Bi-objective Risk-averse Facility Location using a Subset-based

Representation of the Conditional Value-at-Risk

Najmesadat Nazemi

1 a

, Sophie N. Parragh

1 b

and Walter J. Gutjahr

2 c

1

Institute of Production and Logistics Management, Johannes Kepler University Linz, Austria

2

Department of Statistics and Operations Research, University of Vienna, Austria

Keywords:

Risk-averse, Conditional Value-at-Risk, Cutting-plane, Bi-objective Optimization.

Abstract:

For many real-world decision-making problems subject to uncertainty, it may be essential to deal with multiple

and often conﬂicting objectives while taking the decision-makers’ risk preferences into account. Conditional

value-at-risk (CVaR) is a widely applied risk measure to address risk-averseness of the decision-makers. In

this paper, we use the subset-based polyhedral representation of the CVaR to reformulate the bi-objective

two-stage stochastic facility location problem presented in (Nazemi et al., 2021). We propose an approximate

cutting-plane method to deal with this more computationally challenging subset-based formulation. Then,

the cutting plane method is embedded into the ε-constraint method, the balanced-box method, and a recently

developed matheuristic method to address the bi-objective nature of the problem. Our computational results

show the effectiveness of the proposed method. Finally, we discuss how incorporating an approximation of

the subset-based polyhedral formulation affects the obtained solutions.

1 INTRODUCTION

On the one hand, many real-world problems have

multiple conﬂicting objectives, e.g., cost versus cus-

tomer satisfaction or proﬁtability versus environmen-

tal concerns. On the other hand, it is important to

incorporate uncertainty about the considered parame-

ters into the decision-making process. Optimization

under uncertainty and multi-objective optimization

have evolved mostly separately over the past decades.

However, it is desirable to develop optimization tech-

niques to deal with these two domains simultaneously

(Gutjahr and Pichler, 2016).

Due to its sound theoretical background and its

proven algorithmic performance, stochastic program-

ming is a widely applied approach in computational

optimization for handling data uncertainty (see, e.g.,

(Birge and Louveaux, 2011) and (Shapiro et al.,

2014), and references therein). The standard two-

stage stochastic programming problem formulation

considers the expected value as a performance mea-

sure. It assumes a risk-neutral attitude of the decision-

maker and performs well in the context of repet-

a

https://orcid.org/0000-0002-2268-4621

b

https://orcid.org/0000-0002-7428-9770

c

https://orcid.org/0000-0003-1494-2501

itive decision-making problems ((Birge and Lou-

veaux, 2011)). However, decision-makers may have

other risk preferences, especially in non-repetitive ap-

plications, e.g., in large-scale ﬁnancial investments or

in the management of high-impact disasters. In the

case where more robust solutions are of interest, one

of the recently most widely recommended risk mea-

sures is the Conditional Value at Risk (CVaR). It can

be adjusted to the actual degree of risk-averseness of

the decision-maker by specifying a conﬁdence level α

(α ∈ [0, 1)) (CVaR(α)). The closer the value of α to

0, the more risk-averse is the decision-maker. CVaR

has been applied in different streams of the optimiza-

tion literature, such as, e.g., ﬁnancial engineering

(e.g., (Krokhmal et al., 2002), (Roman et al., 2007),

(Huang et al., 2021), (Zheng and Zheng, 2021)) dis-

aster management (e.g., (Noyan, 2012), (Elc¸i and

Noyan, 2018), (Nazemi et al., 2021)), water man-

agement (e.g., (Zhang et al., 2016), (Simic, 2019),

(Chen et al., 2021)), and sustainable supply chain

(e.g., (Rahimi and Ghezavati, 2018), (Rahimi et al.,

2019)). We refer to (Filippi et al., 2020) for a re-

cent survey on application of CVaR to several opti-

mization domains different from ﬁnancial engineer-

ing. (Huang et al., 2021) propose a two-stage distribu-

tionally robust model including CVaR as a risk mea-

sure for two different practical applications, a multi-

Nazemi, N., Parragh, S. and Gutjahr, W.

Bi-objective Risk-averse Facility Location using a Subset-based Representation of the Conditional Value-at-Risk.

DOI: 10.5220/0010914900003117

In Proceedings of the 11th International Conference on Operations Research and Enter prise Systems (ICORES 2022), pages 77-85

ISBN: 978-989-758-548-7; ISSN: 2184-4372

Copyright

c

2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

77

product assembly problem and a portfolio selection

problem. They also extend the model to a multi-

stage one. Two decomposition algorithms based on

the cutting-plane method are developed to solve the

models. Finally, they conclude that the proposed

models show a more robust performance compared

to the risk-neutral model. (Noyan, 2012) incorpo-

rates the classical CVaR representation into the ob-

jective function of a two-stage mean-risk stochastic

problem in disaster management where the demand

and the damage level of the transportation network

are uncertain parameters. She develops decomposi-

tion algorithms to deal with the computationally chal-

lenging nature of the problem. In line with this study,

(Elc¸i and Noyan, 2018) consider a chance-constrained

two-stage mean-risk stochastic relief network design

model. They also develop a Benders decomposition-

based algorithm to solve formulations with alterna-

tive representations of the CVaR. (Zhang et al., 2016)

present a risk-averse multi-stage stochastic water al-

location problem. A mean-CVaR objective is consid-

ered in each stage of the problem. Finally, they pro-

pose a nested L-shaped method to solve the problem.

In addition to the aforementioned works that address

single-objective problems, there are also some stud-

ies in the literature that consider multi-objective opti-

mization under uncertainty incorporating CVaR as the

risk measure into two-stage stochastic programming.

(Zheng and Zheng, 2021) consider a bi-objective port-

folio optimization model with a focus on minimiz-

ing CVaR as a risk measure and maximizing the ex-

pected return rate. They also incorporate transaction

costs into the objective functions. A NSGA-II meta-

heuristic method (Deb et al., 2002) based on sparsity

strategy (Zitzler et al., 2001) (SMP-NSGAII) is de-

veloped to deal with the bi-objective model. (Rahimi

et al., 2019) propose a risk-averse sustainable multi-

objective mixed-integer non-linear model to design

a supply chain network under uncertainty by incor-

porating CVaR into their two-stage stochastic model.

The objectives aim at minimizing the design cost and

maximizing the proﬁt. (Nazemi et al., 2021) incor-

porate a risk-neutral, a risk-averse, and the CVaR

measure into a bi-objective two-stage facility loca-

tion model in a disaster management context to an-

alyze a wide range of risk preferences, including risk-

neutral and worst-case approaches. They integrate

different uncertain two-stage models into two well-

known exact multi-objective frameworks, namely the

ε-constraint and the balanced-box methods. Addi-

tionally, they also develop a matheuristic approach

and they analyze and evaluate the combination of dif-

ferent uncertainty representations and multi-objective

frameworks. In most of the aforementioned studies,

the classical representation of CVaR has been em-

ployed to model risk-averseness. However, as men-

tioned earlier, alternative variants of CVaR are also

proposed in the literature. In this paper, we focus on

the existing two-stage bi-objective model presented

in (Nazemi et al., 2021) and extend this work by re-

formulating their two-stage risk-averse model using a

subset-based polyhedral representation of the CVaR.

To address this alternative formulation, we propose

a scenario cutting-plane algorithm. We conduct a

computational analysis on the test instances used by

(Nazemi et al., 2021) to illustrate how the subset-

based variant performs in comparison to the classical

formulation on their model.

The remainder of this paper is organized as fol-

lows. In Section 2, we describe the existing bi-

objective facility location problem under uncertainty

presented by (Nazemi et al., 2021) along with its cor-

responding subset-based polyhedral mathematical re-

formulation. In Section 3, we summarize the devel-

oped solution approach to solve the problem. In Sec-

tion 4 we discuss the computational results. Finally,

we present the conclusion and address potential future

work in Section 5.

2 TWO-STAGE RISK-AVERSE

STOCHASTIC OPTIMIZATION

The problem addressed in (Nazemi et al., 2021) is a

bi-objective two-stage location-allocation model un-

der demand uncertainty motivated by last-mile net-

work design in a slow-onset disaster context. A ﬁ-

nite discrete set of demand scenarios (s ∈ S, S =

{1, ..., N}) with equal probabilities (

1

N

) is used to in-

corporate stochastic information. The problem aims

at ﬁnding the best location to position temporary lo-

cal distribution centers (LDC) in the response phase

of a disaster, such that the affected people can walk to

these centers to receive their relief aids. The model

tries to ﬁnd the trade-off between two conﬂicting

objectives, i.e., minimization of operational cost on

opening LDCs and maximization of the coverage or

in other words, minimization of the amount of uncov-

ered demand. We use the same notation as introduced

in (Nazemi et al., 2021) to describe the model. The

problem is formulated on a network, where the sets

of nodes representing the demand and potential LDC

locations are denoted by I and J, respectively, assum-

ing that J ⊆ I. The assumption is that the affected

people in each demand node (i ∈ I) will walk only to

an LDC if their distance (d

i j

) from an opened LDC is

less than a certain distance threshold (d

max

). Uncer-

tainty is assumed on the amount of demand at each

ICORES 2022 - 11th International Conference on Operations Research and Enterprise Systems

78

node (q

(s)

i

). The ﬁrst-stage decision (here-and-now)

concerns the selection of sites to open LDCs. Binary

variables y

j

∈ {0, 1} indicate whether or not an LDC

is opened at node j ∈ J. Each LDC is assumed to

have a limited capacity c

j

and a given ﬁxed cost γ

j

.

Decision variables x

(s)

i j

∈ {0, 1} and u

(s)

j

represent the

second-stage decisions (wait-and-see) that determine

the allocation of demand nodes to the opened LDCs

and the amount of relief items delivered to each of

them, respectively. The second-stage decisions are

determined based on ﬁrst-stage values and realized

uncertain demand information. We next elaborate on

our choice of the alternative subset-based polyhedral

representation of CVaR incorporated into the second

stage of the stochastic model.

2.1 Two-stage Subset-based CVaR

(SSCVaR)

(F

´

abi

´

an, 2008) obtains a subset-based polyhedral rep-

resentation of the classical CVaR for the special case

of scenarios with equal probabilities where α = 1−

k

N

;

k is the cardinality of any subset of scenarios. The car-

dinality of all scenarios is equal to N (i.e.,

¯

S ⊆ S, |

¯

S|=

k and |S|= N):

CVaR

1−

k

N

(X) =

1

k

max

¯

S⊆S:|

¯

S|=k

∑

s∈

¯

S

X

(s)

(1)

where X

(s)

is the value of the random variable

X in scenario s. Equation (1) leads to the follow-

ing subset-based polyhedral representation of CVaR

(F

´

abi

´

an, 2008):

CVaR

1−

k

N

(X) = {minρ : ρ ≥

1

k

∑

s∈

¯

S

X

(s)

, ∀

¯

S ⊆ S : |

¯

S|= k}

(2)

We utilize the subset-based representation (2) to

obtain the MIP formulation of the two-stage model

(MB):

min

∑

j∈V

γ

j

y

j

(3)

min CVaR

α

(

∑

i∈I

q

(s)

i

−

∑

j∈J

u

(s)

j

) = ρ (4)

s.t. ρ ≥

1

k

∑

s∈

¯

S

(

∑

i∈I

q

(s)

i

−

∑

j∈J

u

(s)

j

) ∀

¯

S ⊆ S : |

¯

S|= k (5)

∑

j∈J

ψ(d

i j

)x

(s)

i j

≤ 1 ∀i ∈ I, s ∈ S (6)

u

(s)

j

≤ c

j

y

j

∀ j ∈ J, s ∈ S (7)

u

(s)

j

≤

∑

i∈I

q

(s)

i

ψ(d

i j

)x

(s)

i j

∀ j ∈ J, s ∈ S (8)

x

(s)

i j

≤ y

j

∀i ∈ I, ∀ j ∈ J, s ∈ S (9)

y

j

∈ {0, 1} ∀ j ∈ J (10)

x

(s)

i j

∈ {0, 1} ∀i ∈ I, ∀ j ∈ J, s ∈ S (11)

u

(s)

j

∈ Z

+

∀ j ∈ J, s ∈ S (12)

ρ ≥ 0 (13)

The ﬁrst objective (3) minimizes total opening

costs of LDCs. The second objective (4) minimizes

the CVaR of the uncovered demand of the affected

people. The auxiliary variable ρ represents the solu-

tion value of the r.h.s. of (2), which coincides with the

CVaR as given by (1). Constraints (5) make sure that

ρ is set to the correct value. Constraints (6) ensure that

the demand at node i can only be covered at most once

by an opened LDC if it is located within the coverage

threshold of LDC j (ψ(d

i j

)=1, if d

i j

≤ d

max

and 0 oth-

erwise). Constraints (7) are the capacity constraints,

one for each LDC j. Constraints (8) guarantee that the

amount of demand considered to be covered by LDC

j is at most the actual demand assigned to this node

within its coverage threshold. Constraints (9) guar-

antee that a demand node can only be assigned to an

opened LDC. Constraints (10)-(13) give the domains

of the decision variables.

3 SOLUTION APPROACH

In this section, we describe the designed cutting-plane

(approximation) method to deal with the potentially

large number of scenario subsets in the model. To ad-

dress the bi-objective nature of the problem, we em-

bed the cutting-plane method into the multi-objective

frameworks applied in (Nazemi et al., 2021), namely

the ε-constraint (e) (Laumanns et al., 2006), the

balanced-box (Boland et al., 2015) (BB) and a spe-

ciﬁc matheuristic (Mat) developed in (Nazemi et al.,

2021). The matheuristic method embeds the bi-

objective generalization of two heuristic approaches,

namely local branching (Fischetti and Lodi, 2003)

and relaxation induced neighborhood search (RINS)

(Danna et al., 2005), into the ε-constraint method.

The RINS is a variable ﬁxing scheme employed when

moving along the Pareto frontier from one solution

Bi-objective Risk-averse Facility Location using a Subset-based Representation of the Conditional Value-at-Risk

79

to another in the ε-constraint framework. Addition-

ally, a local branching approach is applied for the so-

lutions, for which only ﬁxing the variables does not

solve the problem to optimality within a certain time

limit. In a bi-objective setting, it is usually not possi-

ble to ﬁnd a solution which simultaneously optimizes

all of the considered objectives, but a set of optimal

or efﬁcient trade-off solutions exists, which are in-

comparable among each other and together dominate

all other feasible solutions. The image of an efﬁcient

solution in objective space is called non-dominated

point (NDP) and all NDPs together form the Pareto

frontier. For further background on multi-objective

optimization, we refer to (Ehrgott, 2005).

3.1 Scenario Subset-based

Approximation

Formulation (MB) contains constraints corresponding

to subsets of S. The size of this set of constraints could

grow fast, depending on the cardinality of the subsets

(k) and the set of scenarios S.

As mentioned in Section 1, different

decomposition-based techniques have been pro-

posed to solve two-stage stochastic programming

models with a single objective (e.g., (Shapiro et al.,

2014)). (Elc¸i and Noyan, 2018) propose a Benders

decomposition-based framework for the subset-based

variant of CVaR with a mean-risk objective function.

In the single objective two-stage stochastic model, the

model is decomposed by scenario once the ﬁrst-stage

variables are ﬁxed.

In our scenario subset cutting-plane approach, we

iteratively solve the so-called single objective prob-

lem which relaxes the subset-based constraints (5),

and dynamically generates them using a delayed cut

generation algorithm. To be more precise, the single

objective problem contains the objective function (4)

as the main objective, where the value of the other

objective (3) is ﬁxed.

Given an optimal solution ( ˆy,

ˆ

ρ, ˆu) of the relaxed

model in each iteration, we check whether there is

any violated constraint of the type (5), and identify

a subset

ˆ

S if there is a violation. To solve this so-

called separation problem, a few simple calculations

(steps 3-8 of Algorithm 1) are required. At each itera-

tion, we have the exact optimal second stage objective

values associated with the current ( ˆy,

ˆ

ρ, ˆu) vector; in

particular, these objective values can be expressed as

ρ

s

=

1

k

(

∑

i∈I

q

(s)

i

−

∑

j∈J

ˆu

(s)

j

) for each s ∈ S. Let S

∗

=

{s ∈ S :

∑

s∈S

∗

ρ

s

>

ˆ

ρ}. If S

∗

= {s ∈ S :

∑

s∈S

∗

ρ

s

≤

ˆ

ρ}

then the candidate solution is labeled as the new in-

cumbent. Otherwise, the identiﬁed violated constraint

of the form (5) associated with S

∗

is introduced as

a cutting-plane. The pseudo-code of the delayed cut

generation algorithm is presented in Algorithm 1.

Initial Cut. We propose a heuristic algorithm in or-

der to initialize the relaxed model. This procedure is

based on the following steps:

1. Rank the scenarios in each demand point based

on their corresponding demand value (q

s

j

) in de-

scending order.

2. Sum the ranks for each scenario over I, the set of

demand points.

3. Sort the scenarios based on their cumulative total

rank in descending order.

4. Select the ﬁrst k scenarios of the sorted set.

Algorithm 1: Delayed cut generation.

1: initialize the relaxed model with the initial cut,

t ← 1

2: while

∑

s∈

ˆ

S

ρ

s

>

ˆ

ρ do

3: for each s ∈ S do

4: ρ

s

←

1

k

(

∑

i∈V

q

(s)

i

−

∑

j∈V

ˆu

(s)

j

)

5: (sort ρ

s

in descending order)

6: end for

7: S

∗

← the ﬁrst k values (largest) of the sorted ρ

s

values

8:

ˆ

S ← S

∗

9: t ← t + 1

10: end while

In our bi-objective setting, due to the presence of

integer variables not only in the ﬁrst stage, but also

in the second stage, using cutting-plane technique for

every single point along the Pareto frontier causes

computationally no beneﬁt. However, preliminary

tests showed that, employing the cutting-plane algo-

rithm in order to calculate the ﬁrst extreme point of

the Pareto frontier and, consequently, ﬁxing the gen-

erated subset cuts to the model, leads to a high-quality

approximation of the entire Pareto frontier. We refer

to this approach towards solving model MB by MB in

the following.

4 COMPUTATIONAL STUDY

In this section, we present computational results for

the instances used by (Nazemi et al., 2021) to show

the performance of the proposed method with respect

to the alternative subset-based polyhedral reformula-

tion of the model in comparison to the classical rep-

resentation of CVaR.

ICORES 2022 - 11th International Conference on Operations Research and Enterprise Systems

80

All experiments are executed on a single thread

cluster where each node consists of two Intel Xeon

X5570 CPUs at 2.93 GHz and 8 cores, with 48GB

RAM. The model is implemented in C++ using

CPLEX 12.9. As in (Nazemi et al., 2021) a time

limit (TL) of 7200 s is considered. We note that

we use the generic callback feature of CPLEX in

order to generate the cuts in the MB model. Un-

like lazy constraint callbacks, which prevent CPLEX

from utilizing “dynamic search”, generic callbacks al-

low CPLEX to choose among the options of dynamic

search and classical branch-and-cut as it suits.

4.1 Test Instances

Here, we brieﬂy explain the key points of the test in-

stances and refer the reader to (Nazemi et al., 2021)

for further details. They are derived from a real-world

data set from a drought case presented in (Tricoire

et al., 2012) for the region of Thies in western Sene-

gal. The region is divided into 32 rural areas where

each has between 9 and 31 villages. In total, the re-

gion contains 500 nodes. Opening costs for LDCs are

assumed to be identical for all locations (5000 cost

units). In addition, it is assumed that the affected peo-

ple in a demand node walk to the closest LDC if the

distance is less than 6km. For each instance, ﬁrst, 10

sample scenarios (|S|= 10) are generated to cope with

demand uncertainty. Then, in order to compare the

algorithms for a larger number of scenarios, samples

with sizes of |S|= 100, 500, and 1000 are generated.

4.2 Solution Quality Indicators

In order to assess the performance of heuristic meth-

ods for bi- and multi-objective optimization, several

different quality indicators have been proposed in the

literature (see, (Zitzler et al., 2003)). In this study,

instead of the hypervolume indicator (I

H

) used by

(Nazemi et al., 2021), we employ the hypervolume

gap (gH) and the multiplicative ε-indicators.

For a bi-objective framework with minimization

objective functions, assume X is the set of feasible

solutions in decision space and Z = f (X) is the image

of this set in objective space. We assume A ⊆ Z is an

approximation set of a Pareto frontier, and R ⊆ Z is a

reference set.

The hypervolume gap (gH) deﬁnition is as fol-

lows:

gH% =

I

H

(R) − I

H

(A)

I

H

(R)

· 100 (14)

The closer the value of gH% to 0, the higher is the

quality of the approximated Pareto frontier.

The multiplicative epsilon indicator (I

ε

) ((Zitzler

et al., 2003)) gives the minimum factor ε that the ap-

proximation set (A) in the objective space has to be

multiplied with, such that it dominates the reference

set (R). The closer the value of I

ε

to 1, the better is the

approximated Pareto frontier. The ε value calculated

here can be interpreted as the gap between A and R in

the objective space.

4.3 Results

We compare all combinations of multi-objective cri-

terion space approaches used in (Nazemi et al., 2021)

(e, BB, and Mat) with two linear counterparts of

the classical (MA) and the approximation of subset-

based polyhedral (MB) representations of the two-

stage CVaR model for different levels of risk. {e-

MA, BB-MA, Mat-MA, e-MB, BB-MB, Mat-MB}

denotes the set of all methods evaluated in this study.

The run time for the two different representations

of the CVaR approach, MA, and MB, is reported in

Table 1. MA is solved to optimality, whereas MB is

the MB which is solved using the explained cutting-

plane technique to approximate its Pareto frontier. It

is worth mentioning that the cutting-plane method

solves the ﬁrst extreme point of the Pareto frontier to

optimality and all other non-dominated solutions are

computed without generating additional cuts. Results

for different levels of α are reported: the two special

cases identical with the expected value (risk-neutral)

and the worst-case (risk-averse) point of view, and

one ”middle” level and their corresponding k-values

(α = 1 −

k

N

). The results in this table show that the

combination of the BB with MB solves the instances

slightly faster than its combination with MA. In the

following, we compare the quality of the obtained so-

lutions using MA and MB.

After analyzing the two exact multi-objective

techniques (e and BB), we apply the Mat method in

combination with the MA and MB models to solve

the test instances. As shown in Table 2, the MB-Mat

method is slightly faster than MA-Mat. Additionally,

we solve a few instances with a larger number of sce-

narios. We report the run time and the number of

found non-dominated points (NDP) in Table 3. The

results show that the approximately solved subset-

based representation of the CVaR method (MB) pays

off when the number of scenarios increases.

Performance comparison of the Mat method and

the ε-constraint method for small instances where the

latter could ﬁnd the optimal Pareto frontier is shown

in Tables 4 and 5. To measure the quality, we report

the hypervolume gap (gH%) and the value of the I

ε

indicator. Analyzing the results in Table 4 conﬁrms

Bi-objective Risk-averse Facility Location using a Subset-based Representation of the Conditional Value-at-Risk

81

Table 1: Run time comparison of two representations of CVaR, MA, and MB, for different α-level/k-value for instances of

size 21-500 with sample size 10.

α/k value

0/10 0.7/3 0.9/1

#Node e-MA BB-MA e-MB BB-MB e-MA BB-MA e-MB BB-MB e-MA BB-MA e-MB BB-MB

21 1 2 1 2 2 1 1 2 2 2 1 1

44 16 22 15 19 17 23 13 14 20 26 10 17

56 24 31 19 23 18 22 15 23 19 33 14 16

72 36 51 29 35 45 48 30 41 64 103 36 40

90 62 110 59 80 73 78 59 75 99 179 64 73

106 123 229 118 133 151 161 111 132 210 380 141 175

120 143 241 135 163 144 158 121 160 138 262 100 115

163 445 818 392 552 417 506 369 503 636 1365 TL 596

182 577 1550 481 738 932 1053 701 776 1326 2548 1010 1192

203 956 2078 879 1208 616 699 552 716 515 TL TL 558

254 6103 TL TL TL TL TL TL TL TL TL TL TL

264 3903 6800 TL 2449 6453 TL TL TL TL TL TL 4007

275 7133 TL 6585 TL TL TL TL TL TL TL TL TL

295 TL TL TL TL TL TL TL TL TL TL TL TL

326 TL TL TL TL TL TL TL TL TL TL TL TL

355 TL TL TL TL TL TL TL TL TL TL TL TL

388 TL TL TL TL TL TL TL TL TL TL TL TL

410 TL TL TL TL TL TL TL TL TL TL TL TL

436 TL TL TL TL TL TL TL TL TL TL TL TL

449 TL TL TL TL TL TL TL TL TL TL TL TL

472 TL TL TL TL TL TL TL TL TL TL TL TL

482 TL TL TL TL TL TL TL TL TL TL TL TL

500 TL TL TL TL TL TL TL TL TL TL TL TL

Table 2: Run time comparison of a combination of ε-constraint method (e), and the proposed matheuristic (Mat) method with

MA and MB for different α/k-values for test instances of size 21-500 with sample size of 10 scenarios.

α/k value

0/10 0.7/3 0.9/1

#Node e-MA Mat-MA e-MB Mat-MB e-MA Mat-MA e-MB Mat-MB e-MA Mat-MA e-MB Mat-MB

21 1 0.51 1 0.42 2 0.43 1 0.43 2 0.58 1 0.46

44 16 3 15 3 17 3 13 2 20 3 10 3

56 24 4 19 4 18 4 15 3 19 5 14 3

72 36 11 29 8 45 9 30 6 64 11 36 7

90 62 18 59 16 73 23 59 19 99 21 64 14

106 123 32 118 37 151 36 111 28 210 44 141 29

120 143 37 135 40 144 39 121 36 138 43 100 38

163 445 183 392 102 417 132 369 98 636 167 392 104

182 577 212 481 140 932 178 701 128 1326 172 1010 136

203 956 206 879 198 616 217 552 172 515 210 TL 194

254 6103 414 TL 462 TL 589 TL 356 TL 568 TL 514

264 3903 479 TL 652 6453 470 TL 405 TL 478 TL 445

275 7133 686 6585 1082 TL 781 TL 711 TL 819 TL 636

295 TL 726 TL 1068 TL 729 TL 733 TL 836 TL 775

326 TL 893 TL 1027 TL 2407 TL 856 TL 1182 TL 914

355 TL 1447 TL 1378 TL 1499 TL 1339 TL 1619 TL 1333

388 TL 1611 TL 1711 TL 1835 TL 1726 TL 1931 TL 2115

410 TL 2133 TL 2251 TL 4062 TL 2167 TL 2847 TL 1901

436 TL 2439 TL 2413 TL 2531 TL 2340 TL 3608 TL 2220

449 TL 2633 TL 2857 TL 2953 TL 2546 TL 2903 TL 2534

472 TL 2955 TL 3185 TL 3167 TL 2971 TL 3194 TL 3155

482 TL 5054 TL 4044 TL 3715 TL 3216 TL 3885 TL 3162

500 TL 6238 TL 4442 TL 5653 TL 2955 TL 4278 TL 2358

that the combination of Mat and MA gives an upper

bound for the Pareto frontier (as Mat is a heuristic),

whereas the combination of the ε-constraint method

with MB ﬁnds a lower bound for the Pareto frontier.

A lower bound is obtained because the ε-constraint

method computes exact solutions of the approximated

problem, but approximates the CVaR objective from

below, as the MB only works with a reduced number

of subsets that are not necessarily worst for each non-

dominated point. Therefore, we obtain negative gH%

values and I

ε

values ≤ 1. Nonetheless, both methods

provide high-quality approximations of Pareto fron-

ICORES 2022 - 11th International Conference on Operations Research and Enterprise Systems

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Table 3: Performance comparison of a combination of ε-constraint method (e), and the proposed matheuristic (Mat) method

with MA and MB for α-level 0.7 and its corresponding k-value for larger number of scenarios. T[s] indicates the run time in

seconds.

α/k value: 0.7/3

e-MA Mat-MA e-MB Mat- MB

#Node #Scenario T[s] #NDP T[s] #NDP T[s] #NDP T[s] #NDP

21

100 161 17 27 17 56 17 5 17

500 6476 16 348 16 322 16 52 16

1000 TL 6 1668 17 TL 13 282 17

44

100 1076 30 178 30 316 30 34 30

500 TL 5 1374 32 TL 19 365 32

1000 TL 3 TL 3 TL 7 1239 34

56

100 2267 39 234 39 583 39 51 39

500 TL 3 2675 40 TL 22 776 38

1000 TL 3 TL 2 TL 3 1747 41

72

100 4186 50 180 50 826 50 107 50

500 TL 3 TL 2 TL 15 983 51

1000 TL 2 TL 2 TL 2 TL 2

tiers as both gH%, and I

ε

represent a gap between the

approximated sets and the reference set.

To measure the performance of the combination

of the Mat method and MB, we re-evaluate the second

stage of MB by solving the MA model, where the ﬁrst

stage decisions are ﬁxed to the values obtained from

the

MB. In other words, in order to assess its true

quality, the solution provided by MB is re-evaluated

by using the exact second-stage objective value as ob-

tained through MA. As can be seen in Table 5, the e-

MB and Mat-MB frameworks generate high-quality

Pareto frontiers.

After that, we also assess the performance of Mat-

MB (with re-evaluated second stage) on large-size

instances where the optimal Pareto frontier is not

known. For this purpose, we consider a union of the

Pareto frontiers (U ) obtained by e-MA, BB-MA, and

Mat-MA as the reference set. Table 6 shows the hy-

pervolume gap (gH%) values. Negative values indi-

cate that Mat-MB with re-evaluated second stage val-

ues obtains better approximations than the union of

the other methods.

5 CONCLUSIONS

In this paper, we investigate the subset-based poly-

hedral representation of CVaR to capture risk in a

two-stage bi-objective location-allocation model in-

troduced in (Nazemi et al., 2021), the purpose of

which is the optimal design of a relief network. The

model aims at ﬁnding the trade-off between a de-

terministic objective (minimization of location cost)

and an uncertain one (minimization of uncovered

demand) where uncertainty is assumed in demand

values. We introduce an approximate cutting-plane

method to deal with the computationally challenging

subset-based formulation of the two-stage stochastic

optimization problem containing the CVaR.

We evaluate the performance of the proposed

method by conducting a computational study on the

instances used by (Nazemi et al., 2021). Our exper-

iments show that the proposed approximate cutting-

plane approach pays off for a large number of scenar-

ios and they illustrate how incorporating an approx-

imation of the subset-based formulation affects the

quality of the produced solutions. In future research,

we would like to investigate whether the performance

of the cutting-plane method can be further improved

by employing additional enhancements.

ACKNOWLEDGEMENTS

The authors want to thank (Tricoire et al., 2012) for

providing us with the real-world data set. This re-

search was funded in whole, or in part, by the Aus-

trian Science Fund (FWF) [P 31366]. For the purpose

of open access, the author has applied a CC BY public

copyright licence to any Author Accepted Manuscript

version arising from this submission.

Bi-objective Risk-averse Facility Location using a Subset-based Representation of the Conditional Value-at-Risk

83

Table 4: Quality comparison using I

ε

and hypervolume gap (gH%) for approximated Pareto frontiers where the optimal Pareto

frontier is known from solutions of solved instances using ε-constraint method (e), with the sample size of 10 scenarios.

α/k value: 0.7/3

e-MA (Reference set) Mat-MA e-MB

#Node gH% I

ε

gH% I

ε

gH% I

ε

21 0 1 0.00 1.00 -0.01 0.99

44 0 1 0.00 1.00 0.00 1.00

56 0 1 0.00 1.00 0.00 0.91

72 0 1 0.42 1.22 -0.02 0.95

90 0 1 0.05 1.32 0.00 0.87

106 0 1 0.06 1.33 -0.01 0.94

120 0 1 0.02 1.35 0.00 1.00

163 0 1 0.00 1.00 0.00 0.96

182 0 1 0.00 1.00 0.00 0.94

203 0 1 0.02 1.02 0.00 0.97

Table 5: Quality comparison using I

ε

and hypervolume gap (gH%) for re-evaluated approximated Pareto frontiers using e-MB

and Mat-MB, where the optimal Pareto frontier is known from solutions of solved instances using ε-constraint method (e),

with the sample size of 10 scenarios.

α/k value: 0.7/3

e-MA (Reference set) e-MB Mat-MB

#Node gH% I

ε

gH% I

ε

gH% I

ε

21 0 1 0.00 1.00 0.00 1.00

44 0 1 0.00 1.00 0.00 1.00

56 0 1 0.00 1.00 0.00 1.00

72 0 1 0.01 1.03 0.01 1.03

90 0 1 0.00 1.04 0.00 1.04

106 0 1 0.01 1.00 0.01 1.00

120 0 1 0.00 1.00 0.00 1.00

163 0 1 0.00 1.00 0.00 1.00

182 0 1 0.00 1.04 0.00 1.04

203 0 1 0.00 1.00 0.00 1.00

Table 6: Performance of Mat-MB on large-size instances,

where the optimal Pareto-frontier is not known. The union

of the Pareto frontiers obtained by e-MA, BB-MA, and Mat-

MB is considered as a reference set. The sample size of

scenarios is 10.

α/k value: 0.7/3

U (Reference set) Mat-MB

#Node gH% gH%

254 0.00 0.00

275 0.00 0.00

295 0.00 -0.02

326 0.00 -0.06

355 0.00 0.01

388 0.00 -0.10

410 0.00 -0.05

436 0.00 -0.02

449 0.00 -0.03

472 0.00 -0.08

482 0.00 -0.02

500 0.00 0.00

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