Data-Driven Stochastic Robust Optimization: General Computational
Framework and Algorithm Leveraging Machine Learning for
Optimization under Uncertainty in the Big Data Era
Chao Ning, Fengqi You *
Robert Frederick Smith School of Chemical and Biomolecular Engineering,
Cornell University, Ithaca, New York 14853, USA
December 29, 2017
Submitted to Computers & Chemical Engineering

Abstract
A novel data-driven stochastic robust optimization (DDSRO) framework is proposed for
optimization under uncertainty leveraging labeled multi-class uncertainty data. Uncertainty data
in large datasets are often collected from various conditions, which are encoded by class labels.
Machine learning methods including Dirichlet process mixture model and maximum likelihood
estimation are employed for uncertainty modeling. A DDSRO framework is further proposed
based on the data-driven uncertainty model through a bi-level optimization structure. The outer
optimization problem follows a two-stage stochastic programming approach to optimize the
expected objective across different data classes; adaptive robust optimization is nested as the inner
problem to ensure the robustness of the solution while maintaining computational tractability. A
decomposition-based algorithm is further developed to solve the resulting multi-level optimization
problem efficiently. Case studies on process network design and planning are presented to
demonstrate the applicability of the proposed framework and algorithm.

Key words: big data, optimization under uncertainty, Bayesian model, machine learning, process
design and operations

*

Corresponding author. Phone: (607) 255-1162; Fax: (607) 255-9166; E-mail: fengqi.you@cornell.edu

1

1. Introduction
Labeled multi-class data are ubiquitous in a variety of areas and disciplines, such as process
monitoring [1], multimedia studies [2, 3], and machine learning [4]. For example, process data are
labeled with the operating modes of chemical plants [1], and text documents are tagged with labels
to indicate their topics [5]. Due to the massive amount of available uncertainty data (realizations
of uncertain parameters) and dramatic progress in big data analytics [6], data-driven optimization
emerges as a promising paradigm for decision making under uncertainty [7-14]. Most existing
data-driven optimization methods are restricted to unlabeled uncertainty data. However, recent
work has revealed the significance of leveraging labeled uncertainty data in decision-making under
uncertainty [15]. Applying the existing methods to labeled uncertainty data cannot take advantage
of useful information embedded in labels. Consequently, novel modeling frameworks and
computational algorithms are needed to leverage labeled multi-class uncertainty data in datadriven decision making under uncertainty.
Optimization under uncertainty has attracted tremendous attention from both academia and
industry [16-21]. Uncertain parameters, if not being accounted for, could render the solution of an
optimization problem suboptimal or even infeasible [22]. To this end, a plethora of mathematical
programming techniques, such as stochastic programming and robust optimization [23-25], have
been proposed for decision making under uncertainty. These techniques have their respective
strengths and weaknesses, which lead to different application scopes [26-41]. Stochastic
programming focuses on the expected performance of a solution by leveraging the scenarios of
uncertainty realization and their probability distribution [42-44]. However, this approach requires
accurate information on the probability distribution, and the resulting optimization problem could
become computationally challenging as the number of scenarios increases [40, 45, 46]. Robust
optimization provides an alternative approach that does not require accurate knowledge on
probability distributions of uncertain parameters [47-51]. It has attracted considerable interest
owing to the merit of feasibility guarantee and computational tractability [50]. Nevertheless, robust
optimization does not take advantage of available probability distribution information, and its
solution usually suffers from the conservatism issue. The state-of-the-art approaches for
optimization under uncertainty leverage the synergy of different optimization methods to inherit
their corresponding strengths and complement respective weaknesses [52-54]. However, these
approaches do not take advantage of the recent advances in machine learning and big data analytics
2

to leverage uncertainty data for optimization under uncertainty. Therefore, the research objective
of our work is to propose a novel data-driven decision-making framework that organically
integrates machine learning methods for labeled uncertainty data with the state-of-the-art
optimization approach.
There are a number of research challenges towards a general data-driven stochastic robust
optimization (DDSRO) framework. First, uncertainty data are often collected from a wide
spectrum of conditions, which are indicated by categorical labels. For example, uncertain power
generation data of a solar farm are labeled with weather conditions, such as cloudy, sunny and
rainy [55]. The occurrence probability of each weather condition is known in the data collection
process, and is typically embedded in the datasets. Moreover, the uncertainty data with the same
label could exhibit very complicated distribution, which is impossible to fit into some commonlyknown probability distributions. Therefore, a major research challenge is how to develop machine
learning methods to accurately extract useful information from labeled multi-class uncertainty data
for constructing the uncertainty model in the data-driven optimization framework. Second, we are
confronted with the challenge of how to integrate different approaches for optimization under
uncertainty in a holistic framework to leverage their respective advantages. The third challenge is
to develop an efficient computational algorithm for the resulting large-scale multi-level
optimization problem, which cannot be solved directly by any off-the-shelf optimization solvers.
This paper proposes a novel DDSRO framework that leverages machine learning for decision
making under uncertainty. In large datasets, uncertainty data are often attached with labels to
indicate their data classes [6]. The proposed framework takes advantage of machine learning
methods to accurately and truthfully extract uncertainty information, including probability
distributions of data classes and uncertainty sets. The probability distribution of data classes is
learned from labeled uncertainty data through maximum likelihood estimation for the multinoulli
distribution [4, 56]. To accurately capture the structure and complexity of uncertainty data, a group
of Dirichlet process mixture models are employed to construct uncertainty sets with a variational
inference algorithm [57, 58]. These two pieces of uncertainty information are incorporated into the
DDSRO framework through a bi-level optimization structure. Two-stage stochastic programming
is nested in the outer problem to optimize the expected objective over categorical data classes;
adaptive robust optimization (ARO) is nested as the inner problem to hedge against uncertainty
and ensure computational tractability. Estimating a categorical distribution is much more
3

computationally tractable than estimating a joint probability distribution of high-dimensional
uncertainties [4]. Therefore, the outer optimization problem follows a two-stage stochastic
programming structure to take advantage of the available probability information on data classes.
Robust optimization, which uses an uncertainty set instead of accurate knowledge on probability
distributions, is more suitable to be nested as the inner problem to tackle high-dimensional
uncertainties. The DDSRO problem is formulated as a multi-level mixed-integer linear program
(MILP). We further develop a decomposition-based algorithm to efficiently solve the resulting
multi-level MILP problem. To demonstrate the advantages of the proposed framework, two case
studies on design and planning of process networks under uncertainty are presented.
The major novelties of this paper are summarized as follows.
•

A novel uncertainty modeling framework based on machine learning methods that
accurately extract different types of uncertainty information for optimization under
uncertainty;

•

A general data-driven decision making under uncertainty framework that combines
machine learning with mathematical programming and that leverages the strengths of
both stochastic programming and robust optimization;

•

An efficient decomposition-based algorithm to solve the resulting data-driven stochastic
robust four-level MILP problems.

The rest of this paper is organized as follows. Section 2 proposes the general DDSRO
framework, which includes data-driven uncertainty model, optimization model and computational
algorithm. A motivating example is presented in Section 3. It is followed by case studies on process
network design and planning under uncertainty in Section 4. Conclusions are drawn in Section 5.

2. Data-Driven Stochastic Robust Optimization Framework
In this section, we first provide a brief introduction to stochastic robust optimization. The
proposed data-driven uncertainty model and the DDSRO framework are presented next. A
decomposition-based algorithm is further developed to efficiently solve the resulting multi-level
optimization problem.

4

2.1. Stochastic robust optimization
The stochastic robust optimization framework combines two-stage stochastic programming
and ARO [53]. In this subsection, we first analyze the strengths and weaknesses of two-stage
stochastic programming and two-stage ARO. They are then integrated in a novel form to leverage
their corresponding advantages and complement respective shortcomings.
A general two-stage stochastic MILP in its compact form is given as follows [23].

min cT x + u Q ( x, u ) 
x

s.t.

Ax ≥ d

(1)

x ∈ R+n1 × Z n2
The recourse function Q(x, u) is defined as follows.
Q ( x, u ) = min b T y
y

s.t. Wy ≥ h − Tx − Mu

(2)

y∈R

n3
+

where x is first-stage decisions made “here-and-now” before the uncertainty u is realized, while
the second-stage decisions or recourse decisions y are postponed in a “wait-and-see” manner after
the uncertainties are revealed. x can include both continuous and integer variables, while y only
includes continuous variables. The objective of (1) includes two parts: the first-stage objective cTx
and the expectation of the second-stage objective bTy. The constraints associated with the firststage decisions are Ax ≥ d, x ∈ R+n × Z n , and the constraints of the second-stage decisions are
1

2

Wy ≥ h − Tx − Mu, y ∈ R+n3 .

Two-stage stochastic programming makes full use of probability distribution information, and
aims to find a solution that performs well on average under all scenarios. However, an accurate
joint probability distribution of high-dimensional uncertainty u is required to calculate the
expectation term in (1). Besides, the resulting two-stage stochastic programming problem could
become computationally challenging as the number of scenarios increases [45].
Another popular approach for optimization under uncertainty is two-stage ARO, which
hedges against the worst-case uncertainty realization within an uncertainty set. A general twostage ARO problem in its compact form is given as follows [59].

5

min cT x + max
x

min b T y

y∈Ω ( x ,u )

u∈U

(3)

s.t. Ax ≥ d, x ∈ R+n1 × Z n2

}

{

Ω ( x, u ) = y ∈ R+n3 : Wy ≥ h − Tx − Mu

where x is the vector of first-stage decision variables, and y represents the vector of second-stage
decisions. Note that x includes both continuous and integer variables, and y is the vector of
continuous recourse variables. U is an uncertainty set that characterizes the region in which
uncertainty realization resides. Note that the two-stage ARO problem in (3) is a tri-level
optimization problem.
The stochastic robust optimization framework is able to organically integrate the two-stage
stochastic programming approach with the ARO method and leverage their strengths. A general
form of the stochastic robust MILP is given by [53],
min cT x + σ ∈Π C σ ( x ) 
x

s.t.

Ax ≥ d
x ∈ R+n1 × Z n2
C σ ( x ) = max min b T yσ

yσ
u∈Uσ




≥
−
−
Wy
h
Tx
Mu
s.t.

 , ∀σ ∈ Π
σ


yσ ∈ R+n3



(4)

where σ is an uncertain scenario that influences the uncertainty set, and Π is a set of the scenarios.
Two-stage stochastic programming approach is nested in the outer problem to optimize the
expectation of objectives for different scenarios, and robust optimization is nested inside to hedge
against the worst-case uncertainty realization. Stochastic robust optimization is capable of
handling different types of uncertainties in a holistic framework.
In stochastic programming, an accurate joint probability distribution of high-dimensional
uncertainty u is required. However, from a statistical inference perspective, estimating a joint
probability distribution of high-dimensional uncertainty is far more challenging than estimating
accurate probability distributions of individual scenarios [4]. Therefore, it is better to use a coarsegrained uncertainty set to describe high-dimensional uncertainty u, and to use fine-grained
probability distributions to model scenarios. As a result, two-stage stochastic programming
approach is nested outside to leverage the available probability distributions. Robust optimization
is more suitable to be nested inside for computational tractability.
6

The stochastic robust optimization method typically assumes the uncertainty probability
distributions and uncertainty sets are given and known a priori, rather than deriving them from the
uncertainty data. However, the predefined probability distribution and manually constructed
uncertainty sets might not correctly capture the fine-grained information from the available
uncertainty data. Thus, we propose a novel data-driven uncertainty modeling framework for
stochastic robust optimization with labeled multi-class uncertainty data to fill this knowledge gap.

2.2. Data-driven uncertainty modeling
In this subsection, we propose a data-driven uncertainty modeling framework based on
machine learning techniques to seamlessly integrate data-driven system and model-based system
in the proposed decision-making framework. This uncertainty modeling framework includes two
parts, i.e. probability distribution estimation for data classes, and uncertainty set construction from
uncertainty data in the same data class. For the first part, the probability distributions of data classes
are learned from the label information in multi-class uncertainty data through maximum likelihood
estimation. For the second part, a group of Dirichlet process mixture models is employed to
construct uncertainty sets with a variational inference algorithm [58].

{

We consider the multi-class uncertainty data with labels u(i ) , c(i )

}

L

i =1

, where u(i) is i th

uncertainty data point, c(i) is its corresponding label, and L is the total number of uncertainty data.
Although the labeled uncertainty data is of practical relevance [6, 15], existing literature on data-

{ }

driven optimization under uncertainty are restricted to unlabeled uncertainty data u(i )

L

[8, 12].

i=1

The uncertainty information on the probability of different data classes can be extracted from
labeled uncertainty data by leveraging their label information. The occurrence probabilities of data
classes are modeled with a multinoulli distribution. The probability of each data class can be
calculated through maximum likelihood estimation, as given by [4, 56],

∑  ( c( ) = s )
i

ps =

i

(5)

L

where ps represents the occurrence probability of data class s, c(i) is the label associated with the i
th uncertainty data point, c(i)=s indicates that the i th uncertainty data point is from data class s,
and the indicator function is defined as follows.

7

1 if c(i ) = s
i
 c( =) s= 
0 else

(

)

(6)

As will be shown in the next subsection, the extracted probability distribution information of
data classes is incorporated into the two-stage stochastic programming framework nested as the
outer optimization problem of DDSRO.
The second part of the uncertainty modeling is the construction of data-driven uncertainty sets
from uncertainty data in the same data class. The information on uncertainty sets is incorporated
into robust optimization nested as the inner optimization problem of DDSRO. To handle the
labeled multi-class uncertainty data, a group of Dirichlet process mixture models is employed, and
uncertainty data with the same label are modeled using one separate Dirichlet process mixture
model [58]. The details of Dirichlet process mixture model are given in Appendix A.
Based on the extracted information from Dirichlet process mixture model, we construct a
data-driven uncertainty set for data class s using both l1 and l∞ norms [8], as shown below,
Us
=

=
 U s,i U s,1 U s,2 U s,m( s)

(7)

{u =u

(8)

i: γ s ,i ≥γ *

U s=
,i

μ s ,i + κ s ,i Ψ1/2
z
s ,i Λ s ,i z ,

∞

≤ 1, z 1 ≤ Φ s ,i }

where Us is the data-driven uncertainty set for data class s, Us,i is the basic uncertainty set
corresponding to the i th component of data class s, Λs,i is a scaling factor, γs,i is the weight of the
i th component of data class s, γ s ,i =

τ s ,i
τ s ,i + vs ,i

i −1

∏τ
j =1

vs , j
s, j

+ vs , j

and γ s , M = 1 −

∑

M −1
i =1

γ s ,i . The weight γs,i

indicates the probability of the corresponding component. γ* is a threshold value. m(s) is the total
number of components for data class s. κ s ,i =

λs ,i + 1
[57], τs,i, νs,i, μs,i, λs,i, ωs,i, ψs,i
λs ,i (ωs ,i + 1 − dim ( u ) )

are the inference results of the i th component learned from uncertainty data of data class s, and
Φs,i is its uncertainty budget. It is worth noting that the number of basic uncertainty sets m(s) for
each data class varies depending on the structure and complexity of the uncertainty data.

2.3. Data-driven stochastic robust optimization model
The proposed DDSRO framework is shown as follows.

8


min cT x +  s∈Ξ 
max
x
 u∈U s ,1 U s ,2 U s ,m( s )
s.t. Ax ≥ d, x ∈ R+n1 × Z n2

{u =u μ + κ Ψ
Ω ( x, u )= {y ∈ R : Wy

U s=
,i

s ,i

s

s ,i

n3
+

1/2
s ,i
s


min b T y s 


y s ∈Ω ( x ,u )

Λ s ,i z , z

∞

≤ 1, z 1 ≤ Φ s ,i }

(9)

}

≥ h − Tx − Mu

where Ξ ={1, 2, ,C} is the set of data classes, C is the total number of data classes, m(s) is the
total number of mixture components for data class s. Each data class corresponds to a scenario in
the two-stage stochastic program nested outside the DDSRO framework. It is worth noting that
the DDSRO problem in (9) is a multi-level MILP that involves mixed-integer first-stage decisions,
and continuous recourse decisions.
In the DDSRO framework, the outer optimization problem follows a two-stage stochastic
programming structure, and robust optimization is nested as the inner optimization problem to
hedge against high-dimensional uncertainty u. DDSRO aims to find a solution that performs well
on average across all data classes by making full use of their occurrence probabilities. Instead of
treating all the uncertainty data as a whole, the DDSRO framework explicitly accounts for multiclass uncertainty data by leveraging their label information. Stochastic robust optimization
structure is employed because it is more suitable to leverage different types of uncertainty
information in a holistic framework. Note that DDSRO is a general framework for data-driven
optimization under uncertainty, while stochastic robust optimization methods assume uncertainty
information is given a priori rather than learning it from uncertainty data.
There are two reasons that two-stage stochastic programming is nested as the outer
optimization problem while robust optimization is nested as the inner optimization problem. First,
the proposed DDSRO framework aims to balance all data classes instead of focusing only on the
worst-case data class, and it also ensures the robustness of the solution. Second, estimating the
probability distribution of data classes accurately is much more computationally tractable than
estimating an accurate joint probability distribution [4]. Consequently, it is better to employ
uncertainty set to characterize high-dimensional uncertain parameters in the inner optimization
problem. On the other hand, stochastic programming is more suitable as the outer optimization
problem to leverage the probability information of data classes.
The DDSRO framework leverages machine learning techniques to truthfully capture the
structure and complexity of uncertainty data for decision making under uncertainty. It enjoys the
9

computational tractability of robust optimization, while accounting for exact probability
distribution of data class in the same way as stochastic programming approach.
Note that the proposed data-driven uncertainty set for each data class in (7) is a union of
several basic uncertainty sets. Therefore, we reformulate the DDSRO model in (9) into (10), as
given below.

min cT x +  s∈Ξ  max max
x
 i∈{1,,m( s )} u∈U s ,i
s.t. Ax ≥ d, x ∈ R+n1 × Z n2

{u =u μ + κ Ψ
Ω ( x, u )= {y ∈ R : Wy

U s=
,i

s ,i

s

s ,i

n3
+


min b T y s 


y s ∈Ω ( x ,u )

1/2
s ,i
s

Λ s ,i z , z

∞

≤ 1, z 1 ≤ Φ s ,i }

}

≥ h − Tx − Mu

Fig. 1. An overview of the proposed DDSRO framework.
10

(10)

The DDSRO model explicitly accounts for the label information within multi-class
uncertainty data, and balances different data classes by using the weighted sum based on their
occurrence probabilities. Robust optimization is nested as the inner optimization problem to handle
uncertainty u, thus providing a good balance between solution quality and computational
tractability. Moreover, multiple basic uncertainty sets Us,i are unified to capture the structure and
complexity of uncertainty data, and this scheme leads to the inner max-max optimization problem
in (15). An overview of the proposed DDSRO framework for labeled multi-class uncertainty data
is shown in Fig. 1. Uncertainty data points are depicted using circles with different colors to
indicate its associated data class, and they are treated separately using a group of Dirichlet process
mixture models. The probability distribution of data classes can be inferred from uncertainty data
using eq. (5). The proposed framework leverages the strengths of stochastic programming and
robust optimization, and uses the uncertainty information embedded in uncertainty data.
The resulting DDSRO problem has a multi-level optimization structure that cannot be solved
by any general-purpose optimization solvers directly. An efficient computational algorithm is
further proposed in the next subsection to address this computational challenge.

2.4. The decomposition-based algorithm
To address computational challenge of solving the DDSRO problem, we extend the algorithm
proposed in [8]. The main idea is to first decompose the multi-level MILP shown in (15) into a
master problem and a number of groups of sub-problems, where each group corresponds to a data
class. Moreover, there are a number of sub-problems in each group, and each sub-problem
corresponds to a component in the Dirichlet process mixture model.
The proposed decomposition-based algorithm iteratively solves the master problem and the
sub-problems, and adds cuts to the master problem after iteration, until the relative optimality gap
is reduced to the predefined tolerance ζ. The pseudocode of the algorithm is given in Fig. 2, where
C represents the total number of data classes. The algorithm is guaranteed to terminate within finite
iterations due to the finite number of extreme points of basic uncertainty sets [53]. The details of
the decomposition-based algorithm, including the formulations of master problem (MP) and subproblem (SUPs,i), are given in Appendix B.

11

Decomposition Algorithm
1:
2:
3:
4:
5:

Set LB ← −∞ , UB ← +∞ , k ← 0 , and ζ ;
UB − LB
while
> ζ do
UB
Solve (MP) to obtain x*k +1 ,η k*+1 , y1*s , , y ks * ;
Update LB ← cT x*k +1 + ηk*+1 ;
for s=1 to C do
for i = 1 to m ( s ) do

6:

Solve (SUPs,i) to obtain u s ,i and Qs ,i ( x*k +1 ) ;
k +1

7:
8:

end
i ← arg max Qs ,i ( x*k +1 ) ;
*

9:

i

u

10:
11:

k +1
s

←u

k +1
s , i*

and Qs ( x*k +1 ) ← Qs ,i* ( x*k +1 ) ;

end



Update UB ← min UB, cT x*k +1 + ∑ ps ⋅ Qs x*k +1  ;
s =1



(

12:

Create

second-stage

variables

)

y ks +1 and

add

cuts

k +1
k +1
η ≥ ∑ ps ( b T y ks +1 ) and Tx + Wy s ≥ h − Mu s to (MP);

13:

s

k ← k +1;

14:
15:
16:

end
return UB ;

Fig. 2. The pseudocode of the decomposition-based algorithm.

2.5. Relationship with the existing data-driven adaptive nested robust
optimization framework
In this subsection, we discuss the relationship between the DDSRO framework proposed in
this work and the data-driven adaptive nested robust optimization (DDANRO) framework
proposed earlier [8].
The DDANRO framework in its general form has a four-level optimization structure, as given
below.

12

min cT x + max max min b T y
i∈{1,, m} u∈U i

x

y∈Ω ( x ,u )

s.t. Ax ≥ d, x ∈ R+n1 × Z n2
U i=

{u u= μi + κ i Ψ1/2i Λi z, z

{

∞

≤ 1, z 1 ≤ ∆ i }

(11)

}

Ω ( x, u ) = y ∈ R+n3 : Wy ≥ h − Tx − Mu

It is worth noting that the DDANRO framework treats uncertainty data homogeneously
without considering the label information [8]. DDSRO greatly enhances the capabilities of
DDANRO, and expands its application scope to more complex uncertainty data structures. There
are some similarities between DDSRO and DDANRO. For example, they both employ data-driven
uncertainty sets, and have multi-level optimization structures. Moreover, DDANRO can be
considered as a special case of DDSRO when there is one only one class of uncertainty data.
Specifically, if there is only one data class, we can simplify the expectation term as a single term
in (10), such that the DDSRO model shown in (10) reduces to the DDANRO model in (11). The
DDANRO framework is more appropriate to handle single-class or unlabeled uncertainty data,
while the proposed DDSRO framework, on the other hand, is suitable for handling labeled multiclass uncertainty data systematically. Therefore, the DDSRO framework proposed in this paper
could be considered as a generalization of the existing DDANRO approach.

3. Motivating Example
In this section, a motivating example is first presented to illustrate different characteristics of
optimization methods, and to show that accounting for label information in optimization problem
is critical.
The deterministic formulation of the motivating example is shown in (12).
min 3x1 + 5 x2 + 6 x3 + 6y1 + 10 y2 + 12 y3
x,y

s.t. x1 + x2 + x3 ≤ 200
x1 + y1 ≥ u1

(12)

x2 + y2 ≥ u2
x3 + y3 ≥ u3
xi , yi ≥ 0, i =
1, 2,3

where x1, x2, x3, y1, y2 and y3 are decision variables, u1, u2 and u3 are parameters. In the deterministic
optimization problem, u1, u2 and u3 are assumed to be known exactly.
13

Under the uncertain environment, the parameters u1, u2 and u3 are subject to uncertainty. In a
data-driven optimization setting, what decision makers know about these uncertain parameters are
their realizations, also known as uncertainty data [60]. The scatter plot of 1,000 labeled uncertainty
data is shown in Fig. 1, where the points represent uncertainty realizations of u1, u2 and u3. There
are 4 labels within uncertainty data, which means that there are 4 data classes in total. Note that
uncertainty data points from different data classes are indicated with disparate shapes in Fig. 1.
Moreover, uncertainty data in the same data class could exhibit complex characteristics, such as
correlation, asymmetry and multimode.

Fig. 3. The scatter plot of labeled uncertainty data in the motivating example.
One conventional way to hedging against uncertainty is the two-stage stochastic programming
approach [23]. However, it is computationally challenging to fit the uncertainty data points in Fig.
3 into a reasonable probability distribution. If a scenario-based stochastic programming approach
is applied to this problem, each data point in Fig. 3 corresponds to a scenario. There are 1,000
scenarios in total, and the probability of each scenario is given as 0.001. The scenario-based twostage stochastic programming model is shown as follows.

14

min 3 x1 + 5 x2 + 6 x3 + ∑ pω ⋅ 6yω ,1 + 10 yω ,2 + 12 yω ,3 
x , yω
ω∈Θ

s.t. x1 + x2 + x3 ≤ 200
x1 + yω ,1 ≥ uω ,1 ∀ω

(13)

x2 + yω ,2 ≥ uω ,2 ∀ω
x3 + yω ,3 ≥ uω ,3 ∀ω
∀ω , i =
1, 2,3

xi , yω ,i ≥ 0,

where the notation Θ represents the set of scenarios, and ω is the scenario index. The probability
of each scenario is pω. x1, x2 and x3 are the first-stage or “here-and-now” decision variables, uω,1,
uω,2 and uω,3 are the uncertainty realizations, and yω,1, yω,2 and yω,3 are the second-stage or “waitand-see” decision variables corresponding to scenario ω. Note that the “here-and-now” decisions
are made prior to the resolution of uncertainties, whereas “wait-and-see” decisions could be made
after the uncertainties are revealed. The deterministic equivalent of this stochastic program will be
a large-scale linear program with 3,003 variables and 6,004 constraints based on the 1,000
scenarios. We note that the size of the stochastic programming problem grows exponentially as
the number of uncertainty data points increases. While decomposition-based optimization
algorithms, such as multi-cut Benders decomposition algorithm, can take advantage of the problem
structure and improve the computational efficiency of solving this problem [61], it remains
intractable to handle problems with “big” data for uncertain parameters (e.g. those with millions
of data points).
An alternative way to optimization under uncertainty is the two-stage ARO method [50], of
which the problem size might be less sensitive to the amount of uncertainty data. The two-stage
ARO model with a box uncertainty set is given as follows.
min 3x1 + 5 x2 + 6 x3 + max min 6y1 + 10 y2 + 12 y3
u∈U

x

y

s.t. x1 + x2 + x3 ≤ 200
x1 + y1 ≥ u1
x2 + y2 ≥ u2
x3 + y3 ≥ u3
U
=

{u u

L
i

(14)

i 1, 2,3}
≤ ui ≤ uiU ,=

1, 2,3
xi , yi ≥ 0, i =

where uiL and uiU are the lower and upper bounds of uncertain parameter ui, respectively. These
two parameters can be readily derived from the uncertainty data in Fig. 3.
15

The numerical results are shown in Table 1. In the deterministic optimization method, u1, u2
and u3 are set to be their mean values of the uncertainty data. Although the deterministic approach
achieves a low objective value of 455.3 in the minimization problem, its optimal solution could
become infeasible when the parameters u1, u2 and u3 are subject to uncertainty. The two-stage
stochastic programming approach focuses on the expected objective value, while the two-stage
ARO method aims to minimize the worst-case objective value. Therefore, the objective value of
the two-stage stochastic programming approach is lower than that of the two-stage ARO method.
Compared with the two-stage stochastic programming approach, the two-stage ARO method is
more computationally efficient.
A large amount of uncertainty data implies there is much uncertainty information.
Nevertheless, the two-stage stochastic programming problem could become computationally
intractable as the size of uncertainty data increases. Moreover, the stochastic programming
approach could not well capture uncertainty data if we fit the uncertainty data into an inaccurate
probability distribution, thereby resulting in poor-quality solutions. Consequently, it might be
unsuitable to apply the conventional two-stage stochastic programming approach to the
optimization problem where large amounts of uncertainty data are available. Despite its
computational efficiency, the conventional two-stage ARO method only extracts coarse-grained
uncertainty information on the bounds of uncertain parameters. Neglecting relevant uncertainty
information makes the conventional ARO method unsuitable for the data-driven optimization
setting either.
The DDANRO framework was recently proposed to overcome the drawback of the
conventional ARO method [8] by integrating machine learning methods with ARO. The
DDANRO model of this motivating example is given by
min 3x1 + 5 x2 + 6 x3 + max max min 6y1 + 10 y2 + 12 y3
j∈{1,, m} u∈U j

x

y

s.t. x1 + x2 + x3 ≤ 200
x1 + y1 ≥ u1
x2 + y2 ≥ u2

(15)

x3 + y3 ≥ u3
xi , yi ≥ 0, i =
1, 2,3
U=
j

{u u=

μ j + κ j Ψ1/2
z
j Λ j z,

16

∞

≤ 1, z 1 ≤ ∆ j }

where Uj is a basic uncertainty set, μj, κj, ψj, and Λj are the parameters derived from uncertainty
data [8]. Δj is an uncertainty budget, and m represents the total number of the basic uncertainty
sets.
Nevertheless, one limitation of the DDANRO method is that it does not leverage the label
information in Fig. 3 to infer the occurrence probability of each data class. This information
derived from labels should inform the decision-making process. Besides, DDANRO does not
leverage the strengths of stochastic programming for tackling uncertainty in optimization. These
motivate us to propose the DDSRO framework in this paper as an extension of the DDANRO
method for optimization under labeled multi-class uncertainty data.
The DDSRO model formulation of this motivating example is given in (16).
min 3x1 + 5 x2 + 6 x3 +  s∈Ξ  max min 6y1 + 10 y2 + 12 y3 
 u∈U s y

x
s.t. x1 + x2 + x3 ≤ 200
x1 + y1 ≥ u1

(16)

x2 + y2 ≥ u2
x3 + y3 ≥ u3
xi , yi ≥ 0, i =
1, 2,3

where the notation Ξ represents the set of data classes, and s is the corresponding element. The
data-driven uncertainty set Us for data class s is defined in (7).
Following the proposed data-driven uncertainty modeling framework in Section 2, the
occurrence probabilities of different data classes are inferred from the label information embedded
within uncertainty data using maximum likelihood estimation. The computational results show
that the probabilities of data classes 1-4 are 0.2, 0.4, 0.3 and 0.1, respectively. Moreover, the
uncertainty modeling results also show that there are two components in the Dirichlet process
mixture models for data class 1-3, while there is only one component for data class 4. The
occurrence probabilities of different data classes are listed in Fig. 4. The data-driven uncertainty
sets with uncertainty budget Φ=1.8 are also constructed for each data class, and are depicted using
different colors in Fig. 4. From Fig. 4, we can see that the structure and complexity of uncertainty
data are accurately and truthfully captured by the proposed data-driven uncertainty model. For
example, the uncertainty set for data class 2 accurately captures the bimodal feature in uncertainty
data, and the correlations among uncertain parameters u1, u2 and u3. The computational results of
the DDANRO and DDSRO methods are also provided in Table 1. By leveraging the label
17

information, the DDSRO approach is less conservative than the DDANRO method. From Table 1,
we can see that the objective value of DDANRO is 18.8% larger than that of DDSRO, while the
computational times of both problems are comparable.

Fig. 4. The data-driven uncertainty modeling results for the four data classes in the motivating
example.

Table 1. Computational results of different optimization methods in the motivating example.
Deterministic
optimization
Min.obj.
Iterations
Total CPU (s)
Decision x

Two-stage
ARO

DDANRO
Δ=1.8*

DDSRO
Φ=1.8#

455.3
N/A
0.1

Two-stage
stochastic
programming
646.4
7
477

x1 = 38.43

964.0
2
1

x1 = 66.64

944.6
5
9

x1 = 55.48

795.4
6
12

x1 = 25.18

x2 = 29.55

x2 = 27.55

x2 = 62.49

x2 = 52.56

x2 = 33.62

x3 = 34.30

x3 = 39.92

x3 = 70.87

x3 = 67.60

x3 = 53.14

x1 = 33.93

* Δ is the data-driven uncertainty budget in the DDANRO approach.
# Φ is the data-driven uncertainty budget in the DDSRO approach.

18

4. Application: Data-Driven Stochastic Robust Planning of Chemical Process
Networks under Uncertainty
The application of the proposed DDSRO framework to a multi-period process network
planning problem is presented in this section. Large integrated chemical complexes consist of
interconnected processes and various chemicals (see Fig. 9 for an example) [62, 63]. These
interconnections allow the chemical production to make the most of different processes [64, 65].
The chemicals in the process network include raw materials, intermediates and final products. In
the process network planning problem, purchase levels of raw materials, sales of products, capacity
expansions, and production profiles of processes at each time period should be determined in order
to maximize the net present value (NPV) during the entire planning horizon [66].

4.1. Data-driven stochastic robust planning of process networks model
The DDSRO model for process network planning under uncertainty is formulated as follows.
The objective is to maximize the NPV, which is given in (17). The constraints include capacity
expansion constraints (18)-(19), budget constraints (20)-(21), production level constraint (22),
mass balance constraint (23), supply and demand constraints (24)-(25), non-negativity constraints
(26)-(27), and integrity constraint (28). A list of indices/sets, parameters and variables is given in
Nomenclature, where all the parameters are in lower-case symbols, and all the variables are
denoted in upper-case symbols.
The objective function is defined by (17).





max − ∑∑ ( c1it ⋅ QEit + c 2it ⋅ Yit ) +  s∈Ξ
min
max
ν jt ⋅ SAsjt − ∑∑ c3it ⋅ Wsit − ∑∑ c 4 jt ⋅ Psjt 
 du jt ∈U sdem , su jt ∈U ssup Psjt ,Qsit ,  ∑∑

QEit ,Yit
i∈I t∈T
j∈J t∈T
i∈I t∈T
j∈J t∈T


SAsjt ,Wsit



(17)
where QEit is a decision variable for capacity expansion of process i at period t, Yit is a binary
decision variable to determine on whether process i is expanded at period t, SAsjt is the amount of
chemical j sold to the market at period t for data class s, Wsit represents the operating level of
process i at period t for data class s, and Psjt is a decision variable on the amount of chemical j
purchased at period t for data class s. c1it, c2it, c3it and c4jt are the coefficients associated with
variable investment cost, fixed investment cost, operating cost and purchase cost, respectively. νjt
is the sale price of chemical j in time period t.
19

Constraint (18) specifies the lower and upper bounds of the capacity expansions. Specifically,
if Yit=1, i.e. process i is selected to be expanded at period t, the expanded capacity should be within
the range [qeitL, qeitU].
qeitL ⋅ Yit ≤ QEit ≤ qeitU ⋅ Yit ,

∀i, t

(18)

Constraint (19) depicts the update of capacity of process i.
Qit =
Qit −1 + QEit ,

∀i, t

(19)

where Qit is a decision variable for total capacity of process i at period t.
Constraints (20) and (21) enforce the largest number of capacity expansions for process and
investment budget, respectively.

∑Y

it

≤ cei ,

∀i

(20)

t

∑ ( c1

it

⋅ QEit + c 2it ⋅ Yit ) ≤ cbt ,

∀t

(21)

i

where cei is the largest possible expansion times of process i, and cbt is the investment budget for
period t.
Constraint (22) specifies that the production level of a process cannot exceed its total capacity.
Wsit ≤ Qit ,

∀s, i, t

(22)

Constraint (23) specifies the mass balance for chemicals.
Psjt − ∑ κ ij ⋅ Wsit − SAsjt = 0,

∀s, j , t

(23)

i

where κij represents the mass balance coefficient of chemical j in process i.
Constraint (24) specifies that the purchase amount of a feedstock cannot exceed its available
amount. The sale amount of a product and its market demand are specified in Constraint (25).
Psjt ≤ su jt ,
SAsjt ≤ du jt ,

∀s, j , t

(24)

∀s, j , t

(25)

where sujt is the market availability and dujt is the demand of chemical j in time period t.
Constraints (26)-(27) enforces the non-negativity of continuous decisions.

QEit ≥ 0, Qit ≥ 0 ∀i, t

(26)

Psjt , SAsjt , Wsit ≥ 0,

(27)

∀s, j , t

Constraint (28) ensures that Yit is a binary decision variable.
Yit ∈ {0,1} ,

∀i, t

(28)
20

Uncertainty information for demand and supply is shown in (29)-(30).
dem
U=
s



*
k : γ sdem
,k ≥ γ

du
{du =

dem
dem
dem
μ dem
z
s ,k + κ s ,k ⋅ Ψ s ,k ⋅ Λ s ,k ⋅ z,

∞

≤ 1, z 1 ≤ Φ dem
s ,k

}

(29)

dem
dem
dem
where μ s ,k , κ s , k , Ψ s ,k are the inference results learned from uncertain product demand data

using the variational inference algorithm.
sup
U=
s



*
k : γ ssup
,k ≥ γ

su
{su =

sup
sup
sup
μ sup
z
s , k + ss , k ⋅ Ψ s , k ⋅ Λ s , k ⋅ z ,

∞

≤ 1, z 1 ≤ Φ sup
s ,k

}

(30)

sup
sup
sup
where μ s ,k , κ s ,k , Ψ s ,k are the inference results learned from uncertain supply data employing the

variational inference algorithm. The probabilities of different data classes are extracted using
maximum likelihood estimation.
The resulting problem on strategic planning of process network under uncertainty is a
stochastic robust MILP. The proposed computational algorithm is employed to efficiently solve
the large-scale multi-level optimization problem. In the next two subsections, two case studies are
presented to demonstrate the applicability and advantages of the proposed approach.

4.2. Case study 1
A small-scale case study is provided in this subsection. The process network, which is shown
in Fig. 5, consists of five chemicals and three processes [63]. In Fig. 5, chemicals A-C represent
raw materials, which can be either purchased from suppliers or produced by certain processes. For
example, Chemical C can be either manufactured by Process 3 or purchased from a supplier.
Chemicals D and E are final products, which are sold to the markets. In this case study, we consider
10 time periods over the 20-year planning horizon, and the duration of each time period is 2 years.
It is assumed that processes do not have initial capacities, and they can be installed at the beginning
of the planning horizon.

Fig. 5. The chemical process network of case study 1.
21

The supplies of feedstocks A-C, and demands of products D-E are subject to uncertainty. A
total of 160 demand uncertainty data points are used in this case study, and each data point
corresponds to a combination of all product demand realizations. For the supply uncertainty, 200
uncertainty data points are used. Each data point represents a combination of supply uncertainty
realizations for all raw materials. Demand and supply uncertainty data are attached with the labels
of different government policies [67]. Government could encourage or discourage a certain
industry by the means of subsidies and tax rates [68]. There are two labels within demand
uncertainty data, which indicates two data classes. There are also two data classes within the supply
uncertainty data. The two different labels represent two kinds of government policies that
encourage or discourage the industry related to the materials and products of this process network
[67, 68].
The results of the proposed data-driven uncertainty modeling framework are given in Fig.6
and Fig. 7. In these two figures, the red dots and red stars represent uncertainty data from different
data classes, and the polytopes represent data-driven uncertainty sets with uncertainty budgets
Φdem=1 and Φsup=1. Due to the settings of uncertainty budgets, some data points are not included
in the uncertainty sets. The occurrence probabilities of different data classes are inferred from the
label information, and the corresponding numerical values are listed in these two figures. From
Fig. 6, we can see that the probabilities of demand data classes 1 and 2 are both 0.5. There is only
one component in the Dirichlet process mixture model for both data classes. The proposed
uncertainty model accurately captures the correlations among the two uncertain parameters for
demands. Fig. 7 shows that the occurrence probabilities for the two supply data classes are the
same. There is one component in each of these data classes. The DDSRO problem for the planning
of process network can be solved using the proposed computational algorithm.

22

Fig. 6. The data-driven demand uncertainty modeling results (Φdem=1) for the two data classes in
case study 1.

Fig. 7. The data-driven supply uncertainty modeling results (Φsup=1) for the two data classes in
case study 1.
We solve the optimization problem of process network planning using the deterministic
optimization method, the two-stage stochastic programming method, the two-stage ARO method
with a box set, the DDANRO approach, and the DDSRO approach. All optimization problems are
modeled in GAMS 24.7.3 [69], solved with CPLEX 12.6.3, and are implemented on a computer
with an Intel (R) Core (TM) i7-6700 CPU @ 3.40 GHz and 32 GB RAM. The relative optimality

23

gaps for both the decomposition-based algorithm in Section 2.4 and the multi-cut Benders
decomposition algorithm are set to be 0.1%.
The problem sizes and computational results are summarized in Table 2. For the two-stage
stochastic programming problem, each scenario corresponds to a demand uncertainty data point
and a supply uncertainty data point. There are 32,000 (160×200) scenarios in the two-stage
stochastic programming problem, because there are 160 and 200 data points for demand and supply
uncertainty, respectively. Two-stage stochastic programming approach aims to find a solution that
performs well on average, so its objective value is only 3.7% higher than that of the DDSRO
approach. However, it takes 28 iterations and 106,140 seconds to solve the resulting two-stage
stochastic programming problem with the multi-cut Benders decomposition [61]. We note that the
deterministic equivalent MILP of this two-stage stochastic program is not suitable to be solved
directly due to its large-scale problem size. By contrast, the DDSRO problem can be solved within
only 3 seconds, which demonstrates its computational advantage. The two-stage ARO problem
with box uncertainty sets is solved using the bounds of demand and supply derived from
uncertainty data points in Fig. 6 and Fig. 7, respectively. Data-driven uncertainty sets for demand
and supply are constructed based on all the data points in Fig. 6 and Fig. 7, respectively. The
uncertainty budgets for demand and supply are both 1. The computational times of the two-stage
ARO problem and the DDANRO problem are 1 CPU second and 7 CPU seconds, respectively. In
spite of its computational tractability, the two-stage ARO method is conservative and only
generates a NPV of $1,372.5MM. Compared with the two-stage ARO method, the DDANRO
method ameliorates the conservatism issue and generates $244.6MM higher NPV.

Table 2. Comparisons of problem sizes and computational results in case study 1.
Deterministic
Planning
Bin. Var.

30

Two-stage
stochastic
programming
30

Two-stage
ARO

246

Cont. Var.

196

4,160,060

Constraints

276

10,560,193

376

396

551

Max. NPV ($MM)

1,809.5

1,802.6

1,372.5

1,671.7

1,739.1

Total CPU (s)

0.2

106,140

1

7

3

Iterations

N/A

28

2

4

2

30

DDANRO
(Δdem=1, Δsup=1)*

DDSRO
(Φdem=1, Φsup=1)#

30

30

246

468

* Δdem and Δsup are data-driven uncertainty budgets in the DDANRO approach for demand and supply, respectively.
# Φdem and Φsup are data-driven uncertainty budgets in the DDSRO approach for demand and supply, respectively.

24

The optimal capacity expansion decisions determined by the two-stage stochastic
programming approach, the DDANRO method and DDSRO approach are shown in Fig. 8 (a), (b)
and (c), respectively. As can be observed from Fig. 8, Process 1 is expanded at time period 5, and
Process 2 starts the expansion at time period 2 for all the three methods. Moreover, Process 2 is
further expanded at the 6-th time period in the solution determined by the DDSRO approach. By
contrast, Process 2 is not selected to be expanded from time period 2 in the DDANRO solution.
The optimal process capacities at the end of the planning horizon determined by the two-stage
stochastic programming approach, the DDANRO method and the DDSRO approach are listed in
Table 3.

Table 3. The optimal capacity of processes at the end of the planning horizon.
Process capacity (kt/y)
Process 1
Process 2
Process 3

Two-stage stochastic
programming
100
227
51

25

DDANRO

DDSRO

114
207
39

114
225
39

Fig. 8. Optimal capacity expansion decisions over the entire planning horizon determined by (a)
the two-stage stochastic programming approach, (b) the DDANRO approach, and (c) the DDSRO
approach in case study 1.

4.3. Case study 2
A large-scale case study is presented in this subsection to demonstrate the advantages of the
DDSRO approach. The process network in this case study consists of 28 chemicals and 38
processes [63], as shown in Fig. 9. Chemicals A-J represent raw materials, which can be purchased
26

from suppliers or manufactured by some processes. Chemicals K-Z are final products, which could
be sold to markets. There are 2 intermediates AA and AB. This complex process network has such
flexibility that many process technologies are available. For example, Chemical L can be
manufactured by Process 2, Process 15 and Process 17. In this case study, we consider 10 time
periods over the planning horizon, and the duration of each time period is 2 years. It is assumed
that processes 12, 13, 16 and 38 have initial capacities of 40, 25, 300 and 200 kt/y at the beginning
of the planning horizon. These 4 processes cannot be expanded until time period 2. The other
processes can be built at the beginning of time period 1.
As in the previous case study, supplies of all raw materials and demands of all final products
are subject to uncertainty. For the supply uncertainty, a set of 40,000 uncertainty data are used for
uncertainty modeling. Each data point has 10 dimensions for a combination of all raw materials.
For the demand uncertainty, 40,000 uncertainty data points are used, and each data point is for a
combination of all the 16 products. There are 3 labels within demand uncertainty data, which
indicates different data classes. There are also 3 data classes within the supply uncertainty data.
The 3 labels are different kinds of government policies regarding the industry associated with the
process network [67, 68]: (1) encouragement policy, (2) neutral policy and (3) discouragement
policy. These policies not only influence the willingness of suppliers to provide related raw
materials but also the consumptions of products via subsidies or tax rates. The occurrence
probabilities of different data classes are inferred from the label information. The data-driven
uncertainty modeling results show that the probabilities of supply data classes 1-3 are 0.25, 0.50,
and 0.25, respectively. The probabilities of demand data classes 1-3 are calculated to be 0.25, 0.50,
and 0.25, respectively. The number of components in the Dirichlet process mixture models is 1 for
all demand data classes. The extracted uncertainty information is incorporated into the DDSRO
based process network planning problem.
To demonstrate the advantages of the proposed approach, we solve the process network
planning problem using deterministic method, the two-stage stochastic programming approach,
the two-stage ARO method, the DDANRO method and the DDSRO approach. In the two-stage
stochastic programming problem, there are 1.6 billion scenarios (40,000×40,000), as there are
40,000 data points for both demand and supply uncertainty. In the deterministic equivalent of the
two-stage stochastic programming problem, there are 380 integer variables, 1,472,000,001,000
continuous variables, and 1,024,000,004,000 constraints. The size of this two-stage stochastic
27

programming problem is too large to be solved within the computational time limit of 48 hours
using the multi-cut Benders decomposition algorithm. While multi-cut Benders decomposition
algorithm can take advantage of the problem structure and improve the computational efficiency,
it is still intractable to handle this process network planning problem with “big” data for uncertain
parameters. Therefore, we only report the problem sizes and computational results of the
deterministic approach, the two-stage ARO method, the DDANRO method and the DDSRO
approach in Table 4.

28

Fig. 9. The chemical process network of case study 2.

29

Table 4. Comparisons of problem sizes and computational results in case study 2.
Deterministic
Planning

Two-stage ARO

DDANRO
(Δdem=5, Δsup=3)*

380

DDSRO
(Φdem=5, Φsup=3)#

Bin. Var.

380

380

380

Cont. Var.

1,706

1,986

1,986

3,678

Constraints

2,366

2,926

2,946

6,462

Max. NPV ($MM)

2,204.5

698.4

761.0

1,831.2

Total CPU (s)

0.4

5

3,232

505

Iterations

N/A

6

12

6

* Δdem and Δsup are data-driven uncertainty budgets in the DDANRO approach for demand and supply, respective.
# Φdem and Φsup are data-driven uncertainty budgets in the DDSRO approach for demand and supply, respective.

From Table 4, we can see that the deterministic planning method consumes less computational
time, and generates the highest NPV. However, its solution suffers from infeasibility issue if the
supply of raw materials and demand of final products are uncertain. By leveraging the
corresponding strengths of two-stage stochastic programming approach and the two-stage ARO
method, the DDSRO approach generates $1,070.2MM higher NPV than the solution determined
by the DDANRO method, while still maintaining computational tractability. The DDSRO problem
can be solved using only 505 seconds. The optimal design and planning decisions at the end of the
planning horizon determined by the deterministic optimization method, DDANRO, and DDSRO
are shown in Fig. 10, Fig. 11 and Fig. 12, respectively. In these 3 figures, the optimal total
production capacities are displayed under operating processes.

30

Fig. 10. The optimal design and planning decisions at the end of the planning horizon determined
by the deterministic optimization method in case study 2.

31

Fig. 11. The optimal design and planning decisions at the end of the planning horizon determined
by the DDANRO approach in case study 2.

32

Fig. 12. The optimal design and planning decisions at the end of the planning horizon determined
by the proposed DDSRO approach in case study 2.

33

The optimal capacity expansion decisions determined by the DDANRO method and the
DDSRO approach are shown in Fig. 13 and Fig. 14, respectively. From Fig. 13, we can see that a
total of 15 processes are selected in the optimal process network determined by the DDANRO
method. As shown in Fig. 14, a total of 18 processes are chosen in the optimal process network
determined by the DDSRO approach. Specifically, Processes 2, 6 and 11 are not selected in the
optimal process network determined by the DDANRO method. Besides, the optimal expansion
frequencies and total capacities of Process 28 determined by the two methods are different.

Fig. 13. Optimal capacity expansion decisions over the entire planning horizon determined by the
DDANRO approach in case study 2.

34

Fig. 14. Optimal capacity expansion decisions over the entire planning horizon determined by the
proposed DDSRO approach in case study 2.

Fig. 15 displays the upper and lower bounds in each iteration of the proposed algorithm. The
X-axis and Y-axis denote the iteration number and the objective function values, respectively. In
Fig. 15, the green dots stand for the upper bounds, while the yellow circles represent the lower
bounds. The proposed solution algorithm requires only 6 iterations to converge. During the first 3
iterations, the relative optimality gap decreases significantly from 88.1% to 0.6%.

Fig. 15. Upper and lower bounds in each iteration of the decomposition-based algorithm in case
study 2.
35

5. Conclusions
This paper proposed a novel DDSRO framework that leveraged machine learning methods to
extract accurate uncertainty information from multi-class uncertainty data for decision making
under uncertainty. The probability distribution of data classes was learned through maximum
likelihood estimation. A group of Dirichlet process mixture models were used to construct
uncertainty sets for each data class. Stochastic programming approach was nested as the outer
optimization problem to leverage available probability distribution, while ARO was nested as the
inner optimization problem for computational tractability. The proposed DDSRO framework
leveraged the strengths of both stochastic programming and robust optimization. A decompositionbased algorithm was further developed to efficiently solve the resulting DDSRO problem. Twostage stochastic programming approach was computationally intractable for the large-scale
optimization problem with huge amounts of uncertainty data. The DDSRO approach was much
more computationally efficient than two-stage stochastic programming approach. The size of the
stochastic programming problem grew exponentially as the number of uncertainty data points
increased. By contrast, the problem size of DDSRO was less sensitive to the amount of uncertainty
data. Therefore, DDSRO remained tractable to handle problems with “big” data for uncertainties.
Moreover, DDSRO generated less conservative solutions compared with two-stage ARO method.
Moreover, the DDSRO approach could be considered as a generalization of the DDANRO method.
The results showed that the DDSRO approach was advantageous over the DDANRO method for
labeled multi-class uncertainty data.

Acknowledgements
The authors acknowledge ﬁnancial support from the National Science Foundation (NSF)
CAREER Award (CBET-1643244).

Appendix A. Dirichlet Process Mixture Model
The Dirichlet process is a widely used stochastic process, especially in the Dirichlet process
mixture model. Suppose a random distribution G follows a Dirichlet process, denoted as G ~ DP

36

(α, G0), (G(Θ1),…, G(Θr)) is then Dirichlet distributed for any measurable partition (Θ1, ..., Θr) of
Θ [70].

( G (Θ1 ), , G(Θr ) ) ~ Dir (α G0 (Θ1 ), , α G0 (Θr ) )

(A1)

where α and G0 are the concentration parameter and base distribution, respectively. Dir in (A1)
represents the Dirichlet distribution. A random draw from the Dirichlet process follows the stickbreaking construction in (A2) [71].

G = ∑ k =1 π k δθk
∞

=
π k βk
where

∏ (1 − β ) , β
k −1
j =1

j

k

(A2)

is the proportion of stick broken off from the remaining stick, and

β k ~ Beta (1, α). θ k is the random variable drawn from the base distribution G0. The delta function
δθ is equal to 1 at the location of θ k , and becomes 0 elsewhere.
k

In the Dirichlet process mixture model, θ k is treated as the parameters of the data distribution.
A significant feature of the Dirichlet process mixture model is that it has a potentially infinite
number of components. This novel feature grants it with great flexibility to model intricate data
distribution. The Dirichlet process mixture model is given in (A3) [58].

β k α ~ Beta(1, α )
θ k G0 ~ G0
ti {β1 , β 2 ,} ~ Mult (π ( β ) )

(A3)

oi ti ~ p (oi θli )
where Beta and Mult denote Beta distribution and multinomial distribution, respectively. The
observation is denoted as oi drawn from the ti th component.

Appendix B. Details of The Decomposition Algorithm
The multi-level optimization problem (10) can be reformulated to an equivalent single-level
optimization problem by enumerating all the extreme points [72]. However, this single-level
optimization problem could be a large-scale optimization problem due to the potentially large
number of extreme points. Therefore, we partially enumerate the extreme points, and construct the
master problem following the literature [8]. Because master problem (MP) contains a subset of the

37

extreme points, it is a relaxation of the original single-level optimization problem. The master
problem (MP) is shown below.
min cT x + η

x ,η , y ls

s.t. Ax ≥ d
1, , r
η ≥ ∑ ps ( b T y ls ), l =

(MP)

s

Tx + Wy ls ≥ h − Muls =
, l 1, , r , ∀s
x ∈ R+n1 × Z n2 , y ls ∈ R+n3 , =
l 1, , r , ∀s

where s is the index for data class, and uls is the enumerated uncertainty realization at the l th
iteration for data class s, y ls is its corresponding recourse variable, and r stands for the current
number of iterations.
To enumerate the important uncertainty realization on-the-fly, the sub-problem in (B1) is
solved in each iteration [8].
Qs ,i ( x ) = max min b T y s
u∈U s ,i

ys

s.t. Wy s ≥ h − Tx − Mu

(B1)

y s ∈ R+n3

It is worth noting that (B1) is a max-min optimization problem, which can be transformed to
a single-level optimization problem by employing strong duality or KKT conditions. To make the
sub-problem computationally tractable, we dualize the inner optimization problem of (B1), which
is a linear program with regard to ys, and merge the dual problem with the maximization problem
with respect to u. The resulting problem is shown below.
Qs ,i ( =
x ) max ( h − Tx − Mμ s ,i ) φ − ∑∑ (κ s ,i MΨ1/2
s ,i Λ s ,i ) z j φ t
T

z ,φ

(SUPs,i)

t

j

tj

s.t. W T φ ≤ b
φ≥0
z

∞

≤ 1, z 1 ≤ Φ s ,i

where φ is the vector of dual variables corresponding to the constraint Wy ≥ h − Tx − Mu , φt
represents the t th component in vector φ, and zj is the j th component in vector z. Problem (SUPs,i)
is reformulated to handle the bilinear term zjφt in its objective function.
To facilitate the reformulation, zj is divided into two parts z=
z +j − z −j . Thus, we have
j
+
−
u =μ s ,i + κ s ,i Ψ1/2
s ,i Λ s ,i ( z − z )

38

(B2)

Following the existing literature [73, 74], the uncertainty budget parameter Φs,i is typically set
as an integer value due to its physical meaning. For the integer uncertainty budget, the optimal z +j
and z −j take the values of 0 or 1, and these two variables can be restricted to binary variables. We
employ Glover’s linearization for the bilinear terms G tj+ = z +j φt and G tj− = z −j φt [75], which are
the products of a binary variable z +j (and z −j ) and a continuous variable φt, and reformulate (SUPs,i)
into (B3) following the procedure introduced in [8].
max+
Qs ,i (=
x)
+ −

φ , z , z ,G ,G

−

( h − Tx − Mμ )

T

s ,i

(

φ − Tr (κ s ,i MΨ1/s ,i2 Λ s ,i )

T

(G

+

− G− )

)

s.t. WT φ ≤ b
φ≥0
0 ≤ G + ≤ φ ⋅ eT
0 ≤ G − ≤ φ ⋅ eT
G + ≤ e ⋅ ( M 0 ⋅ z + )
G − ≤ e ⋅ ( M 0 ⋅ z − )

T

T

(
)
− e ⋅ ( M ⋅ ( e − z ) )

G + ≥ φ ⋅ eT − e ⋅ M 0 ⋅ ( e − z + )

T

G − ≥ φ ⋅ eT

T

−

0

e T ( z + + z − ) ≤ Φ s ,i
z + + z − ≤ e,

z + , z − ∈ {0,1}

K

(B3)

where e and e denote vectors with all elements being one. The t th row and j th column elements
of matrices G+ and G- are G tj+ and G tj− , respectively.

Nomenclature
DDSRO Framework
The main notations used in the DDSRO modeling framework are listed below.
b

vector of objective coefficients corresponding to second-stage decisions

c

vector of objective coefficients corresponding to first-stage decisions

u

vector of uncertainties

x

vector of first-stage decisions

ys

vector of second-stage decisions for data class s
39

C

total number of data classes in uncertainty data

m(s)

total number of basic uncertainty sets for data class s

M

truncation level in the Variational inference

M0

constant with a very large value

Us

uncertainty set for data class s

s

index of data class

Λi

scaling factor in the data-driven uncertainty set

τi

a hyper-parameter for Beta distribution in the Variational inference

νi

a hyper-parameter for Beta distribution in the Variational inference

γi

weight of mixture component

γ*

threshold value for the weight of mixture component

Φs,i

uncertainty budget in data-driven uncertainty set

A

coefficient matrix corresponding to the first-stage decisions

M

coefficient matrix corresponding to the uncertainties

T

technology matrix in the DDSRO framework

W

recourse matrix in the DDSRO framework

Application to planning of process networks
The sets, parameters, and variables used in the application part are summarized below. Note
that all parameters are denoted in lower-case symbols, and all variables are denoted in upper-case
symbols.
Sets/indices
I

set of processes indexed by i

J

set of chemicals indexed by j

T

set of time periods indexed by t

Ξ

set of data classes indexed by s
Parameters

c1it

variable investment cost for process i in time period t
40

c2it

fixed investment cost for process i in time period t

c3it

unit operating cost for process i in time period t

c4it

purchase price of chemical j in time period t

cbt

maximum allowable investment in time period t

cei

maximum number of expansions for process i over the planning horizon

dujt

demand of chemical j in time period t

qeitL

lower bound for capacity expansion of process i in time period t

qeitU

upper bound for capacity expansion of process i in time period t

sujt

supply of chemical j in time period t

vjt

sale price of chemical j in time period t

κij

mass balance coefficient for chemical j in process i
Binary variables

Yit

binary variable that indicates whether process i is chosen for expansion in time
period t
Continuous variables

Psjt

purchase amount of chemical j in time period t for data class s

Qit

total capacity of process i in time period t

QEit

capacity expansion of process i in time period t

SAsjt

sale amount of chemical j in time period t for data class s

Wsit

operation level of process i in time period t for data class s

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