Bounds on risk-averse mixed-integer multi-stage stochastic programming problems with mean-CVaR

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Contents lists available at ScienceDirect

European

Journal

of

Operational

Research

journal homepage: www.elsevier.com/locate/ejor

Stochastics

and

Statistics

Bounds

on

risk-averse

mixed-integer

multi-stage

stochastic

programming

problems

with

mean-CVaR

Ali

˙Irfan

Mahmuto

˘gulları,

Özlem

Çavu

¸s

∗

,

M. Selim

Aktürk

Department of Industrial Engineering, Bilkent University, Ankara 06800, Turkey

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 2 July 2016 Accepted 17 October 2017 Available online 13 November 2017 Keywords:

Stochastic programming

Mixed-integer multi-stage stochastic programming

Dynamic measures of risk CVaR

Bounding

a

b

s

t

r

a

c

t

Risk-averse mixed-integermulti-stage stochastic programming formsa classof extremely challenging problemssincetheproblemsize growsexponentiallywiththe numberofstages,theproblemis non-convexduetointegralityrestrictions,and theobjectivefunctionisnonlinearingeneral.Weproposea scenariotreedecompositionapproach, namelygroupsubproblem approach,toobtainboundsforsuch problemswithanobjectiveofdynamicmeanconditionalvalue-at-risk(mean-CVaR).Ourapproachdoes notrequireanyspecialproblemstructuresuchasconvexityandlinearity,thereforeitcanbeappliedto awiderangeofproblems. Weobtainlowerboundsbyusingdifferentconvolution ofmean-CVaR risk measuresanddifferentscenariopartitionstrategies.Theupperboundsareobtainedthroughtheuseof optimal solutionsofgroup subproblems.Usingtheselower andupper bounds, weproposeasolution algorithmforrisk-aversemixed-integermulti-stagestochasticproblemswithmean-CVaRriskmeasures. Wetesttheperformanceoftheproposedalgorithmonamulti-stagestochasticlotsizingproblemand comparedifferentchoicesoflower boundsand partitionstrategies.Comparisonoftheproposed algo-rithmtoacommercialsolverrevealedthat,ontheaverage,theproposedalgorithmyields1.13%stronger bounds.Thecommercialsolverrequiresadditionalrunningtimemorethanafactorofﬁve,onthe aver-age,toreachthesameoptimalitygapobtainedbytheproposedalgorithm.

1. Introduction

In risk-averse stochastic optimization problems, risk measures are used to assess the risk involved in the decisions made. Due to the structural properties of risk measures, risk-averse models are more challenging than their risk-neutral counterparts. The multi- stage risk-averse stochastic models are even more complicated due to their dynamic nature and excessive amount of decision variables. Both the risk-neutral and risk-averse multi-stage stochastic problems are non-convex when some of the decision variables are required to be integer valued. Therefore, the solution methods suggested for convex multi-stage stochastic problems cannot be used to solve these problems.

In this study, we consider risk-averse mixed-integer multi-stage stochastic problems with an objective function of dynamic mean conditional value-at-risk (mean-CVaR). Both CVaR and mean-CVaR are coherent measures of risk that have been used in the literature extensively (see, Rockafellar&Uryasev, 2002). Coherent measures of risk and their axiomatic properties are introduced in the pio-

∗ _{Corresponding author.}

E-mail address: ozlem.cavus@bilkent.edu.tr (Ö. Çavu ¸s ).

neering paper by Artzner,Delbaen, Eber,andHeath (1999). Later, the theory of coherent risk measures is extended by Ruszczynski andShapiro(2006a, 2006b), and references therein.

In a multi-stage decision horizon, risk involved in a stream of random outcomes is considered. Therefore, dynamic coherent risk measures are introduced to quantify the risk in multi-stage models (see, Artzner,Delbaen, Eber, Heath, andKu, 2007; Kovacevic and Pﬂug, 2009; Pﬂug and Römisch, 2007; Ruszczynski and Shapiro, 2006a;2006b, and references therein).

For the multi-stage stochastic optimization problems with dynamic measures of risk, some exact solution techniques are suggested under the assumption that the decision variables are con- tinuous. These techniques, such as stochastic dual dynamic programming (SDDP), which is first suggested by Pereira and Pinto (1991)for risk-neutral problems and then extended to risk-averse problems by Shapiro(2011), Shapiro,Tekaya, daCosta,andSoares (2013), Philpott, de Matos, and Finardi (2013), and Lagrangian relaxation of nonanticipativity constraints suggested by Collado, Papp,andRuszczyński(2012) rely on the convex structure of the problem, therefore, they cannot be used to find an exact solution when some of the decision variables are integer valued. On the other hand, these methods can be used to obtain lower bounds on the optimal value of multi-stage stochastic integer problems. https://doi.org/10.1016/j.ejor.2017.10.038

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Bonnans,Cen, andChristel (2012) propose an extension of SDDP method for the risk-neutral problems with integer variables by re- laxing the integrality requirements in the backward steps of the algorithm. Later, Bruno,Ahmed,Shapiro,andStreet(2016) extend this approach to risk-averse integer problems. Zou, Ahmed, and Sun(2016)consider SDDP method to solve risk-neutral multi-stage mixed-integer problems with binary state variables. They prove that SDDP method provides an exact solution to the problem in ﬁnite number of iterations when the cuts satisfy some suﬃcient conditions. Similarly, Schultz (2003) uses Lagrangian relaxation of nonanticipativity constraints to obtain lower bounds within a branch-and-bound procedure for risk-neutral multi-stage problems with integer variables. However, these approaches rely on some restrictive assumptions. SDDP method requires stagewise indepen- dency of random process and the branch-and-bound procedure requires complete recourse assumptions. Therefore, they cannot be applicable to a wide range of problems.

A recent stream of research proposes an alternative way of obtaining bounds for mixed-integer multi-stage stochastic problems via a scenario tree decomposition. In that approach, the sample space is partitioned into subspaces called as groups, and the problem is solved for the scenarios in a group instead of the original sample space. These smaller problems are called as group subproblems. Sandıkçı, Kong, and Schaefer (2013) propose a group subproblem approach for risk-neutral mixed-integer two- stage stochastic problems. They show that the expected value of the optimal values of group subproblems gives a lower bound on the optimal value of the original problem. Later, this approach is extended to the risk-neutral multi-stage problems by Sandıkçı and Özaltın (2014), Zenarosa, Prokopyev, and Schaefer (2014), and Maggioni, Allevi, and Bertocchi (2016). Recently, Maggioni andPﬂug (2016)apply group subproblem approach to risk-averse mixed-integer multi-stage stochastic problems where the objective is a concave utility function applied to the total cost over the planning horizon. Although, group subproblems include less number of scenarios than the original problem, the length of the decision horizon in group subproblems and the original problem is the same. Therefore, one may argue that scalability is a drawback of this approach when the decision horizon is too long.

In this study, we propose a scenario tree decomposition algorithm for risk-averse mixed-integer multi-stage stochastic problems with a dynamic objective function defined via mean-CVaR. The suggested algorithm is based on group subproblem approach and is used to find lower and upper bounds on the optimal value of the problem. We propose infinitely many valid lower bounds on mean-CVaR risk measure that can be used within the frame of the algorithm. We also investigate the effect of scenario partitioning strategies on the quality of the different lower bounds by considering different partitioning strategies based on the structure of the scenario tree and disparateness of scenario realizations.

As outlined earlier, our approach does not require any special structural property such as convexity and linearity of the feasible set. Moreover, it does not require complete recourse or stagewise independence assumptions, therefore, it can be applied to a wide range of problems. We conduct computational experiments on a multi-stage lot sizing problem by considering different choices of bounds and scenario tree partitions. The experiments reveal that the obtained bounds are tight and require reasonable CPU times. Our approach yields 1.13% stronger bounds than solving the problem with IBM ILOG CPLEX. On the other hand, CPLEX requires more than 5.45 times of CPU time to obtain the same optimality gaps of our approach.

The organization of the paper is as follows: In Section 2, we present problem deﬁnition and some preliminaries. Section 3 in- cludes our main results on obtaining different lower bounds for mean-CVaR via a scenario grouping approach. We consider the

application of these lower bounds to a risk-averse mixed-integer multi-stage stochastic problem with a dynamic objective function deﬁned via mean-CVaR. We also suggest a method to obtain an upper bound. The computational study conducted on a multi- stage lot sizing problem and related discussions are presented in Section 4. Section 5 is devoted to concluding remarks and future research directions.

2. Risk-aversemixed-integermulti-stagestochasticproblems withdynamicmean-CVaRobjective

We consider a multi-stage discrete decision horizon where the decisions at stage t ∈

{

1 ,...,T

}

are made based on the available information up to that stage. Let

be a ﬁnite sample space and

{

0 ,∅

}

=F1 ⊂ F2 ⊂ · · · ⊂ FT =F be a ﬁltration, that is, an ordered

set of sigma algebras on

, representing gradually increasing information through stages. We use

ξ

tand xtto denote the vector of

problem parameters and decisions at stage t∈

{

1 ,...,T

}

, respectively. For each t_∈

{

1 _,2 _,_._._._,T

}

_,

ξ

t and xt are Ft−measurable. At

ﬁrst stage, the vector of problem parameters

ξ

1 and decisions x1

are deterministic, since F1 =

{

0 ,∅

}

. At stage t∈

{

2 ,...,T

}

, some

or all problem parameters are random.

An element

ω

of

is called as a scenario. A scenario

ω

∈

corresponds to a realization of a sequence of random parameters

ξ

2

(

ω

)

,. . .,

ξ

T

(

ω

)

in stages 2 ,...,T.

Our main interest is a risk-averse mixed-integer multi-stage stochastic problem with an objective of dynamic risk measure ϱ1,T( ·) over the horizon 1 ,...,T. The problem can be deﬁned as:

min

x∈X

1,T

(

f1

(

x1

)

,f2

(

x2,

ξ

2

)

,...,fT

(

xT,

ξ

T

))

, (1)

where _X₌_X1× X2

(

x1,

ξ

2

)

× · · · × XT

(

xT−1,

ξ

T

)

is the abstract rep-

resentation of possibly nonlinear feasibility set. Let R and Z denote the set of real numbers and integers, respectively. X1 ⊆ Rn1× Zm1

is a mixed-integer deterministic set and, for t_∈

{

2 _,_._._._,T

}

_, _Xt:

R n_t₋₁_{× Z}m_t₋₁_×

_{⇒ R}nt× Z mt _areF_t−measurable _{mixed-integer}

point-to-set mappings. The cost in the ﬁrst stage is deterministic and represented by a possibly nonlinear, real-valued function

f1: R n1× Z m1→ R . The cost functions ft: R nt× Z mt×

→ R ,t ∈

{

2 ,...,T

}

are Ft−measurable, real-valued, and may be nonlinear.

Classical solution methods such as SDDP and Lagrangian relaxation of nonanticipativity constraints cannot be used to solve problem (1)due to integrality restrictions of some decision variables. Therefore, our focus is to obtain bounds on (1)where the objective function _ϱ1,T( ·) is a dynamic risk measure deﬁned via mean-CVaR.

Now, we present some necessary concepts and notation on coherent, conditional, and dynamic risk measures to exploit the structure of problem (1).

2.1. Coherentmeasuresofrisk

Let Z :=L∞

(

,F,P

)

be the space of bounded and F-

measurable random variables with respect to sample space

and probability distribution P. Let Z_,W_∈_{Z represent} uncertain outcomes for which lower realizations are preferable. Also, let Z_ωbe the value that the random variable Z takes under scenario

ω

∈

. As deﬁned in Artzneretal.(1999), a function

ρ

: _Z_→_R is called a coherent measure of risk if it satisﬁes:

(A1) Convexity:

ρ

(

α

Z+

(

1 −

α

)

W

)

≤

αρ

(

Z

)

+

(

1 −

α

)

ρ

(

W

)

for all

Z_,W_∈_Z and

α

_∈[0 _,1] _,

(A2) Monotonicity: Z W implies

ρ

(

Z

)

≥

ρ

(

W

)

for all Z,W ∈ Z,

(A3) Translational Equivariance:

ρ

(

Z+t

)

=

ρ

(

Z

)

+t for all t∈ R and Z∈Z,

(A4) Positive Homogeneity:

ρ

(

tZ

)

= t

ρ

(

Z

)

for all t> 0 and Z∈

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Fig. 1. An example of four-stage scenario tree. (a) 1 , 2 , 3 and 4 are the set of nodes at stages 1, 2, 3 and 4, respectively. (b) C(v) is the set of children nodes of node v , a (v) is the ancestor node of node v and p vu is the conditional probability of node u given v .

where ZW indicates pointwise partial ordering such that Z_ω≥ Wω for a.e.

ω

∈

.

We assume that cardinality of

is ﬁnite and F is the set of all events deﬁned on

. Then, the probability of a scenario

ω

∈

can be speciﬁed as p_ω>0. In this case, elements of both Z and its dual space _Z∗can be represented as elements of _R||since both _{Z and}

Z∗_{are isomorphic to}_R||_.

Let

μ

_ωbe the value that

μ

∈Z∗ _{takes under scenario}

_ω

_∈

_.

For Z_∈_{Z and}

μ

_∈_Z∗_, the scalar product

_·,_·

is deﬁned as

μ

,Z

:=

ω∈

p_ω

μ

_ωZ_ω.

The following fact is known as dual representation of coherent measures of risk (see, RuszczynskiandShapiro,2006b, for example): if

ρ

( ·) is a coherent measure of risk, then, under some assumptions, for every random variable Z_∈_Z,

ρ

(

Z

)

= max

μ∈A

μ

,Z

, (2)

where A⊆ Z∗_{is a compact and convex set. We call this set as the}

dual set of the risk measure

ρ

( ·). A coherent measure of risk can be characterized via its dual set. The reader is referred to Ruszczynski andShapiro (2006b) for a detailed discussion on the dual representation of coherent measures of risk.

2.2. Conditionalanddynamicriskmeasures

When a multi-stage stochastic process is considered, all realizations of the process form a scenario tree in the ﬁnite distribution case. In this section, we follow the notation used by Collado et al. (2012) to represent the scenario tree. Let

t be the set

of nodes at stage t∈

{

1 ,...,T

}

. At stage t= 1 , there is only one node, called as root node and it is represented by

v

1. The nodes

at stages t ∈

{

2 ,...,T

}

represent elementary events in Ft, that is Ft =

σ

(

t

)

, a sigma algebra on

t.

The set

T corresponds to all possible scenarios, that is

T =

. Each node

v

∈

t,t ∈

{

2 ,...,T

}

has a unique ancestor at stage t− 1 and this ancestor node is called as a

(

v

)

. Also, each node

v

∈

t,t∈

{

1 ,...,T− 1

}

has a set of children nodes C

(

v

)

such

that C

(

v

)

=

{

u ∈

t+1: a

(

u

)

=

v

}

. The probability measure P can be

speciﬁed by conditional probabilities

pv u:= P[ u

|

v

] ,

v

∈

t,u∈ C

(

v

)

, t ∈

{

1 ,...,T − 1

}

,

and probability of a scenario

ω

_∈

Tcan be computed as pω = pv 1v 2pv 2v 3...pv t−1ω ,

where

v

1,

v

2,...,

v

t−1,

ω

is the unique path from root node

v

1 to

node

ω

.

The notation mentioned above is depicted in Fig.1for a four- stage scenario tree.

For a multi-stage decision horizon with stages t∈

{

1 ,...,T

}

, let

Zt:=L∞

(

,Ft,P

)

. The mapping

ρ

_F_t₊₁_|_F_t: Zt+1 →Ztis called as

one-step conditional risk measure if it satisﬁes axioms (A1)–(A4) for corresponding spaces Zt and Zt+1 for all t∈

{

1 ,2 ,...,T− 1

}

.

The risk involved in a sequence of random variables Zt ∈Zt,t∈

{

1 _,_._._._,T

}

adopted to the ﬁltration _Ft,t∈

{

1 ,...,T

}

can be evalu-

ated by a time consistent dynamic measure of risk ϱ1,T( ·), that is,

1,T

(

Z1,Z2,...,ZT

)

= Z1 +

ρ

F2|F1

(

Z2 +

ρ

F3|F2

Z3+ · · · +

ρ

FT|FT−1

(

ZT

)

...

)

. (3)

The structure (3)is presented in RuszczynskiandShapiro(2006a). Later, Ruszczy´nski(2010)shows that the representation (3)can be constructed using monotonicity of conditional risk measures and the concept of time consistency. A time consistent dynamic risk measure ϱ1,T( ·) is not, in general, law invariant, even in the case

it is a composition of law invariant conditional risk measures (see, Shapiro, 2012). The reader is referred to Ruszczy´nski (2010) and Eckstein,Eskandani,andFan(2016)for the deﬁnition of time consistency and law invariance.

Colladoetal.(2012)show that the dual representation of coherent risk measures can be extended to dynamic measures of risk. If ϱ1,T( ·) is a dynamic risk measure given as in (3), then for every

sequence of random variables

{

Zt ∈Zt

}

tT=1,

1,T

(

Z1,Z2,...,ZT

)

= max qT∈QT

qT,Z1 + Z2 + · · · + ZT

, (4)

where

QT = At−1 ◦ · · · ◦ A 2 ◦ A 1, (5)

and At,t∈

{

2 ,...,T

}

is a convex and compact set used in the dual

representation of

ρ

_F

t+1|Ft

(·)

. The operator “◦” deﬁnes convolution

of probability measures, that is,

(

μ

t ◦ q t

)(

u

)

= qt

(

a

(

u

))

μ

t

(

a

(

u

)

,u

)

,

∀

u ∈

t+1,

and

At ◦ Q t =

{

μ

t ◦ q t: qt ∈ Qt,

μ

t ∈ At

}

,

for all t_∈

{

1 _,2 _,_._._._,T_{− 1}

}

. Recall that a( u) is the ancestor node of

u.

In this study, we use conditional mean-CVaR as one-step conditional risk measure. Therefore, the next section is devoted to the deﬁnition of mean-CVaR.

2.3.CVaRandmean-CVaR

An important and extensively used example of coherent measures of risk is Conditional Value-at-Risk (CVaR). CVaR of Z ∈ Z at

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CVaR _α

(

Z

)

:= inf η∈R

η

+ 1 1 −

α

E [

(

Z−

η

)

+]

, (6)

where

(

a

)

₊:=max

{

a,0

}

for a∈R.

Given a level parameter

α

_∈[0, 1) and a weight parameter

1∈ [0, 1], mean-CVaR of Z ∈ Z is deﬁned as

ρ

(

Z

)

:=

(

1 −

1

)

E [ Z]+

1CVaR α

(

Z

)

. (7)

As seen in (7), despite CVaR, mean-CVaR risk measure conveys the expected value information of a random variable, as well. As

α

or

1increase, the decision-maker gets more risk-averse.

The expression in (7)can equivalently be represented as following linear program for ﬁnite probability spaces.

ρ

(

Z

)

= minimize ϑ,η

(

1 −

1

)

ω∈ p_ωZω +

1

η

+ 1 1 −

α

ω∈ p_ω

ϑ

_ω

subject to

ϑ

ω ≥ Z ω −

η

,

∀

ω

∈

ϑ

ω ≥ 0 ,

∀

ω

∈

.

When the sample space is ﬁnite, the dual representation (2) holds for mean-CVaR with the set A represented as (see, Ruszczynski&Shapiro,2006b):

A=

{

μ

∈ Z∗_{: 1}₋

1 ≤

μ

ω ≤ 1 +

2,

∀

ω

∈

and E [

μ

] = 1

}

, (8)

where

2:= ₁₋

α

_α

1≥ 0 ,

and E [

μ

] = _ω_∈pω

μ

ω .

For any Zt+1 ∈Zt+1, the one-step conditional mean-CVaR risk

measure

ρ

_F

t+1|Ft

(

Zt+1

)

with parameters

α

t∈[0, 1) and

1t∈[0, 1]

and its dual set _At are deﬁned similar to (7)and (8). However, in

(6), the inﬁmum is over

η

t ∈Zt and the expectation operators in

(6)–(8)are replaced with conditional expectations with respect to

Ft .

For the remainder of the paper, we will focus on mean-CVaR risk measure. Hence, we will use

ρ

( ·) to refer to mean-CVaR and

ρ

Ft+1|Ft

(·)

, t∈

{

1 ,2 ...,T− 1

}

to refer to one-step conditional

mean-CVaR.

3. Bounds

The main motivation of this section is to propose lower and upper bounds for problem (1)with an objective of dynamic mean- CVaR. Therefore, in Section 3.1, we ﬁrst propose a continuum of time consistent lower bounds for mean-CVaR risk measure by scenario grouping. Some possible lower bounds are presented in Section3.2. The application of these time consistent bounds to a risk-averse mixed-integer multi-stage stochastic problems with an objective of (3) is presented in Section 3.3Extension of the proposed lower bounds to other dynamic mean-CVaR risk measures is discussed in Section3.4. In Section 3.5, we propose a method for obtaining an upper bound to the problem. The proposed algorithm beneﬁts from these results and yields lower and upper bounds for problem (1).

3.1.Lowerboundsformean-CVaRriskmeasure

Let

ρ

( _·) be a mean-CVaR risk measure with dual set _A. We would like to construct another coherent risk measure

ρ

(·)

which provides a time consistent lower bound for

ρ

( ·). The risk measure

ρ

(·)

, or equivalently its dual set _A_, can be constructed in different ways. When the cardinality of the sample space is large, due to computational concerns, one may think of dealing with subsets of sample space separately and then obtaining a lower bound for

ρ

( ·). For such construction, we need the deﬁnition of scenario groups and partition.

A subset of scenarios S⊆is called as a group. Let S=

{

Sj

}

Jj=1

be a collection of groups that forms a partition of

, that is, J

j=1Sj =

and SjSj=∅ for all j,j∈

{

1 ,2 ,...,J

}

such that j=j. Note that the groups may not be necessarily disjoint (see, Sandıkçı &Özaltın,2014), i.e. S_jSj₌_∅_, but for the ease of representation, we partition the sample space into disjoint groups. Let G be a

σ

−algebra generated by partition S where each group

Sj ∈S,j∈

{

1 ,2 ,...,J

}

corresponds to an elementary event j of G.

The probability of an elementary event j is pj= _ω_∈S_jpω which

is the total probability of scenarios in Sj. We also deﬁne the ad-

justed probability of each scenario

ω

as pjω =pω /pj for all

ω

∈Sj

and j ∈

{

1 ,2 ,...,J

}

. Note that, G is a sub

σ

−algebra of F. Once a partition of the sample space

is given, one way to construct

ρ

(·)

is to deﬁne it as a convolution of a coherent risk measure

ρ

G : L∞

(

,G,P

)

→R with dual set AG and a one-step

conditional risk measure

ρ

_F_|_G : Z→L∞

(

,G,P

)

with dual set

AF|G. That is,

ρ

(·)

=

(

ρ

G ◦

ρ

F|G

)(·)

, and its dual set is the con-

volution of the sets _A_G and _A_F_|_G such that _A₌_A_F_|_G_◦_A_G. Note that,

ρ

_F_|_G

(·)

can be represented in terms of

ρ

S_j

(·)

,j∈

{

1 ,2 ...,J

}

, that is,

ρ

F|G

(·)

j=

ρ

Sj

(·)

(see, Miller and

Ruszczy´nski, 2011, for example) where

ρ

S_j: L∞

(

,

σ

(

Sj

)

,P

)

→R

is a coherent risk measure and

σ

( S_j) is the

σ

_−algebra on S_j. Fig.2 depicts aforementioned notation for a given partition of a scenario tree with ﬁve scenarios.

For mean-CVaR,

ρ

(·)

or equivalently its dual set _A_, can be ex- plicitly stated. Let parameters of

ρ

G be

α

1∈ [0, 1),

11∈ [0 ,1] , and

1 2= α 1 1−α1

11, and parameters of

ρ

F|G be

α

2∈[0, 1),

21∈ [0 ,1] and

2 2= α 2

1−α2

12. Consider the convolution

ρ

=

ρ

G ◦

ρ

F|G : F→

R and its dual set

A = AF|G ◦ AG =

{

μ

∈ Z∗:

μ

=

μ

1◦

μ

2,

μ

1 ∈ AG,

μ

2 ∈ AF|G

}

=

{

μ

∈ Z∗_:

_μ

₌

_μ

1_◦

_μ

2_,₁₋

1 1≤

μ

1j≤ 1 +

21,

∀

j ∈ 1 ,2 ...,J and E [

μ

1] = 1 ,1 −

2 1≤

μ

2ω ≤ 1 +

22,

∀

ω

∈

and E [

μ

2

|

G] = 1

}

, (9) where E [

μ

1_]₌ j∈{1,...,J}pj

μ

1j,

E [

μ

2

_|

_G_]

j= ω∈S_jpjω

μ

2_ωfor j ∈

{

1 ,...,J

}

, and 1 is a G-measurable random variable that takes value of one in all realizations. Construction of the set A for the example in Fig.2can be seen in AppendixA.

Now, we are ready to prove that a lower bound for mean-CVaR risk measure

ρ

( ·) can be obtained by

ρ

(·)

=

(

ρ

G ◦

ρ

F|G

)(·)

. Proposition1. Let

ρ

( ·) be a mean-CVaRrisk measurewith param-eters

α

∈[0, 1),

1∈[0, 1],

2 = 1−αα

1 ≥ 0,anddual setA.Alsolet

ρ

(·)

=

(

ρ

G ◦

ρ

F|G

)(·)

where

ρ

G isamean-CVaRriskmeasurewith parameters

α

1_∈_{[0, 1),}

1

1∈[0 ,1] ,

21= α

1

1−α1

11, and dual setAG;

and

ρ

_F_|_G is a one-step conditional mean-CVaR risk measure with parameters

α

2_∈_{[0, 1),}

2

1∈ [0 ,1] ,

22= α

2

1−α2

21,anddualsetAF|G.

Then,

ρ

(

Z

)

≤

ρ

(

Z

)

forallZ∈Z if

1 −

1 ≤

(

1 −

11

)(

1 −

12

)

and 1 +

α

1 1 −

α

1

1 1

1 +

α

2 1 −

α

2

2 1

≤ 1 +

α

1 −

α

1. (10)

Proof. Let

μ

_∈_A₌_A_F_|_G_◦_A_G. Then, from (9), there exist

μ

1_∈ _A

G and

μ

2∈ AF|G such that

μ

=

μ

1◦

μ

2 with E[

μ

1] =1

and _E[

μ

2

|

_G_]₌₁_{. Properties of conditional expectation implies that}

E [

μ

] = E [ E [

μ|

G]] = E [ E [

μ

1_◦

_μ

2

_|

_G_]]₌_E_[

_μ

1_{◦ E}_[

_μ

2

_|

_G_]]₌

E[

μ

1_{◦ 1}_]₌_E_[

_μ

1_]₌₁_.

From the deﬁnition of

2,

21 and

22, second part of (10)im-

plies

(

1 +

1

2

)(

1 +

22

)

≤ 1 +

2. Moreover, by (9),

(

1 −

11

)(

1 −

2

(5)

Fig. 2. (a) An example partition for a two-stage scenario tree: There are ﬁve scenarios 1,2,3,4, and 5 with probabilities p 1 , p 2 , p 3 , p 4 , and p 5 , respectively. (b) S = { S a , S b} is a partition of where S a = { 1 , 2 , 3 } and S b = { 4 , 5 } . Nodes a and b correspond to groups S a and S b with probabilities p a = p 1 + p 2 + p 3 and p b = p 4 + p 5 , respectively. (c) ρ: Z → R is the original risk measure. (d) G is a sub σ−algebra of F. ρG : L ∞(, G , P) → R is a coherent risk measure and ρF|G : Z → L ∞(, G , P) is a one-step conditional risk measure that can be represented via ρSa : L ∞(, σ(Sa) , P) → R and ρSb : L ∞(, σ(Sb) , P) → R as [ ρF|G(·)] a = ρSa(·) and [ ρF|G(·)] b = ρSb(·) .

(

1 −

2

1

)

and

(

1 +

21

)(

1 +

22

)

≤ 1 +

2, then 1 −

1 ≤

μ

ω ≤ 1 +

2,

for all

ω

∈

which implies,

μ

∈A. Since

μ

is arbitrary, A⊆ A. For any Z_∈_Z, let

ρ

(

Z

)

₌max _μ_∈_A

μ

_,Z

and

μ

∗_∈arg max _μ_∈_A

μ

,Z

. If A⊆ A, then

μ

∗∈A and

ρ

(

Z

)

=

μ

∗_,_Z

_{≤ max}_μ

∈A

μ

,

Z

₌

ρ

(

Z

)

. Since Z is arbitrary,

ρ

(

Z

)

_≤

ρ

(

Z

)

for all Z_∈_Z.

Proposition 1 partially extends Theorem 8 and Corollary 6 of Iancu,Petrik,andSubramanian(2015)to mean-CVaR risk measure. It implies that, under conditions (10),

ρ

(·)

=

(

ρ

G ◦

ρ

F|G

)(·)

is a

valid lower bound for

ρ

( ·) for any partition S of

. If

ρ

( ·) is a conditional mean-CVaR risk measure, Proposition1still applies. In this case, the expectations in the proof are replaced with corresponding conditional expectations.

3.2. Possiblelowerbounds

We have shown that a lower bound for

ρ

( ·) can be obtained by convolutions of mean-CVaR risk measures whose parameters

satisfy condition (10). Due to Proposition 1, we can generate in- ﬁnitely many lower bounds. Under the settings on Proposition 1, Table 1 presents some special cases of parameters of

ρ

G

(·)

and

ρ

F|G

(·)

such that they can be used to obtain a lower bound for

ρ

( ·).

Bounds

ρ

_G ◦ EF|G and EG ◦

ρ

F|G represent the extreme cases

where either

ρ

G

(·)

or

ρ

F|G

(·)

is the expectation operator. Bound

ρ

s

G◦

ρ

Fs |G is an intermediate case where both

ρ

G

(·)

and

ρ

F|G

(·)

have the same parameters, that is,

α

1₌

α

2_,

1

1=

12 and

21=

22.

Under these conditions, in order to construct the largest set A, the inequalities in (10)are forced to hold at equality.

An interesting question is whether one of the possible lower bounds presented above is always preferable among others. Fol- lowing example reveals that

ρ

s

G◦

ρ

Fs |G is not necessarily the

tightest bound among others.

Example1. Consider a random variable Z with sample space

=

{

ω

i

}

4i=1. All four realizations have equal probabilities, that is, pω i=

(6)

Table 1

Possible choices of ρG(·) and ρF|G(·) that can be used to obtain lower bound on mean-CVaR risk measure ρ( ·).

Values of different lower bounds (LB’s) for Example 1 .

LB Choice S S

ρG ◦ E F|G 3.5 2.5

ρs

G ◦ρsF|G 3.12 3

E G ◦ρF|G 3 3.5

1 /4 for all i∈ {1, 2, 3, 4}. The value that Z takes under scenario

ω

i

is i, that is, Z_ω_i=i for i∈{1, 2, 3, 4}.

Let

1 =1 and

α

= 0 .5 , then (7)reduces to CVaR value at

α

=

0 _.5 and then

ρ

(

Z

)

₌ 3 _.5 .

Two different partitions of scenarios are S =

{{

ω

1,

ω

2

}

,

{

ω

3,

ω

4

}}

and S=

{

ω

1,

ω

4

}

,

{

ω

2,

ω

3

}

. Values of the three

bounds for partitions _{S and}_Sare given in Table2.

As seen in Table2, the tightest bounds for partitions S andS

are bounds

ρ

_G ◦ EF|G and EG ◦

ρ

F|G, respectively. Another ob-

servation is the fact that

ρ

s

G◦

ρ

Fs |G is not necessarily the tight-

est bound among others. In Example 1, although either

ρ

_G ◦ E F|G or E G ◦

ρ

F|G can be the tightest bound among others un-

der different scenario partitions, the computational experiments in Section 4 reveal that EG ◦

ρ

F|G is the most promising lower

bound choice.

Although Shapiro, Dentcheva, and Ruszczy´nski (2009) show that, under some assumptions, the lower bound

ρ

_G ◦ EF|G can

be extended to any coherent risk measures, the other bounds pro- vided in Table1may not be applicable for all coherent risk measures. Example2reveals that EG ◦

ρ

F|G is not necessarily a valid

lower bound for an arbitrary coherent risk measure.

Example2. Consider a random variable Z that takes values Z_ω₁= 100 ,Z_ω₂ =0 ,Z_ω₃= 1 and Z_ω₄= 500 with probabilities 0.3, 0.2, 0.4 and 0.1, respectively. We use the ﬁrst-order mean semi-deviation as a risk measure, that is:

ρ

(

Z

)

= E [ Z]+

κ

E [

(

Z− E [ Z]

)

₊] ,

κ

∈ [0 ,1] . (11) Let

ρ

_F_|_G

(·)

be the one-step conditional ﬁrst-order mean semi- deviation with the same

κ

value as in (11). Set

κ

= 0 .5 . For partition S=

{

ω

1,

ω

2

}

,

{

ω

3,

ω

4

}

,

ρ

(

Z

)

= 104 .32 but

(E

G ◦

ρ

F|G

)(

Z

)

=106 .36 .

Therefore, E G ◦

ρ

F|G is not necessarily a valid lower bound for

all coherent risk measures.

3.3.Lowerboundforoptimizationproblem

In this section, we extend the lower bound proposed in Proposition1to a risk-averse mixed-integer multi-stage stochastic problem with an objective of dynamic mean-CVaR risk measure. Using the structure presented in (3), the problem (1)can be writ- ten as (P) min x1∈X1 f1

(

x1

)

+

ρ

(

Q

(

x1,

ξ

))

, (12) where Q

(

x1,

ξ

)

= min xt∈Xt,t∈{2,...,T}

2,T

(

f2

(

x2,

ξ

2

)

,...,fT

(

xT,

ξ

T

)

, (13)

ξ

=

{

ξ

t

}

T_t₌₂,

ρ

( ·) is a mean-CVaR risk measure with parameters

α

∈[0, 1) and

1∈[0, 1], and ϱ2,T( ·) is a dynamic mean-CVaR. Let x∗₁ and z∗ be an optimal ﬁrst stage solution and the optimal value of (P), respectively.

Recall the partition _S₌

{

S_j

}

J_j

=1 of

and sigma algebra G in-

duced by this partition. Then, the jth group subproblem is just problem (P) with sample space Sj and adjusted probabilities pjω ,

ω

∈Sj. Additionally, the risk measure

ρ

( ·) in (12) is replaced by

ρ

S_j

(·)

. For j∈

{

1 ,2 ,...,J

}

, let zjbe the optimal value of jth group

subproblem. Also let ZLB be a G-measurable random variable that

takes value of zj_{with probability}_p

j = ω∈S_jpω .

In Theorem 1, we show that a lower bound for risk-averse mixed-integer multi-stage stochastic problem (P) can be obtained by using optimal values of group subproblems.

Theorem 1. Let

ρ

_G : L∞

(

,G,P

)

→ R be a mean-CVaR risk

mea-sure with parameters

α

1_∈_[0,₁₎_and

1

1∈[0 ,1] ; and

ρ

F|G :

L∞

(

,F,P

)

→L∞

(

,G,P

)

bea conditional mean-CVaRrisk

mea-sure with parameters

α

2_∈_{[0, 1)}_and

2

1∈ [0 ,1] satisfying 1 −

1 ≤

(

1 −

1 1

)(

1 −

12

)

and

(

1 + α 1 1−α1

11

)(

1 + α 2 1−α2

12

)

≤ 1+1−αα

1.Then, z∗_≥

ρ

_G

(

ZLB

)

.

Proof. Recall that x∗₁ is an optimal ﬁrst stage solution of (P). Note that, it is a feasible ﬁrst stage solution for each group subproblem. By optimality of each group subproblem, we have

f1

(

x∗1

)

+

ρ

Sj

(

Q

(

x ∗

1,

ξ

))

≥ z j,

∀

j ∈

{

1 ,...,J

}

and

f1

(

x∗1

)

+

ρ

F|G

(

Q

(

x∗1,

ξ

))

Z LB. (14)

The values on the both sides of inequality (14)are G−measurable. Since,

ρ

_G

(·)

is a coherent risk measure and it satisﬁes monotonicity axiom (A2), we get

ρ

G

(

f1

(

x∗1

)

+

ρ

F|G

(

Q

(

x∗1,

ξ

)))

≥

ρ

G

(

ZLB

)

. (15)

Note that, f1

(

x∗1

)

is an F−measurable cost. Since G is a sub

σ

−algebra of F, f1

(

x∗1

)

is G−measurable, as well. Applying trans-

lational equivariance axiom (A3) to the left hand side of (15), we get

ρ

G

(

ρ

F|G

(

f1

(

x∗1

)

+ Q

(

x∗1,

ξ

)))

≥

ρ

G

(

ZLB

)

. (16)

Since conditions in (10)are satisﬁed, we can apply Proposition1to the left hand side of inequality (16)and obtain:

ρ

(

f1

(

x∗1

)

+ Q

(

x∗1,

ξ

))

≥

ρ

G

(

ZLB

)

.

Finally, using translational equivariance axiom (A3), we get

z∗= f1

(

x∗1

)

+

ρ

(

Q

(

x∗1,

ξ

))

≥

ρ

G

(

ZLB

)

.

Theorem1implies that a lower bound on the optimal value of

(P) can be obtained by solving group subproblems and then applying

ρ

G

(·)

to the optimal values of these group subproblems. Since

group subproblems include smaller number of scenarios compared to the original problem, they are computationally less challenging. Moreover, applying

ρ

G

(·)

to the optimal values of group subprob-

(7)

calculation of value of a risk measure

ρ

G

(·)

for a given random

cost.

Although the nested structure presented in (3)is widely used in the literature, there are other risk measures that can be used to evaluate the risk of a sequence of random variables. We show that our approach can also be applied to the risk-averse mixed-integer multi-stage stochastic problems with different dynamic extensions of mean-CVaR.

3.4. Extensiontootherdynamicmeasuresofrisk

Some examples of dynamic risk measures apart from the nested structure in (3) are multiperiod mean-CVaR and sum of mean-CVaR (see, PﬂugandRömisch,2007;EichhornandRömisch, 2005, respectively). For a sequence of random variables Zt ∈Zt,t∈

{

1 ,...,T

}

adopted to the ﬁltration Ft,t∈

{

1 ,...,T

}

, multiperiod

mean-CVaR is deﬁned as

ρ

multi

₍

_{

_Z t

}

Tt=2

)

= T t=2

λ

tE [

ρ

Ft|Ft−1

(

Zt

)

] , (17)

and sum of mean-CVaR is represented as

ρ

sum

₍

_{

_Z t

}

Tt=2

)

= T t=2

λ

t

ρ

t

(

Zt

)

, (18) with T_t₌₂

λ

t = 1 ,

λ

t ≥ 0 for t∈

{

2 ,3 ,...,T

}

.

Our approach is also applicable for the case where the risk measure is applied to whole scenario cost as a time inconsistent objective function, that is,

ρ

whole

₍

_{

_Z

t

}

Tt=1

)

=

ρ

(

Z1 + Z2 + · · · + ZT

)

. (19)

Although the risk measure (19)can be applied to a sequence of random variables, it is not a dynamic measure of risk.

The risk measure deﬁned in (17)is a time consistent dynamic measure of risk whereas the risk measures (18)and (19)are not time consistent.

In the following three propositions, we show that a lower bound for these three risk measures can be obtained by scenario grouping. Therefore, our approach is still valid for Problem (P) with an objective of one of these risk measures.

Consider an arbitrary sequence of random variables Zt ∈Zt,t∈

{

1 ,...,T

}

adopted to the ﬁltration Ft,t∈

{

1 ,...,T

}

. To avoid no-

tational ambiguity, expectation operators and risk measures are given without reference sigma algebras.

Proposition 2. Fora multiperiod mean-CVaR risk measure

ρ

multi₍_·) asdeﬁnedin (17),E◦

ρ

multi

_(·)

_is_a_valid_lower_bound.

Proof. If multiperiod mean-CVaR risk measure (17) is applied to the sequence Zt ∈Zt,t∈

{

1 ,...,T

}

, then

ρ

multi

₍

_{

_Z t

}

Tt=2

)

= T t=2

λ

tE

ρ

Ft|Ft−1

(

Zt

)

. Since

ρ

_F

t|Ft−1

(·)

is a conditional mean-CVaR risk measure, the lower bound E◦

ρ

Ft|Ft−1

(·)

applies for t∈

{

2 ,3 ,...,T

}

. Then,

ρ

multi

₍

_{

_Z t

}

Tt=2

)

≥ T t=2

λ

tE

E

ρ

Ft|Ft−1

(

Zt

)

.

Since expectation is a linear operator, we get

ρ

multi

₍

_{

_Z t

}

Tt=2

)

≥ E

T t=2

λ

tE [

ρ

Ft|Ft−1

(

Zt

)

]

, or equivalently,

ρ

multi

₍

_{

_Z t

}

tT=2

)

≥ E

ρ

multi

₍

_{

_Z t

}

Tt=2

)

.

Since the sequence Zt ∈Zt,t∈

{

1 ,...,T

}

is arbitrary, the desired

result follows.

Proposition3. Forasumofmean-CVaRriskmeasure

ρ

sum₍_·)_as de-ﬁnedin (18), E ◦

ρ

sum

_(·)

_is_a_valid_lower_bound.

Proof. If sum of mean-CVaR risk measure (18) is applied to the sequence Zt ∈Zt,t∈

{

1 ,...,T

}

, then

ρ

sum

₍

_{

_Z t

}

Tt=2

)

= T t=2

λ

t

ρ

t

(

Zt

)

.

Similarly, E ◦

ρ

t

(·)

applies for t ∈

{

2 ,3 ,...,T

}

. Then,

ρ

sum

₍

_{

_Z t

}

Tt=2

)

≥ T t=2

λ

tE [

ρ

t

(

Zt

)

] , and

ρ

sum

₍

_{

_Z t

}

Tt=2

)

≥ E

T t=2

λ

t

ρ

t

(

Zt

)

, or equivalently,

ρ

sum

₍

_{

_Z t

}

Tt=2

)

≥ E

ρ

sum

₍

_{

_Z t

}

Tt=2

)

.

Since the sequence Zt ∈ Zt,t ∈

{

1 ,...,T

}

is arbitrary, the desired

result follows.

Proposition 4. For the risk measure

ρ

whole

_(·)

_as _deﬁned _in ₍₁₉₎_,

ρ

G ◦

ρ

F|G

(·)

is a valid lower bound if parameters of

ρ

G

(·)

and

ρ

F|G

(·)

satisfyconditionsin (10).

Proof. Follows from Proposition1.

As shown above, our proposed lower bound is quite general and can be applied to other dynamic mean-CVaR measures.

3.5.Upperboundforoptimizationproblem

Obtaining an upper bound, or equivalently ﬁnding a feasible solution of a minimization problem, is crucial for the instances where an optimal solution is not available. A good quality feasible solution gives the decision maker an action to be taken and measures the quality of obtained lower bound when an optimal solution is not available.

An upper bound for the optimal value of (P) can be obtained by using optimal solutions of group subproblems. Once jth group subproblem is solved, an optimal solution of it, namely xj_{, is obtained.}

Let UBj be the optimal value of (P) where (some of) the variables

appearing in jth group subproblem are set to xj_{. We call this prob-}

lem as restricted problem. Since some of the problem variables are ﬁxed, solving the restricted problem is easier than the original one and the resulting scenario tree can become decomposable.

If the restricted problem does not have a feasible solution, then corresponding upper bound UBjis set to inﬁnity. The best available

upper bound UB is obtained by taking minimum of UB_jvalues over all j∈

{

1 ,...,J

}

, that is,

UB = min

j∈{1,...,J}UBj. (20)

In Algorithm 1, we present how group subproblem approach can be used to obtain lower and upper bounds for the multi-stage risk-averse mixed-integer problem (P) with dynamic mean-CVaR objective. The algorithm can be easily adopted to the other risk measures given in Section3.4.

(8)

Algorithm1 Lower and upper bounds for (P).

Input: A risk-averse mixed-integer multi-stage stochastic problem (P) and a partition S =

{

Sj

}

Jj=1 of sample space

.

Initialize:LB←−∞ and UB←+∞ Lower Bounding:

forall j ∈

{

1 ,2 ,...,J

}

do

Solve the jth group subproblem.

xj_←_{an optimal solution of}_jth_{group subproblem} zj_{← optimal value of}_jth_{group subproblem} endfor

Let ZLBbe a random variable that takes value zjwith probability pj = _ω_∈S_jpω

LB←

ρ

G

(

ZLB

)

Upper Bounding:

forall j∈

{

1 ,2 ,...,J

}

do

UBj ← the optimal value of the original problem with the ad-

ditional constraint where (some of) the variables appearing in

jth group subproblem are set to xj_. endfor

UB_← min _j_∈_{₁_,₂_..._,J_}UB_j Return:LB and UB

4. Computationalexperiments

In this section, we conduct our numerical experiments on a multi-stage lot sizing problem studied in Guan, Ahmed, and Nemhauser(2009). All computational experiments are performed on an Intel(R) Core(TM) i7-4790 CPU@3.60 gigahertz computer with 8.00 gigabyte of RAM with Java 1.8.0.31 and IBM ILOG CPLEX 12.6. We ﬁrst introduce risk-averse multi-stage lot sizing problem (RAMLSP) with dynamic mean-CVaR deﬁned in (3). Then, we compare the results obtained via usage of different scenario partition strategies and lower bound choices. We also compare the proposed algorithm and CPLEX in terms of solution quality and required CPU time.

4.1.Risk-aversemulti-stagelotsizingproblemwithmean-CVaR

The objective of RAMLSP is to minimize the dynamic mean- CVaR risk measure over T periods subject to demand satisfaction and capacity constraints. RAMLSP-T-r represents an RAMLSP instance with T stages in which random components can take r different values at each stage. Therefore, total number of scenarios in an RAMLSP-T-r instance is rT−1_{. We generate random test in-}

stances as in Guanetal.(2009). The same setting of the parameters is also used by Sandıkçı andÖzaltın(2014), that is, htu∼ U[0,

10],

α

tu∼ U[3.2, 4.8] E[ h],

β

tu∼ U[320, 480] E[ h], dtu∼ U[0, 100] and Mtu∼ U [40 T, 60 T], where U[ a, b] represents uniform distribution

between a and b.

Using the scenario tree representation given in Section 2.2, RAMLSP can be stated as follows:

(RAMLSP) minimize Z1 +

ρ

F2|F1

(

Z2 +

ρ

F3|F2

Z3 + · · · +

ρ

FT|FT−1

(

ZT

)

...

)

(21) subject to Ztu =

α

tuxtu+

β

tuytu + htustu,

∀

u ∈

t, t ∈

{

1 ,...,T

}

(22) s₍t−1) a(u) + xtu = dtu+ stu,

∀

u ∈

t, t ∈

{

1 ,...,T

}

(23) xtu ≤ M tuytu,

∀

u ∈

t, t ∈

{

1 ,...,T

}

(24) xtu,stu ≥ 0 and integer , ytu ∈

{

0 ,1

}

,

∀

u∈

t, t ∈

{

1 ,...,T

}

(25) s0a(v 1) = 0 .

Here xtu is the production level, ytu is the setup indicator

and stu is the inventory level variables at node u∈

t in period t∈

{

1 ,...,T

}

.

α

tu,

β

tu, htu, dtu and Mtu denote unit production

cost, setup cost, inventory holding cost, demand and production capacity parameters, respectively. Z1 is the sum of deterministic

production, setup and inventory holding costs incurred in the ﬁrst stage. Similarly, Ztu is the cost incurred at node u∈

t at stage t_∈

{

2 _,_._._._,T

}

. Zt represents the random variable that takes values

of Ztu, u∈

t with respective probabilities. The objective (21) is

the dynamic risk value over the planning horizon. Constraint (22) calculate the cost incurred at each node of the scenario tree. Constraints (23)and (24)are inventory balance and capacity constraints, respectively. Constraint (25) are domain constraints. Unlike Guan et al. (2009) and Sandıkçı and Özaltın (2014), we assume that production and inventory levels are required to be integer valued. Although this assumption increases the problem complexity, we have a more realistic representation to evaluate the performance of the algorithm. In order to linearize RAMLSP, the linearization of mean-CVaR presented in Section2.3is used.

For the computational experiments, we use three different values of weight parameter

1∈{0.8, 0.5, 0.3} and level parameter

α

∈ {0.9, 0.8, 0.7} of mean-CVaR. Therefore, we have nine different risk-aversion settings.

4.2. Choicesofscenariopartitionsandlowerbounds

As seen in Example1, the value of each lower bound highly depends on chosen scenario partition. We consider four possible scenario partition strategies obtained by grouping the scenarios in different ways, namely index1, index2, similar and different. For each strategy, we can also specify the number of scenarios in each group as a function of the number of scenarios |

| and the number of groups J. Let a% b be the remainder after the division of a∈R by

b∈R,

·

be the ceiling function, and

·

be the ﬂoor function. Then, each scenario grouping strategy yields a scenario partition that has J groups, where |

|% J groups have cardinality

|

|/ J

and

J−

(

|

% J

)

groups have cardinality

|

|/ J

. For example, if

|

= 32 and J₌5 then the cardinality of two groups will be seven and the other three groups will have cardinality of six.

Partition strategies index1 and index2 are based on the structure of scenario tree. In index1, the last stage nodes sharing the high- est number of common nodes are placed into the same group. On the other hand, index2 is obtained by placing the last stage nodes sharing the least number of common nodes into the same group.

If a priori information on the cost of each single scenario under an optimal solution is available, the groups can also be obtained with respect to similarity and diversity of individual scenarios. Since this information is not available before solving the original problem, the deterministic version of the original problem apriori can be solved for each scenario separately, and the corresponding single scenario costs can be used to obtain two different scenario partition strategies named as similar and different. Note that, for both strategies, an additional computational effort is required to obtain single scenario costs.

In strategy similar, we assign

|

|/ J

scenarios that have the largest single scenario costs to the ﬁrst group. Then, depending on the cardinality of the second group,

|

|/ J

or

|

|/ J

scenarios that have the second largest single scenario costs are assigned to the second group, and so on.

Bounds on risk-averse mixed-integer multi-stage stochastic programming problems with mean-CVaR

European

Journal

of

Operational

Research

Stochastics

and

Statistics

Bounds

on

risk-averse

mixed-integer

multi-stage

stochastic

programming

problems

with

mean-CVaR

Ali

˙Irfan

Mahmuto

˘gulları,

Özlem

Çavu

¸s

,

M. Selim

Aktürk

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