.77 74.24 75.89 77.50
2 000 0.85 0 1.17 1.39 1.61 74.27 76.24 78.05
2 000 0.85 1 0.66 0.90 1.14 74.58 76.32 78.16
2 000 0.85 2 1.28 1.53 1.77 74.61 76.25 78.74
2 000 0.85 3 1.19 1.35 1.56 74.06 75.60 77.28
2 000 0.85 4 0.95 1.20 1.49 74.03 75.54 76.94
2 000 0.85 5 1.32 1.66 1.93 73.92 75.21 77.05
2 000 0.85 6 0.96 1.10 1.24 73.61 74.95 76.17
2 000 0.85 7 1.05 1.50 1.77 73.75 74.74 75.98
2 000 0.85 8 0.85 1.05 1.21 73.66 74.67 75.94
2 000 0.85 9 0.95 1.25 1.55 73.58 74.62 75.64
¯x 1.19 1.46 1.73 73.79 74.78 76.01
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Table 13: Running times and gaps for Local Search considering LS parameter in G instances
n m #
gap (%) time (sec.)
2 320 0 0.10 2.03 5.35 29.97 36.11 39.00
2 320 1 1.13 3.55 7.72 35.42 37.28 39.52
2 320 2 0.14 1.48 3.98 35.48 37.17 39.39
2 320 3 0.20 1.72 4.21 35.50 37.02 38.27
2 320 4 -0.05 1.66 4.49 35.05 36.97 38.55
2 320 5 0.72 2.60 7.39 35.53 37.00 38.94
2 320 6 0.05 1.13 3.57 35.33 36.82 38.55
2 320 7 0.58 2.32 4.80 35.67 36.94 38.88
2 320 8 0.56 2.92 7.08 35.56 37.05 38.75
2 320 9 0.27 2.12 5.79 35.81 36.98 38.06
¯x 0.37 2.15 5.44 34.93 36.93 38.79
32 320 0 1.53 1.53 1.53 524.62 543.43 690.44
32 320 1 1.05 1.05 1.05 527.66 546.10 570.54
32 320 2 0.50 0.50 0.50 530.74 542.35 560.60
32 320 3 0.00 0.00 0.00 525.15 544.01 568.63
32 320 4 0.02 0.02 0.02 425.49 514.53 562.55
32 320 5 1.14 1.14 1.14 527.57 539.22 558.19
32 320 6 0.30 0.30 0.30 519.00 539.68 564.13
32 320 7 0.68 0.68 0.68 522.90 535.56 557.27
32 320 8 0.00 0.00 0.00 523.02 533.74 560.29
32 320 9 0.12 0.12 0.12 517.04 537.62 562.82
¯x 0.54 0.54 0.54 514.32 537.62 575.55
Table 14: Running times and gaps for Simulated Annealing considering SA1 parameters in G instances
n m #
gap (%) time (sec.)
2 320 0 0.30 1.46 5.06 87.30 89.33 91.58
2 320 1 0.33 2.32 5.72 87.17 89.13 91.06
2 320 2 0.00 1.22 3.61 86.44 88.62 91.41
2 320 3 0.00 1.52 6.97 86.38 88.92 91.45
2 320 4 -0.12 1.53 5.21 86.38 89.13 92.05
2 320 5 0.09 1.81 4.76 85.77 89.07 91.80
2 320 6 0.01 1.02 2.83 85.47 87.80 90.52
2 320 7 0.52 2.11 5.64 87.13 89.71 91.77
2 320 8 0.11 1.66 5.02 88.58 90.50 92.83
2 320 9 0.00 1.59 5.30 87.78 90.20 93.52
¯x 0.12 1.62 5.01 86.84 89.24 91.80
32 320 0 1.53 1.53 1.53 426.19 437.98 453.05
32 320 1 0.78 1.03 1.05 426.82 443.25 463.80
32 320 2 0.50 0.50 0.50 433.61 447.09 464.36
32 320 3 0.00 0.00 0.00 433.52 445.93 458.96
32 320 4 0.02 0.02 0.02 421.72 435.41 459.54
32 320 5 1.14 1.14 1.14 425.99 441.10 458.79
32 320 6 0.30 0.30 0.30 418.83 431.42 455.00
32 320 7 0.68 0.68 0.68 421.77 430.09 455.68
32 320 8 0.00 0.00 0.00 421.32 437.69 457.27
32 320 9 0.12 0.12 0.12 421.93 441.90 579.79
0 ¯x 0.51 0.53 0.54 425.17 439.19 470.62
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Table 15: Running times and gaps for Simulated Annealing considering SA2 parameters in G instances
n m #
gap (%) time (sec.)
2 320 0 0.34 1.87 4.55 27.30 29.10 30.39
2 320 1 1.02 3.55 8.18 27.70 29.22 30.64
2 320 2 0.08 1.79 5.63 28.08 29.31 30.77
2 320 3 0.00 2.26 6.53 27.84 29.17 30.64
2 320 4 0.21 1.80 6.52 27.17 29.08 30.48
2 320 5 0.67 2.29 6.56 27.56 29.15 30.81
2 320 6 0.08 1.35 3.09 27.27 29.05 30.83
2 320 7 1.30 2.60 5.08 27.55 29.03 30.61
2 320 8 0.56 2.59 6.29 27.94 29.08 30.64
2 320 9 0.26 2.23 5.38 27.61 28.95 30.44
¯x 0.45 2.23 5.78 27.60 29.11 30.63
32 320 0 1.53 1.53 1.53 209.47 214.18 220.53
32 320 1 1.05 1.05 1.05 205.10 211.21 217.42
32 320 2 0.50 0.50 0.50 210.32 213.02 221.89
32 320 3 0.00 0.00 0.00 209.20 212.28 218.96
32 320 4 0.02 0.02 0.02 206.83 211.54 218.61
32 320 5 0.61 1.12 1.14 254.61 262.52 272.08
32 320 6 0.30 0.30 0.30 253.83 260.54 267.74
32 320 7 0.68 0.68 0.68 251.63 257.48 265.92
32 320 8 0.00 0.00 0.00 205.00 227.80 273.21
32 320 9 0.12 0.12 0.12 203.92 208.37 214.70
¯x 0.48 0.53 0.54 220.99 227.89 239.11
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Table 16: Running times and gaps for Local Search considering LS parameters in L2 instances. n is the number
of nodes, d manages the interval length and i is an identiﬁer for each instance.
gap(%) time (sec.)
n d i min av max min av max
1 000 0.15 0 0.21 0.49 0.97 33.05 35.86 41.56
1 000 0.15 1 0.16 0.55 1.08 34.20 36.13 39.41
1 000 0.15 2 3.43 3.43 3.43 34.08 36.14 40.20
1 000 0.15 3 2.97 2.97 2.97 34.17 36.03 38.04
1 000 0.15 4 0.32 3.09 3.56 34.03 36.06 39.84
1 000 0.15 5 0.28 0.70 1.48 34.17 36.19 39.39
1 000 0.15 6 0.07 0.49 1.34 34.12 36.05 38.33
1 000 0.15 7 0.18 0.58 1.33 34.25 36.18 38.64
1 000 0.15 8 0.30 1.29 3.22 34.08 35.85 39.27
1 000 0.15 9 0.81 3.45 3.55 34.02 36.05 39.78
1 000 0.50 0 0.19 0.50 0.88 34.53 36.10 38.75
1 000 0.50 1 0.02 0.48 1.09 34.16 36.13 38.41
1 000 0.50 2 3.31 3.31 3.31 34.08 35.91 39.39
1 000 0.50 3 2.74 2.74 2.74 34.2 36.10 39.19
1 000 0.50 4 0.18 1.51 3.03 34.41 36.10 39.63
1 000 0.50 5 0.21 0.71 1.14 34.19 35.99 38.02
1 000 0.50 6 0.13 0.51 0.97 34.03 35.70 38.88
1 000 0.50 7 0.11 0.53 0.98 34.08 35.81 38.52
1 000 0.50 8 0.12 0.42 0.84 34.16 35.86 38.55
1 000 0.50 9 3.04 3.04 3.04 34.14 35.71 38.48
1 000 0.85 0 0.11 0.43 0.93 34.28 35.99 38.20
1 000 0.85 1 0.18 0.51 1.23 34.23 36.02 38.17
1 000 0.85 2 3.54 3.54 3.54 34.30 35.61 38.09
1 000 0.85 3 2.74 2.74 2.74 34.31 35.41 37.33
1 000 0.85 4 0.26 2.54 3.08 34.36 35.43 37.28
1 000 0.85 5 0.03 0.59 1.20 34.38 35.65 37.58
1 000 0.85 6 0.00 0.35 1.01 34.20 35.66 37.91
1 000 0.85 7 0.04 0.40 0.94 34.31 35.36 36.86
1 000 0.85 8 0.02 0.32 0.77 34.28 35.55 37.86
1 000 0.85 9 3.01 3.01 3.01 34.11 35.19 37.38
¯x 0.96 1.51 1.98 34.16 35.86 38.63
2 000 0.15 0 0.11 0.39 0.76 69.97 72.00 73.84
2 000 0.15 1 3.06 3.06 3.06 70.94 72.02 73.71
2 000 0.15 2 0.32 0.55 1.15 71.07 72.41 74.91
2 000 0.15 3 0.00 0.28 0.67 72.39 73.58 75.22
2 000 0.15 4 3.39 3.39 3.39 72.12 73.31 74.79
2 000 0.15 5 0.26 0.53 0.99 72.56 73.47 75.41
2 000 0.15 6 0.08 0.42 0.69 72.57 73.37 76.57
2 000 0.15 7 0.19 0.40 0.70 72.25 73.03 74.48
2 000 0.15 8 0.03 0.30 0.64 72.40 73.22 75.62
2 000 0.15 9 0.11 0.35 0.83 72.30 73.23 76.04
2 000 0.50 0 0.15 0.38 0.58 71.33 72.54 76.08
2 000 0.50 1 2.94 2.94 2.94 71.33 72.14 72.81
2 000 0.50 2 0.04 0.39 0.83 71.49 72.37 73.42
2 000 0.50 3 0.06 0.33 0.59 71.49 72.41 73.54
2 000 0.50 4 2.92 2.92 2.92 71.83 72.46 73.49
2 000 0.50 5 0.19 0.52 0.84 71.70 72.73 74.19
2 000 0.50 6 0.14 0.39 0.81 72.26 73.07 74.08
2 000 0.50 7 0.34 3.05 3.10 72.29 73.21 74.27
2 000 0.50 8 0.05 0.26 0.61 72.38 73.55 74.54
2 000 0.50 9 0.09 0.34 0.64 72.75 73.75 74.67
2 000 0.85 0 0.04 0.32 0.70 71.97 73.59 76.05
2 000 0.85 1 3.07 3.07 3.07 71.79 73.09 79.53
2 000 0.85 2 0.13 0.52 2.57 71.33 72.81 75.81
2 000 0.85 3 -0.04 0.21 0.56 71.82 75.02 79.83
2 000 0.85 4 2.80 2.80 2.80 71.72 72.54 74.12
2 000 0.85 5 0.13 0.39 0.86 70.44 72.14 77.82
2 000 0.85 6 0.13 0.39 0.85 70.48 71.75 74.26
2 000 0.85 7 3.08 3.08 3.08 70.88 71.76 73.98
2 000 0.85 8 -0.43 -0.24 0.00 70.42 71.90 74.19
2 000 0.85 9 -0.10 0.13 0.44 70.48 71.93 75.64
¯x 0.78 1.06 1.39 71.63 72.81 75.10
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Table 17: Running times and gaps for Simulated Annealing considering SA1 parameters in L2 instances. n is
the number of nodes, d manages the interval length and i is an identiﬁer for each instance.
gap (%) time (sec.)
n p i min av max min av max
1 000 0.15 0 0.00 0.19 3.14 82.97 87.87 97.66
1 000 0.15 1 0.00 0.03 0.15 82.6 86.38 91.78
1 000 0.15 2 3.43 3.43 3.43 82.42 85.04 89.45
1 000 0.15 3 2.97 2.97 2.97 81.34 84.95 90.84
1 000 0.15 4 0.00 0.58 3.56 82.14 85.77 89.97
1 000 0.15 5 0.00 0.00 0.02 82.38 86.35 90.80
1 000 0.15 6 0.00 0.00 0.01 82.17 86.12 92.64
1 000 0.15 7 0.00 0.01 0.05 82.19 85.96 91.36
1 000 0.15 8 0.00 0.04 0.11 82.97 85.91 91.10
1 000 0.15 9 0.09 3.25 3.55 81.58 83.89 88.13
1 000 0.50 0 0.00 0.04 0.17 82.42 85.31 88.50
1 000 0.50 1 0.00 0.01 0.08 82.80 85.03 88.64
1 000 0.50 2 3.31 3.31 3.31 80.94 83.62 88.92
1 000 0.50 3 2.74 2.74 2.74 81.44 84.09 88.88
1 000 0.50 4 -0.04 0.05 3.03 81.92 85.49 89.52
1 000 0.50 5 0.00 0.03 0.23 83.20 85.88 89.67
1 000 0.50 6 0.00 0.00 0.05 82.86 85.95 89.34
1 000 0.50 7 0.00 0.01 0.11 82.74 85.56 89.70
1 000 0.50 8 0.00 0.03 0.13 82.27 85.37 89.38
1 000 0.50 9 3.04 3.04 3.04 80.74 83.66 87.19
1 000 0.85 0 0.00 0.04 0.16 82.39 84.89 89.39
1 000 0.85 1 0.00 0.01 0.07 82.53 85.12 88.81
1 000 0.85 2 3.54 3.54 3.54 81.03 83.92 88.16
1 000 0.85 3 2.74 2.74 2.74 81.06 83.52 88.02
1 000 0.85 4 0.00 1.62 3.08 81.27 83.62 85.97
1 000 0.85 5 0.00 0.04 0.23 82.64 84.57 89.22
1 000 0.85 6 0.00 0.00 0.04 82.61 84.38 88.70
1 000 0.85 7 0.00 0.02 0.11 82.47 84.34 86.97
1 000 0.85 8 0.00 0.04 0.10 82.61 84.83 88.30
1 000 0.85 9 3.01 3.01 3.01 81.16 82.80 86.31
¯x 0.83 1.03 1.43 82.13 85.01 89.44
2 000 0.15 0 -0.03 0.01 0.08 164.00 175.10 178.50
2 000 0.15 1 3.06 3.06 3.06 168.78 170.30 172.98
2 000 0.15 2 -0.02 0.01 0.12 171.61 173.97 177.95
2 000 0.15 3 -0.07 -0.06 0.01 172.50 174.00 176.80
2 000 0.15 4 3.39 3.39 3.39 167.34 170.19 172.65
2 000 0.15 5 0.00 0.03 0.16 170.71 173.05 174.66
2 000 0.15 6 -0.03 0.00 0.04 170.41 173.31 176.82
2 000 0.15 7 -0.03 0.00 0.06 172.06 175.73 180.17
2 000 0.15 8 -0.01 0.00 0.09 172.29 176.36 182.99
2 000 0.15 9 0.00 0.02 0.07 162.89 174.83 178.81
2 000 0.50 0 -0.02 0.02 0.09 167.48 173.82 177.45
2 000 0.50 1 2.94 2.94 2.94 167.50 172.12 174.74
2 000 0.50 2 -0.11 -0.07 0.08 175.80 177.25 179.37
2 000 0.50 3 -0.04 -0.02 0.06 175.18 176.55 178.68
2 000 0.50 4 2.92 2.92 2.92 168.91 171.23 176.18
2 000 0.50 5 0.00 0.04 0.11 173.16 174.75 176.36
2 000 0.50 6 -0.02 0.01 0.10 173.60 175.56 177.48
2 000 0.50 7 -0.02 2.98 3.10 170.63 172.89 174.87
2 000 0.50 8 -0.02 -0.01 0.07 174.97 177.99 187.02
2 000 0.50 9 -0.01 0.01 0.10 173.83 177.79 190.40
2 000 0.85 0 -0.06 -0.02 0.08 171.84 175.04 178.73
2 000 0.85 1 3.07 3.07 3.07 167.10 169.08 172.49
2 000 0.85 2 0.00 0.06 0.16 160.72 170.03 176.83
2 000 0.85 3 -0.11 -0.09 -0.04 161.22 161.73 166.16
2 000 0.85 4 2.80 2.80 2.80 157.30 157.89 164.16
2 000 0.85 5 -0.05 -0.02 0.04 161.58 161.86 162.27
2 000 0.85 6 -0.01 0.03 0.09 161.47 161.87 162.25
2 000 0.85 7 3.08 3.08 3.08 157.36 157.80 158.30
2 000 0.85 8 -0.45 -0.44 -0.36 161.36 161.86 162.49
2 000 0.85 9 -0.21 -0.18 -0.13 161.33 161.80 162.17
¯x 0.66 0.78 0.85 167.83 170.86 174.36
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Table 18: Running times and gaps for Simulated Annealing considering SA2 parameters in L2 instances. n is
the number of nodes, d manages the interval length and i is an identiﬁer for each instance.
gap (%) time (sec.)
n p i min av max min av max
1 000 0.15 0 0.00 0.10 0.29 26.86 27.51 28.92
1 000 0.15 1 0.00 0.09 0.33 26.84 27.69 29.33
1 000 0.15 2 3.43 3.43 3.43 26.48 27.25 28.73
1 000 0.15 3 2.97 2.97 2.97 26.50 27.18 28.53
1 000 0.15 4 0.00 2.41 3.56 26.50 27.23 28.80
1 000 0.15 5 0.00 0.14 0.42 26.84 27.42 28.70
1 000 0.15 6 0.00 0.08 0.30 26.83 27.59 28.80
1 000 0.15 7 0.00 0.12 0.49 26.84 27.70 29.11
1 000 0.15 8 0.00 0.84 3.22 26.41 27.40 28.95
1 000 0.15 9 0.01 3.33 3.55 26.39 27.37 28.77
1 000 0.5 0 0.01 0.15 0.35 26.86 27.81 29.20
1 000 0.5 1 0.00 0.06 0.29 26.81 27.75 29.14
1 000 0.5 2 3.31 3.31 3.31 26.50 27.37 28.97
1 000 0.5 3 2.74 2.74 2.74 26.50 27.48 30.28
1 000 0.5 4 -0.04 1.27 3.03 26.55 27.48 29.09
1 000 0.5 5 0.00 0.22 0.59 26.83 27.77 29.42
1 000 0.5 6 0.00 0.10 0.41 26.89 27.82 29.39
1 000 0.5 7 0.00 0.12 0.38 26.86 27.61 29.20
1 000 0.5 8 0.00 0.09 0.29 26.81 27.77 29.06
1 000 0.5 9 3.04 3.04 3.04 26.44 27.25 28.67
1 000 0.85 0 0.00 0.12 0.30 26.92 27.77 29.20
1 000 0.85 1 0.00 0.07 0.27 26.98 27.72 29.11
1 000 0.85 2 3.54 3.54 3.54 26.61 27.46 28.77
1 000 0.85 3 2.74 2.74 2.74 26.58 27.43 28.78
1 000 0.85 4 0.04 2.51 3.08 26.53 27.27 28.84
1 000 0.85 5 0.00 0.20 0.81 27.02 27.84 28.91
1 000 0.85 6 0.00 0.07 0.33 27.11 27.71 28.92
1 000 0.85 7 0.00 0.07 0.29 27.03 27.70 28.92
1 000 0.85 8 0.00 0.08 0.26 27.00 27.71 29.17
1 000 0.85 9 3.01 3.01 3.01 26.56 27.32 28.58
¯x 0.83 1.23 1.59 26.73 27.55 29.01
2 000 0.15 0 0.04 0.21 0.52 56.03 57.77 61.22
2 000 0.15 1 3.06 3.06 3.06 54.99 56.41 57.66
2 000 0.15 2 0.07 0.34 1.23 56.19 56.97 59.72
2 000 0.15 3 0.01 0.14 0.41 56.27 57.26 58.86
2 000 0.15 4 3.39 3.39 3.39 55.25 56.04 57.00
2 000 0.15 5 0.06 0.28 0.74 56.20 57.15 58.23
2 000 0.15 6 0.01 0.18 0.49 56.36 57.22 58.25
2 000 0.15 7 0.07 0.23 0.46 56.10 56.94 57.92
2 000 0.15 8 0.03 0.19 0.64 56.10 56.99 59.89
2 000 0.15 9 0.02 0.26 0.67 56.45 56.98 57.89
2 000 0.5 0 0.02 0.17 0.39 56.31 57.11 58.34
2 000 0.5 1 2.94 2.94 2.94 54.91 55.72 57.23
2 000 0.5 2 -0.05 0.17 0.43 56.17 57.25 58.26
2 000 0.5 3 -0.01 0.16 0.50 56.02 57.02 58.13
2 000 0.5 4 2.92 2.92 2.92 55.01 56.13 58.26
2 000 0.5 5 0.07 0.20 0.41 56.02 56.83 57.47
2 000 0.5 6 0.04 0.18 0.33 55.72 56.55 57.47
2 000 0.5 7 0.24 3.04 3.10 54.88 55.65 57.21
2 000 0.5 8 0.00 0.12 0.27 56.21 56.63 57.74
2 000 0.5 9 0.02 0.18 0.51 55.96 56.64 57.20
2 000 0.85 0 -0.06 0.11 0.27 55.89 56.77 58.10
2 000 0.85 1 3.07 3.07 3.07 55.33 56.54 58.61
2 000 0.85 2 0.08 0.51 3.14 56.30 57.30 58.43
2 000 0.85 3 -0.09 0.04 0.22 56.20 57.47 59.02
2 000 0.85 4 2.80 2.80 2.80 55.44 56.50 58.23
2 000 0.85 5 0.00 0.17 0.45 56.22 57.85 60.12
2 000 0.85 6 0.02 0.15 0.34 56.12 57.70 59.17
2 000 0.85 7 3.08 3.08 3.08 55.70 56.72 59.52
2 000 0.85 8 -0.44 -0.33 -0.20 56.72 57.73 59.65
2 000 0.85 9 -0.17 -0.02 0.18 56.50 57.42 58.72
¯x 0.71 0.93 1.22 55.92 56.91 58.45
26

2018 algorithms for the minmax regret path problem with interval data

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