A Two-stage Stochastic Programming Approach for the

Traveling Salesman Problem

Pablo Adasme

, Rafael Andrade

, Janny Leung

and Abdel Lisser

Departamento de Ingenier´ıa El´ectrica, Universidad de Santiago de Chile, Avenida Ecuador 3519, Santiago, Chile

Departamento de Estat´ıstica e Matem´atica Aplicada, Universidade Federal do Cear´a, Campus do Pici, BL 910,

CEP 60.455-760, Fortaleza, Cear´a, Brazil

Department of Systems Engineering & Engineering Management, Chinese University of Hong Kong, Shatin,

Hong Kong, China

Laboratoire de Recherche en Informatique, Universit´e de Paris-Sud 11, Bˆat. 650 Ada Lovelace, Paris, France

Keywords:

Two-stage Stochastic Programming, Traveling Salesman Problem, Compact Formulations, Iterative Algorith-

mic Approach.

Abstract:

In the context of combinatorial optimization, recently some efforts have been made by extending classical

optimization problems under the two-stage stochastic programming framework. In this paper, we introduce

the two-stage stochastic traveling salesman problem (STSP). Let G = (V, E

∪E

) be a non directed complete

graph with set of nodes V and set of weighted edges E

∪ E

where E

∩ E

0. The edges in E

and E

have deterministic and uncertain weights, respectively. Let K = {1,2, ·· · ,|K|} be a given set of scenarios

referred to the uncertain weights of the edges in E

. The STSP consists in determining Hamiltonian cycles

of G, one for each scenario s ∈ K, sharing the same deterministic edges while minimizing the sum of the

deterministic weights plus the expected weight over all scenarios associated with the uncertain edges. We

propose two compact models and a formulation with an exponential number of constraints which are adapted

from the classic TSP. One of the compact models allows to solve instances with up to 40 nodes and 5 scenarios

to optimality. Finally, we propose an iterative procedure that allows to compute optimal solutions and tight

lower bounds within very small CPU time.

1 INTRODUCTION

Stochastic programming is an optimization frame-

work which allows to deal with the uncertainty of the

input parameters of a mathematical program (Shapiro

et al., 2009). Thus, it is commonly assumed that

probability distributions take values within a discrete

and ﬁnite space which allows to consider sets of sce-

narios for the input parameters. A well known sce-

nario based approach is the “recourse model” or “two-

stage stochastic programming approach” (Gaivoron-

ski et al., 2011; Shapiro et al., 2009). In the con-

text of combinatorial optimization, recently some ef-

forts have been made by extending classical combi-

natorial optimization problems (e.g., Knapsack prob-

lems (Gaivoronski et al., 2011), the maximum weight

matching problem (Escofﬁer et al., 2010), maximal

and minimal spanning tree problems (Flaxman et al.,

2006; Escofﬁer et al., 2010), the stochastic maxi-

mum weight forest problem (Adasme et al., 2013;

Adasme et al., 2015)) under the two-stage stochas-

tic programming framework. In this paper, we in-

troduce the two-stage stochastic traveling salesman

problem (STSP) which can be described as follows.

Let G = (V, E

∪E

) be a non directed complete graph

with a set of nodes V and a set of weighted edges

∪ E

where E

∩ E

0. The edges in E

and

have deterministic and uncertain weights, respec-

tively. Let K = {1, 2, ··· , |K|} be a given set of sce-

narios referred to the uncertain weights of the edges

in E

. The STSP problem consists in determining

Hamiltonian cycles of G, one for each scenario s ∈ K,

sharing the same deterministic edges while minimiz-

ing the sum of the deterministic weights plus the ex-

pected weight over all scenarios associated with the

uncertain edges. For |K| = 1, the problem reduces

to the classic traveling salesman problem. We pro-

pose two compact polynomial models and a formula-

tion with an exponential number of constraints. These

models are based on the classic TSP (Miller et al.,

1960; Gavish and Graves, 1978; Letchford et al.,

2013). Stochastic programming variants of the travel-

Adasme, P., Andrade, R., Leung, J. and Lisser, A.

A Two-stage Stochastic Programming Approach for the Traveling Salesman Problem.

DOI: 10.5220/0005738801630169

In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 163-169

ISBN: 978-989-758-171-7

163

ing salesman problem have been previously studied,

see for instance (Maggioni et al., 2014; Bertazzi and

Maggioni, 2014). In particular, the two-stage stochas-

tic problem we present in this paper can be seen as a

particular case of the stochastic capacitated traveling

salesmen location problem with recourse (Bertazzi

and Maggioni, 2014). As far as we know, this spe-

cial case has never been studied before in the liter-

ature. Notice that all the applications of the classic

traveling salesman problem can be extended to the

models we present in this paper. We compare numer-

ically the exponential model versus the two compact

polynomial formulations for randomly generated in-

stances. For this purpose, we solve the exponential

model by generating all cycle elimination constraints

at once and also by using a simple iterative algorith-

mic approach which consists of adding violated cycle

elimination constraints within each iteration until no

cycle is found in the current solution. Finally, we use

the iterative algorithm in order to compute tight lower

bounds in signiﬁcantly short CPU time.

The remaining of the paper is organized as fol-

lows. In section 2, we present the two-stage stochastic

formulations of the problem. Then, in section 3, we

present the iterativealgorithm to solve the exponential

formulation alternatively. Subsequently, in section 4

we conduct numerical results in order to compare all

the proposed models and the algorithmic approach.

Finally, in section 5 we give the main conclusions of

the paper.

2 TWO-STAGE STOCHASTIC

FORMULATIONS

In this section, we propose three stochastic formu-

lations for the STSP that we adapt from the classic

TSP (Miller et al., 1960; Gavish and Graves, 1978;

Letchford et al., 2013). The ﬁrst one is an exponen-

tial model that contains an exponential numberof sub-

tour elimination constraints (SECs). The second one

is adapted from (Miller et al., 1960), and the third one

corresponds to an extension of the single ﬂow com-

modity model proposed in (Gavish and Graves, 1978).

Consider the non directed complete graph G and the

set of discrete scenarios K as deﬁned in section 1. An

exponential model for the STSP can be written

STSP

min

{x,y}

(

∑

(i, j)∈E

|K|

∑

s=1

∑

(i, j)∈E

)

(1)

subject to :

∑

j:(i, j)∈ E

∑

j:(i, j)∈ E

= 1, ∀i ∈ V, s ∈ K (2)

∑

i:(i, j)∈E

∑

i:(i, j)∈E

= 1, ∀ j ∈ V, s ∈ K (3)

∑

(i, j)∈E(S)∩E

∑

(i, j)∈E(S)∩E

≤ |S| − 1,

S ⊂ V, s ∈ K (4)

∈ {0, 1}, ∀(i, j) ∈ E

, (5)

∈ {0, 1}, ∀(i, j) ∈ E

, s ∈ K (6)

In (1), we minimize the sum of the deterministic

edge weights plus the expected cost of uncertain edge

weights obtained over all scenarios. The parame-

ter p

, ∀s ∈ K, represents the probability for scenario

s ∈ K where

∑

s∈K

= 1. Constraints (2)-(3) ensure

that the salesman arrives at and departs from each

node exactly once for each scenario s ∈ K. Con-

straints (4) are sub-tour elimination constraints for

each S ⊂ V, s ∈ K. Finally, (5)-(6) are the domain con-

straints for the binary decision variables x

, ∀(i, j) ∈

and y

, ∀(i, j) ∈ E

, s ∈ K. The variable x

= 1 if

the deterministic edge (i, j) ∈ E

is selected in each

Hamiltonian cycle, ∀s ∈ K, otherwise x

= 0. Simi-

larly, the variable y

= 1 if the edge (i, j) ∈ E

is se-

lected in the Hamiltonian cycle associated to the sce-

nario s ∈ K, and y

= 0 otherwise.

Now let A

and A

represent the sets of arcs ob-

tained from E

and E

, respectively where an edge

(i, j) is replaced by two arcs (i, j), ( j, i) of same cost

in each corresponding set. A polynomial compact for-

mulation based on (Miller et al., 1960) is

STSP

min

{x,y,u}

(

∑

(i, j)∈A

|K|

∑

s=1

∑

(i, j)∈A

)

subject to:

∑

j:(i, j)∈ A

∑

j:(i, j)∈ A

= 1, ∀i ∈ V, s ∈ K

∑

i:(i, j)∈A

∑

i:(i, j)∈A

= 1, ∀ j ∈ V, s ∈ K

= 1, ∀s ∈ K (7)

2 ≤ u

≤ |V|, ∀i ∈ |V|, (i 6= 1), ∀s ∈ K (8)

− u

+ 1 ≤

(|V| − 1)(1− x

ij:(i, j)∈A

− y

ij:(i, j)∈A

∀i, j ∈ V, (i, j 6= 1), s ∈ K (9)

∈ {0, 1}, ∀(i, j) ∈ A

, (10)

∈ {0, 1}, ∀(i, j) ∈ A

, s ∈ K (11)

∈ R

, ∀i ∈ V, s ∈ K (12)

where the constraints (9) ensure that, if the salesman

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

164

travels from i to j, then the nodes i and j are arranged

sequentially for each s ∈ K. These constraints to-

gether with (7) and with the bounds (8) ensure that

each node is in a unique position. Finally, (10)-(12)

are the domain constraints for the decision variables.

A third formulation can be obtained by extending

the classic single commodity ﬂow formulation for the

TSP (Gavish and Graves, 1978). For this purpose,

we assume that the salesman carries |V| − 1 units of

a commodity when he leaves node 1, and delivers 1

unit of this commodity to each node. Using the sets

and A

, we can deﬁne additional continuous vari-

ables g

, ∀(i, j) ∈ A

and w

, ∀(i, j) ∈ A

, s ∈ K, rep-

resenting the amount of the commodity (if any) routed

directly from node i to node j, for all s ∈ K. The new

formulation is

STSP

min

{x,y,g,w}

(

∑

(i, j)∈A

|K|

∑

s=1

∑

(i, j)∈A

)

subject to:

∑

j:(i, j)∈ A

∑

j:(i, j)∈ A

= 1, ∀i ∈ V, s ∈ K

∑

i:(i, j)∈A

∑

i:(i, j)∈A

= 1, ∀ j ∈ V, s ∈ K

∑

j:( j,i)∈ A

∑

j:( j,i)∈ A

−

∑

j>1:(i, j)∈A

−

∑

j>1:(i, j)∈A

= 1

∀i ∈ { 2, . . . , |V|}, s ∈ K (13)

0 ≤ g

≤ (|V| − 1)x

, ∀(i, j) ∈ A

(14)

0 ≤ w

≤ (|V| − 1)y

, ∀(i, j) ∈ A

, s ∈ K (15)

∈ {0, 1}, ∀(i, j) ∈ A

, (16)

∈ {0, 1}, ∀(i, j) ∈ A

, s ∈ K (17)

The constraints (13) ensure that one unit of the com-

modity is delivered to each node, ∀s ∈ K. The bounds

(14)-(15) ensure that the commodity can ﬂow only

along arcs in the solution.

In the next section, we propose an iterative algo-

rithmic procedure that allows to obtain optimal solu-

tions and lower bounds for the STSP while using the

exponential formulation.

3 ITERATIVE PROCEDURE FOR

GENERATING SECS

The procedure to generate SECs is quite general and

it can be adapted straightforwardly using Algorithms

4.1 and 4.2 from (Adasme et al., 2015) to the STSP.

The idea is as follows. If we remove constraints (4)

from STSP

and solve the resulting integer linear pro-

gramming problem, then the underlying optimal so-

lution induces a graph G

for each s ∈ K that may

contain a cycle with at least three or up to |V| − 1

nodes. In this case, it can be detected by a depth-ﬁrst

search procedure (Cormen et al., 2009). We refer the

reader to the Algorithm 4.1 in (Adasme et al., 2015)

for a deeper understanding on how we obtain cycles

for each G

, s ∈ K. In particular, if the cardinality of

a subset of nodes found with Algorithm 4.1 inducing

a cycle equals |V|, we do not generate the SEC, oth-

erwise Hamiltonian cycles would be infeasible for the

problem. The Algorithm 4.1 is used iteratively by the

Algorithm 1: Iterative procedure to compute lower

bounds for STSP

Data: A problem instance of STSP

Result: A lower bound with solution (x, y) for

STSP

with objective function value z

Step 0

: Set ν = 1;

Let STSP

be the problem obtained from STSP

removing the constraints (4) at iteration ν;

Solve the LP relaxation of problem STSP

and let

, y

) be its optimal solution of value z

iteration ν;

Let z

= inf;

Step 1

: while |z

ν−1

− z

| > ε do

foreach s ∈ K do

Construct the graph G

= (V, E

∪ E

) for

scenario s with the rounded solution

( ˜x

, ˜y

) obtained from (x

, y

);

= searchCycles(G

,V);

foreach cycle ∈ C

Add the corresponding constraint (4) to

STSP

;

Set ν = ν + 1;

Solve the LP relaxation of problem STSP

and

let (x

, y

) be its optimal solution of value z

iteration ν;

return the solution (x

, y

, z

);

Algorithm 4.2 in (Adasme et al., 2015) that we adapt

to solve problem STSP

. First, we remove constraints

(4) from STSP

and solve the resulting integer opti-

mization problem. Consider the underlying optimal

solution undirected graph G

= (V, E

∪ E

) where V

is the set of nodes and E

∪ E

is the set of edges such

that E

⊆ E

and E

⊆ E

for a given scenario s. If G

contains a cycle with three or up to |V|−1 nodes, then

the Algorithm 4.1 detects it. A subset of nodes induc-

ing a cycle deﬁnes a new constraint (4) which cuts off

this cycle from the solution space. Problem STSP

re-optimized taking into account the new added con-

straints. This iterative process goes on until the under-

lying current optimal solution of STSP

has no more

A Two-stage Stochastic Programming Approach for the Traveling Salesman Problem

165

cycles. Since the number of cycles is ﬁnite, so is the

number of constraints (4) that can be added to prob-

lem STSP

. Notice that each subset of nodes contain-

ing at least three or up to |V| − 1 nodes generates one

cycle elimination constraint for a particular scenario.

Also notice that the number of SECs of type (4) that

can be added to problem STSP

is at most O(|K|2

|V|

Consequently, Algorithm 4.2 adapted to solve STSP

converges to the optimal solution of the problem in

at most O(|K|2

|V|

) outer iterations. The proof can be

directly deduced from Theorem 2 in (Adasme et al.,

2015). The aforementioned procedure can also be

used to compute lower bounds for STSP

. This proce-

dure is depicted in Algorithm 1 and it is described as

follows. First, we remove constraints (4) from STSP

and solve the resulting linear programming (LP) re-

laxation of STSP

at step 1. Next, we enter into a

while loop searching cycles in the current rounded LP

solution for each s ∈ K. If G

contains a cycle with

three or more nodes up to |V| − 1, then the Algorithm

4.1 referred to as “searchCycles(G

,V)” in (Adasme

et al., 2015) detects it. A subset of nodes inducing

a cycle deﬁnes a new constraint (4). The LP relax-

ation of problem STSP

is re-optimized taking into

account the new added constraints. This iterative pro-

cess goes on until the difference between the current

optimal objective function value z

and the previous

one z

ν−1

is less than a small positive value ε.

4 NUMERICAL RESULTS

In this section, we present preliminary numerical re-

sults. A Matlab (R2012a) program is developed using

CPLEX 12.6 to solve STSP

, STSP

and their

corresponding LP relaxations. The numerical exper-

iments have been carried out on an Intel(R) 64 bits

core (TM) with 3.4 Ghz and 8G of RAM. CPLEX

solver is used with default options. We generate the

input data as follows. The edges in E

and E

are

chosen randomly with 50% of probability. The values

of p

, ∀s ∈ K are drawn randomly from the interval

[0;1] such that

∑

s∈K

= 1. Without loss of general-

ity, we assume that the set of scenarios K is ﬁnite and

known. Deterministic and uncertain edge costs are

randomly drawn from the interval [0;5]. We set the

parameter ε = 10

−8

in Algorithm 1. Finally, we men-

tion that we solve STSP

with up to 15 nodes while

generating all cycle elimination constraints. In par-

ticular, for the instances 1-25, we set the maximum

CPU time to solve the linear models with CPLEX to

at most 2 hours. Except for the last instance where we

set the maximum CPU time to 12 hours. The legend

of Table 1 is as follows. Column 1 shows the instance

number. Columns 2-3 present the number of nodes

|V| and the number of scenarios |K| of each instance,

respectively. Columns 4-8, 9-13 and 14-18 present

the optimal solution of STSP

, STSP

, the

number of branch and bound nodes used by CPLEX,

the CPU time in seconds to solve the mixed integer

programs and their corresponding LP relaxations to-

gether with their CPU time in seconds, respectively.

Finally, in columns 19-21, we present gaps we com-

pute as

Opt−LP

Opt

∗ 100 for STSP

, STSP

and STSP

respectively.

From Table 1, we observe that the optimal objec-

tive function values for STSP

and STSP

are exactly

the same and slightly larger for STSP

. We also see

that the CPU times are signiﬁcantly lower for STSP

Regarding the number of branch and bound nodes,

we observe that CPLEX requires less nodes for solv-

ing STSP

than STSP

and STSP

, and less nodes for

solving STSP

than for STSP

. Finally, the LP relax-

ations of STSP

can be solved faster than for STSP

and STSP

. However, we see that the gaps of the LP

relaxations are tighter for the exponential model. Fi-

nally, we observe that all the instances (e.g. 1-24,26)

are solved to optimality with the exception of instance

number 25. In particular, all these optimal solutions

are obtained with STSP

that shows a signiﬁcantly

better performance. Finally, we mention that we can-

not solve, with the exponential model, instances with

more than 15 nodes due to the large number of sub-

tour elimination constraints involved. The instances

in Table 2 are the same as in Table 1. In Table 2,

the legend is as follows. In column 1, we show the

instance number. In columns 2-5, we show the op-

timal solution obtained with the adapted version of

Algorithm 4.2 (Adasme et al., 2015), its CPU time in

seconds, the number of cycles found with this algo-

rithm and the number of iterations, respectively. In

columns 6-10, we present the lower bound obtained

with Algorithm 1, its CPU time in seconds, the num-

ber of cycles found with it, the number of iterations,

and the optimal solution found with STSP

while us-

ing all the cycle elimination constraints found with

Algorithm 1, respectively. For the latter, we do not

report the CPU time required by CPLEX. However,

we mention that for most of the instances (e.g. 1-21)

these CPU times are less than 2 seconds. For the in-

stances 22-26, we limit CPLEX to a maximum CPU

time of 1 hour. In particular, for the instance num-

ber 25 we cannot ﬁnd a feasible solution in 1 hour.

Finally, in columns 11-12, we provide gaps that we

compute by

Opt

−Opt

Opt

∗ 100 and



Opt

−Opt

Opt



∗ 100,

respectively. From Table 2, we observe that Algo-

rithm 4.2 can ﬁnd the optimal solutions for all the

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

166

Table 1: Numerical results obtained with the Mixed integer linear models.

Inst. Dim. STSP

STSP

Gaps

|V| |K| Opt B&Bn Time (s) LP Time (s) Opt B&Bn Time (s) LP Time (s) Opt B&Bn Time (s) LP Time (s) Gap

% Gap

1 4 5 7.50 0 0.27 7.50 0.14 7.50 0 0.14 6.81 0.13 7.50 0 0.14 6.81 0.13 0 9.12 9.12

6 5 10.98 0 0.16 10.98 0.14 10.98 28 0.16 9.28 0.14 10.98 0 0.14 10.61 0.13 0 15.49 3.38

8 5 7.66 0 0.38 7.65 0.20 7.66 0 0.19 6.90 0.14 7.66 0 0.25 7.32 0.14 0.08 9.90 4.46

10 5 12.22 9 1.39 11.16 0.36 12.22 47 0.25 9.42 0.16 12.38 95 0.48 9.89 0.16 8.67 22.93 20.09

12 5 10.60 7 6.10 10.30 1.33 10.60 33 0.42 8.09 0.14 10.60 23 0.75 9.01 0.14 2.88 23.69 15.08

14 5 12.40 74 47.33 11.29 7.04 12.40 153 0.62 10.28 0.17 12.89 412 3.35 10.45 0.17 8.89 17.10 18.95

15 5 9.79 36 87.10 8.65 16.57 9.79 974 1.98 7.43 0.14 10.49 1298 11.72 7.60 0.16 11.69 24.05 27.51

8 4 10 10.16 0 0.27 10.16 0.16 10.16 0 0.16 9.74 0.16 10.16 0 0.16 9.74 0.16 0 4.11 4.11

6 10 12.02 7 0.30 11.28 0.14 12.02 31 0.27 10.15 0.28 12.60 34 0.33 10.33 0.12 6.15 15.57 17.95

8 10 12.39 15 1.70 11.43 0.20 12.39 136 0.34 9.32 0.16 12.46 97 0.56 10.30 0.16 7.74 24.72 17.36

10 10 11.47 27 3.62 11.04 0.50 11.47 596 1.31 10.39 0.17 12.19 508533 828.00 10.66 0.39 3.74 9.41 12.54

12 10 11.63 7 12.65 11.51 2.29 11.63 50 0.59 8.64 0.17 11.63 125 1.90 9.88 0.19 0.99 25.73 15.07

14 10 13.61 224 214.91 11.90 11.57 13.61 25584 81.73 10.85 0.16 17.57 656991 7200.51 11.42 0.42 12.62 20.32 34.99

15 10 12.32 171 439.33 10.62 29.20 12.32 8720 30.97 9.56 0.20 14.00 96228 1377.95 9.79 0.33 13.80 22.42 30.05

15 4 25 10.35 0 0.41 10.35 0.20 10.35 0 0.20 10.15 0.14 10.35 0 0.20 10.15 0.16 0 1.99 1.99

6 25 8.64 0 0.36 8.64 0.19 8.64 7 0.23 6.70 0.16 8.64 0 0.20 6.96 0.14 0 22.48 19.51

8 25 10.13 0 0.47 10.13 0.44 10.13 15 0.30 9.10 0.17 10.13 0 0.19 9.96 0.17 0 10.17 1.70

10 25 12.32 40 10.64 11.48 0.95 12.32 460 2.39 11.25 0.22 13.13 1884 22.00 11.44 0.23 6.79 8.65 12.85

12 25 16.70 151 135.99 15.06 5.20 16.70 2473 20.26 13.82 0.22 16.89 1237 42.28 14.11 0.30 9.82 17.27 16.48

14 25 13.77 126 1163.72 12.91 29.80 13.77 944 14.17 11.46 0.28 14.13 566 35.79 11.83 0.37 6.27 16.81 16.26

15 25 10.30 80 3469.84 8.69 81.57 10.30 614 12.48 7.20 0.95 10.30 413 69.05 7.67 0.98 15.61 30.10 25.55

22 20 5 - - - - - 12.63 34329 139.25 10.47 0.31 13.79 198011 2824.50 11.08 0.39 - 17.06 19.65

25 5 - - - - - 12.89 50541 443.95 9.97 0.30 12.98 147422 6962.65 10.11 0.64 - 22.64 22.13

30 5 - - - - - 16.76 380096 3254.34 14.67 0.41 27.77 121894 7200.36 15.17 1.11 - 12.48 45.39

35 5 - - - - - 13.07 314434 7231.83 10.59 0.56 22.81 55442 7200.55 10.76 1.31 - 19.01 52.80

40 5 - - - - - 13.05 625722 29773.38 10.76 0.72 19.88 213290 39599.72 10.96 3.40 - 17.58 44.90

Table 2: Numerical results obtained with the iterative algorithmic procedures.

Algorithms 4.1 and 4.2 adapted from (Adasme et al., 2015) Algorithm 1 Gaps

Opt

Time (s) #Cycles #Iter Opt

Gap

% Gap

1 7.50 0.48 10 2 7.50 0.42 10 2 7.50 0 0

10.98 0.28 15 2 10.98 0.30 15 2 10.98 0 0

7.66 0.31 20 2 7.65 0.58 97 3 7.66 0.08 0

12.22 1.11 43 6 11.16 0.59 94 3 11.64 8.67 4.80

10.60 0.52 36 3 10.30 0.61 150 3 10.52 2.88 0.76

12.40 1.58 75 6 11.20 0.72 261 3 11.77 9.64 5.03

9.79 1.34 53 5 8.54 0.70 233 3 9.69 12.81 0.96

8 10.16 0.34 20 2 10.16 0.33 20 2 10.16 0 0

12.02 0.92 64 5 11.28 0.61 133 3 11.88 6.15 1.12

12.39 1.72 78 7 11.43 0.80 280 4 11.89 7.74 4.01

11.47 2.65 129 8 11.04 1.30 502 6 11.43 3.74 0.39

11.63 0.69 60 2 11.51 0.98 439 4 11.63 0.99 0

13.61 10.73 227 17 11.88 1.19 655 4 12.96 12.71 4.82

12.32 10.65 188 17 10.62 1.72 886 5 11.69 13.86 5.14

15 10.35 0.39 50 2 10.35 0.33 50 2 10.35 0 0

8.64 0.33 75 2 8.64 0.36 75 2 8.64 0 0

10.13 0.38 100 2 10.13 0.36 100 2 10.13 0 0

12.32 3.28 210 5 11.48 1.33 1054 4 12.13 6.79 1.54

16.70 17.29 535 14 15.06 1.64 1530 4 16.27 9.86 2.57

13.77 12.04 460 11 12.88 1.75 1361 4 13.47 6.49 2.16

10.30 11.26 268 5 8.67 3.17 2285 4 10.22 15.81 0.82

22 12.63 57.99 329 41 11.27 1.64 583 5 12.01 10.79 4.89

12.89 1706.91 409 43 10.84 2.96 985 5 12.13 15.96 5.90

16.76 438.17 325 34 15.64 1.84 599 3 16.29 6.65 2.77

12.93 36135.64 799 84 11.23 1.61 488 2 - 13.18 -

13.05 56379.58 959 81 11.60 3.38 817 3 12.58 11.16 3.64

A Two-stage Stochastic Programming Approach for the Traveling Salesman Problem

167

2 4 6 8 10 12 14 16

a) Number of scenarios

Optimal Solutions

and lower bounds

2 4 6 8 10 12 14 16

200

400

600

b) Number of scenarios

CPU time (s)

2 4 6 8 10 12 14 16

−1000

1000

2000

3000

c) Number of scenarios

Number of cycles

and iterations

2 4 6 8 10 12 14 16

d) Number of scenarios

Gaps (%)

Opt(STSP

)

LP(ST SP

)

Opt

ST SP

LP(ST SP

)

Opt

Cycles

Iter.

LP(ST SP

)

Gap

Figure 1: Numerical results for STSP

varying the number of scenarios using |V| = 16.

instances. In particular, the optimal solution of in-

stance 25 is also found, although at a higher CPU time

when compared to STSP

. The number of cycles and

iterations are not so large, e.g., less than 1000 and

100, respectively, for the largest instance. Notice that

the total number of cycles for most of the instances

is huge. However, the number of cycles required by

Algorithm 4.2 to ﬁnd the optimal solution is signiﬁ-

cantly small. In general, we see that the Algorithm 1

can ﬁnd tight lower bounds in very short CPU time,

i.e., in less than 4 seconds for all the instances. In this

case, the number of cycles are slightly larger when

compared to Algorithm 4.2. On the opposite, we see

that Algorithm 1 requires less iterations. Finally, we

observe that solving STSP

with all the cycle elimi-

nation constraints found with Algorithm 1 allows to

compute tight bounds when compared to the optimal

solution of the problem and optimal solutions in many

cases (e.g. instances 1-3, 8, 12, 15-17). More pre-

cisely, these bounds are computed with gaps which

are lower than 6% for most of the instances.

In order to give more insight with respect to

the performances obtained with the compact model

STSP

and with Algorithm 1, in Figure 1, we solve

several instances for ﬁxed |V| = 16 while varying

the number of scenarios from 2 to 16. More pre-

cisely, in Figure 1a, we show the optimal solution

of STSP

we denote by Opt(STSP

), its LP relax-

ation LP(STSP

), the lower bound Opt

obtained

with Algorithm 1 and the lower bound Opt

obtained

with STSP

while using all the cycle elimination con-

straints found with Algorithm 1. In Figure 1b, we

present the CPU time in seconds for STSP

, for its

LP relaxation LP(STSP

), and for Opt

. In the lat-

ter, we include the CPU time required to solve Al-

gorithm 1 and the time required to solve STSP

with

CPLEX. In Figure 1c, we show the number of cycles

and iterations required by Algorithm 1. Finally, in

Figure 1d, we present gaps as deﬁned for Tables 1

and 2. From Figure 1a, we mainly observe that the

lower bounds Opt

remain tight when incrementing

the number of scenarios. In general, we see that the

bounds LP(STSP

), Opt

and Opt

do not seem to

be affected by the increase in the number of scenarios.

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

168

Finally, we observe that the bounds Opt

are tighter

than Opt

, and Opt

are tighter than LP(STSP

From Figure 1b, we conﬁrm that the Algorithm 1

can ﬁnd these lower bounds in very short CPU time.

Similarly, the LP relaxation of STSP

is obtained very

fast. In Figure 1c, we observe that the number of iter-

ations are very low and that the number of cycles used

to ﬁnd the lower bounds Opt

slightly grows with the

number of scenarios. Finally, in Figure 1d, we con-

ﬁrm with the gaps, the quality and order of the bounds

presented in Figure 1a.

5 CONCLUSIONS

In the context of combinatorial optimization, recently

some efforts have been made by extending classical

optimization problems under the two-stage stochastic

programming framework (Gaivoronski et al., 2011;

Flaxman et al., 2006; Escofﬁer et al., 2010; Adasme

et al., 2013; Adasme et al., 2015). In this paper, we

introduce a deterministic two-stage stochastic travel-

ing salesman problem and propose two compact mod-

els and a formulation with an exponential number of

constraints that we adapt from the classic TSP. Subse-

quently, we adapt the iterative algorithmic procedure

proposed in (Adasme et al., 2015) and compute opti-

mal solutions and tight lower bounds for the stochas-

tic traveling salesman problem. Our preliminary nu-

merical results indicate that one of the compact mod-

els allows to solve instances with up to 40 nodes and 5

scenarios to optimality. Finally, the lower bounds are

obtained within a small CPU time for all the tested

instances.

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