A Distributionally Robust Formulation for Stochastic Quadratic Bi-level

Programming

Pablo Adasme

, Abdel Lisser

and Chen Wang

Departamento de Ingenieria Electrica, Universidad de Santiago de Chile, Avenida Ecuador 3519, Santiago, Chile

Laboratoire de Recherche en Informatique, Universite Paris-Sud XI, Batiment 650, 91405 Orsay Cedex, France

Keywords:

Distributionally Robust Optimization, Stochastic Programming, Binary Quadratic Bi-level Programming,

Mixed Integer Programming.

Abstract:

In this paper, we propose a distributionally robust model for a (0-1) stochastic quadratic bi-level programming

problem. To this purpose, we ﬁrst transform the stochastic bi-level problem into an equivalent deterministic

formulation. Then, we use this formulation to derive a bi-level distributionally robust model (Liao, 2011).

The latter is accomplished while taking into account the set of all possible distributions for the input random

parameters. Finally, we transform both, the deterministic and the distributionally robust models into single

level optimization problems (Audet et al., 1997). This allows comparing the optimal solutions of the proposed

models. Our preliminary numerical results indicate that slight conservative solutions can be obtained when the

number of binary variables in the upper level problem is larger than the number of variables in the follower.

1 INTRODUCTION

Bi-level programming(BP) is a hierarchical optimiza-

tion framework. It consists in optimizing an objective

function subject to a constrained set where another

optimization problem is embedded. The ﬁrst level op-

timization problem is referred to as the leader prob-

lem while the lower level, as the follower problem.

Formally, a BP problem can be written as follows

min

{x∈X,y}

F(x, y)

s.t. G(x, y) ≤ 0

min

{y}

f(x, y)

s.t. g(x, y) ≤ 0

where x ∈ R

, y ∈ R

, F : R

×R

→ R and f :

×R

→ R are the decision variables and the ob-

jective functions for the upper and lower level prob-

lems, respectively. The functions G : R

×R

→R

and g : R

×R

→R

denote upper and lower level

constraints. The goal is to ﬁnd an optimal point such

that the leader and the follower minimizes their re-

spective objective functions subject to their respec-

tive linking constraints (Audet et al., 1997). Ap-

plications of BP include transportation, network de-

sign, management and planning among others. For

more application domains, see for instance (Floudas

and Pardalos, 2001). It has been shown that bi-level

problems are strongly NP-Hard, even for the simplest

case where all the involved functions are afﬁne (Au-

det et al., 1997).

As far as we know, robust optimization ap-

proaches have not yet been reported in the literature

for bi-level programming. Some preliminary works

concerning pure stochastic programming approaches

can be found, for instance, in (Audestad et al., 2006;

Ozaltin et al., 2010; Carrion et al., 2009; Kalashnikov

et al., 2010; Wynter, 2009). In (Carrion et al., 2009),

an application for retailer futures market trading is

considered whereas a natural gas cash-out problem is

studied in (Kalashnikov et al., 2010).

Stochastic programming (SP) as well as robust op-

timization (RO) are well known optimization tech-

niques to deal with mathematical problems involving

uncertainty in the input parameters. In SP, it is usually

assumed that the probability distributions are discrete

and known or that they can be estimated (Shapiro

et al., 2009). There are two well known scenario ap-

proachesin SP, the recourse model and the probabilis-

tic constrained approach. See for instance (Schultz

et al., 1996; Birge and Louveaux, 1997). Different

from the SP approach, the RO framework assumes

that the input random parameters lie within a convex

uncertainty set and that the robust solutions must re-

main feasible for all possible realizations of the in-

222

Adasme P., Lisser A. and Wang C..

A Distributionally Robust Formulation for Stochastic Quadratic Bi-level Programming.

DOI: 10.5220/0004207100240031

In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 24-31

ISBN: 978-989-8565-40-2

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

put parameters. Thus, the optimization is performed

over the worst case realization of the input param-

eters. In compensation, we obtain robust solutions

which are protected from undesired ﬂuctuations in the

input parameters. In this case, the objective function

provides more conservative solutions. We refer the

reader to (Bertsimas and Sim, 2004) and (Bertsimas

et al., 2010) for a more general understanding on RO.

In this paper, we propose a distributionally RO

model for a (0-1) stochastic quadratic bi-level prob-

lem with expectation in the objective and probabilis-

tic knapsack constraints in the leader. To this pur-

pose, we ﬁrst transform the stochastic problem into an

equivalent deterministic problem (Gaivoronski et al.,

2011). Subsequently, we apply a novel and simple

distributionally robust approach proposed by (Liao,

2011) to derive a distributionally robust formulation

for our stochastic bi-level problem. The latter allows

optimizing the objective function over the set of all

possible distributions in the input random parameters.

Finally, we compute optimal solutions by transform-

ing both problems, the deterministic as well as the

distributionally models into single level optimization

problems (Audet et al., 1997). Preliminary numeri-

cal comparisons are given. The paper is organized

as follows. Section 2, presents the stochastic model

under study and the equivalent deterministic formula-

tion. In section 3, we derive the distributionally robust

formulation. In section 4, we transform the determin-

istic and robust models into single level optimization

problems. Then, in section 5, we provide preliminary

numerical comparisons. Finally, section 6 concludes

the paper.

2 PROBLEM FORMULATION

We consider the following (0-1) stochastic quadratic

bi-level problem we denote hereby Q

as follows

max

{x}

(

∑

i=1

∑

j=1

i, j

(ξ)x

)

(1)

s.t. P

(

∑

j=1

(ξ)x

∑

j=1

(ξ)y

≤ c(ξ)

)

≥

(1−α) (2)

∈{0, 1}, j = 1 : n

(3)

y ∈ argmax

{y}

{

∑

j=1

} (4)

s.t.

∑

j=1

i, j

∑

j=1

i, j

≤ h

, i = 1 : m

(5)

0 ≤ y

≤ 1, j = 1 : n

(6)

where x ∈ {0, 1}

and y ∈ [0, 1]

are the leader and

the follower decision variables respectively. In Q

(1)-(3) correspond to the leader problem while (4)-

(6) represent the follower problem. The term E{·}

denotes mathematical expectation while P{·} repre-

sents a probability imposed on the upper level knap-

sack constraint. This probability should be satisﬁed

at least for (1−α)% of the cases where α ∈ (0, 0.5]

represents the risk. The matrices D, F,G and vectors

a, b, d, h, c are input nonnegative real matrices/vectors

deﬁned accordingly. We assume that the matrix

D = D(ξ), vectors a = a(ξ), b = b(ξ) and c = c(ξ)

are random variables distributed according to a dis-

crete probability distribution Ω. As such, one may

suppose that a

(ξ), b

(ξ) and c(ξ) are concentrated

on a ﬁnite set of scenarios as a

(ξ) = {a

, .., a

(ξ) = {b

, .., b

} and c(ξ) = {c

, .., c

} respec-

tively, with probability vector q

= (q

, .., q

) such

that

∑

k=1

= 1 and q

≥ 0. In (Gaivoronski et al.,

2011), the authors propose a deterministic equivalent

formulation for Q

by replacing the probabilistic con-

straint (2) with the following deterministic constraints

∑

j=1

∑

j=1

≤ c

+ M

, z

∈ {0, 1}∀k

∑

k=1

≤ α (7)

where M

is deﬁned for each k = 1 : K by M

∑

j=1

∑

j=1

−c

. The variable z

for each k is

a binary variable used to decide whether a particular

constraint is discarded. This is handled by taking the

risk α in constraint (7).

Analogously, the random variables D

i, j

(ξ) are dis-

cretely distributed, i.e. D

i, j

(ξ) = (D

i, j

, ..., D

i, j

), ∀i, j

such that

∑

k=1

= 1 and ρ

≥0 where ρ is the prob-

ability vector. Thus, the expectation in the objective

function (1) can be written as

max

{x}

∑

k=1

∑

i=1

∑

j=1

i, j

This yields the following deterministic equivalent

problem we denote by Q

as follows

max

{x,z}

∑

k=1

∑

i=1

∑

j=1

i, j

s.t.

∑

j=1

∑

j=1

≤ c

+ M

, ∀k

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223

∑

k=1

≤ α

∈ {0, 1}∀k

∈ {0, 1}, j = 1 : n

y ∈ argmax

{y}

{

∑

j=1

}

s.t.

∑

j=1

i, j

∑

j=1

i, j

≤ h

, i = 1 : m

0 ≤ y

≤ 1, j = 1 : n

This model is a deterministic equivalent formulation

for Q

provided the assumption on the discrete prob-

ability space Ω holds.

3 THE DISTRIBUTIONALLY

ROBUST FORMULATION

In this section, we derive a distributionally RO model

for Q

. For this, we assume that the probability dis-

tribution of the random vectors ρ

= (ρ

, .., ρ

) and

= (q

, .., q

) are not known and that they can be

estimated by some statistical mean from some avail-

able historical data. Thus, we consider the maximum

likelihood estimator of the probability vectors ρ

and

to be the observed frequency vectors.

3.0.1 The Distributionally Robust Model

In order to formulate a robust model for Q

, we write

its objective function as follows

min

{x}

max

{π∈H

}

∑

k=1

∑

i=1

∑

j=1

−D

i, j

(8)

and the left hand side of constraint (7) as the maxi-

mization problem

max

{p∈H

}

∑

k=1

(9)

where the sets H

and H

are deﬁned respectively as

(

≥ 0, ∀k :

∑

k=1

= 1,

∑

k=1

|π

−ρ

√

≤ β

)

and

(

≥ 0, ∀k :

∑

k=1

= 1,

∑

k=1

−q

√

≤ γ

)

where β, γ ∈ [0, ∞). Now, let δ

= π

−ρ

, then the

inner max problem in (8) can be written as

max

{δ}

∑

k=1

(δ

+ ρ

)

∑

i=1

∑

j=1

−D

i, j

s.t.

∑

k=1

|δ

√

≤ β (10)

∑

k=1

= 0 (11)

≥ −ρ

, k = 1 : K (12)

The associated dual problem is

min

,ϕ

}

∑

k=1

∑

i=1

∑

j=1

−D

i, j

∑

k=1

+ βϕ

s.t. ϕ

≥

√

+ w

−

∑

i=1

∑

j=1

i, j

, ∀k

≥ −

√

+ w

−

∑

i=1

∑

j=1

i, j

, ∀k

≥ 0, ∀k

and ϕ

, v

, w

are Lagrangian multipliers for con-

straints (10)-(12), respectively. Similarly, we obtain

a dual formulation for (9) as follows

min

,ϕ

}

∑

k=1

∑

k=1

+ γϕ

s.t. ϕ

≥

√



+ w

+ z



, ∀k

≥ −

√



+ w

+ z



, ∀k

≥ 0, ∀k

where ϕ

, v

, w

are Lagrangian multipliers associ-

ated with its primal constraints. Now, replacing these

dual problems in Q

gives rise to the following distri-

butionally robust formulation we denote by Q

max

,ϕ

,x,z}

∑

k=1

∑

i=1

∑

j=1

i, j

−

∑

k=1

−βϕ

s.t.ϕ

≥

√

+ w

−

∑

i=1

∑

j=1

i, j

, ∀k

≥ −

√

+ w

−

∑

i=1

∑

j=1

i, j

, ∀k

≥ 0, ∀k (13)

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∑

j=1

∑

j=1

≤ c

+ M

, k = 1 : K

∈ {0, 1} k = 1 : K

∑

k=1

∑

k=1

+ γϕ

≤α

≥

√

+ v

+ w

), ∀k

≥ −

√

+ v

+ w

), ∀k

≥ 0, ∀k (14)

∈ {0, 1}, j = 1 : n

y ∈ argmax

{y}

{

∑

j=1

}

s.t.

∑

j=1

i, j

∑

j=1

i, j

≤ h

, i = 1 : m

0 ≤ y

≤ 1, j = 1 : n

In the next section we transform both models: Q

and

into single level optimization problems. More

precisely, we obtain Mixed Integer Linear program-

ming problems (MILP) (Audet et al., 1997).

4 EQUIVALENT MILP

FORMULATIONS

Since the follower problem is the same for both Q

and Q

, we derive equivalent MILPs by replacing the

follower problem with its primal, dual and comple-

mentarity slackness conditions. These conditions can

be written as

∑

j=1

i, j

∑

j=1

i, j

≤ h

, i = 1 : m

(15)

0 ≤ y

≤ 1, j = 1 : n

(16)

∑

i=1

i, j

+ µ

≥ d

, j = 1 : n

(17)

≥ 0, i = 1 : m

(18)

≥ 0, j = 1 : n

(19)

−

∑

j=1

i, j

−

∑

j=1

i, j

= 0,

i = 1 : m

(20)

(1−y

) = 0, j = 1 : n

(21)

∑

i=1

i, j

+ µ

−d

= 0,

j = 1 : n

(22)

where (15)-(16) and (17)-(19) are the primal and dual

follower constraints, respectively. Note that con-

straints (20)-(22) are quadratic constraints. In (Audet

et al., 1997), the authors propose a splitting scheme

to linearize these complementarity constraints. The

approach introduces binary variables as follows

−

∑

j=1

i, j

−

∑

j=1

i, j

+ ν

L ≤L,

i = 1 : m

(23)

≤ν

L, i = 1 : m

(24)

∈ {0, 1}, i = 1 : m

(25)

1−y

+ ν

L ≤L, j = 1 : n

(26)

≤ ν

L, j = 1 : n

(27)

∈ {0, 1}, j = 1 : n

(28)

∑

i=1

i, j

+ µ

−d

+ ν

L ≤L,

j = 1 : n

(29)

≤ ν

L, j = 1 : n

(30)

∈ {0, 1}, j = 1 : n

(31)

where constraints (23)-(25), (26)-(28) and (29)-(31)

replace the single constraints (20), (21) and (22), re-

spectively. The parameter L is a large positive num-

ber.

Finally, let ψ

i, j

= x

be a linearization variable

for each quadratic term in Q

and Q

(Fortet, 1960).

Thus, a MILP formulation for Q

can be written as

max

{x,y,z,ψ,λ,µ,ν

,ν

}

∑

k=1

∑

i=1

∑

j=1

i, j

s.t.

∑

j=1

∑

j=1

≤c

+ M

, ∀k

∑

k=1

≤ α

∈{0, 1}∀k

i, j

≤ x

, i, j = 1 : n

(32)

i, j

≤ x

, i, j = 1 : n

(33)

i, j

≥ x

+ x

−1, i, j = 1 : n

(34)

i, j

∈ {0, 1}, i, j = 1 : n

(35)

∈{0, 1}, j = 1 : n

∑

j=1

i, j

∑

j=1

i, j

≤ h

, i = 1 : m

0 ≤ y

≤ 1, j = 1 : n

ADistributionallyRobustFormulationforStochasticQuadraticBi-levelProgramming

225

∑

i=1

i, j

+ µ

≥ d

, j = 1 : n

≥ 0, i = 1 : m

≥ 0, j = 1 : n

−

∑

j=1

i, j

−

∑

j=1

i, j

+ ν

L ≤L,

i = 1 : m

≤ ν

L, i = 1 : m

∈ {0, 1}, i = 1 : m

1−y

+ ν

L ≤L, j = 1 : n

≤ ν

L, j = 1 : n

∈ {0, 1}, j = 1 : n

∑

i=1

i, j

+ µ

−d

+ ν

L ≤L,

j = 1 : n

≤ ν

L, j = 1 : n

∈ {0, 1}, j = 1 : n

where constraints (32)-(35) are Fortet linearization

constraints. We denote this model by MIP

. Con-

sequently, a MILP distributionally robust model for

can be written as follows

max

{ϒ}

∑

k=1

∑

i=1

∑

j=1

i, j

−

∑

k=1

−βϕ

s.t.ϕ

≥

√

+ w

−

∑

i=1

∑

j=1

i, j

, ∀k

≥ −

√

+ w

−

∑

i=1

∑

j=1

i, j

, ∀k

≥ 0, ∀k

∑

j=1

∑

j=1

≤ c

+ M

, k = 1 : K

∑

k=1

∑

k=1

+ γϕ

≤ α

∈{0, 1} k = 1 : K

≥

√

+ v

+ w

), ∀k

≥ −

√

+ v

+ w

), ∀k

≥ 0, ∀k

i, j

≤ x

, i, j = 1 : n

i, j

≤ x

, i, j = 1 : n

i, j

≥ x

+ x

−1, i, j = 1 : n

i, j

∈ {0, 1}, i, j = 1 : n

∈{0, 1}, j = 1 : n

∑

j=1

i, j

∑

j=1

i, j

≤ h

, i = 1 : m

0 ≤ y

≤ 1, j = 1 : n

∑

i=1

i, j

+ µ

≥ d

, j = 1 : n

≥ 0, i = 1 : m

≥ 0, j = 1 : n

−

∑

j=1

i, j

−

∑

j=1

i, j

+ ν

L ≤L,

i = 1 : m

≤ ν

L, i = 1 : m

∈ {0, 1}, i = 1 : m

1−y

+ ν

L ≤L, j = 1 : n

≤ ν

L, j = 1 : n

∈ {0, 1}, j = 1 : n

∑

i=1

i, j

+ µ

−d

+ ν

L ≤L,

j = 1 : n

≤ ν

L, j = 1 : n

∈ {0, 1}, j = 1 : n

where ϒ = {w

, ϕ

, v

, w

, ϕ

, v

, x, y, z, ψ, λ, µ, ν

, ν

}. We denote this model by MIP

In the next section, we provide numerical com-

parisons between MIP

and MIP

. This allows mea-

suring the conservatism level of MIP

with respect

to MIP

. The conservatism level can be measured by

the loss in optimality in exchange for a robust solution

which is more protected against uncertainty (Bertsi-

mas and Sim, 2004). This means, the less conser-

vative the robust solutions are, the better the RO ap-

proach.

5 NUMERICAL RESULTS

In this section, we present preliminary numerical re-

sults. A Matlab program is developed using Cplex

12.3 for solving MIP

and MIP

. The numerical ex-

periments have been carried out on a Pentium IV, 1.9

GHz with 2 GB of RAM under windows XP. The in-

put data is generated as follows. The probability vec-

tors ρ and q are uniformly distributed in [0, 1] such

that the sums are equal to one. The parameter α is

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226

set to 0.1. Matrices F, G and vectors a

, b

, ∀k are

uniformly distributed in [0, 1]. The symmetric matri-

ces D

, ∀k and vector d are uniformly distributed in

[0, 10]. The scalars c

, ∀k and the vector h are gener-

ated respectively as

∑

j=1

∑

j=1

, ∀k

and

∑

j=1

i, j

∑

j=1

i, j

, ∀i = 1 : m

In table 1, columns 1-4 give the size of the instances.

Columns 5-6 provide the average optimal solutions

over 25 different sample instances. Finally, column 7

gives the average gaps we compute for each instance

(MIP

−MIP

)

MIP

·100%. These results are calculated

for different values of β and γ. From table 1, we

Table 1: Average comparisons over 25 instances.

Instance size Avg. Opt. Sol.

Avg. Gap

K m

MIP

β = 50 and γ = 50

10 10 10 5 300.09 267.31 10.85 %

10 10 30 5 283.95 229.39 21.88 %

10 10 10 10 322.94 284.46 11.98 %

20 10 10 5 985.82 917.55 6.95 %

10 20 10 5 152.09 115.25 22.12 %

β = 100 and γ = 50

10 10 10 5 313.29 258.47 17.74 %

10 10 30 5 272.49 212.07 22.05 %

10 10 10 10 320.94 290.30 9.29 %

20 10 10 5 990.99 931.64 5.93 %

10 20 10 5 138.99 100.97 27.53 %

β = 50 and γ = 100

10 10 10 5 290.98 255.61 12.06 %

10 10 30 5 278.32 197.80 28.66 %

10 10 10 10 311.54 282.26 9.08 %

20 10 10 5 1013.41 958.89 5.23 %

10 20 10 5 169.78 89.12 47.33 %

mainly observe that the solutions tend to be more con-

servative when a) the number of scenarios K is larger

than n

, n

and m

and b) when the number of vari-

ables of the follower problem: n

is larger than n

, K

and m

. On the opposite, we see slight conservative

solutions when the number of binary variables: n

larger than n

, K and m

. The variations of β and γ do

not seem to affect these trends. However, they seem to

affect the conservatism level in each case. For exam-

ple, the average increases signiﬁcantly up to 47.33%

when β < γ and n

is large. Same remarks when K is

large.

In order to see how the parameters β and γ af-

fect the conservatism levels, we solve one instance

for each row in table 1 while varying only β and γ.

These results are shown in tables, 2, 3, 4, 5 and 6,

respectively. All columns in these tables provide the

same information for each instance. In columns 1-2,

we give the values of β and γ. Columns 3-4 give the

optimal solutions for MIP

and MIP

, respectively.

Finally, in column 5, we give the gap we compute

(MIP

−MIP

)

MIP

·100%. In table 2, we observe that

Table 2: Instance # 1: n

= n

= 10, m

= 5, K = 10.

Robustness Optimal Solutions

Gap

β γ MIP

MIP

0 0

328.37

328.37 0 %

0 30 328.37 0 %

0 60 328.37 0 %

0 90 328.37 0 %

30 0 301.18 8.28 %

30 30 311.27 5.21 %

30 60 315.48 3.93 %

30 90 315.48 3.93 %

60 0 290.70 11.47 %

60 30 291.79 11.14 %

60 60 311.04 5.28 %

60 90 311.04 5.28 %

90 0 302.53 7.87 %

90 30 309.27 5.82 %

90 60 309.27 5.82 %

90 90 290.54 11.52 %

when β = 0, then augmenting the values of γ does not

affect the optimal solutions. This is not the case when

γ = 0 and β > 0. Next, when both β > 0 and γ > 0,

the optimal solutions are affected. In particular, we

observe that the parameter β affects more the optimal

solutions than γ does. For example, when β goes from

30 to 60, we observe an increment of 5.93% while

from 30 to 90, we observe an increment of 0.61%.

This is not the case when γ increases. In this partic-

ular case, we observe a decrement of 1.28% in each

case. The increase of γ seems to produce the opposite

effect than incrementing β. For example, we notice

that when β = 30, 60, 90 and γ goes from 0 to 30, 60

or 90, the gaps are decremented except in the worst

case when both, β = γ = 90.

Similar observations are obtained for instances 3

and 5 in tables 4 and 6, respectively. Instances 2 and

4 in tables 3 and 5 respectively, provide additional in-

formation. Table 3 corresponds to the case where the

number of scenarios K is larger compared to n

, n

and m

. In this case, increasing γ when β = 0 af-

fects the optimal solutions. In particular, when β = 0

and γ goes from 60 to 90, we have a large increase of

31.04% in the conservatism level. This is repeated for

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Table 3: Instance # 2: n

= n

= 10, m

= 5, K = 30.

Robustness Optimal Solutions

Gap

β γ MIP

MIP

0 0

181.14

181.14 0 %

0 30 181.03 0.06 %

0 60 179.85 0.71 %

0 90 123.63 31.75 %

30 0 178.82 1.28 %

30 30 177.12 2.22 %

30 60 177.12 2.22 %

30 90 123.67 31.73 %

60 0 176.63 2.49 %

60 30 176.63 2.49 %

60 60 175.07 3.35 %

60 90 123.08 32.05 %

90 0 174.60 3.61 %

90 30 173.15 4.41 %

90 60 173.15 4.41 %

90 90 121.96 32.67 %

Table 4: Instance # 3: n

= n

= 10, m

= 10, K = 10.

Robustness Optimal Solutions

Gap

β γ MIP

MIP

0 0

331.48

331.48 0 %

0 30 331.48 0 %

0 60 331.48 0 %

0 90 331.48 0 %

30 0 316.51 4.52 %

30 30 316.51 4.52 %

30 60 316.51 4.52 %

30 90 311.11 6.15 %

60 0 306.65 7.49 %

60 30 306.65 7.49 %

60 60 306.65 7.49 %

60 90 309.91 6.51 %

90 0 308.84 6.83 %

90 30 308.84 6.83 %

90 60 308.84 6.83 %

90 90 308.84 6.83 %

each value of β = 0, 30, 60, 90 when γ goes from 60 to

90. The worst gap occurs when β = γ = 90.

Finally, in table 5 we observe weak conservatism

levels in all cases. In fact, they are lower than 10%.

This instance corresponds to the case when the binary

variables of the leader problem, i.e. n

are largerwhen

compared to n

, m

and K. Notice that when β = 0

and γ grows, then the optimal solutions are slightly

affected.

6 CONCLUSIONS

In this paper, we proposed a distributionally robust

model for a (0-1) stochastic quadratic bi-level pro-

Table 5: Instance # 4: n

= 20, n

= 10, m

= 5, K = 10.

Robustness Optimal Solutions

Gap

β γ MIP

MIP

0 0

982.24

982.24 0 %

0 30 965.06 1.75 %

0 60 973.95 0.84 %

0 90 982.24 0 %

30 0 923.13 6.02 %

30 30 934.96 4.81 %

30 60 940.78 4.22 %

30 90 940.78 4.22 %

60 0 940.38 4.26 %

60 30 943.63 3.93 %

60 60 931.84 5.13 %

60 90 902.04 8.16 %

90 0 936.32 4.67 %

90 30 926.40 5.68 %

90 60 929.28 5.39 %

90 90 895.58 8.82 %

Table 6: Instance # 5: n

= 10, n

= 20, m

= 5, K = 10.

Robustness Optimal Solutions

Gap

β γ MIP

MIP

0 0

257.00

257.00 0 %

0 30 257.00 0 %

0 60 257.00 0 %

0 90 257.00 0 %

30 0 241.17 6.16 %

30 30 241.17 6.16 %

30 60 241.17 6.16 %

30 90 241.17 6.16 %

60 0 230.29 10.39 %

60 30 230.29 10.39 %

60 60 230.29 10.39 %

60 90 230.29 10.39 %

90 0 223.45 13.06 %

90 30 223.45 13.06 %

90 60 223.45 13.06 %

90 90 223.45 13.06 %

gramming problem. To this end, we transformed the

stochastic bi-level problem into an equivalent deter-

ministic model. Afterward, we derived a bi-level dis-

tributionally robust model using the deterministic for-

mulation. In particular, we applied a distributionally

robust approach proposed in (Liao, 2011). This al-

lows optimizing the problem when taking into ac-

count the set of all possible distributions of the in-

put random parameters. Thus, we derived Mixed In-

teger Linear Programming formulations using Fortet

linearization method (Fortet, 1960) and the approach

proposed by (Audet et al., 1997). Finally, we com-

pared the optimal solutions of this model to measure

the conservatism level of the proposed robust model.

Our preliminary numerical results show that slight

conservative solutions are obtained for the case when

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

228

the numberof binary variables in the upper level prob-

lem is larger than the number of variables in the fol-

lower problem.

ACKNOWLEDGEMENTS

The ﬁrst author is grateful for the ﬁnancial support

given by Conicyt Chilean government through the In-

sertion project number: 79100020.

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